:: GROUP_9 semantic presentation
begin
definition
let O, E be set ;
let A be Action of O,E;
let IT be set ;
predIT is_stable_under_the_action_of A means :Def1: :: GROUP_9:def 1
for o being Element of O
for f being Function of E,E st o in O & f = A . o holds
f .: IT c= IT;
end;
:: deftheorem Def1 defines is_stable_under_the_action_of GROUP_9:def_1_:_
for O, E being set
for A being Action of O,E
for IT being set holds
( IT is_stable_under_the_action_of A iff for o being Element of O
for f being Function of E,E st o in O & f = A . o holds
f .: IT c= IT );
Lm1: for O, E being set
for A being Action of O,E holds [#] E is_stable_under_the_action_of A
proof
let O, E be set ; ::_thesis: for A being Action of O,E holds [#] E is_stable_under_the_action_of A
let A be Action of O,E; ::_thesis: [#] E is_stable_under_the_action_of A
for o being Element of O
for f being Function of E,E st o in O & f = A . o holds
f .: ([#] E) c= [#] E ;
hence [#] E is_stable_under_the_action_of A by Def1; ::_thesis: verum
end;
definition
let O, E be set ;
let A be Action of O,E;
let X be Subset of E;
func the_stable_subset_generated_by (X,A) -> Subset of E means :Def2: :: GROUP_9:def 2
( X c= it & it is_stable_under_the_action_of A & ( for Y being Subset of E st Y is_stable_under_the_action_of A & X c= Y holds
it c= Y ) );
existence
ex b1 being Subset of E st
( X c= b1 & b1 is_stable_under_the_action_of A & ( for Y being Subset of E st Y is_stable_under_the_action_of A & X c= Y holds
b1 c= Y ) )
proof
defpred S1[ set ] means ex B being Subset of E st
( $1 = B & X c= $1 & B is_stable_under_the_action_of A );
consider XX being set such that
A1: for Y being set holds
( Y in XX iff ( Y in bool E & S1[Y] ) ) from XBOOLE_0:sch_1();
set M = meet XX;
[#] E is_stable_under_the_action_of A by Lm1;
then A2: E in XX by A1;
then for x being set st x in meet XX holds
x in E by SETFAM_1:def_1;
then reconsider M = meet XX as Subset of E by TARSKI:def_3;
take M ; ::_thesis: ( X c= M & M is_stable_under_the_action_of A & ( for Y being Subset of E st Y is_stable_under_the_action_of A & X c= Y holds
M c= Y ) )
now__::_thesis:_for_x_being_set_st_x_in_X_holds_
x_in_M
let x be set ; ::_thesis: ( x in X implies x in M )
assume A3: x in X ; ::_thesis: x in M
now__::_thesis:_for_Y_being_set_st_Y_in_XX_holds_
x_in_Y
let Y be set ; ::_thesis: ( Y in XX implies x in Y )
assume Y in XX ; ::_thesis: x in Y
then ex B being Subset of E st
( Y = B & X c= Y & B is_stable_under_the_action_of A ) by A1;
hence x in Y by A3; ::_thesis: verum
end;
hence x in M by A2, SETFAM_1:def_1; ::_thesis: verum
end;
hence X c= M by TARSKI:def_3; ::_thesis: ( M is_stable_under_the_action_of A & ( for Y being Subset of E st Y is_stable_under_the_action_of A & X c= Y holds
M c= Y ) )
now__::_thesis:_for_o_being_Element_of_O
for_f_being_Function_of_E,E_st_o_in_O_&_f_=_A_._o_holds_
f_.:_M_c=_M
let o be Element of O; ::_thesis: for f being Function of E,E st o in O & f = A . o holds
f .: M c= M
let f be Function of E,E; ::_thesis: ( o in O & f = A . o implies f .: M c= M )
assume A4: o in O ; ::_thesis: ( f = A . o implies f .: M c= M )
assume A5: f = A . o ; ::_thesis: f .: M c= M
now__::_thesis:_for_y_being_set_st_y_in_f_.:_M_holds_
y_in_M
let y be set ; ::_thesis: ( y in f .: M implies y in M )
assume A6: y in f .: M ; ::_thesis: y in M
now__::_thesis:_for_Y_being_set_st_Y_in_XX_holds_
y_in_Y
let Y be set ; ::_thesis: ( Y in XX implies y in Y )
assume A7: Y in XX ; ::_thesis: y in Y
then ex B being Subset of E st
( Y = B & X c= Y & B is_stable_under_the_action_of A ) by A1;
then A8: f .: Y c= Y by A4, A5, Def1;
f .: M c= f .: Y by A7, RELAT_1:123, SETFAM_1:3;
then f .: M c= Y by A8, XBOOLE_1:1;
hence y in Y by A6; ::_thesis: verum
end;
hence y in M by A2, SETFAM_1:def_1; ::_thesis: verum
end;
hence f .: M c= M by TARSKI:def_3; ::_thesis: verum
end;
hence M is_stable_under_the_action_of A by Def1; ::_thesis: for Y being Subset of E st Y is_stable_under_the_action_of A & X c= Y holds
M c= Y
now__::_thesis:_for_Y_being_Subset_of_E_st_Y_is_stable_under_the_action_of_A_&_X_c=_Y_holds_
M_c=_Y
let Y be Subset of E; ::_thesis: ( Y is_stable_under_the_action_of A & X c= Y implies M c= Y )
assume ( Y is_stable_under_the_action_of A & X c= Y ) ; ::_thesis: M c= Y
then Y in XX by A1;
hence M c= Y by SETFAM_1:3; ::_thesis: verum
end;
hence for Y being Subset of E st Y is_stable_under_the_action_of A & X c= Y holds
M c= Y ; ::_thesis: verum
end;
uniqueness
for b1, b2 being Subset of E st X c= b1 & b1 is_stable_under_the_action_of A & ( for Y being Subset of E st Y is_stable_under_the_action_of A & X c= Y holds
b1 c= Y ) & X c= b2 & b2 is_stable_under_the_action_of A & ( for Y being Subset of E st Y is_stable_under_the_action_of A & X c= Y holds
b2 c= Y ) holds
b1 = b2
proof
let B1, B2 be Subset of E; ::_thesis: ( X c= B1 & B1 is_stable_under_the_action_of A & ( for Y being Subset of E st Y is_stable_under_the_action_of A & X c= Y holds
B1 c= Y ) & X c= B2 & B2 is_stable_under_the_action_of A & ( for Y being Subset of E st Y is_stable_under_the_action_of A & X c= Y holds
B2 c= Y ) implies B1 = B2 )
assume ( X c= B1 & B1 is_stable_under_the_action_of A & ( for Y being Subset of E st Y is_stable_under_the_action_of A & X c= Y holds
B1 c= Y ) & X c= B2 & B2 is_stable_under_the_action_of A & ( for Y being Subset of E st Y is_stable_under_the_action_of A & X c= Y holds
B2 c= Y ) ) ; ::_thesis: B1 = B2
then ( B1 c= B2 & B2 c= B1 ) ;
hence B1 = B2 by XBOOLE_0:def_10; ::_thesis: verum
end;
end;
:: deftheorem Def2 defines the_stable_subset_generated_by GROUP_9:def_2_:_
for O, E being set
for A being Action of O,E
for X, b5 being Subset of E holds
( b5 = the_stable_subset_generated_by (X,A) iff ( X c= b5 & b5 is_stable_under_the_action_of A & ( for Y being Subset of E st Y is_stable_under_the_action_of A & X c= Y holds
b5 c= Y ) ) );
definition
let O, E be set ;
let A be Action of O,E;
let F be FinSequence of O;
func Product (F,A) -> Function of E,E means :Def3: :: GROUP_9:def 3
it = id E if len F = 0
otherwise ex PF being FinSequence of Funcs (E,E) st
( it = PF . (len F) & len PF = len F & PF . 1 = A . (F . 1) & ( for n being Nat st n <> 0 & n < len F holds
ex f, g being Function of E,E st
( f = PF . n & g = A . (F . (n + 1)) & PF . (n + 1) = f * g ) ) );
existence
( ( len F = 0 implies ex b1 being Function of E,E st b1 = id E ) & ( not len F = 0 implies ex b1 being Function of E,E ex PF being FinSequence of Funcs (E,E) st
( b1 = PF . (len F) & len PF = len F & PF . 1 = A . (F . 1) & ( for n being Nat st n <> 0 & n < len F holds
ex f, g being Function of E,E st
( f = PF . n & g = A . (F . (n + 1)) & PF . (n + 1) = f * g ) ) ) ) )
proof
percases ( len F = 0 or len F <> 0 ) ;
suppose len F = 0 ; ::_thesis: ( ( len F = 0 implies ex b1 being Function of E,E st b1 = id E ) & ( not len F = 0 implies ex b1 being Function of E,E ex PF being FinSequence of Funcs (E,E) st
( b1 = PF . (len F) & len PF = len F & PF . 1 = A . (F . 1) & ( for n being Nat st n <> 0 & n < len F holds
ex f, g being Function of E,E st
( f = PF . n & g = A . (F . (n + 1)) & PF . (n + 1) = f * g ) ) ) ) )
hence ( ( len F = 0 implies ex b1 being Function of E,E st b1 = id E ) & ( not len F = 0 implies ex b1 being Function of E,E ex PF being FinSequence of Funcs (E,E) st
( b1 = PF . (len F) & len PF = len F & PF . 1 = A . (F . 1) & ( for n being Nat st n <> 0 & n < len F holds
ex f, g being Function of E,E st
( f = PF . n & g = A . (F . (n + 1)) & PF . (n + 1) = f * g ) ) ) ) ) ; ::_thesis: verum
end;
supposeA1: len F <> 0 ; ::_thesis: ( ( len F = 0 implies ex b1 being Function of E,E st b1 = id E ) & ( not len F = 0 implies ex b1 being Function of E,E ex PF being FinSequence of Funcs (E,E) st
( b1 = PF . (len F) & len PF = len F & PF . 1 = A . (F . 1) & ( for n being Nat st n <> 0 & n < len F holds
ex f, g being Function of E,E st
( f = PF . n & g = A . (F . (n + 1)) & PF . (n + 1) = f * g ) ) ) ) )
defpred S1[ Element of NAT ] means for F being FinSequence of O st len F = $1 & len F <> 0 holds
ex PF being FinSequence of Funcs (E,E) ex IT being Function of E,E st
( IT = PF . (len PF) & len PF = len F & PF . 1 = A . (F . 1) & ( for k being Nat st k <> 0 & k < len F holds
ex f, g being Function of E,E st
( f = PF . k & g = A . (F . (k + 1)) & PF . (k + 1) = f * g ) ) );
A2: for k being Element of NAT st S1[k] holds
S1[k + 1]
proof
let k be Element of NAT ; ::_thesis: ( S1[k] implies S1[k + 1] )
assume A3: S1[k] ; ::_thesis: S1[k + 1]
thus S1[k + 1] ::_thesis: verum
proof
let F be FinSequence of O; ::_thesis: ( len F = k + 1 & len F <> 0 implies ex PF being FinSequence of Funcs (E,E) ex IT being Function of E,E st
( IT = PF . (len PF) & len PF = len F & PF . 1 = A . (F . 1) & ( for k being Nat st k <> 0 & k < len F holds
ex f, g being Function of E,E st
( f = PF . k & g = A . (F . (k + 1)) & PF . (k + 1) = f * g ) ) ) )
assume that
A4: len F = k + 1 and
len F <> 0 ; ::_thesis: ex PF being FinSequence of Funcs (E,E) ex IT being Function of E,E st
( IT = PF . (len PF) & len PF = len F & PF . 1 = A . (F . 1) & ( for k being Nat st k <> 0 & k < len F holds
ex f, g being Function of E,E st
( f = PF . k & g = A . (F . (k + 1)) & PF . (k + 1) = f * g ) ) )
reconsider G = F | (Seg k) as FinSequence of O by FINSEQ_1:18;
A5: len G = k by A4, FINSEQ_3:53;
percases ( len G = 0 or len G <> 0 ) ;
supposeA6: len G = 0 ; ::_thesis: ex PF being FinSequence of Funcs (E,E) ex IT being Function of E,E st
( IT = PF . (len PF) & len PF = len F & PF . 1 = A . (F . 1) & ( for k being Nat st k <> 0 & k < len F holds
ex f, g being Function of E,E st
( f = PF . k & g = A . (F . (k + 1)) & PF . (k + 1) = f * g ) ) )
set IT = A . (F . 1);
1 in Seg (len F) by A4, A5, A6;
then 1 in dom F by FINSEQ_1:def_3;
then F . 1 in rng F by FUNCT_1:3;
then F . 1 in O ;
then F . 1 in dom A by FUNCT_2:def_1;
then A7: A . (F . 1) in rng A by FUNCT_1:3;
set f = the Function of E,E;
reconsider IT = A . (F . 1) as Element of Funcs (E,E) by A7;
set PF = <*IT*>;
ex f being Function st
( IT = f & dom f = E & rng f c= E ) by FUNCT_2:def_2;
then reconsider IT = IT as Function of E,E by FUNCT_2:2;
take <*IT*> ; ::_thesis: ex IT being Function of E,E st
( IT = <*IT*> . (len <*IT*>) & len <*IT*> = len F & <*IT*> . 1 = A . (F . 1) & ( for k being Nat st k <> 0 & k < len F holds
ex f, g being Function of E,E st
( f = <*IT*> . k & g = A . (F . (k + 1)) & <*IT*> . (k + 1) = f * g ) ) )
take IT ; ::_thesis: ( IT = <*IT*> . (len <*IT*>) & len <*IT*> = len F & <*IT*> . 1 = A . (F . 1) & ( for k being Nat st k <> 0 & k < len F holds
ex f, g being Function of E,E st
( f = <*IT*> . k & g = A . (F . (k + 1)) & <*IT*> . (k + 1) = f * g ) ) )
len <*IT*> = 1 by FINSEQ_1:40;
hence IT = <*IT*> . (len <*IT*>) by FINSEQ_1:40; ::_thesis: ( len <*IT*> = len F & <*IT*> . 1 = A . (F . 1) & ( for k being Nat st k <> 0 & k < len F holds
ex f, g being Function of E,E st
( f = <*IT*> . k & g = A . (F . (k + 1)) & <*IT*> . (k + 1) = f * g ) ) )
thus len <*IT*> = len F by A4, A5, A6, FINSEQ_1:40; ::_thesis: ( <*IT*> . 1 = A . (F . 1) & ( for k being Nat st k <> 0 & k < len F holds
ex f, g being Function of E,E st
( f = <*IT*> . k & g = A . (F . (k + 1)) & <*IT*> . (k + 1) = f * g ) ) )
thus <*IT*> . 1 = A . (F . 1) by FINSEQ_1:40; ::_thesis: for k being Nat st k <> 0 & k < len F holds
ex f, g being Function of E,E st
( f = <*IT*> . k & g = A . (F . (k + 1)) & <*IT*> . (k + 1) = f * g )
let k be Nat; ::_thesis: ( k <> 0 & k < len F implies ex f, g being Function of E,E st
( f = <*IT*> . k & g = A . (F . (k + 1)) & <*IT*> . (k + 1) = f * g ) )
assume A8: ( k <> 0 & k < len F ) ; ::_thesis: ex f, g being Function of E,E st
( f = <*IT*> . k & g = A . (F . (k + 1)) & <*IT*> . (k + 1) = f * g )
take the Function of E,E ; ::_thesis: ex g being Function of E,E st
( the Function of E,E = <*IT*> . k & g = A . (F . (k + 1)) & <*IT*> . (k + 1) = the Function of E,E * g )
take the Function of E,E ; ::_thesis: ( the Function of E,E = <*IT*> . k & the Function of E,E = A . (F . (k + 1)) & <*IT*> . (k + 1) = the Function of E,E * the Function of E,E )
thus ( the Function of E,E = <*IT*> . k & the Function of E,E = A . (F . (k + 1)) ) by A4, A5, A6, A8, NAT_1:14; ::_thesis: <*IT*> . (k + 1) = the Function of E,E * the Function of E,E
thus <*IT*> . (k + 1) = the Function of E,E * the Function of E,E by A4, A5, A6, A8, NAT_1:14; ::_thesis: verum
end;
supposeA9: len G <> 0 ; ::_thesis: ex PF being FinSequence of Funcs (E,E) ex IT being Function of E,E st
( IT = PF . (len PF) & len PF = len F & PF . 1 = A . (F . 1) & ( for k being Nat st k <> 0 & k < len F holds
ex f, g being Function of E,E st
( f = PF . k & g = A . (F . (k + 1)) & PF . (k + 1) = f * g ) ) )
set g = A . (F . (k + 1));
A10: 0 + k <= k + 1 by XREAL_1:6;
A11: 0 + 1 < k + 1 by A5, A9, XREAL_1:6;
then A12: 1 <= k by NAT_1:13;
then 1 in Seg k ;
then 1 in (Seg (k + 1)) /\ (Seg k) by A10, FINSEQ_1:7;
then A13: 1 in (dom F) /\ (Seg k) by A4, FINSEQ_1:def_3;
k + 1 in Seg (len F) by A4, A11;
then k + 1 in dom F by FINSEQ_1:def_3;
then F . (k + 1) in rng F by FUNCT_1:3;
then F . (k + 1) in O ;
then F . (k + 1) in dom A by FUNCT_2:def_1;
then A . (F . (k + 1)) in rng A by FUNCT_1:3;
then ex f being Function st
( A . (F . (k + 1)) = f & dom f = E & rng f c= E ) by FUNCT_2:def_2;
then reconsider g = A . (F . (k + 1)) as Function of E,E by FUNCT_2:2;
consider PFk being FinSequence of Funcs (E,E), ITk being Function of E,E such that
ITk = PFk . (len PFk) and
A14: len PFk = len G and
A15: PFk . 1 = A . (G . 1) and
A16: for k being Nat st k <> 0 & k < len G holds
ex f, g being Function of E,E st
( f = PFk . k & g = A . (G . (k + 1)) & PFk . (k + 1) = f * g ) by A3, A4, A9, FINSEQ_3:53;
set f = PFk . k;
k in Seg (len PFk) by A5, A14, A12;
then A17: k in dom PFk by FINSEQ_1:def_3;
then PFk . k in Funcs (E,E) by FINSEQ_2:11;
then ex f being Function st
( PFk . k = f & dom f = E & rng f c= E ) by FUNCT_2:def_2;
then reconsider f = PFk . k as Function of E,E by FUNCT_2:2;
set IT = f * g;
set PF = PFk ^ <*(f * g)*>;
f * g in Funcs (E,E) by FUNCT_2:9;
then <*(f * g)*> is FinSequence of Funcs (E,E) by FINSEQ_1:74;
then reconsider PF = PFk ^ <*(f * g)*> as FinSequence of Funcs (E,E) by FINSEQ_1:75;
take PF ; ::_thesis: ex IT being Function of E,E st
( IT = PF . (len PF) & len PF = len F & PF . 1 = A . (F . 1) & ( for k being Nat st k <> 0 & k < len F holds
ex f, g being Function of E,E st
( f = PF . k & g = A . (F . (k + 1)) & PF . (k + 1) = f * g ) ) )
take f * g ; ::_thesis: ( f * g = PF . (len PF) & len PF = len F & PF . 1 = A . (F . 1) & ( for k being Nat st k <> 0 & k < len F holds
ex f, g being Function of E,E st
( f = PF . k & g = A . (F . (k + 1)) & PF . (k + 1) = f * g ) ) )
A18: len PF = (len G) + (len <*(f * g)*>) by A14, FINSEQ_1:22
.= k + 1 by A5, FINSEQ_1:39 ;
then len PF = (len PFk) + 1 by A4, A14, FINSEQ_3:53;
hence A19: ( f * g = PF . (len PF) & len PF = len F ) by A4, A18, FINSEQ_1:42; ::_thesis: ( PF . 1 = A . (F . 1) & ( for k being Nat st k <> 0 & k < len F holds
ex f, g being Function of E,E st
( f = PF . k & g = A . (F . (k + 1)) & PF . (k + 1) = f * g ) ) )
0 + 1 < (len G) + 1 by A9, XREAL_1:6;
then 1 <= len G by NAT_1:13;
then 1 in Seg (len PFk) by A14;
then 1 in dom PFk by FINSEQ_1:def_3;
then PF . 1 = A . (G . 1) by A15, FINSEQ_1:def_7;
hence PF . 1 = A . (F . 1) by A13, FUNCT_1:48; ::_thesis: for k being Nat st k <> 0 & k < len F holds
ex f, g being Function of E,E st
( f = PF . k & g = A . (F . (k + 1)) & PF . (k + 1) = f * g )
let n be Nat; ::_thesis: ( n <> 0 & n < len F implies ex f, g being Function of E,E st
( f = PF . n & g = A . (F . (n + 1)) & PF . (n + 1) = f * g ) )
assume A20: n <> 0 ; ::_thesis: ( not n < len F or ex f, g being Function of E,E st
( f = PF . n & g = A . (F . (n + 1)) & PF . (n + 1) = f * g ) )
assume n < len F ; ::_thesis: ex f, g being Function of E,E st
( f = PF . n & g = A . (F . (n + 1)) & PF . (n + 1) = f * g )
then A21: n <= k by A4, NAT_1:13;
percases ( n >= k or n < k ) ;
supposeA22: n >= k ; ::_thesis: ex f, g being Function of E,E st
( f = PF . n & g = A . (F . (n + 1)) & PF . (n + 1) = f * g )
then A . (F . (n + 1)) = g by A21, XXREAL_0:1;
then reconsider g9 = A . (F . (n + 1)) as Function of E,E ;
A23: n = k by A21, A22, XXREAL_0:1;
then reconsider f9 = PF . n as Function of E,E by A17, FINSEQ_1:def_7;
take f9 ; ::_thesis: ex g being Function of E,E st
( f9 = PF . n & g = A . (F . (n + 1)) & PF . (n + 1) = f9 * g )
take g9 ; ::_thesis: ( f9 = PF . n & g9 = A . (F . (n + 1)) & PF . (n + 1) = f9 * g9 )
thus ( f9 = PF . n & g9 = A . (F . (n + 1)) ) ; ::_thesis: PF . (n + 1) = f9 * g9
thus PF . (n + 1) = f9 * g9 by A17, A18, A19, A23, FINSEQ_1:def_7; ::_thesis: verum
end;
supposeA24: n < k ; ::_thesis: ex f, g being Function of E,E st
( f = PF . n & g = A . (F . (n + 1)) & PF . (n + 1) = f * g )
A25: 0 + 1 < n + 1 by A20, XREAL_1:6;
then 1 <= n by NAT_1:13;
then n in Seg (len PFk) by A5, A14, A24, FINSEQ_1:1;
then A26: n in dom PFk by FINSEQ_1:def_3;
consider f9, g9 being Function of E,E such that
A27: ( f9 = PFk . n & g9 = A . (G . (n + 1)) ) and
A28: PFk . (n + 1) = f9 * g9 by A5, A16, A20, A24;
take f9 ; ::_thesis: ex g being Function of E,E st
( f9 = PF . n & g = A . (F . (n + 1)) & PF . (n + 1) = f9 * g )
take g9 ; ::_thesis: ( f9 = PF . n & g9 = A . (F . (n + 1)) & PF . (n + 1) = f9 * g9 )
A29: 0 + k <= 1 + k by XREAL_1:6;
A30: n + 1 <= k by A24, NAT_1:13;
then n + 1 in Seg k by A25;
then n + 1 in (Seg (k + 1)) /\ (Seg k) by A29, FINSEQ_1:7;
then n + 1 in (dom F) /\ (Seg k) by A4, FINSEQ_1:def_3;
hence ( f9 = PF . n & g9 = A . (F . (n + 1)) ) by A27, A26, FINSEQ_1:def_7, FUNCT_1:48; ::_thesis: PF . (n + 1) = f9 * g9
n + 1 in Seg (len PFk) by A5, A14, A25, A30;
then n + 1 in dom PFk by FINSEQ_1:def_3;
hence PF . (n + 1) = f9 * g9 by A28, FINSEQ_1:def_7; ::_thesis: verum
end;
end;
end;
end;
end;
end;
A31: S1[ 0 ] ;
for k being Element of NAT holds S1[k] from NAT_1:sch_1(A31, A2);
then ex PF being FinSequence of Funcs (E,E) ex IT being Function of E,E st
( IT = PF . (len PF) & len PF = len F & PF . 1 = A . (F . 1) & ( for k being Nat st k <> 0 & k < len F holds
ex f, g being Function of E,E st
( f = PF . k & g = A . (F . (k + 1)) & PF . (k + 1) = f * g ) ) ) by A1;
hence ( ( len F = 0 implies ex b1 being Function of E,E st b1 = id E ) & ( not len F = 0 implies ex b1 being Function of E,E ex PF being FinSequence of Funcs (E,E) st
( b1 = PF . (len F) & len PF = len F & PF . 1 = A . (F . 1) & ( for n being Nat st n <> 0 & n < len F holds
ex f, g being Function of E,E st
( f = PF . n & g = A . (F . (n + 1)) & PF . (n + 1) = f * g ) ) ) ) ) ; ::_thesis: verum
end;
end;
end;
uniqueness
for b1, b2 being Function of E,E holds
( ( len F = 0 & b1 = id E & b2 = id E implies b1 = b2 ) & ( not len F = 0 & ex PF being FinSequence of Funcs (E,E) st
( b1 = PF . (len F) & len PF = len F & PF . 1 = A . (F . 1) & ( for n being Nat st n <> 0 & n < len F holds
ex f, g being Function of E,E st
( f = PF . n & g = A . (F . (n + 1)) & PF . (n + 1) = f * g ) ) ) & ex PF being FinSequence of Funcs (E,E) st
( b2 = PF . (len F) & len PF = len F & PF . 1 = A . (F . 1) & ( for n being Nat st n <> 0 & n < len F holds
ex f, g being Function of E,E st
( f = PF . n & g = A . (F . (n + 1)) & PF . (n + 1) = f * g ) ) ) implies b1 = b2 ) )
proof
now__::_thesis:_for_IT1,_IT2_being_Function_of_E,E_st_ex_PF1_being_FinSequence_of_Funcs_(E,E)_st_
(_IT1_=_PF1_._(len_F)_&_len_PF1_=_len_F_&_PF1_._1_=_A_._(F_._1)_&_(_for_k_being_Nat_st_k_<>_0_&_k_<_len_F_holds_
ex_f,_g_being_Function_of_E,E_st_
(_f_=_PF1_._k_&_g_=_A_._(F_._(k_+_1))_&_PF1_._(k_+_1)_=_f_*_g_)_)_)_&_ex_PF2_being_FinSequence_of_Funcs_(E,E)_st_
(_IT2_=_PF2_._(len_F)_&_len_PF2_=_len_F_&_PF2_._1_=_A_._(F_._1)_&_(_for_k_being_Nat_st_k_<>_0_&_k_<_len_F_holds_
ex_f,_g_being_Function_of_E,E_st_
(_f_=_PF2_._k_&_g_=_A_._(F_._(k_+_1))_&_PF2_._(k_+_1)_=_f_*_g_)_)_)_holds_
IT1_=_IT2
let IT1, IT2 be Function of E,E; ::_thesis: ( ex PF1 being FinSequence of Funcs (E,E) st
( IT1 = PF1 . (len F) & len PF1 = len F & PF1 . 1 = A . (F . 1) & ( for k being Nat st k <> 0 & k < len F holds
ex f, g being Function of E,E st
( f = PF1 . k & g = A . (F . (k + 1)) & PF1 . (k + 1) = f * g ) ) ) & ex PF2 being FinSequence of Funcs (E,E) st
( IT2 = PF2 . (len F) & len PF2 = len F & PF2 . 1 = A . (F . 1) & ( for k being Nat st k <> 0 & k < len F holds
ex f, g being Function of E,E st
( f = PF2 . k & g = A . (F . (k + 1)) & PF2 . (k + 1) = f * g ) ) ) implies IT1 = IT2 )
given PF1 being FinSequence of Funcs (E,E) such that A32: IT1 = PF1 . (len F) and
A33: len PF1 = len F and
A34: PF1 . 1 = A . (F . 1) and
A35: for k being Nat st k <> 0 & k < len F holds
ex f, g being Function of E,E st
( f = PF1 . k & g = A . (F . (k + 1)) & PF1 . (k + 1) = f * g ) ; ::_thesis: ( ex PF2 being FinSequence of Funcs (E,E) st
( IT2 = PF2 . (len F) & len PF2 = len F & PF2 . 1 = A . (F . 1) & ( for k being Nat st k <> 0 & k < len F holds
ex f, g being Function of E,E st
( f = PF2 . k & g = A . (F . (k + 1)) & PF2 . (k + 1) = f * g ) ) ) implies IT1 = IT2 )
given PF2 being FinSequence of Funcs (E,E) such that A36: ( IT2 = PF2 . (len F) & len PF2 = len F ) and
A37: PF2 . 1 = A . (F . 1) and
A38: for k being Nat st k <> 0 & k < len F holds
ex f, g being Function of E,E st
( f = PF2 . k & g = A . (F . (k + 1)) & PF2 . (k + 1) = f * g ) ; ::_thesis: IT1 = IT2
defpred S1[ Nat] means ( 1 <= $1 & $1 <= len PF1 implies PF1 . $1 = PF2 . $1 );
A39: for k being Nat st S1[k] holds
S1[k + 1]
proof
let k be Nat; ::_thesis: ( S1[k] implies S1[k + 1] )
assume A40: S1[k] ; ::_thesis: S1[k + 1]
now__::_thesis:_(_1_<=_k_+_1_&_k_+_1_<=_len_PF1_implies_PF1_._(k_+_1)_=_PF2_._(k_+_1)_)
assume 1 <= k + 1 ; ::_thesis: ( k + 1 <= len PF1 implies PF1 . (k + 1) = PF2 . (k + 1) )
assume A41: k + 1 <= len PF1 ; ::_thesis: PF1 . (k + 1) = PF2 . (k + 1)
then A42: k < len PF1 by NAT_1:13;
percases ( k = 0 or k <> 0 ) ;
suppose k = 0 ; ::_thesis: PF1 . (k + 1) = PF2 . (k + 1)
hence PF1 . (k + 1) = PF2 . (k + 1) by A34, A37; ::_thesis: verum
end;
supposeA43: k <> 0 ; ::_thesis: PF1 . (k + 1) = PF2 . (k + 1)
then A44: 0 + 1 < k + 1 by XREAL_1:6;
( ex f1, g1 being Function of E,E st
( f1 = PF1 . k & g1 = A . (F . (k + 1)) & PF1 . (k + 1) = f1 * g1 ) & ex f2, g2 being Function of E,E st
( f2 = PF2 . k & g2 = A . (F . (k + 1)) & PF2 . (k + 1) = f2 * g2 ) ) by A33, A35, A38, A42, A43;
hence PF1 . (k + 1) = PF2 . (k + 1) by A40, A41, A44, NAT_1:13; ::_thesis: verum
end;
end;
end;
hence S1[k + 1] ; ::_thesis: verum
end;
A45: S1[ 0 ] ;
for k being Nat holds S1[k] from NAT_1:sch_2(A45, A39);
hence IT1 = IT2 by A32, A33, A36, FINSEQ_1:14; ::_thesis: verum
end;
hence for b1, b2 being Function of E,E holds
( ( len F = 0 & b1 = id E & b2 = id E implies b1 = b2 ) & ( not len F = 0 & ex PF being FinSequence of Funcs (E,E) st
( b1 = PF . (len F) & len PF = len F & PF . 1 = A . (F . 1) & ( for n being Nat st n <> 0 & n < len F holds
ex f, g being Function of E,E st
( f = PF . n & g = A . (F . (n + 1)) & PF . (n + 1) = f * g ) ) ) & ex PF being FinSequence of Funcs (E,E) st
( b2 = PF . (len F) & len PF = len F & PF . 1 = A . (F . 1) & ( for n being Nat st n <> 0 & n < len F holds
ex f, g being Function of E,E st
( f = PF . n & g = A . (F . (n + 1)) & PF . (n + 1) = f * g ) ) ) implies b1 = b2 ) ) ; ::_thesis: verum
end;
consistency
for b1 being Function of E,E holds verum ;
end;
:: deftheorem Def3 defines Product GROUP_9:def_3_:_
for O, E being set
for A being Action of O,E
for F being FinSequence of O
for b5 being Function of E,E holds
( ( len F = 0 implies ( b5 = Product (F,A) iff b5 = id E ) ) & ( not len F = 0 implies ( b5 = Product (F,A) iff ex PF being FinSequence of Funcs (E,E) st
( b5 = PF . (len F) & len PF = len F & PF . 1 = A . (F . 1) & ( for n being Nat st n <> 0 & n < len F holds
ex f, g being Function of E,E st
( f = PF . n & g = A . (F . (n + 1)) & PF . (n + 1) = f * g ) ) ) ) ) );
definition
let O be set ;
let G be Group;
let IT be Action of O, the carrier of G;
attrIT is distributive means :Def4: :: GROUP_9:def 4
for o being Element of O st o in O holds
IT . o is Homomorphism of G,G;
end;
:: deftheorem Def4 defines distributive GROUP_9:def_4_:_
for O being set
for G being Group
for IT being Action of O, the carrier of G holds
( IT is distributive iff for o being Element of O st o in O holds
IT . o is Homomorphism of G,G );
definition
let O be set ;
attrc2 is strict ;
struct HGrWOpStr over O -> multMagma ;
aggrHGrWOpStr(# carrier, multF, action #) -> HGrWOpStr over O;
sel action c2 -> Action of O, the carrier of c2;
end;
registration
let O be set ;
cluster non empty for HGrWOpStr over O;
existence
not for b1 being HGrWOpStr over O holds b1 is empty
proof
set A = the non empty set ;
set m = the BinOp of the non empty set ;
set h = the Action of O, the non empty set ;
take HGrWOpStr(# the non empty set , the BinOp of the non empty set , the Action of O, the non empty set #) ; ::_thesis: not HGrWOpStr(# the non empty set , the BinOp of the non empty set , the Action of O, the non empty set #) is empty
thus not HGrWOpStr(# the non empty set , the BinOp of the non empty set , the Action of O, the non empty set #) is empty ; ::_thesis: verum
end;
end;
definition
let O be set ;
let IT be non empty HGrWOpStr over O;
attrIT is distributive means :Def5: :: GROUP_9:def 5
for G being Group
for a being Action of O, the carrier of G st a = the action of IT & multMagma(# the carrier of G, the multF of G #) = multMagma(# the carrier of IT, the multF of IT #) holds
a is distributive ;
end;
:: deftheorem Def5 defines distributive GROUP_9:def_5_:_
for O being set
for IT being non empty HGrWOpStr over O holds
( IT is distributive iff for G being Group
for a being Action of O, the carrier of G st a = the action of IT & multMagma(# the carrier of G, the multF of G #) = multMagma(# the carrier of IT, the multF of IT #) holds
a is distributive );
Lm2: for O, E being set holds [:O,{(id E)}:] is Action of O,E
proof
let O, E be set ; ::_thesis: [:O,{(id E)}:] is Action of O,E
set h = [:O,{(id E)}:];
now__::_thesis:_for_x_being_set_st_x_in_{(id_E)}_holds_
x_in_Funcs_(E,E)
let x be set ; ::_thesis: ( x in {(id E)} implies x in Funcs (E,E) )
assume x in {(id E)} ; ::_thesis: x in Funcs (E,E)
then reconsider f = x as Function of E,E by TARSKI:def_1;
f in Funcs (E,E) by FUNCT_2:9;
hence x in Funcs (E,E) ; ::_thesis: verum
end;
then {(id E)} c= Funcs (E,E) by TARSKI:def_3;
then reconsider h = [:O,{(id E)}:] as Relation of O,(Funcs (E,E)) by ZFMISC_1:95;
A1: now__::_thesis:_(_(_(_Funcs_(E,E)_=_{}_implies_O_=_{}_)_implies_O_=_dom_h_)_&_(_O_=_{}_implies_h_=_{}_)_)
thus ( ( Funcs (E,E) = {} implies O = {} ) implies O = dom h ) ::_thesis: ( O = {} implies h = {} )
proof
assume ( Funcs (E,E) = {} implies O = {} ) ; ::_thesis: O = dom h
now__::_thesis:_for_x_being_set_st_x_in_O_holds_
ex_y_being_set_st_[x,y]_in_h
let x be set ; ::_thesis: ( x in O implies ex y being set st [x,y] in h )
assume A2: x in O ; ::_thesis: ex y being set st [x,y] in h
consider y being set such that
A3: y = id E ;
take y = y; ::_thesis: [x,y] in h
y in {(id E)} by A3, TARSKI:def_1;
hence [x,y] in h by A2, ZFMISC_1:def_2; ::_thesis: verum
end;
hence O = dom h by RELSET_1:9; ::_thesis: verum
end;
assume O = {} ; ::_thesis: h = {}
hence h = {} ; ::_thesis: verum
end;
now__::_thesis:_for_x,_y1,_y2_being_set_st_[x,y1]_in_h_&_[x,y2]_in_h_holds_
y1_=_y2
let x, y1, y2 be set ; ::_thesis: ( [x,y1] in h & [x,y2] in h implies y1 = y2 )
assume that
A4: [x,y1] in h and
A5: [x,y2] in h ; ::_thesis: y1 = y2
consider x9, y9 being set such that
x9 in O and
A6: ( y9 in {(id E)} & [x,y1] = [x9,y9] ) by A4, ZFMISC_1:def_2;
A7: ( y9 = id E & y1 = y9 ) by A6, TARSKI:def_1, XTUPLE_0:1;
consider x99, y99 being set such that
x99 in O and
A8: y99 in {(id E)} and
A9: [x,y2] = [x99,y99] by A5, ZFMISC_1:def_2;
y99 = id E by A8, TARSKI:def_1;
hence y1 = y2 by A9, A7, XTUPLE_0:1; ::_thesis: verum
end;
then reconsider h = h as PartFunc of O,(Funcs (E,E)) by FUNCT_1:def_1;
h is Action of O,E by A1, FUNCT_2:def_1;
hence [:O,{(id E)}:] is Action of O,E ; ::_thesis: verum
end;
Lm3: for O being set
for G being strict Group ex H being non empty HGrWOpStr over O st
( H is strict & H is distributive & H is Group-like & H is associative & G = multMagma(# the carrier of H, the multF of H #) )
proof
let O be set ; ::_thesis: for G being strict Group ex H being non empty HGrWOpStr over O st
( H is strict & H is distributive & H is Group-like & H is associative & G = multMagma(# the carrier of H, the multF of H #) )
let G be strict Group; ::_thesis: ex H being non empty HGrWOpStr over O st
( H is strict & H is distributive & H is Group-like & H is associative & G = multMagma(# the carrier of H, the multF of H #) )
reconsider h = [:O,{(id the carrier of G)}:] as Action of O, the carrier of G by Lm2;
set A = the carrier of G;
set m = the multF of G;
set GO = HGrWOpStr(# the carrier of G, the multF of G,h #);
reconsider GO = HGrWOpStr(# the carrier of G, the multF of G,h #) as non empty HGrWOpStr over O ;
reconsider G9 = GO as non empty multMagma ;
A1: now__::_thesis:_ex_e9_being_Element_of_G9_st_
for_h9_being_Element_of_G9_holds_
(_h9_*_e9_=_h9_&_e9_*_h9_=_h9_&_ex_g9_being_Element_of_G9_st_
(_h9_*_g9_=_e9_&_g9_*_h9_=_e9_)_)
set e = 1_ G;
reconsider e9 = 1_ G as Element of G9 ;
take e9 = e9; ::_thesis: for h9 being Element of G9 holds
( h9 * e9 = h9 & e9 * h9 = h9 & ex g9 being Element of G9 st
( h9 * g9 = e9 & g9 * h9 = e9 ) )
let h9 be Element of G9; ::_thesis: ( h9 * e9 = h9 & e9 * h9 = h9 & ex g9 being Element of G9 st
( h9 * g9 = e9 & g9 * h9 = e9 ) )
reconsider h = h9 as Element of G ;
set g = h " ;
reconsider g9 = h " as Element of G9 ;
h9 * e9 = h * (1_ G)
.= h by GROUP_1:def_4 ;
hence h9 * e9 = h9 ; ::_thesis: ( e9 * h9 = h9 & ex g9 being Element of G9 st
( h9 * g9 = e9 & g9 * h9 = e9 ) )
e9 * h9 = (1_ G) * h
.= h by GROUP_1:def_4 ;
hence e9 * h9 = h9 ; ::_thesis: ex g9 being Element of G9 st
( h9 * g9 = e9 & g9 * h9 = e9 )
take g9 = g9; ::_thesis: ( h9 * g9 = e9 & g9 * h9 = e9 )
h9 * g9 = h * (h ")
.= 1_ G by GROUP_1:def_5 ;
hence h9 * g9 = e9 ; ::_thesis: g9 * h9 = e9
g9 * h9 = (h ") * h
.= 1_ G by GROUP_1:def_5 ;
hence g9 * h9 = e9 ; ::_thesis: verum
end;
take GO ; ::_thesis: ( GO is strict & GO is distributive & GO is Group-like & GO is associative & G = multMagma(# the carrier of GO, the multF of GO #) )
A2: now__::_thesis:_for_G99_being_Group
for_a_being_Action_of_O,_the_carrier_of_G99_st_a_=_the_action_of_GO_&_multMagma(#_the_carrier_of_G99,_the_multF_of_G99_#)_=_multMagma(#_the_carrier_of_GO,_the_multF_of_GO_#)_holds_
a_is_distributive
let G99 be Group; ::_thesis: for a being Action of O, the carrier of G99 st a = the action of GO & multMagma(# the carrier of G99, the multF of G99 #) = multMagma(# the carrier of GO, the multF of GO #) holds
a is distributive
let a be Action of O, the carrier of G99; ::_thesis: ( a = the action of GO & multMagma(# the carrier of G99, the multF of G99 #) = multMagma(# the carrier of GO, the multF of GO #) implies a is distributive )
assume A3: a = the action of GO ; ::_thesis: ( multMagma(# the carrier of G99, the multF of G99 #) = multMagma(# the carrier of GO, the multF of GO #) implies a is distributive )
assume A4: multMagma(# the carrier of G99, the multF of G99 #) = multMagma(# the carrier of GO, the multF of GO #) ; ::_thesis: a is distributive
now__::_thesis:_for_o_being_Element_of_O_st_o_in_O_holds_
a_._o_is_Homomorphism_of_G99,G99
let o be Element of O; ::_thesis: ( o in O implies a . o is Homomorphism of G99,G99 )
assume o in O ; ::_thesis: a . o is Homomorphism of G99,G99
then o in dom h by FUNCT_2:def_1;
then [o,(h . o)] in [:O,{(id the carrier of G99)}:] by A4, FUNCT_1:1;
then consider x, y being set such that
x in O and
A5: ( y in {(id the carrier of G99)} & [o,(h . o)] = [x,y] ) by ZFMISC_1:def_2;
( y = id the carrier of G99 & h . o = y ) by A5, TARSKI:def_1, XTUPLE_0:1;
hence a . o is Homomorphism of G99,G99 by A3, GROUP_6:38; ::_thesis: verum
end;
hence a is distributive by Def4; ::_thesis: verum
end;
now__::_thesis:_for_x9,_y9,_z9_being_Element_of_G9_holds_(x9_*_y9)_*_z9_=_x9_*_(y9_*_z9)
let x9, y9, z9 be Element of G9; ::_thesis: (x9 * y9) * z9 = x9 * (y9 * z9)
reconsider x = x9, y = y9, z = z9 as Element of G ;
(x9 * y9) * z9 = (x * y) * z
.= x * (y * z) by GROUP_1:def_3 ;
hence (x9 * y9) * z9 = x9 * (y9 * z9) ; ::_thesis: verum
end;
hence ( GO is strict & GO is distributive & GO is Group-like & GO is associative & G = multMagma(# the carrier of GO, the multF of GO #) ) by A1, A2, Def5, GROUP_1:def_2, GROUP_1:def_3; ::_thesis: verum
end;
registration
let O be set ;
cluster non empty Group-like associative strict distributive for HGrWOpStr over O;
existence
ex b1 being non empty HGrWOpStr over O st
( b1 is strict & b1 is distributive & b1 is Group-like & b1 is associative )
proof
set G = the strict Group;
consider H being non empty HGrWOpStr over O such that
A1: ( H is strict & H is distributive & H is Group-like & H is associative ) and
multMagma(# the carrier of H, the multF of H #) = the strict Group by Lm3;
take H ; ::_thesis: ( H is strict & H is distributive & H is Group-like & H is associative )
thus ( H is strict & H is distributive & H is Group-like & H is associative ) by A1; ::_thesis: verum
end;
end;
definition
let O be set ;
mode GroupWithOperators of O is non empty Group-like associative distributive HGrWOpStr over O;
end;
definition
let O be set ;
let G be GroupWithOperators of O;
let o be Element of O;
funcG ^ o -> Homomorphism of G,G equals :Def6: :: GROUP_9:def 6
the action of G . o if o in O
otherwise id the carrier of G;
correctness
coherence
( ( o in O implies the action of G . o is Homomorphism of G,G ) & ( not o in O implies id the carrier of G is Homomorphism of G,G ) );
consistency
for b1 being Homomorphism of G,G holds verum;
proof
now__::_thesis:_(_o_in_O_implies_the_action_of_G_._o_is_Homomorphism_of_G,G_)
assume A1: o in O ; ::_thesis: the action of G . o is Homomorphism of G,G
consider G9 being Group such that
A2: multMagma(# the carrier of G9, the multF of G9 #) = multMagma(# the carrier of G, the multF of G #) ;
reconsider a = the action of G as Action of O, the carrier of G9 by A2;
a is distributive by A2, Def5;
then reconsider f9 = a . o as Homomorphism of G9,G9 by A1, Def4;
reconsider f = f9 as Function of G,G by A2;
now__::_thesis:_for_g1,_g2_being_Element_of_G_holds_f_._(g1_*_g2)_=_(f_._g1)_*_(f_._g2)
let g1, g2 be Element of G; ::_thesis: f . (g1 * g2) = (f . g1) * (f . g2)
reconsider g19 = g1, g29 = g2 as Element of G9 by A2;
f . (g1 * g2) = f9 . (g19 * g29) by A2
.= (f9 . g19) * (f9 . g29) by GROUP_6:def_6
.= the multF of G . ((f . g1),(f . g2)) by A2 ;
hence f . (g1 * g2) = (f . g1) * (f . g2) ; ::_thesis: verum
end;
hence the action of G . o is Homomorphism of G,G by GROUP_6:def_6; ::_thesis: verum
end;
hence ( ( o in O implies the action of G . o is Homomorphism of G,G ) & ( not o in O implies id the carrier of G is Homomorphism of G,G ) & ( for b1 being Homomorphism of G,G holds verum ) ) by GROUP_6:38; ::_thesis: verum
end;
end;
:: deftheorem Def6 defines ^ GROUP_9:def_6_:_
for O being set
for G being GroupWithOperators of O
for o being Element of O holds
( ( o in O implies G ^ o = the action of G . o ) & ( not o in O implies G ^ o = id the carrier of G ) );
definition
let O be set ;
let G be GroupWithOperators of O;
mode StableSubgroup of G -> non empty Group-like associative distributive HGrWOpStr over O means :Def7: :: GROUP_9:def 7
( it is Subgroup of G & ( for o being Element of O holds it ^ o = (G ^ o) | the carrier of it ) );
correctness
existence
ex b1 being non empty Group-like associative distributive HGrWOpStr over O st
( b1 is Subgroup of G & ( for o being Element of O holds b1 ^ o = (G ^ o) | the carrier of b1 ) );
proof
set H = G;
take G ; ::_thesis: ( G is Subgroup of G & ( for o being Element of O holds G ^ o = (G ^ o) | the carrier of G ) )
thus ( G is Subgroup of G & ( for o being Element of O holds G ^ o = (G ^ o) | the carrier of G ) ) by GROUP_2:54; ::_thesis: verum
end;
end;
:: deftheorem Def7 defines StableSubgroup GROUP_9:def_7_:_
for O being set
for G being GroupWithOperators of O
for b3 being non empty Group-like associative distributive HGrWOpStr over O holds
( b3 is StableSubgroup of G iff ( b3 is Subgroup of G & ( for o being Element of O holds b3 ^ o = (G ^ o) | the carrier of b3 ) ) );
Lm4: for O being set
for G being GroupWithOperators of O holds HGrWOpStr(# the carrier of G, the multF of G, the action of G #) is StableSubgroup of G
proof
let O be set ; ::_thesis: for G being GroupWithOperators of O holds HGrWOpStr(# the carrier of G, the multF of G, the action of G #) is StableSubgroup of G
let G be GroupWithOperators of O; ::_thesis: HGrWOpStr(# the carrier of G, the multF of G, the action of G #) is StableSubgroup of G
reconsider G9 = HGrWOpStr(# the carrier of G, the multF of G, the action of G #) as non empty multMagma ;
A1: now__::_thesis:_ex_e9_being_Element_of_G9_st_
for_h9_being_Element_of_G9_holds_
(_h9_*_e9_=_h9_&_e9_*_h9_=_h9_&_ex_g9_being_Element_of_G9_st_
(_h9_*_g9_=_e9_&_g9_*_h9_=_e9_)_)
set e = 1_ G;
reconsider e9 = 1_ G as Element of G9 ;
take e9 = e9; ::_thesis: for h9 being Element of G9 holds
( h9 * e9 = h9 & e9 * h9 = h9 & ex g9 being Element of G9 st
( h9 * g9 = e9 & g9 * h9 = e9 ) )
let h9 be Element of G9; ::_thesis: ( h9 * e9 = h9 & e9 * h9 = h9 & ex g9 being Element of G9 st
( h9 * g9 = e9 & g9 * h9 = e9 ) )
reconsider h = h9 as Element of G ;
set g = h " ;
reconsider g9 = h " as Element of G9 ;
h9 * e9 = h * (1_ G)
.= h by GROUP_1:def_4 ;
hence h9 * e9 = h9 ; ::_thesis: ( e9 * h9 = h9 & ex g9 being Element of G9 st
( h9 * g9 = e9 & g9 * h9 = e9 ) )
e9 * h9 = (1_ G) * h
.= h by GROUP_1:def_4 ;
hence e9 * h9 = h9 ; ::_thesis: ex g9 being Element of G9 st
( h9 * g9 = e9 & g9 * h9 = e9 )
take g9 = g9; ::_thesis: ( h9 * g9 = e9 & g9 * h9 = e9 )
h9 * g9 = h * (h ")
.= 1_ G by GROUP_1:def_5 ;
hence h9 * g9 = e9 ; ::_thesis: g9 * h9 = e9
g9 * h9 = (h ") * h
.= 1_ G by GROUP_1:def_5 ;
hence g9 * h9 = e9 ; ::_thesis: verum
end;
now__::_thesis:_for_x9,_y9,_z9_being_Element_of_G9_holds_(x9_*_y9)_*_z9_=_x9_*_(y9_*_z9)
let x9, y9, z9 be Element of G9; ::_thesis: (x9 * y9) * z9 = x9 * (y9 * z9)
reconsider x = x9, y = y9, z = z9 as Element of G ;
(x9 * y9) * z9 = (x * y) * z
.= x * (y * z) by GROUP_1:def_3 ;
hence (x9 * y9) * z9 = x9 * (y9 * z9) ; ::_thesis: verum
end;
then reconsider G9 = G9 as non empty Group-like associative strict HGrWOpStr over O by A1, GROUP_1:def_2, GROUP_1:def_3;
for G being Group
for a being Action of O, the carrier of G st a = the action of G9 & multMagma(# the carrier of G, the multF of G #) = multMagma(# the carrier of G9, the multF of G9 #) holds
a is distributive by Def5;
then reconsider G9 = G9 as non empty Group-like associative distributive HGrWOpStr over O by Def5;
A2: now__::_thesis:_for_o_being_Element_of_O_holds_G9_^_o_=_(G_^_o)_|_the_carrier_of_G9
let o be Element of O; ::_thesis: G9 ^ o = (G ^ o) | the carrier of G9
A3: now__::_thesis:_G9_^_o_=_G_^_o
percases ( o in O or not o in O ) ;
supposeA4: o in O ; ::_thesis: G9 ^ o = G ^ o
then G9 ^ o = the action of G9 . o by Def6;
hence G9 ^ o = G ^ o by A4, Def6; ::_thesis: verum
end;
supposeA5: not o in O ; ::_thesis: G9 ^ o = G ^ o
then G9 ^ o = id the carrier of G9 by Def6;
hence G9 ^ o = G ^ o by A5, Def6; ::_thesis: verum
end;
end;
end;
dom (G9 ^ o) = the carrier of G by FUNCT_2:def_1;
hence G9 ^ o = (G ^ o) | the carrier of G9 by A3, RELAT_1:69; ::_thesis: verum
end;
dom the multF of G = [: the carrier of G9, the carrier of G9:] by FUNCT_2:def_1;
then the multF of G9 = the multF of G || the carrier of G9 by RELAT_1:68;
then G9 is Subgroup of G by GROUP_2:def_5;
hence HGrWOpStr(# the carrier of G, the multF of G, the action of G #) is StableSubgroup of G by A2, Def7; ::_thesis: verum
end;
registration
let O be set ;
let G be GroupWithOperators of O;
cluster non empty unital Group-like associative strict distributive for StableSubgroup of G;
correctness
existence
ex b1 being StableSubgroup of G st b1 is strict ;
proof
reconsider G9 = HGrWOpStr(# the carrier of G, the multF of G, the action of G #) as StableSubgroup of G by Lm4;
take G9 ; ::_thesis: G9 is strict
thus G9 is strict ; ::_thesis: verum
end;
end;
Lm5: for O being set
for G being GroupWithOperators of O
for H1, H2 being strict StableSubgroup of G st the carrier of H1 = the carrier of H2 holds
H1 = H2
proof
let O be set ; ::_thesis: for G being GroupWithOperators of O
for H1, H2 being strict StableSubgroup of G st the carrier of H1 = the carrier of H2 holds
H1 = H2
let G be GroupWithOperators of O; ::_thesis: for H1, H2 being strict StableSubgroup of G st the carrier of H1 = the carrier of H2 holds
H1 = H2
let H1, H2 be strict StableSubgroup of G; ::_thesis: ( the carrier of H1 = the carrier of H2 implies H1 = H2 )
reconsider H19 = H1, H29 = H2 as Subgroup of G by Def7;
A1: dom the action of H2 = O by FUNCT_2:def_1
.= dom the action of H1 by FUNCT_2:def_1 ;
assume A2: the carrier of H1 = the carrier of H2 ; ::_thesis: H1 = H2
A3: now__::_thesis:_for_x_being_set_st_x_in_dom_the_action_of_H2_holds_
the_action_of_H1_._x_=_the_action_of_H2_._x
let x be set ; ::_thesis: ( x in dom the action of H2 implies the action of H1 . x = the action of H2 . x )
assume A4: x in dom the action of H2 ; ::_thesis: the action of H1 . x = the action of H2 . x
then reconsider o = x as Element of O ;
A5: H1 ^ o = the action of H1 . o by A4, Def6;
H1 ^ o = (G ^ o) | the carrier of H2 by A2, Def7
.= H2 ^ o by Def7 ;
hence the action of H1 . x = the action of H2 . x by A4, A5, Def6; ::_thesis: verum
end;
multMagma(# the carrier of H19, the multF of H19 #) = multMagma(# the carrier of H29, the multF of H29 #) by A2, GROUP_2:59;
hence H1 = H2 by A1, A3, FUNCT_1:2; ::_thesis: verum
end;
definition
let O be set ;
let G be GroupWithOperators of O;
func (1). G -> strict StableSubgroup of G means :Def8: :: GROUP_9:def 8
the carrier of it = {(1_ G)};
existence
ex b1 being strict StableSubgroup of G st the carrier of b1 = {(1_ G)}
proof
set G9 = (1). G;
consider H being non empty HGrWOpStr over O such that
A1: ( H is strict & H is distributive & H is Group-like & H is associative ) and
A2: (1). G = multMagma(# the carrier of H, the multF of H #) by Lm3;
reconsider H = H as strict GroupWithOperators of O by A1;
A3: the carrier of H c= the carrier of G by A2, GROUP_2:def_5;
the multF of H = the multF of G || the carrier of H by A2, GROUP_2:def_5;
then A4: H is Subgroup of G by A3, GROUP_2:def_5;
now__::_thesis:_for_o_being_Element_of_O_holds_H_^_o_=_(G_^_o)_|_the_carrier_of_H
let o be Element of O; ::_thesis: H ^ o = (G ^ o) | the carrier of H
reconsider f9 = H ^ o, f = (G ^ o) | the carrier of H as Function ;
A5: dom f = dom ((G ^ o) * (id the carrier of H)) by RELAT_1:65
.= (dom (G ^ o)) /\ the carrier of H by FUNCT_1:19
.= the carrier of G /\ the carrier of H by FUNCT_2:def_1
.= the carrier of ((1). G) by A2, A3, XBOOLE_1:28 ;
A6: now__::_thesis:_for_x_being_set_st_x_in_dom_f_holds_
f_._x_=_f9_._x
let x be set ; ::_thesis: ( x in dom f implies f . x = f9 . x )
assume A7: x in dom f ; ::_thesis: f . x = f9 . x
then A8: x in dom (id the carrier of H) by A2, A5;
x in {(1_ G)} by A5, A7, GROUP_2:def_7;
then A9: x = 1_ G by TARSKI:def_1;
then x = 1_ H by A4, GROUP_2:44;
then A10: f9 . x = 1_ H by GROUP_6:31;
f . x = ((G ^ o) * (id the carrier of H)) . x by RELAT_1:65
.= (G ^ o) . ((id the carrier of H) . x) by A8, FUNCT_1:13
.= (G ^ o) . x by A2, A5, A7, FUNCT_1:18
.= 1_ G by A9, GROUP_6:31 ;
hence f . x = f9 . x by A4, A10, GROUP_2:44; ::_thesis: verum
end;
dom f9 = the carrier of ((1). G) by A2, FUNCT_2:def_1;
hence H ^ o = (G ^ o) | the carrier of H by A5, A6, FUNCT_1:2; ::_thesis: verum
end;
then reconsider H = H as strict StableSubgroup of G by A4, Def7;
take H ; ::_thesis: the carrier of H = {(1_ G)}
thus the carrier of H = {(1_ G)} by A2, GROUP_2:def_7; ::_thesis: verum
end;
uniqueness
for b1, b2 being strict StableSubgroup of G st the carrier of b1 = {(1_ G)} & the carrier of b2 = {(1_ G)} holds
b1 = b2 by Lm5;
end;
:: deftheorem Def8 defines (1). GROUP_9:def_8_:_
for O being set
for G being GroupWithOperators of O
for b3 being strict StableSubgroup of G holds
( b3 = (1). G iff the carrier of b3 = {(1_ G)} );
definition
let O be set ;
let G be GroupWithOperators of O;
func (Omega). G -> strict StableSubgroup of G equals :: GROUP_9:def 9
HGrWOpStr(# the carrier of G, the multF of G, the action of G #);
correctness
coherence
HGrWOpStr(# the carrier of G, the multF of G, the action of G #) is strict StableSubgroup of G;
by Lm4;
end;
:: deftheorem defines (Omega). GROUP_9:def_9_:_
for O being set
for G being GroupWithOperators of O holds (Omega). G = HGrWOpStr(# the carrier of G, the multF of G, the action of G #);
definition
let O be set ;
let G be GroupWithOperators of O;
let IT be StableSubgroup of G;
attrIT is normal means :Def10: :: GROUP_9:def 10
for H being strict Subgroup of G st H = multMagma(# the carrier of IT, the multF of IT #) holds
H is normal ;
end;
:: deftheorem Def10 defines normal GROUP_9:def_10_:_
for O being set
for G being GroupWithOperators of O
for IT being StableSubgroup of G holds
( IT is normal iff for H being strict Subgroup of G st H = multMagma(# the carrier of IT, the multF of IT #) holds
H is normal );
registration
let O be set ;
let G be GroupWithOperators of O;
cluster non empty unital Group-like associative strict distributive normal for StableSubgroup of G;
existence
ex b1 being StableSubgroup of G st
( b1 is strict & b1 is normal )
proof
set H = (1). G;
set H9 = (1). G;
reconsider H = (1). G as StableSubgroup of G ;
take (1). G ; ::_thesis: ( (1). G is strict & (1). G is normal )
now__::_thesis:_for_H99_being_strict_Subgroup_of_G_st_H99_=_multMagma(#_the_carrier_of_H,_the_multF_of_H_#)_holds_
H99_is_normal
reconsider G9 = G as Group ;
let H99 be strict Subgroup of G; ::_thesis: ( H99 = multMagma(# the carrier of H, the multF of H #) implies H99 is normal )
assume A1: H99 = multMagma(# the carrier of H, the multF of H #) ; ::_thesis: H99 is normal
A2: the multF of ((1). G9) = the multF of G9 || the carrier of ((1). G9) by GROUP_2:def_5;
the carrier of ((1). G9) = {(1_ G9)} by GROUP_2:def_7
.= the carrier of ((1). G) by Def8 ;
hence H99 is normal by A1, A2, GROUP_2:def_5; ::_thesis: verum
end;
hence ( (1). G is strict & (1). G is normal ) by Def10; ::_thesis: verum
end;
end;
registration
let O be set ;
let G be GroupWithOperators of O;
let H be StableSubgroup of G;
cluster non empty unital Group-like associative distributive normal for StableSubgroup of H;
existence
ex b1 being StableSubgroup of H st b1 is normal
proof
reconsider H9 = (1). H as GroupWithOperators of O ;
reconsider H9 = H9 as StableSubgroup of H ;
take H9 ; ::_thesis: H9 is normal
now__::_thesis:_for_H99_being_strict_Subgroup_of_H_st_multMagma(#_the_carrier_of_H9,_the_multF_of_H9_#)_=_H99_holds_
H99_is_normal
let H99 be strict Subgroup of H; ::_thesis: ( multMagma(# the carrier of H9, the multF of H9 #) = H99 implies H99 is normal )
reconsider H = H as Group ;
assume multMagma(# the carrier of H9, the multF of H9 #) = H99 ; ::_thesis: H99 is normal
then the carrier of H99 = {(1_ H)} by Def8;
then H99 = (1). H by GROUP_2:def_7;
hence H99 is normal ; ::_thesis: verum
end;
hence H9 is normal by Def10; ::_thesis: verum
end;
end;
registration
let O be set ;
let G be GroupWithOperators of O;
cluster (1). G -> strict normal ;
correctness
coherence
(1). G is normal ;
proof
now__::_thesis:_for_H_being_strict_Subgroup_of_G_st_H_=_multMagma(#_the_carrier_of_((1)._G),_the_multF_of_((1)._G)_#)_holds_
H_is_normal
reconsider G9 = G as Group ;
let H be strict Subgroup of G; ::_thesis: ( H = multMagma(# the carrier of ((1). G), the multF of ((1). G) #) implies H is normal )
reconsider H9 = H as strict Subgroup of G9 ;
assume H = multMagma(# the carrier of ((1). G), the multF of ((1). G) #) ; ::_thesis: H is normal
then the carrier of H = {(1_ G)} by Def8;
then H9 = (1). G9 by GROUP_2:def_7;
hence H is normal ; ::_thesis: verum
end;
hence (1). G is normal by Def10; ::_thesis: verum
end;
cluster (Omega). G -> strict normal ;
correctness
coherence
(Omega). G is normal ;
proof
now__::_thesis:_for_H_being_strict_Subgroup_of_G_st_H_=_multMagma(#_the_carrier_of_((Omega)._G),_the_multF_of_((Omega)._G)_#)_holds_
H_is_normal
reconsider G9 = G as Group ;
let H be strict Subgroup of G; ::_thesis: ( H = multMagma(# the carrier of ((Omega). G), the multF of ((Omega). G) #) implies H is normal )
reconsider H9 = H as strict Subgroup of G9 ;
assume H = multMagma(# the carrier of ((Omega). G), the multF of ((Omega). G) #) ; ::_thesis: H is normal
then H9 = (Omega). G9 ;
hence H is normal ; ::_thesis: verum
end;
hence (Omega). G is normal by Def10; ::_thesis: verum
end;
end;
definition
let O be set ;
let G be GroupWithOperators of O;
func the_stable_subgroups_of G -> set means :Def11: :: GROUP_9:def 11
for x being set holds
( x in it iff x is strict StableSubgroup of G );
existence
ex b1 being set st
for x being set holds
( x in b1 iff x is strict StableSubgroup of G )
proof
defpred S1[ set , set ] means ex H being strict StableSubgroup of G st
( $2 = H & $1 = the carrier of H );
defpred S2[ set ] means ex H being strict StableSubgroup of G st $1 = the carrier of H;
consider B being set such that
A1: for x being set holds
( x in B iff ( x in bool the carrier of G & S2[x] ) ) from XBOOLE_0:sch_1();
A2: for x, y1, y2 being set st S1[x,y1] & S1[x,y2] holds
y1 = y2 by Lm5;
consider f being Function such that
A3: for x, y being set holds
( [x,y] in f iff ( x in B & S1[x,y] ) ) from FUNCT_1:sch_1(A2);
for x being set holds
( x in B iff ex y being set st [x,y] in f )
proof
let x be set ; ::_thesis: ( x in B iff ex y being set st [x,y] in f )
thus ( x in B implies ex y being set st [x,y] in f ) ::_thesis: ( ex y being set st [x,y] in f implies x in B )
proof
assume A4: x in B ; ::_thesis: ex y being set st [x,y] in f
then consider H being strict StableSubgroup of G such that
A5: x = the carrier of H by A1;
reconsider y = H as set ;
take y ; ::_thesis: [x,y] in f
thus [x,y] in f by A3, A4, A5; ::_thesis: verum
end;
given y being set such that A6: [x,y] in f ; ::_thesis: x in B
thus x in B by A3, A6; ::_thesis: verum
end;
then A7: B = dom f by XTUPLE_0:def_12;
for y being set holds
( y in rng f iff y is strict StableSubgroup of G )
proof
let y be set ; ::_thesis: ( y in rng f iff y is strict StableSubgroup of G )
thus ( y in rng f implies y is strict StableSubgroup of G ) ::_thesis: ( y is strict StableSubgroup of G implies y in rng f )
proof
assume y in rng f ; ::_thesis: y is strict StableSubgroup of G
then consider x being set such that
A8: ( x in dom f & y = f . x ) by FUNCT_1:def_3;
[x,y] in f by A8, FUNCT_1:def_2;
then ex H being strict StableSubgroup of G st
( y = H & x = the carrier of H ) by A3;
hence y is strict StableSubgroup of G ; ::_thesis: verum
end;
assume y is strict StableSubgroup of G ; ::_thesis: y in rng f
then reconsider H = y as strict StableSubgroup of G ;
reconsider x = the carrier of H as set ;
H is Subgroup of G by Def7;
then the carrier of H c= the carrier of G by GROUP_2:def_5;
then A9: x in dom f by A1, A7;
then [x,y] in f by A3, A7;
then y = f . x by A9, FUNCT_1:def_2;
hence y in rng f by A9, FUNCT_1:def_3; ::_thesis: verum
end;
hence ex b1 being set st
for x being set holds
( x in b1 iff x is strict StableSubgroup of G ) ; ::_thesis: verum
end;
uniqueness
for b1, b2 being set st ( for x being set holds
( x in b1 iff x is strict StableSubgroup of G ) ) & ( for x being set holds
( x in b2 iff x is strict StableSubgroup of G ) ) holds
b1 = b2
proof
defpred S1[ set ] means $1 is strict StableSubgroup of G;
let A1, A2 be set ; ::_thesis: ( ( for x being set holds
( x in A1 iff x is strict StableSubgroup of G ) ) & ( for x being set holds
( x in A2 iff x is strict StableSubgroup of G ) ) implies A1 = A2 )
assume A10: for x being set holds
( x in A1 iff S1[x] ) ; ::_thesis: ( ex x being set st
( ( x in A2 implies x is strict StableSubgroup of G ) implies ( x is strict StableSubgroup of G & not x in A2 ) ) or A1 = A2 )
assume A11: for x being set holds
( x in A2 iff S1[x] ) ; ::_thesis: A1 = A2
thus A1 = A2 from XBOOLE_0:sch_2(A10, A11); ::_thesis: verum
end;
end;
:: deftheorem Def11 defines the_stable_subgroups_of GROUP_9:def_11_:_
for O being set
for G being GroupWithOperators of O
for b3 being set holds
( b3 = the_stable_subgroups_of G iff for x being set holds
( x in b3 iff x is strict StableSubgroup of G ) );
registration
let O be set ;
let G be GroupWithOperators of O;
cluster the_stable_subgroups_of G -> non empty ;
correctness
coherence
not the_stable_subgroups_of G is empty ;
proof
(1). G in the_stable_subgroups_of G by Def11;
hence not the_stable_subgroups_of G is empty ; ::_thesis: verum
end;
end;
definition
let IT be Group;
attrIT is simple means :Def12: :: GROUP_9:def 12
( not IT is trivial & ( for H being strict normal Subgroup of IT holds
( not H <> (Omega). IT or not H <> (1). IT ) ) );
end;
:: deftheorem Def12 defines simple GROUP_9:def_12_:_
for IT being Group holds
( IT is simple iff ( not IT is trivial & ( for H being strict normal Subgroup of IT holds
( not H <> (Omega). IT or not H <> (1). IT ) ) ) );
Lm6: Group_of_Perm 2 is simple
proof
set G = Group_of_Perm 2;
A1: now__::_thesis:_for_H_being_strict_normal_Subgroup_of_Group_of_Perm_2_st_H_<>_(Omega)._(Group_of_Perm_2)_holds_
not_H_<>_(1)._(Group_of_Perm_2)
let H be strict normal Subgroup of Group_of_Perm 2; ::_thesis: ( H <> (Omega). (Group_of_Perm 2) implies not H <> (1). (Group_of_Perm 2) )
assume A2: H <> (Omega). (Group_of_Perm 2) ; ::_thesis: not H <> (1). (Group_of_Perm 2)
assume A3: H <> (1). (Group_of_Perm 2) ; ::_thesis: contradiction
1_ (Group_of_Perm 2) in H by GROUP_2:46;
then 1_ (Group_of_Perm 2) in the carrier of H by STRUCT_0:def_5;
then {(1_ (Group_of_Perm 2))} c= the carrier of H by ZFMISC_1:31;
then {<*1,2*>} c= the carrier of H by FINSEQ_2:52, MATRIX_2:24;
then A4: <*1,2*> in the carrier of H by ZFMISC_1:31;
the carrier of H c= the carrier of (Group_of_Perm 2) by GROUP_2:def_5;
then A5: the carrier of H c= {<*1,2*>,<*2,1*>} by MATRIX_2:def_10, MATRIX_7:3;
percases ( the carrier of H = {} or the carrier of H = {<*1,2*>} or the carrier of H = {<*2,1*>} or the carrier of H = {<*1,2*>,<*2,1*>} ) by A5, ZFMISC_1:36;
suppose the carrier of H = {} ; ::_thesis: contradiction
hence contradiction ; ::_thesis: verum
end;
suppose the carrier of H = {<*1,2*>} ; ::_thesis: contradiction
then {(1_ (Group_of_Perm 2))} = the carrier of H by FINSEQ_2:52, MATRIX_2:24;
hence contradiction by A3, GROUP_2:def_7; ::_thesis: verum
end;
suppose the carrier of H = {<*2,1*>} ; ::_thesis: contradiction
then <*2,1*> . 1 = <*1,2*> . 1 by A4, TARSKI:def_1;
then 2 = <*1,2*> . 1 by FINSEQ_1:44;
hence contradiction by FINSEQ_1:44; ::_thesis: verum
end;
suppose the carrier of H = {<*1,2*>,<*2,1*>} ; ::_thesis: contradiction
then the carrier of H = the carrier of (Group_of_Perm 2) by MATRIX_2:def_10, MATRIX_7:3;
hence contradiction by A2, GROUP_2:61; ::_thesis: verum
end;
end;
end;
now__::_thesis:_not_Group_of_Perm_2_is_trivial
assume Group_of_Perm 2 is trivial ; ::_thesis: contradiction
then consider e being set such that
A6: the carrier of (Group_of_Perm 2) = {e} by GROUP_6:def_2;
Permutations 2 = {e} by A6, MATRIX_2:def_10;
then <*2,1*> = <*1,2*> by MATRIX_7:3, ZFMISC_1:5;
then 2 = <*1,2*> . 1 by FINSEQ_1:44;
hence contradiction by FINSEQ_1:44; ::_thesis: verum
end;
hence Group_of_Perm 2 is simple by A1, Def12; ::_thesis: verum
end;
registration
cluster non empty strict unital Group-like associative simple for multMagma ;
existence
ex b1 being Group st
( b1 is strict & b1 is simple ) by Lm6;
end;
definition
let O be set ;
let IT be GroupWithOperators of O;
attrIT is simple means :Def13: :: GROUP_9:def 13
( not IT is trivial & ( for H being strict normal StableSubgroup of IT holds
( not H <> (Omega). IT or not H <> (1). IT ) ) );
end;
:: deftheorem Def13 defines simple GROUP_9:def_13_:_
for O being set
for IT being GroupWithOperators of O holds
( IT is simple iff ( not IT is trivial & ( for H being strict normal StableSubgroup of IT holds
( not H <> (Omega). IT or not H <> (1). IT ) ) ) );
Lm7: for O being set
for G being GroupWithOperators of O
for N being normal StableSubgroup of G holds multMagma(# the carrier of N, the multF of N #) is strict normal Subgroup of G
proof
let O be set ; ::_thesis: for G being GroupWithOperators of O
for N being normal StableSubgroup of G holds multMagma(# the carrier of N, the multF of N #) is strict normal Subgroup of G
let G be GroupWithOperators of O; ::_thesis: for N being normal StableSubgroup of G holds multMagma(# the carrier of N, the multF of N #) is strict normal Subgroup of G
let N be normal StableSubgroup of G; ::_thesis: multMagma(# the carrier of N, the multF of N #) is strict normal Subgroup of G
set H = multMagma(# the carrier of N, the multF of N #);
reconsider H = multMagma(# the carrier of N, the multF of N #) as non empty multMagma ;
now__::_thesis:_ex_e9_being_Element_of_H_st_
for_h9_being_Element_of_H_holds_
(_h9_*_e9_=_h9_&_e9_*_h9_=_h9_&_ex_g9_being_Element_of_H_st_
(_h9_*_g9_=_e9_&_g9_*_h9_=_e9_)_)
set e = 1_ N;
reconsider e9 = 1_ N as Element of H ;
take e9 = e9; ::_thesis: for h9 being Element of H holds
( h9 * e9 = h9 & e9 * h9 = h9 & ex g9 being Element of H st
( h9 * g9 = e9 & g9 * h9 = e9 ) )
let h9 be Element of H; ::_thesis: ( h9 * e9 = h9 & e9 * h9 = h9 & ex g9 being Element of H st
( h9 * g9 = e9 & g9 * h9 = e9 ) )
reconsider h = h9 as Element of N ;
set g = h " ;
reconsider g9 = h " as Element of H ;
h9 * e9 = h * (1_ N)
.= h by GROUP_1:def_4 ;
hence h9 * e9 = h9 ; ::_thesis: ( e9 * h9 = h9 & ex g9 being Element of H st
( h9 * g9 = e9 & g9 * h9 = e9 ) )
e9 * h9 = (1_ N) * h
.= h by GROUP_1:def_4 ;
hence e9 * h9 = h9 ; ::_thesis: ex g9 being Element of H st
( h9 * g9 = e9 & g9 * h9 = e9 )
take g9 = g9; ::_thesis: ( h9 * g9 = e9 & g9 * h9 = e9 )
h9 * g9 = h * (h ")
.= 1_ N by GROUP_1:def_5 ;
hence h9 * g9 = e9 ; ::_thesis: g9 * h9 = e9
g9 * h9 = (h ") * h
.= 1_ N by GROUP_1:def_5 ;
hence g9 * h9 = e9 ; ::_thesis: verum
end;
then reconsider H = H as non empty Group-like multMagma by GROUP_1:def_2;
N is Subgroup of G by Def7;
then ( the carrier of H c= the carrier of G & the multF of H = the multF of G || the carrier of H ) by GROUP_2:def_5;
then reconsider H = H as Subgroup of G by GROUP_2:def_5;
H is normal by Def10;
hence multMagma(# the carrier of N, the multF of N #) is strict normal Subgroup of G ; ::_thesis: verum
end;
Lm8: for G1, G2 being Group
for A1 being Subset of G1
for A2 being Subset of G2
for H1 being strict Subgroup of G1
for H2 being strict Subgroup of G2 st multMagma(# the carrier of G1, the multF of G1 #) = multMagma(# the carrier of G2, the multF of G2 #) & A1 = A2 & H1 = H2 holds
( A1 * H1 = A2 * H2 & H1 * A1 = H2 * A2 )
proof
let G1, G2 be Group; ::_thesis: for A1 being Subset of G1
for A2 being Subset of G2
for H1 being strict Subgroup of G1
for H2 being strict Subgroup of G2 st multMagma(# the carrier of G1, the multF of G1 #) = multMagma(# the carrier of G2, the multF of G2 #) & A1 = A2 & H1 = H2 holds
( A1 * H1 = A2 * H2 & H1 * A1 = H2 * A2 )
let A1 be Subset of G1; ::_thesis: for A2 being Subset of G2
for H1 being strict Subgroup of G1
for H2 being strict Subgroup of G2 st multMagma(# the carrier of G1, the multF of G1 #) = multMagma(# the carrier of G2, the multF of G2 #) & A1 = A2 & H1 = H2 holds
( A1 * H1 = A2 * H2 & H1 * A1 = H2 * A2 )
let A2 be Subset of G2; ::_thesis: for H1 being strict Subgroup of G1
for H2 being strict Subgroup of G2 st multMagma(# the carrier of G1, the multF of G1 #) = multMagma(# the carrier of G2, the multF of G2 #) & A1 = A2 & H1 = H2 holds
( A1 * H1 = A2 * H2 & H1 * A1 = H2 * A2 )
let H1 be strict Subgroup of G1; ::_thesis: for H2 being strict Subgroup of G2 st multMagma(# the carrier of G1, the multF of G1 #) = multMagma(# the carrier of G2, the multF of G2 #) & A1 = A2 & H1 = H2 holds
( A1 * H1 = A2 * H2 & H1 * A1 = H2 * A2 )
let H2 be strict Subgroup of G2; ::_thesis: ( multMagma(# the carrier of G1, the multF of G1 #) = multMagma(# the carrier of G2, the multF of G2 #) & A1 = A2 & H1 = H2 implies ( A1 * H1 = A2 * H2 & H1 * A1 = H2 * A2 ) )
assume A1: multMagma(# the carrier of G1, the multF of G1 #) = multMagma(# the carrier of G2, the multF of G2 #) ; ::_thesis: ( not A1 = A2 or not H1 = H2 or ( A1 * H1 = A2 * H2 & H1 * A1 = H2 * A2 ) )
A2: now__::_thesis:_for_A1,_B1_being_Subset_of_G1
for_A2,_B2_being_Subset_of_G2_st_A1_=_A2_&_B1_=_B2_holds_
{__(g_*_h)_where_g,_h_is_Element_of_G1_:_(_g_in_A1_&_h_in_B1_)__}__=__{__(g_*_h)_where_g,_h_is_Element_of_G2_:_(_g_in_A2_&_h_in_B2_)__}_
let A1, B1 be Subset of G1; ::_thesis: for A2, B2 being Subset of G2 st A1 = A2 & B1 = B2 holds
{ (g * h) where g, h is Element of G1 : ( g in A1 & h in B1 ) } = { (g * h) where g, h is Element of G2 : ( g in A2 & h in B2 ) }
let A2, B2 be Subset of G2; ::_thesis: ( A1 = A2 & B1 = B2 implies { (g * h) where g, h is Element of G1 : ( g in A1 & h in B1 ) } = { (g * h) where g, h is Element of G2 : ( g in A2 & h in B2 ) } )
set X = { (g * h) where g, h is Element of G1 : ( g in A1 & h in B1 ) } ;
set Y = { (g * h) where g, h is Element of G2 : ( g in A2 & h in B2 ) } ;
assume A3: ( A1 = A2 & B1 = B2 ) ; ::_thesis: { (g * h) where g, h is Element of G1 : ( g in A1 & h in B1 ) } = { (g * h) where g, h is Element of G2 : ( g in A2 & h in B2 ) }
A4: now__::_thesis:_for_x_being_set_st_x_in__{__(g_*_h)_where_g,_h_is_Element_of_G1_:_(_g_in_A1_&_h_in_B1_)__}__holds_
x_in__{__(g_*_h)_where_g,_h_is_Element_of_G2_:_(_g_in_A2_&_h_in_B2_)__}_
let x be set ; ::_thesis: ( x in { (g * h) where g, h is Element of G1 : ( g in A1 & h in B1 ) } implies x in { (g * h) where g, h is Element of G2 : ( g in A2 & h in B2 ) } )
assume x in { (g * h) where g, h is Element of G1 : ( g in A1 & h in B1 ) } ; ::_thesis: x in { (g * h) where g, h is Element of G2 : ( g in A2 & h in B2 ) }
then consider g, h being Element of G1 such that
A5: ( x = g * h & g in A1 & h in B1 ) ;
set h9 = h;
set g9 = g;
reconsider g9 = g, h9 = h as Element of G2 by A1;
g * h = g9 * h9 by A1;
hence x in { (g * h) where g, h is Element of G2 : ( g in A2 & h in B2 ) } by A3, A5; ::_thesis: verum
end;
now__::_thesis:_for_x_being_set_st_x_in__{__(g_*_h)_where_g,_h_is_Element_of_G2_:_(_g_in_A2_&_h_in_B2_)__}__holds_
x_in__{__(g_*_h)_where_g,_h_is_Element_of_G1_:_(_g_in_A1_&_h_in_B1_)__}_
let x be set ; ::_thesis: ( x in { (g * h) where g, h is Element of G2 : ( g in A2 & h in B2 ) } implies x in { (g * h) where g, h is Element of G1 : ( g in A1 & h in B1 ) } )
assume x in { (g * h) where g, h is Element of G2 : ( g in A2 & h in B2 ) } ; ::_thesis: x in { (g * h) where g, h is Element of G1 : ( g in A1 & h in B1 ) }
then consider g, h being Element of G2 such that
A6: ( x = g * h & g in A2 & h in B2 ) ;
reconsider g9 = g, h9 = h as Element of G1 by A1;
g * h = g9 * h9 by A1;
hence x in { (g * h) where g, h is Element of G1 : ( g in A1 & h in B1 ) } by A3, A6; ::_thesis: verum
end;
hence { (g * h) where g, h is Element of G1 : ( g in A1 & h in B1 ) } = { (g * h) where g, h is Element of G2 : ( g in A2 & h in B2 ) } by A4, TARSKI:1; ::_thesis: verum
end;
assume A7: A1 = A2 ; ::_thesis: ( not H1 = H2 or ( A1 * H1 = A2 * H2 & H1 * A1 = H2 * A2 ) )
assume A8: H1 = H2 ; ::_thesis: ( A1 * H1 = A2 * H2 & H1 * A1 = H2 * A2 )
hence A1 * H1 = A2 * H2 by A7, A2; ::_thesis: H1 * A1 = H2 * A2
thus H1 * A1 = H2 * A2 by A7, A8, A2; ::_thesis: verum
end;
registration
let O be set ;
cluster non empty unital Group-like associative strict distributive simple for HGrWOpStr over O;
existence
ex b1 being GroupWithOperators of O st
( b1 is strict & b1 is simple )
proof
set Gp2 = Group_of_Perm 2;
consider G being non empty HGrWOpStr over O such that
A1: ( G is strict & G is distributive & G is Group-like & G is associative ) and
A2: Group_of_Perm 2 = multMagma(# the carrier of G, the multF of G #) by Lm3;
reconsider G = G as strict GroupWithOperators of O by A1;
take G ; ::_thesis: ( G is strict & G is simple )
now__::_thesis:_G_is_simple
assume A3: not G is simple ; ::_thesis: contradiction
percases ( G is trivial or ex H being strict normal StableSubgroup of G st
( H <> (Omega). G & H <> (1). G ) ) by A3, Def13;
suppose G is trivial ; ::_thesis: contradiction
hence contradiction by A2, Def12, Lm6; ::_thesis: verum
end;
supposeA4: ex H being strict normal StableSubgroup of G st
( H <> (Omega). G & H <> (1). G ) ; ::_thesis: contradiction
reconsider G9 = G as Group ;
consider H being strict normal StableSubgroup of G such that
A5: H <> (Omega). G and
A6: H <> (1). G by A4;
reconsider H9 = multMagma(# the carrier of H, the multF of H #) as strict normal Subgroup of G by Lm7;
reconsider H9 = H9 as strict normal Subgroup of G9 ;
set H99 = H9;
( the carrier of H9 c= the carrier of G9 & the multF of H9 = the multF of G9 || the carrier of H9 ) by GROUP_2:def_5;
then reconsider H99 = H9 as strict Subgroup of Group_of_Perm 2 by A2, GROUP_2:def_5;
now__::_thesis:_for_A_being_Subset_of_(Group_of_Perm_2)_holds_A_*_H99_=_H99_*_A
let A be Subset of (Group_of_Perm 2); ::_thesis: A * H99 = H99 * A
reconsider A9 = A as Subset of G9 by A2;
A * H99 = A9 * H9 by A2, Lm8
.= H9 * A9 by GROUP_3:120 ;
hence A * H99 = H99 * A by A2, Lm8; ::_thesis: verum
end;
then reconsider H99 = H99 as strict normal Subgroup of Group_of_Perm 2 by GROUP_3:120;
A7: now__::_thesis:_not_H99_=_(1)._(Group_of_Perm_2)
reconsider e = 1_ (Group_of_Perm 2) as Element of G by A2;
A8: now__::_thesis:_for_h_being_Element_of_G_holds_
(_h_*_e_=_h_&_e_*_h_=_h_)
let h be Element of G; ::_thesis: ( h * e = h & e * h = h )
reconsider h9 = h as Element of (Group_of_Perm 2) by A2;
h * e = h9 * (1_ (Group_of_Perm 2)) by A2
.= h9 by GROUP_1:def_4 ;
hence h * e = h ; ::_thesis: e * h = h
e * h = (1_ (Group_of_Perm 2)) * h9 by A2
.= h9 by GROUP_1:def_4 ;
hence e * h = h ; ::_thesis: verum
end;
assume H99 = (1). (Group_of_Perm 2) ; ::_thesis: contradiction
then the carrier of H99 = {(1_ (Group_of_Perm 2))} by GROUP_2:def_7;
then the carrier of H = {(1_ G)} by A8, GROUP_1:def_4;
hence contradiction by A6, Def8; ::_thesis: verum
end;
H99 <> (Omega). (Group_of_Perm 2) by A2, A5, Lm5;
hence contradiction by A7, Def12, Lm6; ::_thesis: verum
end;
end;
end;
hence ( G is strict & G is simple ) ; ::_thesis: verum
end;
end;
definition
let O be set ;
let G be GroupWithOperators of O;
let N be normal StableSubgroup of G;
func Cosets N -> set means :Def14: :: GROUP_9:def 14
for H being strict normal Subgroup of G st H = multMagma(# the carrier of N, the multF of N #) holds
it = Cosets H;
existence
ex b1 being set st
for H being strict normal Subgroup of G st H = multMagma(# the carrier of N, the multF of N #) holds
b1 = Cosets H
proof
reconsider H = multMagma(# the carrier of N, the multF of N #) as strict normal Subgroup of G by Lm7;
set x = Cosets H;
take Cosets H ; ::_thesis: for H being strict normal Subgroup of G st H = multMagma(# the carrier of N, the multF of N #) holds
Cosets H = Cosets H
let H be strict normal Subgroup of G; ::_thesis: ( H = multMagma(# the carrier of N, the multF of N #) implies Cosets H = Cosets H )
assume H = multMagma(# the carrier of N, the multF of N #) ; ::_thesis: Cosets H = Cosets H
hence Cosets H = Cosets H ; ::_thesis: verum
end;
uniqueness
for b1, b2 being set st ( for H being strict normal Subgroup of G st H = multMagma(# the carrier of N, the multF of N #) holds
b1 = Cosets H ) & ( for H being strict normal Subgroup of G st H = multMagma(# the carrier of N, the multF of N #) holds
b2 = Cosets H ) holds
b1 = b2
proof
reconsider H = multMagma(# the carrier of N, the multF of N #) as strict normal Subgroup of G by Lm7;
let y1, y2 be set ; ::_thesis: ( ( for H being strict normal Subgroup of G st H = multMagma(# the carrier of N, the multF of N #) holds
y1 = Cosets H ) & ( for H being strict normal Subgroup of G st H = multMagma(# the carrier of N, the multF of N #) holds
y2 = Cosets H ) implies y1 = y2 )
assume for H being strict normal Subgroup of G st H = multMagma(# the carrier of N, the multF of N #) holds
y1 = Cosets H ; ::_thesis: ( ex H being strict normal Subgroup of G st
( H = multMagma(# the carrier of N, the multF of N #) & not y2 = Cosets H ) or y1 = y2 )
then A1: y1 = Cosets H ;
assume for H being strict normal Subgroup of G st H = multMagma(# the carrier of N, the multF of N #) holds
y2 = Cosets H ; ::_thesis: y1 = y2
hence y1 = y2 by A1; ::_thesis: verum
end;
end;
:: deftheorem Def14 defines Cosets GROUP_9:def_14_:_
for O being set
for G being GroupWithOperators of O
for N being normal StableSubgroup of G
for b4 being set holds
( b4 = Cosets N iff for H being strict normal Subgroup of G st H = multMagma(# the carrier of N, the multF of N #) holds
b4 = Cosets H );
definition
let O be set ;
let G be GroupWithOperators of O;
let N be normal StableSubgroup of G;
func CosOp N -> BinOp of (Cosets N) means :Def15: :: GROUP_9:def 15
for H being strict normal Subgroup of G st H = multMagma(# the carrier of N, the multF of N #) holds
it = CosOp H;
existence
ex b1 being BinOp of (Cosets N) st
for H being strict normal Subgroup of G st H = multMagma(# the carrier of N, the multF of N #) holds
b1 = CosOp H
proof
reconsider H = multMagma(# the carrier of N, the multF of N #) as strict normal Subgroup of G by Lm7;
Cosets N = Cosets H by Def14;
then reconsider x = CosOp H as BinOp of (Cosets N) ;
take x ; ::_thesis: for H being strict normal Subgroup of G st H = multMagma(# the carrier of N, the multF of N #) holds
x = CosOp H
let H be strict normal Subgroup of G; ::_thesis: ( H = multMagma(# the carrier of N, the multF of N #) implies x = CosOp H )
assume H = multMagma(# the carrier of N, the multF of N #) ; ::_thesis: x = CosOp H
hence x = CosOp H ; ::_thesis: verum
end;
uniqueness
for b1, b2 being BinOp of (Cosets N) st ( for H being strict normal Subgroup of G st H = multMagma(# the carrier of N, the multF of N #) holds
b1 = CosOp H ) & ( for H being strict normal Subgroup of G st H = multMagma(# the carrier of N, the multF of N #) holds
b2 = CosOp H ) holds
b1 = b2
proof
reconsider H = multMagma(# the carrier of N, the multF of N #) as strict normal Subgroup of G by Lm7;
let y1, y2 be BinOp of (Cosets N); ::_thesis: ( ( for H being strict normal Subgroup of G st H = multMagma(# the carrier of N, the multF of N #) holds
y1 = CosOp H ) & ( for H being strict normal Subgroup of G st H = multMagma(# the carrier of N, the multF of N #) holds
y2 = CosOp H ) implies y1 = y2 )
assume for H being strict normal Subgroup of G st H = multMagma(# the carrier of N, the multF of N #) holds
y1 = CosOp H ; ::_thesis: ( ex H being strict normal Subgroup of G st
( H = multMagma(# the carrier of N, the multF of N #) & not y2 = CosOp H ) or y1 = y2 )
then A1: y1 = CosOp H ;
assume for H being strict normal Subgroup of G st H = multMagma(# the carrier of N, the multF of N #) holds
y2 = CosOp H ; ::_thesis: y1 = y2
hence y1 = y2 by A1; ::_thesis: verum
end;
end;
:: deftheorem Def15 defines CosOp GROUP_9:def_15_:_
for O being set
for G being GroupWithOperators of O
for N being normal StableSubgroup of G
for b4 being BinOp of (Cosets N) holds
( b4 = CosOp N iff for H being strict normal Subgroup of G st H = multMagma(# the carrier of N, the multF of N #) holds
b4 = CosOp H );
Lm9: for G being Group
for N being normal Subgroup of G
for A being Element of Cosets N
for g being Element of G holds
( g in A iff A = g * N )
proof
let G be Group; ::_thesis: for N being normal Subgroup of G
for A being Element of Cosets N
for g being Element of G holds
( g in A iff A = g * N )
let N be normal Subgroup of G; ::_thesis: for A being Element of Cosets N
for g being Element of G holds
( g in A iff A = g * N )
let A be Element of Cosets N; ::_thesis: for g being Element of G holds
( g in A iff A = g * N )
let g be Element of G; ::_thesis: ( g in A iff A = g * N )
hereby ::_thesis: ( A = g * N implies g in A )
consider a being Element of G such that
A1: A = a * N by GROUP_2:def_15;
assume g in A ; ::_thesis: A = g * N
then consider h being Element of G such that
A2: g = a * h and
A3: h in N by A1, GROUP_2:103;
(g ") * a = ((h ") * (a ")) * a by A2, GROUP_1:17
.= (h ") * ((a ") * a) by GROUP_1:def_3
.= (h ") * (1_ G) by GROUP_1:def_5
.= h " by GROUP_1:def_4 ;
then (g ") * a in N by A3, GROUP_2:51;
hence A = g * N by A1, GROUP_2:114; ::_thesis: verum
end;
( g = g * (1_ G) & 1_ G in N ) by GROUP_1:def_4, GROUP_2:46;
hence ( A = g * N implies g in A ) by GROUP_2:103; ::_thesis: verum
end;
Lm10: for O being set
for o being Element of O
for G being GroupWithOperators of O
for H being StableSubgroup of G
for g being Element of G st g in H holds
(G ^ o) . g in H
proof
let O be set ; ::_thesis: for o being Element of O
for G being GroupWithOperators of O
for H being StableSubgroup of G
for g being Element of G st g in H holds
(G ^ o) . g in H
let o be Element of O; ::_thesis: for G being GroupWithOperators of O
for H being StableSubgroup of G
for g being Element of G st g in H holds
(G ^ o) . g in H
let G be GroupWithOperators of O; ::_thesis: for H being StableSubgroup of G
for g being Element of G st g in H holds
(G ^ o) . g in H
let H be StableSubgroup of G; ::_thesis: for g being Element of G st g in H holds
(G ^ o) . g in H
let g be Element of G; ::_thesis: ( g in H implies (G ^ o) . g in H )
set f = G ^ o;
assume g in H ; ::_thesis: (G ^ o) . g in H
then A1: g in the carrier of H by STRUCT_0:def_5;
then (G ^ o) . g = ((G ^ o) | the carrier of H) . g by FUNCT_1:49;
then A2: (G ^ o) . g = (H ^ o) . g by Def7;
(H ^ o) . g in the carrier of H by A1, FUNCT_2:5;
hence (G ^ o) . g in H by A2, STRUCT_0:def_5; ::_thesis: verum
end;
definition
let O be set ;
let G be GroupWithOperators of O;
let N be normal StableSubgroup of G;
func CosAc N -> Action of O,(Cosets N) means :Def16: :: GROUP_9:def 16
for o being Element of O holds it . o = { [A,B] where A, B is Element of Cosets N : ex g, h being Element of G st
( g in A & h in B & h = (G ^ o) . g ) } if not O is empty
otherwise it = [:{},{(id (Cosets N))}:];
existence
( ( not O is empty implies ex b1 being Action of O,(Cosets N) st
for o being Element of O holds b1 . o = { [A,B] where A, B is Element of Cosets N : ex g, h being Element of G st
( g in A & h in B & h = (G ^ o) . g ) } ) & ( O is empty implies ex b1 being Action of O,(Cosets N) st b1 = [:{},{(id (Cosets N))}:] ) )
proof
A1: now__::_thesis:_(_not_O_is_empty_implies_ex_IT_being_Action_of_O,(Cosets_N)_st_
for_o_being_Element_of_O_holds_IT_._o_=__{__[A,B]_where_A,_B_is_Element_of_Cosets_N_:_ex_g,_h_being_Element_of_G_st_
(_g_in_A_&_h_in_B_&_h_=_(G_^_o)_._g_)__}__)
deffunc H1( set ) -> set = { [A,B] where A, B is Element of Cosets N : for o being Element of O st $1 = o holds
ex g, h being Element of G st
( g in A & h in B & h = (G ^ o) . g ) } ;
reconsider H = multMagma(# the carrier of N, the multF of N #) as strict normal Subgroup of G by Lm7;
assume A2: not O is empty ; ::_thesis: ex IT being Action of O,(Cosets N) st
for o being Element of O holds IT . o = { [A,B] where A, B is Element of Cosets N : ex g, h being Element of G st
( g in A & h in B & h = (G ^ o) . g ) }
A3: Cosets N = Cosets H by Def14;
A4: now__::_thesis:_for_x_being_set_st_x_in_O_holds_
H1(x)_in_Funcs_((Cosets_N),(Cosets_N))
let x be set ; ::_thesis: ( x in O implies H1(x) in Funcs ((Cosets N),(Cosets N)) )
set f = H1(x);
A5: now__::_thesis:_for_y_being_set_st_y_in_H1(x)_holds_
ex_A,_B_being_set_st_y_=_[A,B]
let y be set ; ::_thesis: ( y in H1(x) implies ex A, B being set st y = [A,B] )
assume y in H1(x) ; ::_thesis: ex A, B being set st y = [A,B]
then consider A, B being Element of Cosets N such that
A6: y = [A,B] and
for o being Element of O st x = o holds
ex g, h being Element of G st
( g in A & h in B & h = (G ^ o) . g ) ;
reconsider A = A, B = B as set ;
take A = A; ::_thesis: ex B being set st y = [A,B]
take B = B; ::_thesis: y = [A,B]
thus y = [A,B] by A6; ::_thesis: verum
end;
assume A7: x in O ; ::_thesis: H1(x) in Funcs ((Cosets N),(Cosets N))
now__::_thesis:_for_y,_y1,_y2_being_set_st_[y,y1]_in_H1(x)_&_[y,y2]_in_H1(x)_holds_
y1_=_y2
reconsider o = x as Element of O by A7;
let y, y1, y2 be set ; ::_thesis: ( [y,y1] in H1(x) & [y,y2] in H1(x) implies y1 = y2 )
assume [y,y1] in H1(x) ; ::_thesis: ( [y,y2] in H1(x) implies y1 = y2 )
then consider A1, B1 being Element of Cosets N such that
A8: [y,y1] = [A1,B1] and
A9: for o being Element of O st x = o holds
ex g, h being Element of G st
( g in A1 & h in B1 & h = (G ^ o) . g ) ;
assume [y,y2] in H1(x) ; ::_thesis: y1 = y2
then consider A2, B2 being Element of Cosets N such that
A10: [y,y2] = [A2,B2] and
A11: for o being Element of O st x = o holds
ex g, h being Element of G st
( g in A2 & h in B2 & h = (G ^ o) . g ) ;
A12: y1 = B1 by A8, XTUPLE_0:1;
A13: y2 = B2 by A10, XTUPLE_0:1;
A14: y = A2 by A10, XTUPLE_0:1;
set f = G ^ o;
A15: y = A1 by A8, XTUPLE_0:1;
consider g1, h1 being Element of G such that
A16: g1 in A1 and
A17: h1 in B1 and
A18: h1 = (G ^ o) . g1 by A9;
consider g2, h2 being Element of G such that
A19: g2 in A2 and
A20: h2 in B2 and
A21: h2 = (G ^ o) . g2 by A11;
reconsider A1 = A1, A2 = A2, B1 = B1, B2 = B2 as Element of Cosets H by Def14;
A22: A2 = g2 * H by A19, Lm9;
A1 = g1 * H by A16, Lm9;
then (g2 ") * g1 in H by A15, A14, A22, GROUP_2:114;
then (g2 ") * g1 in the carrier of H by STRUCT_0:def_5;
then (g2 ") * g1 in N by STRUCT_0:def_5;
then (G ^ o) . ((g2 ") * g1) in N by Lm10;
then ((G ^ o) . (g2 ")) * ((G ^ o) . g1) in N by GROUP_6:def_6;
then (h2 ") * h1 in N by A18, A21, GROUP_6:32;
then (h2 ") * h1 in the carrier of N by STRUCT_0:def_5;
then A23: (h2 ") * h1 in H by STRUCT_0:def_5;
A24: B2 = h2 * H by A20, Lm9;
B1 = h1 * H by A17, Lm9;
hence y1 = y2 by A12, A13, A23, A24, GROUP_2:114; ::_thesis: verum
end;
then reconsider f = H1(x) as Function by A5, FUNCT_1:def_1, RELAT_1:def_1;
now__::_thesis:_for_y1_being_set_holds_
(_(_y1_in_Cosets_N_implies_ex_y2_being_set_st_[y1,y2]_in_f_)_&_(_ex_y2_being_set_st_[y1,y2]_in_f_implies_y1_in_Cosets_N_)_)
let y1 be set ; ::_thesis: ( ( y1 in Cosets N implies ex y2 being set st [y1,y2] in f ) & ( ex y2 being set st [y1,y2] in f implies y1 in Cosets N ) )
hereby ::_thesis: ( ex y2 being set st [y1,y2] in f implies y1 in Cosets N )
reconsider o = x as Element of O by A7;
assume A25: y1 in Cosets N ; ::_thesis: ex y2 being set st [y1,y2] in f
then reconsider A = y1 as Element of Cosets N ;
y1 in Cosets H by A25, Def14;
then consider g being Element of G such that
A26: y1 = g * H and
y1 = H * g by GROUP_6:13;
set h = (G ^ o) . g;
reconsider B = ((G ^ o) . g) * H as Element of Cosets N by A3, GROUP_2:def_15;
reconsider y2 = B as set ;
take y2 = y2; ::_thesis: [y1,y2] in f
now__::_thesis:_for_o_being_Element_of_O_st_x_=_o_holds_
ex_g_being_Element_of_G_ex_h_being_Element_of_the_carrier_of_G_st_
(_g_in_A_&_h_in_B_&_h_=_(G_^_o)_._g_)
let o be Element of O; ::_thesis: ( x = o implies ex g being Element of G ex h being Element of the carrier of G st
( g in A & h in B & h = (G ^ o) . g ) )
assume A27: x = o ; ::_thesis: ex g being Element of G ex h being Element of the carrier of G st
( g in A & h in B & h = (G ^ o) . g )
take g = g; ::_thesis: ex h being Element of the carrier of G st
( g in A & h in B & h = (G ^ o) . g )
take h = (G ^ o) . g; ::_thesis: ( g in A & h in B & h = (G ^ o) . g )
thus g in A by A3, A26, Lm9; ::_thesis: ( h in B & h = (G ^ o) . g )
thus h in B by A3, Lm9; ::_thesis: h = (G ^ o) . g
thus h = (G ^ o) . g by A27; ::_thesis: verum
end;
hence [y1,y2] in f ; ::_thesis: verum
end;
given y2 being set such that A28: [y1,y2] in f ; ::_thesis: y1 in Cosets N
consider A, B being Element of Cosets N such that
A29: [y1,y2] = [A,B] and
for o being Element of O st x = o holds
ex g, h being Element of G st
( g in A & h in B & h = (G ^ o) . g ) by A28;
A in Cosets N by A3;
hence y1 in Cosets N by A29, XTUPLE_0:1; ::_thesis: verum
end;
then A30: dom f = Cosets N by XTUPLE_0:def_12;
now__::_thesis:_for_y2_being_set_st_y2_in_rng_f_holds_
y2_in_Cosets_N
let y2 be set ; ::_thesis: ( y2 in rng f implies y2 in Cosets N )
assume y2 in rng f ; ::_thesis: y2 in Cosets N
then consider y1 being set such that
A31: [y1,y2] in f by XTUPLE_0:def_13;
consider A, B being Element of Cosets N such that
A32: [y1,y2] = [A,B] and
for o being Element of O st x = o holds
ex g, h being Element of G st
( g in A & h in B & h = (G ^ o) . g ) by A31;
B in Cosets N by A3;
hence y2 in Cosets N by A32, XTUPLE_0:1; ::_thesis: verum
end;
then rng f c= Cosets N by TARSKI:def_3;
hence H1(x) in Funcs ((Cosets N),(Cosets N)) by A30, FUNCT_2:def_2; ::_thesis: verum
end;
ex f being Function of O,(Funcs ((Cosets N),(Cosets N))) st
for x being set st x in O holds
f . x = H1(x) from FUNCT_2:sch_2(A4);
then consider IT being Function of O,(Funcs ((Cosets N),(Cosets N))) such that
A33: for x being set st x in O holds
IT . x = H1(x) ;
reconsider IT = IT as Action of O,(Cosets N) ;
take IT = IT; ::_thesis: for o being Element of O holds IT . o = { [A,B] where A, B is Element of Cosets N : ex g, h being Element of G st
( g in A & h in B & h = (G ^ o) . g ) }
let o be Element of O; ::_thesis: IT . o = { [A,B] where A, B is Element of Cosets N : ex g, h being Element of G st
( g in A & h in B & h = (G ^ o) . g ) }
reconsider x = o as set ;
set X = { [A,B] where A, B is Element of Cosets N : for o being Element of O st x = o holds
ex g, h being Element of G st
( g in A & h in B & h = (G ^ o) . g ) } ;
set Y = { [A,B] where A, B is Element of Cosets N : ex g, h being Element of G st
( g in A & h in B & h = (G ^ o) . g ) } ;
A34: now__::_thesis:_for_y_being_set_holds_
(_(_y_in__{__[A,B]_where_A,_B_is_Element_of_Cosets_N_:_for_o_being_Element_of_O_st_x_=_o_holds_
ex_g,_h_being_Element_of_G_st_
(_g_in_A_&_h_in_B_&_h_=_(G_^_o)_._g_)__}__implies_y_in__{__[A,B]_where_A,_B_is_Element_of_Cosets_N_:_ex_g,_h_being_Element_of_G_st_
(_g_in_A_&_h_in_B_&_h_=_(G_^_o)_._g_)__}__)_&_(_y_in__{__[A,B]_where_A,_B_is_Element_of_Cosets_N_:_ex_g,_h_being_Element_of_G_st_
(_g_in_A_&_h_in_B_&_h_=_(G_^_o)_._g_)__}__implies_y_in__{__[A,B]_where_A,_B_is_Element_of_Cosets_N_:_for_o_being_Element_of_O_st_x_=_o_holds_
ex_g,_h_being_Element_of_G_st_
(_g_in_A_&_h_in_B_&_h_=_(G_^_o)_._g_)__}__)_)
let y be set ; ::_thesis: ( ( y in { [A,B] where A, B is Element of Cosets N : for o being Element of O st x = o holds
ex g, h being Element of G st
( g in A & h in B & h = (G ^ o) . g ) } implies y in { [A,B] where A, B is Element of Cosets N : ex g, h being Element of G st
( g in A & h in B & h = (G ^ o) . g ) } ) & ( y in { [A,B] where A, B is Element of Cosets N : ex g, h being Element of G st
( g in A & h in B & h = (G ^ o) . g ) } implies y in { [A,B] where A, B is Element of Cosets N : for o being Element of O st x = o holds
ex g, h being Element of G st
( g in A & h in B & h = (G ^ o) . g ) } ) )
hereby ::_thesis: ( y in { [A,B] where A, B is Element of Cosets N : ex g, h being Element of G st
( g in A & h in B & h = (G ^ o) . g ) } implies y in { [A,B] where A, B is Element of Cosets N : for o being Element of O st x = o holds
ex g, h being Element of G st
( g in A & h in B & h = (G ^ o) . g ) } )
assume y in { [A,B] where A, B is Element of Cosets N : for o being Element of O st x = o holds
ex g, h being Element of G st
( g in A & h in B & h = (G ^ o) . g ) } ; ::_thesis: y in { [A,B] where A, B is Element of Cosets N : ex g, h being Element of G st
( g in A & h in B & h = (G ^ o) . g ) }
then consider A, B being Element of Cosets N such that
A35: y = [A,B] and
A36: for o being Element of O st x = o holds
ex g, h being Element of G st
( g in A & h in B & h = (G ^ o) . g ) ;
ex g, h being Element of G st
( g in A & h in B & h = (G ^ o) . g ) by A36;
hence y in { [A,B] where A, B is Element of Cosets N : ex g, h being Element of G st
( g in A & h in B & h = (G ^ o) . g ) } by A35; ::_thesis: verum
end;
assume y in { [A,B] where A, B is Element of Cosets N : ex g, h being Element of G st
( g in A & h in B & h = (G ^ o) . g ) } ; ::_thesis: y in { [A,B] where A, B is Element of Cosets N : for o being Element of O st x = o holds
ex g, h being Element of G st
( g in A & h in B & h = (G ^ o) . g ) }
then consider A, B being Element of Cosets N such that
A37: y = [A,B] and
A38: ex g, h being Element of G st
( g in A & h in B & h = (G ^ o) . g ) ;
for o being Element of O st x = o holds
ex g, h being Element of G st
( g in A & h in B & h = (G ^ o) . g ) by A38;
hence y in { [A,B] where A, B is Element of Cosets N : for o being Element of O st x = o holds
ex g, h being Element of G st
( g in A & h in B & h = (G ^ o) . g ) } by A37; ::_thesis: verum
end;
IT . o = { [A,B] where A, B is Element of Cosets N : for o being Element of O st x = o holds
ex g, h being Element of G st
( g in A & h in B & h = (G ^ o) . g ) } by A2, A33;
hence IT . o = { [A,B] where A, B is Element of Cosets N : ex g, h being Element of G st
( g in A & h in B & h = (G ^ o) . g ) } by A34, TARSKI:1; ::_thesis: verum
end;
now__::_thesis:_(_O_is_empty_implies_ex_IT_being_Action_of_O,(Cosets_N)_st_IT_=_[:{},{(id_(Cosets_N))}:]_)
assume O is empty ; ::_thesis: ex IT being Action of O,(Cosets N) st IT = [:{},{(id (Cosets N))}:]
then reconsider IT = [:{},{(id (Cosets N))}:] as Action of O,(Cosets N) by Lm2;
take IT = IT; ::_thesis: IT = [:{},{(id (Cosets N))}:]
thus IT = [:{},{(id (Cosets N))}:] ; ::_thesis: verum
end;
hence ( ( not O is empty implies ex b1 being Action of O,(Cosets N) st
for o being Element of O holds b1 . o = { [A,B] where A, B is Element of Cosets N : ex g, h being Element of G st
( g in A & h in B & h = (G ^ o) . g ) } ) & ( O is empty implies ex b1 being Action of O,(Cosets N) st b1 = [:{},{(id (Cosets N))}:] ) ) by A1; ::_thesis: verum
end;
uniqueness
for b1, b2 being Action of O,(Cosets N) holds
( ( not O is empty & ( for o being Element of O holds b1 . o = { [A,B] where A, B is Element of Cosets N : ex g, h being Element of G st
( g in A & h in B & h = (G ^ o) . g ) } ) & ( for o being Element of O holds b2 . o = { [A,B] where A, B is Element of Cosets N : ex g, h being Element of G st
( g in A & h in B & h = (G ^ o) . g ) } ) implies b1 = b2 ) & ( O is empty & b1 = [:{},{(id (Cosets N))}:] & b2 = [:{},{(id (Cosets N))}:] implies b1 = b2 ) )
proof
now__::_thesis:_(_not_O_is_empty_implies_for_IT1,_IT2_being_Action_of_O,(Cosets_N)_st_(_for_o_being_Element_of_O_holds_IT1_._o_=__{__[A,B]_where_A,_B_is_Element_of_Cosets_N_:_ex_g,_h_being_Element_of_G_st_
(_g_in_A_&_h_in_B_&_h_=_(G_^_o)_._g_)__}__)_&_(_for_o_being_Element_of_O_holds_IT2_._o_=__{__[A,B]_where_A,_B_is_Element_of_Cosets_N_:_ex_g,_h_being_Element_of_G_st_
(_g_in_A_&_h_in_B_&_h_=_(G_^_o)_._g_)__}__)_holds_
IT1_=_IT2_)
assume not O is empty ; ::_thesis: for IT1, IT2 being Action of O,(Cosets N) st ( for o being Element of O holds IT1 . o = { [A,B] where A, B is Element of Cosets N : ex g, h being Element of G st
( g in A & h in B & h = (G ^ o) . g ) } ) & ( for o being Element of O holds IT2 . o = { [A,B] where A, B is Element of Cosets N : ex g, h being Element of G st
( g in A & h in B & h = (G ^ o) . g ) } ) holds
IT1 = IT2
let IT1, IT2 be Action of O,(Cosets N); ::_thesis: ( ( for o being Element of O holds IT1 . o = { [A,B] where A, B is Element of Cosets N : ex g, h being Element of G st
( g in A & h in B & h = (G ^ o) . g ) } ) & ( for o being Element of O holds IT2 . o = { [A,B] where A, B is Element of Cosets N : ex g, h being Element of G st
( g in A & h in B & h = (G ^ o) . g ) } ) implies IT1 = IT2 )
assume A39: for o being Element of O holds IT1 . o = { [A,B] where A, B is Element of Cosets N : ex g, h being Element of G st
( g in A & h in B & h = (G ^ o) . g ) } ; ::_thesis: ( ( for o being Element of O holds IT2 . o = { [A,B] where A, B is Element of Cosets N : ex g, h being Element of G st
( g in A & h in B & h = (G ^ o) . g ) } ) implies IT1 = IT2 )
assume A40: for o being Element of O holds IT2 . o = { [A,B] where A, B is Element of Cosets N : ex g, h being Element of G st
( g in A & h in B & h = (G ^ o) . g ) } ; ::_thesis: IT1 = IT2
A41: now__::_thesis:_for_x_being_set_st_x_in_dom_IT1_holds_
IT1_._x_=_IT2_._x
let x be set ; ::_thesis: ( x in dom IT1 implies IT1 . x = IT2 . x )
assume x in dom IT1 ; ::_thesis: IT1 . x = IT2 . x
then reconsider o = x as Element of O ;
IT1 . o = { [A,B] where A, B is Element of Cosets N : ex g, h being Element of G st
( g in A & h in B & h = (G ^ o) . g ) } by A39;
hence IT1 . x = IT2 . x by A40; ::_thesis: verum
end;
( dom IT1 = O & dom IT2 = O ) by FUNCT_2:def_1;
hence IT1 = IT2 by A41, FUNCT_1:2; ::_thesis: verum
end;
hence for b1, b2 being Action of O,(Cosets N) holds
( ( not O is empty & ( for o being Element of O holds b1 . o = { [A,B] where A, B is Element of Cosets N : ex g, h being Element of G st
( g in A & h in B & h = (G ^ o) . g ) } ) & ( for o being Element of O holds b2 . o = { [A,B] where A, B is Element of Cosets N : ex g, h being Element of G st
( g in A & h in B & h = (G ^ o) . g ) } ) implies b1 = b2 ) & ( O is empty & b1 = [:{},{(id (Cosets N))}:] & b2 = [:{},{(id (Cosets N))}:] implies b1 = b2 ) ) ; ::_thesis: verum
end;
correctness
consistency
for b1 being Action of O,(Cosets N) holds verum;
;
end;
:: deftheorem Def16 defines CosAc GROUP_9:def_16_:_
for O being set
for G being GroupWithOperators of O
for N being normal StableSubgroup of G
for b4 being Action of O,(Cosets N) holds
( ( not O is empty implies ( b4 = CosAc N iff for o being Element of O holds b4 . o = { [A,B] where A, B is Element of Cosets N : ex g, h being Element of G st
( g in A & h in B & h = (G ^ o) . g ) } ) ) & ( O is empty implies ( b4 = CosAc N iff b4 = [:{},{(id (Cosets N))}:] ) ) );
definition
let O be set ;
let G be GroupWithOperators of O;
let N be normal StableSubgroup of G;
funcG ./. N -> HGrWOpStr over O equals :: GROUP_9:def 17
HGrWOpStr(# (Cosets N),(CosOp N),(CosAc N) #);
correctness
coherence
HGrWOpStr(# (Cosets N),(CosOp N),(CosAc N) #) is HGrWOpStr over O;
;
end;
:: deftheorem defines ./. GROUP_9:def_17_:_
for O being set
for G being GroupWithOperators of O
for N being normal StableSubgroup of G holds G ./. N = HGrWOpStr(# (Cosets N),(CosOp N),(CosAc N) #);
registration
let O be set ;
let G be GroupWithOperators of O;
let N be normal StableSubgroup of G;
clusterG ./. N -> non empty ;
correctness
coherence
not G ./. N is empty ;
proof
reconsider H = multMagma(# the carrier of N, the multF of N #) as strict normal Subgroup of G by Lm7;
Cosets N = Cosets H by Def14;
hence not G ./. N is empty ; ::_thesis: verum
end;
clusterG ./. N -> Group-like associative distributive ;
correctness
coherence
( G ./. N is distributive & G ./. N is Group-like & G ./. N is associative );
proof
reconsider H = multMagma(# the carrier of N, the multF of N #) as strict normal Subgroup of G by Lm7;
set G9 = multMagma(# the carrier of (G ./. N), the multF of (G ./. N) #);
A1: now__::_thesis:_ex_e_being_Element_of_(G_./._N)_st_
for_h_being_Element_of_(G_./._N)_holds_
(_h_*_e_=_h_&_e_*_h_=_h_&_ex_g_being_Element_of_(G_./._N)_st_
(_h_*_g_=_e_&_g_*_h_=_e_)_)
set e9 = 1_ (G ./. H);
reconsider e = 1_ (G ./. H) as Element of (G ./. N) by Def14;
take e = e; ::_thesis: for h being Element of (G ./. N) holds
( h * e = h & e * h = h & ex g being Element of (G ./. N) st
( h * g = e & g * h = e ) )
let h be Element of (G ./. N); ::_thesis: ( h * e = h & e * h = h & ex g being Element of (G ./. N) st
( h * g = e & g * h = e ) )
reconsider h9 = h as Element of (G ./. H) by Def14;
set g = h9 " ;
set g9 = h9 " ;
h * e = h9 * (1_ (G ./. H)) by Def15
.= h9 by GROUP_1:def_4 ;
hence h * e = h ; ::_thesis: ( e * h = h & ex g being Element of (G ./. N) st
( h * g = e & g * h = e ) )
e * h = (1_ (G ./. H)) * h9 by Def15
.= h9 by GROUP_1:def_4 ;
hence e * h = h ; ::_thesis: ex g being Element of (G ./. N) st
( h * g = e & g * h = e )
reconsider g = h9 " as Element of (G ./. N) by Def14;
take g = g; ::_thesis: ( h * g = e & g * h = e )
h * g = h9 * (h9 ") by Def15
.= 1_ (G ./. H) by GROUP_1:def_5 ;
hence h * g = e ; ::_thesis: g * h = e
g * h = (h9 ") * h9 by Def15
.= 1_ (G ./. H) by GROUP_1:def_5 ;
hence g * h = e ; ::_thesis: verum
end;
A2: now__::_thesis:_for_G9_being_Group
for_a_being_Action_of_O,_the_carrier_of_G9_st_a_=_the_action_of_(G_./._N)_&_multMagma(#_the_carrier_of_G9,_the_multF_of_G9_#)_=_multMagma(#_the_carrier_of_(G_./._N),_the_multF_of_(G_./._N)_#)_holds_
a_is_distributive
let G9 be Group; ::_thesis: for a being Action of O, the carrier of G9 st a = the action of (G ./. N) & multMagma(# the carrier of G9, the multF of G9 #) = multMagma(# the carrier of (G ./. N), the multF of (G ./. N) #) holds
a is distributive
let a be Action of O, the carrier of G9; ::_thesis: ( a = the action of (G ./. N) & multMagma(# the carrier of G9, the multF of G9 #) = multMagma(# the carrier of (G ./. N), the multF of (G ./. N) #) implies a is distributive )
assume A3: a = the action of (G ./. N) ; ::_thesis: ( multMagma(# the carrier of G9, the multF of G9 #) = multMagma(# the carrier of (G ./. N), the multF of (G ./. N) #) implies a is distributive )
assume A4: multMagma(# the carrier of G9, the multF of G9 #) = multMagma(# the carrier of (G ./. N), the multF of (G ./. N) #) ; ::_thesis: a is distributive
now__::_thesis:_for_o_being_Element_of_O_st_o_in_O_holds_
a_._o_is_Homomorphism_of_G9,G9
let o be Element of O; ::_thesis: ( o in O implies a . o is Homomorphism of G9,G9 )
assume A5: o in O ; ::_thesis: a . o is Homomorphism of G9,G9
then A6: a . o = { [A,B] where A, B is Element of Cosets N : ex g, h being Element of G st
( g in A & h in B & h = (G ^ o) . g ) } by A3, Def16;
a . o in Funcs ((Cosets N),(Cosets N)) by A3, A5, FUNCT_2:5;
then consider f being Function such that
A7: a . o = f and
A8: dom f = Cosets N and
A9: rng f c= Cosets N by FUNCT_2:def_2;
reconsider f = f as Function of the carrier of G9, the carrier of G9 by A4, A8, A9, FUNCT_2:2;
now__::_thesis:_for_A1,_A2_being_Element_of_G9_holds_f_._(A1_*_A2)_=_(f_._A1)_*_(f_._A2)
let A1, A2 be Element of G9; ::_thesis: f . (A1 * A2) = (f . A1) * (f . A2)
set A3 = A1 * A2;
set B1 = f . A1;
set B2 = f . A2;
set B3 = f . (A1 * A2);
[A1,(f . A1)] in f by A4, A8, FUNCT_1:1;
then consider A19, B19 being Element of Cosets N such that
A10: [A1,(f . A1)] = [A19,B19] and
A11: ex g1, h1 being Element of G st
( g1 in A19 & h1 in B19 & h1 = (G ^ o) . g1 ) by A6, A7;
[A2,(f . A2)] in f by A4, A8, FUNCT_1:1;
then consider A29, B29 being Element of Cosets N such that
A12: [A2,(f . A2)] = [A29,B29] and
A13: ex g2, h2 being Element of G st
( g2 in A29 & h2 in B29 & h2 = (G ^ o) . g2 ) by A6, A7;
[(A1 * A2),(f . (A1 * A2))] in f by A4, A8, FUNCT_1:1;
then consider A39, B39 being Element of Cosets N such that
A14: [(A1 * A2),(f . (A1 * A2))] = [A39,B39] and
A15: ex g3, h3 being Element of G st
( g3 in A39 & h3 in B39 & h3 = (G ^ o) . g3 ) by A6, A7;
consider g3, h3 being Element of G such that
A16: g3 in A39 and
A17: h3 in B39 and
A18: h3 = (G ^ o) . g3 by A15;
consider g2, h2 being Element of G such that
A19: g2 in A29 and
A20: h2 in B29 and
A21: h2 = (G ^ o) . g2 by A13;
consider g1, h1 being Element of G such that
A22: g1 in A19 and
A23: h1 in B19 and
A24: h1 = (G ^ o) . g1 by A11;
A25: ( @ ((nat_hom H) . g1) = (nat_hom H) . g1 & @ ((nat_hom H) . g2) = (nat_hom H) . g2 ) ;
A26: ( (nat_hom H) . g1 = g1 * H & (nat_hom H) . g2 = g2 * H ) by GROUP_6:def_8;
reconsider A19 = A19, A29 = A29, A39 = A39, B19 = B19, B29 = B29, B39 = B39 as Element of Cosets H by Def14;
A27: A29 = g2 * H by A19, Lm9;
A28: A39 = g3 * H by A16, Lm9;
A29: B29 = h2 * H by A20, Lm9;
reconsider A19 = A19, A29 = A29, B19 = B19, B29 = B29 as Element of (G ./. H) ;
A2 = g2 * H by A12, A27, XTUPLE_0:1;
then A1 * A2 = the multF of G9 . (A19,A29) by A10, A27, XTUPLE_0:1
.= @ (A19 * A29) by A4, Def15
.= (@ A19) * (@ A29) by GROUP_6:20 ;
then A1 * A2 = (g1 * H) * (g2 * H) by A22, A27, Lm9
.= ((nat_hom H) . g1) * ((nat_hom H) . g2) by A25, A26, GROUP_6:19
.= (nat_hom H) . (g1 * g2) by GROUP_6:def_6
.= (g1 * g2) * H by GROUP_6:def_8 ;
then g3 * H = (g1 * g2) * H by A14, A28, XTUPLE_0:1;
then (g3 ") * (g1 * g2) in H by GROUP_2:114;
then (g3 ") * (g1 * g2) in the carrier of H by STRUCT_0:def_5;
then (g3 ") * (g1 * g2) in N by STRUCT_0:def_5;
then (G ^ o) . ((g3 ") * (g1 * g2)) in N by Lm10;
then ((G ^ o) . (g3 ")) * ((G ^ o) . (g1 * g2)) in N by GROUP_6:def_6;
then ((G ^ o) . (g3 ")) * (((G ^ o) . g1) * ((G ^ o) . g2)) in N by GROUP_6:def_6;
then (h3 ") * (h1 * h2) in N by A24, A21, A18, GROUP_6:32;
then (h3 ") * (h1 * h2) in the carrier of N by STRUCT_0:def_5;
then A30: (h3 ") * (h1 * h2) in H by STRUCT_0:def_5;
A31: ( (nat_hom H) . h1 = h1 * H & (nat_hom H) . h2 = h2 * H ) by GROUP_6:def_8;
B39 = h3 * H by A17, Lm9;
then A32: f . (A1 * A2) = h3 * H by A14, XTUPLE_0:1;
A33: ( @ ((nat_hom H) . h1) = (nat_hom H) . h1 & @ ((nat_hom H) . h2) = (nat_hom H) . h2 ) ;
f . A2 = h2 * H by A12, A29, XTUPLE_0:1;
then (f . A1) * (f . A2) = the multF of G9 . (B19,B29) by A10, A29, XTUPLE_0:1
.= @ (B19 * B29) by A4, Def15
.= (@ B19) * (@ B29) by GROUP_6:20 ;
then (f . A1) * (f . A2) = (h1 * H) * (h2 * H) by A23, A29, Lm9
.= ((nat_hom H) . h1) * ((nat_hom H) . h2) by A33, A31, GROUP_6:19
.= (nat_hom H) . (h1 * h2) by GROUP_6:def_6
.= (h1 * h2) * H by GROUP_6:def_8 ;
hence f . (A1 * A2) = (f . A1) * (f . A2) by A32, A30, GROUP_2:114; ::_thesis: verum
end;
hence a . o is Homomorphism of G9,G9 by A7, GROUP_6:def_6; ::_thesis: verum
end;
hence a is distributive by Def4; ::_thesis: verum
end;
the carrier of (G ./. N) = the carrier of (G ./. H) by Def14;
then A34: ( multMagma(# the carrier of (G ./. N), the multF of (G ./. N) #) is Group-like & multMagma(# the carrier of (G ./. N), the multF of (G ./. N) #) is associative ) by Def15;
now__::_thesis:_for_x,_y,_z_being_Element_of_(G_./._N)_holds_(x_*_y)_*_z_=_x_*_(y_*_z)
let x, y, z be Element of (G ./. N); ::_thesis: (x * y) * z = x * (y * z)
reconsider x9 = x, y9 = y, z9 = z as Element of multMagma(# the carrier of (G ./. N), the multF of (G ./. N) #) ;
( (x9 * y9) * z9 = (x * y) * z & x9 * (y9 * z9) = x * (y * z) ) ;
hence (x * y) * z = x * (y * z) by A34, GROUP_1:def_3; ::_thesis: verum
end;
hence ( G ./. N is distributive & G ./. N is Group-like & G ./. N is associative ) by A1, A2, Def5, GROUP_1:def_2, GROUP_1:def_3; ::_thesis: verum
end;
end;
definition
let O be set ;
let G, H be GroupWithOperators of O;
let f be Function of G,H;
attrf is homomorphic means :Def18: :: GROUP_9:def 18
for o being Element of O
for g being Element of G holds f . ((G ^ o) . g) = (H ^ o) . (f . g);
end;
:: deftheorem Def18 defines homomorphic GROUP_9:def_18_:_
for O being set
for G, H being GroupWithOperators of O
for f being Function of G,H holds
( f is homomorphic iff for o being Element of O
for g being Element of G holds f . ((G ^ o) . g) = (H ^ o) . (f . g) );
registration
let O be set ;
let G, H be GroupWithOperators of O;
cluster Relation-like the carrier of G -defined the carrier of H -valued Function-like non empty total quasi_total multiplicative homomorphic for Element of bool [: the carrier of G, the carrier of H:];
existence
ex b1 being Function of G,H st
( b1 is multiplicative & b1 is homomorphic )
proof
set f = 1: (G,H);
reconsider f = 1: (G,H) as Function of G,H ;
take f ; ::_thesis: ( f is multiplicative & f is homomorphic )
thus f is multiplicative ; ::_thesis: f is homomorphic
let o be Element of O; :: according to GROUP_9:def_18 ::_thesis: for g being Element of G holds f . ((G ^ o) . g) = (H ^ o) . (f . g)
let g be Element of G; ::_thesis: f . ((G ^ o) . g) = (H ^ o) . (f . g)
(H ^ o) . (f . g) = (H ^ o) . (1_ H) by GROUP_6:def_7
.= 1_ H by GROUP_6:31 ;
hence f . ((G ^ o) . g) = (H ^ o) . (f . g) by GROUP_6:def_7; ::_thesis: verum
end;
end;
definition
let O be set ;
let G, H be GroupWithOperators of O;
mode Homomorphism of G,H is multiplicative homomorphic Function of G,H;
end;
Lm11: for O being set
for G, H, I being GroupWithOperators of O
for h being Homomorphism of G,H
for h1 being Homomorphism of H,I holds h1 * h is Homomorphism of G,I
proof
let O be set ; ::_thesis: for G, H, I being GroupWithOperators of O
for h being Homomorphism of G,H
for h1 being Homomorphism of H,I holds h1 * h is Homomorphism of G,I
let G, H, I be GroupWithOperators of O; ::_thesis: for h being Homomorphism of G,H
for h1 being Homomorphism of H,I holds h1 * h is Homomorphism of G,I
let h be Homomorphism of G,H; ::_thesis: for h1 being Homomorphism of H,I holds h1 * h is Homomorphism of G,I
let h1 be Homomorphism of H,I; ::_thesis: h1 * h is Homomorphism of G,I
reconsider f = h1 * h as Function of G,I ;
now__::_thesis:_for_o_being_Element_of_O
for_g_being_Element_of_G_holds_f_._((G_^_o)_._g)_=_(I_^_o)_._(f_._g)
let o be Element of O; ::_thesis: for g being Element of G holds f . ((G ^ o) . g) = (I ^ o) . (f . g)
let g be Element of G; ::_thesis: f . ((G ^ o) . g) = (I ^ o) . (f . g)
thus f . ((G ^ o) . g) = h1 . (h . ((G ^ o) . g)) by FUNCT_2:15
.= h1 . ((H ^ o) . (h . g)) by Def18
.= (I ^ o) . (h1 . (h . g)) by Def18
.= (I ^ o) . (f . g) by FUNCT_2:15 ; ::_thesis: verum
end;
hence h1 * h is Homomorphism of G,I by Def18; ::_thesis: verum
end;
definition
let O be set ;
let G, H, I be GroupWithOperators of O;
let h be Homomorphism of G,H;
let h1 be Homomorphism of H,I;
:: original: *
redefine funch1 * h -> Homomorphism of G,I;
correctness
coherence
h * h1 is Homomorphism of G,I;
by Lm11;
end;
definition
let O be set ;
let G, H be GroupWithOperators of O;
predG,H are_isomorphic means :Def19: :: GROUP_9:def 19
ex h being Homomorphism of G,H st h is bijective ;
reflexivity
for G being GroupWithOperators of O ex h being Homomorphism of G,G st h is bijective
proof
let G be GroupWithOperators of O; ::_thesis: ex h being Homomorphism of G,G st h is bijective
reconsider G9 = G as Group ;
set h = id the carrier of G9;
now__::_thesis:_for_o_being_Element_of_O
for_g_being_Element_of_G_holds_(id_the_carrier_of_G9)_._((G_^_o)_._g)_=_(G_^_o)_._((id_the_carrier_of_G9)_._g)
let o be Element of O; ::_thesis: for g being Element of G holds (id the carrier of G9) . ((G ^ o) . g) = (G ^ o) . ((id the carrier of G9) . g)
let g be Element of G; ::_thesis: (id the carrier of G9) . ((G ^ o) . g) = (G ^ o) . ((id the carrier of G9) . g)
(id the carrier of G9) . ((G ^ o) . g) = (G ^ o) . g by FUNCT_1:17;
hence (id the carrier of G9) . ((G ^ o) . g) = (G ^ o) . ((id the carrier of G9) . g) by FUNCT_1:17; ::_thesis: verum
end;
then reconsider h = id the carrier of G9 as Homomorphism of G,G by Def18, GROUP_6:38;
take h ; ::_thesis: h is bijective
rng h = the carrier of G by RELAT_1:45;
then h is onto by FUNCT_2:def_3;
hence h is bijective ; ::_thesis: verum
end;
end;
:: deftheorem Def19 defines are_isomorphic GROUP_9:def_19_:_
for O being set
for G, H being GroupWithOperators of O holds
( G,H are_isomorphic iff ex h being Homomorphism of G,H st h is bijective );
Lm12: for O being set
for G, H being GroupWithOperators of O st G,H are_isomorphic holds
H,G are_isomorphic
proof
let O be set ; ::_thesis: for G, H being GroupWithOperators of O st G,H are_isomorphic holds
H,G are_isomorphic
let G, H be GroupWithOperators of O; ::_thesis: ( G,H are_isomorphic implies H,G are_isomorphic )
assume G,H are_isomorphic ; ::_thesis: H,G are_isomorphic
then consider f being Homomorphism of G,H such that
A1: f is bijective by Def19;
set f9 = f " ;
A2: rng f = the carrier of H by A1, FUNCT_2:def_3;
then A3: dom (f ") = the carrier of H by A1, FUNCT_1:33;
A4: dom f = the carrier of G by FUNCT_2:def_1;
then A5: rng (f ") = the carrier of G by A1, FUNCT_1:33;
then reconsider f9 = f " as Function of H,G by A3, FUNCT_2:1;
A6: now__::_thesis:_for_o_being_Element_of_O
for_h_being_Element_of_H_holds_f9_._((H_^_o)_._h)_=_(G_^_o)_._(f9_._h)
let o be Element of O; ::_thesis: for h being Element of H holds f9 . ((H ^ o) . h) = (G ^ o) . (f9 . h)
let h be Element of H; ::_thesis: f9 . ((H ^ o) . h) = (G ^ o) . (f9 . h)
set g = f9 . h;
thus f9 . ((H ^ o) . h) = f9 . ((H ^ o) . (f . (f9 . h))) by A1, A2, FUNCT_1:35
.= f9 . (f . ((G ^ o) . (f9 . h))) by Def18
.= (G ^ o) . (f9 . h) by A1, A4, FUNCT_1:34 ; ::_thesis: verum
end;
now__::_thesis:_for_h1,_h2_being_Element_of_H_holds_f9_._(h1_*_h2)_=_(f9_._h1)_*_(f9_._h2)
let h1, h2 be Element of H; ::_thesis: f9 . (h1 * h2) = (f9 . h1) * (f9 . h2)
set g1 = f9 . h1;
set g2 = f9 . h2;
( f . (f9 . h1) = h1 & f . (f9 . h2) = h2 ) by A1, A2, FUNCT_1:35;
hence f9 . (h1 * h2) = f9 . (f . ((f9 . h1) * (f9 . h2))) by GROUP_6:def_6
.= (f9 . h1) * (f9 . h2) by A1, A4, FUNCT_1:34 ;
::_thesis: verum
end;
then reconsider f9 = f9 as Homomorphism of H,G by A6, Def18, GROUP_6:def_6;
take f9 ; :: according to GROUP_9:def_19 ::_thesis: f9 is bijective
f9 is onto by A5, FUNCT_2:def_3;
hence f9 is bijective by A1; ::_thesis: verum
end;
definition
let O be set ;
let G, H be GroupWithOperators of O;
:: original: are_isomorphic
redefine predG,H are_isomorphic ;
symmetry
for G, H being GroupWithOperators of O st (O,b1,b2) holds
(O,b2,b1) by Lm12;
end;
definition
let O be set ;
let G be GroupWithOperators of O;
let N be normal StableSubgroup of G;
func nat_hom N -> Homomorphism of G,(G ./. N) means :Def20: :: GROUP_9:def 20
for H being strict normal Subgroup of G st H = multMagma(# the carrier of N, the multF of N #) holds
it = nat_hom H;
existence
ex b1 being Homomorphism of G,(G ./. N) st
for H being strict normal Subgroup of G st H = multMagma(# the carrier of N, the multF of N #) holds
b1 = nat_hom H
proof
set H = multMagma(# the carrier of N, the multF of N #);
reconsider H = multMagma(# the carrier of N, the multF of N #) as strict normal Subgroup of G by Lm7;
set IT = nat_hom H;
reconsider K = G ./. N as GroupWithOperators of O ;
reconsider IT9 = nat_hom H as Function of G,K by Def14;
A1: now__::_thesis:_for_a,_b_being_Element_of_G_holds_IT9_._(a_*_b)_=_(IT9_._a)_*_(IT9_._b)
let a, b be Element of G; ::_thesis: IT9 . (a * b) = (IT9 . a) * (IT9 . b)
IT9 . (a * b) = ((nat_hom H) . a) * ((nat_hom H) . b) by GROUP_6:def_6
.= (IT9 . a) * (IT9 . b) by Def15 ;
hence IT9 . (a * b) = (IT9 . a) * (IT9 . b) ; ::_thesis: verum
end;
now__::_thesis:_for_o_being_Element_of_O
for_g_being_Element_of_G_holds_IT9_._((G_^_o)_._g)_=_(K_^_o)_._(IT9_._g)
let o be Element of O; ::_thesis: for g being Element of G holds IT9 . ((G ^ b2) . b3) = (K ^ b2) . (IT9 . b3)
let g be Element of G; ::_thesis: IT9 . ((G ^ b1) . b2) = (K ^ b1) . (IT9 . b2)
percases ( O <> {} or O = {} ) ;
supposeA2: O <> {} ; ::_thesis: IT9 . ((G ^ b1) . b2) = (K ^ b1) . (IT9 . b2)
then the action of K . o in Funcs ( the carrier of K, the carrier of K) by FUNCT_2:5;
then consider f being Function such that
A3: f = the action of K . o and
A4: dom f = the carrier of K and
rng f c= the carrier of K by FUNCT_2:def_2;
A5: f = { [A,B] where A, B is Element of Cosets N : ex g, h being Element of G st
( g in A & h in B & h = (G ^ o) . g ) } by A2, A3, Def16;
[(IT9 . g),(f . (IT9 . g))] in f by A4, FUNCT_1:def_2;
then consider A, B being Element of Cosets N such that
A6: [(IT9 . g),(f . (IT9 . g))] = [A,B] and
A7: ex g, h being Element of G st
( g in A & h in B & h = (G ^ o) . g ) by A5;
A8: IT9 . g = A by A6, XTUPLE_0:1;
consider g9, h9 being Element of G such that
A9: g9 in A and
A10: ( h9 in B & h9 = (G ^ o) . g9 ) by A7;
A11: (G ^ o) . ((g9 ") * g) = ((G ^ o) . (g9 ")) * ((G ^ o) . g) by GROUP_6:def_6
.= (((G ^ o) . g9) ") * ((G ^ o) . g) by GROUP_6:32 ;
reconsider A = A, B = B as Element of Cosets H by Def14;
A = g9 * H by A9, Lm9;
then g * H = g9 * H by A8, GROUP_6:def_8;
then (g9 ") * g in H by GROUP_2:114;
then (g9 ") * g in the carrier of N by STRUCT_0:def_5;
then (g9 ") * g in N by STRUCT_0:def_5;
then (G ^ o) . ((g9 ") * g) in N by Lm10;
then (G ^ o) . ((g9 ") * g) in the carrier of N by STRUCT_0:def_5;
then A12: (G ^ o) . ((g9 ") * g) in H by STRUCT_0:def_5;
A13: (K ^ o) . (IT9 . g) = f . (IT9 . g) by A2, A3, Def6;
IT9 . ((G ^ o) . g) = ((G ^ o) . g) * H by GROUP_6:def_8
.= ((G ^ o) . g9) * H by A12, A11, GROUP_2:114
.= B by A10, Lm9 ;
hence IT9 . ((G ^ o) . g) = (K ^ o) . (IT9 . g) by A13, A6, XTUPLE_0:1; ::_thesis: verum
end;
supposeA14: O = {} ; ::_thesis: IT9 . ((G ^ b1) . b2) = (K ^ b1) . (IT9 . b2)
then G ^ o = id the carrier of G by Def6;
then A15: (G ^ o) . g = g by FUNCT_1:18;
K ^ o = id the carrier of K by A14, Def6;
hence IT9 . ((G ^ o) . g) = (K ^ o) . (IT9 . g) by A15, FUNCT_1:18; ::_thesis: verum
end;
end;
end;
then reconsider IT9 = IT9 as Homomorphism of G,K by A1, Def18, GROUP_6:def_6;
reconsider IT9 = IT9 as Homomorphism of G,(G ./. N) ;
take IT9 ; ::_thesis: for H being strict normal Subgroup of G st H = multMagma(# the carrier of N, the multF of N #) holds
IT9 = nat_hom H
let H be strict normal Subgroup of G; ::_thesis: ( H = multMagma(# the carrier of N, the multF of N #) implies IT9 = nat_hom H )
assume H = multMagma(# the carrier of N, the multF of N #) ; ::_thesis: IT9 = nat_hom H
hence IT9 = nat_hom H ; ::_thesis: verum
end;
uniqueness
for b1, b2 being Homomorphism of G,(G ./. N) st ( for H being strict normal Subgroup of G st H = multMagma(# the carrier of N, the multF of N #) holds
b1 = nat_hom H ) & ( for H being strict normal Subgroup of G st H = multMagma(# the carrier of N, the multF of N #) holds
b2 = nat_hom H ) holds
b1 = b2
proof
reconsider H = multMagma(# the carrier of N, the multF of N #) as strict normal Subgroup of G by Lm7;
let IT1, IT2 be Homomorphism of G,(G ./. N); ::_thesis: ( ( for H being strict normal Subgroup of G st H = multMagma(# the carrier of N, the multF of N #) holds
IT1 = nat_hom H ) & ( for H being strict normal Subgroup of G st H = multMagma(# the carrier of N, the multF of N #) holds
IT2 = nat_hom H ) implies IT1 = IT2 )
assume for H being strict normal Subgroup of G st H = multMagma(# the carrier of N, the multF of N #) holds
IT1 = nat_hom H ; ::_thesis: ( ex H being strict normal Subgroup of G st
( H = multMagma(# the carrier of N, the multF of N #) & not IT2 = nat_hom H ) or IT1 = IT2 )
then A16: IT1 = nat_hom H ;
assume for H being strict normal Subgroup of G st H = multMagma(# the carrier of N, the multF of N #) holds
IT2 = nat_hom H ; ::_thesis: IT1 = IT2
hence IT1 = IT2 by A16; ::_thesis: verum
end;
end;
:: deftheorem Def20 defines nat_hom GROUP_9:def_20_:_
for O being set
for G being GroupWithOperators of O
for N being normal StableSubgroup of G
for b4 being Homomorphism of G,(G ./. N) holds
( b4 = nat_hom N iff for H being strict normal Subgroup of G st H = multMagma(# the carrier of N, the multF of N #) holds
b4 = nat_hom H );
Lm13: for O being set
for G, H being GroupWithOperators of O
for g being Homomorphism of G,H holds g . (1_ G) = 1_ H
proof
let O be set ; ::_thesis: for G, H being GroupWithOperators of O
for g being Homomorphism of G,H holds g . (1_ G) = 1_ H
let G, H be GroupWithOperators of O; ::_thesis: for g being Homomorphism of G,H holds g . (1_ G) = 1_ H
let g be Homomorphism of G,H; ::_thesis: g . (1_ G) = 1_ H
g . (1_ G) = g . ((1_ G) * (1_ G)) by GROUP_1:def_4
.= (g . (1_ G)) * (g . (1_ G)) by GROUP_6:def_6 ;
hence g . (1_ G) = 1_ H by GROUP_1:7; ::_thesis: verum
end;
Lm14: for O being set
for G, H being GroupWithOperators of O
for a being Element of G
for g being Homomorphism of G,H holds g . (a ") = (g . a) "
proof
let O be set ; ::_thesis: for G, H being GroupWithOperators of O
for a being Element of G
for g being Homomorphism of G,H holds g . (a ") = (g . a) "
let G, H be GroupWithOperators of O; ::_thesis: for a being Element of G
for g being Homomorphism of G,H holds g . (a ") = (g . a) "
let a be Element of G; ::_thesis: for g being Homomorphism of G,H holds g . (a ") = (g . a) "
let g be Homomorphism of G,H; ::_thesis: g . (a ") = (g . a) "
(g . (a ")) * (g . a) = g . ((a ") * a) by GROUP_6:def_6
.= g . (1_ G) by GROUP_1:def_5
.= 1_ H by Lm13 ;
hence g . (a ") = (g . a) " by GROUP_1:12; ::_thesis: verum
end;
Lm15: for O being set
for G being GroupWithOperators of O
for A being Subset of G st A <> {} & ( for g1, g2 being Element of G st g1 in A & g2 in A holds
g1 * g2 in A ) & ( for g being Element of G st g in A holds
g " in A ) & ( for o being Element of O
for g being Element of G st g in A holds
(G ^ o) . g in A ) holds
ex H being strict StableSubgroup of G st the carrier of H = A
proof
let O be set ; ::_thesis: for G being GroupWithOperators of O
for A being Subset of G st A <> {} & ( for g1, g2 being Element of G st g1 in A & g2 in A holds
g1 * g2 in A ) & ( for g being Element of G st g in A holds
g " in A ) & ( for o being Element of O
for g being Element of G st g in A holds
(G ^ o) . g in A ) holds
ex H being strict StableSubgroup of G st the carrier of H = A
let G be GroupWithOperators of O; ::_thesis: for A being Subset of G st A <> {} & ( for g1, g2 being Element of G st g1 in A & g2 in A holds
g1 * g2 in A ) & ( for g being Element of G st g in A holds
g " in A ) & ( for o being Element of O
for g being Element of G st g in A holds
(G ^ o) . g in A ) holds
ex H being strict StableSubgroup of G st the carrier of H = A
let A be Subset of G; ::_thesis: ( A <> {} & ( for g1, g2 being Element of G st g1 in A & g2 in A holds
g1 * g2 in A ) & ( for g being Element of G st g in A holds
g " in A ) & ( for o being Element of O
for g being Element of G st g in A holds
(G ^ o) . g in A ) implies ex H being strict StableSubgroup of G st the carrier of H = A )
assume A1: A <> {} ; ::_thesis: ( ex g1, g2 being Element of G st
( g1 in A & g2 in A & not g1 * g2 in A ) or ex g being Element of G st
( g in A & not g " in A ) or ex o being Element of O ex g being Element of G st
( g in A & not (G ^ o) . g in A ) or ex H being strict StableSubgroup of G st the carrier of H = A )
assume A2: for g1, g2 being Element of G st g1 in A & g2 in A holds
g1 * g2 in A ; ::_thesis: ( ex g being Element of G st
( g in A & not g " in A ) or ex o being Element of O ex g being Element of G st
( g in A & not (G ^ o) . g in A ) or ex H being strict StableSubgroup of G st the carrier of H = A )
assume for g being Element of G st g in A holds
g " in A ; ::_thesis: ( ex o being Element of O ex g being Element of G st
( g in A & not (G ^ o) . g in A ) or ex H being strict StableSubgroup of G st the carrier of H = A )
then consider H9 being strict Subgroup of G such that
A3: the carrier of H9 = A by A1, A2, GROUP_2:52;
set m9 = the multF of H9;
set A9 = the carrier of H9;
assume A4: for o being Element of O
for g being Element of G st g in A holds
(G ^ o) . g in A ; ::_thesis: ex H being strict StableSubgroup of G st the carrier of H = A
A5: now__::_thesis:_for_H_being_non_empty_HGrWOpStr_over_O
for_a9_being_Action_of_O,_the_carrier_of_H9_st_H_=_HGrWOpStr(#_the_carrier_of_H9,_the_multF_of_H9,a9_#)_holds_
(_H_is_associative_&_H_is_Group-like_)
let H be non empty HGrWOpStr over O; ::_thesis: for a9 being Action of O, the carrier of H9 st H = HGrWOpStr(# the carrier of H9, the multF of H9,a9 #) holds
( H is associative & H is Group-like )
let a9 be Action of O, the carrier of H9; ::_thesis: ( H = HGrWOpStr(# the carrier of H9, the multF of H9,a9 #) implies ( H is associative & H is Group-like ) )
assume A6: H = HGrWOpStr(# the carrier of H9, the multF of H9,a9 #) ; ::_thesis: ( H is associative & H is Group-like )
now__::_thesis:_for_x,_y,_z_being_Element_of_H_holds_(x_*_y)_*_z_=_x_*_(y_*_z)
let x, y, z be Element of H; ::_thesis: (x * y) * z = x * (y * z)
reconsider x9 = x, y9 = y, z9 = z as Element of H9 by A6;
(x * y) * z = (x9 * y9) * z9 by A6
.= x9 * (y9 * z9) by GROUP_1:def_3 ;
hence (x * y) * z = x * (y * z) by A6; ::_thesis: verum
end;
hence H is associative by GROUP_1:def_3; ::_thesis: H is Group-like
now__::_thesis:_ex_e_being_Element_of_H_st_
for_h_being_Element_of_H_holds_
(_h_*_e_=_h_&_e_*_h_=_h_&_ex_g_being_Element_of_H_st_
(_h_*_g_=_e_&_g_*_h_=_e_)_)
set e9 = 1_ H9;
reconsider e = 1_ H9 as Element of H by A6;
take e = e; ::_thesis: for h being Element of H holds
( h * e = h & e * h = h & ex g being Element of H st
( h * g = e & g * h = e ) )
let h be Element of H; ::_thesis: ( h * e = h & e * h = h & ex g being Element of H st
( h * g = e & g * h = e ) )
reconsider h9 = h as Element of H9 by A6;
set g9 = h9 " ;
h * e = h9 * (1_ H9) by A6
.= h9 by GROUP_1:def_4 ;
hence h * e = h ; ::_thesis: ( e * h = h & ex g being Element of H st
( h * g = e & g * h = e ) )
e * h = (1_ H9) * h9 by A6
.= h9 by GROUP_1:def_4 ;
hence e * h = h ; ::_thesis: ex g being Element of H st
( h * g = e & g * h = e )
reconsider g = h9 " as Element of H by A6;
take g = g; ::_thesis: ( h * g = e & g * h = e )
h * g = h9 * (h9 ") by A6
.= 1_ H9 by GROUP_1:def_5 ;
hence h * g = e ; ::_thesis: g * h = e
g * h = (h9 ") * h9 by A6
.= 1_ H9 by GROUP_1:def_5 ;
hence g * h = e ; ::_thesis: verum
end;
hence H is Group-like by GROUP_1:def_2; ::_thesis: verum
end;
percases ( O is empty or not O is empty ) ;
supposeA7: O is empty ; ::_thesis: ex H being strict StableSubgroup of G st the carrier of H = A
set a9 = [:{},{(id the carrier of H9)}:];
reconsider a9 = [:{},{(id the carrier of H9)}:] as Action of O, the carrier of H9 by A7, Lm2;
set H = HGrWOpStr(# the carrier of H9, the multF of H9,a9 #);
reconsider H = HGrWOpStr(# the carrier of H9, the multF of H9,a9 #) as non empty HGrWOpStr over O ;
now__::_thesis:_for_G9_being_Group
for_a_being_Action_of_O,_the_carrier_of_G9_st_a_=_the_action_of_H_&_multMagma(#_the_carrier_of_G9,_the_multF_of_G9_#)_=_multMagma(#_the_carrier_of_H,_the_multF_of_H_#)_holds_
a_is_distributive
let G9 be Group; ::_thesis: for a being Action of O, the carrier of G9 st a = the action of H & multMagma(# the carrier of G9, the multF of G9 #) = multMagma(# the carrier of H, the multF of H #) holds
a is distributive
let a be Action of O, the carrier of G9; ::_thesis: ( a = the action of H & multMagma(# the carrier of G9, the multF of G9 #) = multMagma(# the carrier of H, the multF of H #) implies a is distributive )
assume a = the action of H ; ::_thesis: ( multMagma(# the carrier of G9, the multF of G9 #) = multMagma(# the carrier of H, the multF of H #) implies a is distributive )
assume multMagma(# the carrier of G9, the multF of G9 #) = multMagma(# the carrier of H, the multF of H #) ; ::_thesis: a is distributive
for o being Element of O st o in O holds
a . o is Homomorphism of G9,G9 by A7;
hence a is distributive by Def4; ::_thesis: verum
end;
then reconsider H = H as GroupWithOperators of O by A5, Def5;
A8: the carrier of H c= the carrier of G by GROUP_2:def_5;
A9: now__::_thesis:_for_o_being_Element_of_O_holds_H_^_o_=_(G_^_o)_|_the_carrier_of_H
let o be Element of O; ::_thesis: H ^ o = (G ^ o) | the carrier of H
A10: now__::_thesis:_for_x,_y_being_set_st_[x,y]_in_(id_the_carrier_of_G)_|_the_carrier_of_H_holds_
[x,y]_in_id_the_carrier_of_H
let x, y be set ; ::_thesis: ( [x,y] in (id the carrier of G) | the carrier of H implies [x,y] in id the carrier of H )
assume A11: [x,y] in (id the carrier of G) | the carrier of H ; ::_thesis: [x,y] in id the carrier of H
then [x,y] in id the carrier of G by RELAT_1:def_11;
then A12: x = y by RELAT_1:def_10;
x in the carrier of H by A11, RELAT_1:def_11;
hence [x,y] in id the carrier of H by A12, RELAT_1:def_10; ::_thesis: verum
end;
A13: now__::_thesis:_for_x,_y_being_set_st_[x,y]_in_id_the_carrier_of_H_holds_
[x,y]_in_(id_the_carrier_of_G)_|_the_carrier_of_H
let x, y be set ; ::_thesis: ( [x,y] in id the carrier of H implies [x,y] in (id the carrier of G) | the carrier of H )
assume A14: [x,y] in id the carrier of H ; ::_thesis: [x,y] in (id the carrier of G) | the carrier of H
then A15: x in the carrier of H by RELAT_1:def_10;
x = y by A14, RELAT_1:def_10;
then [x,y] in id the carrier of G by A8, A15, RELAT_1:def_10;
hence [x,y] in (id the carrier of G) | the carrier of H by A15, RELAT_1:def_11; ::_thesis: verum
end;
H ^ o = id the carrier of H by A7, Def6
.= (id the carrier of G) | the carrier of H by A13, A10, RELAT_1:def_2 ;
hence H ^ o = (G ^ o) | the carrier of H by A7, Def6; ::_thesis: verum
end;
the multF of H = the multF of G || the carrier of H by GROUP_2:def_5;
then H is Subgroup of G by A8, GROUP_2:def_5;
then reconsider H = H as strict StableSubgroup of G by A9, Def7;
take H ; ::_thesis: the carrier of H = A
thus the carrier of H = A by A3; ::_thesis: verum
end;
supposeA16: not O is empty ; ::_thesis: ex H being strict StableSubgroup of G st the carrier of H = A
set a9 = { [o,((G ^ o) | the carrier of H9)] where o is Element of O : verum } ;
now__::_thesis:_for_x_being_set_st_x_in__{__[o,((G_^_o)_|_the_carrier_of_H9)]_where_o_is_Element_of_O_:_verum__}__holds_
ex_y1,_y2_being_set_st_x_=_[y1,y2]
let x be set ; ::_thesis: ( x in { [o,((G ^ o) | the carrier of H9)] where o is Element of O : verum } implies ex y1, y2 being set st x = [y1,y2] )
assume x in { [o,((G ^ o) | the carrier of H9)] where o is Element of O : verum } ; ::_thesis: ex y1, y2 being set st x = [y1,y2]
then ex o being Element of O st x = [o,((G ^ o) | the carrier of H9)] ;
hence ex y1, y2 being set st x = [y1,y2] ; ::_thesis: verum
end;
then reconsider a9 = { [o,((G ^ o) | the carrier of H9)] where o is Element of O : verum } as Relation by RELAT_1:def_1;
A17: now__::_thesis:_for_x_being_set_st_x_in_O_holds_
ex_y_being_set_st_[x,y]_in_a9
let x be set ; ::_thesis: ( x in O implies ex y being set st [x,y] in a9 )
assume x in O ; ::_thesis: ex y being set st [x,y] in a9
then reconsider o = x as Element of O ;
reconsider y = (G ^ o) | the carrier of H9 as set ;
take y = y; ::_thesis: [x,y] in a9
thus [x,y] in a9 ; ::_thesis: verum
end;
now__::_thesis:_for_x_being_set_st_ex_y_being_set_st_[x,y]_in_a9_holds_
x_in_O
let x be set ; ::_thesis: ( ex y being set st [x,y] in a9 implies x in O )
given y being set such that A18: [x,y] in a9 ; ::_thesis: x in O
consider o being Element of O such that
A19: [x,y] = [o,((G ^ o) | the carrier of H9)] by A18;
o in O by A16;
hence x in O by A19, XTUPLE_0:1; ::_thesis: verum
end;
then A20: dom a9 = O by A17, XTUPLE_0:def_12;
now__::_thesis:_for_x,_y1,_y2_being_set_st_[x,y1]_in_a9_&_[x,y2]_in_a9_holds_
y1_=_y2
let x, y1, y2 be set ; ::_thesis: ( [x,y1] in a9 & [x,y2] in a9 implies y1 = y2 )
assume [x,y1] in a9 ; ::_thesis: ( [x,y2] in a9 implies y1 = y2 )
then consider o1 being Element of O such that
A21: [x,y1] = [o1,((G ^ o1) | the carrier of H9)] ;
A22: x = o1 by A21, XTUPLE_0:1;
assume [x,y2] in a9 ; ::_thesis: y1 = y2
then consider o2 being Element of O such that
A23: [x,y2] = [o2,((G ^ o2) | the carrier of H9)] ;
x = o2 by A23, XTUPLE_0:1;
hence y1 = y2 by A21, A23, A22, XTUPLE_0:1; ::_thesis: verum
end;
then reconsider a9 = a9 as Function by FUNCT_1:def_1;
now__::_thesis:_for_y_being_set_st_y_in_rng_a9_holds_
y_in_Funcs_(_the_carrier_of_H9,_the_carrier_of_H9)
let y be set ; ::_thesis: ( y in rng a9 implies y in Funcs ( the carrier of H9, the carrier of H9) )
assume y in rng a9 ; ::_thesis: y in Funcs ( the carrier of H9, the carrier of H9)
then consider x being set such that
A24: ( x in dom a9 & y = a9 . x ) by FUNCT_1:def_3;
[x,y] in a9 by A24, FUNCT_1:1;
then consider o being Element of O such that
A25: [x,y] = [o,((G ^ o) | the carrier of H9)] ;
now__::_thesis:_ex_f_being_Function_st_
(_y_=_f_&_dom_f_=_the_carrier_of_H9_&_rng_f_c=_the_carrier_of_H9_)
reconsider f = (G ^ o) | the carrier of H9 as Function ;
take f = f; ::_thesis: ( y = f & dom f = the carrier of H9 & rng f c= the carrier of H9 )
A26: dom ((G ^ o) | the carrier of H9) = dom ((G ^ o) * (id the carrier of H9)) by RELAT_1:65
.= (dom (G ^ o)) /\ the carrier of H9 by FUNCT_1:19
.= the carrier of G /\ the carrier of H9 by FUNCT_2:def_1 ;
thus y = f by A25, XTUPLE_0:1; ::_thesis: ( dom f = the carrier of H9 & rng f c= the carrier of H9 )
the carrier of H9 c= the carrier of G by GROUP_2:def_5;
hence A27: dom f = the carrier of H9 by A26, XBOOLE_1:28; ::_thesis: rng f c= the carrier of H9
now__::_thesis:_for_y_being_set_st_y_in_rng_f_holds_
y_in_the_carrier_of_H9
let y be set ; ::_thesis: ( y in rng f implies y in the carrier of H9 )
A28: dom f = dom ((G ^ o) * (id the carrier of H9)) by RELAT_1:65;
assume y in rng f ; ::_thesis: y in the carrier of H9
then consider x being set such that
A29: x in dom f and
A30: y = f . x by FUNCT_1:def_3;
y = ((G ^ o) * (id the carrier of H9)) . x by A30, RELAT_1:65
.= (G ^ o) . ((id the carrier of H9) . x) by A28, A29, FUNCT_1:12
.= (G ^ o) . x by A27, A29, FUNCT_1:18 ;
hence y in the carrier of H9 by A4, A3, A27, A29; ::_thesis: verum
end;
hence rng f c= the carrier of H9 by TARSKI:def_3; ::_thesis: verum
end;
hence y in Funcs ( the carrier of H9, the carrier of H9) by FUNCT_2:def_2; ::_thesis: verum
end;
then rng a9 c= Funcs ( the carrier of H9, the carrier of H9) by TARSKI:def_3;
then reconsider a9 = a9 as Action of O, the carrier of H9 by A20, FUNCT_2:2;
reconsider H = HGrWOpStr(# the carrier of H9, the multF of H9,a9 #) as non empty HGrWOpStr over O ;
A31: the multF of H = the multF of G || the carrier of H by GROUP_2:def_5;
( H is Group-like & the carrier of H c= the carrier of G ) by A5, GROUP_2:def_5;
then A32: H is Subgroup of G by A31, GROUP_2:def_5;
now__::_thesis:_for_G9_being_Group
for_a_being_Action_of_O,_the_carrier_of_G9_st_a_=_the_action_of_H_&_multMagma(#_the_carrier_of_G9,_the_multF_of_G9_#)_=_multMagma(#_the_carrier_of_H,_the_multF_of_H_#)_holds_
a_is_distributive
let G9 be Group; ::_thesis: for a being Action of O, the carrier of G9 st a = the action of H & multMagma(# the carrier of G9, the multF of G9 #) = multMagma(# the carrier of H, the multF of H #) holds
a is distributive
let a be Action of O, the carrier of G9; ::_thesis: ( a = the action of H & multMagma(# the carrier of G9, the multF of G9 #) = multMagma(# the carrier of H, the multF of H #) implies a is distributive )
assume A33: a = the action of H ; ::_thesis: ( multMagma(# the carrier of G9, the multF of G9 #) = multMagma(# the carrier of H, the multF of H #) implies a is distributive )
assume A34: multMagma(# the carrier of G9, the multF of G9 #) = multMagma(# the carrier of H, the multF of H #) ; ::_thesis: a is distributive
now__::_thesis:_for_o_being_Element_of_O_st_o_in_O_holds_
a_._o_is_Homomorphism_of_G9,G9
let o be Element of O; ::_thesis: ( o in O implies a . o is Homomorphism of G9,G9 )
assume o in O ; ::_thesis: a . o is Homomorphism of G9,G9
then A35: o in dom a by FUNCT_2:def_1;
then a . o in rng a by FUNCT_1:3;
then consider f being Function such that
A36: a . o = f and
A37: ( dom f = the carrier of G9 & rng f c= the carrier of G9 ) by FUNCT_2:def_2;
reconsider f = f as Function of G9,G9 by A37, FUNCT_2:2;
[o,(a . o)] in a9 by A33, A35, FUNCT_1:1;
then consider o9 being Element of O such that
A38: [o,(a . o)] = [o9,((G ^ o9) | the carrier of H9)] ;
A39: ( o = o9 & a . o = (G ^ o9) | the carrier of H9 ) by A38, XTUPLE_0:1;
now__::_thesis:_for_a9,_b9_being_Element_of_G9_holds_f_._(a9_*_b9)_=_(f_._a9)_*_(f_._b9)
let a9, b9 be Element of G9; ::_thesis: f . (a9 * b9) = (f . a9) * (f . b9)
b9 in the carrier of H9 by A34;
then A40: b9 in dom (id the carrier of H9) ;
reconsider a = a9, b = b9 as Element of H by A34;
reconsider g1 = a, g2 = b as Element of G by GROUP_2:42;
a9 in the carrier of H9 by A34;
then A41: a9 in dom (id the carrier of H9) ;
reconsider h1 = (G ^ o) . g1, h2 = (G ^ o) . g2 as Element of H by A4, A3;
a9 * b9 in the carrier of H9 by A34;
then A42: a9 * b9 in dom (id the carrier of H9) ;
A43: f . b9 = ((G ^ o) * (id the carrier of H9)) . b9 by A36, A39, RELAT_1:65
.= (G ^ o) . ((id the carrier of H9) . b9) by A40, FUNCT_1:13
.= h2 by FUNCT_1:18 ;
A44: f . a9 = ((G ^ o) * (id the carrier of H9)) . a9 by A36, A39, RELAT_1:65
.= (G ^ o) . ((id the carrier of H9) . a9) by A41, FUNCT_1:13
.= h1 by FUNCT_1:18 ;
thus f . (a9 * b9) = ((G ^ o) * (id the carrier of H9)) . (a9 * b9) by A36, A39, RELAT_1:65
.= (G ^ o) . ((id the carrier of H9) . (a9 * b9)) by A42, FUNCT_1:13
.= (G ^ o) . (a * b) by A34, FUNCT_1:18
.= (G ^ o) . (g1 * g2) by A32, GROUP_2:43
.= ((G ^ o) . g1) * ((G ^ o) . g2) by GROUP_6:def_6
.= h1 * h2 by A32, GROUP_2:43
.= (f . a9) * (f . b9) by A34, A44, A43 ; ::_thesis: verum
end;
hence a . o is Homomorphism of G9,G9 by A36, GROUP_6:def_6; ::_thesis: verum
end;
hence a is distributive by Def4; ::_thesis: verum
end;
then reconsider H = H as GroupWithOperators of O by A5, Def5;
now__::_thesis:_for_o_being_Element_of_O_holds_H_^_o_=_(G_^_o)_|_the_carrier_of_H
let o be Element of O; ::_thesis: H ^ o = (G ^ o) | the carrier of H
o in O by A16;
then o in dom a9 by FUNCT_2:def_1;
then [o,(a9 . o)] in a9 by FUNCT_1:1;
then consider o9 being Element of O such that
A45: [o,(a9 . o)] = [o9,((G ^ o9) | the carrier of H9)] ;
( o = o9 & a9 . o = (G ^ o9) | the carrier of H9 ) by A45, XTUPLE_0:1;
hence H ^ o = (G ^ o) | the carrier of H by A16, Def6; ::_thesis: verum
end;
then reconsider H = H as strict StableSubgroup of G by A32, Def7;
take H ; ::_thesis: the carrier of H = A
thus the carrier of H = A by A3; ::_thesis: verum
end;
end;
end;
definition
let O be set ;
let G, H be GroupWithOperators of O;
let g be Homomorphism of G,H;
func Ker g -> strict StableSubgroup of G means :Def21: :: GROUP_9:def 21
the carrier of it = { a where a is Element of G : g . a = 1_ H } ;
existence
ex b1 being strict StableSubgroup of G st the carrier of b1 = { a where a is Element of G : g . a = 1_ H }
proof
defpred S1[ set ] means g . $1 = 1_ H;
reconsider A = { a where a is Element of G : S1[a] } as Subset of G from DOMAIN_1:sch_7();
A1: now__::_thesis:_for_a,_b_being_Element_of_G_st_a_in_A_&_b_in_A_holds_
a_*_b_in_A
let a, b be Element of G; ::_thesis: ( a in A & b in A implies a * b in A )
assume ( a in A & b in A ) ; ::_thesis: a * b in A
then A2: ( ex a1 being Element of G st
( a1 = a & g . a1 = 1_ H ) & ex b1 being Element of G st
( b1 = b & g . b1 = 1_ H ) ) ;
g . (a * b) = (g . a) * (g . b) by GROUP_6:def_6
.= 1_ H by A2, GROUP_1:def_4 ;
hence a * b in A ; ::_thesis: verum
end;
A3: now__::_thesis:_for_a_being_Element_of_G_st_a_in_A_holds_
a_"_in_A
let a be Element of G; ::_thesis: ( a in A implies a " in A )
assume a in A ; ::_thesis: a " in A
then ex a1 being Element of G st
( a1 = a & g . a1 = 1_ H ) ;
then g . (a ") = (1_ H) " by Lm14
.= 1_ H by GROUP_1:8 ;
hence a " in A ; ::_thesis: verum
end;
A4: now__::_thesis:_for_o_being_Element_of_O
for_a_being_Element_of_G_st_a_in_A_holds_
(G_^_o)_._a_in_A
let o be Element of O; ::_thesis: for a being Element of G st a in A holds
(G ^ o) . a in A
let a be Element of G; ::_thesis: ( a in A implies (G ^ o) . a in A )
assume a in A ; ::_thesis: (G ^ o) . a in A
then ex a1 being Element of G st
( a1 = a & g . a1 = 1_ H ) ;
then g . ((G ^ o) . a) = (H ^ o) . (1_ H) by Def18
.= 1_ H by GROUP_6:31 ;
hence (G ^ o) . a in A ; ::_thesis: verum
end;
g . (1_ G) = 1_ H by Lm13;
then 1_ G in A ;
then consider B being strict StableSubgroup of G such that
A5: the carrier of B = A by A1, A3, A4, Lm15;
take B ; ::_thesis: the carrier of B = { a where a is Element of G : g . a = 1_ H }
thus the carrier of B = { a where a is Element of G : g . a = 1_ H } by A5; ::_thesis: verum
end;
uniqueness
for b1, b2 being strict StableSubgroup of G st the carrier of b1 = { a where a is Element of G : g . a = 1_ H } & the carrier of b2 = { a where a is Element of G : g . a = 1_ H } holds
b1 = b2 by Lm5;
end;
:: deftheorem Def21 defines Ker GROUP_9:def_21_:_
for O being set
for G, H being GroupWithOperators of O
for g being Homomorphism of G,H
for b5 being strict StableSubgroup of G holds
( b5 = Ker g iff the carrier of b5 = { a where a is Element of G : g . a = 1_ H } );
registration
let O be set ;
let G, H be GroupWithOperators of O;
let g be Homomorphism of G,H;
cluster Ker g -> strict normal ;
correctness
coherence
Ker g is normal ;
proof
now__::_thesis:_for_N_being_strict_Subgroup_of_G_st_N_=_multMagma(#_the_carrier_of_(Ker_g),_the_multF_of_(Ker_g)_#)_holds_
N_is_normal
reconsider G9 = G, H9 = H as Group ;
let N be strict Subgroup of G; ::_thesis: ( N = multMagma(# the carrier of (Ker g), the multF of (Ker g) #) implies N is normal )
reconsider g9 = g as Homomorphism of G9,H9 ;
A1: the carrier of (Ker g9) = { a where a is Element of G : g . a = 1_ H } by GROUP_6:def_9;
assume N = multMagma(# the carrier of (Ker g), the multF of (Ker g) #) ; ::_thesis: N is normal
then the carrier of (Ker g9) = the carrier of N by A1, Def21;
hence N is normal by GROUP_2:59; ::_thesis: verum
end;
hence Ker g is normal by Def10; ::_thesis: verum
end;
end;
Lm16: for O being set
for G being GroupWithOperators of O
for H being StableSubgroup of G holds multMagma(# the carrier of H, the multF of H #) is strict Subgroup of G
proof
let O be set ; ::_thesis: for G being GroupWithOperators of O
for H being StableSubgroup of G holds multMagma(# the carrier of H, the multF of H #) is strict Subgroup of G
let G be GroupWithOperators of O; ::_thesis: for H being StableSubgroup of G holds multMagma(# the carrier of H, the multF of H #) is strict Subgroup of G
let H be StableSubgroup of G; ::_thesis: multMagma(# the carrier of H, the multF of H #) is strict Subgroup of G
reconsider H9 = multMagma(# the carrier of H, the multF of H #) as non empty multMagma ;
now__::_thesis:_ex_e9_being_Element_of_H9_st_
for_h9_being_Element_of_H9_holds_
(_h9_*_e9_=_h9_&_e9_*_h9_=_h9_&_ex_g9_being_Element_of_H9_st_
(_h9_*_g9_=_e9_&_g9_*_h9_=_e9_)_)
set e = 1_ H;
reconsider e9 = 1_ H as Element of H9 ;
take e9 = e9; ::_thesis: for h9 being Element of H9 holds
( h9 * e9 = h9 & e9 * h9 = h9 & ex g9 being Element of H9 st
( h9 * g9 = e9 & g9 * h9 = e9 ) )
let h9 be Element of H9; ::_thesis: ( h9 * e9 = h9 & e9 * h9 = h9 & ex g9 being Element of H9 st
( h9 * g9 = e9 & g9 * h9 = e9 ) )
reconsider h = h9 as Element of H ;
set g = h " ;
reconsider g9 = h " as Element of H9 ;
h9 * e9 = h * (1_ H)
.= h by GROUP_1:def_4 ;
hence h9 * e9 = h9 ; ::_thesis: ( e9 * h9 = h9 & ex g9 being Element of H9 st
( h9 * g9 = e9 & g9 * h9 = e9 ) )
e9 * h9 = (1_ H) * h
.= h by GROUP_1:def_4 ;
hence e9 * h9 = h9 ; ::_thesis: ex g9 being Element of H9 st
( h9 * g9 = e9 & g9 * h9 = e9 )
take g9 = g9; ::_thesis: ( h9 * g9 = e9 & g9 * h9 = e9 )
h9 * g9 = h * (h ")
.= 1_ H by GROUP_1:def_5 ;
hence h9 * g9 = e9 ; ::_thesis: g9 * h9 = e9
g9 * h9 = (h ") * h
.= 1_ H by GROUP_1:def_5 ;
hence g9 * h9 = e9 ; ::_thesis: verum
end;
then reconsider H9 = H9 as non empty Group-like multMagma by GROUP_1:def_2;
H is Subgroup of G by Def7;
then ( the carrier of H9 c= the carrier of G & the multF of H9 = the multF of G || the carrier of H9 ) by GROUP_2:def_5;
hence multMagma(# the carrier of H, the multF of H #) is strict Subgroup of G by GROUP_2:def_5; ::_thesis: verum
end;
Lm17: for O being set
for G, H being GroupWithOperators of O
for G9 being strict StableSubgroup of G
for f being Homomorphism of G,H ex H9 being strict StableSubgroup of H st the carrier of H9 = f .: the carrier of G9
proof
let O be set ; ::_thesis: for G, H being GroupWithOperators of O
for G9 being strict StableSubgroup of G
for f being Homomorphism of G,H ex H9 being strict StableSubgroup of H st the carrier of H9 = f .: the carrier of G9
let G, H be GroupWithOperators of O; ::_thesis: for G9 being strict StableSubgroup of G
for f being Homomorphism of G,H ex H9 being strict StableSubgroup of H st the carrier of H9 = f .: the carrier of G9
let G9 be strict StableSubgroup of G; ::_thesis: for f being Homomorphism of G,H ex H9 being strict StableSubgroup of H st the carrier of H9 = f .: the carrier of G9
reconsider G99 = multMagma(# the carrier of G9, the multF of G9 #) as strict Subgroup of G by Lm16;
let f be Homomorphism of G,H; ::_thesis: ex H9 being strict StableSubgroup of H st the carrier of H9 = f .: the carrier of G9
set A = { (f . g) where g is Element of G : g in G99 } ;
1_ G in G99 by GROUP_2:46;
then f . (1_ G) in { (f . g) where g is Element of G : g in G99 } ;
then reconsider A = { (f . g) where g is Element of G : g in G99 } as non empty set ;
now__::_thesis:_for_x_being_set_st_x_in_A_holds_
x_in_the_carrier_of_H
let x be set ; ::_thesis: ( x in A implies x in the carrier of H )
assume x in A ; ::_thesis: x in the carrier of H
then ex g being Element of G st
( x = f . g & g in G99 ) ;
hence x in the carrier of H ; ::_thesis: verum
end;
then reconsider A = A as Subset of H by TARSKI:def_3;
A1: now__::_thesis:_for_h1,_h2_being_Element_of_H_st_h1_in_A_&_h2_in_A_holds_
h1_*_h2_in_A
let h1, h2 be Element of H; ::_thesis: ( h1 in A & h2 in A implies h1 * h2 in A )
assume that
A2: h1 in A and
A3: h2 in A ; ::_thesis: h1 * h2 in A
consider a being Element of G such that
A4: ( h1 = f . a & a in G99 ) by A2;
consider b being Element of G such that
A5: ( h2 = f . b & b in G99 ) by A3;
( f . (a * b) = h1 * h2 & a * b in G99 ) by A4, A5, GROUP_2:50, GROUP_6:def_6;
hence h1 * h2 in A ; ::_thesis: verum
end;
A6: now__::_thesis:_for_o_being_Element_of_O
for_h_being_Element_of_H_st_h_in_A_holds_
(H_^_o)_._h_in_A
let o be Element of O; ::_thesis: for h being Element of H st h in A holds
(H ^ o) . h in A
let h be Element of H; ::_thesis: ( h in A implies (H ^ o) . h in A )
assume h in A ; ::_thesis: (H ^ o) . h in A
then consider g being Element of G such that
A7: h = f . g and
A8: g in G99 ;
g in the carrier of G99 by A8, STRUCT_0:def_5;
then g in G9 by STRUCT_0:def_5;
then (G ^ o) . g in G9 by Lm10;
then (G ^ o) . g in the carrier of G9 by STRUCT_0:def_5;
then A9: (G ^ o) . g in G99 by STRUCT_0:def_5;
(H ^ o) . h = f . ((G ^ o) . g) by A7, Def18;
hence (H ^ o) . h in A by A9; ::_thesis: verum
end;
now__::_thesis:_for_h_being_Element_of_H_st_h_in_A_holds_
h_"_in_A
let h be Element of H; ::_thesis: ( h in A implies h " in A )
assume h in A ; ::_thesis: h " in A
then consider g being Element of G such that
A10: ( h = f . g & g in G99 ) ;
( g " in G99 & h " = f . (g ") ) by A10, Lm14, GROUP_2:51;
hence h " in A ; ::_thesis: verum
end;
then consider H99 being strict StableSubgroup of H such that
A11: the carrier of H99 = A by A1, A6, Lm15;
take H99 ; ::_thesis: the carrier of H99 = f .: the carrier of G9
now__::_thesis:_for_h_being_Element_of_H_holds_
(_(_h_in_A_implies_h_in_f_.:_the_carrier_of_G9_)_&_(_h_in_f_.:_the_carrier_of_G9_implies_h_in_A_)_)
set R = f;
let h be Element of H; ::_thesis: ( ( h in A implies h in f .: the carrier of G9 ) & ( h in f .: the carrier of G9 implies h in A ) )
reconsider R = f as Relation of the carrier of G, the carrier of H ;
hereby ::_thesis: ( h in f .: the carrier of G9 implies h in A )
assume h in A ; ::_thesis: h in f .: the carrier of G9
then consider g being Element of G such that
A12: h = f . g and
A13: g in G99 ;
A14: g in the carrier of G9 by A13, STRUCT_0:def_5;
dom f = the carrier of G by FUNCT_2:def_1;
then [g,h] in f by A12, FUNCT_1:1;
hence h in f .: the carrier of G9 by A14, RELSET_1:29; ::_thesis: verum
end;
assume h in f .: the carrier of G9 ; ::_thesis: h in A
then consider g being Element of G such that
A15: ( [g,h] in R & g in the carrier of G9 ) by RELSET_1:29;
( f . g = h & g in G99 ) by A15, FUNCT_1:1, STRUCT_0:def_5;
hence h in A ; ::_thesis: verum
end;
hence the carrier of H99 = f .: the carrier of G9 by A11, SUBSET_1:3; ::_thesis: verum
end;
definition
let O be set ;
let G, H be GroupWithOperators of O;
let g be Homomorphism of G,H;
func Image g -> strict StableSubgroup of H means :Def22: :: GROUP_9:def 22
the carrier of it = g .: the carrier of G;
existence
ex b1 being strict StableSubgroup of H st the carrier of b1 = g .: the carrier of G
proof
reconsider G9 = HGrWOpStr(# the carrier of G, the multF of G, the action of G #) as strict StableSubgroup of G by Lm4;
consider H9 being strict StableSubgroup of H such that
A1: the carrier of H9 = g .: the carrier of G9 by Lm17;
take H9 ; ::_thesis: the carrier of H9 = g .: the carrier of G
thus the carrier of H9 = g .: the carrier of G by A1; ::_thesis: verum
end;
uniqueness
for b1, b2 being strict StableSubgroup of H st the carrier of b1 = g .: the carrier of G & the carrier of b2 = g .: the carrier of G holds
b1 = b2 by Lm5;
end;
:: deftheorem Def22 defines Image GROUP_9:def_22_:_
for O being set
for G, H being GroupWithOperators of O
for g being Homomorphism of G,H
for b5 being strict StableSubgroup of H holds
( b5 = Image g iff the carrier of b5 = g .: the carrier of G );
definition
let O be set ;
let G be GroupWithOperators of O;
let H be StableSubgroup of G;
func carr H -> Subset of G equals :: GROUP_9:def 23
the carrier of H;
coherence
the carrier of H is Subset of G
proof
reconsider H9 = H as Subgroup of G by Def7;
carr H9 is Subset of G ;
hence the carrier of H is Subset of G ; ::_thesis: verum
end;
end;
:: deftheorem defines carr GROUP_9:def_23_:_
for O being set
for G being GroupWithOperators of O
for H being StableSubgroup of G holds carr H = the carrier of H;
definition
let O be set ;
let G be GroupWithOperators of O;
let H1, H2 be StableSubgroup of G;
funcH1 * H2 -> Subset of G equals :: GROUP_9:def 24
(carr H1) * (carr H2);
coherence
(carr H1) * (carr H2) is Subset of G ;
end;
:: deftheorem defines * GROUP_9:def_24_:_
for O being set
for G being GroupWithOperators of O
for H1, H2 being StableSubgroup of G holds H1 * H2 = (carr H1) * (carr H2);
Lm18: for O being set
for G being GroupWithOperators of O
for H being StableSubgroup of G holds 1_ G in H
proof
let O be set ; ::_thesis: for G being GroupWithOperators of O
for H being StableSubgroup of G holds 1_ G in H
let G be GroupWithOperators of O; ::_thesis: for H being StableSubgroup of G holds 1_ G in H
let H be StableSubgroup of G; ::_thesis: 1_ G in H
H is Subgroup of G by Def7;
hence 1_ G in H by GROUP_2:46; ::_thesis: verum
end;
Lm19: for O being set
for G being GroupWithOperators of O
for H being StableSubgroup of G
for g1, g2 being Element of G st g1 in H & g2 in H holds
g1 * g2 in H
proof
let O be set ; ::_thesis: for G being GroupWithOperators of O
for H being StableSubgroup of G
for g1, g2 being Element of G st g1 in H & g2 in H holds
g1 * g2 in H
let G be GroupWithOperators of O; ::_thesis: for H being StableSubgroup of G
for g1, g2 being Element of G st g1 in H & g2 in H holds
g1 * g2 in H
let H be StableSubgroup of G; ::_thesis: for g1, g2 being Element of G st g1 in H & g2 in H holds
g1 * g2 in H
let g1, g2 be Element of G; ::_thesis: ( g1 in H & g2 in H implies g1 * g2 in H )
assume A1: ( g1 in H & g2 in H ) ; ::_thesis: g1 * g2 in H
H is Subgroup of G by Def7;
hence g1 * g2 in H by A1, GROUP_2:50; ::_thesis: verum
end;
Lm20: for O being set
for G being GroupWithOperators of O
for H being StableSubgroup of G
for g being Element of G st g in H holds
g " in H
proof
let O be set ; ::_thesis: for G being GroupWithOperators of O
for H being StableSubgroup of G
for g being Element of G st g in H holds
g " in H
let G be GroupWithOperators of O; ::_thesis: for H being StableSubgroup of G
for g being Element of G st g in H holds
g " in H
let H be StableSubgroup of G; ::_thesis: for g being Element of G st g in H holds
g " in H
let g be Element of G; ::_thesis: ( g in H implies g " in H )
assume A1: g in H ; ::_thesis: g " in H
H is Subgroup of G by Def7;
hence g " in H by A1, GROUP_2:51; ::_thesis: verum
end;
definition
let O be set ;
let G be GroupWithOperators of O;
let H1, H2 be StableSubgroup of G;
funcH1 /\ H2 -> strict StableSubgroup of G means :Def25: :: GROUP_9:def 25
the carrier of it = (carr H1) /\ (carr H2);
existence
ex b1 being strict StableSubgroup of G st the carrier of b1 = (carr H1) /\ (carr H2)
proof
set A = (carr H1) /\ (carr H2);
1_ G in H2 by Lm18;
then A1: 1_ G in the carrier of H2 by STRUCT_0:def_5;
A2: now__::_thesis:_for_g1,_g2_being_Element_of_G_st_g1_in_(carr_H1)_/\_(carr_H2)_&_g2_in_(carr_H1)_/\_(carr_H2)_holds_
g1_*_g2_in_(carr_H1)_/\_(carr_H2)
let g1, g2 be Element of G; ::_thesis: ( g1 in (carr H1) /\ (carr H2) & g2 in (carr H1) /\ (carr H2) implies g1 * g2 in (carr H1) /\ (carr H2) )
assume that
A3: g1 in (carr H1) /\ (carr H2) and
A4: g2 in (carr H1) /\ (carr H2) ; ::_thesis: g1 * g2 in (carr H1) /\ (carr H2)
g2 in carr H2 by A4, XBOOLE_0:def_4;
then A5: g2 in H2 by STRUCT_0:def_5;
g1 in carr H2 by A3, XBOOLE_0:def_4;
then g1 in H2 by STRUCT_0:def_5;
then g1 * g2 in H2 by A5, Lm19;
then A6: g1 * g2 in carr H2 by STRUCT_0:def_5;
g2 in carr H1 by A4, XBOOLE_0:def_4;
then A7: g2 in H1 by STRUCT_0:def_5;
g1 in carr H1 by A3, XBOOLE_0:def_4;
then g1 in H1 by STRUCT_0:def_5;
then g1 * g2 in H1 by A7, Lm19;
then g1 * g2 in carr H1 by STRUCT_0:def_5;
hence g1 * g2 in (carr H1) /\ (carr H2) by A6, XBOOLE_0:def_4; ::_thesis: verum
end;
A8: now__::_thesis:_for_o_being_Element_of_O
for_a_being_Element_of_G_st_a_in_(carr_H1)_/\_(carr_H2)_holds_
(G_^_o)_._a_in_(carr_H1)_/\_(carr_H2)
let o be Element of O; ::_thesis: for a being Element of G st a in (carr H1) /\ (carr H2) holds
(G ^ o) . a in (carr H1) /\ (carr H2)
let a be Element of G; ::_thesis: ( a in (carr H1) /\ (carr H2) implies (G ^ o) . a in (carr H1) /\ (carr H2) )
assume A9: a in (carr H1) /\ (carr H2) ; ::_thesis: (G ^ o) . a in (carr H1) /\ (carr H2)
then a in carr H2 by XBOOLE_0:def_4;
then a in H2 by STRUCT_0:def_5;
then (G ^ o) . a in H2 by Lm10;
then A10: (G ^ o) . a in carr H2 by STRUCT_0:def_5;
a in carr H1 by A9, XBOOLE_0:def_4;
then a in H1 by STRUCT_0:def_5;
then (G ^ o) . a in H1 by Lm10;
then (G ^ o) . a in carr H1 by STRUCT_0:def_5;
hence (G ^ o) . a in (carr H1) /\ (carr H2) by A10, XBOOLE_0:def_4; ::_thesis: verum
end;
A11: now__::_thesis:_for_g_being_Element_of_G_st_g_in_(carr_H1)_/\_(carr_H2)_holds_
g_"_in_(carr_H1)_/\_(carr_H2)
let g be Element of G; ::_thesis: ( g in (carr H1) /\ (carr H2) implies g " in (carr H1) /\ (carr H2) )
assume A12: g in (carr H1) /\ (carr H2) ; ::_thesis: g " in (carr H1) /\ (carr H2)
then g in carr H2 by XBOOLE_0:def_4;
then g in H2 by STRUCT_0:def_5;
then g " in H2 by Lm20;
then A13: g " in carr H2 by STRUCT_0:def_5;
g in carr H1 by A12, XBOOLE_0:def_4;
then g in H1 by STRUCT_0:def_5;
then g " in H1 by Lm20;
then g " in carr H1 by STRUCT_0:def_5;
hence g " in (carr H1) /\ (carr H2) by A13, XBOOLE_0:def_4; ::_thesis: verum
end;
1_ G in H1 by Lm18;
then 1_ G in the carrier of H1 by STRUCT_0:def_5;
then (carr H1) /\ (carr H2) <> {} by A1, XBOOLE_0:def_4;
hence ex b1 being strict StableSubgroup of G st the carrier of b1 = (carr H1) /\ (carr H2) by A2, A11, A8, Lm15; ::_thesis: verum
end;
uniqueness
for b1, b2 being strict StableSubgroup of G st the carrier of b1 = (carr H1) /\ (carr H2) & the carrier of b2 = (carr H1) /\ (carr H2) holds
b1 = b2 by Lm5;
commutativity
for b1 being strict StableSubgroup of G
for H1, H2 being StableSubgroup of G st the carrier of b1 = (carr H1) /\ (carr H2) holds
the carrier of b1 = (carr H2) /\ (carr H1) ;
end;
:: deftheorem Def25 defines /\ GROUP_9:def_25_:_
for O being set
for G being GroupWithOperators of O
for H1, H2 being StableSubgroup of G
for b5 being strict StableSubgroup of G holds
( b5 = H1 /\ H2 iff the carrier of b5 = (carr H1) /\ (carr H2) );
Lm21: for O being set
for G being GroupWithOperators of O
for H1, H2 being StableSubgroup of G st the carrier of H1 c= the carrier of H2 holds
H1 is StableSubgroup of H2
proof
let O be set ; ::_thesis: for G being GroupWithOperators of O
for H1, H2 being StableSubgroup of G st the carrier of H1 c= the carrier of H2 holds
H1 is StableSubgroup of H2
let G be GroupWithOperators of O; ::_thesis: for H1, H2 being StableSubgroup of G st the carrier of H1 c= the carrier of H2 holds
H1 is StableSubgroup of H2
let H1, H2 be StableSubgroup of G; ::_thesis: ( the carrier of H1 c= the carrier of H2 implies H1 is StableSubgroup of H2 )
reconsider H19 = H1, H29 = H2 as Subgroup of G by Def7;
assume A1: the carrier of H1 c= the carrier of H2 ; ::_thesis: H1 is StableSubgroup of H2
A2: now__::_thesis:_for_o_being_Element_of_O_holds_H1_^_o_=_(H2_^_o)_|_the_carrier_of_H1
let o be Element of O; ::_thesis: H1 ^ o = (H2 ^ o) | the carrier of H1
thus H1 ^ o = (G ^ o) | the carrier of H1 by Def7
.= ((G ^ o) | the carrier of H2) | the carrier of H1 by A1, RELAT_1:74
.= (H2 ^ o) | the carrier of H1 by Def7 ; ::_thesis: verum
end;
H19 is Subgroup of H29 by A1, GROUP_2:57;
hence H1 is StableSubgroup of H2 by A2, Def7; ::_thesis: verum
end;
definition
let O be set ;
let G be GroupWithOperators of O;
let A be Subset of G;
func the_stable_subgroup_of A -> strict StableSubgroup of G means :Def26: :: GROUP_9:def 26
( A c= the carrier of it & ( for H being strict StableSubgroup of G st A c= the carrier of H holds
it is StableSubgroup of H ) );
existence
ex b1 being strict StableSubgroup of G st
( A c= the carrier of b1 & ( for H being strict StableSubgroup of G st A c= the carrier of H holds
b1 is StableSubgroup of H ) )
proof
defpred S1[ set ] means ex H being strict StableSubgroup of G st
( $1 = carr H & A c= $1 );
consider X being set such that
A1: for Y being set holds
( Y in X iff ( Y in bool the carrier of G & S1[Y] ) ) from XBOOLE_0:sch_1();
set M = meet X;
A2: carr ((Omega). G) = the carrier of ((Omega). G) ;
then A3: X <> {} by A1;
A4: the carrier of G in X by A1, A2;
A5: meet X c= the carrier of G
proof
let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in meet X or y in the carrier of G )
consider x being set such that
A6: x in X by A4;
consider H being strict StableSubgroup of G such that
A7: x = carr H and
A c= x by A1, A6;
assume y in meet X ; ::_thesis: y in the carrier of G
then y in carr H by A6, A7, SETFAM_1:def_1;
hence y in the carrier of G ; ::_thesis: verum
end;
now__::_thesis:_for_Y_being_set_st_Y_in_X_holds_
1__G_in_Y
let Y be set ; ::_thesis: ( Y in X implies 1_ G in Y )
assume Y in X ; ::_thesis: 1_ G in Y
then consider H being strict StableSubgroup of G such that
A8: Y = carr H and
A c= Y by A1;
1_ G in H by Lm18;
hence 1_ G in Y by A8, STRUCT_0:def_5; ::_thesis: verum
end;
then A9: meet X <> {} by A3, SETFAM_1:def_1;
reconsider M = meet X as Subset of G by A5;
A10: now__::_thesis:_for_o_being_Element_of_O
for_a_being_Element_of_G_st_a_in_M_holds_
(G_^_o)_._a_in_M
let o be Element of O; ::_thesis: for a being Element of G st a in M holds
(G ^ o) . a in M
let a be Element of G; ::_thesis: ( a in M implies (G ^ o) . a in M )
assume A11: a in M ; ::_thesis: (G ^ o) . a in M
now__::_thesis:_for_Y_being_set_st_Y_in_X_holds_
(G_^_o)_._a_in_Y
let Y be set ; ::_thesis: ( Y in X implies (G ^ o) . a in Y )
assume A12: Y in X ; ::_thesis: (G ^ o) . a in Y
then consider H being strict StableSubgroup of G such that
A13: Y = carr H and
A c= Y by A1;
a in carr H by A11, A12, A13, SETFAM_1:def_1;
then a in H by STRUCT_0:def_5;
then (G ^ o) . a in H by Lm10;
hence (G ^ o) . a in Y by A13, STRUCT_0:def_5; ::_thesis: verum
end;
hence (G ^ o) . a in M by A3, SETFAM_1:def_1; ::_thesis: verum
end;
A14: now__::_thesis:_for_a,_b_being_Element_of_G_st_a_in_M_&_b_in_M_holds_
a_*_b_in_M
let a, b be Element of G; ::_thesis: ( a in M & b in M implies a * b in M )
assume that
A15: a in M and
A16: b in M ; ::_thesis: a * b in M
now__::_thesis:_for_Y_being_set_st_Y_in_X_holds_
a_*_b_in_Y
let Y be set ; ::_thesis: ( Y in X implies a * b in Y )
assume A17: Y in X ; ::_thesis: a * b in Y
then consider H being strict StableSubgroup of G such that
A18: Y = carr H and
A c= Y by A1;
b in carr H by A16, A17, A18, SETFAM_1:def_1;
then A19: b in H by STRUCT_0:def_5;
a in carr H by A15, A17, A18, SETFAM_1:def_1;
then a in H by STRUCT_0:def_5;
then a * b in H by A19, Lm19;
hence a * b in Y by A18, STRUCT_0:def_5; ::_thesis: verum
end;
hence a * b in M by A3, SETFAM_1:def_1; ::_thesis: verum
end;
now__::_thesis:_for_a_being_Element_of_G_st_a_in_M_holds_
a_"_in_M
let a be Element of G; ::_thesis: ( a in M implies a " in M )
assume A20: a in M ; ::_thesis: a " in M
now__::_thesis:_for_Y_being_set_st_Y_in_X_holds_
a_"_in_Y
let Y be set ; ::_thesis: ( Y in X implies a " in Y )
assume A21: Y in X ; ::_thesis: a " in Y
then consider H being strict StableSubgroup of G such that
A22: Y = carr H and
A c= Y by A1;
a in carr H by A20, A21, A22, SETFAM_1:def_1;
then a in H by STRUCT_0:def_5;
then a " in H by Lm20;
hence a " in Y by A22, STRUCT_0:def_5; ::_thesis: verum
end;
hence a " in M by A3, SETFAM_1:def_1; ::_thesis: verum
end;
then consider H being strict StableSubgroup of G such that
A23: the carrier of H = M by A9, A14, A10, Lm15;
take H ; ::_thesis: ( A c= the carrier of H & ( for H being strict StableSubgroup of G st A c= the carrier of H holds
H is StableSubgroup of H ) )
now__::_thesis:_for_Y_being_set_st_Y_in_X_holds_
A_c=_Y
let Y be set ; ::_thesis: ( Y in X implies A c= Y )
assume Y in X ; ::_thesis: A c= Y
then ex H being strict StableSubgroup of G st
( Y = carr H & A c= Y ) by A1;
hence A c= Y ; ::_thesis: verum
end;
hence A c= the carrier of H by A3, A23, SETFAM_1:5; ::_thesis: for H being strict StableSubgroup of G st A c= the carrier of H holds
H is StableSubgroup of H
let H1 be strict StableSubgroup of G; ::_thesis: ( A c= the carrier of H1 implies H is StableSubgroup of H1 )
A24: the carrier of H1 = carr H1 ;
assume A c= the carrier of H1 ; ::_thesis: H is StableSubgroup of H1
then the carrier of H1 in X by A1, A24;
hence H is StableSubgroup of H1 by A23, Lm21, SETFAM_1:3; ::_thesis: verum
end;
uniqueness
for b1, b2 being strict StableSubgroup of G st A c= the carrier of b1 & ( for H being strict StableSubgroup of G st A c= the carrier of H holds
b1 is StableSubgroup of H ) & A c= the carrier of b2 & ( for H being strict StableSubgroup of G st A c= the carrier of H holds
b2 is StableSubgroup of H ) holds
b1 = b2
proof
let H1, H2 be strict StableSubgroup of G; ::_thesis: ( A c= the carrier of H1 & ( for H being strict StableSubgroup of G st A c= the carrier of H holds
H1 is StableSubgroup of H ) & A c= the carrier of H2 & ( for H being strict StableSubgroup of G st A c= the carrier of H holds
H2 is StableSubgroup of H ) implies H1 = H2 )
assume that
A25: A c= the carrier of H1 and
A26: ( ( for H being strict StableSubgroup of G st A c= the carrier of H holds
H1 is StableSubgroup of H ) & A c= the carrier of H2 ) and
A27: for H being strict StableSubgroup of G st A c= the carrier of H holds
H2 is StableSubgroup of H ; ::_thesis: H1 = H2
H1 is StableSubgroup of H2 by A26;
then H1 is Subgroup of H2 by Def7;
then A28: the carrier of H1 c= the carrier of H2 by GROUP_2:def_5;
H2 is StableSubgroup of H1 by A25, A27;
then H2 is Subgroup of H1 by Def7;
then the carrier of H2 c= the carrier of H1 by GROUP_2:def_5;
then the carrier of H1 = the carrier of H2 by A28, XBOOLE_0:def_10;
hence H1 = H2 by Lm5; ::_thesis: verum
end;
end;
:: deftheorem Def26 defines the_stable_subgroup_of GROUP_9:def_26_:_
for O being set
for G being GroupWithOperators of O
for A being Subset of G
for b4 being strict StableSubgroup of G holds
( b4 = the_stable_subgroup_of A iff ( A c= the carrier of b4 & ( for H being strict StableSubgroup of G st A c= the carrier of H holds
b4 is StableSubgroup of H ) ) );
definition
let O be set ;
let G be GroupWithOperators of O;
let H1, H2 be StableSubgroup of G;
funcH1 "\/" H2 -> strict StableSubgroup of G equals :: GROUP_9:def 27
the_stable_subgroup_of ((carr H1) \/ (carr H2));
correctness
coherence
the_stable_subgroup_of ((carr H1) \/ (carr H2)) is strict StableSubgroup of G;
;
end;
:: deftheorem defines "\/" GROUP_9:def_27_:_
for O being set
for G being GroupWithOperators of O
for H1, H2 being StableSubgroup of G holds H1 "\/" H2 = the_stable_subgroup_of ((carr H1) \/ (carr H2));
begin
theorem Th1: :: GROUP_9:1
for O, x being set
for G being GroupWithOperators of O
for H1 being StableSubgroup of G st x in H1 holds
x in G
proof
let O, x be set ; ::_thesis: for G being GroupWithOperators of O
for H1 being StableSubgroup of G st x in H1 holds
x in G
let G be GroupWithOperators of O; ::_thesis: for H1 being StableSubgroup of G st x in H1 holds
x in G
let H1 be StableSubgroup of G; ::_thesis: ( x in H1 implies x in G )
assume A1: x in H1 ; ::_thesis: x in G
H1 is Subgroup of G by Def7;
hence x in G by A1, GROUP_2:40; ::_thesis: verum
end;
theorem Th2: :: GROUP_9:2
for O being set
for G being GroupWithOperators of O
for H1 being StableSubgroup of G
for h1 being Element of H1 holds h1 is Element of G
proof
let O be set ; ::_thesis: for G being GroupWithOperators of O
for H1 being StableSubgroup of G
for h1 being Element of H1 holds h1 is Element of G
let G be GroupWithOperators of O; ::_thesis: for H1 being StableSubgroup of G
for h1 being Element of H1 holds h1 is Element of G
let H1 be StableSubgroup of G; ::_thesis: for h1 being Element of H1 holds h1 is Element of G
let h1 be Element of H1; ::_thesis: h1 is Element of G
H1 is Subgroup of G by Def7;
hence h1 is Element of G by GROUP_2:42; ::_thesis: verum
end;
theorem Th3: :: GROUP_9:3
for O being set
for G being GroupWithOperators of O
for H1 being StableSubgroup of G
for g1, g2 being Element of G
for h1, h2 being Element of H1 st h1 = g1 & h2 = g2 holds
h1 * h2 = g1 * g2
proof
let O be set ; ::_thesis: for G being GroupWithOperators of O
for H1 being StableSubgroup of G
for g1, g2 being Element of G
for h1, h2 being Element of H1 st h1 = g1 & h2 = g2 holds
h1 * h2 = g1 * g2
let G be GroupWithOperators of O; ::_thesis: for H1 being StableSubgroup of G
for g1, g2 being Element of G
for h1, h2 being Element of H1 st h1 = g1 & h2 = g2 holds
h1 * h2 = g1 * g2
let H1 be StableSubgroup of G; ::_thesis: for g1, g2 being Element of G
for h1, h2 being Element of H1 st h1 = g1 & h2 = g2 holds
h1 * h2 = g1 * g2
let g1, g2 be Element of G; ::_thesis: for h1, h2 being Element of H1 st h1 = g1 & h2 = g2 holds
h1 * h2 = g1 * g2
let h1, h2 be Element of H1; ::_thesis: ( h1 = g1 & h2 = g2 implies h1 * h2 = g1 * g2 )
assume A1: ( h1 = g1 & h2 = g2 ) ; ::_thesis: h1 * h2 = g1 * g2
H1 is Subgroup of G by Def7;
hence h1 * h2 = g1 * g2 by A1, GROUP_2:43; ::_thesis: verum
end;
theorem Th4: :: GROUP_9:4
for O being set
for G being GroupWithOperators of O
for H1 being StableSubgroup of G holds 1_ G = 1_ H1
proof
let O be set ; ::_thesis: for G being GroupWithOperators of O
for H1 being StableSubgroup of G holds 1_ G = 1_ H1
let G be GroupWithOperators of O; ::_thesis: for H1 being StableSubgroup of G holds 1_ G = 1_ H1
let H1 be StableSubgroup of G; ::_thesis: 1_ G = 1_ H1
reconsider H19 = H1 as Subgroup of G by Def7;
1_ H1 = 1_ H19 ;
hence 1_ G = 1_ H1 by GROUP_2:44; ::_thesis: verum
end;
theorem :: GROUP_9:5
for O being set
for G being GroupWithOperators of O
for H1 being StableSubgroup of G holds 1_ G in H1 by Lm18;
theorem Th6: :: GROUP_9:6
for O being set
for G being GroupWithOperators of O
for H1 being StableSubgroup of G
for g1 being Element of G
for h1 being Element of H1 st h1 = g1 holds
h1 " = g1 "
proof
let O be set ; ::_thesis: for G being GroupWithOperators of O
for H1 being StableSubgroup of G
for g1 being Element of G
for h1 being Element of H1 st h1 = g1 holds
h1 " = g1 "
let G be GroupWithOperators of O; ::_thesis: for H1 being StableSubgroup of G
for g1 being Element of G
for h1 being Element of H1 st h1 = g1 holds
h1 " = g1 "
let H1 be StableSubgroup of G; ::_thesis: for g1 being Element of G
for h1 being Element of H1 st h1 = g1 holds
h1 " = g1 "
let g1 be Element of G; ::_thesis: for h1 being Element of H1 st h1 = g1 holds
h1 " = g1 "
let h1 be Element of H1; ::_thesis: ( h1 = g1 implies h1 " = g1 " )
reconsider g9 = h1 " as Element of G by Th2;
A1: h1 * (h1 ") = 1_ H1 by GROUP_1:def_5;
assume h1 = g1 ; ::_thesis: h1 " = g1 "
then g1 * g9 = 1_ H1 by A1, Th3
.= 1_ G by Th4 ;
hence h1 " = g1 " by GROUP_1:12; ::_thesis: verum
end;
theorem :: GROUP_9:7
for O being set
for G being GroupWithOperators of O
for H1 being StableSubgroup of G
for g1, g2 being Element of G st g1 in H1 & g2 in H1 holds
g1 * g2 in H1 by Lm19;
theorem :: GROUP_9:8
for O being set
for G being GroupWithOperators of O
for H1 being StableSubgroup of G
for g1 being Element of G st g1 in H1 holds
g1 " in H1 by Lm20;
theorem :: GROUP_9:9
for O being set
for G being GroupWithOperators of O
for A being Subset of G st A <> {} & ( for g1, g2 being Element of G st g1 in A & g2 in A holds
g1 * g2 in A ) & ( for g1 being Element of G st g1 in A holds
g1 " in A ) & ( for o being Element of O
for g1 being Element of G st g1 in A holds
(G ^ o) . g1 in A ) holds
ex H being strict StableSubgroup of G st the carrier of H = A by Lm15;
theorem Th10: :: GROUP_9:10
for O being set
for G being GroupWithOperators of O holds G is StableSubgroup of G
proof
let O be set ; ::_thesis: for G being GroupWithOperators of O holds G is StableSubgroup of G
let G be GroupWithOperators of O; ::_thesis: G is StableSubgroup of G
A1: now__::_thesis:_for_o_being_Element_of_O_holds_G_^_o_=_(G_^_o)_|_the_carrier_of_G
let o be Element of O; ::_thesis: G ^ o = (G ^ o) | the carrier of G
thus G ^ o = (G ^ o) | the carrier of G ; ::_thesis: verum
end;
G is Subgroup of G by GROUP_2:54;
hence G is StableSubgroup of G by A1, Def7; ::_thesis: verum
end;
theorem Th11: :: GROUP_9:11
for O being set
for G1, G2, G3 being GroupWithOperators of O st G1 is StableSubgroup of G2 & G2 is StableSubgroup of G3 holds
G1 is StableSubgroup of G3
proof
let O be set ; ::_thesis: for G1, G2, G3 being GroupWithOperators of O st G1 is StableSubgroup of G2 & G2 is StableSubgroup of G3 holds
G1 is StableSubgroup of G3
let G1, G2, G3 be GroupWithOperators of O; ::_thesis: ( G1 is StableSubgroup of G2 & G2 is StableSubgroup of G3 implies G1 is StableSubgroup of G3 )
assume that
A1: G1 is StableSubgroup of G2 and
A2: G2 is StableSubgroup of G3 ; ::_thesis: G1 is StableSubgroup of G3
A3: G1 is Subgroup of G2 by A1, Def7;
A4: now__::_thesis:_for_o_being_Element_of_O_holds_G1_^_o_=_(G3_^_o)_|_the_carrier_of_G1
let o be Element of O; ::_thesis: G1 ^ o = (G3 ^ o) | the carrier of G1
A5: the carrier of G1 c= the carrier of G2 by A3, GROUP_2:def_5;
G1 ^ o = (G2 ^ o) | the carrier of G1 by A1, Def7
.= ((G3 ^ o) | the carrier of G2) | the carrier of G1 by A2, Def7
.= (G3 ^ o) | ( the carrier of G2 /\ the carrier of G1) by RELAT_1:71 ;
hence G1 ^ o = (G3 ^ o) | the carrier of G1 by A5, XBOOLE_1:28; ::_thesis: verum
end;
G2 is Subgroup of G3 by A2, Def7;
then G1 is Subgroup of G3 by A3, GROUP_2:56;
hence G1 is StableSubgroup of G3 by A4, Def7; ::_thesis: verum
end;
theorem :: GROUP_9:12
for O being set
for G being GroupWithOperators of O
for H1, H2 being StableSubgroup of G st the carrier of H1 c= the carrier of H2 holds
H1 is StableSubgroup of H2 by Lm21;
theorem Th13: :: GROUP_9:13
for O being set
for G being GroupWithOperators of O
for H1, H2 being StableSubgroup of G st ( for g being Element of G st g in H1 holds
g in H2 ) holds
H1 is StableSubgroup of H2
proof
let O be set ; ::_thesis: for G being GroupWithOperators of O
for H1, H2 being StableSubgroup of G st ( for g being Element of G st g in H1 holds
g in H2 ) holds
H1 is StableSubgroup of H2
let G be GroupWithOperators of O; ::_thesis: for H1, H2 being StableSubgroup of G st ( for g being Element of G st g in H1 holds
g in H2 ) holds
H1 is StableSubgroup of H2
let H1, H2 be StableSubgroup of G; ::_thesis: ( ( for g being Element of G st g in H1 holds
g in H2 ) implies H1 is StableSubgroup of H2 )
assume A1: for g being Element of G st g in H1 holds
g in H2 ; ::_thesis: H1 is StableSubgroup of H2
the carrier of H1 c= the carrier of H2
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in the carrier of H1 or x in the carrier of H2 )
assume x in the carrier of H1 ; ::_thesis: x in the carrier of H2
then reconsider g = x as Element of H1 ;
reconsider g = g as Element of G by Th2;
g in H1 by STRUCT_0:def_5;
then g in H2 by A1;
hence x in the carrier of H2 by STRUCT_0:def_5; ::_thesis: verum
end;
hence H1 is StableSubgroup of H2 by Lm21; ::_thesis: verum
end;
theorem :: GROUP_9:14
for O being set
for G being GroupWithOperators of O
for H1, H2 being strict StableSubgroup of G st the carrier of H1 = the carrier of H2 holds
H1 = H2 by Lm5;
theorem Th15: :: GROUP_9:15
for O being set
for G being GroupWithOperators of O
for H1 being StableSubgroup of G holds (1). G = (1). H1
proof
let O be set ; ::_thesis: for G being GroupWithOperators of O
for H1 being StableSubgroup of G holds (1). G = (1). H1
let G be GroupWithOperators of O; ::_thesis: for H1 being StableSubgroup of G holds (1). G = (1). H1
let H1 be StableSubgroup of G; ::_thesis: (1). G = (1). H1
A1: 1_ H1 = 1_ G by Th4;
( (1). H1 is StableSubgroup of G & the carrier of ((1). H1) = {(1_ H1)} ) by Def8, Th11;
hence (1). G = (1). H1 by A1, Def8; ::_thesis: verum
end;
theorem Th16: :: GROUP_9:16
for O being set
for G being GroupWithOperators of O
for H1 being StableSubgroup of G holds (1). G is StableSubgroup of H1
proof
let O be set ; ::_thesis: for G being GroupWithOperators of O
for H1 being StableSubgroup of G holds (1). G is StableSubgroup of H1
let G be GroupWithOperators of O; ::_thesis: for H1 being StableSubgroup of G holds (1). G is StableSubgroup of H1
let H1 be StableSubgroup of G; ::_thesis: (1). G is StableSubgroup of H1
(1). G = (1). H1 by Th15;
hence (1). G is StableSubgroup of H1 ; ::_thesis: verum
end;
theorem Th17: :: GROUP_9:17
for O being set
for G being GroupWithOperators of O
for H1, H2 being StableSubgroup of G st (carr H1) * (carr H2) = (carr H2) * (carr H1) holds
ex H being strict StableSubgroup of G st the carrier of H = (carr H1) * (carr H2)
proof
let O be set ; ::_thesis: for G being GroupWithOperators of O
for H1, H2 being StableSubgroup of G st (carr H1) * (carr H2) = (carr H2) * (carr H1) holds
ex H being strict StableSubgroup of G st the carrier of H = (carr H1) * (carr H2)
let G be GroupWithOperators of O; ::_thesis: for H1, H2 being StableSubgroup of G st (carr H1) * (carr H2) = (carr H2) * (carr H1) holds
ex H being strict StableSubgroup of G st the carrier of H = (carr H1) * (carr H2)
let H1, H2 be StableSubgroup of G; ::_thesis: ( (carr H1) * (carr H2) = (carr H2) * (carr H1) implies ex H being strict StableSubgroup of G st the carrier of H = (carr H1) * (carr H2) )
assume A1: (carr H1) * (carr H2) = (carr H2) * (carr H1) ; ::_thesis: ex H being strict StableSubgroup of G st the carrier of H = (carr H1) * (carr H2)
A2: now__::_thesis:_for_o_being_Element_of_O
for_g_being_Element_of_G_st_g_in_(carr_H1)_*_(carr_H2)_holds_
(G_^_o)_._g_in_(carr_H1)_*_(carr_H2)
let o be Element of O; ::_thesis: for g being Element of G st g in (carr H1) * (carr H2) holds
(G ^ o) . g in (carr H1) * (carr H2)
let g be Element of G; ::_thesis: ( g in (carr H1) * (carr H2) implies (G ^ o) . g in (carr H1) * (carr H2) )
assume g in (carr H1) * (carr H2) ; ::_thesis: (G ^ o) . g in (carr H1) * (carr H2)
then consider a, b being Element of G such that
A3: g = a * b and
A4: a in carr H1 and
A5: b in carr H2 ;
a in H1 by A4, STRUCT_0:def_5;
then (G ^ o) . a in H1 by Lm10;
then A6: (G ^ o) . a in carr H1 by STRUCT_0:def_5;
b in H2 by A5, STRUCT_0:def_5;
then (G ^ o) . b in H2 by Lm10;
then (G ^ o) . b in carr H2 by STRUCT_0:def_5;
then ((G ^ o) . a) * ((G ^ o) . b) in (carr H1) * (carr H2) by A6;
hence (G ^ o) . g in (carr H1) * (carr H2) by A3, GROUP_6:def_6; ::_thesis: verum
end;
A7: H2 is Subgroup of G by Def7;
A8: H1 is Subgroup of G by Def7;
A9: now__::_thesis:_for_g_being_Element_of_G_st_g_in_(carr_H1)_*_(carr_H2)_holds_
g_"_in_(carr_H1)_*_(carr_H2)
let g be Element of G; ::_thesis: ( g in (carr H1) * (carr H2) implies g " in (carr H1) * (carr H2) )
assume A10: g in (carr H1) * (carr H2) ; ::_thesis: g " in (carr H1) * (carr H2)
then consider a, b being Element of G such that
A11: g = a * b and
a in carr H1 and
b in carr H2 ;
consider b1, a1 being Element of G such that
A12: a * b = b1 * a1 and
A13: b1 in carr H2 and
A14: a1 in carr H1 by A1, A10, A11;
b1 in H2 by A13, STRUCT_0:def_5;
then b1 " in H2 by A7, GROUP_2:51;
then A15: b1 " in carr H2 by STRUCT_0:def_5;
a1 in H1 by A14, STRUCT_0:def_5;
then a1 " in H1 by A8, GROUP_2:51;
then A16: a1 " in carr H1 by STRUCT_0:def_5;
g " = (a1 ") * (b1 ") by A11, A12, GROUP_1:17;
hence g " in (carr H1) * (carr H2) by A16, A15; ::_thesis: verum
end;
A17: now__::_thesis:_for_g1,_g2_being_Element_of_G_st_g1_in_(carr_H1)_*_(carr_H2)_&_g2_in_(carr_H1)_*_(carr_H2)_holds_
g1_*_g2_in_(carr_H1)_*_(carr_H2)
let g1, g2 be Element of G; ::_thesis: ( g1 in (carr H1) * (carr H2) & g2 in (carr H1) * (carr H2) implies g1 * g2 in (carr H1) * (carr H2) )
assume that
A18: g1 in (carr H1) * (carr H2) and
A19: g2 in (carr H1) * (carr H2) ; ::_thesis: g1 * g2 in (carr H1) * (carr H2)
consider a1, b1 being Element of G such that
A20: g1 = a1 * b1 and
A21: a1 in carr H1 and
A22: b1 in carr H2 by A18;
consider a2, b2 being Element of G such that
A23: g2 = a2 * b2 and
A24: a2 in carr H1 and
A25: b2 in carr H2 by A19;
b1 * a2 in (carr H1) * (carr H2) by A1, A22, A24;
then consider a, b being Element of G such that
A26: b1 * a2 = a * b and
A27: a in carr H1 and
A28: b in carr H2 ;
A29: a in H1 by A27, STRUCT_0:def_5;
A30: b in H2 by A28, STRUCT_0:def_5;
b2 in H2 by A25, STRUCT_0:def_5;
then b * b2 in H2 by A7, A30, GROUP_2:50;
then A31: b * b2 in carr H2 by STRUCT_0:def_5;
a1 in H1 by A21, STRUCT_0:def_5;
then a1 * a in H1 by A8, A29, GROUP_2:50;
then A32: a1 * a in carr H1 by STRUCT_0:def_5;
g1 * g2 = ((a1 * b1) * a2) * b2 by A20, A23, GROUP_1:def_3
.= (a1 * (b1 * a2)) * b2 by GROUP_1:def_3 ;
then g1 * g2 = ((a1 * a) * b) * b2 by A26, GROUP_1:def_3
.= (a1 * a) * (b * b2) by GROUP_1:def_3 ;
hence g1 * g2 in (carr H1) * (carr H2) by A32, A31; ::_thesis: verum
end;
(carr H1) * (carr H2) <> {} by GROUP_2:9;
hence ex H being strict StableSubgroup of G st the carrier of H = (carr H1) * (carr H2) by A17, A9, A2, Lm15; ::_thesis: verum
end;
theorem Th18: :: GROUP_9:18
for O being set
for G being GroupWithOperators of O
for H1, H2 being StableSubgroup of G holds
( ( for H being StableSubgroup of G st H = H1 /\ H2 holds
the carrier of H = the carrier of H1 /\ the carrier of H2 ) & ( for H being strict StableSubgroup of G st the carrier of H = the carrier of H1 /\ the carrier of H2 holds
H = H1 /\ H2 ) )
proof
let O be set ; ::_thesis: for G being GroupWithOperators of O
for H1, H2 being StableSubgroup of G holds
( ( for H being StableSubgroup of G st H = H1 /\ H2 holds
the carrier of H = the carrier of H1 /\ the carrier of H2 ) & ( for H being strict StableSubgroup of G st the carrier of H = the carrier of H1 /\ the carrier of H2 holds
H = H1 /\ H2 ) )
let G be GroupWithOperators of O; ::_thesis: for H1, H2 being StableSubgroup of G holds
( ( for H being StableSubgroup of G st H = H1 /\ H2 holds
the carrier of H = the carrier of H1 /\ the carrier of H2 ) & ( for H being strict StableSubgroup of G st the carrier of H = the carrier of H1 /\ the carrier of H2 holds
H = H1 /\ H2 ) )
let H1, H2 be StableSubgroup of G; ::_thesis: ( ( for H being StableSubgroup of G st H = H1 /\ H2 holds
the carrier of H = the carrier of H1 /\ the carrier of H2 ) & ( for H being strict StableSubgroup of G st the carrier of H = the carrier of H1 /\ the carrier of H2 holds
H = H1 /\ H2 ) )
A1: ( the carrier of H1 = carr H1 & the carrier of H2 = carr H2 ) ;
thus for H being StableSubgroup of G st H = H1 /\ H2 holds
the carrier of H = the carrier of H1 /\ the carrier of H2 ::_thesis: for H being strict StableSubgroup of G st the carrier of H = the carrier of H1 /\ the carrier of H2 holds
H = H1 /\ H2
proof
let H be StableSubgroup of G; ::_thesis: ( H = H1 /\ H2 implies the carrier of H = the carrier of H1 /\ the carrier of H2 )
assume H = H1 /\ H2 ; ::_thesis: the carrier of H = the carrier of H1 /\ the carrier of H2
hence the carrier of H = (carr H1) /\ (carr H2) by Def25
.= the carrier of H1 /\ the carrier of H2 ;
::_thesis: verum
end;
let H be strict StableSubgroup of G; ::_thesis: ( the carrier of H = the carrier of H1 /\ the carrier of H2 implies H = H1 /\ H2 )
assume the carrier of H = the carrier of H1 /\ the carrier of H2 ; ::_thesis: H = H1 /\ H2
hence H = H1 /\ H2 by A1, Def25; ::_thesis: verum
end;
theorem Th19: :: GROUP_9:19
for O being set
for G being GroupWithOperators of O
for H being strict StableSubgroup of G holds H /\ H = H
proof
let O be set ; ::_thesis: for G being GroupWithOperators of O
for H being strict StableSubgroup of G holds H /\ H = H
let G be GroupWithOperators of O; ::_thesis: for H being strict StableSubgroup of G holds H /\ H = H
let H be strict StableSubgroup of G; ::_thesis: H /\ H = H
the carrier of (H /\ H) = (carr H) /\ (carr H) by Def25
.= the carrier of H ;
hence H /\ H = H by Lm5; ::_thesis: verum
end;
theorem Th20: :: GROUP_9:20
for O being set
for G being GroupWithOperators of O
for H1, H2, H3 being StableSubgroup of G holds (H1 /\ H2) /\ H3 = H1 /\ (H2 /\ H3)
proof
let O be set ; ::_thesis: for G being GroupWithOperators of O
for H1, H2, H3 being StableSubgroup of G holds (H1 /\ H2) /\ H3 = H1 /\ (H2 /\ H3)
let G be GroupWithOperators of O; ::_thesis: for H1, H2, H3 being StableSubgroup of G holds (H1 /\ H2) /\ H3 = H1 /\ (H2 /\ H3)
let H1, H2, H3 be StableSubgroup of G; ::_thesis: (H1 /\ H2) /\ H3 = H1 /\ (H2 /\ H3)
the carrier of ((H1 /\ H2) /\ H3) = (carr (H1 /\ H2)) /\ (carr H3) by Def25
.= ((carr H1) /\ (carr H2)) /\ (carr H3) by Def25
.= (carr H1) /\ ((carr H2) /\ (carr H3)) by XBOOLE_1:16
.= (carr H1) /\ (carr (H2 /\ H3)) by Def25
.= the carrier of (H1 /\ (H2 /\ H3)) by Def25 ;
hence (H1 /\ H2) /\ H3 = H1 /\ (H2 /\ H3) by Lm5; ::_thesis: verum
end;
Lm22: for O being set
for G being GroupWithOperators of O
for H2 being StableSubgroup of G
for H1 being strict StableSubgroup of G holds
( H1 is StableSubgroup of H2 iff H1 /\ H2 = H1 )
proof
let O be set ; ::_thesis: for G being GroupWithOperators of O
for H2 being StableSubgroup of G
for H1 being strict StableSubgroup of G holds
( H1 is StableSubgroup of H2 iff H1 /\ H2 = H1 )
let G be GroupWithOperators of O; ::_thesis: for H2 being StableSubgroup of G
for H1 being strict StableSubgroup of G holds
( H1 is StableSubgroup of H2 iff H1 /\ H2 = H1 )
let H2 be StableSubgroup of G; ::_thesis: for H1 being strict StableSubgroup of G holds
( H1 is StableSubgroup of H2 iff H1 /\ H2 = H1 )
let H1 be strict StableSubgroup of G; ::_thesis: ( H1 is StableSubgroup of H2 iff H1 /\ H2 = H1 )
thus ( H1 is StableSubgroup of H2 implies H1 /\ H2 = H1 ) ::_thesis: ( H1 /\ H2 = H1 implies H1 is StableSubgroup of H2 )
proof
assume H1 is StableSubgroup of H2 ; ::_thesis: H1 /\ H2 = H1
then H1 is Subgroup of H2 by Def7;
then A1: the carrier of H1 c= the carrier of H2 by GROUP_2:def_5;
the carrier of (H1 /\ H2) = (carr H1) /\ (carr H2) by Def25;
hence H1 /\ H2 = H1 by A1, Lm5, XBOOLE_1:28; ::_thesis: verum
end;
assume H1 /\ H2 = H1 ; ::_thesis: H1 is StableSubgroup of H2
then the carrier of H1 = (carr H1) /\ (carr H2) by Def25
.= the carrier of H1 /\ the carrier of H2 ;
hence H1 is StableSubgroup of H2 by Lm21, XBOOLE_1:17; ::_thesis: verum
end;
theorem Th21: :: GROUP_9:21
for O being set
for G being GroupWithOperators of O
for H1 being StableSubgroup of G holds
( ((1). G) /\ H1 = (1). G & H1 /\ ((1). G) = (1). G )
proof
let O be set ; ::_thesis: for G being GroupWithOperators of O
for H1 being StableSubgroup of G holds
( ((1). G) /\ H1 = (1). G & H1 /\ ((1). G) = (1). G )
let G be GroupWithOperators of O; ::_thesis: for H1 being StableSubgroup of G holds
( ((1). G) /\ H1 = (1). G & H1 /\ ((1). G) = (1). G )
let H1 be StableSubgroup of G; ::_thesis: ( ((1). G) /\ H1 = (1). G & H1 /\ ((1). G) = (1). G )
A1: (1). G is StableSubgroup of H1 by Th16;
hence ((1). G) /\ H1 = (1). G by Lm22; ::_thesis: H1 /\ ((1). G) = (1). G
thus H1 /\ ((1). G) = (1). G by A1, Lm22; ::_thesis: verum
end;
theorem Th22: :: GROUP_9:22
for O being set
for G being GroupWithOperators of O
for N being normal StableSubgroup of G holds union (Cosets N) = the carrier of G
proof
let O be set ; ::_thesis: for G being GroupWithOperators of O
for N being normal StableSubgroup of G holds union (Cosets N) = the carrier of G
let G be GroupWithOperators of O; ::_thesis: for N being normal StableSubgroup of G holds union (Cosets N) = the carrier of G
let N be normal StableSubgroup of G; ::_thesis: union (Cosets N) = the carrier of G
reconsider H = multMagma(# the carrier of N, the multF of N #) as strict normal Subgroup of G by Lm7;
now__::_thesis:_for_x_being_set_st_x_in_the_carrier_of_G_holds_
x_in_union_(Cosets_H)
set h = the Element of H;
let x be set ; ::_thesis: ( x in the carrier of G implies x in union (Cosets H) )
reconsider g = the Element of H as Element of G by GROUP_2:42;
assume x in the carrier of G ; ::_thesis: x in union (Cosets H)
then reconsider a = x as Element of G ;
A1: a = a * (1_ G) by GROUP_1:def_4
.= a * ((g ") * g) by GROUP_1:def_5
.= (a * (g ")) * g by GROUP_1:def_3 ;
A2: (a * (g ")) * H in Cosets H by GROUP_2:def_15;
the Element of H in H by STRUCT_0:def_5;
then a in (a * (g ")) * H by A1, GROUP_2:103;
hence x in union (Cosets H) by A2, TARSKI:def_4; ::_thesis: verum
end;
then A3: the carrier of G c= union (Cosets H) by TARSKI:def_3;
Cosets N = Cosets H by Def14;
hence union (Cosets N) = the carrier of G by A3, XBOOLE_0:def_10; ::_thesis: verum
end;
theorem Th23: :: GROUP_9:23
for O being set
for G being GroupWithOperators of O
for N1, N2 being strict normal StableSubgroup of G ex N being strict normal StableSubgroup of G st the carrier of N = (carr N1) * (carr N2)
proof
let O be set ; ::_thesis: for G being GroupWithOperators of O
for N1, N2 being strict normal StableSubgroup of G ex N being strict normal StableSubgroup of G st the carrier of N = (carr N1) * (carr N2)
let G be GroupWithOperators of O; ::_thesis: for N1, N2 being strict normal StableSubgroup of G ex N being strict normal StableSubgroup of G st the carrier of N = (carr N1) * (carr N2)
let N1, N2 be strict normal StableSubgroup of G; ::_thesis: ex N being strict normal StableSubgroup of G st the carrier of N = (carr N1) * (carr N2)
set N19 = multMagma(# the carrier of N1, the multF of N1 #);
set N29 = multMagma(# the carrier of N2, the multF of N2 #);
reconsider N19 = multMagma(# the carrier of N1, the multF of N1 #), N29 = multMagma(# the carrier of N2, the multF of N2 #) as strict normal Subgroup of G by Lm7;
set A = (carr N19) * (carr N29);
set B = carr N19;
set C = carr N29;
(carr N19) * (carr N29) = (carr N29) * (carr N19) by GROUP_3:125;
then consider H9 being strict Subgroup of G such that
A1: the carrier of H9 = (carr N19) * (carr N29) by GROUP_2:78;
A2: now__::_thesis:_for_o_being_Element_of_O
for_g_being_Element_of_G_st_g_in_(carr_N19)_*_(carr_N29)_holds_
(G_^_o)_._g_in_(carr_N19)_*_(carr_N29)
let o be Element of O; ::_thesis: for g being Element of G st g in (carr N19) * (carr N29) holds
(G ^ o) . g in (carr N19) * (carr N29)
let g be Element of G; ::_thesis: ( g in (carr N19) * (carr N29) implies (G ^ o) . g in (carr N19) * (carr N29) )
assume g in (carr N19) * (carr N29) ; ::_thesis: (G ^ o) . g in (carr N19) * (carr N29)
then consider a, b being Element of G such that
A3: g = a * b and
A4: a in carr N1 and
A5: b in carr N2 ;
a in N1 by A4, STRUCT_0:def_5;
then (G ^ o) . a in N1 by Lm10;
then A6: (G ^ o) . a in carr N1 by STRUCT_0:def_5;
b in N2 by A5, STRUCT_0:def_5;
then (G ^ o) . b in N2 by Lm10;
then (G ^ o) . b in carr N2 by STRUCT_0:def_5;
then ((G ^ o) . a) * ((G ^ o) . b) in (carr N1) * (carr N2) by A6;
hence (G ^ o) . g in (carr N19) * (carr N29) by A3, GROUP_6:def_6; ::_thesis: verum
end;
A7: now__::_thesis:_for_g_being_Element_of_G_st_g_in_(carr_N19)_*_(carr_N29)_holds_
g_"_in_(carr_N19)_*_(carr_N29)
let g be Element of G; ::_thesis: ( g in (carr N19) * (carr N29) implies g " in (carr N19) * (carr N29) )
assume g in (carr N19) * (carr N29) ; ::_thesis: g " in (carr N19) * (carr N29)
then g in H9 by A1, STRUCT_0:def_5;
then g " in H9 by GROUP_2:51;
hence g " in (carr N19) * (carr N29) by A1, STRUCT_0:def_5; ::_thesis: verum
end;
now__::_thesis:_for_g1,_g2_being_Element_of_G_st_g1_in_(carr_N19)_*_(carr_N29)_&_g2_in_(carr_N19)_*_(carr_N29)_holds_
g1_*_g2_in_(carr_N19)_*_(carr_N29)
let g1, g2 be Element of G; ::_thesis: ( g1 in (carr N19) * (carr N29) & g2 in (carr N19) * (carr N29) implies g1 * g2 in (carr N19) * (carr N29) )
assume ( g1 in (carr N19) * (carr N29) & g2 in (carr N19) * (carr N29) ) ; ::_thesis: g1 * g2 in (carr N19) * (carr N29)
then ( g1 in H9 & g2 in H9 ) by A1, STRUCT_0:def_5;
then g1 * g2 in H9 by GROUP_2:50;
hence g1 * g2 in (carr N19) * (carr N29) by A1, STRUCT_0:def_5; ::_thesis: verum
end;
then consider H being strict StableSubgroup of G such that
A8: the carrier of H = (carr N19) * (carr N29) by A1, A7, A2, Lm15;
now__::_thesis:_for_a_being_Element_of_G_holds_a_*_H9_=_H9_*_a
let a be Element of G; ::_thesis: a * H9 = H9 * a
thus a * H9 = (a * N19) * (carr N29) by A1, GROUP_2:29
.= (N19 * a) * (carr N29) by GROUP_3:117
.= (carr N19) * (a * N29) by GROUP_2:30
.= (carr N19) * (N29 * a) by GROUP_3:117
.= H9 * a by A1, GROUP_2:31 ; ::_thesis: verum
end;
then H9 is normal Subgroup of G by GROUP_3:117;
then for H99 being strict Subgroup of G st H99 = multMagma(# the carrier of H, the multF of H #) holds
H99 is normal by A1, A8, GROUP_2:59;
then H is normal by Def10;
hence ex N being strict normal StableSubgroup of G st the carrier of N = (carr N1) * (carr N2) by A8; ::_thesis: verum
end;
Lm23: for F1 being FinSequence
for y being Element of NAT st y in dom F1 holds
( ((len F1) - y) + 1 is Element of NAT & ((len F1) - y) + 1 >= 1 & ((len F1) - y) + 1 <= len F1 )
proof
let F1 be FinSequence; ::_thesis: for y being Element of NAT st y in dom F1 holds
( ((len F1) - y) + 1 is Element of NAT & ((len F1) - y) + 1 >= 1 & ((len F1) - y) + 1 <= len F1 )
let y be Element of NAT ; ::_thesis: ( y in dom F1 implies ( ((len F1) - y) + 1 is Element of NAT & ((len F1) - y) + 1 >= 1 & ((len F1) - y) + 1 <= len F1 ) )
assume A1: y in dom F1 ; ::_thesis: ( ((len F1) - y) + 1 is Element of NAT & ((len F1) - y) + 1 >= 1 & ((len F1) - y) + 1 <= len F1 )
now__::_thesis:_not_((len_F1)_-_y)_+_1_<_0
assume ((len F1) - y) + 1 < 0 ; ::_thesis: contradiction
then 1 < 0 - ((len F1) - y) by XREAL_1:20;
then 1 < y - (len F1) ;
then A2: (len F1) + 1 < y by XREAL_1:20;
y <= len F1 by A1, FINSEQ_3:25;
hence contradiction by A2, NAT_1:12; ::_thesis: verum
end;
then reconsider n = ((len F1) - y) + 1 as Element of NAT by INT_1:3;
y >= 1 by A1, FINSEQ_3:25;
then (n - 1) - y <= ((len F1) - y) - 1 by XREAL_1:13;
then A3: n - (y + 1) <= (len F1) - (y + 1) ;
y + 0 <= len F1 by A1, FINSEQ_3:25;
then ( 0 + 1 = 1 & 0 <= (len F1) - y ) by XREAL_1:19;
hence ( ((len F1) - y) + 1 is Element of NAT & ((len F1) - y) + 1 >= 1 & ((len F1) - y) + 1 <= len F1 ) by A3, XREAL_1:6, XREAL_1:9; ::_thesis: verum
end;
Lm24: for G, H being Group
for F1 being FinSequence of the carrier of G
for F2 being FinSequence of the carrier of H
for I being FinSequence of INT
for f being Homomorphism of G,H st ( for k being Nat st k in dom F1 holds
F2 . k = f . (F1 . k) ) & len F1 = len I & len F2 = len I holds
f . (Product (F1 |^ I)) = Product (F2 |^ I)
proof
defpred S1[ Nat] means for G, H being Group
for F1 being FinSequence of the carrier of G
for F2 being FinSequence of the carrier of H
for I being FinSequence of INT
for f being Homomorphism of G,H st ( for k being Nat st k in dom F1 holds
F2 . k = f . (F1 . k) ) & len F1 = len I & len F2 = len I & $1 = len I holds
f . (Product (F1 |^ I)) = Product (F2 |^ I);
let G, H be Group; ::_thesis: for F1 being FinSequence of the carrier of G
for F2 being FinSequence of the carrier of H
for I being FinSequence of INT
for f being Homomorphism of G,H st ( for k being Nat st k in dom F1 holds
F2 . k = f . (F1 . k) ) & len F1 = len I & len F2 = len I holds
f . (Product (F1 |^ I)) = Product (F2 |^ I)
let F1 be FinSequence of the carrier of G; ::_thesis: for F2 being FinSequence of the carrier of H
for I being FinSequence of INT
for f being Homomorphism of G,H st ( for k being Nat st k in dom F1 holds
F2 . k = f . (F1 . k) ) & len F1 = len I & len F2 = len I holds
f . (Product (F1 |^ I)) = Product (F2 |^ I)
let F2 be FinSequence of the carrier of H; ::_thesis: for I being FinSequence of INT
for f being Homomorphism of G,H st ( for k being Nat st k in dom F1 holds
F2 . k = f . (F1 . k) ) & len F1 = len I & len F2 = len I holds
f . (Product (F1 |^ I)) = Product (F2 |^ I)
let I be FinSequence of INT ; ::_thesis: for f being Homomorphism of G,H st ( for k being Nat st k in dom F1 holds
F2 . k = f . (F1 . k) ) & len F1 = len I & len F2 = len I holds
f . (Product (F1 |^ I)) = Product (F2 |^ I)
let f be Homomorphism of G,H; ::_thesis: ( ( for k being Nat st k in dom F1 holds
F2 . k = f . (F1 . k) ) & len F1 = len I & len F2 = len I implies f . (Product (F1 |^ I)) = Product (F2 |^ I) )
assume A1: ( ( for k being Nat st k in dom F1 holds
F2 . k = f . (F1 . k) ) & len F1 = len I & len F2 = len I ) ; ::_thesis: f . (Product (F1 |^ I)) = Product (F2 |^ I)
A2: now__::_thesis:_for_n_being_Nat_st_S1[n]_holds_
S1[n_+_1]
let n be Nat; ::_thesis: ( S1[n] implies S1[n + 1] )
assume A3: S1[n] ; ::_thesis: S1[n + 1]
thus S1[n + 1] ::_thesis: verum
proof
let G, H be Group; ::_thesis: for F1 being FinSequence of the carrier of G
for F2 being FinSequence of the carrier of H
for I being FinSequence of INT
for f being Homomorphism of G,H st ( for k being Nat st k in dom F1 holds
F2 . k = f . (F1 . k) ) & len F1 = len I & len F2 = len I & n + 1 = len I holds
f . (Product (F1 |^ I)) = Product (F2 |^ I)
let F1 be FinSequence of the carrier of G; ::_thesis: for F2 being FinSequence of the carrier of H
for I being FinSequence of INT
for f being Homomorphism of G,H st ( for k being Nat st k in dom F1 holds
F2 . k = f . (F1 . k) ) & len F1 = len I & len F2 = len I & n + 1 = len I holds
f . (Product (F1 |^ I)) = Product (F2 |^ I)
let F2 be FinSequence of the carrier of H; ::_thesis: for I being FinSequence of INT
for f being Homomorphism of G,H st ( for k being Nat st k in dom F1 holds
F2 . k = f . (F1 . k) ) & len F1 = len I & len F2 = len I & n + 1 = len I holds
f . (Product (F1 |^ I)) = Product (F2 |^ I)
let I be FinSequence of INT ; ::_thesis: for f being Homomorphism of G,H st ( for k being Nat st k in dom F1 holds
F2 . k = f . (F1 . k) ) & len F1 = len I & len F2 = len I & n + 1 = len I holds
f . (Product (F1 |^ I)) = Product (F2 |^ I)
let f be Homomorphism of G,H; ::_thesis: ( ( for k being Nat st k in dom F1 holds
F2 . k = f . (F1 . k) ) & len F1 = len I & len F2 = len I & n + 1 = len I implies f . (Product (F1 |^ I)) = Product (F2 |^ I) )
assume A4: for k being Nat st k in dom F1 holds
F2 . k = f . (F1 . k) ; ::_thesis: ( not len F1 = len I or not len F2 = len I or not n + 1 = len I or f . (Product (F1 |^ I)) = Product (F2 |^ I) )
assume that
A5: len F1 = len I and
A6: len F2 = len I and
A7: n + 1 = len I ; ::_thesis: f . (Product (F1 |^ I)) = Product (F2 |^ I)
consider F1n being FinSequence of the carrier of G, g being Element of G such that
A8: F1 = F1n ^ <*g*> by A5, A7, FINSEQ_2:19;
A9: len F1 = (len F1n) + (len <*g*>) by A8, FINSEQ_1:22;
then A10: n + 1 = (len F1n) + 1 by A5, A7, FINSEQ_1:40;
consider F2n being FinSequence of the carrier of H, h being Element of H such that
A11: F2 = F2n ^ <*h*> by A6, A7, FINSEQ_2:19;
A12: ( dom F1 = dom F2 & dom F2 = dom I ) by A5, A6, FINSEQ_3:29;
1 <= n + 1 by NAT_1:11;
then A13: n + 1 in dom I by A7, FINSEQ_3:25;
set F21 = <*h*>;
set F11 = <*g*>;
consider In being FinSequence of INT , i being Element of INT such that
A14: I = In ^ <*i*> by A7, FINSEQ_2:19;
set I1 = <*i*>;
len I = (len In) + (len <*i*>) by A14, FINSEQ_1:22;
then A15: n + 1 = (len In) + 1 by A7, FINSEQ_1:40;
A16: len F2 = (len F2n) + (len <*h*>) by A11, FINSEQ_1:22;
then A17: n + 1 = (len F2n) + 1 by A6, A7, FINSEQ_1:40;
A18: now__::_thesis:_for_k_being_Nat_st_k_in_dom_F1n_holds_
F2n_._k_=_f_._(F1n_._k)
let k be Nat; ::_thesis: ( k in dom F1n implies F2n . k = f . (F1n . k) )
0 + n <= 1 + n by XREAL_1:6;
then A19: dom F1n c= dom F1 by A5, A7, A10, FINSEQ_3:30;
assume A20: k in dom F1n ; ::_thesis: F2n . k = f . (F1n . k)
then k in dom F2n by A10, A17, FINSEQ_3:29;
hence F2n . k = F2 . k by A11, FINSEQ_1:def_7
.= f . (F1 . k) by A4, A20, A19
.= f . (F1n . k) by A8, A20, FINSEQ_1:def_7 ;
::_thesis: verum
end;
A21: F2 . (n + 1) = (F2n ^ <*h*>) . ((len F2n) + 1) by A6, A7, A11, A16, FINSEQ_1:40
.= h by FINSEQ_1:42 ;
A22: F1 . (n + 1) = (F1n ^ <*g*>) . ((len F1n) + 1) by A5, A7, A8, A9, FINSEQ_1:40
.= g by FINSEQ_1:42 ;
len <*h*> = 1 by FINSEQ_1:40
.= len <*i*> by FINSEQ_1:40 ;
then A23: Product (F2 |^ I) = Product ((F2n |^ In) ^ (<*h*> |^ <*i*>)) by A14, A11, A15, A17, GROUP_4:19
.= (Product (F2n |^ In)) * (Product (<*h*> |^ <*i*>)) by GROUP_4:5 ;
A24: f . (Product (<*g*> |^ <*i*>)) = f . (Product (<*g*> |^ <*(@ i)*>))
.= f . (Product <*(g |^ i)*>) by GROUP_4:22
.= f . (g |^ i) by GROUP_4:9
.= (f . g) |^ i by GROUP_6:37
.= h |^ i by A4, A13, A12, A22, A21
.= Product <*(h |^ i)*> by GROUP_4:9
.= Product (<*h*> |^ <*(@ i)*>) by GROUP_4:22
.= Product (<*h*> |^ <*i*>) ;
len <*g*> = 1 by FINSEQ_1:40
.= len <*i*> by FINSEQ_1:40 ;
then Product (F1 |^ I) = Product ((F1n |^ In) ^ (<*g*> |^ <*i*>)) by A14, A8, A15, A10, GROUP_4:19
.= (Product (F1n |^ In)) * (Product (<*g*> |^ <*i*>)) by GROUP_4:5 ;
then f . (Product (F1 |^ I)) = (f . (Product (F1n |^ In))) * (f . (Product (<*g*> |^ <*i*>))) by GROUP_6:def_6
.= (Product (F2n |^ In)) * (Product (<*h*> |^ <*i*>)) by A3, A15, A10, A17, A18, A24 ;
hence f . (Product (F1 |^ I)) = Product (F2 |^ I) by A23; ::_thesis: verum
end;
end;
A25: S1[ 0 ]
proof
let G, H be Group; ::_thesis: for F1 being FinSequence of the carrier of G
for F2 being FinSequence of the carrier of H
for I being FinSequence of INT
for f being Homomorphism of G,H st ( for k being Nat st k in dom F1 holds
F2 . k = f . (F1 . k) ) & len F1 = len I & len F2 = len I & 0 = len I holds
f . (Product (F1 |^ I)) = Product (F2 |^ I)
let F1 be FinSequence of the carrier of G; ::_thesis: for F2 being FinSequence of the carrier of H
for I being FinSequence of INT
for f being Homomorphism of G,H st ( for k being Nat st k in dom F1 holds
F2 . k = f . (F1 . k) ) & len F1 = len I & len F2 = len I & 0 = len I holds
f . (Product (F1 |^ I)) = Product (F2 |^ I)
let F2 be FinSequence of the carrier of H; ::_thesis: for I being FinSequence of INT
for f being Homomorphism of G,H st ( for k being Nat st k in dom F1 holds
F2 . k = f . (F1 . k) ) & len F1 = len I & len F2 = len I & 0 = len I holds
f . (Product (F1 |^ I)) = Product (F2 |^ I)
let I be FinSequence of INT ; ::_thesis: for f being Homomorphism of G,H st ( for k being Nat st k in dom F1 holds
F2 . k = f . (F1 . k) ) & len F1 = len I & len F2 = len I & 0 = len I holds
f . (Product (F1 |^ I)) = Product (F2 |^ I)
let f be Homomorphism of G,H; ::_thesis: ( ( for k being Nat st k in dom F1 holds
F2 . k = f . (F1 . k) ) & len F1 = len I & len F2 = len I & 0 = len I implies f . (Product (F1 |^ I)) = Product (F2 |^ I) )
assume for k being Nat st k in dom F1 holds
F2 . k = f . (F1 . k) ; ::_thesis: ( not len F1 = len I or not len F2 = len I or not 0 = len I or f . (Product (F1 |^ I)) = Product (F2 |^ I) )
assume that
A26: len F1 = len I and
A27: len F2 = len I and
A28: 0 = len I ; ::_thesis: f . (Product (F1 |^ I)) = Product (F2 |^ I)
len (F2 |^ I) = 0 by A27, A28, GROUP_4:def_3;
then F2 |^ I = <*> the carrier of H ;
then A29: Product (F2 |^ I) = 1_ H by GROUP_4:8;
len (F1 |^ I) = 0 by A26, A28, GROUP_4:def_3;
then F1 |^ I = <*> the carrier of G ;
then Product (F1 |^ I) = 1_ G by GROUP_4:8;
hence f . (Product (F1 |^ I)) = Product (F2 |^ I) by A29, GROUP_6:31; ::_thesis: verum
end;
for n being Nat holds S1[n] from NAT_1:sch_2(A25, A2);
hence f . (Product (F1 |^ I)) = Product (F2 |^ I) by A1; ::_thesis: verum
end;
theorem Th24: :: GROUP_9:24
for O being set
for G being GroupWithOperators of O
for A being Subset of G
for g1 being Element of G holds
( g1 in the_stable_subgroup_of A iff ex F being FinSequence of the carrier of G ex I being FinSequence of INT ex C being Subset of G st
( C = the_stable_subset_generated_by (A, the action of G) & len F = len I & rng F c= C & Product (F |^ I) = g1 ) )
proof
let O be set ; ::_thesis: for G being GroupWithOperators of O
for A being Subset of G
for g1 being Element of G holds
( g1 in the_stable_subgroup_of A iff ex F being FinSequence of the carrier of G ex I being FinSequence of INT ex C being Subset of G st
( C = the_stable_subset_generated_by (A, the action of G) & len F = len I & rng F c= C & Product (F |^ I) = g1 ) )
let G be GroupWithOperators of O; ::_thesis: for A being Subset of G
for g1 being Element of G holds
( g1 in the_stable_subgroup_of A iff ex F being FinSequence of the carrier of G ex I being FinSequence of INT ex C being Subset of G st
( C = the_stable_subset_generated_by (A, the action of G) & len F = len I & rng F c= C & Product (F |^ I) = g1 ) )
let A be Subset of G; ::_thesis: for g1 being Element of G holds
( g1 in the_stable_subgroup_of A iff ex F being FinSequence of the carrier of G ex I being FinSequence of INT ex C being Subset of G st
( C = the_stable_subset_generated_by (A, the action of G) & len F = len I & rng F c= C & Product (F |^ I) = g1 ) )
let g1 be Element of G; ::_thesis: ( g1 in the_stable_subgroup_of A iff ex F being FinSequence of the carrier of G ex I being FinSequence of INT ex C being Subset of G st
( C = the_stable_subset_generated_by (A, the action of G) & len F = len I & rng F c= C & Product (F |^ I) = g1 ) )
set H9 = the_stable_subgroup_of A;
set Y = the carrier of (the_stable_subgroup_of A);
A1: A c= the carrier of (the_stable_subgroup_of A) by Def26;
thus ( g1 in the_stable_subgroup_of A implies ex F being FinSequence of the carrier of G ex I being FinSequence of INT ex C being Subset of G st
( C = the_stable_subset_generated_by (A, the action of G) & len F = len I & rng F c= C & Product (F |^ I) = g1 ) ) ::_thesis: ( ex F being FinSequence of the carrier of G ex I being FinSequence of INT ex C being Subset of G st
( C = the_stable_subset_generated_by (A, the action of G) & len F = len I & rng F c= C & Product (F |^ I) = g1 ) implies g1 in the_stable_subgroup_of A )
proof
defpred S1[ set ] means ex F being FinSequence of the carrier of G ex I being FinSequence of INT ex C being Subset of G st
( C = the_stable_subset_generated_by (A, the action of G) & $1 = Product (F |^ I) & len F = len I & rng F c= C );
assume A2: g1 in the_stable_subgroup_of A ; ::_thesis: ex F being FinSequence of the carrier of G ex I being FinSequence of INT ex C being Subset of G st
( C = the_stable_subset_generated_by (A, the action of G) & len F = len I & rng F c= C & Product (F |^ I) = g1 )
reconsider B = { b where b is Element of G : S1[b] } as Subset of G from DOMAIN_1:sch_7();
A3: now__::_thesis:_for_c,_d_being_Element_of_G_st_c_in_B_&_d_in_B_holds_
c_*_d_in_B
let c, d be Element of G; ::_thesis: ( c in B & d in B implies c * d in B )
assume that
A4: c in B and
A5: d in B ; ::_thesis: c * d in B
ex d1 being Element of G st
( c = d1 & ex F being FinSequence of the carrier of G ex I being FinSequence of INT ex C being Subset of G st
( C = the_stable_subset_generated_by (A, the action of G) & d1 = Product (F |^ I) & len F = len I & rng F c= C ) ) by A4;
then consider F1 being FinSequence of the carrier of G, I1 being FinSequence of INT , C being Subset of G such that
A6: C = the_stable_subset_generated_by (A, the action of G) and
A7: c = Product (F1 |^ I1) and
A8: len F1 = len I1 and
A9: rng F1 c= C ;
ex d2 being Element of G st
( d = d2 & ex F being FinSequence of the carrier of G ex I being FinSequence of INT ex C being Subset of G st
( C = the_stable_subset_generated_by (A, the action of G) & d2 = Product (F |^ I) & len F = len I & rng F c= C ) ) by A5;
then consider F2 being FinSequence of the carrier of G, I2 being FinSequence of INT , C being Subset of G such that
A10: C = the_stable_subset_generated_by (A, the action of G) and
A11: d = Product (F2 |^ I2) and
A12: len F2 = len I2 and
A13: rng F2 c= C ;
A14: len (F1 ^ F2) = (len I1) + (len I2) by A8, A12, FINSEQ_1:22
.= len (I1 ^ I2) by FINSEQ_1:22 ;
rng (F1 ^ F2) = (rng F1) \/ (rng F2) by FINSEQ_1:31;
then A15: rng (F1 ^ F2) c= C by A6, A9, A10, A13, XBOOLE_1:8;
c * d = Product ((F1 |^ I1) ^ (F2 |^ I2)) by A7, A11, GROUP_4:5
.= Product ((F1 ^ F2) |^ (I1 ^ I2)) by A8, A12, GROUP_4:19 ;
hence c * d in B by A10, A15, A14; ::_thesis: verum
end;
A16: now__::_thesis:_for_o_being_Element_of_O
for_c_being_Element_of_G_st_c_in_B_holds_
(G_^_o)_._c_in_B
let o be Element of O; ::_thesis: for c being Element of G st c in B holds
(G ^ o) . c in B
let c be Element of G; ::_thesis: ( c in B implies (G ^ o) . c in B )
assume c in B ; ::_thesis: (G ^ o) . c in B
then ex d1 being Element of G st
( c = d1 & ex F being FinSequence of the carrier of G ex I being FinSequence of INT ex C being Subset of G st
( C = the_stable_subset_generated_by (A, the action of G) & d1 = Product (F |^ I) & len F = len I & rng F c= C ) ) ;
then consider F1 being FinSequence of the carrier of G, I1 being FinSequence of INT , C being Subset of G such that
A17: C = the_stable_subset_generated_by (A, the action of G) and
A18: c = Product (F1 |^ I1) and
A19: len F1 = len I1 and
A20: rng F1 c= C ;
deffunc H1( Nat) -> set = (G ^ o) . (F1 . $1);
consider F2 being FinSequence such that
A21: len F2 = len F1 and
A22: for k being Nat st k in dom F2 holds
F2 . k = H1(k) from FINSEQ_1:sch_2();
A23: dom F2 = dom F1 by A21, FINSEQ_3:29;
A24: Seg (len F1) = dom F1 by FINSEQ_1:def_3;
now__::_thesis:_for_y_being_set_st_y_in_rng_F2_holds_
y_in_C
A25: C is_stable_under_the_action_of the action of G by A17, Def2;
let y be set ; ::_thesis: ( y in rng F2 implies b1 in C )
assume y in rng F2 ; ::_thesis: b1 in C
then consider x being set such that
A26: x in dom F2 and
A27: y = F2 . x by FUNCT_1:def_3;
A28: x in Seg (len F1) by A21, A26, FINSEQ_1:def_3;
reconsider x = x as Element of NAT by A26;
A29: F2 . x = (G ^ o) . (F1 . x) by A22, A26;
A30: F1 . x in rng F1 by A24, A28, FUNCT_1:3;
percases ( O <> {} or O = {} ) ;
supposeA31: O <> {} ; ::_thesis: b1 in C
set f = the action of G . o;
A32: G ^ o = the action of G . o by A31, Def6;
then reconsider f = the action of G . o as Function of G,G ;
dom f = the carrier of G by FUNCT_2:def_1;
then A33: y in f .: C by A20, A27, A29, A30, A32, FUNCT_1:def_6;
f .: C c= C by A25, A31, Def1;
hence y in C by A33; ::_thesis: verum
end;
suppose O = {} ; ::_thesis: b1 in C
then G ^ o = id the carrier of G by Def6;
then (G ^ o) . (F1 . x) = F1 . x by A30, FUNCT_1:18;
hence y in C by A20, A27, A29, A30; ::_thesis: verum
end;
end;
end;
then A34: rng F2 c= C by TARSKI:def_3;
then rng F2 c= the carrier of G by XBOOLE_1:1;
then reconsider F2 = F2 as FinSequence of the carrier of G by FINSEQ_1:def_4;
(G ^ o) . c = Product (F2 |^ I1) by A18, A19, A21, A22, A23, Lm24;
hence (G ^ o) . c in B by A17, A19, A21, A34; ::_thesis: verum
end;
A35: now__::_thesis:_for_c_being_Element_of_G_st_c_in_B_holds_
c_"_in_B
let c be Element of G; ::_thesis: ( c in B implies c " in B )
assume c in B ; ::_thesis: c " in B
then ex d1 being Element of G st
( c = d1 & ex F being FinSequence of the carrier of G ex I being FinSequence of INT ex C being Subset of G st
( C = the_stable_subset_generated_by (A, the action of G) & d1 = Product (F |^ I) & len F = len I & rng F c= C ) ) ;
then consider F1 being FinSequence of the carrier of G, I1 being FinSequence of INT , C being Subset of G such that
A36: ( C = the_stable_subset_generated_by (A, the action of G) & c = Product (F1 |^ I1) ) and
A37: len F1 = len I1 and
A38: rng F1 c= C ;
deffunc H1( Nat) -> set = F1 . (((len F1) - $1) + 1);
consider F2 being FinSequence such that
A39: len F2 = len F1 and
A40: for k being Nat st k in dom F2 holds
F2 . k = H1(k) from FINSEQ_1:sch_2();
A41: Seg (len I1) = dom I1 by FINSEQ_1:def_3;
A42: rng F2 c= rng F1
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in rng F2 or x in rng F1 )
assume x in rng F2 ; ::_thesis: x in rng F1
then consider y being set such that
A43: y in dom F2 and
A44: F2 . y = x by FUNCT_1:def_3;
reconsider y = y as Element of NAT by A43;
reconsider n = ((len F1) - y) + 1 as Element of NAT by A39, A43, Lm23;
( 1 <= n & n <= len F1 ) by A39, A43, Lm23;
then A45: n in dom F1 by FINSEQ_3:25;
x = F1 . (((len F1) - y) + 1) by A40, A43, A44;
hence x in rng F1 by A45, FUNCT_1:def_3; ::_thesis: verum
end;
then A46: rng F2 c= C by A38, XBOOLE_1:1;
set p = F1 |^ I1;
A47: Seg (len F1) = dom F1 by FINSEQ_1:def_3;
A48: len (F1 |^ I1) = len F1 by GROUP_4:def_3;
defpred S2[ Nat, set ] means ex i being Integer st
( i = I1 . (((len I1) - $1) + 1) & $2 = - i );
A49: for k being Nat st k in Seg (len I1) holds
ex x being set st S2[k,x]
proof
let k be Nat; ::_thesis: ( k in Seg (len I1) implies ex x being set st S2[k,x] )
assume k in Seg (len I1) ; ::_thesis: ex x being set st S2[k,x]
then A50: k in dom I1 by FINSEQ_1:def_3;
then reconsider n = ((len I1) - k) + 1 as Element of NAT by Lm23;
( 1 <= n & n <= len I1 ) by A50, Lm23;
then n in dom I1 by FINSEQ_3:25;
then I1 . n in rng I1 by FUNCT_1:def_3;
then reconsider i = I1 . n as Element of INT ;
reconsider i = i as Integer ;
reconsider x = - i as set ;
take x ; ::_thesis: S2[k,x]
take i ; ::_thesis: ( i = I1 . (((len I1) - k) + 1) & x = - i )
thus ( i = I1 . (((len I1) - k) + 1) & x = - i ) ; ::_thesis: verum
end;
consider I2 being FinSequence such that
A51: dom I2 = Seg (len I1) and
A52: for k being Nat st k in Seg (len I1) holds
S2[k,I2 . k] from FINSEQ_1:sch_1(A49);
A53: len F2 = len I2 by A37, A39, A51, FINSEQ_1:def_3;
A54: rng I2 c= INT
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in rng I2 or x in INT )
assume x in rng I2 ; ::_thesis: x in INT
then consider y being set such that
A55: y in dom I2 and
A56: x = I2 . y by FUNCT_1:def_3;
reconsider y = y as Element of NAT by A55;
ex i being Integer st
( i = I1 . (((len I1) - y) + 1) & x = - i ) by A51, A52, A55, A56;
hence x in INT by INT_1:def_2; ::_thesis: verum
end;
A57: rng F2 c= the carrier of G by A42, XBOOLE_1:1;
A58: dom F2 = dom I2 by A37, A39, A51, FINSEQ_1:def_3;
reconsider I2 = I2 as FinSequence of INT by A54, FINSEQ_1:def_4;
reconsider F2 = F2 as FinSequence of the carrier of G by A57, FINSEQ_1:def_4;
set q = F2 |^ I2;
A59: len (F2 |^ I2) = len F2 by GROUP_4:def_3;
then A60: dom (F2 |^ I2) = dom F2 by FINSEQ_3:29;
A61: dom F1 = dom I1 by A37, FINSEQ_3:29;
now__::_thesis:_for_k_being_Element_of_NAT_st_k_in_dom_(F2_|^_I2)_holds_
((F2_|^_I2)_/._k)_"_=_(F1_|^_I1)_._(((len_(F1_|^_I1))_-_k)_+_1)
let k be Element of NAT ; ::_thesis: ( k in dom (F2 |^ I2) implies ((F2 |^ I2) /. k) " = (F1 |^ I1) . (((len (F1 |^ I1)) - k) + 1) )
A62: I2 /. k = @ (I2 /. k) ;
assume A63: k in dom (F2 |^ I2) ; ::_thesis: ((F2 |^ I2) /. k) " = (F1 |^ I1) . (((len (F1 |^ I1)) - k) + 1)
then reconsider n = ((len (F1 |^ I1)) - k) + 1 as Element of NAT by A39, A48, A59, Lm23;
A64: ( I1 /. n = @ (I1 /. n) & (F2 |^ I2) /. k = (F2 |^ I2) . k ) by A63, PARTFUN1:def_6;
A65: ( F2 /. k = F2 . k & F2 . k = F1 . n ) by A40, A48, A60, A63, PARTFUN1:def_6;
( 1 <= n & len (F1 |^ I1) >= n ) by A39, A48, A59, A63, Lm23;
then A66: n in dom I2 by A37, A51, A48;
then A67: I1 . n = I1 /. n by A51, A41, PARTFUN1:def_6;
dom (F2 |^ I2) = dom I1 by A37, A39, A59, FINSEQ_3:29;
then consider i being Integer such that
A68: i = I1 . n and
A69: I2 . k = - i by A37, A52, A41, A48, A63;
I2 . k = I2 /. k by A58, A60, A63, PARTFUN1:def_6;
then A70: (F2 |^ I2) . k = (F2 /. k) |^ (- i) by A60, A63, A69, A62, GROUP_4:def_3;
F1 /. n = F1 . n by A37, A47, A51, A66, PARTFUN1:def_6;
then (F2 |^ I2) . k = ((F1 /. n) |^ i) " by A70, A65, GROUP_1:36;
hence ((F2 |^ I2) /. k) " = (F1 |^ I1) . (((len (F1 |^ I1)) - k) + 1) by A61, A51, A41, A66, A68, A67, A64, GROUP_4:def_3; ::_thesis: verum
end;
then (Product (F1 |^ I1)) " = Product (F2 |^ I2) by A39, A48, A59, GROUP_4:14;
hence c " in B by A36, A53, A46; ::_thesis: verum
end;
A71: len {} = 0 ;
A72: ( rng (<*> the carrier of G) = {} & {} c= the_stable_subset_generated_by (A, the action of G) ) by XBOOLE_1:2;
( 1_ G = Product (<*> the carrier of G) & (<*> the carrier of G) |^ (<*> INT) = {} ) by GROUP_4:8, GROUP_4:21;
then 1_ G in B by A72, A71;
then consider H being strict StableSubgroup of G such that
A73: the carrier of H = B by A3, A35, A16, Lm15;
A c= B
proof
set C = the_stable_subset_generated_by (A, the action of G);
reconsider p = 1 as Integer ;
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in A or x in B )
assume A74: x in A ; ::_thesis: x in B
then reconsider a = x as Element of G ;
A c= the_stable_subset_generated_by (A, the action of G) by Def2;
then A75: ( rng <*a*> = {a} & {a} c= the_stable_subset_generated_by (A, the action of G) ) by A74, FINSEQ_1:39, ZFMISC_1:31;
A76: Product (<*a*> |^ <*(@ p)*>) = Product <*(a |^ 1)*> by GROUP_4:22
.= a |^ 1 by GROUP_4:9
.= a by GROUP_1:26 ;
( len <*a*> = 1 & len <*(@ p)*> = 1 ) by FINSEQ_1:39;
hence x in B by A76, A75; ::_thesis: verum
end;
then the_stable_subgroup_of A is StableSubgroup of H by A73, Def26;
then the_stable_subgroup_of A is Subgroup of H by Def7;
then g1 in H by A2, GROUP_2:40;
then g1 in B by A73, STRUCT_0:def_5;
then ex b being Element of G st
( b = g1 & ex F being FinSequence of the carrier of G ex I being FinSequence of INT ex C being Subset of G st
( C = the_stable_subset_generated_by (A, the action of G) & b = Product (F |^ I) & len F = len I & rng F c= C ) ) ;
hence ex F being FinSequence of the carrier of G ex I being FinSequence of INT ex C being Subset of G st
( C = the_stable_subset_generated_by (A, the action of G) & len F = len I & rng F c= C & Product (F |^ I) = g1 ) ; ::_thesis: verum
end;
given F being FinSequence of the carrier of G, I being FinSequence of INT , C being Subset of G such that A77: C = the_stable_subset_generated_by (A, the action of G) and
len F = len I and
A78: rng F c= C and
A79: Product (F |^ I) = g1 ; ::_thesis: g1 in the_stable_subgroup_of A
the_stable_subgroup_of A is Subgroup of G by Def7;
then reconsider Y = the carrier of (the_stable_subgroup_of A) as Subset of G by GROUP_2:def_5;
now__::_thesis:_for_o_being_Element_of_O
for_f_being_Function_of_G,G_st_o_in_O_&_f_=_the_action_of_G_._o_holds_
f_.:_Y_c=_Y
let o be Element of O; ::_thesis: for f being Function of G,G st o in O & f = the action of G . o holds
f .: Y c= Y
let f be Function of G,G; ::_thesis: ( o in O & f = the action of G . o implies f .: Y c= Y )
assume A80: o in O ; ::_thesis: ( f = the action of G . o implies f .: Y c= Y )
assume A81: f = the action of G . o ; ::_thesis: f .: Y c= Y
now__::_thesis:_for_y_being_set_st_y_in_f_.:_Y_holds_
y_in_Y
let y be set ; ::_thesis: ( y in f .: Y implies y in Y )
assume y in f .: Y ; ::_thesis: y in Y
then consider x being set such that
A82: x in dom f and
A83: x in Y and
A84: y = f . x by FUNCT_1:def_6;
reconsider x = x as Element of G by A82;
x in the_stable_subgroup_of A by A83, STRUCT_0:def_5;
then (G ^ o) . x in the_stable_subgroup_of A by Lm10;
then f . x in the_stable_subgroup_of A by A80, A81, Def6;
hence y in Y by A84, STRUCT_0:def_5; ::_thesis: verum
end;
hence f .: Y c= Y by TARSKI:def_3; ::_thesis: verum
end;
then A85: Y is_stable_under_the_action_of the action of G by Def1;
reconsider H9 = the_stable_subgroup_of A as Subgroup of G by Def7;
C c= the carrier of H9 by A77, A1, A85, Def2;
then rng F c= carr H9 by A78, XBOOLE_1:1;
hence g1 in the_stable_subgroup_of A by A79, GROUP_4:20; ::_thesis: verum
end;
Lm25: for O being set
for G being GroupWithOperators of O
for A being Subset of G st A is empty holds
the_stable_subgroup_of A = (1). G
proof
let O be set ; ::_thesis: for G being GroupWithOperators of O
for A being Subset of G st A is empty holds
the_stable_subgroup_of A = (1). G
let G be GroupWithOperators of O; ::_thesis: for A being Subset of G st A is empty holds
the_stable_subgroup_of A = (1). G
let A be Subset of G; ::_thesis: ( A is empty implies the_stable_subgroup_of A = (1). G )
A1: now__::_thesis:_for_H_being_strict_StableSubgroup_of_G_st_A_c=_the_carrier_of_H_holds_
(1)._G_is_StableSubgroup_of_H
let H be strict StableSubgroup of G; ::_thesis: ( A c= the carrier of H implies (1). G is StableSubgroup of H )
assume A c= the carrier of H ; ::_thesis: (1). G is StableSubgroup of H
(1). G = (1). H by Th15;
hence (1). G is StableSubgroup of H ; ::_thesis: verum
end;
assume A is empty ; ::_thesis: the_stable_subgroup_of A = (1). G
then A c= the carrier of ((1). G) by XBOOLE_1:2;
hence the_stable_subgroup_of A = (1). G by A1, Def26; ::_thesis: verum
end;
Lm26: for O being non empty set
for E being set
for o being Element of O
for A being Action of O,E holds Product (<*o*>,A) = A . o
proof
let O be non empty set ; ::_thesis: for E being set
for o being Element of O
for A being Action of O,E holds Product (<*o*>,A) = A . o
let E be set ; ::_thesis: for o being Element of O
for A being Action of O,E holds Product (<*o*>,A) = A . o
let o be Element of O; ::_thesis: for A being Action of O,E holds Product (<*o*>,A) = A . o
let A be Action of O,E; ::_thesis: Product (<*o*>,A) = A . o
( len <*o*> = 1 & ex PF being FinSequence of Funcs (E,E) st
( Product (<*o*>,A) = PF . (len <*o*>) & len PF = len <*o*> & PF . 1 = A . (<*o*> . 1) & ( for k being Nat st k <> 0 & k < len <*o*> holds
ex f, g being Function of E,E st
( f = PF . k & g = A . (<*o*> . (k + 1)) & PF . (k + 1) = f * g ) ) ) ) by Def3, FINSEQ_1:39;
hence Product (<*o*>,A) = A . o by FINSEQ_1:40; ::_thesis: verum
end;
Lm27: for O being non empty set
for E being set
for o being Element of O
for F being FinSequence of O
for A being Action of O,E holds Product ((F ^ <*o*>),A) = (Product (F,A)) * (Product (<*o*>,A))
proof
let O be non empty set ; ::_thesis: for E being set
for o being Element of O
for F being FinSequence of O
for A being Action of O,E holds Product ((F ^ <*o*>),A) = (Product (F,A)) * (Product (<*o*>,A))
let E be set ; ::_thesis: for o being Element of O
for F being FinSequence of O
for A being Action of O,E holds Product ((F ^ <*o*>),A) = (Product (F,A)) * (Product (<*o*>,A))
let o be Element of O; ::_thesis: for F being FinSequence of O
for A being Action of O,E holds Product ((F ^ <*o*>),A) = (Product (F,A)) * (Product (<*o*>,A))
let F be FinSequence of O; ::_thesis: for A being Action of O,E holds Product ((F ^ <*o*>),A) = (Product (F,A)) * (Product (<*o*>,A))
let A be Action of O,E; ::_thesis: Product ((F ^ <*o*>),A) = (Product (F,A)) * (Product (<*o*>,A))
set F1 = F ^ <*o*>;
A1: len (F ^ <*o*>) = (len F) + (len <*o*>) by FINSEQ_1:22
.= (len F) + 1 by FINSEQ_1:39 ;
consider PF1 being FinSequence of Funcs (E,E) such that
A2: Product ((F ^ <*o*>),A) = PF1 . (len (F ^ <*o*>)) and
A3: len PF1 = len (F ^ <*o*>) and
A4: PF1 . 1 = A . ((F ^ <*o*>) . 1) and
A5: for k being Nat st k <> 0 & k < len (F ^ <*o*>) holds
ex f, g being Function of E,E st
( f = PF1 . k & g = A . ((F ^ <*o*>) . (k + 1)) & PF1 . (k + 1) = f * g ) by Def3;
percases ( len F <> 0 or len F = 0 ) ;
supposeA6: len F <> 0 ; ::_thesis: Product ((F ^ <*o*>),A) = (Product (F,A)) * (Product (<*o*>,A))
reconsider PF = PF1 | (Seg (len F)) as FinSequence of Funcs (E,E) by FINSEQ_1:18;
set IT = PF . (len F);
A7: Product (<*o*>,A) = A . o by Lm26
.= A . ((F ^ <*o*>) . ((len F) + 1)) by FINSEQ_1:42 ;
A8: now__::_thesis:_for_k_being_Nat_st_k_<>_0_&_k_<_len_F_holds_
ex_f,_g_being_Function_of_E,E_st_
(_f_=_PF_._k_&_g_=_A_._(F_._(k_+_1))_&_PF_._(k_+_1)_=_f_*_g_)
let k be Nat; ::_thesis: ( k <> 0 & k < len F implies ex f, g being Function of E,E st
( f = PF . k & g = A . (F . (k + 1)) & PF . (k + 1) = f * g ) )
assume A9: k <> 0 ; ::_thesis: ( k < len F implies ex f, g being Function of E,E st
( f = PF . k & g = A . (F . (k + 1)) & PF . (k + 1) = f * g ) )
then A10: 0 + 1 < k + 1 by XREAL_1:6;
assume A11: k < len F ; ::_thesis: ex f, g being Function of E,E st
( f = PF . k & g = A . (F . (k + 1)) & PF . (k + 1) = f * g )
then k < len (F ^ <*o*>) by A1, NAT_1:13;
then consider f, g being Function of E,E such that
A12: f = PF1 . k and
A13: g = A . ((F ^ <*o*>) . (k + 1)) and
A14: PF1 . (k + 1) = f * g by A5, A9;
take f = f; ::_thesis: ex g being Function of E,E st
( f = PF . k & g = A . (F . (k + 1)) & PF . (k + 1) = f * g )
take g = g; ::_thesis: ( f = PF . k & g = A . (F . (k + 1)) & PF . (k + 1) = f * g )
1 <= k by A10, NAT_1:13;
then k in Seg (len F) by A11, FINSEQ_1:1;
hence f = PF . k by A12, FUNCT_1:49; ::_thesis: ( g = A . (F . (k + 1)) & PF . (k + 1) = f * g )
k + 1 <= len F by A11, NAT_1:13;
then A15: k + 1 in Seg (len F) by A10;
then k + 1 in dom F by FINSEQ_1:def_3;
hence g = A . (F . (k + 1)) by A13, FINSEQ_1:def_7; ::_thesis: PF . (k + 1) = f * g
thus PF . (k + 1) = f * g by A14, A15, FUNCT_1:49; ::_thesis: verum
end;
A16: len F < len (F ^ <*o*>) by A1, NAT_1:13;
then A17: ex f, g being Function of E,E st
( f = PF1 . (len F) & g = A . ((F ^ <*o*>) . ((len F) + 1)) & PF1 . ((len F) + 1) = f * g ) by A5, A6;
0 + 1 < (len F) + 1 by A6, XREAL_1:6;
then 1 <= len F by NAT_1:13;
then A18: 1 in Seg (len F) ;
then A19: 1 in dom F by FINSEQ_1:def_3;
A20: len F in Seg (len F) by A6, FINSEQ_1:3;
Seg (len F) c= Seg (len PF1) by A3, A16, FINSEQ_1:5;
then len F in Seg (len PF1) by A20;
then len F in dom PF1 by FINSEQ_1:def_3;
then len F in (dom PF1) /\ (Seg (len F)) by A20, XBOOLE_0:def_4;
then len F in dom PF by RELAT_1:61;
then PF . (len F) in rng PF by FUNCT_1:3;
then ex f being Function st
( PF . (len F) = f & dom f = E & rng f c= E ) by FUNCT_2:def_2;
then reconsider IT = PF . (len F) as Function of E,E by FUNCT_2:2;
A21: len PF = len F by A3, A16, FINSEQ_1:17;
PF . 1 = A . ((F ^ <*o*>) . 1) by A4, A18, FUNCT_1:49
.= A . (F . 1) by A19, FINSEQ_1:def_7 ;
then IT = Product (F,A) by A6, A21, A8, Def3;
hence Product ((F ^ <*o*>),A) = (Product (F,A)) * (Product (<*o*>,A)) by A1, A2, A6, A17, A7, FINSEQ_1:3, FUNCT_1:49; ::_thesis: verum
end;
supposeA22: len F = 0 ; ::_thesis: Product ((F ^ <*o*>),A) = (Product (F,A)) * (Product (<*o*>,A))
then F = <*> O ;
hence Product ((F ^ <*o*>),A) = Product (((<*> O) ^ <*o*>),A)
.= Product (<*o*>,A) by FINSEQ_1:34
.= (id E) * (Product (<*o*>,A)) by FUNCT_2:17
.= (Product (F,A)) * (Product (<*o*>,A)) by A22, Def3 ;
::_thesis: verum
end;
end;
end;
Lm28: for O being non empty set
for E being set
for o being Element of O
for F being FinSequence of O
for A being Action of O,E holds Product ((<*o*> ^ F),A) = (Product (<*o*>,A)) * (Product (F,A))
proof
let O be non empty set ; ::_thesis: for E being set
for o being Element of O
for F being FinSequence of O
for A being Action of O,E holds Product ((<*o*> ^ F),A) = (Product (<*o*>,A)) * (Product (F,A))
let E be set ; ::_thesis: for o being Element of O
for F being FinSequence of O
for A being Action of O,E holds Product ((<*o*> ^ F),A) = (Product (<*o*>,A)) * (Product (F,A))
let o be Element of O; ::_thesis: for F being FinSequence of O
for A being Action of O,E holds Product ((<*o*> ^ F),A) = (Product (<*o*>,A)) * (Product (F,A))
let F be FinSequence of O; ::_thesis: for A being Action of O,E holds Product ((<*o*> ^ F),A) = (Product (<*o*>,A)) * (Product (F,A))
let A be Action of O,E; ::_thesis: Product ((<*o*> ^ F),A) = (Product (<*o*>,A)) * (Product (F,A))
defpred S1[ Element of NAT ] means for F being FinSequence of O st len F = $1 holds
Product ((<*o*> ^ F),A) = (Product (<*o*>,A)) * (Product (F,A));
reconsider k = len F as Element of NAT ;
A1: k = len F ;
A2: for k being Element of NAT st S1[k] holds
S1[k + 1]
proof
let k be Element of NAT ; ::_thesis: ( S1[k] implies S1[k + 1] )
assume A3: S1[k] ; ::_thesis: S1[k + 1]
now__::_thesis:_for_F_being_FinSequence_of_O_st_len_F_=_k_+_1_holds_
Product_((<*o*>_^_F),A)_=_(Product_(<*o*>,A))_*_(Product_(F,A))
let F be FinSequence of O; ::_thesis: ( len F = k + 1 implies Product ((<*o*> ^ F),A) = (Product (<*o*>,A)) * (Product (F,A)) )
assume A4: len F = k + 1 ; ::_thesis: Product ((<*o*> ^ F),A) = (Product (<*o*>,A)) * (Product (F,A))
then consider Fk being FinSequence of O, o9 being Element of O such that
A5: F = Fk ^ <*o9*> by FINSEQ_2:19;
len F = (len Fk) + (len <*o9*>) by A5, FINSEQ_1:22;
then A6: k + 1 = (len Fk) + 1 by A4, FINSEQ_1:39;
set F2k = <*o*> ^ Fk;
thus Product ((<*o*> ^ F),A) = Product (((<*o*> ^ Fk) ^ <*o9*>),A) by A5, FINSEQ_1:32
.= (Product ((<*o*> ^ Fk),A)) * (Product (<*o9*>,A)) by Lm27
.= ((Product (<*o*>,A)) * (Product (Fk,A))) * (Product (<*o9*>,A)) by A3, A6
.= (Product (<*o*>,A)) * ((Product (Fk,A)) * (Product (<*o9*>,A))) by RELAT_1:36
.= (Product (<*o*>,A)) * (Product (F,A)) by A5, Lm27 ; ::_thesis: verum
end;
hence S1[k + 1] ; ::_thesis: verum
end;
A7: S1[ 0 ]
proof
let F be FinSequence of O; ::_thesis: ( len F = 0 implies Product ((<*o*> ^ F),A) = (Product (<*o*>,A)) * (Product (F,A)) )
assume A8: len F = 0 ; ::_thesis: Product ((<*o*> ^ F),A) = (Product (<*o*>,A)) * (Product (F,A))
then F = <*> O ;
hence Product ((<*o*> ^ F),A) = Product ((<*o*> ^ (<*> O)),A)
.= Product (<*o*>,A) by FINSEQ_1:34
.= (Product (<*o*>,A)) * (id E) by FUNCT_2:17
.= (Product (<*o*>,A)) * (Product (F,A)) by A8, Def3 ;
::_thesis: verum
end;
for k being Element of NAT holds S1[k] from NAT_1:sch_1(A7, A2);
hence Product ((<*o*> ^ F),A) = (Product (<*o*>,A)) * (Product (F,A)) by A1; ::_thesis: verum
end;
Lm29: for O being non empty set
for E being set
for F1, F2 being FinSequence of O
for A being Action of O,E holds Product ((F1 ^ F2),A) = (Product (F1,A)) * (Product (F2,A))
proof
let O be non empty set ; ::_thesis: for E being set
for F1, F2 being FinSequence of O
for A being Action of O,E holds Product ((F1 ^ F2),A) = (Product (F1,A)) * (Product (F2,A))
let E be set ; ::_thesis: for F1, F2 being FinSequence of O
for A being Action of O,E holds Product ((F1 ^ F2),A) = (Product (F1,A)) * (Product (F2,A))
let F1, F2 be FinSequence of O; ::_thesis: for A being Action of O,E holds Product ((F1 ^ F2),A) = (Product (F1,A)) * (Product (F2,A))
let A be Action of O,E; ::_thesis: Product ((F1 ^ F2),A) = (Product (F1,A)) * (Product (F2,A))
defpred S1[ Element of NAT ] means for F1, F2 being FinSequence of O st len F1 = $1 holds
Product ((F1 ^ F2),A) = (Product (F1,A)) * (Product (F2,A));
reconsider k = len F1 as Element of NAT ;
A1: k = len F1 ;
A2: for k being Element of NAT st S1[k] holds
S1[k + 1]
proof
let k be Element of NAT ; ::_thesis: ( S1[k] implies S1[k + 1] )
assume A3: S1[k] ; ::_thesis: S1[k + 1]
now__::_thesis:_for_F1,_F2_being_FinSequence_of_O_st_len_F1_=_k_+_1_holds_
Product_((F1_^_F2),A)_=_(Product_(F1,A))_*_(Product_(F2,A))
let F1, F2 be FinSequence of O; ::_thesis: ( len F1 = k + 1 implies Product ((F1 ^ F2),A) = (Product (F1,A)) * (Product (F2,A)) )
assume A4: len F1 = k + 1 ; ::_thesis: Product ((F1 ^ F2),A) = (Product (F1,A)) * (Product (F2,A))
then consider F1k being FinSequence of O, o being Element of O such that
A5: F1 = F1k ^ <*o*> by FINSEQ_2:19;
set F2k = <*o*> ^ F2;
len F1 = (len F1k) + (len <*o*>) by A5, FINSEQ_1:22;
then A6: k + 1 = (len F1k) + 1 by A4, FINSEQ_1:39;
thus Product ((F1 ^ F2),A) = Product ((F1k ^ (<*o*> ^ F2)),A) by A5, FINSEQ_1:32
.= (Product (F1k,A)) * (Product ((<*o*> ^ F2),A)) by A3, A6
.= (Product (F1k,A)) * ((Product (<*o*>,A)) * (Product (F2,A))) by Lm28
.= ((Product (F1k,A)) * (Product (<*o*>,A))) * (Product (F2,A)) by RELAT_1:36
.= (Product (F1,A)) * (Product (F2,A)) by A3, A5, A6 ; ::_thesis: verum
end;
hence S1[k + 1] ; ::_thesis: verum
end;
A7: S1[ 0 ]
proof
let F1, F2 be FinSequence of O; ::_thesis: ( len F1 = 0 implies Product ((F1 ^ F2),A) = (Product (F1,A)) * (Product (F2,A)) )
assume A8: len F1 = 0 ; ::_thesis: Product ((F1 ^ F2),A) = (Product (F1,A)) * (Product (F2,A))
then F1 = <*> O ;
hence Product ((F1 ^ F2),A) = Product (((<*> O) ^ F2),A)
.= Product (F2,A) by FINSEQ_1:34
.= (id E) * (Product (F2,A)) by FUNCT_2:17
.= (Product (F1,A)) * (Product (F2,A)) by A8, Def3 ;
::_thesis: verum
end;
for k being Element of NAT holds S1[k] from NAT_1:sch_1(A7, A2);
hence Product ((F1 ^ F2),A) = (Product (F1,A)) * (Product (F2,A)) by A1; ::_thesis: verum
end;
Lm30: for O, E being set
for F being FinSequence of O
for Y being Subset of E
for A being Action of O,E st Y is_stable_under_the_action_of A holds
(Product (F,A)) .: Y c= Y
proof
let O, E be set ; ::_thesis: for F being FinSequence of O
for Y being Subset of E
for A being Action of O,E st Y is_stable_under_the_action_of A holds
(Product (F,A)) .: Y c= Y
let F be FinSequence of O; ::_thesis: for Y being Subset of E
for A being Action of O,E st Y is_stable_under_the_action_of A holds
(Product (F,A)) .: Y c= Y
let Y be Subset of E; ::_thesis: for A being Action of O,E st Y is_stable_under_the_action_of A holds
(Product (F,A)) .: Y c= Y
let A be Action of O,E; ::_thesis: ( Y is_stable_under_the_action_of A implies (Product (F,A)) .: Y c= Y )
assume A1: Y is_stable_under_the_action_of A ; ::_thesis: (Product (F,A)) .: Y c= Y
percases ( O = {} or O <> {} ) ;
suppose O = {} ; ::_thesis: (Product (F,A)) .: Y c= Y
then len F = 0 ;
then Product (F,A) = id E by Def3;
hence (Product (F,A)) .: Y c= Y by FUNCT_1:92; ::_thesis: verum
end;
supposeA2: O <> {} ; ::_thesis: (Product (F,A)) .: Y c= Y
defpred S1[ Element of NAT ] means for F being FinSequence of O st len F = $1 holds
(Product (F,A)) .: Y c= Y;
A3: for k being Element of NAT st S1[k] holds
S1[k + 1]
proof
let k be Element of NAT ; ::_thesis: ( S1[k] implies S1[k + 1] )
assume A4: S1[k] ; ::_thesis: S1[k + 1]
now__::_thesis:_for_F_being_FinSequence_of_O_st_len_F_=_k_+_1_holds_
(Product_(F,A))_.:_Y_c=_Y
let F be FinSequence of O; ::_thesis: ( len F = k + 1 implies (Product (F,A)) .: Y c= Y )
assume A5: len F = k + 1 ; ::_thesis: (Product (F,A)) .: Y c= Y
then consider Fk being FinSequence of O, o being Element of O such that
A6: F = Fk ^ <*o*> by FINSEQ_2:19;
len F = (len Fk) + (len <*o*>) by A6, FINSEQ_1:22;
then k + 1 = (len Fk) + 1 by A5, FINSEQ_1:39;
then A7: (Product (Fk,A)) .: Y c= Y by A4;
reconsider F1 = <*o*> as FinSequence of O by A6, FINSEQ_1:36;
Product (F,A) = (Product (Fk,A)) * (Product (F1,A)) by A2, A6, Lm29;
then A8: (Product (F,A)) .: Y = (Product (Fk,A)) .: ((Product (F1,A)) .: Y) by RELAT_1:126;
Product (F1,A) = A . o by A2, Lm26;
then (Product (F1,A)) .: Y c= Y by A1, A2, Def1;
then (Product (F,A)) .: Y c= (Product (Fk,A)) .: Y by A8, RELAT_1:123;
hence (Product (F,A)) .: Y c= Y by A7, XBOOLE_1:1; ::_thesis: verum
end;
hence S1[k + 1] ; ::_thesis: verum
end;
reconsider k = len F as Element of NAT ;
A9: k = len F ;
A10: S1[ 0 ]
proof
let F be FinSequence of O; ::_thesis: ( len F = 0 implies (Product (F,A)) .: Y c= Y )
assume len F = 0 ; ::_thesis: (Product (F,A)) .: Y c= Y
then Product (F,A) = id E by Def3;
hence (Product (F,A)) .: Y c= Y by FUNCT_1:92; ::_thesis: verum
end;
for k being Element of NAT holds S1[k] from NAT_1:sch_1(A10, A3);
hence (Product (F,A)) .: Y c= Y by A9; ::_thesis: verum
end;
end;
end;
Lm31: for O being set
for E being non empty set
for A being Action of O,E
for X being Subset of E
for a being Element of E st not X is empty holds
( a in the_stable_subset_generated_by (X,A) iff ex F being FinSequence of O ex x being Element of X st (Product (F,A)) . x = a )
proof
let O be set ; ::_thesis: for E being non empty set
for A being Action of O,E
for X being Subset of E
for a being Element of E st not X is empty holds
( a in the_stable_subset_generated_by (X,A) iff ex F being FinSequence of O ex x being Element of X st (Product (F,A)) . x = a )
let E be non empty set ; ::_thesis: for A being Action of O,E
for X being Subset of E
for a being Element of E st not X is empty holds
( a in the_stable_subset_generated_by (X,A) iff ex F being FinSequence of O ex x being Element of X st (Product (F,A)) . x = a )
let A be Action of O,E; ::_thesis: for X being Subset of E
for a being Element of E st not X is empty holds
( a in the_stable_subset_generated_by (X,A) iff ex F being FinSequence of O ex x being Element of X st (Product (F,A)) . x = a )
let X be Subset of E; ::_thesis: for a being Element of E st not X is empty holds
( a in the_stable_subset_generated_by (X,A) iff ex F being FinSequence of O ex x being Element of X st (Product (F,A)) . x = a )
let a be Element of E; ::_thesis: ( not X is empty implies ( a in the_stable_subset_generated_by (X,A) iff ex F being FinSequence of O ex x being Element of X st (Product (F,A)) . x = a ) )
defpred S1[ set ] means ex F being FinSequence of O ex x being Element of X st (Product (F,A)) . x = $1;
set B = { e where e is Element of E : S1[e] } ;
reconsider B = { e where e is Element of E : S1[e] } as Subset of E from DOMAIN_1:sch_7();
assume A1: not X is empty ; ::_thesis: ( a in the_stable_subset_generated_by (X,A) iff ex F being FinSequence of O ex x being Element of X st (Product (F,A)) . x = a )
A2: now__::_thesis:_for_Y_being_Subset_of_E_st_Y_is_stable_under_the_action_of_A_&_X_c=_Y_holds_
B_c=_Y
let Y be Subset of E; ::_thesis: ( Y is_stable_under_the_action_of A & X c= Y implies B c= Y )
assume A3: Y is_stable_under_the_action_of A ; ::_thesis: ( X c= Y implies B c= Y )
assume A4: X c= Y ; ::_thesis: B c= Y
now__::_thesis:_for_x_being_set_st_x_in_B_holds_
x_in_Y
let x be set ; ::_thesis: ( x in B implies x in Y )
assume x in B ; ::_thesis: x in Y
then consider e being Element of E such that
A5: x = e and
A6: ex F being FinSequence of O ex x9 being Element of X st (Product (F,A)) . x9 = e ;
consider F being FinSequence of O, x9 being Element of X such that
A7: (Product (F,A)) . x9 = e by A6;
A8: x9 in X by A1;
then x9 in E ;
then x9 in dom (Product (F,A)) by FUNCT_2:def_1;
then A9: (Product (F,A)) . x9 in (Product (F,A)) .: Y by A4, A8, FUNCT_1:def_6;
(Product (F,A)) .: Y c= Y by A3, Lm30;
hence x in Y by A5, A7, A9; ::_thesis: verum
end;
hence B c= Y by TARSKI:def_3; ::_thesis: verum
end;
now__::_thesis:_for_o_being_Element_of_O
for_f_being_Function_of_E,E_st_o_in_O_&_f_=_A_._o_holds_
f_.:_B_c=_B
let o be Element of O; ::_thesis: for f being Function of E,E st o in O & f = A . o holds
b3 .: B c= B
let f be Function of E,E; ::_thesis: ( o in O & f = A . o implies b2 .: B c= B )
assume A10: o in O ; ::_thesis: ( f = A . o implies b2 .: B c= B )
assume A11: f = A . o ; ::_thesis: b2 .: B c= B
percases ( O = {} or O <> {} ) ;
suppose O = {} ; ::_thesis: b2 .: B c= B
hence f .: B c= B by A10; ::_thesis: verum
end;
supposeA12: O <> {} ; ::_thesis: b2 .: B c= B
now__::_thesis:_for_y_being_set_st_y_in_f_.:_B_holds_
y_in_B
reconsider o = o as Element of O ;
reconsider F99 = <*o*> as FinSequence of O by A12, FINSEQ_1:74;
let y be set ; ::_thesis: ( y in f .: B implies y in B )
assume y in f .: B ; ::_thesis: y in B
then consider x being set such that
A13: x in dom f and
A14: x in B and
A15: y = f . x by FUNCT_1:def_6;
y in rng f by A13, A15, FUNCT_1:3;
then reconsider e = y as Element of E ;
consider e9 being Element of E such that
A16: e9 = x and
A17: ex F9 being FinSequence of O ex x9 being Element of X st (Product (F9,A)) . x9 = e9 by A14;
consider F9 being FinSequence of O, x9 being Element of X such that
A18: (Product (F9,A)) . x9 = e9 by A17;
reconsider F = F99 ^ F9 as FinSequence of O ;
x9 in X by A1;
then x9 in E ;
then A19: x9 in dom (Product (F9,A)) by FUNCT_2:def_1;
(Product (F,A)) . x9 = ((Product (F99,A)) * (Product (F9,A))) . x9 by A12, Lm29
.= (Product (F99,A)) . ((Product (F9,A)) . x9) by A19, FUNCT_1:13
.= e by A11, A12, A15, A16, A18, Lm26 ;
hence y in B ; ::_thesis: verum
end;
hence f .: B c= B by TARSKI:def_3; ::_thesis: verum
end;
end;
end;
then A20: B is_stable_under_the_action_of A by Def1;
now__::_thesis:_for_x_being_set_st_x_in_X_holds_
x_in_B
set F = <*> O;
let x be set ; ::_thesis: ( x in X implies x in B )
assume A21: x in X ; ::_thesis: x in B
then reconsider e = x as Element of E ;
reconsider x9 = e as Element of X by A21;
len (<*> O) = 0 ;
then (Product ((<*> O),A)) . x = (id E) . x by Def3
.= x by A21, FUNCT_1:18 ;
then (Product ((<*> O),A)) . x9 = e ;
hence x in B ; ::_thesis: verum
end;
then X c= B by TARSKI:def_3;
then A22: B = the_stable_subset_generated_by (X,A) by A20, A2, Def2;
hereby ::_thesis: ( ex F being FinSequence of O ex x being Element of X st (Product (F,A)) . x = a implies a in the_stable_subset_generated_by (X,A) )
assume a in the_stable_subset_generated_by (X,A) ; ::_thesis: ex F being FinSequence of O ex x being Element of X st (Product (F,A)) . x = a
then consider e being Element of E such that
A23: a = e and
A24: ex F being FinSequence of O ex x being Element of X st (Product (F,A)) . x = e by A22;
consider F being FinSequence of O, x being Element of X such that
A25: (Product (F,A)) . x = e by A24;
take F = F; ::_thesis: ex x being Element of X st (Product (F,A)) . x = a
take x = x; ::_thesis: (Product (F,A)) . x = a
thus (Product (F,A)) . x = a by A23, A25; ::_thesis: verum
end;
given F being FinSequence of O, x being Element of X such that A26: (Product (F,A)) . x = a ; ::_thesis: a in the_stable_subset_generated_by (X,A)
thus a in the_stable_subset_generated_by (X,A) by A22, A26; ::_thesis: verum
end;
theorem Th25: :: GROUP_9:25
for O being set
for G being GroupWithOperators of O
for H being strict StableSubgroup of G holds the_stable_subgroup_of (carr H) = H
proof
let O be set ; ::_thesis: for G being GroupWithOperators of O
for H being strict StableSubgroup of G holds the_stable_subgroup_of (carr H) = H
let G be GroupWithOperators of O; ::_thesis: for H being strict StableSubgroup of G holds the_stable_subgroup_of (carr H) = H
let H be strict StableSubgroup of G; ::_thesis: the_stable_subgroup_of (carr H) = H
for H1 being strict StableSubgroup of G st carr H c= the carrier of H1 holds
H is StableSubgroup of H1 by Lm21;
hence the_stable_subgroup_of (carr H) = H by Def26; ::_thesis: verum
end;
theorem Th26: :: GROUP_9:26
for O being set
for G being GroupWithOperators of O
for A, B being Subset of G st A c= B holds
the_stable_subgroup_of A is StableSubgroup of the_stable_subgroup_of B
proof
let O be set ; ::_thesis: for G being GroupWithOperators of O
for A, B being Subset of G st A c= B holds
the_stable_subgroup_of A is StableSubgroup of the_stable_subgroup_of B
let G be GroupWithOperators of O; ::_thesis: for A, B being Subset of G st A c= B holds
the_stable_subgroup_of A is StableSubgroup of the_stable_subgroup_of B
let A, B be Subset of G; ::_thesis: ( A c= B implies the_stable_subgroup_of A is StableSubgroup of the_stable_subgroup_of B )
assume A1: A c= B ; ::_thesis: the_stable_subgroup_of A is StableSubgroup of the_stable_subgroup_of B
percases ( A is empty or not A is empty ) ;
supposeA2: A is empty ; ::_thesis: the_stable_subgroup_of A is StableSubgroup of the_stable_subgroup_of B
reconsider H1 = (1). G, H2 = (1). (the_stable_subgroup_of B) as strict StableSubgroup of G by Th11;
the carrier of H1 = {(1_ G)} by Def8
.= {(1_ (the_stable_subgroup_of B))} by Th4
.= the carrier of H2 by Def8 ;
then (1). G = (1). (the_stable_subgroup_of B) by Lm5;
hence the_stable_subgroup_of A is StableSubgroup of the_stable_subgroup_of B by A2, Lm25; ::_thesis: verum
end;
supposeA3: not A is empty ; ::_thesis: the_stable_subgroup_of A is StableSubgroup of the_stable_subgroup_of B
now__::_thesis:_for_a_being_Element_of_G_st_a_in_the_stable_subgroup_of_A_holds_
a_in_the_stable_subgroup_of_B
set D = the_stable_subset_generated_by (B, the action of G);
let a be Element of G; ::_thesis: ( a in the_stable_subgroup_of A implies a in the_stable_subgroup_of B )
assume a in the_stable_subgroup_of A ; ::_thesis: a in the_stable_subgroup_of B
then consider F being FinSequence of the carrier of G, I being FinSequence of INT , C being Subset of G such that
A4: C = the_stable_subset_generated_by (A, the action of G) and
A5: len F = len I and
A6: rng F c= C and
A7: Product (F |^ I) = a by Th24;
now__::_thesis:_for_y_being_set_st_y_in_C_holds_
y_in_the_stable_subset_generated_by_(B,_the_action_of_G)
let y be set ; ::_thesis: ( y in C implies y in the_stable_subset_generated_by (B, the action of G) )
assume A8: y in C ; ::_thesis: y in the_stable_subset_generated_by (B, the action of G)
then reconsider b = y as Element of G ;
consider F1 being FinSequence of O, x being Element of A such that
A9: (Product (F1, the action of G)) . x = b by A3, A4, A8, Lm31;
x in A by A3;
hence y in the_stable_subset_generated_by (B, the action of G) by A1, A9, Lm31; ::_thesis: verum
end;
then C c= the_stable_subset_generated_by (B, the action of G) by TARSKI:def_3;
then rng F c= the_stable_subset_generated_by (B, the action of G) by A6, XBOOLE_1:1;
hence a in the_stable_subgroup_of B by A5, A7, Th24; ::_thesis: verum
end;
hence the_stable_subgroup_of A is StableSubgroup of the_stable_subgroup_of B by Th13; ::_thesis: verum
end;
end;
end;
scheme :: GROUP_9:sch 1
MeetSbgWOpEx{ F1() -> set , F2() -> GroupWithOperators of F1(), P1[ set ] } :
ex H being strict StableSubgroup of F2() st the carrier of H = meet { A where A is Subset of F2() : ex K being strict StableSubgroup of F2() st
( A = the carrier of K & P1[K] ) }
provided
A1: ex H being strict StableSubgroup of F2() st P1[H]
proof
set X = { A where A is Subset of F2() : ex K being strict StableSubgroup of F2() st
( A = the carrier of K & P1[K] ) } ;
consider T being strict StableSubgroup of F2() such that
A2: P1[T] by A1;
A3: carr T in { A where A is Subset of F2() : ex K being strict StableSubgroup of F2() st
( A = the carrier of K & P1[K] ) } by A2;
then reconsider Y = meet { A where A is Subset of F2() : ex K being strict StableSubgroup of F2() st
( A = the carrier of K & P1[K] ) } as Subset of F2() by SETFAM_1:7;
A4: now__::_thesis:_for_a_being_Element_of_F2()_st_a_in_Y_holds_
a_"_in_Y
let a be Element of F2(); ::_thesis: ( a in Y implies a " in Y )
assume A5: a in Y ; ::_thesis: a " in Y
now__::_thesis:_for_Z_being_set_st_Z_in__{__A_where_A_is_Subset_of_F2()_:_ex_K_being_strict_StableSubgroup_of_F2()_st_
(_A_=_the_carrier_of_K_&_P1[K]_)__}__holds_
a_"_in_Z
let Z be set ; ::_thesis: ( Z in { A where A is Subset of F2() : ex K being strict StableSubgroup of F2() st
( A = the carrier of K & P1[K] ) } implies a " in Z )
assume A6: Z in { A where A is Subset of F2() : ex K being strict StableSubgroup of F2() st
( A = the carrier of K & P1[K] ) } ; ::_thesis: a " in Z
then consider A being Subset of F2() such that
A7: A = Z and
A8: ex H being strict StableSubgroup of F2() st
( A = the carrier of H & P1[H] ) ;
consider H being StableSubgroup of F2() such that
A9: A = the carrier of H and
P1[H] by A8;
a in the carrier of H by A5, A6, A7, A9, SETFAM_1:def_1;
then a in H by STRUCT_0:def_5;
then a " in H by Lm20;
hence a " in Z by A7, A9, STRUCT_0:def_5; ::_thesis: verum
end;
hence a " in Y by A3, SETFAM_1:def_1; ::_thesis: verum
end;
A10: now__::_thesis:_for_a,_b_being_Element_of_F2()_st_a_in_Y_&_b_in_Y_holds_
a_*_b_in_Y
let a, b be Element of F2(); ::_thesis: ( a in Y & b in Y implies a * b in Y )
assume that
A11: a in Y and
A12: b in Y ; ::_thesis: a * b in Y
now__::_thesis:_for_Z_being_set_st_Z_in__{__A_where_A_is_Subset_of_F2()_:_ex_K_being_strict_StableSubgroup_of_F2()_st_
(_A_=_the_carrier_of_K_&_P1[K]_)__}__holds_
a_*_b_in_Z
let Z be set ; ::_thesis: ( Z in { A where A is Subset of F2() : ex K being strict StableSubgroup of F2() st
( A = the carrier of K & P1[K] ) } implies a * b in Z )
assume A13: Z in { A where A is Subset of F2() : ex K being strict StableSubgroup of F2() st
( A = the carrier of K & P1[K] ) } ; ::_thesis: a * b in Z
then consider A being Subset of F2() such that
A14: A = Z and
A15: ex H being strict StableSubgroup of F2() st
( A = the carrier of H & P1[H] ) ;
consider H being StableSubgroup of F2() such that
A16: A = the carrier of H and
P1[H] by A15;
b in the carrier of H by A12, A13, A14, A16, SETFAM_1:def_1;
then A17: b in H by STRUCT_0:def_5;
a in the carrier of H by A11, A13, A14, A16, SETFAM_1:def_1;
then a in H by STRUCT_0:def_5;
then a * b in H by A17, Lm19;
hence a * b in Z by A14, A16, STRUCT_0:def_5; ::_thesis: verum
end;
hence a * b in Y by A3, SETFAM_1:def_1; ::_thesis: verum
end;
A18: now__::_thesis:_for_o_being_Element_of_F1()
for_a_being_Element_of_F2()_st_a_in_Y_holds_
(F2()_^_o)_._a_in_Y
let o be Element of F1(); ::_thesis: for a being Element of F2() st a in Y holds
(F2() ^ o) . a in Y
let a be Element of F2(); ::_thesis: ( a in Y implies (F2() ^ o) . a in Y )
assume A19: a in Y ; ::_thesis: (F2() ^ o) . a in Y
now__::_thesis:_for_Z_being_set_st_Z_in__{__A_where_A_is_Subset_of_F2()_:_ex_K_being_strict_StableSubgroup_of_F2()_st_
(_A_=_the_carrier_of_K_&_P1[K]_)__}__holds_
(F2()_^_o)_._a_in_Z
let Z be set ; ::_thesis: ( Z in { A where A is Subset of F2() : ex K being strict StableSubgroup of F2() st
( A = the carrier of K & P1[K] ) } implies (F2() ^ o) . a in Z )
assume A20: Z in { A where A is Subset of F2() : ex K being strict StableSubgroup of F2() st
( A = the carrier of K & P1[K] ) } ; ::_thesis: (F2() ^ o) . a in Z
then consider A being Subset of F2() such that
A21: A = Z and
A22: ex H being strict StableSubgroup of F2() st
( A = the carrier of H & P1[H] ) ;
consider H being StableSubgroup of F2() such that
A23: A = the carrier of H and
P1[H] by A22;
a in the carrier of H by A19, A20, A21, A23, SETFAM_1:def_1;
then a in H by STRUCT_0:def_5;
then (F2() ^ o) . a in H by Lm10;
hence (F2() ^ o) . a in Z by A21, A23, STRUCT_0:def_5; ::_thesis: verum
end;
hence (F2() ^ o) . a in Y by A3, SETFAM_1:def_1; ::_thesis: verum
end;
now__::_thesis:_for_Z_being_set_st_Z_in__{__A_where_A_is_Subset_of_F2()_:_ex_K_being_strict_StableSubgroup_of_F2()_st_
(_A_=_the_carrier_of_K_&_P1[K]_)__}__holds_
1__F2()_in_Z
let Z be set ; ::_thesis: ( Z in { A where A is Subset of F2() : ex K being strict StableSubgroup of F2() st
( A = the carrier of K & P1[K] ) } implies 1_ F2() in Z )
assume Z in { A where A is Subset of F2() : ex K being strict StableSubgroup of F2() st
( A = the carrier of K & P1[K] ) } ; ::_thesis: 1_ F2() in Z
then consider A being Subset of F2() such that
A24: Z = A and
A25: ex K being strict StableSubgroup of F2() st
( A = the carrier of K & P1[K] ) ;
consider H being StableSubgroup of F2() such that
A26: A = the carrier of H and
P1[H] by A25;
1_ F2() in H by Lm18;
hence 1_ F2() in Z by A24, A26, STRUCT_0:def_5; ::_thesis: verum
end;
then Y <> {} by A3, SETFAM_1:def_1;
hence ex H being strict StableSubgroup of F2() st the carrier of H = meet { A where A is Subset of F2() : ex K being strict StableSubgroup of F2() st
( A = the carrier of K & P1[K] ) } by A10, A4, A18, Lm15; ::_thesis: verum
end;
theorem Th27: :: GROUP_9:27
for O being set
for G being GroupWithOperators of O
for A being Subset of G holds the carrier of (the_stable_subgroup_of A) = meet { B where B is Subset of G : ex H being strict StableSubgroup of G st
( B = the carrier of H & A c= carr H ) }
proof
let O be set ; ::_thesis: for G being GroupWithOperators of O
for A being Subset of G holds the carrier of (the_stable_subgroup_of A) = meet { B where B is Subset of G : ex H being strict StableSubgroup of G st
( B = the carrier of H & A c= carr H ) }
let G be GroupWithOperators of O; ::_thesis: for A being Subset of G holds the carrier of (the_stable_subgroup_of A) = meet { B where B is Subset of G : ex H being strict StableSubgroup of G st
( B = the carrier of H & A c= carr H ) }
let A be Subset of G; ::_thesis: the carrier of (the_stable_subgroup_of A) = meet { B where B is Subset of G : ex H being strict StableSubgroup of G st
( B = the carrier of H & A c= carr H ) }
defpred S1[ StableSubgroup of G] means A c= carr $1;
set X = { B where B is Subset of G : ex H being strict StableSubgroup of G st
( B = the carrier of H & A c= carr H ) } ;
A1: now__::_thesis:_for_Y_being_set_st_Y_in__{__B_where_B_is_Subset_of_G_:_ex_H_being_strict_StableSubgroup_of_G_st_
(_B_=_the_carrier_of_H_&_A_c=_carr_H_)__}__holds_
A_c=_Y
let Y be set ; ::_thesis: ( Y in { B where B is Subset of G : ex H being strict StableSubgroup of G st
( B = the carrier of H & A c= carr H ) } implies A c= Y )
assume Y in { B where B is Subset of G : ex H being strict StableSubgroup of G st
( B = the carrier of H & A c= carr H ) } ; ::_thesis: A c= Y
then ex B being Subset of G st
( Y = B & ex H being strict StableSubgroup of G st
( B = the carrier of H & A c= carr H ) ) ;
hence A c= Y ; ::_thesis: verum
end;
the carrier of ((Omega). G) = carr ((Omega). G) ;
then A2: ex H being strict StableSubgroup of G st S1[H] ;
consider H being strict StableSubgroup of G such that
A3: the carrier of H = meet { B where B is Subset of G : ex H being strict StableSubgroup of G st
( B = the carrier of H & S1[H] ) } from GROUP_9:sch_1(A2);
A4: now__::_thesis:_for_H1_being_strict_StableSubgroup_of_G_st_A_c=_the_carrier_of_H1_holds_
H_is_StableSubgroup_of_H1
let H1 be strict StableSubgroup of G; ::_thesis: ( A c= the carrier of H1 implies H is StableSubgroup of H1 )
A5: the carrier of H1 = carr H1 ;
assume A c= the carrier of H1 ; ::_thesis: H is StableSubgroup of H1
then the carrier of H1 in { B where B is Subset of G : ex H being strict StableSubgroup of G st
( B = the carrier of H & A c= carr H ) } by A5;
hence H is StableSubgroup of H1 by A3, Lm21, SETFAM_1:3; ::_thesis: verum
end;
carr ((Omega). G) in { B where B is Subset of G : ex H being strict StableSubgroup of G st
( B = the carrier of H & A c= carr H ) } ;
then A c= the carrier of H by A3, A1, SETFAM_1:5;
hence the carrier of (the_stable_subgroup_of A) = meet { B where B is Subset of G : ex H being strict StableSubgroup of G st
( B = the carrier of H & A c= carr H ) } by A3, A4, Def26; ::_thesis: verum
end;
Lm32: for O being set
for G being GroupWithOperators of O
for B, A being Subset of G st B = the carrier of (gr A) holds
the_stable_subgroup_of A = the_stable_subgroup_of B
proof
let O be set ; ::_thesis: for G being GroupWithOperators of O
for B, A being Subset of G st B = the carrier of (gr A) holds
the_stable_subgroup_of A = the_stable_subgroup_of B
let G be GroupWithOperators of O; ::_thesis: for B, A being Subset of G st B = the carrier of (gr A) holds
the_stable_subgroup_of A = the_stable_subgroup_of B
let B, A be Subset of G; ::_thesis: ( B = the carrier of (gr A) implies the_stable_subgroup_of A = the_stable_subgroup_of B )
A1: A c= the carrier of (gr A) by GROUP_4:def_4;
assume A2: B = the carrier of (gr A) ; ::_thesis: the_stable_subgroup_of A = the_stable_subgroup_of B
A3: now__::_thesis:_for_H_being_strict_StableSubgroup_of_G_st_A_c=_the_carrier_of_H_holds_
the_stable_subgroup_of_B_is_StableSubgroup_of_H
let H be strict StableSubgroup of G; ::_thesis: ( A c= the carrier of H implies the_stable_subgroup_of B is StableSubgroup of H )
reconsider H9 = multMagma(# the carrier of H, the multF of H #) as strict Subgroup of G by Lm16;
assume A c= the carrier of H ; ::_thesis: the_stable_subgroup_of B is StableSubgroup of H
then gr A is Subgroup of H9 by GROUP_4:def_4;
then B c= the carrier of H9 by A2, GROUP_2:def_5;
hence the_stable_subgroup_of B is StableSubgroup of H by Def26; ::_thesis: verum
end;
the carrier of (gr A) c= the carrier of (the_stable_subgroup_of B) by A2, Def26;
then A c= the carrier of (the_stable_subgroup_of B) by A1, XBOOLE_1:1;
hence the_stable_subgroup_of A = the_stable_subgroup_of B by A3, Def26; ::_thesis: verum
end;
theorem Th28: :: GROUP_9:28
for O being set
for G being GroupWithOperators of O
for N1, N2 being strict normal StableSubgroup of G holds N1 * N2 = N2 * N1
proof
let O be set ; ::_thesis: for G being GroupWithOperators of O
for N1, N2 being strict normal StableSubgroup of G holds N1 * N2 = N2 * N1
let G be GroupWithOperators of O; ::_thesis: for N1, N2 being strict normal StableSubgroup of G holds N1 * N2 = N2 * N1
let N1, N2 be strict normal StableSubgroup of G; ::_thesis: N1 * N2 = N2 * N1
reconsider N19 = multMagma(# the carrier of N1, the multF of N1 #), N29 = multMagma(# the carrier of N2, the multF of N2 #) as strict normal Subgroup of G by Lm7;
thus N1 * N2 = (carr N29) * (carr N19) by GROUP_3:125
.= N2 * N1 ; ::_thesis: verum
end;
theorem Th29: :: GROUP_9:29
for O being set
for G being GroupWithOperators of O
for H1, H2 being StableSubgroup of G holds H1 "\/" H2 = the_stable_subgroup_of (H1 * H2)
proof
let O be set ; ::_thesis: for G being GroupWithOperators of O
for H1, H2 being StableSubgroup of G holds H1 "\/" H2 = the_stable_subgroup_of (H1 * H2)
let G be GroupWithOperators of O; ::_thesis: for H1, H2 being StableSubgroup of G holds H1 "\/" H2 = the_stable_subgroup_of (H1 * H2)
let H1, H2 be StableSubgroup of G; ::_thesis: H1 "\/" H2 = the_stable_subgroup_of (H1 * H2)
reconsider H19 = H1, H29 = H2 as Subgroup of G by Def7;
reconsider Y = the carrier of (H19 "\/" H29) as Subset of G by GROUP_2:def_5;
A1: Y = the carrier of (gr (H19 * H29)) by GROUP_4:50;
H1 "\/" H2 = the_stable_subgroup_of Y by Lm32
.= the_stable_subgroup_of (H19 * H29) by A1, Lm32 ;
hence H1 "\/" H2 = the_stable_subgroup_of (H1 * H2) ; ::_thesis: verum
end;
theorem Th30: :: GROUP_9:30
for O being set
for G being GroupWithOperators of O
for H1, H2 being StableSubgroup of G st H1 * H2 = H2 * H1 holds
the carrier of (H1 "\/" H2) = H1 * H2
proof
let O be set ; ::_thesis: for G being GroupWithOperators of O
for H1, H2 being StableSubgroup of G st H1 * H2 = H2 * H1 holds
the carrier of (H1 "\/" H2) = H1 * H2
let G be GroupWithOperators of O; ::_thesis: for H1, H2 being StableSubgroup of G st H1 * H2 = H2 * H1 holds
the carrier of (H1 "\/" H2) = H1 * H2
let H1, H2 be StableSubgroup of G; ::_thesis: ( H1 * H2 = H2 * H1 implies the carrier of (H1 "\/" H2) = H1 * H2 )
assume H1 * H2 = H2 * H1 ; ::_thesis: the carrier of (H1 "\/" H2) = H1 * H2
then consider H being strict StableSubgroup of G such that
A1: the carrier of H = (carr H1) * (carr H2) by Th17;
now__::_thesis:_for_a_being_Element_of_G_st_a_in_H_holds_
a_in_H1_"\/"_H2
set A = (carr H1) \/ (carr H2);
let a be Element of G; ::_thesis: ( a in H implies a in H1 "\/" H2 )
set X = { B where B is Subset of G : ex H being strict StableSubgroup of G st
( B = the carrier of H & (carr H1) \/ (carr H2) c= carr H ) } ;
assume a in H ; ::_thesis: a in H1 "\/" H2
then a in (carr H1) * (carr H2) by A1, STRUCT_0:def_5;
then consider b, c being Element of G such that
A2: a = b * c and
A3: b in carr H1 and
A4: c in carr H2 ;
A5: now__::_thesis:_for_Y_being_set_st_Y_in__{__B_where_B_is_Subset_of_G_:_ex_H_being_strict_StableSubgroup_of_G_st_
(_B_=_the_carrier_of_H_&_(carr_H1)_\/_(carr_H2)_c=_carr_H_)__}__holds_
a_in_Y
let Y be set ; ::_thesis: ( Y in { B where B is Subset of G : ex H being strict StableSubgroup of G st
( B = the carrier of H & (carr H1) \/ (carr H2) c= carr H ) } implies a in Y )
assume Y in { B where B is Subset of G : ex H being strict StableSubgroup of G st
( B = the carrier of H & (carr H1) \/ (carr H2) c= carr H ) } ; ::_thesis: a in Y
then consider B being Subset of G such that
A6: Y = B and
A7: ex H being strict StableSubgroup of G st
( B = the carrier of H & (carr H1) \/ (carr H2) c= carr H ) ;
consider H9 being strict StableSubgroup of G such that
A8: B = the carrier of H9 and
A9: (carr H1) \/ (carr H2) c= carr H9 by A7;
c in (carr H1) \/ (carr H2) by A4, XBOOLE_0:def_3;
then A10: c in H9 by A9, STRUCT_0:def_5;
A11: H9 is Subgroup of G by Def7;
b in (carr H1) \/ (carr H2) by A3, XBOOLE_0:def_3;
then b in H9 by A9, STRUCT_0:def_5;
then b * c in H9 by A11, A10, GROUP_2:50;
hence a in Y by A2, A6, A8, STRUCT_0:def_5; ::_thesis: verum
end;
carr ((Omega). G) in { B where B is Subset of G : ex H being strict StableSubgroup of G st
( B = the carrier of H & (carr H1) \/ (carr H2) c= carr H ) } ;
then a in meet { B where B is Subset of G : ex H being strict StableSubgroup of G st
( B = the carrier of H & (carr H1) \/ (carr H2) c= carr H ) } by A5, SETFAM_1:def_1;
then a in the carrier of (the_stable_subgroup_of ((carr H1) \/ (carr H2))) by Th27;
hence a in H1 "\/" H2 by STRUCT_0:def_5; ::_thesis: verum
end;
then H is StableSubgroup of H1 "\/" H2 by Th13;
then H is Subgroup of H1 "\/" H2 by Def7;
then A12: the carrier of H c= the carrier of (H1 "\/" H2) by GROUP_2:def_5;
(carr H1) \/ (carr H2) c= (carr H1) * (carr H2)
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in (carr H1) \/ (carr H2) or x in (carr H1) * (carr H2) )
assume A13: x in (carr H1) \/ (carr H2) ; ::_thesis: x in (carr H1) * (carr H2)
then reconsider a = x as Element of G ;
now__::_thesis:_x_in_(carr_H1)_*_(carr_H2)
percases ( x in carr H1 or x in carr H2 ) by A13, XBOOLE_0:def_3;
supposeA14: x in carr H1 ; ::_thesis: x in (carr H1) * (carr H2)
1_ G in H2 by Lm18;
then A15: 1_ G in carr H2 by STRUCT_0:def_5;
a * (1_ G) = a by GROUP_1:def_4;
hence x in (carr H1) * (carr H2) by A14, A15; ::_thesis: verum
end;
supposeA16: x in carr H2 ; ::_thesis: x in (carr H1) * (carr H2)
1_ G in H1 by Lm18;
then A17: 1_ G in carr H1 by STRUCT_0:def_5;
(1_ G) * a = a by GROUP_1:def_4;
hence x in (carr H1) * (carr H2) by A16, A17; ::_thesis: verum
end;
end;
end;
hence x in (carr H1) * (carr H2) ; ::_thesis: verum
end;
then H1 "\/" H2 is StableSubgroup of H by A1, Def26;
then H1 "\/" H2 is Subgroup of H by Def7;
then the carrier of (H1 "\/" H2) c= the carrier of H by GROUP_2:def_5;
hence the carrier of (H1 "\/" H2) = H1 * H2 by A1, A12, XBOOLE_0:def_10; ::_thesis: verum
end;
theorem Th31: :: GROUP_9:31
for O being set
for G being GroupWithOperators of O
for N1, N2 being strict normal StableSubgroup of G holds the carrier of (N1 "\/" N2) = N1 * N2
proof
let O be set ; ::_thesis: for G being GroupWithOperators of O
for N1, N2 being strict normal StableSubgroup of G holds the carrier of (N1 "\/" N2) = N1 * N2
let G be GroupWithOperators of O; ::_thesis: for N1, N2 being strict normal StableSubgroup of G holds the carrier of (N1 "\/" N2) = N1 * N2
let N1, N2 be strict normal StableSubgroup of G; ::_thesis: the carrier of (N1 "\/" N2) = N1 * N2
N1 * N2 = N2 * N1 by Th28;
hence the carrier of (N1 "\/" N2) = N1 * N2 by Th30; ::_thesis: verum
end;
theorem Th32: :: GROUP_9:32
for O being set
for G being GroupWithOperators of O
for N1, N2 being strict normal StableSubgroup of G holds N1 "\/" N2 is normal StableSubgroup of G
proof
let O be set ; ::_thesis: for G being GroupWithOperators of O
for N1, N2 being strict normal StableSubgroup of G holds N1 "\/" N2 is normal StableSubgroup of G
let G be GroupWithOperators of O; ::_thesis: for N1, N2 being strict normal StableSubgroup of G holds N1 "\/" N2 is normal StableSubgroup of G
let N1, N2 be strict normal StableSubgroup of G; ::_thesis: N1 "\/" N2 is normal StableSubgroup of G
( ex N being strict normal StableSubgroup of G st the carrier of N = (carr N1) * (carr N2) & the carrier of (N1 "\/" N2) = N1 * N2 ) by Th23, Th31;
hence N1 "\/" N2 is normal StableSubgroup of G by Lm5; ::_thesis: verum
end;
theorem Th33: :: GROUP_9:33
for O being set
for G being GroupWithOperators of O
for H being strict StableSubgroup of G holds
( ((1). G) "\/" H = H & H "\/" ((1). G) = H )
proof
let O be set ; ::_thesis: for G being GroupWithOperators of O
for H being strict StableSubgroup of G holds
( ((1). G) "\/" H = H & H "\/" ((1). G) = H )
let G be GroupWithOperators of O; ::_thesis: for H being strict StableSubgroup of G holds
( ((1). G) "\/" H = H & H "\/" ((1). G) = H )
let H be strict StableSubgroup of G; ::_thesis: ( ((1). G) "\/" H = H & H "\/" ((1). G) = H )
1_ G in H by Lm18;
then 1_ G in carr H by STRUCT_0:def_5;
then {(1_ G)} c= carr H by ZFMISC_1:31;
then A1: {(1_ G)} \/ (carr H) = carr H by XBOOLE_1:12;
carr ((1). G) = {(1_ G)} by Def8;
hence ( ((1). G) "\/" H = H & H "\/" ((1). G) = H ) by A1, Th25; ::_thesis: verum
end;
theorem Th34: :: GROUP_9:34
for O being set
for G being GroupWithOperators of O
for H1 being StableSubgroup of G holds
( ((Omega). G) "\/" H1 = (Omega). G & H1 "\/" ((Omega). G) = (Omega). G )
proof
let O be set ; ::_thesis: for G being GroupWithOperators of O
for H1 being StableSubgroup of G holds
( ((Omega). G) "\/" H1 = (Omega). G & H1 "\/" ((Omega). G) = (Omega). G )
let G be GroupWithOperators of O; ::_thesis: for H1 being StableSubgroup of G holds
( ((Omega). G) "\/" H1 = (Omega). G & H1 "\/" ((Omega). G) = (Omega). G )
let H1 be StableSubgroup of G; ::_thesis: ( ((Omega). G) "\/" H1 = (Omega). G & H1 "\/" ((Omega). G) = (Omega). G )
the carrier of ((Omega). G) \/ (carr H1) = [#] the carrier of G by SUBSET_1:11;
hence ( ((Omega). G) "\/" H1 = (Omega). G & H1 "\/" ((Omega). G) = (Omega). G ) by Th25; ::_thesis: verum
end;
Lm33: for O being set
for G being GroupWithOperators of O
for H1, H2 being StableSubgroup of G holds H1 is StableSubgroup of H1 "\/" H2
proof
let O be set ; ::_thesis: for G being GroupWithOperators of O
for H1, H2 being StableSubgroup of G holds H1 is StableSubgroup of H1 "\/" H2
let G be GroupWithOperators of O; ::_thesis: for H1, H2 being StableSubgroup of G holds H1 is StableSubgroup of H1 "\/" H2
let H1, H2 be StableSubgroup of G; ::_thesis: H1 is StableSubgroup of H1 "\/" H2
( carr H1 c= (carr H1) \/ (carr H2) & (carr H1) \/ (carr H2) c= the carrier of (the_stable_subgroup_of ((carr H1) \/ (carr H2))) ) by Def26, XBOOLE_1:7;
hence H1 is StableSubgroup of H1 "\/" H2 by Lm21, XBOOLE_1:1; ::_thesis: verum
end;
theorem Th35: :: GROUP_9:35
for O being set
for G being GroupWithOperators of O
for H1, H2 being StableSubgroup of G holds
( H1 is StableSubgroup of H1 "\/" H2 & H2 is StableSubgroup of H1 "\/" H2 )
proof
let O be set ; ::_thesis: for G being GroupWithOperators of O
for H1, H2 being StableSubgroup of G holds
( H1 is StableSubgroup of H1 "\/" H2 & H2 is StableSubgroup of H1 "\/" H2 )
let G be GroupWithOperators of O; ::_thesis: for H1, H2 being StableSubgroup of G holds
( H1 is StableSubgroup of H1 "\/" H2 & H2 is StableSubgroup of H1 "\/" H2 )
let H1, H2 be StableSubgroup of G; ::_thesis: ( H1 is StableSubgroup of H1 "\/" H2 & H2 is StableSubgroup of H1 "\/" H2 )
H1 "\/" H2 = H2 "\/" H1 ;
hence ( H1 is StableSubgroup of H1 "\/" H2 & H2 is StableSubgroup of H1 "\/" H2 ) by Lm33; ::_thesis: verum
end;
theorem Th36: :: GROUP_9:36
for O being set
for G being GroupWithOperators of O
for H1 being StableSubgroup of G
for H2 being strict StableSubgroup of G holds
( H1 is StableSubgroup of H2 iff H1 "\/" H2 = H2 )
proof
let O be set ; ::_thesis: for G being GroupWithOperators of O
for H1 being StableSubgroup of G
for H2 being strict StableSubgroup of G holds
( H1 is StableSubgroup of H2 iff H1 "\/" H2 = H2 )
let G be GroupWithOperators of O; ::_thesis: for H1 being StableSubgroup of G
for H2 being strict StableSubgroup of G holds
( H1 is StableSubgroup of H2 iff H1 "\/" H2 = H2 )
let H1 be StableSubgroup of G; ::_thesis: for H2 being strict StableSubgroup of G holds
( H1 is StableSubgroup of H2 iff H1 "\/" H2 = H2 )
let H2 be strict StableSubgroup of G; ::_thesis: ( H1 is StableSubgroup of H2 iff H1 "\/" H2 = H2 )
thus ( H1 is StableSubgroup of H2 implies H1 "\/" H2 = H2 ) ::_thesis: ( H1 "\/" H2 = H2 implies H1 is StableSubgroup of H2 )
proof
assume H1 is StableSubgroup of H2 ; ::_thesis: H1 "\/" H2 = H2
then H1 is Subgroup of H2 by Def7;
then the carrier of H1 c= the carrier of H2 by GROUP_2:def_5;
hence H1 "\/" H2 = the_stable_subgroup_of (carr H2) by XBOOLE_1:12
.= H2 by Th25 ;
::_thesis: verum
end;
thus ( H1 "\/" H2 = H2 implies H1 is StableSubgroup of H2 ) by Th35; ::_thesis: verum
end;
theorem Th37: :: GROUP_9:37
for O being set
for G being GroupWithOperators of O
for H1, H2 being StableSubgroup of G
for H3 being strict StableSubgroup of G st H1 is StableSubgroup of H3 & H2 is StableSubgroup of H3 holds
H1 "\/" H2 is StableSubgroup of H3
proof
let O be set ; ::_thesis: for G being GroupWithOperators of O
for H1, H2 being StableSubgroup of G
for H3 being strict StableSubgroup of G st H1 is StableSubgroup of H3 & H2 is StableSubgroup of H3 holds
H1 "\/" H2 is StableSubgroup of H3
let G be GroupWithOperators of O; ::_thesis: for H1, H2 being StableSubgroup of G
for H3 being strict StableSubgroup of G st H1 is StableSubgroup of H3 & H2 is StableSubgroup of H3 holds
H1 "\/" H2 is StableSubgroup of H3
let H1, H2 be StableSubgroup of G; ::_thesis: for H3 being strict StableSubgroup of G st H1 is StableSubgroup of H3 & H2 is StableSubgroup of H3 holds
H1 "\/" H2 is StableSubgroup of H3
let H3 be strict StableSubgroup of G; ::_thesis: ( H1 is StableSubgroup of H3 & H2 is StableSubgroup of H3 implies H1 "\/" H2 is StableSubgroup of H3 )
assume that
A1: H1 is StableSubgroup of H3 and
A2: H2 is StableSubgroup of H3 ; ::_thesis: H1 "\/" H2 is StableSubgroup of H3
H2 is Subgroup of H3 by A2, Def7;
then A3: carr H2 c= carr H3 by GROUP_2:def_5;
H1 is Subgroup of H3 by A1, Def7;
then carr H1 c= carr H3 by GROUP_2:def_5;
then the_stable_subgroup_of ((carr H1) \/ (carr H2)) is StableSubgroup of the_stable_subgroup_of (carr H3) by A3, Th26, XBOOLE_1:8;
hence H1 "\/" H2 is StableSubgroup of H3 by Th25; ::_thesis: verum
end;
theorem Th38: :: GROUP_9:38
for O being set
for G being GroupWithOperators of O
for H1 being StableSubgroup of G
for H2, H3 being strict StableSubgroup of G st H1 is StableSubgroup of H2 holds
H1 "\/" H3 is StableSubgroup of H2 "\/" H3
proof
let O be set ; ::_thesis: for G being GroupWithOperators of O
for H1 being StableSubgroup of G
for H2, H3 being strict StableSubgroup of G st H1 is StableSubgroup of H2 holds
H1 "\/" H3 is StableSubgroup of H2 "\/" H3
let G be GroupWithOperators of O; ::_thesis: for H1 being StableSubgroup of G
for H2, H3 being strict StableSubgroup of G st H1 is StableSubgroup of H2 holds
H1 "\/" H3 is StableSubgroup of H2 "\/" H3
let H1 be StableSubgroup of G; ::_thesis: for H2, H3 being strict StableSubgroup of G st H1 is StableSubgroup of H2 holds
H1 "\/" H3 is StableSubgroup of H2 "\/" H3
let H2, H3 be strict StableSubgroup of G; ::_thesis: ( H1 is StableSubgroup of H2 implies H1 "\/" H3 is StableSubgroup of H2 "\/" H3 )
assume H1 is StableSubgroup of H2 ; ::_thesis: H1 "\/" H3 is StableSubgroup of H2 "\/" H3
then H1 is Subgroup of H2 by Def7;
then carr H1 c= carr H2 by GROUP_2:def_5;
hence H1 "\/" H3 is StableSubgroup of H2 "\/" H3 by Th26, XBOOLE_1:9; ::_thesis: verum
end;
theorem Th39: :: GROUP_9:39
for O being set
for G being GroupWithOperators of O
for H1 being StableSubgroup of G
for X, Y being StableSubgroup of H1
for X9, Y9 being StableSubgroup of G st X = X9 & Y = Y9 holds
X9 /\ Y9 = X /\ Y
proof
let O be set ; ::_thesis: for G being GroupWithOperators of O
for H1 being StableSubgroup of G
for X, Y being StableSubgroup of H1
for X9, Y9 being StableSubgroup of G st X = X9 & Y = Y9 holds
X9 /\ Y9 = X /\ Y
let G be GroupWithOperators of O; ::_thesis: for H1 being StableSubgroup of G
for X, Y being StableSubgroup of H1
for X9, Y9 being StableSubgroup of G st X = X9 & Y = Y9 holds
X9 /\ Y9 = X /\ Y
let H1 be StableSubgroup of G; ::_thesis: for X, Y being StableSubgroup of H1
for X9, Y9 being StableSubgroup of G st X = X9 & Y = Y9 holds
X9 /\ Y9 = X /\ Y
let X, Y be StableSubgroup of H1; ::_thesis: for X9, Y9 being StableSubgroup of G st X = X9 & Y = Y9 holds
X9 /\ Y9 = X /\ Y
reconsider Z = X /\ Y as StableSubgroup of G by Th11;
let X9, Y9 be StableSubgroup of G; ::_thesis: ( X = X9 & Y = Y9 implies X9 /\ Y9 = X /\ Y )
assume A1: ( X = X9 & Y = Y9 ) ; ::_thesis: X9 /\ Y9 = X /\ Y
the carrier of (X /\ Y) = (carr X) /\ (carr Y) by Def25;
then X9 /\ Y9 = Z by A1, Th18;
hence X9 /\ Y9 = X /\ Y ; ::_thesis: verum
end;
theorem Th40: :: GROUP_9:40
for O being set
for G being GroupWithOperators of O
for N being normal StableSubgroup of G
for H1 being StableSubgroup of G st N is StableSubgroup of H1 holds
N is normal StableSubgroup of H1
proof
let O be set ; ::_thesis: for G being GroupWithOperators of O
for N being normal StableSubgroup of G
for H1 being StableSubgroup of G st N is StableSubgroup of H1 holds
N is normal StableSubgroup of H1
let G be GroupWithOperators of O; ::_thesis: for N being normal StableSubgroup of G
for H1 being StableSubgroup of G st N is StableSubgroup of H1 holds
N is normal StableSubgroup of H1
let N be normal StableSubgroup of G; ::_thesis: for H1 being StableSubgroup of G st N is StableSubgroup of H1 holds
N is normal StableSubgroup of H1
let H1 be StableSubgroup of G; ::_thesis: ( N is StableSubgroup of H1 implies N is normal StableSubgroup of H1 )
assume N is StableSubgroup of H1 ; ::_thesis: N is normal StableSubgroup of H1
then reconsider N9 = N as StableSubgroup of H1 ;
now__::_thesis:_for_H_being_strict_Subgroup_of_H1_st_H_=_multMagma(#_the_carrier_of_N9,_the_multF_of_N9_#)_holds_
H_is_normal
reconsider N99 = multMagma(# the carrier of N, the multF of N #) as normal Subgroup of G by Lm7;
let H be strict Subgroup of H1; ::_thesis: ( H = multMagma(# the carrier of N9, the multF of N9 #) implies H is normal )
assume A1: H = multMagma(# the carrier of N9, the multF of N9 #) ; ::_thesis: H is normal
reconsider N = N as Subgroup of G by Def7;
( H1 is Subgroup of G & N99 is Subgroup of N ) by Def7, GROUP_2:57;
hence H is normal by A1, GROUP_6:8; ::_thesis: verum
end;
hence N is normal StableSubgroup of H1 by Def10; ::_thesis: verum
end;
Lm34: for O being set
for G being GroupWithOperators of O
for H1, H2 being StableSubgroup of G holds H1 /\ H2 is StableSubgroup of H1
proof
let O be set ; ::_thesis: for G being GroupWithOperators of O
for H1, H2 being StableSubgroup of G holds H1 /\ H2 is StableSubgroup of H1
let G be GroupWithOperators of O; ::_thesis: for H1, H2 being StableSubgroup of G holds H1 /\ H2 is StableSubgroup of H1
let H1, H2 be StableSubgroup of G; ::_thesis: H1 /\ H2 is StableSubgroup of H1
the carrier of (H1 /\ H2) = the carrier of H1 /\ the carrier of H2 by Th18;
hence H1 /\ H2 is StableSubgroup of H1 by Lm21, XBOOLE_1:17; ::_thesis: verum
end;
theorem Th41: :: GROUP_9:41
for O being set
for G being GroupWithOperators of O
for N being normal StableSubgroup of G
for H1 being StableSubgroup of G holds
( H1 /\ N is normal StableSubgroup of H1 & N /\ H1 is normal StableSubgroup of H1 )
proof
let O be set ; ::_thesis: for G being GroupWithOperators of O
for N being normal StableSubgroup of G
for H1 being StableSubgroup of G holds
( H1 /\ N is normal StableSubgroup of H1 & N /\ H1 is normal StableSubgroup of H1 )
let G be GroupWithOperators of O; ::_thesis: for N being normal StableSubgroup of G
for H1 being StableSubgroup of G holds
( H1 /\ N is normal StableSubgroup of H1 & N /\ H1 is normal StableSubgroup of H1 )
let N be normal StableSubgroup of G; ::_thesis: for H1 being StableSubgroup of G holds
( H1 /\ N is normal StableSubgroup of H1 & N /\ H1 is normal StableSubgroup of H1 )
let H1 be StableSubgroup of G; ::_thesis: ( H1 /\ N is normal StableSubgroup of H1 & N /\ H1 is normal StableSubgroup of H1 )
thus H1 /\ N is normal StableSubgroup of H1 ::_thesis: N /\ H1 is normal StableSubgroup of H1
proof
reconsider A = H1 /\ N as StableSubgroup of H1 by Lm34;
now__::_thesis:_for_H_being_strict_Subgroup_of_H1_st_H_=_multMagma(#_the_carrier_of_A,_the_multF_of_A_#)_holds_
H_is_normal
reconsider N9 = multMagma(# the carrier of N, the multF of N #) as normal Subgroup of G by Lm7;
let H be strict Subgroup of H1; ::_thesis: ( H = multMagma(# the carrier of A, the multF of A #) implies H is normal )
assume A1: H = multMagma(# the carrier of A, the multF of A #) ; ::_thesis: H is normal
now__::_thesis:_for_b_being_Element_of_H1_holds_b_*_H_c=_H_*_b
let b be Element of H1; ::_thesis: b * H c= H * b
thus b * H c= H * b ::_thesis: verum
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in b * H or x in H * b )
assume x in b * H ; ::_thesis: x in H * b
then consider a being Element of H1 such that
A2: x = b * a and
A3: a in H by GROUP_2:103;
reconsider a9 = a, b9 = b as Element of G by Th2;
reconsider x9 = x as Element of H1 by A2;
A4: b9 " = b " by Th6;
a in the carrier of A by A1, A3, STRUCT_0:def_5;
then a in (carr H1) /\ (carr N) by Def25;
then a in carr N9 by XBOOLE_0:def_4;
then A5: a in N9 by STRUCT_0:def_5;
x = b9 * a9 by A2, Th3;
then A6: x in b9 * N9 by A5, GROUP_2:103;
b9 * N9 c= N9 * b9 by GROUP_3:118;
then consider b1 being Element of G such that
A7: x = b1 * b9 and
A8: b1 in N9 by A6, GROUP_2:104;
reconsider x99 = x as Element of G by A7;
b1 = x99 * (b9 ") by A7, GROUP_1:14;
then A9: b1 = x9 * (b ") by A4, Th3;
then reconsider b19 = b1 as Element of H1 ;
b1 in the carrier of N by A8, STRUCT_0:def_5;
then b1 in (carr H1) /\ (carr N) by A9, XBOOLE_0:def_4;
then b19 in the carrier of A by Def25;
then A10: b19 in H by A1, STRUCT_0:def_5;
b19 * b = x by A7, Th3;
hence x in H * b by A10, GROUP_2:104; ::_thesis: verum
end;
end;
hence H is normal by GROUP_3:118; ::_thesis: verum
end;
hence H1 /\ N is normal StableSubgroup of H1 by Def10; ::_thesis: verum
end;
hence N /\ H1 is normal StableSubgroup of H1 ; ::_thesis: verum
end;
theorem Th42: :: GROUP_9:42
for O being set
for G being strict GroupWithOperators of O st G is trivial holds
(1). G = G
proof
let O be set ; ::_thesis: for G being strict GroupWithOperators of O st G is trivial holds
(1). G = G
let G be strict GroupWithOperators of O; ::_thesis: ( G is trivial implies (1). G = G )
reconsider H = G as StableSubgroup of G by Lm4;
assume G is trivial ; ::_thesis: (1). G = G
then ex x being set st the carrier of G = {x} by ZFMISC_1:131;
then the carrier of H = {(1_ G)} by TARSKI:def_1;
hence (1). G = G by Def8; ::_thesis: verum
end;
Lm35: for O being set
for G being GroupWithOperators of O
for N being normal StableSubgroup of G
for N9 being normal Subgroup of G st N9 = multMagma(# the carrier of N, the multF of N #) holds
( G ./. N9 = multMagma(# the carrier of (G ./. N), the multF of (G ./. N) #) & 1_ (G ./. N9) = 1_ (G ./. N) )
proof
let O be set ; ::_thesis: for G being GroupWithOperators of O
for N being normal StableSubgroup of G
for N9 being normal Subgroup of G st N9 = multMagma(# the carrier of N, the multF of N #) holds
( G ./. N9 = multMagma(# the carrier of (G ./. N), the multF of (G ./. N) #) & 1_ (G ./. N9) = 1_ (G ./. N) )
let G be GroupWithOperators of O; ::_thesis: for N being normal StableSubgroup of G
for N9 being normal Subgroup of G st N9 = multMagma(# the carrier of N, the multF of N #) holds
( G ./. N9 = multMagma(# the carrier of (G ./. N), the multF of (G ./. N) #) & 1_ (G ./. N9) = 1_ (G ./. N) )
let N be normal StableSubgroup of G; ::_thesis: for N9 being normal Subgroup of G st N9 = multMagma(# the carrier of N, the multF of N #) holds
( G ./. N9 = multMagma(# the carrier of (G ./. N), the multF of (G ./. N) #) & 1_ (G ./. N9) = 1_ (G ./. N) )
let N9 be normal Subgroup of G; ::_thesis: ( N9 = multMagma(# the carrier of N, the multF of N #) implies ( G ./. N9 = multMagma(# the carrier of (G ./. N), the multF of (G ./. N) #) & 1_ (G ./. N9) = 1_ (G ./. N) ) )
assume A1: N9 = multMagma(# the carrier of N, the multF of N #) ; ::_thesis: ( G ./. N9 = multMagma(# the carrier of (G ./. N), the multF of (G ./. N) #) & 1_ (G ./. N9) = 1_ (G ./. N) )
then reconsider e = 1_ (G ./. N9) as Element of (G ./. N) by Def14;
Cosets N9 = Cosets N by A1, Def14;
hence G ./. N9 = multMagma(# the carrier of (G ./. N), the multF of (G ./. N) #) by A1, Def15; ::_thesis: 1_ (G ./. N9) = 1_ (G ./. N)
now__::_thesis:_for_h_being_Element_of_(G_./._N)_holds_
(_h_*_e_=_h_&_e_*_h_=_h_)
let h be Element of (G ./. N); ::_thesis: ( h * e = h & e * h = h )
reconsider h9 = h as Element of (G ./. N9) by A1, Def14;
thus h * e = h9 * (1_ (G ./. N9)) by A1, Def15
.= h by GROUP_1:def_4 ; ::_thesis: e * h = h
thus e * h = (1_ (G ./. N9)) * h9 by A1, Def15
.= h by GROUP_1:def_4 ; ::_thesis: verum
end;
hence 1_ (G ./. N9) = 1_ (G ./. N) by GROUP_1:4; ::_thesis: verum
end;
theorem Th43: :: GROUP_9:43
for O being set
for G being GroupWithOperators of O
for N being normal StableSubgroup of G holds 1_ (G ./. N) = carr N
proof
let O be set ; ::_thesis: for G being GroupWithOperators of O
for N being normal StableSubgroup of G holds 1_ (G ./. N) = carr N
let G be GroupWithOperators of O; ::_thesis: for N being normal StableSubgroup of G holds 1_ (G ./. N) = carr N
let N be normal StableSubgroup of G; ::_thesis: 1_ (G ./. N) = carr N
reconsider N9 = multMagma(# the carrier of N, the multF of N #) as normal Subgroup of G by Lm7;
1_ (G ./. N9) = carr N9 by GROUP_6:24;
hence 1_ (G ./. N) = carr N by Lm35; ::_thesis: verum
end;
theorem Th44: :: GROUP_9:44
for O being set
for G being GroupWithOperators of O
for M, N being strict normal StableSubgroup of G
for MN being normal StableSubgroup of N st MN = M & M is StableSubgroup of N holds
N ./. MN is normal StableSubgroup of G ./. M
proof
let O be set ; ::_thesis: for G being GroupWithOperators of O
for M, N being strict normal StableSubgroup of G
for MN being normal StableSubgroup of N st MN = M & M is StableSubgroup of N holds
N ./. MN is normal StableSubgroup of G ./. M
let G be GroupWithOperators of O; ::_thesis: for M, N being strict normal StableSubgroup of G
for MN being normal StableSubgroup of N st MN = M & M is StableSubgroup of N holds
N ./. MN is normal StableSubgroup of G ./. M
let M, N be strict normal StableSubgroup of G; ::_thesis: for MN being normal StableSubgroup of N st MN = M & M is StableSubgroup of N holds
N ./. MN is normal StableSubgroup of G ./. M
reconsider M9 = multMagma(# the carrier of M, the multF of M #) as normal Subgroup of G by Lm7;
reconsider N9 = multMagma(# the carrier of N, the multF of N #) as normal Subgroup of G by Lm7;
let MN be normal StableSubgroup of N; ::_thesis: ( MN = M & M is StableSubgroup of N implies N ./. MN is normal StableSubgroup of G ./. M )
assume A1: MN = M ; ::_thesis: ( not M is StableSubgroup of N or N ./. MN is normal StableSubgroup of G ./. M )
reconsider MN99 = (N9,M9) `*` as normal Subgroup of N9 ;
reconsider MN9 = multMagma(# the carrier of MN, the multF of MN #) as normal Subgroup of N by Lm7;
assume M is StableSubgroup of N ; ::_thesis: N ./. MN is normal StableSubgroup of G ./. M
then M is Subgroup of N by Def7;
then ( the carrier of M c= the carrier of N & the multF of M = the multF of N || the carrier of M ) by GROUP_2:def_5;
then A2: M9 is Subgroup of N9 by GROUP_2:def_5;
then A3: (N9,M9) `*` = MN9 by A1, GROUP_6:def_1;
reconsider K = N9 ./. ((N9,M9) `*`) as normal Subgroup of G ./. M9 by A2, GROUP_6:29;
A4: now__::_thesis:_for_x_being_set_holds_
(_(_x_in_Cosets_MN9_implies_x_in_Cosets_MN99_)_&_(_x_in_Cosets_MN99_implies_x_in_Cosets_MN9_)_)
let x be set ; ::_thesis: ( ( x in Cosets MN9 implies x in Cosets MN99 ) & ( x in Cosets MN99 implies x in Cosets MN9 ) )
hereby ::_thesis: ( x in Cosets MN99 implies x in Cosets MN9 )
assume x in Cosets MN9 ; ::_thesis: x in Cosets MN99
then consider a being Element of N such that
A5: x = a * MN9 and
x = MN9 * a by GROUP_6:13;
reconsider a9 = a as Element of N9 ;
reconsider A = {a} as Subset of N by ZFMISC_1:31;
reconsider A9 = {a9} as Subset of N9 by ZFMISC_1:31;
now__::_thesis:_for_y_being_set_holds_
(_(_y_in__{__(g_*_h)_where_g,_h_is_Element_of_N_:_(_g_in_A_&_h_in_carr_MN9_)__}__implies_y_in__{__(g99_*_h99)_where_g99,_h99_is_Element_of_N9_:_(_g99_in_A9_&_h99_in_carr_MN99_)__}__)_&_(_y_in__{__(g_*_h)_where_g,_h_is_Element_of_N9_:_(_g_in_A9_&_h_in_carr_MN99_)__}__implies_y_in__{__(g99_*_h99)_where_g99,_h99_is_Element_of_N_:_(_g99_in_A_&_h99_in_carr_MN9_)__}__)_)
let y be set ; ::_thesis: ( ( y in { (g * h) where g, h is Element of N : ( g in A & h in carr MN9 ) } implies y in { (g99 * h99) where g99, h99 is Element of N9 : ( g99 in A9 & h99 in carr MN99 ) } ) & ( y in { (g * h) where g, h is Element of N9 : ( g in A9 & h in carr MN99 ) } implies y in { (g99 * h99) where g99, h99 is Element of N : ( g99 in A & h99 in carr MN9 ) } ) )
hereby ::_thesis: ( y in { (g * h) where g, h is Element of N9 : ( g in A9 & h in carr MN99 ) } implies y in { (g99 * h99) where g99, h99 is Element of N : ( g99 in A & h99 in carr MN9 ) } )
assume y in { (g * h) where g, h is Element of N : ( g in A & h in carr MN9 ) } ; ::_thesis: y in { (g99 * h99) where g99, h99 is Element of N9 : ( g99 in A9 & h99 in carr MN99 ) }
then consider g, h being Element of N such that
A6: y = g * h and
A7: ( g in A & h in carr MN9 ) ;
reconsider h9 = h as Element of N9 ;
reconsider g9 = g as Element of N9 ;
y = g9 * h9 by A6;
hence y in { (g99 * h99) where g99, h99 is Element of N9 : ( g99 in A9 & h99 in carr MN99 ) } by A3, A7; ::_thesis: verum
end;
assume y in { (g * h) where g, h is Element of N9 : ( g in A9 & h in carr MN99 ) } ; ::_thesis: y in { (g99 * h99) where g99, h99 is Element of N : ( g99 in A & h99 in carr MN9 ) }
then consider g, h being Element of N9 such that
A8: y = g * h and
A9: ( g in A9 & h in carr MN99 ) ;
reconsider h9 = h as Element of N ;
reconsider g9 = g as Element of N ;
y = g9 * h9 by A8;
hence y in { (g99 * h99) where g99, h99 is Element of N : ( g99 in A & h99 in carr MN9 ) } by A3, A9; ::_thesis: verum
end;
then x = a9 * MN99 by A5, TARSKI:1;
hence x in Cosets MN99 by GROUP_6:14; ::_thesis: verum
end;
assume x in Cosets MN99 ; ::_thesis: x in Cosets MN9
then consider a9 being Element of N9 such that
A10: x = a9 * MN99 and
x = MN99 * a9 by GROUP_6:13;
reconsider a = a9 as Element of N ;
reconsider A9 = {a9} as Subset of N9 by ZFMISC_1:31;
reconsider A = {a} as Subset of N by ZFMISC_1:31;
now__::_thesis:_for_y_being_set_holds_
(_(_y_in__{__(g_*_h)_where_g,_h_is_Element_of_N_:_(_g_in_A_&_h_in_carr_MN9_)__}__implies_y_in__{__(g99_*_h99)_where_g99,_h99_is_Element_of_N9_:_(_g99_in_A9_&_h99_in_carr_MN99_)__}__)_&_(_y_in__{__(g_*_h)_where_g,_h_is_Element_of_N9_:_(_g_in_A9_&_h_in_carr_MN99_)__}__implies_y_in__{__(g99_*_h99)_where_g99,_h99_is_Element_of_N_:_(_g99_in_A_&_h99_in_carr_MN9_)__}__)_)
let y be set ; ::_thesis: ( ( y in { (g * h) where g, h is Element of N : ( g in A & h in carr MN9 ) } implies y in { (g99 * h99) where g99, h99 is Element of N9 : ( g99 in A9 & h99 in carr MN99 ) } ) & ( y in { (g * h) where g, h is Element of N9 : ( g in A9 & h in carr MN99 ) } implies y in { (g99 * h99) where g99, h99 is Element of N : ( g99 in A & h99 in carr MN9 ) } ) )
hereby ::_thesis: ( y in { (g * h) where g, h is Element of N9 : ( g in A9 & h in carr MN99 ) } implies y in { (g99 * h99) where g99, h99 is Element of N : ( g99 in A & h99 in carr MN9 ) } )
assume y in { (g * h) where g, h is Element of N : ( g in A & h in carr MN9 ) } ; ::_thesis: y in { (g99 * h99) where g99, h99 is Element of N9 : ( g99 in A9 & h99 in carr MN99 ) }
then consider g, h being Element of N such that
A11: y = g * h and
A12: ( g in A & h in carr MN9 ) ;
reconsider h9 = h as Element of N9 ;
reconsider g9 = g as Element of N9 ;
y = g9 * h9 by A11;
hence y in { (g99 * h99) where g99, h99 is Element of N9 : ( g99 in A9 & h99 in carr MN99 ) } by A3, A12; ::_thesis: verum
end;
assume y in { (g * h) where g, h is Element of N9 : ( g in A9 & h in carr MN99 ) } ; ::_thesis: y in { (g99 * h99) where g99, h99 is Element of N : ( g99 in A & h99 in carr MN9 ) }
then consider g, h being Element of N9 such that
A13: y = g * h and
A14: ( g in A9 & h in carr MN99 ) ;
reconsider h9 = h as Element of N ;
reconsider g9 = g as Element of N ;
y = g9 * h9 by A13;
hence y in { (g99 * h99) where g99, h99 is Element of N : ( g99 in A & h99 in carr MN9 ) } by A3, A14; ::_thesis: verum
end;
then x = a * MN9 by A10, TARSKI:1;
hence x in Cosets MN9 by GROUP_6:14; ::_thesis: verum
end;
then A15: the carrier of K = Cosets MN9 by TARSKI:1
.= the carrier of (N ./. MN) by Def14 ;
A16: now__::_thesis:_for_H_being_strict_Subgroup_of_G_./._M_st_H_=_multMagma(#_the_carrier_of_(N_./._MN),_the_multF_of_(N_./._MN)_#)_holds_
H_is_normal
let H be strict Subgroup of G ./. M; ::_thesis: ( H = multMagma(# the carrier of (N ./. MN), the multF of (N ./. MN) #) implies H is normal )
assume A17: H = multMagma(# the carrier of (N ./. MN), the multF of (N ./. MN) #) ; ::_thesis: H is normal
now__::_thesis:_for_a_being_Element_of_(G_./._M)_holds_a_*_H_c=_H_*_a
let a be Element of (G ./. M); ::_thesis: a * H c= H * a
reconsider a9 = a as Element of (G ./. M9) by Def14;
now__::_thesis:_for_x_being_set_st_x_in_a_*_(carr_H)_holds_
x_in_(carr_H)_*_a
let x be set ; ::_thesis: ( x in a * (carr H) implies x in (carr H) * a )
assume x in a * (carr H) ; ::_thesis: x in (carr H) * a
then consider b being Element of (G ./. M) such that
A18: x = a * b and
A19: b in carr H by GROUP_2:27;
reconsider b9 = b as Element of (G ./. M9) by Def14;
A20: x = a9 * b9 by A18, Def15;
then reconsider x9 = x as Element of (G ./. M9) ;
( a9 * K c= K * a9 & x9 in a9 * (carr K) ) by A15, A17, A19, A20, GROUP_2:27, GROUP_3:118;
then consider c9 being Element of (G ./. M9) such that
A21: x9 = c9 * a9 and
A22: c9 in carr K by GROUP_2:28;
reconsider c = c9 as Element of (G ./. M) by Def14;
x = c * a by A21, Def15;
hence x in (carr H) * a by A15, A17, A22, GROUP_2:28; ::_thesis: verum
end;
hence a * H c= H * a by TARSKI:def_3; ::_thesis: verum
end;
hence H is normal by GROUP_3:118; ::_thesis: verum
end;
A23: the carrier of (G ./. M) = the carrier of (G ./. M9) by Def14;
then A24: the carrier of (N ./. MN) c= the carrier of (G ./. M) by A15, GROUP_2:def_5;
A25: now__::_thesis:_for_o_being_Element_of_O_holds_(N_./._MN)_^_o_=_((G_./._M)_^_o)_|_the_carrier_of_(N_./._MN)
let o be Element of O; ::_thesis: (N ./. MN) ^ b1 = ((G ./. M) ^ b1) | the carrier of (N ./. MN)
percases ( not o in O or o in O ) ;
supposeA26: not o in O ; ::_thesis: (N ./. MN) ^ b1 = ((G ./. M) ^ b1) | the carrier of (N ./. MN)
A27: the carrier of (N ./. MN) c= the carrier of (G ./. M) by A23, A15, GROUP_2:def_5;
A28: now__::_thesis:_for_x,_y_being_set_st_[x,y]_in_id_the_carrier_of_(N_./._MN)_holds_
[x,y]_in_(id_the_carrier_of_(G_./._M))_|_the_carrier_of_(N_./._MN)
let x, y be set ; ::_thesis: ( [x,y] in id the carrier of (N ./. MN) implies [x,y] in (id the carrier of (G ./. M)) | the carrier of (N ./. MN) )
assume A29: [x,y] in id the carrier of (N ./. MN) ; ::_thesis: [x,y] in (id the carrier of (G ./. M)) | the carrier of (N ./. MN)
then A30: x in the carrier of (N ./. MN) by RELAT_1:def_10;
x = y by A29, RELAT_1:def_10;
then [x,y] in id the carrier of (G ./. M) by A27, A30, RELAT_1:def_10;
hence [x,y] in (id the carrier of (G ./. M)) | the carrier of (N ./. MN) by A30, RELAT_1:def_11; ::_thesis: verum
end;
A31: now__::_thesis:_for_x,_y_being_set_st_[x,y]_in_(id_the_carrier_of_(G_./._M))_|_the_carrier_of_(N_./._MN)_holds_
[x,y]_in_id_the_carrier_of_(N_./._MN)
let x, y be set ; ::_thesis: ( [x,y] in (id the carrier of (G ./. M)) | the carrier of (N ./. MN) implies [x,y] in id the carrier of (N ./. MN) )
assume A32: [x,y] in (id the carrier of (G ./. M)) | the carrier of (N ./. MN) ; ::_thesis: [x,y] in id the carrier of (N ./. MN)
then [x,y] in id the carrier of (G ./. M) by RELAT_1:def_11;
then A33: x = y by RELAT_1:def_10;
x in the carrier of (N ./. MN) by A32, RELAT_1:def_11;
hence [x,y] in id the carrier of (N ./. MN) by A33, RELAT_1:def_10; ::_thesis: verum
end;
thus (N ./. MN) ^ o = id the carrier of (N ./. MN) by A26, Def6
.= (id the carrier of (G ./. M)) | the carrier of (N ./. MN) by A28, A31, RELAT_1:def_2
.= ((G ./. M) ^ o) | the carrier of (N ./. MN) by A26, Def6 ; ::_thesis: verum
end;
supposeA34: o in O ; ::_thesis: (N ./. MN) ^ b1 = ((G ./. M) ^ b1) | the carrier of (N ./. MN)
then the action of (G ./. M) . o in Funcs ( the carrier of (G ./. M), the carrier of (G ./. M)) by FUNCT_2:5;
then consider f being Function such that
A35: f = the action of (G ./. M) . o and
A36: dom f = the carrier of (G ./. M) and
rng f c= the carrier of (G ./. M) by FUNCT_2:def_2;
A37: f = { [A,B] where A, B is Element of Cosets M : ex a, b being Element of G st
( a in A & b in B & b = (G ^ o) . a ) } by A34, A35, Def16;
the action of (N ./. MN) . o in Funcs ( the carrier of (N ./. MN), the carrier of (N ./. MN)) by A34, FUNCT_2:5;
then consider g being Function such that
A38: g = the action of (N ./. MN) . o and
A39: dom g = the carrier of (N ./. MN) and
rng g c= the carrier of (N ./. MN) by FUNCT_2:def_2;
A40: dom g = (dom f) /\ the carrier of (N ./. MN) by A24, A36, A39, XBOOLE_1:28;
A41: g = { [A,B] where A, B is Element of Cosets MN : ex a, b being Element of N st
( a in A & b in B & b = (N ^ o) . a ) } by A34, A38, Def16;
A42: now__::_thesis:_for_x_being_set_st_x_in_dom_g_holds_
g_._x_=_f_._x
let x be set ; ::_thesis: ( x in dom g implies g . x = f . x )
assume A43: x in dom g ; ::_thesis: g . x = f . x
then [x,(g . x)] in g by FUNCT_1:1;
then consider A2, B2 being Element of Cosets MN such that
A44: [x,(g . x)] = [A2,B2] and
A45: ex a, b being Element of N st
( a in A2 & b in B2 & b = (N ^ o) . a ) by A41;
A46: A2 = x by A44, XTUPLE_0:1;
[x,(f . x)] in f by A24, A36, A39, A43, FUNCT_1:1;
then consider A1, B1 being Element of Cosets M such that
A47: [x,(f . x)] = [A1,B1] and
A48: ex a, b being Element of G st
( a in A1 & b in B1 & b = (G ^ o) . a ) by A37;
A49: A1 = x by A47, XTUPLE_0:1;
reconsider A29 = A2, B29 = B2 as Element of Cosets MN9 by Def14;
reconsider A19 = A1, B19 = B1 as Element of Cosets M9 by Def14;
set fo = G ^ o;
N is Subgroup of G by Def7;
then A50: the carrier of N c= the carrier of G by GROUP_2:def_5;
consider a2, b2 being Element of N such that
A51: a2 in A2 and
A52: b2 in B2 and
A53: b2 = (N ^ o) . a2 by A45;
A54: B29 = b2 * MN9 by A52, Lm9;
( a2 in the carrier of N & b2 in the carrier of N ) ;
then reconsider a29 = a2, b29 = b2 as Element of G by A50;
consider a1, b1 being Element of G such that
A55: a1 in A1 and
A56: b1 in B1 and
A57: b1 = (G ^ o) . a1 by A48;
A58: A19 = a1 * M9 by A55, Lm9;
now__::_thesis:_for_x_being_set_holds_
(_(_x_in_b2_*_(carr_MN9)_implies_x_in_b29_*_(carr_M9)_)_&_(_x_in_b29_*_(carr_M9)_implies_x_in_b2_*_(carr_MN9)_)_)
let x be set ; ::_thesis: ( ( x in b2 * (carr MN9) implies x in b29 * (carr M9) ) & ( x in b29 * (carr M9) implies x in b2 * (carr MN9) ) )
hereby ::_thesis: ( x in b29 * (carr M9) implies x in b2 * (carr MN9) )
assume x in b2 * (carr MN9) ; ::_thesis: x in b29 * (carr M9)
then consider h being Element of N such that
A59: x = b2 * h and
A60: h in carr MN9 by GROUP_2:27;
h in the carrier of N ;
then reconsider h9 = h as Element of G by A50;
x = b29 * h9 by A59, Th3;
hence x in b29 * (carr M9) by A1, A60, GROUP_2:27; ::_thesis: verum
end;
assume x in b29 * (carr M9) ; ::_thesis: x in b2 * (carr MN9)
then consider h being Element of G such that
A61: x = b29 * h and
A62: h in carr M9 by GROUP_2:27;
h in carr MN9 by A1, A62;
then reconsider h9 = h as Element of N ;
x = b2 * h9 by A61, Th3;
hence x in b2 * (carr MN9) by A1, A62, GROUP_2:27; ::_thesis: verum
end;
then A63: b29 * M9 = b2 * MN9 by TARSKI:1;
A64: B2 = g . x by A44, XTUPLE_0:1;
A65: B1 = f . x by A47, XTUPLE_0:1;
now__::_thesis:_for_x_being_set_holds_
(_(_x_in_a2_*_(carr_MN9)_implies_x_in_a29_*_(carr_M9)_)_&_(_x_in_a29_*_(carr_M9)_implies_x_in_a2_*_(carr_MN9)_)_)
let x be set ; ::_thesis: ( ( x in a2 * (carr MN9) implies x in a29 * (carr M9) ) & ( x in a29 * (carr M9) implies x in a2 * (carr MN9) ) )
hereby ::_thesis: ( x in a29 * (carr M9) implies x in a2 * (carr MN9) )
assume x in a2 * (carr MN9) ; ::_thesis: x in a29 * (carr M9)
then consider h being Element of N such that
A66: x = a2 * h and
A67: h in carr MN9 by GROUP_2:27;
h in the carrier of N ;
then reconsider h9 = h as Element of G by A50;
x = a29 * h9 by A66, Th3;
hence x in a29 * (carr M9) by A1, A67, GROUP_2:27; ::_thesis: verum
end;
assume x in a29 * (carr M9) ; ::_thesis: x in a2 * (carr MN9)
then consider h being Element of G such that
A68: x = a29 * h and
A69: h in carr M9 by GROUP_2:27;
h in carr MN9 by A1, A69;
then reconsider h9 = h as Element of N ;
x = a2 * h9 by A68, Th3;
hence x in a2 * (carr MN9) by A1, A69, GROUP_2:27; ::_thesis: verum
end;
then A70: a2 * MN9 = a29 * M9 by TARSKI:1;
A29 = a2 * MN9 by A51, Lm9;
then (a1 ") * a29 in M9 by A49, A46, A58, A70, GROUP_2:114;
then (a1 ") * a29 in the carrier of M by STRUCT_0:def_5;
then (a1 ") * a29 in M by STRUCT_0:def_5;
then A71: (G ^ o) . ((a1 ") * a29) in M by Lm10;
A72: b1 " = (G ^ o) . (a1 ") by A57, GROUP_6:32;
b29 = ((G ^ o) | the carrier of N) . a2 by A53, Def7
.= (G ^ o) . a29 by FUNCT_1:49 ;
then (b1 ") * b29 in M by A72, A71, GROUP_6:def_6;
then (b1 ") * b29 in the carrier of M by STRUCT_0:def_5;
then A73: (b1 ") * b29 in M9 by STRUCT_0:def_5;
B19 = b1 * M9 by A56, Lm9;
hence g . x = f . x by A65, A64, A63, A73, A54, GROUP_2:114; ::_thesis: verum
end;
thus (N ./. MN) ^ o = the action of (N ./. MN) . o by A34, Def6
.= f | the carrier of (N ./. MN) by A38, A40, A42, FUNCT_1:46
.= ((G ./. M) ^ o) | the carrier of (N ./. MN) by A34, A35, Def6 ; ::_thesis: verum
end;
end;
end;
Cosets MN99 = Cosets MN9 by A4, TARSKI:1;
then reconsider f = CosOp MN99 as BinOp of (Cosets MN9) ;
now__::_thesis:_for_W1,_W2_being_Element_of_Cosets_MN9
for_A1,_A2_being_Subset_of_N_st_W1_=_A1_&_W2_=_A2_holds_
f_._(W1,W2)_=_A1_*_A2
let W1, W2 be Element of Cosets MN9; ::_thesis: for A1, A2 being Subset of N st W1 = A1 & W2 = A2 holds
f . (W1,W2) = A1 * A2
reconsider W19 = W1, W29 = W2 as Element of Cosets MN99 by A4;
let A1, A2 be Subset of N; ::_thesis: ( W1 = A1 & W2 = A2 implies f . (W1,W2) = A1 * A2 )
assume A74: W1 = A1 ; ::_thesis: ( W2 = A2 implies f . (W1,W2) = A1 * A2 )
reconsider A19 = A1, A29 = A2 as Subset of N9 ;
assume A75: W2 = A2 ; ::_thesis: f . (W1,W2) = A1 * A2
A76: now__::_thesis:_for_x_being_set_holds_
(_(_x_in_A1_*_A2_implies_x_in_A19_*_A29_)_&_(_x_in_A19_*_A29_implies_x_in_A1_*_A2_)_)
let x be set ; ::_thesis: ( ( x in A1 * A2 implies x in A19 * A29 ) & ( x in A19 * A29 implies x in A1 * A2 ) )
hereby ::_thesis: ( x in A19 * A29 implies x in A1 * A2 )
assume x in A1 * A2 ; ::_thesis: x in A19 * A29
then consider g, h being Element of N such that
A77: x = g * h and
A78: ( g in A1 & h in A2 ) ;
reconsider g9 = g, h9 = h as Element of N9 ;
x = g9 * h9 by A77;
hence x in A19 * A29 by A78; ::_thesis: verum
end;
assume x in A19 * A29 ; ::_thesis: x in A1 * A2
then consider g9, h9 being Element of N9 such that
A79: x = g9 * h9 and
A80: ( g9 in A19 & h9 in A29 ) ;
reconsider g = g9, h = h9 as Element of N ;
x = g * h by A79;
hence x in A1 * A2 by A80; ::_thesis: verum
end;
thus f . (W1,W2) = f . (W19,W29)
.= A19 * A29 by A74, A75, GROUP_6:def_3
.= A1 * A2 by A76, TARSKI:1 ; ::_thesis: verum
end;
then the multF of K = CosOp MN9 by GROUP_6:def_3
.= the multF of (N ./. MN) by Def15 ;
then the multF of (N ./. MN) = the multF of (G ./. M9) || the carrier of K by GROUP_2:def_5
.= the multF of (G ./. M) || the carrier of (N ./. MN) by A15, Def15 ;
then N ./. MN is Subgroup of G ./. M by A24, GROUP_2:def_5;
hence N ./. MN is normal StableSubgroup of G ./. M by A16, A25, Def7, Def10; ::_thesis: verum
end;
theorem :: GROUP_9:45
for O being set
for G, H being GroupWithOperators of O
for h being Homomorphism of G,H holds h . (1_ G) = 1_ H by Lm13;
theorem :: GROUP_9:46
for O being set
for G, H being GroupWithOperators of O
for g1 being Element of G
for h being Homomorphism of G,H holds h . (g1 ") = (h . g1) " by Lm14;
theorem Th47: :: GROUP_9:47
for O being set
for G, H being GroupWithOperators of O
for g1 being Element of G
for h being Homomorphism of G,H holds
( g1 in Ker h iff h . g1 = 1_ H )
proof
let O be set ; ::_thesis: for G, H being GroupWithOperators of O
for g1 being Element of G
for h being Homomorphism of G,H holds
( g1 in Ker h iff h . g1 = 1_ H )
let G, H be GroupWithOperators of O; ::_thesis: for g1 being Element of G
for h being Homomorphism of G,H holds
( g1 in Ker h iff h . g1 = 1_ H )
let g1 be Element of G; ::_thesis: for h being Homomorphism of G,H holds
( g1 in Ker h iff h . g1 = 1_ H )
let h be Homomorphism of G,H; ::_thesis: ( g1 in Ker h iff h . g1 = 1_ H )
thus ( g1 in Ker h implies h . g1 = 1_ H ) ::_thesis: ( h . g1 = 1_ H implies g1 in Ker h )
proof
assume g1 in Ker h ; ::_thesis: h . g1 = 1_ H
then g1 in the carrier of (Ker h) by STRUCT_0:def_5;
then g1 in { b where b is Element of G : h . b = 1_ H } by Def21;
then ex b being Element of G st
( g1 = b & h . b = 1_ H ) ;
hence h . g1 = 1_ H ; ::_thesis: verum
end;
assume h . g1 = 1_ H ; ::_thesis: g1 in Ker h
then g1 in { b where b is Element of G : h . b = 1_ H } ;
then g1 in the carrier of (Ker h) by Def21;
hence g1 in Ker h by STRUCT_0:def_5; ::_thesis: verum
end;
theorem Th48: :: GROUP_9:48
for O being set
for G being GroupWithOperators of O
for N being strict normal StableSubgroup of G holds Ker (nat_hom N) = N
proof
let O be set ; ::_thesis: for G being GroupWithOperators of O
for N being strict normal StableSubgroup of G holds Ker (nat_hom N) = N
let G be GroupWithOperators of O; ::_thesis: for N being strict normal StableSubgroup of G holds Ker (nat_hom N) = N
let N be strict normal StableSubgroup of G; ::_thesis: Ker (nat_hom N) = N
reconsider N9 = multMagma(# the carrier of N, the multF of N #) as strict normal Subgroup of G by Lm7;
A1: ( nat_hom N = nat_hom N9 & 1_ (G ./. N) = 1_ (G ./. N9) ) by Def20, Lm35;
the carrier of (Ker (nat_hom N)) = { a where a is Element of G : (nat_hom N) . a = 1_ (G ./. N) } by Def21
.= { a where a is Element of G : (nat_hom N9) . a = 1_ (G ./. N9) } by A1
.= the carrier of (Ker (nat_hom N9)) by GROUP_6:def_9
.= the carrier of N by GROUP_6:43 ;
hence Ker (nat_hom N) = N by Lm5; ::_thesis: verum
end;
theorem Th49: :: GROUP_9:49
for O being set
for G, H being GroupWithOperators of O
for h being Homomorphism of G,H holds rng h = the carrier of (Image h)
proof
let O be set ; ::_thesis: for G, H being GroupWithOperators of O
for h being Homomorphism of G,H holds rng h = the carrier of (Image h)
let G, H be GroupWithOperators of O; ::_thesis: for h being Homomorphism of G,H holds rng h = the carrier of (Image h)
let h be Homomorphism of G,H; ::_thesis: rng h = the carrier of (Image h)
the carrier of (Image h) = h .: the carrier of G by Def22
.= h .: (dom h) by FUNCT_2:def_1
.= rng h by RELAT_1:113 ;
hence rng h = the carrier of (Image h) ; ::_thesis: verum
end;
theorem Th50: :: GROUP_9:50
for O being set
for G being GroupWithOperators of O
for N being normal StableSubgroup of G holds Image (nat_hom N) = G ./. N
proof
let O be set ; ::_thesis: for G being GroupWithOperators of O
for N being normal StableSubgroup of G holds Image (nat_hom N) = G ./. N
let G be GroupWithOperators of O; ::_thesis: for N being normal StableSubgroup of G holds Image (nat_hom N) = G ./. N
let N be normal StableSubgroup of G; ::_thesis: Image (nat_hom N) = G ./. N
reconsider N9 = multMagma(# the carrier of N, the multF of N #) as strict normal Subgroup of G by Lm7;
reconsider H = G ./. N as strict StableSubgroup of G ./. N by Lm4;
A1: G ./. N9 = multMagma(# the carrier of (G ./. N), the multF of (G ./. N) #) by Lm35;
the carrier of (Image (nat_hom N)) = (nat_hom N) .: the carrier of G by Def22
.= (nat_hom N9) .: the carrier of G by Def20
.= the carrier of (Image (nat_hom N9)) by GROUP_6:def_10
.= the carrier of H by A1, GROUP_6:48 ;
hence Image (nat_hom N) = G ./. N by Lm5; ::_thesis: verum
end;
theorem Th51: :: GROUP_9:51
for O being set
for G being GroupWithOperators of O
for H being strict GroupWithOperators of O
for h being Homomorphism of G,H holds
( h is onto iff Image h = H )
proof
let O be set ; ::_thesis: for G being GroupWithOperators of O
for H being strict GroupWithOperators of O
for h being Homomorphism of G,H holds
( h is onto iff Image h = H )
let G be GroupWithOperators of O; ::_thesis: for H being strict GroupWithOperators of O
for h being Homomorphism of G,H holds
( h is onto iff Image h = H )
let H be strict GroupWithOperators of O; ::_thesis: for h being Homomorphism of G,H holds
( h is onto iff Image h = H )
let h be Homomorphism of G,H; ::_thesis: ( h is onto iff Image h = H )
thus ( h is onto implies Image h = H ) ::_thesis: ( Image h = H implies h is onto )
proof
reconsider H9 = H as strict StableSubgroup of H by Lm4;
assume rng h = the carrier of H ; :: according to FUNCT_2:def_3 ::_thesis: Image h = H
then the carrier of H9 = the carrier of (Image h) by Th49;
hence Image h = H by Lm5; ::_thesis: verum
end;
assume A1: Image h = H ; ::_thesis: h is onto
the carrier of H c= rng h
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in the carrier of H or x in rng h )
assume x in the carrier of H ; ::_thesis: x in rng h
then x in h .: the carrier of G by A1, Def22;
then ex y being set st
( y in dom h & y in the carrier of G & h . y = x ) by FUNCT_1:def_6;
hence x in rng h by FUNCT_1:def_3; ::_thesis: verum
end;
then rng h = the carrier of H by XBOOLE_0:def_10;
hence h is onto by FUNCT_2:def_3; ::_thesis: verum
end;
theorem Th52: :: GROUP_9:52
for O being set
for G being GroupWithOperators of O
for H being strict GroupWithOperators of O
for h being Homomorphism of G,H st h is onto holds
for c being Element of H ex a being Element of G st h . a = c
proof
let O be set ; ::_thesis: for G being GroupWithOperators of O
for H being strict GroupWithOperators of O
for h being Homomorphism of G,H st h is onto holds
for c being Element of H ex a being Element of G st h . a = c
let G be GroupWithOperators of O; ::_thesis: for H being strict GroupWithOperators of O
for h being Homomorphism of G,H st h is onto holds
for c being Element of H ex a being Element of G st h . a = c
let H be strict GroupWithOperators of O; ::_thesis: for h being Homomorphism of G,H st h is onto holds
for c being Element of H ex a being Element of G st h . a = c
let h be Homomorphism of G,H; ::_thesis: ( h is onto implies for c being Element of H ex a being Element of G st h . a = c )
assume A1: h is onto ; ::_thesis: for c being Element of H ex a being Element of G st h . a = c
let c be Element of H; ::_thesis: ex a being Element of G st h . a = c
rng h = the carrier of H by A1, FUNCT_2:def_3;
then consider a being set such that
A2: a in dom h and
A3: c = h . a by FUNCT_1:def_3;
reconsider a = a as Element of G by A2;
take a ; ::_thesis: h . a = c
thus h . a = c by A3; ::_thesis: verum
end;
theorem Th53: :: GROUP_9:53
for O being set
for G being GroupWithOperators of O
for N being normal StableSubgroup of G holds nat_hom N is onto
proof
let O be set ; ::_thesis: for G being GroupWithOperators of O
for N being normal StableSubgroup of G holds nat_hom N is onto
let G be GroupWithOperators of O; ::_thesis: for N being normal StableSubgroup of G holds nat_hom N is onto
let N be normal StableSubgroup of G; ::_thesis: nat_hom N is onto
Image (nat_hom N) = G ./. N by Th50;
hence nat_hom N is onto by Th51; ::_thesis: verum
end;
theorem Th54: :: GROUP_9:54
for O being set
for G being GroupWithOperators of O holds nat_hom ((1). G) is bijective
proof
let O be set ; ::_thesis: for G being GroupWithOperators of O holds nat_hom ((1). G) is bijective
let G be GroupWithOperators of O; ::_thesis: nat_hom ((1). G) is bijective
reconsider H = multMagma(# the carrier of ((1). G), the multF of ((1). G) #) as strict normal Subgroup of G by Lm7;
set g = nat_hom ((1). G);
reconsider G9 = G as Group ;
A1: the carrier of H = {(1_ G9)} by Def8;
A2: ( nat_hom ((1). G9) is bijective & nat_hom ((1). G) is onto ) by Th53, GROUP_6:65;
nat_hom ((1). G) = nat_hom H by Def20
.= nat_hom ((1). G9) by A1, GROUP_2:def_7 ;
hence nat_hom ((1). G) is bijective by A2; ::_thesis: verum
end;
theorem Th55: :: GROUP_9:55
for O being set
for G, H, I being GroupWithOperators of O st G,H are_isomorphic & H,I are_isomorphic holds
G,I are_isomorphic
proof
let O be set ; ::_thesis: for G, H, I being GroupWithOperators of O st G,H are_isomorphic & H,I are_isomorphic holds
G,I are_isomorphic
let G, H, I be GroupWithOperators of O; ::_thesis: ( G,H are_isomorphic & H,I are_isomorphic implies G,I are_isomorphic )
assume that
A1: G,H are_isomorphic and
A2: H,I are_isomorphic ; ::_thesis: G,I are_isomorphic
consider g being Homomorphism of G,H such that
A3: g is bijective by A1, Def19;
consider h1 being Homomorphism of H,I such that
A4: h1 is bijective by A2, Def19;
A5: rng h1 = the carrier of I by A4, FUNCT_2:def_3;
rng g = the carrier of H by A3, FUNCT_2:def_3;
then dom h1 = rng g by FUNCT_2:def_1;
then rng (h1 * g) = the carrier of I by A5, RELAT_1:28;
then h1 * g is onto by FUNCT_2:def_3;
hence G,I are_isomorphic by A3, A4, Def19; ::_thesis: verum
end;
theorem Th56: :: GROUP_9:56
for O being set
for G being strict GroupWithOperators of O holds G,G ./. ((1). G) are_isomorphic
proof
let O be set ; ::_thesis: for G being strict GroupWithOperators of O holds G,G ./. ((1). G) are_isomorphic
let G be strict GroupWithOperators of O; ::_thesis: G,G ./. ((1). G) are_isomorphic
nat_hom ((1). G) is bijective by Th54;
hence G,G ./. ((1). G) are_isomorphic by Def19; ::_thesis: verum
end;
theorem Th57: :: GROUP_9:57
for O being set
for G being strict GroupWithOperators of O holds G ./. ((Omega). G) is trivial
proof
let O be set ; ::_thesis: for G being strict GroupWithOperators of O holds G ./. ((Omega). G) is trivial
let G be strict GroupWithOperators of O; ::_thesis: G ./. ((Omega). G) is trivial
reconsider G9 = G as Group ;
reconsider H = multMagma(# the carrier of ((Omega). G), the multF of ((Omega). G) #) as strict normal Subgroup of G by Lm7;
A1: H = (Omega). G9 ;
the carrier of (G ./. ((Omega). G)) = Cosets H by Def14
.= { the carrier of G} by A1, GROUP_2:142 ;
hence G ./. ((Omega). G) is trivial ; ::_thesis: verum
end;
theorem Th58: :: GROUP_9:58
for O being set
for G, H being strict GroupWithOperators of O st G,H are_isomorphic & G is trivial holds
H is trivial
proof
let O be set ; ::_thesis: for G, H being strict GroupWithOperators of O st G,H are_isomorphic & G is trivial holds
H is trivial
let G, H be strict GroupWithOperators of O; ::_thesis: ( G,H are_isomorphic & G is trivial implies H is trivial )
assume that
A1: G,H are_isomorphic and
A2: G is trivial ; ::_thesis: H is trivial
consider e being set such that
A3: the carrier of G = {e} by A2, GROUP_6:def_2;
consider g being Homomorphism of G,H such that
A4: g is bijective by A1, Def19;
e in the carrier of G by A3, TARSKI:def_1;
then A5: e in dom g by FUNCT_2:def_1;
the carrier of H = the carrier of (Image g) by A4, Th51
.= Im (g,e) by A3, Def22
.= {(g . e)} by A5, FUNCT_1:59 ;
hence H is trivial ; ::_thesis: verum
end;
theorem Th59: :: GROUP_9:59
for O being set
for H, G being GroupWithOperators of O
for h being Homomorphism of G,H holds G ./. (Ker h), Image h are_isomorphic
proof
let O be set ; ::_thesis: for H, G being GroupWithOperators of O
for h being Homomorphism of G,H holds G ./. (Ker h), Image h are_isomorphic
let H, G be GroupWithOperators of O; ::_thesis: for h being Homomorphism of G,H holds G ./. (Ker h), Image h are_isomorphic
let h be Homomorphism of G,H; ::_thesis: G ./. (Ker h), Image h are_isomorphic
reconsider G9 = G, H9 = H as Group ;
reconsider h9 = h as Homomorphism of G9,H9 ;
consider g9 being Homomorphism of (G9 ./. (Ker h9)),(Image h9) such that
A1: g9 is bijective and
A2: h9 = g9 * (nat_hom (Ker h9)) by GROUP_6:79;
A3: the carrier of (Image h9) = h9 .: the carrier of G9 by GROUP_6:def_10
.= the carrier of (Image h) by Def22 ;
now__::_thesis:_for_x_being_set_holds_
(_(_x_in_the_carrier_of_(Ker_h)_implies_x_in_the_carrier_of_(Ker_h9)_)_&_(_x_in_the_carrier_of_(Ker_h9)_implies_x_in_the_carrier_of_(Ker_h)_)_)
let x be set ; ::_thesis: ( ( x in the carrier of (Ker h) implies x in the carrier of (Ker h9) ) & ( x in the carrier of (Ker h9) implies x in the carrier of (Ker h) ) )
hereby ::_thesis: ( x in the carrier of (Ker h9) implies x in the carrier of (Ker h) )
assume x in the carrier of (Ker h) ; ::_thesis: x in the carrier of (Ker h9)
then x in { a where a is Element of G : h . a = 1_ H } by Def21;
hence x in the carrier of (Ker h9) by GROUP_6:def_9; ::_thesis: verum
end;
assume x in the carrier of (Ker h9) ; ::_thesis: x in the carrier of (Ker h)
then x in { a9 where a9 is Element of G9 : h9 . a9 = 1_ H9 } by GROUP_6:def_9;
hence x in the carrier of (Ker h) by Def21; ::_thesis: verum
end;
then A4: the carrier of (Ker h9) = the carrier of (Ker h) by TARSKI:1;
Ker h is Subgroup of G by Def7;
then A5: multMagma(# the carrier of (Ker h9), the multF of (Ker h9) #) = multMagma(# the carrier of (Ker h), the multF of (Ker h) #) by A4, GROUP_2:59;
then the carrier of (G9 ./. (Ker h9)) = the carrier of (G ./. (Ker h)) by Def14;
then reconsider g = g9 as Function of (G ./. (Ker h)),(Image h) by A3;
Image h is Subgroup of H by Def7;
then A6: multMagma(# the carrier of (Image h9), the multF of (Image h9) #) = multMagma(# the carrier of (Image h), the multF of (Image h) #) by A3, GROUP_2:59;
A7: now__::_thesis:_for_a,_b_being_Element_of_(G_./._(Ker_h))_holds_g_._(a_*_b)_=_(g_._a)_*_(g_._b)
let a, b be Element of (G ./. (Ker h)); ::_thesis: g . (a * b) = (g . a) * (g . b)
reconsider b9 = b as Element of (G9 ./. (Ker h9)) by A5, Def14;
reconsider a9 = a as Element of (G9 ./. (Ker h9)) by A5, Def14;
thus g . (a * b) = g9 . (a9 * b9) by A5, Def15
.= (g9 . a9) * (g9 . b9) by GROUP_6:def_6
.= (g . a) * (g . b) by A6 ; ::_thesis: verum
end;
now__::_thesis:_for_o_being_Element_of_O
for_a_being_Element_of_(G_./._(Ker_h))_holds_g_._(((G_./._(Ker_h))_^_o)_._a)_=_((Image_h)_^_o)_._(g_._a)
let o be Element of O; ::_thesis: for a being Element of (G ./. (Ker h)) holds g . (((G ./. (Ker h)) ^ b2) . b3) = ((Image h) ^ b2) . (g . b3)
let a be Element of (G ./. (Ker h)); ::_thesis: g . (((G ./. (Ker h)) ^ b1) . b2) = ((Image h) ^ b1) . (g . b2)
percases ( O is empty or not O is empty ) ;
supposeA8: O is empty ; ::_thesis: g . (((G ./. (Ker h)) ^ b1) . b2) = ((Image h) ^ b1) . (g . b2)
hence g . (((G ./. (Ker h)) ^ o) . a) = g . ((id the carrier of (G ./. (Ker h))) . a) by Def6
.= g . a by FUNCT_1:18
.= (id the carrier of (Image h)) . (g . a) by FUNCT_1:18
.= ((Image h) ^ o) . (g . a) by A8, Def6 ;
::_thesis: verum
end;
supposeA9: not O is empty ; ::_thesis: g . (((G ./. (Ker h)) ^ b1) . b2) = ((Image h) ^ b1) . (g . b2)
reconsider G99 = G ./. (Ker h) as Group ;
set f = the action of (G ./. (Ker h)) . o;
A10: the action of (G ./. (Ker h)) . o = { [A,B] where A, B is Element of Cosets (Ker h) : ex g, h being Element of G st
( g in A & h in B & h = (G ^ o) . g ) } by A9, Def16;
the action of (G ./. (Ker h)) . o = (G ./. (Ker h)) ^ o by A9, Def6;
then reconsider f = the action of (G ./. (Ker h)) . o as Homomorphism of G99,G99 ;
a in the carrier of G99 ;
then a in dom f by FUNCT_2:def_1;
then [a,(f . a)] in f by FUNCT_1:1;
then consider A, B being Element of Cosets (Ker h) such that
A11: [A,B] = [a,(f . a)] and
A12: ex g1, g2 being Element of G st
( g1 in A & g2 in B & g2 = (G ^ o) . g1 ) by A10;
reconsider A = A, B = B as Element of Cosets (Ker h9) by A5, Def14;
consider g1, g2 being Element of G9 such that
A13: g1 in A and
A14: g2 in B and
A15: g2 = (G ^ o) . g1 by A12;
A16: A = g1 * (Ker h9) by A13, Lm9;
g1 in the carrier of G9 ;
then A17: g1 in dom (nat_hom (Ker h9)) by FUNCT_2:def_1;
g2 in the carrier of G9 ;
then A18: g2 in dom (nat_hom (Ker h9)) by FUNCT_2:def_1;
A19: ((Image h) ^ o) . (g . a) = ((H ^ o) | the carrier of (Image h)) . (g . a) by Def7
.= (H ^ o) . (g . a) by FUNCT_1:49
.= (H ^ o) . (g9 . (g1 * (Ker h9))) by A11, A16, XTUPLE_0:1 ;
A20: B = g2 * (Ker h9) by A14, Lm9;
h9 . g2 = (H ^ o) . (h9 . g1) by A15, Def18;
then g9 . ((nat_hom (Ker h9)) . g2) = (H ^ o) . ((g9 * (nat_hom (Ker h9))) . g1) by A2, A18, FUNCT_1:13;
then g9 . ((nat_hom (Ker h9)) . g2) = (H ^ o) . (g9 . ((nat_hom (Ker h9)) . g1)) by A17, FUNCT_1:13;
then A21: g9 . (g2 * (Ker h9)) = (H ^ o) . (g9 . ((nat_hom (Ker h9)) . g1)) by GROUP_6:def_8;
g . (((G ./. (Ker h)) ^ o) . a) = g . (f . a) by A9, Def6
.= g9 . (g2 * (Ker h9)) by A11, A20, XTUPLE_0:1 ;
hence g . (((G ./. (Ker h)) ^ o) . a) = ((Image h) ^ o) . (g . a) by A19, A21, GROUP_6:def_8; ::_thesis: verum
end;
end;
end;
then reconsider g = g as Homomorphism of (G ./. (Ker h)),(Image h) by A7, Def18, GROUP_6:def_6;
g is onto by A1, A3;
hence G ./. (Ker h), Image h are_isomorphic by A1, Def19; ::_thesis: verum
end;
theorem Th60: :: GROUP_9:60
for O being set
for G being GroupWithOperators of O
for H, F1, F2 being strict StableSubgroup of G st F1 is normal StableSubgroup of F2 holds
H /\ F1 is normal StableSubgroup of H /\ F2
proof
let O be set ; ::_thesis: for G being GroupWithOperators of O
for H, F1, F2 being strict StableSubgroup of G st F1 is normal StableSubgroup of F2 holds
H /\ F1 is normal StableSubgroup of H /\ F2
let G be GroupWithOperators of O; ::_thesis: for H, F1, F2 being strict StableSubgroup of G st F1 is normal StableSubgroup of F2 holds
H /\ F1 is normal StableSubgroup of H /\ F2
let H, F1, F2 be strict StableSubgroup of G; ::_thesis: ( F1 is normal StableSubgroup of F2 implies H /\ F1 is normal StableSubgroup of H /\ F2 )
reconsider F = F2 /\ H as StableSubgroup of F2 by Lm34;
assume A1: F1 is normal StableSubgroup of F2 ; ::_thesis: H /\ F1 is normal StableSubgroup of H /\ F2
then A2: F1 /\ H = (F1 /\ F2) /\ H by Lm22
.= F1 /\ (F2 /\ H) by Th20 ;
reconsider F1 = F1 as normal StableSubgroup of F2 by A1;
F1 /\ F is normal StableSubgroup of F by Th41;
hence H /\ F1 is normal StableSubgroup of H /\ F2 by A2, Th39; ::_thesis: verum
end;
begin
theorem :: GROUP_9:61
for O, E being set
for A being Action of O,E holds [#] E is_stable_under_the_action_of A by Lm1;
theorem :: GROUP_9:62
for O, E being set holds [:O,{(id E)}:] is Action of O,E by Lm2;
theorem :: GROUP_9:63
for O being non empty set
for E being set
for o being Element of O
for A being Action of O,E holds Product (<*o*>,A) = A . o by Lm26;
theorem :: GROUP_9:64
for O being non empty set
for E being set
for F1, F2 being FinSequence of O
for A being Action of O,E holds Product ((F1 ^ F2),A) = (Product (F1,A)) * (Product (F2,A)) by Lm29;
theorem :: GROUP_9:65
for O, E being set
for A being Action of O,E
for F being FinSequence of O
for Y being Subset of E st Y is_stable_under_the_action_of A holds
(Product (F,A)) .: Y c= Y by Lm30;
theorem :: GROUP_9:66
for O being set
for E being non empty set
for A being Action of O,E
for X being Subset of E
for a being Element of E st not X is empty holds
( a in the_stable_subset_generated_by (X,A) iff ex F being FinSequence of O ex x being Element of X st (Product (F,A)) . x = a ) by Lm31;
theorem :: GROUP_9:67
for O being set
for G being strict Group ex H being strict GroupWithOperators of O st G = multMagma(# the carrier of H, the multF of H #)
proof
let O be set ; ::_thesis: for G being strict Group ex H being strict GroupWithOperators of O st G = multMagma(# the carrier of H, the multF of H #)
let G be strict Group; ::_thesis: ex H being strict GroupWithOperators of O st G = multMagma(# the carrier of H, the multF of H #)
consider H being non empty HGrWOpStr over O such that
A1: ( H is strict & H is distributive & H is Group-like & H is associative ) and
A2: G = multMagma(# the carrier of H, the multF of H #) by Lm3;
reconsider H = H as strict GroupWithOperators of O by A1;
take H ; ::_thesis: G = multMagma(# the carrier of H, the multF of H #)
thus G = multMagma(# the carrier of H, the multF of H #) by A2; ::_thesis: verum
end;
theorem :: GROUP_9:68
for O being set
for G being GroupWithOperators of O
for H1 being StableSubgroup of G holds multMagma(# the carrier of H1, the multF of H1 #) is strict Subgroup of G by Lm16;
theorem :: GROUP_9:69
for O being set
for G being GroupWithOperators of O
for N being normal StableSubgroup of G holds multMagma(# the carrier of N, the multF of N #) is strict normal Subgroup of G by Lm7;
theorem :: GROUP_9:70
for O being set
for o being Element of O
for G being GroupWithOperators of O
for H1 being StableSubgroup of G
for g1 being Element of G st g1 in H1 holds
(G ^ o) . g1 in H1 by Lm10;
theorem :: GROUP_9:71
for O being set
for G, H being GroupWithOperators of O
for G9 being strict StableSubgroup of G
for f being Homomorphism of G,H ex H9 being strict StableSubgroup of H st the carrier of H9 = f .: the carrier of G9 by Lm17;
theorem :: GROUP_9:72
for O being set
for G being GroupWithOperators of O
for B being Subset of G st B is empty holds
the_stable_subgroup_of B = (1). G by Lm25;
theorem :: GROUP_9:73
for O being set
for G being GroupWithOperators of O
for B, C being Subset of G st B = the carrier of (gr C) holds
the_stable_subgroup_of C = the_stable_subgroup_of B by Lm32;
theorem :: GROUP_9:74
for O being set
for G being GroupWithOperators of O
for N being normal StableSubgroup of G
for N9 being normal Subgroup of G st N9 = multMagma(# the carrier of N, the multF of N #) holds
( G ./. N9 = multMagma(# the carrier of (G ./. N), the multF of (G ./. N) #) & 1_ (G ./. N9) = 1_ (G ./. N) ) by Lm35;
theorem Th75: :: GROUP_9:75
for O being set
for G being GroupWithOperators of O
for H1, H2 being StableSubgroup of G st the carrier of H1 = the carrier of H2 holds
HGrWOpStr(# the carrier of H1, the multF of H1, the action of H1 #) = HGrWOpStr(# the carrier of H2, the multF of H2, the action of H2 #)
proof
let O be set ; ::_thesis: for G being GroupWithOperators of O
for H1, H2 being StableSubgroup of G st the carrier of H1 = the carrier of H2 holds
HGrWOpStr(# the carrier of H1, the multF of H1, the action of H1 #) = HGrWOpStr(# the carrier of H2, the multF of H2, the action of H2 #)
let G be GroupWithOperators of O; ::_thesis: for H1, H2 being StableSubgroup of G st the carrier of H1 = the carrier of H2 holds
HGrWOpStr(# the carrier of H1, the multF of H1, the action of H1 #) = HGrWOpStr(# the carrier of H2, the multF of H2, the action of H2 #)
let H1, H2 be StableSubgroup of G; ::_thesis: ( the carrier of H1 = the carrier of H2 implies HGrWOpStr(# the carrier of H1, the multF of H1, the action of H1 #) = HGrWOpStr(# the carrier of H2, the multF of H2, the action of H2 #) )
reconsider H19 = H1, H29 = H2 as Subgroup of G by Def7;
A1: dom the action of H2 = O by FUNCT_2:def_1
.= dom the action of H1 by FUNCT_2:def_1 ;
assume A2: the carrier of H1 = the carrier of H2 ; ::_thesis: HGrWOpStr(# the carrier of H1, the multF of H1, the action of H1 #) = HGrWOpStr(# the carrier of H2, the multF of H2, the action of H2 #)
A3: now__::_thesis:_for_x_being_set_st_x_in_dom_the_action_of_H2_holds_
the_action_of_H1_._x_=_the_action_of_H2_._x
let x be set ; ::_thesis: ( x in dom the action of H2 implies the action of H1 . x = the action of H2 . x )
assume A4: x in dom the action of H2 ; ::_thesis: the action of H1 . x = the action of H2 . x
then reconsider o = x as Element of O ;
A5: H1 ^ o = the action of H1 . o by A4, Def6;
H1 ^ o = (G ^ o) | the carrier of H2 by A2, Def7
.= H2 ^ o by Def7 ;
hence the action of H1 . x = the action of H2 . x by A4, A5, Def6; ::_thesis: verum
end;
multMagma(# the carrier of H19, the multF of H19 #) = multMagma(# the carrier of H29, the multF of H29 #) by A2, GROUP_2:59;
hence HGrWOpStr(# the carrier of H1, the multF of H1, the action of H1 #) = HGrWOpStr(# the carrier of H2, the multF of H2, the action of H2 #) by A1, A3, FUNCT_1:2; ::_thesis: verum
end;
theorem Th76: :: GROUP_9:76
for O being set
for G being GroupWithOperators of O
for H1 being StableSubgroup of G
for N1 being normal StableSubgroup of H1 st H1 ./. N1 is trivial holds
HGrWOpStr(# the carrier of H1, the multF of H1, the action of H1 #) = HGrWOpStr(# the carrier of N1, the multF of N1, the action of N1 #)
proof
let O be set ; ::_thesis: for G being GroupWithOperators of O
for H1 being StableSubgroup of G
for N1 being normal StableSubgroup of H1 st H1 ./. N1 is trivial holds
HGrWOpStr(# the carrier of H1, the multF of H1, the action of H1 #) = HGrWOpStr(# the carrier of N1, the multF of N1, the action of N1 #)
let G be GroupWithOperators of O; ::_thesis: for H1 being StableSubgroup of G
for N1 being normal StableSubgroup of H1 st H1 ./. N1 is trivial holds
HGrWOpStr(# the carrier of H1, the multF of H1, the action of H1 #) = HGrWOpStr(# the carrier of N1, the multF of N1, the action of N1 #)
let H1 be StableSubgroup of G; ::_thesis: for N1 being normal StableSubgroup of H1 st H1 ./. N1 is trivial holds
HGrWOpStr(# the carrier of H1, the multF of H1, the action of H1 #) = HGrWOpStr(# the carrier of N1, the multF of N1, the action of N1 #)
let N1 be normal StableSubgroup of H1; ::_thesis: ( H1 ./. N1 is trivial implies HGrWOpStr(# the carrier of H1, the multF of H1, the action of H1 #) = HGrWOpStr(# the carrier of N1, the multF of N1, the action of N1 #) )
reconsider N9 = N1 as StableSubgroup of G by Th11;
set H = H1;
reconsider N = multMagma(# the carrier of N1, the multF of N1 #) as normal Subgroup of H1 by Lm7;
assume A1: H1 ./. N1 is trivial ; ::_thesis: HGrWOpStr(# the carrier of H1, the multF of H1, the action of H1 #) = HGrWOpStr(# the carrier of N1, the multF of N1, the action of N1 #)
Cosets N1 = Cosets N by Def14;
then consider e being set such that
A2: the carrier of (H1 ./. N) = {e} by A1, GROUP_6:def_2;
A3: the carrier of H1 = union {e} by A2, GROUP_2:137;
A4: now__::_thesis:_the_carrier_of_H1_c=_the_carrier_of_N
assume not the carrier of H1 c= the carrier of N ; ::_thesis: contradiction
then the carrier of H1 \ the carrier of N <> {} by XBOOLE_1:37;
then consider x being set such that
A5: x in the carrier of H1 \ the carrier of N by XBOOLE_0:def_1;
reconsider x = x as Element of H1 by A5;
A6: now__::_thesis:_not_x_*_N_=_e
assume x * N = e ; ::_thesis: contradiction
then x * N = the carrier of H1 by A3, ZFMISC_1:25;
then consider x9 being Element of H1 such that
A7: 1_ H1 = x * x9 and
A8: x9 in N by GROUP_2:103;
x9 = x " by A7, GROUP_1:12;
then (x ") " in N by A8, GROUP_2:51;
then x in carr N by STRUCT_0:def_5;
hence contradiction by A5, XBOOLE_0:def_5; ::_thesis: verum
end;
x * N in Cosets N by GROUP_6:14;
hence contradiction by A2, A6, TARSKI:def_1; ::_thesis: verum
end;
the carrier of N c= the carrier of H1 by GROUP_2:def_5;
then the carrier of N9 = the carrier of H1 by A4, XBOOLE_0:def_10;
hence HGrWOpStr(# the carrier of H1, the multF of H1, the action of H1 #) = HGrWOpStr(# the carrier of N1, the multF of N1, the action of N1 #) by Th75; ::_thesis: verum
end;
theorem Th77: :: GROUP_9:77
for O being set
for G being GroupWithOperators of O
for H1 being StableSubgroup of G
for N1 being normal StableSubgroup of H1 st the carrier of H1 = the carrier of N1 holds
H1 ./. N1 is trivial
proof
let O be set ; ::_thesis: for G being GroupWithOperators of O
for H1 being StableSubgroup of G
for N1 being normal StableSubgroup of H1 st the carrier of H1 = the carrier of N1 holds
H1 ./. N1 is trivial
let G be GroupWithOperators of O; ::_thesis: for H1 being StableSubgroup of G
for N1 being normal StableSubgroup of H1 st the carrier of H1 = the carrier of N1 holds
H1 ./. N1 is trivial
let H1 be StableSubgroup of G; ::_thesis: for N1 being normal StableSubgroup of H1 st the carrier of H1 = the carrier of N1 holds
H1 ./. N1 is trivial
let N1 be normal StableSubgroup of H1; ::_thesis: ( the carrier of H1 = the carrier of N1 implies H1 ./. N1 is trivial )
reconsider N19 = multMagma(# the carrier of N1, the multF of N1 #) as strict normal Subgroup of H1 by Lm7;
assume A1: the carrier of H1 = the carrier of N1 ; ::_thesis: H1 ./. N1 is trivial
now__::_thesis:_for_x_being_set_holds_
(_(_x_in_Left_Cosets_N19_implies_x_=_the_carrier_of_H1_)_&_(_x_=_the_carrier_of_H1_implies_x_in_Left_Cosets_N19_)_)
let x be set ; ::_thesis: ( ( x in Left_Cosets N19 implies x = the carrier of H1 ) & ( x = the carrier of H1 implies x in Left_Cosets N19 ) )
hereby ::_thesis: ( x = the carrier of H1 implies x in Left_Cosets N19 )
assume A2: x in Left_Cosets N19 ; ::_thesis: x = the carrier of H1
then reconsider A = x as Subset of H1 ;
consider a being Element of H1 such that
A3: A = a * N19 by A2, GROUP_2:def_15;
A = a * ([#] the carrier of H1) by A1, A3;
hence x = the carrier of H1 by GROUP_2:17; ::_thesis: verum
end;
the carrier of H1 = (1_ H1) * ([#] the carrier of H1) by GROUP_2:17;
then A4: the carrier of H1 = (1_ H1) * N19 by A1;
assume x = the carrier of H1 ; ::_thesis: x in Left_Cosets N19
hence x in Left_Cosets N19 by A4, GROUP_2:def_15; ::_thesis: verum
end;
then A5: { the carrier of H1} = Left_Cosets N19 by TARSKI:def_1;
Cosets N1 = Cosets N19 by Def14;
hence H1 ./. N1 is trivial by A5; ::_thesis: verum
end;
theorem Th78: :: GROUP_9:78
for O being set
for G, H being GroupWithOperators of O
for N being StableSubgroup of G
for H9 being strict StableSubgroup of H
for f being Homomorphism of G,H st N = Ker f holds
ex G9 being strict StableSubgroup of G st
( the carrier of G9 = f " the carrier of H9 & ( H9 is normal implies ( N is normal StableSubgroup of G9 & G9 is normal ) ) )
proof
let O be set ; ::_thesis: for G, H being GroupWithOperators of O
for N being StableSubgroup of G
for H9 being strict StableSubgroup of H
for f being Homomorphism of G,H st N = Ker f holds
ex G9 being strict StableSubgroup of G st
( the carrier of G9 = f " the carrier of H9 & ( H9 is normal implies ( N is normal StableSubgroup of G9 & G9 is normal ) ) )
let G, H be GroupWithOperators of O; ::_thesis: for N being StableSubgroup of G
for H9 being strict StableSubgroup of H
for f being Homomorphism of G,H st N = Ker f holds
ex G9 being strict StableSubgroup of G st
( the carrier of G9 = f " the carrier of H9 & ( H9 is normal implies ( N is normal StableSubgroup of G9 & G9 is normal ) ) )
let N be StableSubgroup of G; ::_thesis: for H9 being strict StableSubgroup of H
for f being Homomorphism of G,H st N = Ker f holds
ex G9 being strict StableSubgroup of G st
( the carrier of G9 = f " the carrier of H9 & ( H9 is normal implies ( N is normal StableSubgroup of G9 & G9 is normal ) ) )
let H9 be strict StableSubgroup of H; ::_thesis: for f being Homomorphism of G,H st N = Ker f holds
ex G9 being strict StableSubgroup of G st
( the carrier of G9 = f " the carrier of H9 & ( H9 is normal implies ( N is normal StableSubgroup of G9 & G9 is normal ) ) )
reconsider H99 = multMagma(# the carrier of H9, the multF of H9 #) as strict Subgroup of H by Lm16;
let f be Homomorphism of G,H; ::_thesis: ( N = Ker f implies ex G9 being strict StableSubgroup of G st
( the carrier of G9 = f " the carrier of H9 & ( H9 is normal implies ( N is normal StableSubgroup of G9 & G9 is normal ) ) ) )
assume A1: N = Ker f ; ::_thesis: ex G9 being strict StableSubgroup of G st
( the carrier of G9 = f " the carrier of H9 & ( H9 is normal implies ( N is normal StableSubgroup of G9 & G9 is normal ) ) )
set A = { g where g is Element of G : f . g in H99 } ;
A2: 1_ H in H99 by GROUP_2:46;
then f . (1_ G) in H99 by Lm13;
then 1_ G in { g where g is Element of G : f . g in H99 } ;
then reconsider A = { g where g is Element of G : f . g in H99 } as non empty set ;
now__::_thesis:_for_x_being_set_st_x_in_A_holds_
x_in_the_carrier_of_G
let x be set ; ::_thesis: ( x in A implies x in the carrier of G )
assume x in A ; ::_thesis: x in the carrier of G
then ex g being Element of G st
( x = g & f . g in H99 ) ;
hence x in the carrier of G ; ::_thesis: verum
end;
then reconsider A = A as Subset of G by TARSKI:def_3;
A3: now__::_thesis:_for_g1,_g2_being_Element_of_G_st_g1_in_A_&_g2_in_A_holds_
g1_*_g2_in_A
let g1, g2 be Element of G; ::_thesis: ( g1 in A & g2 in A implies g1 * g2 in A )
assume that
A4: g1 in A and
A5: g2 in A ; ::_thesis: g1 * g2 in A
consider b being Element of G such that
A6: b = g2 and
A7: f . b in H99 by A5;
consider a being Element of G such that
A8: a = g1 and
A9: f . a in H99 by A4;
set fb = f . b;
set fa = f . a;
( f . (a * b) = (f . a) * (f . b) & (f . a) * (f . b) in H99 ) by A9, A7, GROUP_2:50, GROUP_6:def_6;
hence g1 * g2 in A by A8, A6; ::_thesis: verum
end;
A10: now__::_thesis:_for_o_being_Element_of_O
for_g_being_Element_of_G_st_g_in_A_holds_
(G_^_o)_._g_in_A
let o be Element of O; ::_thesis: for g being Element of G st g in A holds
(G ^ o) . g in A
let g be Element of G; ::_thesis: ( g in A implies (G ^ o) . g in A )
assume g in A ; ::_thesis: (G ^ o) . g in A
then consider a being Element of G such that
A11: a = g and
A12: f . a in H99 ;
f . a in the carrier of H99 by A12, STRUCT_0:def_5;
then f . a in H9 by STRUCT_0:def_5;
then (H ^ o) . (f . g) in H9 by A11, Lm10;
then f . ((G ^ o) . g) in H9 by Def18;
then f . ((G ^ o) . g) in the carrier of H9 by STRUCT_0:def_5;
then f . ((G ^ o) . g) in H99 by STRUCT_0:def_5;
hence (G ^ o) . g in A ; ::_thesis: verum
end;
now__::_thesis:_for_g_being_Element_of_G_st_g_in_A_holds_
g_"_in_A
let g be Element of G; ::_thesis: ( g in A implies g " in A )
assume g in A ; ::_thesis: g " in A
then consider a being Element of G such that
A13: a = g and
A14: f . a in H99 ;
(f . a) " in H99 by A14, GROUP_2:51;
then f . (a ") in H99 by Lm14;
hence g " in A by A13; ::_thesis: verum
end;
then consider G99 being strict StableSubgroup of G such that
A15: the carrier of G99 = A by A3, A10, Lm15;
take G99 ; ::_thesis: ( the carrier of G99 = f " the carrier of H9 & ( H9 is normal implies ( N is normal StableSubgroup of G99 & G99 is normal ) ) )
now__::_thesis:_for_g_being_Element_of_G_holds_
(_(_g_in_A_implies_g_in_f_"_the_carrier_of_H9_)_&_(_g_in_f_"_the_carrier_of_H9_implies_g_in_A_)_)
reconsider R = f as Relation of the carrier of G, the carrier of H ;
let g be Element of G; ::_thesis: ( ( g in A implies g in f " the carrier of H9 ) & ( g in f " the carrier of H9 implies g in A ) )
hereby ::_thesis: ( g in f " the carrier of H9 implies g in A )
assume g in A ; ::_thesis: g in f " the carrier of H9
then ex a being Element of G st
( a = g & f . a in H99 ) ;
then A16: f . g in the carrier of H9 by STRUCT_0:def_5;
dom f = the carrier of G by FUNCT_2:def_1;
then [g,(f . g)] in f by FUNCT_1:1;
hence g in f " the carrier of H9 by A16, RELSET_1:30; ::_thesis: verum
end;
assume g in f " the carrier of H9 ; ::_thesis: g in A
then consider h being Element of H such that
A17: ( [g,h] in R & h in the carrier of H9 ) by RELSET_1:30;
( f . g = h & h in H99 ) by A17, FUNCT_1:1, STRUCT_0:def_5;
hence g in A ; ::_thesis: verum
end;
hence the carrier of G99 = f " the carrier of H9 by A15, SUBSET_1:3; ::_thesis: ( H9 is normal implies ( N is normal StableSubgroup of G99 & G99 is normal ) )
reconsider G9 = multMagma(# the carrier of G99, the multF of G99 #) as strict Subgroup of G by Lm16;
now__::_thesis:_(_H9_is_normal_implies_(_N_is_normal_StableSubgroup_of_G99_&_G99_is_normal_)_)
assume A18: H9 is normal ; ::_thesis: ( N is normal StableSubgroup of G99 & G99 is normal )
now__::_thesis:_for_g_being_Element_of_G_st_g_in_N_holds_
g_in_G99
let g be Element of G; ::_thesis: ( g in N implies g in G99 )
assume g in N ; ::_thesis: g in G99
then f . g = 1_ H by A1, Th47;
then g in the carrier of G99 by A2, A15;
hence g in G99 by STRUCT_0:def_5; ::_thesis: verum
end;
hence N is normal StableSubgroup of G99 by A1, Th13, Th40; ::_thesis: G99 is normal
now__::_thesis:_for_g_being_Element_of_G_holds_g_*_G9_c=_G9_*_g
let g be Element of G; ::_thesis: g * G9 c= G9 * g
now__::_thesis:_for_x_being_set_st_x_in_g_*_G9_holds_
x_in_G9_*_g
H99 is normal by A18, Def10;
then A19: H99 |^ ((f . g) ") = H99 by GROUP_3:def_13;
let x be set ; ::_thesis: ( x in g * G9 implies x in G9 * g )
assume x in g * G9 ; ::_thesis: x in G9 * g
then consider h being Element of G such that
A20: x = g * h and
A21: h in A by A15, GROUP_2:27;
set h9 = (g * h) * (g ");
A22: f . ((g * h) * (g ")) = (f . (g * h)) * (f . (g ")) by GROUP_6:def_6
.= ((f . g) * (f . h)) * (f . (g ")) by GROUP_6:def_6
.= ((f . g) * (f . h)) * ((f . g) ") by Lm14
.= ((((f . g) ") ") * (f . h)) * ((f . g) ")
.= (f . h) |^ ((f . g) ") by GROUP_3:def_2 ;
ex a being Element of G st
( a = h & f . a in H99 ) by A21;
then f . ((g * h) * (g ")) in H99 by A19, A22, GROUP_3:58;
then A23: (g * h) * (g ") in A ;
((g * h) * (g ")) * g = (g * h) * ((g ") * g) by GROUP_1:def_3
.= (g * h) * (1_ G) by GROUP_1:def_5
.= x by A20, GROUP_1:def_4 ;
hence x in G9 * g by A15, A23, GROUP_2:28; ::_thesis: verum
end;
hence g * G9 c= G9 * g by TARSKI:def_3; ::_thesis: verum
end;
then for H being strict Subgroup of G st H = multMagma(# the carrier of G99, the multF of G99 #) holds
H is normal by GROUP_3:118;
hence G99 is normal by Def10; ::_thesis: verum
end;
hence ( H9 is normal implies ( N is normal StableSubgroup of G99 & G99 is normal ) ) ; ::_thesis: verum
end;
theorem Th79: :: GROUP_9:79
for O being set
for G, H being GroupWithOperators of O
for N being StableSubgroup of G
for G9 being strict StableSubgroup of G
for f being Homomorphism of G,H st N = Ker f holds
ex H9 being strict StableSubgroup of H st
( the carrier of H9 = f .: the carrier of G9 & f " the carrier of H9 = the carrier of (G9 "\/" N) & ( f is onto & G9 is normal implies H9 is normal ) )
proof
let O be set ; ::_thesis: for G, H being GroupWithOperators of O
for N being StableSubgroup of G
for G9 being strict StableSubgroup of G
for f being Homomorphism of G,H st N = Ker f holds
ex H9 being strict StableSubgroup of H st
( the carrier of H9 = f .: the carrier of G9 & f " the carrier of H9 = the carrier of (G9 "\/" N) & ( f is onto & G9 is normal implies H9 is normal ) )
let G, H be GroupWithOperators of O; ::_thesis: for N being StableSubgroup of G
for G9 being strict StableSubgroup of G
for f being Homomorphism of G,H st N = Ker f holds
ex H9 being strict StableSubgroup of H st
( the carrier of H9 = f .: the carrier of G9 & f " the carrier of H9 = the carrier of (G9 "\/" N) & ( f is onto & G9 is normal implies H9 is normal ) )
let N be StableSubgroup of G; ::_thesis: for G9 being strict StableSubgroup of G
for f being Homomorphism of G,H st N = Ker f holds
ex H9 being strict StableSubgroup of H st
( the carrier of H9 = f .: the carrier of G9 & f " the carrier of H9 = the carrier of (G9 "\/" N) & ( f is onto & G9 is normal implies H9 is normal ) )
reconsider N9 = multMagma(# the carrier of N, the multF of N #) as strict Subgroup of G by Lm16;
let G9 be strict StableSubgroup of G; ::_thesis: for f being Homomorphism of G,H st N = Ker f holds
ex H9 being strict StableSubgroup of H st
( the carrier of H9 = f .: the carrier of G9 & f " the carrier of H9 = the carrier of (G9 "\/" N) & ( f is onto & G9 is normal implies H9 is normal ) )
reconsider G99 = multMagma(# the carrier of G9, the multF of G9 #) as strict Subgroup of G by Lm16;
let f be Homomorphism of G,H; ::_thesis: ( N = Ker f implies ex H9 being strict StableSubgroup of H st
( the carrier of H9 = f .: the carrier of G9 & f " the carrier of H9 = the carrier of (G9 "\/" N) & ( f is onto & G9 is normal implies H9 is normal ) ) )
set A = { (f . g) where g is Element of G : g in G99 } ;
A1: ( G99 * N9 = G9 * N & N9 * G99 = N * G9 ) ;
1_ G in G99 by GROUP_2:46;
then f . (1_ G) in { (f . g) where g is Element of G : g in G99 } ;
then reconsider A = { (f . g) where g is Element of G : g in G99 } as non empty set ;
now__::_thesis:_for_x_being_set_st_x_in_A_holds_
x_in_the_carrier_of_H
let x be set ; ::_thesis: ( x in A implies x in the carrier of H )
assume x in A ; ::_thesis: x in the carrier of H
then ex g being Element of G st
( x = f . g & g in G99 ) ;
hence x in the carrier of H ; ::_thesis: verum
end;
then reconsider A = A as Subset of H by TARSKI:def_3;
A2: now__::_thesis:_for_h1,_h2_being_Element_of_H_st_h1_in_A_&_h2_in_A_holds_
h1_*_h2_in_A
let h1, h2 be Element of H; ::_thesis: ( h1 in A & h2 in A implies h1 * h2 in A )
assume that
A3: h1 in A and
A4: h2 in A ; ::_thesis: h1 * h2 in A
consider a being Element of G such that
A5: ( h1 = f . a & a in G99 ) by A3;
consider b being Element of G such that
A6: ( h2 = f . b & b in G99 ) by A4;
( f . (a * b) = h1 * h2 & a * b in G99 ) by A5, A6, GROUP_2:50, GROUP_6:def_6;
hence h1 * h2 in A ; ::_thesis: verum
end;
A7: now__::_thesis:_for_o_being_Element_of_O
for_h_being_Element_of_H_st_h_in_A_holds_
(H_^_o)_._h_in_A
let o be Element of O; ::_thesis: for h being Element of H st h in A holds
(H ^ o) . h in A
let h be Element of H; ::_thesis: ( h in A implies (H ^ o) . h in A )
assume h in A ; ::_thesis: (H ^ o) . h in A
then consider g being Element of G such that
A8: h = f . g and
A9: g in G99 ;
g in the carrier of G99 by A9, STRUCT_0:def_5;
then g in G9 by STRUCT_0:def_5;
then (G ^ o) . g in G9 by Lm10;
then (G ^ o) . g in the carrier of G9 by STRUCT_0:def_5;
then A10: (G ^ o) . g in G99 by STRUCT_0:def_5;
(H ^ o) . h = f . ((G ^ o) . g) by A8, Def18;
hence (H ^ o) . h in A by A10; ::_thesis: verum
end;
now__::_thesis:_for_h_being_Element_of_H_st_h_in_A_holds_
h_"_in_A
let h be Element of H; ::_thesis: ( h in A implies h " in A )
assume h in A ; ::_thesis: h " in A
then consider g being Element of G such that
A11: ( h = f . g & g in G99 ) ;
( g " in G99 & h " = f . (g ") ) by A11, Lm14, GROUP_2:51;
hence h " in A ; ::_thesis: verum
end;
then consider H99 being strict StableSubgroup of H such that
A12: the carrier of H99 = A by A2, A7, Lm15;
assume A13: N = Ker f ; ::_thesis: ex H9 being strict StableSubgroup of H st
( the carrier of H9 = f .: the carrier of G9 & f " the carrier of H9 = the carrier of (G9 "\/" N) & ( f is onto & G9 is normal implies H9 is normal ) )
then N9 is normal by Def10;
then A14: (carr G99) * N9 = N9 * (carr G99) by GROUP_3:120;
reconsider H9 = multMagma(# the carrier of H99, the multF of H99 #) as strict Subgroup of H by Lm16;
take H99 ; ::_thesis: ( the carrier of H99 = f .: the carrier of G9 & f " the carrier of H99 = the carrier of (G9 "\/" N) & ( f is onto & G9 is normal implies H99 is normal ) )
A15: now__::_thesis:_for_h_being_Element_of_H_holds_
(_(_h_in_A_implies_h_in_f_.:_the_carrier_of_G9_)_&_(_h_in_f_.:_the_carrier_of_G9_implies_h_in_A_)_)
reconsider R = f as Relation of the carrier of G, the carrier of H ;
let h be Element of H; ::_thesis: ( ( h in A implies h in f .: the carrier of G9 ) & ( h in f .: the carrier of G9 implies h in A ) )
hereby ::_thesis: ( h in f .: the carrier of G9 implies h in A )
assume h in A ; ::_thesis: h in f .: the carrier of G9
then consider g being Element of G such that
A16: h = f . g and
A17: g in G99 ;
A18: g in the carrier of G9 by A17, STRUCT_0:def_5;
dom f = the carrier of G by FUNCT_2:def_1;
then [g,h] in f by A16, FUNCT_1:1;
hence h in f .: the carrier of G9 by A18, RELSET_1:29; ::_thesis: verum
end;
assume h in f .: the carrier of G9 ; ::_thesis: h in A
then consider g being Element of G such that
A19: ( [g,h] in R & g in the carrier of G9 ) by RELSET_1:29;
( f . g = h & g in G99 ) by A19, FUNCT_1:1, STRUCT_0:def_5;
hence h in A ; ::_thesis: verum
end;
hence A20: the carrier of H99 = f .: the carrier of G9 by A12, SUBSET_1:3; ::_thesis: ( f " the carrier of H99 = the carrier of (G9 "\/" N) & ( f is onto & G9 is normal implies H99 is normal ) )
A21: now__::_thesis:_for_x_being_set_st_x_in_f_"_the_carrier_of_H9_holds_
x_in_G99_*_N9
let x be set ; ::_thesis: ( x in f " the carrier of H9 implies x in G99 * N9 )
assume A22: x in f " the carrier of H9 ; ::_thesis: x in G99 * N9
then f . x in the carrier of H9 by FUNCT_1:def_7;
then consider g1 being set such that
A23: g1 in dom f and
A24: g1 in the carrier of G9 and
A25: f . g1 = f . x by A20, FUNCT_1:def_6;
reconsider g1 = g1, g2 = x as Element of G by A22, A23;
consider g3 being Element of G such that
A26: g2 = g1 * g3 by GROUP_1:15;
f . g2 = (f . g2) * (f . g3) by A25, A26, GROUP_6:def_6;
then f . g3 = 1_ H by GROUP_1:7;
then g3 in Ker f by Th47;
then g3 in the carrier of N by A13, STRUCT_0:def_5;
hence x in G99 * N9 by A24, A26; ::_thesis: verum
end;
A27: dom f = the carrier of G by FUNCT_2:def_1;
now__::_thesis:_for_x_being_set_st_x_in_G99_*_N9_holds_
x_in_f_"_the_carrier_of_H9
let x be set ; ::_thesis: ( x in G99 * N9 implies x in f " the carrier of H9 )
assume A28: x in G99 * N9 ; ::_thesis: x in f " the carrier of H9
then consider g1, g2 being Element of G such that
A29: x = g1 * g2 and
A30: g1 in carr G9 and
A31: g2 in carr N9 ;
A32: g2 in Ker f by A13, A31, STRUCT_0:def_5;
f . x = (f . g1) * (f . g2) by A29, GROUP_6:def_6
.= (f . g1) * (1_ H) by A32, Th47
.= f . g1 by GROUP_1:def_4 ;
then f . x in f .: the carrier of G9 by A27, A30, FUNCT_1:def_6;
then x in f " (f .: the carrier of G9) by A27, A28, FUNCT_1:def_7;
hence x in f " the carrier of H9 by A12, A15, SUBSET_1:3; ::_thesis: verum
end;
then f " the carrier of H9 = (carr G9) * (carr N) by A21, TARSKI:1;
hence f " the carrier of H99 = the carrier of (G9 "\/" N) by A14, A1, Th30; ::_thesis: ( f is onto & G9 is normal implies H99 is normal )
now__::_thesis:_(_f_is_onto_&_G9_is_normal_implies_H99_is_normal_)
assume that
A33: f is onto and
A34: G9 is normal ; ::_thesis: H99 is normal
A35: G99 is normal by A34, Def10;
now__::_thesis:_for_h1_being_Element_of_H_holds_h1_*_H9_c=_H9_*_h1
let h1 be Element of H; ::_thesis: h1 * H9 c= H9 * h1
now__::_thesis:_for_x_being_set_st_x_in_h1_*_H9_holds_
x_in_H9_*_h1
let x be set ; ::_thesis: ( x in h1 * H9 implies x in H9 * h1 )
assume x in h1 * H9 ; ::_thesis: x in H9 * h1
then consider h2 being Element of H such that
A36: x = h1 * h2 and
A37: h2 in A by A12, GROUP_2:27;
set h29 = (h1 * h2) * (h1 ");
h2 in f .: the carrier of G9 by A15, A37;
then consider g2 being set such that
A38: g2 in dom f and
A39: g2 in the carrier of G99 and
A40: f . g2 = h2 by FUNCT_1:def_6;
rng f = the carrier of H by A33, FUNCT_2:def_3;
then consider g1 being set such that
A41: g1 in dom f and
A42: h1 = f . g1 by FUNCT_1:def_3;
reconsider g1 = g1, g2 = g2 as Element of G by A38, A41;
set g29 = (g1 * g2) * (g1 ");
(g1 * g2) * (g1 ") = (((g1 ") ") * g2) * (g1 ") ;
then A43: (g1 * g2) * (g1 ") = g2 |^ (g1 ") by GROUP_3:def_2;
g2 in G99 by A39, STRUCT_0:def_5;
then (g1 * g2) * (g1 ") in G99 |^ (g1 ") by A43, GROUP_3:58;
then A44: (g1 * g2) * (g1 ") in the carrier of (G99 |^ (g1 ")) by STRUCT_0:def_5;
G99 |^ (g1 ") is Subgroup of G99 by A35, GROUP_3:122;
then A45: the carrier of (G99 |^ (g1 ")) c= the carrier of G99 by GROUP_2:def_5;
(h1 * h2) * (h1 ") = ((f . g1) * (f . g2)) * (f . (g1 ")) by A40, A42, Lm14
.= (f . (g1 * g2)) * (f . (g1 ")) by GROUP_6:def_6
.= f . ((g1 * g2) * (g1 ")) by GROUP_6:def_6 ;
then (h1 * h2) * (h1 ") in f .: the carrier of G99 by A27, A44, A45, FUNCT_1:def_6;
then A46: (h1 * h2) * (h1 ") in A by A15;
((h1 * h2) * (h1 ")) * h1 = (h1 * h2) * ((h1 ") * h1) by GROUP_1:def_3
.= (h1 * h2) * (1_ H) by GROUP_1:def_5
.= x by A36, GROUP_1:def_4 ;
hence x in H9 * h1 by A12, A46, GROUP_2:28; ::_thesis: verum
end;
hence h1 * H9 c= H9 * h1 by TARSKI:def_3; ::_thesis: verum
end;
then for H1 being strict Subgroup of H st H1 = multMagma(# the carrier of H99, the multF of H99 #) holds
H1 is normal by GROUP_3:118;
hence H99 is normal by Def10; ::_thesis: verum
end;
hence ( f is onto & G9 is normal implies H99 is normal ) ; ::_thesis: verum
end;
theorem Th80: :: GROUP_9:80
for O being set
for G being strict GroupWithOperators of O
for N being strict normal StableSubgroup of G
for H being strict StableSubgroup of G ./. N st the carrier of G = (nat_hom N) " the carrier of H holds
H = (Omega). (G ./. N)
proof
let O be set ; ::_thesis: for G being strict GroupWithOperators of O
for N being strict normal StableSubgroup of G
for H being strict StableSubgroup of G ./. N st the carrier of G = (nat_hom N) " the carrier of H holds
H = (Omega). (G ./. N)
let G be strict GroupWithOperators of O; ::_thesis: for N being strict normal StableSubgroup of G
for H being strict StableSubgroup of G ./. N st the carrier of G = (nat_hom N) " the carrier of H holds
H = (Omega). (G ./. N)
let N be strict normal StableSubgroup of G; ::_thesis: for H being strict StableSubgroup of G ./. N st the carrier of G = (nat_hom N) " the carrier of H holds
H = (Omega). (G ./. N)
reconsider N9 = multMagma(# the carrier of N, the multF of N #) as strict normal Subgroup of G by Lm7;
let H be strict StableSubgroup of G ./. N; ::_thesis: ( the carrier of G = (nat_hom N) " the carrier of H implies H = (Omega). (G ./. N) )
reconsider H9 = multMagma(# the carrier of H, the multF of H #) as strict Subgroup of G ./. N by Lm16;
A1: ( the carrier of H9 c= the carrier of (G ./. N) & the multF of H9 = the multF of (G ./. N) || the carrier of H9 ) by GROUP_2:def_5;
( the carrier of (G ./. N) = the carrier of (G ./. N9) & the multF of (G ./. N) = the multF of (G ./. N9) ) by Def14, Def15;
then reconsider H9 = H9 as strict Subgroup of G ./. N9 by A1, GROUP_2:def_5;
assume the carrier of G = (nat_hom N) " the carrier of H ; ::_thesis: H = (Omega). (G ./. N)
then A2: the carrier of G = (nat_hom N9) " the carrier of H9 by Def20;
now__::_thesis:_for_h_being_Element_of_(G_./._N9)_holds_
(_(_h_in_H9_implies_h_in_(Omega)._(G_./._N9)_)_&_(_h_in_(Omega)._(G_./._N9)_implies_h_in_H9_)_)
reconsider R = nat_hom N9 as Relation of the carrier of G, the carrier of (G ./. N9) ;
let h be Element of (G ./. N9); ::_thesis: ( ( h in H9 implies h in (Omega). (G ./. N9) ) & ( h in (Omega). (G ./. N9) implies h in H9 ) )
thus ( h in H9 implies h in (Omega). (G ./. N9) ) by STRUCT_0:def_5; ::_thesis: ( h in (Omega). (G ./. N9) implies h in H9 )
assume h in (Omega). (G ./. N9) ; ::_thesis: h in H9
h in Left_Cosets N9 ;
then consider g being Element of G such that
A3: h = g * N9 by GROUP_2:def_15;
consider h9 being Element of (G ./. N9) such that
A4: [g,h9] in R and
A5: h9 in the carrier of H9 by A2, RELSET_1:30;
(nat_hom N9) . g = h9 by A4, FUNCT_1:1;
then h in the carrier of H9 by A3, A5, GROUP_6:def_8;
hence h in H9 by STRUCT_0:def_5; ::_thesis: verum
end;
then H9 = (Omega). (G ./. N9) by GROUP_2:def_6;
then the carrier of H = Cosets N by Def14;
hence H = (Omega). (G ./. N) by Lm5; ::_thesis: verum
end;
theorem Th81: :: GROUP_9:81
for O being set
for G being strict GroupWithOperators of O
for N being strict normal StableSubgroup of G
for H being strict StableSubgroup of G ./. N st the carrier of N = (nat_hom N) " the carrier of H holds
H = (1). (G ./. N)
proof
let O be set ; ::_thesis: for G being strict GroupWithOperators of O
for N being strict normal StableSubgroup of G
for H being strict StableSubgroup of G ./. N st the carrier of N = (nat_hom N) " the carrier of H holds
H = (1). (G ./. N)
let G be strict GroupWithOperators of O; ::_thesis: for N being strict normal StableSubgroup of G
for H being strict StableSubgroup of G ./. N st the carrier of N = (nat_hom N) " the carrier of H holds
H = (1). (G ./. N)
let N be strict normal StableSubgroup of G; ::_thesis: for H being strict StableSubgroup of G ./. N st the carrier of N = (nat_hom N) " the carrier of H holds
H = (1). (G ./. N)
reconsider N9 = multMagma(# the carrier of N, the multF of N #) as strict normal Subgroup of G by Lm7;
let H be strict StableSubgroup of G ./. N; ::_thesis: ( the carrier of N = (nat_hom N) " the carrier of H implies H = (1). (G ./. N) )
reconsider H9 = multMagma(# the carrier of H, the multF of H #) as strict Subgroup of G ./. N by Lm16;
A1: ( the carrier of H9 c= the carrier of (G ./. N) & the multF of H9 = the multF of (G ./. N) || the carrier of H9 ) by GROUP_2:def_5;
( the carrier of (G ./. N) = the carrier of (G ./. N9) & the multF of (G ./. N) = the multF of (G ./. N9) ) by Def14, Def15;
then reconsider H9 = H9 as strict Subgroup of G ./. N9 by A1, GROUP_2:def_5;
assume the carrier of N = (nat_hom N) " the carrier of H ; ::_thesis: H = (1). (G ./. N)
then A2: the carrier of N9 = (nat_hom N9) " the carrier of H9 by Def20;
assume not H = (1). (G ./. N) ; ::_thesis: contradiction
then not the carrier of H = {(1_ (G ./. N))} by Def8;
then consider h being set such that
A3: ( ( h in the carrier of H & not h in {(1_ (G ./. N))} ) or ( h in {(1_ (G ./. N))} & not h in the carrier of H ) ) by TARSKI:1;
percases ( ( h in the carrier of H & not h in {(1_ (G ./. N))} ) or ( not h in the carrier of H & h in {(1_ (G ./. N))} ) ) by A3;
supposeA4: ( h in the carrier of H & not h in {(1_ (G ./. N))} ) ; ::_thesis: contradiction
then {h} c= the carrier of H by ZFMISC_1:31;
then A5: (nat_hom N9) " {h} c= the carrier of N9 by A2, RELAT_1:143;
A6: rng (nat_hom N9) = the carrier of (Image (nat_hom N9)) by GROUP_6:44
.= the carrier of (G ./. N9) by GROUP_6:48 ;
the carrier of H9 c= the carrier of (G ./. N9) by GROUP_2:def_5;
then consider x being set such that
A7: x in dom (nat_hom N9) and
A8: (nat_hom N9) . x = h by A4, A6, FUNCT_1:def_3;
(nat_hom N9) . x in {h} by A8, TARSKI:def_1;
then x in (nat_hom N9) " {h} by A7, FUNCT_1:def_7;
then A9: x in N9 by A5, STRUCT_0:def_5;
h <> 1_ (G ./. N) by A4, TARSKI:def_1;
then A10: h <> carr N by Th43;
reconsider x = x as Element of G by A7;
x * N9 = h by A8, GROUP_6:def_8;
hence contradiction by A10, A9, GROUP_2:113; ::_thesis: verum
end;
suppose ( not h in the carrier of H & h in {(1_ (G ./. N))} ) ; ::_thesis: contradiction
then ( h = 1_ (G ./. N) & not h in H ) by STRUCT_0:def_5, TARSKI:def_1;
hence contradiction by Lm18; ::_thesis: verum
end;
end;
end;
theorem Th82: :: GROUP_9:82
for O being set
for G, H being strict GroupWithOperators of O st G,H are_isomorphic & G is simple holds
H is simple
proof
let O be set ; ::_thesis: for G, H being strict GroupWithOperators of O st G,H are_isomorphic & G is simple holds
H is simple
let G, H be strict GroupWithOperators of O; ::_thesis: ( G,H are_isomorphic & G is simple implies H is simple )
assume A1: G,H are_isomorphic ; ::_thesis: ( not G is simple or H is simple )
assume A2: G is simple ; ::_thesis: H is simple
assume A3: not H is simple ; ::_thesis: contradiction
percases ( H is trivial or ex H9 being strict normal StableSubgroup of H st
( H9 <> (Omega). H & H9 <> (1). H ) ) by A3, Def13;
suppose H is trivial ; ::_thesis: contradiction
then G is trivial by A1, Th58;
hence contradiction by A2, Def13; ::_thesis: verum
end;
suppose ex H9 being strict normal StableSubgroup of H st
( H9 <> (Omega). H & H9 <> (1). H ) ; ::_thesis: contradiction
then consider H9 being strict normal StableSubgroup of H such that
A4: H9 <> (Omega). H and
A5: H9 <> (1). H ;
consider f being Homomorphism of G,H such that
A6: f is bijective by A1, Def19;
reconsider H99 = multMagma(# the carrier of H9, the multF of H9 #) as strict normal Subgroup of H by Lm7;
multMagma(# the carrier of H9, the multF of H9 #) <> multMagma(# the carrier of H, the multF of H #) by A4, Lm5;
then consider h being Element of H such that
A7: not h in H99 by GROUP_2:62;
the carrier of H9 <> {(1_ H)} by A5, Def8;
then consider x being set such that
A8: x in the carrier of H9 and
A9: x <> 1_ H by ZFMISC_1:35;
A10: x in H99 by A8, STRUCT_0:def_5;
then x in H by GROUP_2:40;
then reconsider x = x as Element of H by STRUCT_0:def_5;
consider y being Element of G such that
A11: f . y = x by A6, Th52;
set A = { g where g is Element of G : f . g in H99 } ;
consider g being Element of G such that
A12: f . g = h by A6, Th52;
1_ H in H99 by GROUP_2:46;
then f . (1_ G) in H99 by Lm13;
then 1_ G in { g where g is Element of G : f . g in H99 } ;
then reconsider A = { g where g is Element of G : f . g in H99 } as non empty set ;
now__::_thesis:_for_x_being_set_st_x_in_A_holds_
x_in_the_carrier_of_G
let x be set ; ::_thesis: ( x in A implies x in the carrier of G )
assume x in A ; ::_thesis: x in the carrier of G
then ex g being Element of G st
( x = g & f . g in H99 ) ;
hence x in the carrier of G ; ::_thesis: verum
end;
then reconsider A = A as Subset of G by TARSKI:def_3;
A13: now__::_thesis:_for_g1,_g2_being_Element_of_G_st_g1_in_A_&_g2_in_A_holds_
g1_*_g2_in_A
let g1, g2 be Element of G; ::_thesis: ( g1 in A & g2 in A implies g1 * g2 in A )
assume that
A14: g1 in A and
A15: g2 in A ; ::_thesis: g1 * g2 in A
consider b being Element of G such that
A16: b = g2 and
A17: f . b in H99 by A15;
consider a being Element of G such that
A18: a = g1 and
A19: f . a in H99 by A14;
set fb = f . b;
set fa = f . a;
( f . (a * b) = (f . a) * (f . b) & (f . a) * (f . b) in H99 ) by A19, A17, GROUP_2:50, GROUP_6:def_6;
hence g1 * g2 in A by A18, A16; ::_thesis: verum
end;
A20: now__::_thesis:_for_o_being_Element_of_O
for_g_being_Element_of_G_st_g_in_A_holds_
(G_^_o)_._g_in_A
let o be Element of O; ::_thesis: for g being Element of G st g in A holds
(G ^ o) . g in A
let g be Element of G; ::_thesis: ( g in A implies (G ^ o) . g in A )
assume g in A ; ::_thesis: (G ^ o) . g in A
then consider a being Element of G such that
A21: a = g and
A22: f . a in H99 ;
f . a in the carrier of H99 by A22, STRUCT_0:def_5;
then f . a in H9 by STRUCT_0:def_5;
then (H ^ o) . (f . g) in H9 by A21, Lm10;
then f . ((G ^ o) . g) in H9 by Def18;
then f . ((G ^ o) . g) in the carrier of H9 by STRUCT_0:def_5;
then f . ((G ^ o) . g) in H99 by STRUCT_0:def_5;
hence (G ^ o) . g in A ; ::_thesis: verum
end;
now__::_thesis:_for_g_being_Element_of_G_st_g_in_A_holds_
g_"_in_A
let g be Element of G; ::_thesis: ( g in A implies g " in A )
assume g in A ; ::_thesis: g " in A
then consider a being Element of G such that
A23: a = g and
A24: f . a in H99 ;
(f . a) " in H99 by A24, GROUP_2:51;
then f . (a ") in H99 by Lm14;
hence g " in A by A23; ::_thesis: verum
end;
then consider G99 being strict StableSubgroup of G such that
A25: the carrier of G99 = A by A13, A20, Lm15;
reconsider G9 = multMagma(# the carrier of G99, the multF of G99 #) as strict Subgroup of G by Lm16;
now__::_thesis:_for_g_being_Element_of_G_holds_g_*_G9_c=_G9_*_g
let g be Element of G; ::_thesis: g * G9 c= G9 * g
now__::_thesis:_for_x_being_set_st_x_in_g_*_G9_holds_
x_in_G9_*_g
let x be set ; ::_thesis: ( x in g * G9 implies x in G9 * g )
A26: H99 |^ ((f . g) ") = H99 by GROUP_3:def_13;
assume x in g * G9 ; ::_thesis: x in G9 * g
then consider h being Element of G such that
A27: x = g * h and
A28: h in A by A25, GROUP_2:27;
set h9 = (g * h) * (g ");
A29: f . ((g * h) * (g ")) = (f . (g * h)) * (f . (g ")) by GROUP_6:def_6
.= ((f . g) * (f . h)) * (f . (g ")) by GROUP_6:def_6
.= ((f . g) * (f . h)) * ((f . g) ") by Lm14
.= ((((f . g) ") ") * (f . h)) * ((f . g) ")
.= (f . h) |^ ((f . g) ") by GROUP_3:def_2 ;
ex a being Element of G st
( a = h & f . a in H99 ) by A28;
then f . ((g * h) * (g ")) in H99 by A26, A29, GROUP_3:58;
then A30: (g * h) * (g ") in A ;
((g * h) * (g ")) * g = (g * h) * ((g ") * g) by GROUP_1:def_3
.= (g * h) * (1_ G) by GROUP_1:def_5
.= x by A27, GROUP_1:def_4 ;
hence x in G9 * g by A25, A30, GROUP_2:28; ::_thesis: verum
end;
hence g * G9 c= G9 * g by TARSKI:def_3; ::_thesis: verum
end;
then for H being strict Subgroup of G st H = multMagma(# the carrier of G99, the multF of G99 #) holds
H is normal by GROUP_3:118;
then A31: G99 is normal by Def10;
A32: y <> 1_ G by A9, A11, Lm13;
y in the carrier of G99 by A25, A10, A11;
then the carrier of G99 <> {(1_ G)} by A32, TARSKI:def_1;
then A33: G99 <> (1). G by Def8;
now__::_thesis:_not_g_in_A
assume g in A ; ::_thesis: contradiction
then ex g9 being Element of G st
( g9 = g & f . g9 in H99 ) ;
hence contradiction by A7, A12; ::_thesis: verum
end;
then G99 <> (Omega). G by A25;
hence contradiction by A2, A33, A31, Def13; ::_thesis: verum
end;
end;
end;
theorem Th83: :: GROUP_9:83
for O being set
for G being GroupWithOperators of O
for H being StableSubgroup of G
for FG being FinSequence of the carrier of G
for FH being FinSequence of the carrier of H
for I being FinSequence of INT st FG = FH & len FG = len I holds
Product (FG |^ I) = Product (FH |^ I)
proof
let O be set ; ::_thesis: for G being GroupWithOperators of O
for H being StableSubgroup of G
for FG being FinSequence of the carrier of G
for FH being FinSequence of the carrier of H
for I being FinSequence of INT st FG = FH & len FG = len I holds
Product (FG |^ I) = Product (FH |^ I)
let G be GroupWithOperators of O; ::_thesis: for H being StableSubgroup of G
for FG being FinSequence of the carrier of G
for FH being FinSequence of the carrier of H
for I being FinSequence of INT st FG = FH & len FG = len I holds
Product (FG |^ I) = Product (FH |^ I)
let H be StableSubgroup of G; ::_thesis: for FG being FinSequence of the carrier of G
for FH being FinSequence of the carrier of H
for I being FinSequence of INT st FG = FH & len FG = len I holds
Product (FG |^ I) = Product (FH |^ I)
let FG be FinSequence of the carrier of G; ::_thesis: for FH being FinSequence of the carrier of H
for I being FinSequence of INT st FG = FH & len FG = len I holds
Product (FG |^ I) = Product (FH |^ I)
let FH be FinSequence of the carrier of H; ::_thesis: for I being FinSequence of INT st FG = FH & len FG = len I holds
Product (FG |^ I) = Product (FH |^ I)
let I be FinSequence of INT ; ::_thesis: ( FG = FH & len FG = len I implies Product (FG |^ I) = Product (FH |^ I) )
assume A1: ( FG = FH & len FG = len I ) ; ::_thesis: Product (FG |^ I) = Product (FH |^ I)
defpred S1[ Nat] means for FG being FinSequence of the carrier of G
for FH being FinSequence of the carrier of H
for I being FinSequence of INT st len FG = $1 & FG = FH & len FG = len I holds
Product (FG |^ I) = Product (FH |^ I);
A2: now__::_thesis:_for_n_being_Nat_st_S1[n]_holds_
S1[n_+_1]
let n be Nat; ::_thesis: ( S1[n] implies S1[n + 1] )
assume A3: S1[n] ; ::_thesis: S1[n + 1]
thus S1[n + 1] ::_thesis: verum
proof
let FG be FinSequence of the carrier of G; ::_thesis: for FH being FinSequence of the carrier of H
for I being FinSequence of INT st len FG = n + 1 & FG = FH & len FG = len I holds
Product (FG |^ I) = Product (FH |^ I)
let FH be FinSequence of the carrier of H; ::_thesis: for I being FinSequence of INT st len FG = n + 1 & FG = FH & len FG = len I holds
Product (FG |^ I) = Product (FH |^ I)
let I be FinSequence of INT ; ::_thesis: ( len FG = n + 1 & FG = FH & len FG = len I implies Product (FG |^ I) = Product (FH |^ I) )
assume A4: len FG = n + 1 ; ::_thesis: ( not FG = FH or not len FG = len I or Product (FG |^ I) = Product (FH |^ I) )
then consider FGn being FinSequence of the carrier of G, g being Element of G such that
A5: FG = FGn ^ <*g*> by FINSEQ_2:19;
A6: len FG = (len FGn) + (len <*g*>) by A5, FINSEQ_1:22;
then A7: n + 1 = (len FGn) + 1 by A4, FINSEQ_1:40;
assume that
A8: FG = FH and
A9: len FG = len I ; ::_thesis: Product (FG |^ I) = Product (FH |^ I)
consider FHn being FinSequence of the carrier of H, h being Element of H such that
A10: FH = FHn ^ <*h*> by A4, A8, FINSEQ_2:19;
consider In being FinSequence of INT , i being Element of INT such that
A11: I = In ^ <*i*> by A4, A9, FINSEQ_2:19;
set FG1 = <*g*>;
set I1 = <*i*>;
len I = (len In) + (len <*i*>) by A11, FINSEQ_1:22;
then A12: n + 1 = (len In) + 1 by A4, A9, FINSEQ_1:40;
A13: len FH = (len FHn) + (len <*h*>) by A10, FINSEQ_1:22;
then A14: FH . (n + 1) = (FHn ^ <*h*>) . ((len FHn) + 1) by A4, A8, A10, FINSEQ_1:40
.= h by FINSEQ_1:42 ;
A15: n + 1 = (len FHn) + 1 by A4, A8, A13, FINSEQ_1:40;
A16: FG . (n + 1) = (FGn ^ <*g*>) . ((len FGn) + 1) by A4, A5, A6, FINSEQ_1:40
.= g by FINSEQ_1:42 ;
A17: now__::_thesis:_g_|^_i_=_h_|^_i
reconsider H9 = H as Subgroup of G by Def7;
reconsider h9 = h as Element of H9 ;
g |^ i = h9 |^ i by A8, A16, A14, GROUP_4:2;
hence g |^ i = h |^ i ; ::_thesis: verum
end;
len <*g*> = 1 by FINSEQ_1:40
.= len <*i*> by FINSEQ_1:40 ;
then A18: Product (FG |^ I) = Product ((FGn |^ In) ^ (<*g*> |^ <*i*>)) by A11, A5, A12, A7, GROUP_4:19
.= (Product (FGn |^ In)) * (Product (<*g*> |^ <*i*>)) by GROUP_4:5 ;
set FH1 = <*h*>;
A19: len <*h*> = 1 by FINSEQ_1:40
.= len <*i*> by FINSEQ_1:40 ;
A20: Product (<*g*> |^ <*i*>) = Product (<*g*> |^ <*(@ i)*>)
.= Product <*(g |^ i)*> by GROUP_4:22
.= h |^ i by A17, GROUP_4:9
.= Product <*(h |^ i)*> by GROUP_4:9
.= Product (<*h*> |^ <*(@ i)*>) by GROUP_4:22
.= Product (<*h*> |^ <*i*>) ;
FGn = FHn by A8, A5, A10, A16, A14, FINSEQ_1:33;
then Product (FGn |^ In) = Product (FHn |^ In) by A3, A12, A15;
then Product (FG |^ I) = (Product (FHn |^ In)) * (Product (<*h*> |^ <*i*>)) by A18, A20, Th3
.= Product ((FHn |^ In) ^ (<*h*> |^ <*i*>)) by GROUP_4:5
.= Product ((FHn ^ <*h*>) |^ (In ^ <*i*>)) by A12, A15, A19, GROUP_4:19 ;
hence Product (FG |^ I) = Product (FH |^ I) by A11, A10; ::_thesis: verum
end;
end;
A21: S1[ 0 ]
proof
let FG be FinSequence of the carrier of G; ::_thesis: for FH being FinSequence of the carrier of H
for I being FinSequence of INT st len FG = 0 & FG = FH & len FG = len I holds
Product (FG |^ I) = Product (FH |^ I)
let FH be FinSequence of the carrier of H; ::_thesis: for I being FinSequence of INT st len FG = 0 & FG = FH & len FG = len I holds
Product (FG |^ I) = Product (FH |^ I)
let I be FinSequence of INT ; ::_thesis: ( len FG = 0 & FG = FH & len FG = len I implies Product (FG |^ I) = Product (FH |^ I) )
assume A22: len FG = 0 ; ::_thesis: ( not FG = FH or not len FG = len I or Product (FG |^ I) = Product (FH |^ I) )
then len (FG |^ I) = 0 by GROUP_4:def_3;
then FG |^ I = <*> the carrier of G ;
then A23: Product (FG |^ I) = 1_ G by GROUP_4:8;
assume that
A24: FG = FH and
len FG = len I ; ::_thesis: Product (FG |^ I) = Product (FH |^ I)
len (FH |^ I) = 0 by A22, A24, GROUP_4:def_3;
then FH |^ I = <*> the carrier of H ;
then Product (FH |^ I) = 1_ H by GROUP_4:8;
hence Product (FG |^ I) = Product (FH |^ I) by A23, Th4; ::_thesis: verum
end;
for n being Nat holds S1[n] from NAT_1:sch_2(A21, A2);
hence Product (FG |^ I) = Product (FH |^ I) by A1; ::_thesis: verum
end;
theorem Th84: :: GROUP_9:84
for O, E1, E2 being set
for A1 being Action of O,E1
for A2 being Action of O,E2
for F being FinSequence of O st E1 c= E2 & ( for o being Element of O
for f1 being Function of E1,E1
for f2 being Function of E2,E2 st f1 = A1 . o & f2 = A2 . o holds
f1 = f2 | E1 ) holds
Product (F,A1) = (Product (F,A2)) | E1
proof
let O, E1, E2 be set ; ::_thesis: for A1 being Action of O,E1
for A2 being Action of O,E2
for F being FinSequence of O st E1 c= E2 & ( for o being Element of O
for f1 being Function of E1,E1
for f2 being Function of E2,E2 st f1 = A1 . o & f2 = A2 . o holds
f1 = f2 | E1 ) holds
Product (F,A1) = (Product (F,A2)) | E1
let A1 be Action of O,E1; ::_thesis: for A2 being Action of O,E2
for F being FinSequence of O st E1 c= E2 & ( for o being Element of O
for f1 being Function of E1,E1
for f2 being Function of E2,E2 st f1 = A1 . o & f2 = A2 . o holds
f1 = f2 | E1 ) holds
Product (F,A1) = (Product (F,A2)) | E1
let A2 be Action of O,E2; ::_thesis: for F being FinSequence of O st E1 c= E2 & ( for o being Element of O
for f1 being Function of E1,E1
for f2 being Function of E2,E2 st f1 = A1 . o & f2 = A2 . o holds
f1 = f2 | E1 ) holds
Product (F,A1) = (Product (F,A2)) | E1
let F be FinSequence of O; ::_thesis: ( E1 c= E2 & ( for o being Element of O
for f1 being Function of E1,E1
for f2 being Function of E2,E2 st f1 = A1 . o & f2 = A2 . o holds
f1 = f2 | E1 ) implies Product (F,A1) = (Product (F,A2)) | E1 )
defpred S1[ Element of NAT ] means for F being FinSequence of O st len F = $1 holds
Product (F,A1) = (Product (F,A2)) | E1;
assume A1: E1 c= E2 ; ::_thesis: ( ex o being Element of O ex f1 being Function of E1,E1 ex f2 being Function of E2,E2 st
( f1 = A1 . o & f2 = A2 . o & not f1 = f2 | E1 ) or Product (F,A1) = (Product (F,A2)) | E1 )
A2: S1[ 0 ]
proof
let F be FinSequence of O; ::_thesis: ( len F = 0 implies Product (F,A1) = (Product (F,A2)) | E1 )
A3: now__::_thesis:_for_x_being_set_st_x_in_dom_(id_E1)_holds_
(id_E1)_._x_=_(id_E2)_._x
let x be set ; ::_thesis: ( x in dom (id E1) implies (id E1) . x = (id E2) . x )
assume A4: x in dom (id E1) ; ::_thesis: (id E1) . x = (id E2) . x
then A5: x in E1 ;
thus (id E1) . x = x by A4, FUNCT_1:18
.= (id E2) . x by A1, A5, FUNCT_1:18 ; ::_thesis: verum
end;
E1 = E2 /\ E1 by A1, XBOOLE_1:28;
then dom (id E1) = E2 /\ E1 ;
then A6: dom (id E1) = (dom (id E2)) /\ E1 ;
assume A7: len F = 0 ; ::_thesis: Product (F,A1) = (Product (F,A2)) | E1
hence Product (F,A1) = id E1 by Def3
.= (id E2) | E1 by A6, A3, FUNCT_1:46
.= (Product (F,A2)) | E1 by A7, Def3 ;
::_thesis: verum
end;
assume A8: for o being Element of O
for f1 being Function of E1,E1
for f2 being Function of E2,E2 st f1 = A1 . o & f2 = A2 . o holds
f1 = f2 | E1 ; ::_thesis: Product (F,A1) = (Product (F,A2)) | E1
percases ( O is empty or not O is empty ) ;
suppose O is empty ; ::_thesis: Product (F,A1) = (Product (F,A2)) | E1
then len F = 0 ;
hence Product (F,A1) = (Product (F,A2)) | E1 by A2; ::_thesis: verum
end;
supposeA9: not O is empty ; ::_thesis: Product (F,A1) = (Product (F,A2)) | E1
A10: for k being Element of NAT st S1[k] holds
S1[k + 1]
proof
let k be Element of NAT ; ::_thesis: ( S1[k] implies S1[k + 1] )
assume A11: S1[k] ; ::_thesis: S1[k + 1]
now__::_thesis:_for_F_being_FinSequence_of_O_st_len_F_=_k_+_1_holds_
Product_(F,A1)_=_(Product_(F,A2))_|_E1
let F be FinSequence of O; ::_thesis: ( len F = k + 1 implies Product (F,A1) = (Product (F,A2)) | E1 )
assume A12: len F = k + 1 ; ::_thesis: Product (F,A1) = (Product (F,A2)) | E1
then consider Fk being FinSequence of O, o being Element of O such that
A13: F = Fk ^ <*o*> by FINSEQ_2:19;
len F = (len Fk) + (len <*o*>) by A13, FINSEQ_1:22;
then A14: k + 1 = (len Fk) + 1 by A12, FINSEQ_1:39;
A15: now__::_thesis:_for_x_being_set_st_x_in_dom_(Product_(F,A1))_holds_
(Product_(F,A1))_._x_=_((Product_(F,A2))_|_E1)_._x
{o} c= O by A9, ZFMISC_1:31;
then rng <*o*> c= O by FINSEQ_1:38;
then reconsider Fo = <*o*> as FinSequence of O by FINSEQ_1:def_4;
let x be set ; ::_thesis: ( x in dom (Product (F,A1)) implies (Product (F,A1)) . x = ((Product (F,A2)) | E1) . x )
assume A16: x in dom (Product (F,A1)) ; ::_thesis: (Product (F,A1)) . x = ((Product (F,A2)) | E1) . x
then A17: x in E1 ;
A18: o in O by A9;
then o in dom A1 by FUNCT_2:def_1;
then A1 . o in rng A1 by FUNCT_1:3;
then consider f1 being Function such that
A19: f1 = A1 . o and
A20: dom f1 = E1 and
A21: rng f1 c= E1 by FUNCT_2:def_2;
A22: Product (Fo,A1) = f1 by A9, A19, Lm26;
o in dom A2 by A18, FUNCT_2:def_1;
then A2 . o in rng A2 by FUNCT_1:3;
then consider f2 being Function such that
A23: f2 = A2 . o and
A24: dom f2 = E2 and
rng f2 c= E2 by FUNCT_2:def_2;
A25: Product (Fo,A2) = f2 by A9, A23, Lm26;
A26: f1 . x in rng f1 by A16, A20, FUNCT_1:3;
A27: Product (F,A2) = (Product (Fk,A2)) * (Product (Fo,A2)) by A9, A13, Lm29
.= (Product (Fk,A2)) * f2 by A9, A23, Lm26 ;
Product (F,A1) = (Product (Fk,A1)) * (Product (Fo,A1)) by A9, A13, Lm29
.= (Product (Fk,A1)) * f1 by A9, A19, Lm26 ;
hence (Product (F,A1)) . x = (Product (Fk,A1)) . (f1 . x) by A16, A20, FUNCT_1:13
.= ((Product (Fk,A2)) | E1) . (f1 . x) by A11, A14
.= (Product (Fk,A2)) . (f1 . x) by A21, A26, FUNCT_1:49
.= (Product (Fk,A2)) . ((f2 | E1) . x) by A8, A19, A23, A22, A25
.= (Product (Fk,A2)) . (f2 . x) by A16, FUNCT_1:49
.= ((Product (Fk,A2)) * f2) . x by A1, A17, A24, FUNCT_1:13
.= ((Product (F,A2)) | E1) . x by A16, A27, FUNCT_1:49 ;
::_thesis: verum
end;
Product (F,A2) in Funcs (E2,E2) by FUNCT_2:9;
then ex f2 being Function st
( Product (F,A2) = f2 & dom f2 = E2 & rng f2 c= E2 ) by FUNCT_2:def_2;
then A28: dom ((Product (F,A2)) | E1) = E2 /\ E1 by RELAT_1:61
.= E1 by A1, XBOOLE_1:28 ;
Product (F,A1) in Funcs (E1,E1) by FUNCT_2:9;
then ex f1 being Function st
( Product (F,A1) = f1 & dom f1 = E1 & rng f1 c= E1 ) by FUNCT_2:def_2;
hence Product (F,A1) = (Product (F,A2)) | E1 by A28, A15, FUNCT_1:2; ::_thesis: verum
end;
hence S1[k + 1] ; ::_thesis: verum
end;
A29: for k being Element of NAT holds S1[k] from NAT_1:sch_1(A2, A10);
reconsider k = len F as Element of NAT ;
k = len F ;
hence Product (F,A1) = (Product (F,A2)) | E1 by A29; ::_thesis: verum
end;
end;
end;
theorem Th85: :: GROUP_9:85
for O being set
for G being GroupWithOperators of O
for H1 being StableSubgroup of G
for N1, N2 being strict StableSubgroup of H1
for N19, N29 being strict StableSubgroup of G st N1 = N19 & N2 = N29 holds
N19 * N29 = N1 * N2
proof
let O be set ; ::_thesis: for G being GroupWithOperators of O
for H1 being StableSubgroup of G
for N1, N2 being strict StableSubgroup of H1
for N19, N29 being strict StableSubgroup of G st N1 = N19 & N2 = N29 holds
N19 * N29 = N1 * N2
let G be GroupWithOperators of O; ::_thesis: for H1 being StableSubgroup of G
for N1, N2 being strict StableSubgroup of H1
for N19, N29 being strict StableSubgroup of G st N1 = N19 & N2 = N29 holds
N19 * N29 = N1 * N2
let H1 be StableSubgroup of G; ::_thesis: for N1, N2 being strict StableSubgroup of H1
for N19, N29 being strict StableSubgroup of G st N1 = N19 & N2 = N29 holds
N19 * N29 = N1 * N2
let N1, N2 be strict StableSubgroup of H1; ::_thesis: for N19, N29 being strict StableSubgroup of G st N1 = N19 & N2 = N29 holds
N19 * N29 = N1 * N2
let N19, N29 be strict StableSubgroup of G; ::_thesis: ( N1 = N19 & N2 = N29 implies N19 * N29 = N1 * N2 )
set X = { (g * h) where g, h is Element of G : ( g in carr N19 & h in carr N29 ) } ;
set Y = { (g * h) where g, h is Element of H1 : ( g in carr N1 & h in carr N2 ) } ;
assume A1: ( N1 = N19 & N2 = N29 ) ; ::_thesis: N19 * N29 = N1 * N2
A2: now__::_thesis:_for_x_being_set_st_x_in__{__(g_*_h)_where_g,_h_is_Element_of_G_:_(_g_in_carr_N19_&_h_in_carr_N29_)__}__holds_
x_in__{__(g_*_h)_where_g,_h_is_Element_of_H1_:_(_g_in_carr_N1_&_h_in_carr_N2_)__}_
N2 is Subgroup of H1 by Def7;
then A3: the carrier of N2 c= the carrier of H1 by GROUP_2:def_5;
let x be set ; ::_thesis: ( x in { (g * h) where g, h is Element of G : ( g in carr N19 & h in carr N29 ) } implies x in { (g * h) where g, h is Element of H1 : ( g in carr N1 & h in carr N2 ) } )
assume x in { (g * h) where g, h is Element of G : ( g in carr N19 & h in carr N29 ) } ; ::_thesis: x in { (g * h) where g, h is Element of H1 : ( g in carr N1 & h in carr N2 ) }
then consider g, h being Element of G such that
A4: x = g * h and
A5: ( g in carr N19 & h in carr N29 ) ;
N1 is Subgroup of H1 by Def7;
then the carrier of N1 c= the carrier of H1 by GROUP_2:def_5;
then reconsider g = g, h = h as Element of H1 by A1, A5, A3;
x = g * h by A4, Th3;
hence x in { (g * h) where g, h is Element of H1 : ( g in carr N1 & h in carr N2 ) } by A1, A5; ::_thesis: verum
end;
now__::_thesis:_for_x_being_set_st_x_in__{__(g_*_h)_where_g,_h_is_Element_of_H1_:_(_g_in_carr_N1_&_h_in_carr_N2_)__}__holds_
x_in__{__(g_*_h)_where_g,_h_is_Element_of_G_:_(_g_in_carr_N19_&_h_in_carr_N29_)__}_
let x be set ; ::_thesis: ( x in { (g * h) where g, h is Element of H1 : ( g in carr N1 & h in carr N2 ) } implies x in { (g * h) where g, h is Element of G : ( g in carr N19 & h in carr N29 ) } )
assume x in { (g * h) where g, h is Element of H1 : ( g in carr N1 & h in carr N2 ) } ; ::_thesis: x in { (g * h) where g, h is Element of G : ( g in carr N19 & h in carr N29 ) }
then consider g, h being Element of H1 such that
A6: x = g * h and
A7: ( g in carr N1 & h in carr N2 ) ;
reconsider g = g, h = h as Element of G by Th2;
x = g * h by A6, Th3;
hence x in { (g * h) where g, h is Element of G : ( g in carr N19 & h in carr N29 ) } by A1, A7; ::_thesis: verum
end;
hence N19 * N29 = N1 * N2 by A2, TARSKI:1; ::_thesis: verum
end;
theorem Th86: :: GROUP_9:86
for O being set
for G being GroupWithOperators of O
for H1 being StableSubgroup of G
for N1, N2 being strict StableSubgroup of H1
for N19, N29 being strict StableSubgroup of G st N1 = N19 & N2 = N29 holds
N19 "\/" N29 = N1 "\/" N2
proof
let O be set ; ::_thesis: for G being GroupWithOperators of O
for H1 being StableSubgroup of G
for N1, N2 being strict StableSubgroup of H1
for N19, N29 being strict StableSubgroup of G st N1 = N19 & N2 = N29 holds
N19 "\/" N29 = N1 "\/" N2
let G be GroupWithOperators of O; ::_thesis: for H1 being StableSubgroup of G
for N1, N2 being strict StableSubgroup of H1
for N19, N29 being strict StableSubgroup of G st N1 = N19 & N2 = N29 holds
N19 "\/" N29 = N1 "\/" N2
let H1 be StableSubgroup of G; ::_thesis: for N1, N2 being strict StableSubgroup of H1
for N19, N29 being strict StableSubgroup of G st N1 = N19 & N2 = N29 holds
N19 "\/" N29 = N1 "\/" N2
let N1, N2 be strict StableSubgroup of H1; ::_thesis: for N19, N29 being strict StableSubgroup of G st N1 = N19 & N2 = N29 holds
N19 "\/" N29 = N1 "\/" N2
reconsider S2 = the_stable_subgroup_of (N1 * N2) as StableSubgroup of G by Th11;
let N19, N29 be strict StableSubgroup of G; ::_thesis: ( N1 = N19 & N2 = N29 implies N19 "\/" N29 = N1 "\/" N2 )
set S1 = the_stable_subgroup_of (N19 * N29);
set X1 = { B where B is Subset of G : ex H being strict StableSubgroup of G st
( B = the carrier of H & N19 * N29 c= carr H ) } ;
set X2 = { B where B is Subset of H1 : ex H being strict StableSubgroup of H1 st
( B = the carrier of H & N1 * N2 c= carr H ) } ;
A1: ( N19 "\/" N29 = the_stable_subgroup_of (N19 * N29) & N1 "\/" N2 = the_stable_subgroup_of (N1 * N2) ) by Th29;
A2: ( the carrier of (the_stable_subgroup_of (N19 * N29)) = meet { B where B is Subset of G : ex H being strict StableSubgroup of G st
( B = the carrier of H & N19 * N29 c= carr H ) } & the carrier of (the_stable_subgroup_of (N1 * N2)) = meet { B where B is Subset of H1 : ex H being strict StableSubgroup of H1 st
( B = the carrier of H & N1 * N2 c= carr H ) } ) by Th27;
assume A3: ( N1 = N19 & N2 = N29 ) ; ::_thesis: N19 "\/" N29 = N1 "\/" N2
now__::_thesis:_for_x_being_set_st_x_in__{__B_where_B_is_Subset_of_H1_:_ex_H_being_strict_StableSubgroup_of_H1_st_
(_B_=_the_carrier_of_H_&_N1_*_N2_c=_carr_H_)__}__holds_
x_in__{__B_where_B_is_Subset_of_G_:_ex_H_being_strict_StableSubgroup_of_G_st_
(_B_=_the_carrier_of_H_&_N19_*_N29_c=_carr_H_)__}_
let x be set ; ::_thesis: ( x in { B where B is Subset of H1 : ex H being strict StableSubgroup of H1 st
( B = the carrier of H & N1 * N2 c= carr H ) } implies x in { B where B is Subset of G : ex H being strict StableSubgroup of G st
( B = the carrier of H & N19 * N29 c= carr H ) } )
assume x in { B where B is Subset of H1 : ex H being strict StableSubgroup of H1 st
( B = the carrier of H & N1 * N2 c= carr H ) } ; ::_thesis: x in { B where B is Subset of G : ex H being strict StableSubgroup of G st
( B = the carrier of H & N19 * N29 c= carr H ) }
then consider B being Subset of H1 such that
A4: x = B and
A5: ex H being strict StableSubgroup of H1 st
( B = the carrier of H & N1 * N2 c= carr H ) ;
now__::_thesis:_ex_H_being_strict_StableSubgroup_of_G_st_
(_B_=_the_carrier_of_H_&_N19_*_N29_c=_carr_H_)
consider H being strict StableSubgroup of H1 such that
A6: ( B = the carrier of H & N1 * N2 c= carr H ) by A5;
reconsider H = H as strict StableSubgroup of G by Th11;
take H = H; ::_thesis: ( B = the carrier of H & N19 * N29 c= carr H )
thus ( B = the carrier of H & N19 * N29 c= carr H ) by A3, A6, Th85; ::_thesis: verum
end;
hence x in { B where B is Subset of G : ex H being strict StableSubgroup of G st
( B = the carrier of H & N19 * N29 c= carr H ) } by A4; ::_thesis: verum
end;
then A7: { B where B is Subset of H1 : ex H being strict StableSubgroup of H1 st
( B = the carrier of H & N1 * N2 c= carr H ) } c= { B where B is Subset of G : ex H being strict StableSubgroup of G st
( B = the carrier of H & N19 * N29 c= carr H ) } by TARSKI:def_3;
now__::_thesis:_ex_x_being_set_st_x_in__{__B_where_B_is_Subset_of_H1_:_ex_H_being_strict_StableSubgroup_of_H1_st_
(_B_=_the_carrier_of_H_&_N1_*_N2_c=_carr_H_)__}_
set x9 = carr H1;
reconsider x = carr H1 as set ;
take x = x; ::_thesis: x in { B where B is Subset of H1 : ex H being strict StableSubgroup of H1 st
( B = the carrier of H & N1 * N2 c= carr H ) }
now__::_thesis:_ex_H_being_strict_StableSubgroup_of_H1_st_
(_carr_H1_=_the_carrier_of_H_&_N1_*_N2_c=_carr_H_)
set H = (Omega). H1;
take H = (Omega). H1; ::_thesis: ( carr H1 = the carrier of H & N1 * N2 c= carr H )
thus carr H1 = the carrier of H ; ::_thesis: N1 * N2 c= carr H
thus N1 * N2 c= carr H ; ::_thesis: verum
end;
hence x in { B where B is Subset of H1 : ex H being strict StableSubgroup of H1 st
( B = the carrier of H & N1 * N2 c= carr H ) } ; ::_thesis: verum
end;
then A8: meet { B where B is Subset of G : ex H being strict StableSubgroup of G st
( B = the carrier of H & N19 * N29 c= carr H ) } c= meet { B where B is Subset of H1 : ex H being strict StableSubgroup of H1 st
( B = the carrier of H & N1 * N2 c= carr H ) } by A7, SETFAM_1:6;
now__::_thesis:_for_x_being_set_st_x_in_the_carrier_of_(the_stable_subgroup_of_(N1_*_N2))_holds_
x_in_the_carrier_of_(the_stable_subgroup_of_(N19_*_N29))
let x be set ; ::_thesis: ( x in the carrier of (the_stable_subgroup_of (N1 * N2)) implies x in the carrier of (the_stable_subgroup_of (N19 * N29)) )
assume A9: x in the carrier of (the_stable_subgroup_of (N1 * N2)) ; ::_thesis: x in the carrier of (the_stable_subgroup_of (N19 * N29))
the_stable_subgroup_of (N1 * N2) is Subgroup of H1 by Def7;
then the carrier of (the_stable_subgroup_of (N1 * N2)) c= the carrier of H1 by GROUP_2:def_5;
then reconsider g = x as Element of H1 by A9;
g in the_stable_subgroup_of (N1 * N2) by A9, STRUCT_0:def_5;
then consider F being FinSequence of the carrier of H1, I being FinSequence of INT , C being Subset of H1 such that
A10: C = the_stable_subset_generated_by ((N1 * N2), the action of H1) and
A11: len F = len I and
A12: rng F c= C and
A13: Product (F |^ I) = g by Th24;
now__::_thesis:_for_x_being_set_st_x_in_the_stable_subset_generated_by_((N1_*_N2),_the_action_of_H1)_holds_
x_in_the_stable_subset_generated_by_((N19_*_N29),_the_action_of_G)
N2 is Subgroup of H1 by Def7;
then 1_ H1 in N2 by GROUP_2:46;
then A14: 1_ H1 in carr N2 by STRUCT_0:def_5;
let x be set ; ::_thesis: ( x in the_stable_subset_generated_by ((N1 * N2), the action of H1) implies x in the_stable_subset_generated_by ((N19 * N29), the action of G) )
assume A15: x in the_stable_subset_generated_by ((N1 * N2), the action of H1) ; ::_thesis: x in the_stable_subset_generated_by ((N19 * N29), the action of G)
then reconsider a = x as Element of H1 ;
N1 is Subgroup of H1 by Def7;
then 1_ H1 in N1 by GROUP_2:46;
then A16: 1_ H1 in carr N1 by STRUCT_0:def_5;
1_ H1 = (1_ H1) * (1_ H1) by GROUP_1:def_4;
then A17: 1_ H1 in (carr N1) * (carr N2) by A16, A14;
then consider F being FinSequence of O, h being Element of N1 * N2 such that
A18: (Product (F, the action of H1)) . h = a by A15, Lm31;
H1 is Subgroup of G by Def7;
then A19: the carrier of H1 c= the carrier of G by GROUP_2:def_5;
then reconsider a = a as Element of G by TARSKI:def_3;
A20: h in N1 * N2 by A17;
reconsider h = h as Element of N19 * N29 by A3, Th85;
now__::_thesis:_for_o_being_Element_of_O
for_f1_being_Function_of_the_carrier_of_H1,_the_carrier_of_H1
for_f2_being_Function_of_the_carrier_of_G,_the_carrier_of_G_st_f1_=_the_action_of_H1_._o_&_f2_=_the_action_of_G_._o_holds_
f1_=_f2_|_the_carrier_of_H1
let o be Element of O; ::_thesis: for f1 being Function of the carrier of H1, the carrier of H1
for f2 being Function of the carrier of G, the carrier of G st f1 = the action of H1 . o & f2 = the action of G . o holds
b4 = b5 | the carrier of H1
let f1 be Function of the carrier of H1, the carrier of H1; ::_thesis: for f2 being Function of the carrier of G, the carrier of G st f1 = the action of H1 . o & f2 = the action of G . o holds
b3 = b4 | the carrier of H1
let f2 be Function of the carrier of G, the carrier of G; ::_thesis: ( f1 = the action of H1 . o & f2 = the action of G . o implies b2 = b3 | the carrier of H1 )
assume that
A21: f1 = the action of H1 . o and
A22: f2 = the action of G . o ; ::_thesis: b2 = b3 | the carrier of H1
percases ( o in O or not o in O ) ;
suppose o in O ; ::_thesis: b2 = b3 | the carrier of H1
then ( H1 ^ o = f1 & G ^ o = f2 ) by A21, A22, Def6;
hence f1 = f2 | the carrier of H1 by Def7; ::_thesis: verum
end;
suppose not o in O ; ::_thesis: b2 = b3 | the carrier of H1
then not o in dom the action of H1 ;
hence f1 = f2 | the carrier of H1 by A21, FUNCT_1:def_2; ::_thesis: verum
end;
end;
end;
then Product (F, the action of H1) = (Product (F, the action of G)) | the carrier of H1 by A19, Th84;
then A23: (Product (F, the action of G)) . h = a by A18, A20, FUNCT_1:49;
not N19 * N29 is empty by A3, A20, Th85;
hence x in the_stable_subset_generated_by ((N19 * N29), the action of G) by A23, Lm31; ::_thesis: verum
end;
then the_stable_subset_generated_by ((N1 * N2), the action of H1) c= the_stable_subset_generated_by ((N19 * N29), the action of G) by TARSKI:def_3;
then A24: rng F c= the_stable_subset_generated_by ((N19 * N29), the action of G) by A10, A12, XBOOLE_1:1;
reconsider g = g as Element of G by Th2;
H1 is Subgroup of G by Def7;
then the carrier of H1 c= the carrier of G by GROUP_2:def_5;
then rng F c= the carrier of G by XBOOLE_1:1;
then reconsider F = F as FinSequence of the carrier of G by FINSEQ_1:def_4;
Product (F |^ I) = g by A11, A13, Th83;
then A25: g in the_stable_subgroup_of (N19 * N29) by A11, A24, Th24;
assume not x in the carrier of (the_stable_subgroup_of (N19 * N29)) ; ::_thesis: contradiction
hence contradiction by A25, STRUCT_0:def_5; ::_thesis: verum
end;
then meet { B where B is Subset of H1 : ex H being strict StableSubgroup of H1 st
( B = the carrier of H & N1 * N2 c= carr H ) } c= meet { B where B is Subset of G : ex H being strict StableSubgroup of G st
( B = the carrier of H & N19 * N29 c= carr H ) } by A2, TARSKI:def_3;
then the carrier of (the_stable_subgroup_of (N19 * N29)) = the carrier of S2 by A2, A8, XBOOLE_0:def_10;
hence N19 "\/" N29 = N1 "\/" N2 by A1, Lm5; ::_thesis: verum
end;
theorem Th87: :: GROUP_9:87
for O being set
for G being GroupWithOperators of O
for H1 being StableSubgroup of G
for N1, N2 being strict StableSubgroup of G st N1 is normal StableSubgroup of H1 & N2 is normal StableSubgroup of H1 holds
N1 "\/" N2 is normal StableSubgroup of H1
proof
let O be set ; ::_thesis: for G being GroupWithOperators of O
for H1 being StableSubgroup of G
for N1, N2 being strict StableSubgroup of G st N1 is normal StableSubgroup of H1 & N2 is normal StableSubgroup of H1 holds
N1 "\/" N2 is normal StableSubgroup of H1
let G be GroupWithOperators of O; ::_thesis: for H1 being StableSubgroup of G
for N1, N2 being strict StableSubgroup of G st N1 is normal StableSubgroup of H1 & N2 is normal StableSubgroup of H1 holds
N1 "\/" N2 is normal StableSubgroup of H1
let H1 be StableSubgroup of G; ::_thesis: for N1, N2 being strict StableSubgroup of G st N1 is normal StableSubgroup of H1 & N2 is normal StableSubgroup of H1 holds
N1 "\/" N2 is normal StableSubgroup of H1
let N1, N2 be strict StableSubgroup of G; ::_thesis: ( N1 is normal StableSubgroup of H1 & N2 is normal StableSubgroup of H1 implies N1 "\/" N2 is normal StableSubgroup of H1 )
assume A1: ( N1 is normal StableSubgroup of H1 & N2 is normal StableSubgroup of H1 ) ; ::_thesis: N1 "\/" N2 is normal StableSubgroup of H1
then reconsider N19 = N1, N29 = N2 as StableSubgroup of H1 ;
N1 "\/" N2 = N19 "\/" N29 by Th86;
hence N1 "\/" N2 is normal StableSubgroup of H1 by A1, Th32; ::_thesis: verum
end;
theorem Th88: :: GROUP_9:88
for O being set
for G, H, I being GroupWithOperators of O
for f being Homomorphism of G,H
for g being Homomorphism of H,I holds the carrier of (Ker (g * f)) = f " the carrier of (Ker g)
proof
let O be set ; ::_thesis: for G, H, I being GroupWithOperators of O
for f being Homomorphism of G,H
for g being Homomorphism of H,I holds the carrier of (Ker (g * f)) = f " the carrier of (Ker g)
let G, H, I be GroupWithOperators of O; ::_thesis: for f being Homomorphism of G,H
for g being Homomorphism of H,I holds the carrier of (Ker (g * f)) = f " the carrier of (Ker g)
let f be Homomorphism of G,H; ::_thesis: for g being Homomorphism of H,I holds the carrier of (Ker (g * f)) = f " the carrier of (Ker g)
let g be Homomorphism of H,I; ::_thesis: the carrier of (Ker (g * f)) = f " the carrier of (Ker g)
A1: now__::_thesis:_for_x_being_set_st_x_in_f_"_the_carrier_of_(Ker_g)_holds_
x_in_the_carrier_of_(Ker_(g_*_f))
let x be set ; ::_thesis: ( x in f " the carrier of (Ker g) implies x in the carrier of (Ker (g * f)) )
assume A2: x in f " the carrier of (Ker g) ; ::_thesis: x in the carrier of (Ker (g * f))
then f . x in the carrier of (Ker g) by FUNCT_1:def_7;
then f . x in { b where b is Element of H : g . b = 1_ I } by Def21;
then A3: ex b being Element of H st
( b = f . x & g . b = 1_ I ) ;
x in dom f by A2, FUNCT_1:def_7;
then 1_ I = (g * f) . x by A3, FUNCT_1:13;
then x in { a9 where a9 is Element of G : (g * f) . a9 = 1_ I } by A2;
hence x in the carrier of (Ker (g * f)) by Def21; ::_thesis: verum
end;
A4: dom f = the carrier of G by FUNCT_2:def_1;
now__::_thesis:_for_x_being_set_st_x_in_the_carrier_of_(Ker_(g_*_f))_holds_
x_in_f_"_the_carrier_of_(Ker_g)
let x be set ; ::_thesis: ( x in the carrier of (Ker (g * f)) implies x in f " the carrier of (Ker g) )
assume x in the carrier of (Ker (g * f)) ; ::_thesis: x in f " the carrier of (Ker g)
then x in { a where a is Element of G : (g * f) . a = 1_ I } by Def21;
then consider a being Element of G such that
A5: x = a and
A6: (g * f) . a = 1_ I ;
reconsider b = f . a as Element of H ;
g . b = 1_ I by A4, A6, FUNCT_1:13;
then f . x in { b9 where b9 is Element of H : g . b9 = 1_ I } by A5;
then f . x in the carrier of (Ker g) by Def21;
hence x in f " the carrier of (Ker g) by A4, A5, FUNCT_1:def_7; ::_thesis: verum
end;
hence the carrier of (Ker (g * f)) = f " the carrier of (Ker g) by A1, TARSKI:1; ::_thesis: verum
end;
theorem Th89: :: GROUP_9:89
for O being set
for G, H being GroupWithOperators of O
for G9 being StableSubgroup of G
for H9 being StableSubgroup of H
for f being Homomorphism of G,H st ( the carrier of H9 = f .: the carrier of G9 or the carrier of G9 = f " the carrier of H9 ) holds
f | the carrier of G9 is Homomorphism of G9,H9
proof
let O be set ; ::_thesis: for G, H being GroupWithOperators of O
for G9 being StableSubgroup of G
for H9 being StableSubgroup of H
for f being Homomorphism of G,H st ( the carrier of H9 = f .: the carrier of G9 or the carrier of G9 = f " the carrier of H9 ) holds
f | the carrier of G9 is Homomorphism of G9,H9
let G, H be GroupWithOperators of O; ::_thesis: for G9 being StableSubgroup of G
for H9 being StableSubgroup of H
for f being Homomorphism of G,H st ( the carrier of H9 = f .: the carrier of G9 or the carrier of G9 = f " the carrier of H9 ) holds
f | the carrier of G9 is Homomorphism of G9,H9
let G9 be StableSubgroup of G; ::_thesis: for H9 being StableSubgroup of H
for f being Homomorphism of G,H st ( the carrier of H9 = f .: the carrier of G9 or the carrier of G9 = f " the carrier of H9 ) holds
f | the carrier of G9 is Homomorphism of G9,H9
let H9 be StableSubgroup of H; ::_thesis: for f being Homomorphism of G,H st ( the carrier of H9 = f .: the carrier of G9 or the carrier of G9 = f " the carrier of H9 ) holds
f | the carrier of G9 is Homomorphism of G9,H9
let f be Homomorphism of G,H; ::_thesis: ( ( the carrier of H9 = f .: the carrier of G9 or the carrier of G9 = f " the carrier of H9 ) implies f | the carrier of G9 is Homomorphism of G9,H9 )
set g = f | the carrier of G9;
G9 is Subgroup of G by Def7;
then A1: the carrier of G9 c= the carrier of G by GROUP_2:def_5;
then A2: the carrier of G9 c= dom f by FUNCT_2:def_1;
then A3: dom (f | the carrier of G9) = the carrier of G9 by RELAT_1:62;
assume A4: ( the carrier of H9 = f .: the carrier of G9 or the carrier of G9 = f " the carrier of H9 ) ; ::_thesis: f | the carrier of G9 is Homomorphism of G9,H9
A5: for x being set st x in the carrier of G9 holds
f . x in the carrier of H9
proof
let x be set ; ::_thesis: ( x in the carrier of G9 implies f . x in the carrier of H9 )
assume A6: x in the carrier of G9 ; ::_thesis: f . x in the carrier of H9
percases ( the carrier of H9 = f .: the carrier of G9 or the carrier of G9 = f " the carrier of H9 ) by A4;
supposeA7: the carrier of H9 = f .: the carrier of G9 ; ::_thesis: f . x in the carrier of H9
assume not f . x in the carrier of H9 ; ::_thesis: contradiction
hence contradiction by A2, A6, A7, FUNCT_1:def_6; ::_thesis: verum
end;
suppose the carrier of G9 = f " the carrier of H9 ; ::_thesis: f . x in the carrier of H9
hence f . x in the carrier of H9 by A6, FUNCT_1:def_7; ::_thesis: verum
end;
end;
end;
now__::_thesis:_for_y_being_set_st_y_in_rng_(f_|_the_carrier_of_G9)_holds_
y_in_the_carrier_of_H9
let y be set ; ::_thesis: ( y in rng (f | the carrier of G9) implies y in the carrier of H9 )
assume y in rng (f | the carrier of G9) ; ::_thesis: y in the carrier of H9
then consider x being set such that
A8: x in dom (f | the carrier of G9) and
A9: y = (f | the carrier of G9) . x by FUNCT_1:def_3;
A10: x in the carrier of G9 by A2, A8, RELAT_1:62;
then y = f . x by A9, FUNCT_1:49;
hence y in the carrier of H9 by A5, A10; ::_thesis: verum
end;
then rng (f | the carrier of G9) c= the carrier of H9 by TARSKI:def_3;
then reconsider g = f | the carrier of G9 as Function of G9,H9 by A3, RELSET_1:4;
A11: now__::_thesis:_for_a9,_b9_being_Element_of_G9_holds_g_._(a9_*_b9)_=_(g_._a9)_*_(g_._b9)
let a9, b9 be Element of G9; ::_thesis: g . (a9 * b9) = (g . a9) * (g . b9)
reconsider a = a9, b = b9 as Element of G by A1, TARSKI:def_3;
A12: ( f . a = g . a9 & f . b = g . b9 ) by FUNCT_1:49;
thus g . (a9 * b9) = f . (a9 * b9) by FUNCT_1:49
.= f . (a * b) by Th3
.= (f . a) * (f . b) by GROUP_6:def_6
.= (g . a9) * (g . b9) by A12, Th3 ; ::_thesis: verum
end;
now__::_thesis:_for_o_being_Element_of_O
for_a9_being_Element_of_G9_holds_g_._((G9_^_o)_._a9)_=_(H9_^_o)_._(g_._a9)
let o be Element of O; ::_thesis: for a9 being Element of G9 holds g . ((G9 ^ o) . a9) = (H9 ^ o) . (g . a9)
let a9 be Element of G9; ::_thesis: g . ((G9 ^ o) . a9) = (H9 ^ o) . (g . a9)
reconsider a = a9 as Element of G by A1, TARSKI:def_3;
thus g . ((G9 ^ o) . a9) = f . ((G9 ^ o) . a9) by FUNCT_1:49
.= f . (((G ^ o) | the carrier of G9) . a9) by Def7
.= f . ((G ^ o) . a) by FUNCT_1:49
.= (H ^ o) . (f . a) by Def18
.= (H ^ o) . (g . a9) by FUNCT_1:49
.= ((H ^ o) | the carrier of H9) . (g . a9) by FUNCT_1:49
.= (H9 ^ o) . (g . a9) by Def7 ; ::_thesis: verum
end;
hence f | the carrier of G9 is Homomorphism of G9,H9 by A11, Def18, GROUP_6:def_6; ::_thesis: verum
end;
theorem Th90: :: GROUP_9:90
for O being set
for G, H being strict GroupWithOperators of O
for N, L, G9 being strict StableSubgroup of G
for f being Homomorphism of G,H st N = Ker f & L is strict normal StableSubgroup of G9 holds
( L "\/" (G9 /\ N) is normal StableSubgroup of G9 & L "\/" N is normal StableSubgroup of G9 "\/" N & ( for N1 being strict normal StableSubgroup of G9 "\/" N
for N2 being strict normal StableSubgroup of G9 st N1 = L "\/" N & N2 = L "\/" (G9 /\ N) holds
(G9 "\/" N) ./. N1,G9 ./. N2 are_isomorphic ) )
proof
let O be set ; ::_thesis: for G, H being strict GroupWithOperators of O
for N, L, G9 being strict StableSubgroup of G
for f being Homomorphism of G,H st N = Ker f & L is strict normal StableSubgroup of G9 holds
( L "\/" (G9 /\ N) is normal StableSubgroup of G9 & L "\/" N is normal StableSubgroup of G9 "\/" N & ( for N1 being strict normal StableSubgroup of G9 "\/" N
for N2 being strict normal StableSubgroup of G9 st N1 = L "\/" N & N2 = L "\/" (G9 /\ N) holds
(G9 "\/" N) ./. N1,G9 ./. N2 are_isomorphic ) )
let G, H be strict GroupWithOperators of O; ::_thesis: for N, L, G9 being strict StableSubgroup of G
for f being Homomorphism of G,H st N = Ker f & L is strict normal StableSubgroup of G9 holds
( L "\/" (G9 /\ N) is normal StableSubgroup of G9 & L "\/" N is normal StableSubgroup of G9 "\/" N & ( for N1 being strict normal StableSubgroup of G9 "\/" N
for N2 being strict normal StableSubgroup of G9 st N1 = L "\/" N & N2 = L "\/" (G9 /\ N) holds
(G9 "\/" N) ./. N1,G9 ./. N2 are_isomorphic ) )
let N, L, G9 be strict StableSubgroup of G; ::_thesis: for f being Homomorphism of G,H st N = Ker f & L is strict normal StableSubgroup of G9 holds
( L "\/" (G9 /\ N) is normal StableSubgroup of G9 & L "\/" N is normal StableSubgroup of G9 "\/" N & ( for N1 being strict normal StableSubgroup of G9 "\/" N
for N2 being strict normal StableSubgroup of G9 st N1 = L "\/" N & N2 = L "\/" (G9 /\ N) holds
(G9 "\/" N) ./. N1,G9 ./. N2 are_isomorphic ) )
reconsider N9 = G9 /\ N as StableSubgroup of G9 by Lm34;
reconsider Gs9 = multMagma(# the carrier of G9, the multF of G9 #) as strict Subgroup of G by Lm16;
let f be Homomorphism of G,H; ::_thesis: ( N = Ker f & L is strict normal StableSubgroup of G9 implies ( L "\/" (G9 /\ N) is normal StableSubgroup of G9 & L "\/" N is normal StableSubgroup of G9 "\/" N & ( for N1 being strict normal StableSubgroup of G9 "\/" N
for N2 being strict normal StableSubgroup of G9 st N1 = L "\/" N & N2 = L "\/" (G9 /\ N) holds
(G9 "\/" N) ./. N1,G9 ./. N2 are_isomorphic ) ) )
reconsider L99 = L as Subgroup of G by Def7;
assume A1: N = Ker f ; ::_thesis: ( not L is strict normal StableSubgroup of G9 or ( L "\/" (G9 /\ N) is normal StableSubgroup of G9 & L "\/" N is normal StableSubgroup of G9 "\/" N & ( for N1 being strict normal StableSubgroup of G9 "\/" N
for N2 being strict normal StableSubgroup of G9 st N1 = L "\/" N & N2 = L "\/" (G9 /\ N) holds
(G9 "\/" N) ./. N1,G9 ./. N2 are_isomorphic ) ) )
then consider H9 being strict StableSubgroup of H such that
A2: the carrier of H9 = f .: the carrier of G9 and
A3: f " the carrier of H9 = the carrier of (G9 "\/" N) and
( f is onto & G9 is normal implies H9 is normal ) by Th79;
reconsider f99 = f | the carrier of (G9 "\/" N) as Homomorphism of (G9 "\/" N),H9 by A3, Th89;
reconsider Ns = multMagma(# the carrier of N, the multF of N #) as strict normal Subgroup of G by A1, Lm7;
(carr Gs9) * Ns = Ns * (carr Gs9) by GROUP_3:120;
then A4: G9 * N = N * G9 ;
A5: now__::_thesis:_for_y_being_set_st_y_in_f_.:_the_carrier_of_G9_holds_
y_in_f_.:_(G9_*_N)
let y be set ; ::_thesis: ( y in f .: the carrier of G9 implies y in f .: (G9 * N) )
assume y in f .: the carrier of G9 ; ::_thesis: y in f .: (G9 * N)
then consider x being set such that
A6: x in dom f and
A7: x in the carrier of G9 and
A8: y = f . x by FUNCT_1:def_6;
reconsider x = x as Element of G by A6;
consider x9 being set such that
A9: x9 = x * (1_ G) ;
A10: x9 in dom f by A6, A9, GROUP_1:def_4;
A11: y = (f . x) * (1_ H) by A8, GROUP_1:def_4
.= (f . x) * (f . (1_ G)) by Lm13
.= f . x9 by A9, GROUP_6:def_6 ;
f . (1_ G) = 1_ H by Lm13;
then 1_ G in Ker f by Th47;
then 1_ G in carr N by A1, STRUCT_0:def_5;
then x9 in G9 * N by A7, A9;
hence y in f .: (G9 * N) by A10, A11, FUNCT_1:def_6; ::_thesis: verum
end;
A12: dom f = the carrier of G by FUNCT_2:def_1;
now__::_thesis:_for_y_being_set_st_y_in_f_.:_(G9_*_N)_holds_
y_in_f_.:_the_carrier_of_G9
let y be set ; ::_thesis: ( y in f .: (G9 * N) implies y in f .: the carrier of G9 )
assume y in f .: (G9 * N) ; ::_thesis: y in f .: the carrier of G9
then consider x being set such that
A13: x in dom f and
A14: x in G9 * N and
A15: y = f . x by FUNCT_1:def_6;
reconsider x = x as Element of G by A13;
consider g1, g2 being Element of G such that
A16: x = g1 * g2 and
A17: g1 in carr G9 and
A18: g2 in carr N by A14;
A19: g2 in N by A18, STRUCT_0:def_5;
y = (f . g1) * (f . g2) by A15, A16, GROUP_6:def_6
.= (f . g1) * (1_ H) by A1, A19, Th47
.= f . g1 by GROUP_1:def_4 ;
hence y in f .: the carrier of G9 by A12, A17, FUNCT_1:def_6; ::_thesis: verum
end;
then f .: the carrier of G9 = f .: (G9 * N) by A5, TARSKI:1;
then A20: ( f99 .: the carrier of (G9 "\/" N) = f .: the carrier of (G9 "\/" N) & the carrier of H9 = f .: the carrier of (G9 "\/" N) ) by A2, A4, Th30, RELAT_1:129;
A21: now__::_thesis:_for_x_being_set_st_x_in_f99_"_(f_.:_the_carrier_of_L)_holds_
x_in_L_*_N
let x be set ; ::_thesis: ( x in f99 " (f .: the carrier of L) implies x in L * N )
assume x in f99 " (f .: the carrier of L) ; ::_thesis: x in L * N
then A22: x in the carrier of (G9 "\/" N) /\ (f " (f .: the carrier of L)) by FUNCT_1:70;
then x in f " (f .: the carrier of L) by XBOOLE_0:def_4;
then f . x in f .: the carrier of L by FUNCT_1:def_7;
then consider g1 being set such that
A23: g1 in dom f and
A24: g1 in the carrier of L and
A25: f . x = f . g1 by FUNCT_1:def_6;
reconsider g1 = g1, g2 = x as Element of G by A22, A23;
consider g3 being Element of G such that
A26: g2 = g1 * g3 by GROUP_1:15;
f . g2 = (f . g2) * (f . g3) by A25, A26, GROUP_6:def_6;
then f . g3 = 1_ H by GROUP_1:7;
then g3 in Ker f by Th47;
then g3 in the carrier of N by A1, STRUCT_0:def_5;
hence x in L * N by A24, A26; ::_thesis: verum
end;
reconsider f9 = f | the carrier of G9 as Homomorphism of G9,H9 by A2, Th89;
A27: now__::_thesis:_for_x_being_set_st_x_in_the_carrier_of_N9_holds_
x_in__{__a_where_a_is_Element_of_G9_:_f9_._a_=_1__H9__}_
let x be set ; ::_thesis: ( x in the carrier of N9 implies x in { a where a is Element of G9 : f9 . a = 1_ H9 } )
assume x in the carrier of N9 ; ::_thesis: x in { a where a is Element of G9 : f9 . a = 1_ H9 }
then A28: x in (carr G9) /\ (carr N) by Def25;
then reconsider a9 = x as Element of G9 by XBOOLE_0:def_4;
reconsider a99 = a9 as Element of G by Th2;
x in carr N by A28, XBOOLE_0:def_4;
then x in N by STRUCT_0:def_5;
then f . a99 = 1_ H by A1, Th47;
then f . a9 = 1_ H9 by Th4;
then f9 . a9 = 1_ H9 by FUNCT_1:49;
hence x in { a where a is Element of G9 : f9 . a = 1_ H9 } ; ::_thesis: verum
end;
assume A29: L is strict normal StableSubgroup of G9 ; ::_thesis: ( L "\/" (G9 /\ N) is normal StableSubgroup of G9 & L "\/" N is normal StableSubgroup of G9 "\/" N & ( for N1 being strict normal StableSubgroup of G9 "\/" N
for N2 being strict normal StableSubgroup of G9 st N1 = L "\/" N & N2 = L "\/" (G9 /\ N) holds
(G9 "\/" N) ./. N1,G9 ./. N2 are_isomorphic ) )
then reconsider L9 = L as strict StableSubgroup of G9 ;
reconsider N1 = L "\/" N as StableSubgroup of G9 "\/" N by A29, Th38;
(carr L99) * Ns = Ns * (carr L99) by GROUP_3:120;
then A30: L * N = N * L ;
now__::_thesis:_for_x_being_set_st_x_in__{__a_where_a_is_Element_of_G9_:_f9_._a_=_1__H9__}__holds_
x_in_the_carrier_of_N9
let x be set ; ::_thesis: ( x in { a where a is Element of G9 : f9 . a = 1_ H9 } implies x in the carrier of N9 )
assume x in { a where a is Element of G9 : f9 . a = 1_ H9 } ; ::_thesis: x in the carrier of N9
then consider a being Element of G9 such that
A31: x = a and
A32: f9 . a = 1_ H9 ;
reconsider a = a as Element of G by Th2;
f . a = 1_ H9 by A32, FUNCT_1:49;
then f . a = 1_ H by Th4;
then x in N by A1, A31, Th47;
then x in carr N by STRUCT_0:def_5;
then x in (carr G9) /\ (carr N) by A31, XBOOLE_0:def_4;
hence x in the carrier of N9 by Def25; ::_thesis: verum
end;
then the carrier of N9 = { a where a is Element of G9 : f9 . a = 1_ H9 } by A27, TARSKI:1;
then A33: N9 = Ker f9 by Def21;
then consider H99 being strict StableSubgroup of H9 such that
A34: the carrier of H99 = f9 .: the carrier of L9 and
A35: f9 " the carrier of H99 = the carrier of (L9 "\/" N9) and
A36: ( f9 is onto & L9 is normal implies H99 is normal ) by Th79;
consider N2 being strict StableSubgroup of G9 such that
A37: the carrier of N2 = f9 " the carrier of H99 and
A38: ( H99 is normal implies ( N9 is normal StableSubgroup of N2 & N2 is normal ) ) by A33, Th78;
( f9 .: the carrier of G9 = f .: the carrier of G9 & H9 is strict StableSubgroup of H9 ) by Lm4, RELAT_1:129;
then Image f9 = H9 by A2, Def22;
then A39: rng f9 = the carrier of H9 by Th49;
then reconsider H99 = H99 as normal StableSubgroup of H9 by A29, A36, FUNCT_2:def_3;
A40: N2 = L9 "\/" N9 by A35, A37, Lm5;
hence L "\/" (G9 /\ N) is normal StableSubgroup of G9 by A29, A36, A38, A39, Th86, FUNCT_2:def_3; ::_thesis: ( L "\/" N is normal StableSubgroup of G9 "\/" N & ( for N1 being strict normal StableSubgroup of G9 "\/" N
for N2 being strict normal StableSubgroup of G9 st N1 = L "\/" N & N2 = L "\/" (G9 /\ N) holds
(G9 "\/" N) ./. N1,G9 ./. N2 are_isomorphic ) )
set l = nat_hom H99;
set f1 = (nat_hom H99) * f99;
A41: N2 = L "\/" (G9 /\ N) by A40, Th86;
A42: L "\/" N is StableSubgroup of G9 "\/" N by A29, Th38;
A43: now__::_thesis:_for_x_being_set_st_x_in_L_*_N_holds_
x_in_f99_"_(f_.:_the_carrier_of_L)
let x be set ; ::_thesis: ( x in L * N implies x in f99 " (f .: the carrier of L) )
assume A44: x in L * N ; ::_thesis: x in f99 " (f .: the carrier of L)
then consider g1, g2 being Element of G such that
A45: x = g1 * g2 and
A46: g1 in carr L and
A47: g2 in carr N ;
A48: g2 in N by A47, STRUCT_0:def_5;
f . x = (f . g1) * (f . g2) by A45, GROUP_6:def_6
.= (f . g1) * (1_ H) by A1, A48, Th47
.= f . g1 by GROUP_1:def_4 ;
then A49: f . x in f .: the carrier of L by A12, A46, FUNCT_1:def_6;
L "\/" N is Subgroup of G9 "\/" N by A42, Def7;
then A50: the carrier of (L "\/" N) c= the carrier of (G9 "\/" N) by GROUP_2:def_5;
A51: x in the carrier of (L "\/" N) by A30, A44, Th30;
then x in G9 "\/" N by A50, STRUCT_0:def_5;
then x in G by Th1;
then x in dom f by A12, STRUCT_0:def_5;
then x in f " (f .: the carrier of L) by A49, FUNCT_1:def_7;
then x in the carrier of (G9 "\/" N) /\ (f " (f .: the carrier of L)) by A51, A50, XBOOLE_0:def_4;
hence x in f99 " (f .: the carrier of L) by FUNCT_1:70; ::_thesis: verum
end;
L is Subgroup of G9 by A29, Def7;
then the carrier of L c= the carrier of G9 by GROUP_2:def_5;
then f9 .: the carrier of L = f .: the carrier of L by RELAT_1:129;
then f99 " (f9 .: the carrier of L) = L * N by A21, A43, TARSKI:1;
then A52: f99 " the carrier of H99 = the carrier of N1 by A34, A30, Th30;
A53: f99 " the carrier of (Ker (nat_hom H99)) = f99 " the carrier of H99 by Th48;
then the carrier of (Ker ((nat_hom H99) * f99)) = the carrier of N1 by A52, Th88;
hence L "\/" N is normal StableSubgroup of G9 "\/" N by Lm5; ::_thesis: for N1 being strict normal StableSubgroup of G9 "\/" N
for N2 being strict normal StableSubgroup of G9 st N1 = L "\/" N & N2 = L "\/" (G9 /\ N) holds
(G9 "\/" N) ./. N1,G9 ./. N2 are_isomorphic
A54: Ker ((nat_hom H99) * f99) = N1 by A52, A53, Lm5, Th88;
now__::_thesis:_for_N19_being_strict_normal_StableSubgroup_of_G9_"\/"_N
for_N29_being_strict_normal_StableSubgroup_of_G9_st_N19_=_L_"\/"_N_&_N29_=_L_"\/"_(G9_/\_N)_holds_
(G9_"\/"_N)_./._N19,G9_./._N29_are_isomorphic
set f2 = (nat_hom H99) * f9;
let N19 be strict normal StableSubgroup of G9 "\/" N; ::_thesis: for N29 being strict normal StableSubgroup of G9 st N19 = L "\/" N & N29 = L "\/" (G9 /\ N) holds
(G9 "\/" N) ./. N19,G9 ./. N29 are_isomorphic
let N29 be strict normal StableSubgroup of G9; ::_thesis: ( N19 = L "\/" N & N29 = L "\/" (G9 /\ N) implies (G9 "\/" N) ./. N19,G9 ./. N29 are_isomorphic )
assume A55: N19 = L "\/" N ; ::_thesis: ( N29 = L "\/" (G9 /\ N) implies (G9 "\/" N) ./. N19,G9 ./. N29 are_isomorphic )
( f99 .: the carrier of (G9 "\/" N) = f9 .: the carrier of G9 & ((nat_hom H99) * f99) .: the carrier of (G9 "\/" N) = (nat_hom H99) .: (f99 .: the carrier of (G9 "\/" N)) ) by A2, A20, RELAT_1:126, RELAT_1:129;
then A56: ((nat_hom H99) * f99) .: the carrier of (G9 "\/" N) = ((nat_hom H99) * f9) .: the carrier of G9 by RELAT_1:126;
A57: f9 " the carrier of (Ker (nat_hom H99)) = f9 " the carrier of H99 by Th48;
assume N29 = L "\/" (G9 /\ N) ; ::_thesis: (G9 "\/" N) ./. N19,G9 ./. N29 are_isomorphic
then A58: N29 = Ker ((nat_hom H99) * f9) by A37, A41, A57, Lm5, Th88;
the carrier of (Image ((nat_hom H99) * f99)) = ((nat_hom H99) * f99) .: the carrier of (G9 "\/" N) by Def22
.= the carrier of (Image ((nat_hom H99) * f9)) by A56, Def22 ;
then A59: Image ((nat_hom H99) * f99) = Image ((nat_hom H99) * f9) by Lm5;
( (G9 "\/" N) ./. (Ker ((nat_hom H99) * f99)), Image ((nat_hom H99) * f99) are_isomorphic & Image ((nat_hom H99) * f9),G9 ./. (Ker ((nat_hom H99) * f9)) are_isomorphic ) by Th59;
hence (G9 "\/" N) ./. N19,G9 ./. N29 are_isomorphic by A54, A55, A59, A58, Th55; ::_thesis: verum
end;
hence for N1 being strict normal StableSubgroup of G9 "\/" N
for N2 being strict normal StableSubgroup of G9 st N1 = L "\/" N & N2 = L "\/" (G9 /\ N) holds
(G9 "\/" N) ./. N1,G9 ./. N2 are_isomorphic ; ::_thesis: verum
end;
begin
theorem Th91: :: GROUP_9:91
for O being set
for G being GroupWithOperators of O
for H, K, H9, K9 being strict StableSubgroup of G
for JH being normal StableSubgroup of H9 "\/" (H /\ K)
for HK being normal StableSubgroup of H /\ K st H9 is normal StableSubgroup of H & K9 is normal StableSubgroup of K & JH = H9 "\/" (H /\ K9) & HK = (H9 /\ K) "\/" (K9 /\ H) holds
(H9 "\/" (H /\ K)) ./. JH,(H /\ K) ./. HK are_isomorphic
proof
let O be set ; ::_thesis: for G being GroupWithOperators of O
for H, K, H9, K9 being strict StableSubgroup of G
for JH being normal StableSubgroup of H9 "\/" (H /\ K)
for HK being normal StableSubgroup of H /\ K st H9 is normal StableSubgroup of H & K9 is normal StableSubgroup of K & JH = H9 "\/" (H /\ K9) & HK = (H9 /\ K) "\/" (K9 /\ H) holds
(H9 "\/" (H /\ K)) ./. JH,(H /\ K) ./. HK are_isomorphic
let G be GroupWithOperators of O; ::_thesis: for H, K, H9, K9 being strict StableSubgroup of G
for JH being normal StableSubgroup of H9 "\/" (H /\ K)
for HK being normal StableSubgroup of H /\ K st H9 is normal StableSubgroup of H & K9 is normal StableSubgroup of K & JH = H9 "\/" (H /\ K9) & HK = (H9 /\ K) "\/" (K9 /\ H) holds
(H9 "\/" (H /\ K)) ./. JH,(H /\ K) ./. HK are_isomorphic
let H, K, H9, K9 be strict StableSubgroup of G; ::_thesis: for JH being normal StableSubgroup of H9 "\/" (H /\ K)
for HK being normal StableSubgroup of H /\ K st H9 is normal StableSubgroup of H & K9 is normal StableSubgroup of K & JH = H9 "\/" (H /\ K9) & HK = (H9 /\ K) "\/" (K9 /\ H) holds
(H9 "\/" (H /\ K)) ./. JH,(H /\ K) ./. HK are_isomorphic
reconsider GG = H as GroupWithOperators of O ;
set G9 = H /\ K;
set L = H /\ K9;
reconsider G9 = H /\ K as strict StableSubgroup of GG by Lm34;
let JH be normal StableSubgroup of H9 "\/" (H /\ K); ::_thesis: for HK being normal StableSubgroup of H /\ K st H9 is normal StableSubgroup of H & K9 is normal StableSubgroup of K & JH = H9 "\/" (H /\ K9) & HK = (H9 /\ K) "\/" (K9 /\ H) holds
(H9 "\/" (H /\ K)) ./. JH,(H /\ K) ./. HK are_isomorphic
let HK be normal StableSubgroup of H /\ K; ::_thesis: ( H9 is normal StableSubgroup of H & K9 is normal StableSubgroup of K & JH = H9 "\/" (H /\ K9) & HK = (H9 /\ K) "\/" (K9 /\ H) implies (H9 "\/" (H /\ K)) ./. JH,(H /\ K) ./. HK are_isomorphic )
assume that
A1: H9 is normal StableSubgroup of H and
A2: K9 is normal StableSubgroup of K ; ::_thesis: ( not JH = H9 "\/" (H /\ K9) or not HK = (H9 /\ K) "\/" (K9 /\ H) or (H9 "\/" (H /\ K)) ./. JH,(H /\ K) ./. HK are_isomorphic )
A3: H /\ K9 is normal StableSubgroup of G9 by A2, Th60;
reconsider N9 = H9 as normal StableSubgroup of GG by A1;
assume that
A4: JH = H9 "\/" (H /\ K9) and
A5: HK = (H9 /\ K) "\/" (K9 /\ H) ; ::_thesis: (H9 "\/" (H /\ K)) ./. JH,(H /\ K) ./. HK are_isomorphic
reconsider N = N9 as StableSubgroup of GG ;
set N1 = G9 /\ N;
A6: G9 "\/" N = (H /\ K) "\/" H9 by Th86
.= H9 "\/" (H /\ K) ;
reconsider L = H /\ K9 as StableSubgroup of GG by A3, Th11;
G9 /\ N = (H /\ K) /\ H9 by Th39;
then A7: L "\/" (G9 /\ N) = (H /\ K9) "\/" ((H /\ K) /\ H9) by Th86
.= ((H9 /\ H) /\ K) "\/" (K9 /\ H) by Th20
.= HK by A1, A5, Lm22 ;
reconsider HH = GG ./. N9 as GroupWithOperators of O ;
reconsider f = nat_hom N9 as Homomorphism of GG,HH ;
A8: N = Ker f by Th48;
L "\/" N = (H /\ K9) "\/" H9 by Th86
.= JH by A4 ;
hence (H9 "\/" (H /\ K)) ./. JH,(H /\ K) ./. HK are_isomorphic by A3, A7, A8, A6, Th90; ::_thesis: verum
end;
theorem Th92: :: GROUP_9:92
for O being set
for G being GroupWithOperators of O
for H, K, H9, K9 being strict StableSubgroup of G st H9 is normal StableSubgroup of H & K9 is normal StableSubgroup of K holds
H9 "\/" (H /\ K9) is normal StableSubgroup of H9 "\/" (H /\ K)
proof
let O be set ; ::_thesis: for G being GroupWithOperators of O
for H, K, H9, K9 being strict StableSubgroup of G st H9 is normal StableSubgroup of H & K9 is normal StableSubgroup of K holds
H9 "\/" (H /\ K9) is normal StableSubgroup of H9 "\/" (H /\ K)
let G be GroupWithOperators of O; ::_thesis: for H, K, H9, K9 being strict StableSubgroup of G st H9 is normal StableSubgroup of H & K9 is normal StableSubgroup of K holds
H9 "\/" (H /\ K9) is normal StableSubgroup of H9 "\/" (H /\ K)
let H, K, H9, K9 be strict StableSubgroup of G; ::_thesis: ( H9 is normal StableSubgroup of H & K9 is normal StableSubgroup of K implies H9 "\/" (H /\ K9) is normal StableSubgroup of H9 "\/" (H /\ K) )
reconsider GG = H as GroupWithOperators of O ;
reconsider G9 = H /\ K as strict StableSubgroup of GG by Lm34;
assume that
A1: H9 is normal StableSubgroup of H and
A2: K9 is normal StableSubgroup of K ; ::_thesis: H9 "\/" (H /\ K9) is normal StableSubgroup of H9 "\/" (H /\ K)
reconsider N9 = H9 as normal StableSubgroup of GG by A1;
reconsider N = N9 as StableSubgroup of GG ;
reconsider HH = GG ./. N9 as GroupWithOperators of O ;
reconsider f = nat_hom N9 as Homomorphism of GG,HH ;
set L = H /\ K9;
A3: H /\ K9 is strict normal StableSubgroup of G9 by A2, Th60;
then reconsider L = H /\ K9 as strict StableSubgroup of GG by Th11;
A4: N = Ker f by Th48;
A5: G9 "\/" N = (H /\ K) "\/" H9 by Th86
.= H9 "\/" (H /\ K) ;
L "\/" N = (H /\ K9) "\/" H9 by Th86
.= H9 "\/" (H /\ K9) ;
hence H9 "\/" (H /\ K9) is normal StableSubgroup of H9 "\/" (H /\ K) by A3, A4, A5, Th90; ::_thesis: verum
end;
theorem Th93: :: GROUP_9:93
for O being set
for G being GroupWithOperators of O
for H, K, H9, K9 being strict StableSubgroup of G
for JH being normal StableSubgroup of H9 "\/" (H /\ K)
for JK being normal StableSubgroup of K9 "\/" (K /\ H) st JH = H9 "\/" (H /\ K9) & JK = K9 "\/" (K /\ H9) & H9 is normal StableSubgroup of H & K9 is normal StableSubgroup of K holds
(H9 "\/" (H /\ K)) ./. JH,(K9 "\/" (K /\ H)) ./. JK are_isomorphic
proof
let O be set ; ::_thesis: for G being GroupWithOperators of O
for H, K, H9, K9 being strict StableSubgroup of G
for JH being normal StableSubgroup of H9 "\/" (H /\ K)
for JK being normal StableSubgroup of K9 "\/" (K /\ H) st JH = H9 "\/" (H /\ K9) & JK = K9 "\/" (K /\ H9) & H9 is normal StableSubgroup of H & K9 is normal StableSubgroup of K holds
(H9 "\/" (H /\ K)) ./. JH,(K9 "\/" (K /\ H)) ./. JK are_isomorphic
let G be GroupWithOperators of O; ::_thesis: for H, K, H9, K9 being strict StableSubgroup of G
for JH being normal StableSubgroup of H9 "\/" (H /\ K)
for JK being normal StableSubgroup of K9 "\/" (K /\ H) st JH = H9 "\/" (H /\ K9) & JK = K9 "\/" (K /\ H9) & H9 is normal StableSubgroup of H & K9 is normal StableSubgroup of K holds
(H9 "\/" (H /\ K)) ./. JH,(K9 "\/" (K /\ H)) ./. JK are_isomorphic
let H, K, H9, K9 be strict StableSubgroup of G; ::_thesis: for JH being normal StableSubgroup of H9 "\/" (H /\ K)
for JK being normal StableSubgroup of K9 "\/" (K /\ H) st JH = H9 "\/" (H /\ K9) & JK = K9 "\/" (K /\ H9) & H9 is normal StableSubgroup of H & K9 is normal StableSubgroup of K holds
(H9 "\/" (H /\ K)) ./. JH,(K9 "\/" (K /\ H)) ./. JK are_isomorphic
let JH be normal StableSubgroup of H9 "\/" (H /\ K); ::_thesis: for JK being normal StableSubgroup of K9 "\/" (K /\ H) st JH = H9 "\/" (H /\ K9) & JK = K9 "\/" (K /\ H9) & H9 is normal StableSubgroup of H & K9 is normal StableSubgroup of K holds
(H9 "\/" (H /\ K)) ./. JH,(K9 "\/" (K /\ H)) ./. JK are_isomorphic
let JK be normal StableSubgroup of K9 "\/" (K /\ H); ::_thesis: ( JH = H9 "\/" (H /\ K9) & JK = K9 "\/" (K /\ H9) & H9 is normal StableSubgroup of H & K9 is normal StableSubgroup of K implies (H9 "\/" (H /\ K)) ./. JH,(K9 "\/" (K /\ H)) ./. JK are_isomorphic )
assume that
A1: JH = H9 "\/" (H /\ K9) and
A2: JK = K9 "\/" (K /\ H9) ; ::_thesis: ( not H9 is normal StableSubgroup of H or not K9 is normal StableSubgroup of K or (H9 "\/" (H /\ K)) ./. JH,(K9 "\/" (K /\ H)) ./. JK are_isomorphic )
set HK = (H9 /\ K) "\/" (K9 /\ H);
assume A3: H9 is normal StableSubgroup of H ; ::_thesis: ( not K9 is normal StableSubgroup of K or (H9 "\/" (H /\ K)) ./. JH,(K9 "\/" (K /\ H)) ./. JK are_isomorphic )
then A4: H9 /\ K is normal StableSubgroup of H /\ K by Th60;
assume A5: K9 is normal StableSubgroup of K ; ::_thesis: (H9 "\/" (H /\ K)) ./. JH,(K9 "\/" (K /\ H)) ./. JK are_isomorphic
then K9 /\ H is normal StableSubgroup of H /\ K by Th60;
then reconsider HK = (H9 /\ K) "\/" (K9 /\ H) as normal StableSubgroup of H /\ K by A4, Th87;
HK = (K9 /\ H) "\/" (H9 /\ K) ;
then A6: (K9 "\/" (K /\ H)) ./. JK,(H /\ K) ./. HK are_isomorphic by A2, A3, A5, Th91;
(H9 "\/" (H /\ K)) ./. JH,(H /\ K) ./. HK are_isomorphic by A1, A3, A5, Th91;
hence (H9 "\/" (H /\ K)) ./. JH,(K9 "\/" (K /\ H)) ./. JK are_isomorphic by A6, Th55; ::_thesis: verum
end;
begin
definition
let O be set ;
let G be GroupWithOperators of O;
let IT be FinSequence of the_stable_subgroups_of G;
attrIT is composition_series means :Def28: :: GROUP_9:def 28
( IT . 1 = (Omega). G & IT . (len IT) = (1). G & ( for i being Nat st i in dom IT & i + 1 in dom IT holds
for H1, H2 being StableSubgroup of G st H1 = IT . i & H2 = IT . (i + 1) holds
H2 is normal StableSubgroup of H1 ) );
end;
:: deftheorem Def28 defines composition_series GROUP_9:def_28_:_
for O being set
for G being GroupWithOperators of O
for IT being FinSequence of the_stable_subgroups_of G holds
( IT is composition_series iff ( IT . 1 = (Omega). G & IT . (len IT) = (1). G & ( for i being Nat st i in dom IT & i + 1 in dom IT holds
for H1, H2 being StableSubgroup of G st H1 = IT . i & H2 = IT . (i + 1) holds
H2 is normal StableSubgroup of H1 ) ) );
registration
let O be set ;
let G be GroupWithOperators of O;
cluster Relation-like NAT -defined the_stable_subgroups_of G -valued Function-like finite FinSequence-like FinSubsequence-like composition_series for FinSequence of the_stable_subgroups_of G;
existence
ex b1 being FinSequence of the_stable_subgroups_of G st b1 is composition_series
proof
take H = <*((Omega). G),((1). G)*>; ::_thesis: ( H is Element of bool [:NAT,(the_stable_subgroups_of G):] & H is FinSequence of the_stable_subgroups_of G & H is composition_series )
( (Omega). G is Element of the_stable_subgroups_of G & (1). G is Element of the_stable_subgroups_of G ) by Def11;
then reconsider H = H as FinSequence of the_stable_subgroups_of G by FINSEQ_2:13;
A1: H . (len H) = H . 2 by FINSEQ_1:44
.= (1). G by FINSEQ_1:44 ;
A2: for i being Nat st i in dom H & i + 1 in dom H holds
for H1, H2 being StableSubgroup of G st H1 = H . i & H2 = H . (i + 1) holds
H2 is normal StableSubgroup of H1
proof
let i be Nat; ::_thesis: ( i in dom H & i + 1 in dom H implies for H1, H2 being StableSubgroup of G st H1 = H . i & H2 = H . (i + 1) holds
H2 is normal StableSubgroup of H1 )
assume A3: i in dom H ; ::_thesis: ( not i + 1 in dom H or for H1, H2 being StableSubgroup of G st H1 = H . i & H2 = H . (i + 1) holds
H2 is normal StableSubgroup of H1 )
assume A4: i + 1 in dom H ; ::_thesis: for H1, H2 being StableSubgroup of G st H1 = H . i & H2 = H . (i + 1) holds
H2 is normal StableSubgroup of H1
len H = 2 by FINSEQ_1:44;
then A5: dom H = {1,2} by FINSEQ_1:2, FINSEQ_1:def_3;
percases ( i = 1 or i = 2 ) by A3, A5, TARSKI:def_2;
supposeA6: i = 1 ; ::_thesis: for H1, H2 being StableSubgroup of G st H1 = H . i & H2 = H . (i + 1) holds
H2 is normal StableSubgroup of H1
let H1, H2 be StableSubgroup of G; ::_thesis: ( H1 = H . i & H2 = H . (i + 1) implies H2 is normal StableSubgroup of H1 )
assume H1 = H . i ; ::_thesis: ( not H2 = H . (i + 1) or H2 is normal StableSubgroup of H1 )
assume H2 = H . (i + 1) ; ::_thesis: H2 is normal StableSubgroup of H1
then A7: H2 = (1). G by A6, FINSEQ_1:44;
then reconsider H2 = H2 as StableSubgroup of H1 by Th16;
now__::_thesis:_for_H_being_strict_Subgroup_of_H1_st_multMagma(#_the_carrier_of_H2,_the_multF_of_H2_#)_=_H_holds_
H_is_normal
let H be strict Subgroup of H1; ::_thesis: ( multMagma(# the carrier of H2, the multF of H2 #) = H implies H is normal )
reconsider H1 = H1 as Subgroup of G by Def7;
assume multMagma(# the carrier of H2, the multF of H2 #) = H ; ::_thesis: H is normal
then the carrier of H = {(1_ G)} by A7, Def8;
then the carrier of H = {(1_ H1)} by GROUP_2:44;
then H = (1). H1 by GROUP_2:def_7;
hence H is normal ; ::_thesis: verum
end;
hence H2 is normal StableSubgroup of H1 by Def10; ::_thesis: verum
end;
suppose i = 2 ; ::_thesis: for H1, H2 being StableSubgroup of G st H1 = H . i & H2 = H . (i + 1) holds
H2 is normal StableSubgroup of H1
hence for H1, H2 being StableSubgroup of G st H1 = H . i & H2 = H . (i + 1) holds
H2 is normal StableSubgroup of H1 by A4, A5, TARSKI:def_2; ::_thesis: verum
end;
end;
end;
H . 1 = (Omega). G by FINSEQ_1:44;
hence ( H is Element of bool [:NAT,(the_stable_subgroups_of G):] & H is FinSequence of the_stable_subgroups_of G & H is composition_series ) by A1, A2, Def28; ::_thesis: verum
end;
end;
definition
let O be set ;
let G be GroupWithOperators of O;
mode CompositionSeries of G is composition_series FinSequence of the_stable_subgroups_of G;
end;
definition
let O be set ;
let G be GroupWithOperators of O;
let s1, s2 be CompositionSeries of G;
preds1 is_finer_than s2 means :Def29: :: GROUP_9:def 29
ex x being set st
( x c= dom s1 & s2 = s1 * (Sgm x) );
reflexivity
for s1 being CompositionSeries of G ex x being set st
( x c= dom s1 & s1 = s1 * (Sgm x) )
proof
now__::_thesis:_for_s1_being_CompositionSeries_of_G_ex_x_being_set_st_
(_x_c=_dom_s1_&_s1_=_s1_*_(Sgm_x)_)
let s1 be CompositionSeries of G; ::_thesis: ex x being set st
( x c= dom s1 & s1 = s1 * (Sgm x) )
set x = dom s1;
reconsider x = dom s1 as set ;
take x = x; ::_thesis: ( x c= dom s1 & s1 = s1 * (Sgm x) )
thus x c= dom s1 ; ::_thesis: s1 = s1 * (Sgm x)
set i = len s1;
Sgm x = Sgm (Seg (len s1)) by FINSEQ_1:def_3
.= idseq (len s1) by FINSEQ_3:48 ;
hence s1 = s1 * (Sgm x) by FINSEQ_2:54; ::_thesis: verum
end;
hence for s1 being CompositionSeries of G ex x being set st
( x c= dom s1 & s1 = s1 * (Sgm x) ) ; ::_thesis: verum
end;
end;
:: deftheorem Def29 defines is_finer_than GROUP_9:def_29_:_
for O being set
for G being GroupWithOperators of O
for s1, s2 being CompositionSeries of G holds
( s1 is_finer_than s2 iff ex x being set st
( x c= dom s1 & s2 = s1 * (Sgm x) ) );
definition
let O be set ;
let G be GroupWithOperators of O;
let IT be CompositionSeries of G;
attrIT is strictly_decreasing means :Def30: :: GROUP_9:def 30
for i being Nat st i in dom IT & i + 1 in dom IT holds
for H being StableSubgroup of G
for N being normal StableSubgroup of H st H = IT . i & N = IT . (i + 1) holds
not H ./. N is trivial ;
end;
:: deftheorem Def30 defines strictly_decreasing GROUP_9:def_30_:_
for O being set
for G being GroupWithOperators of O
for IT being CompositionSeries of G holds
( IT is strictly_decreasing iff for i being Nat st i in dom IT & i + 1 in dom IT holds
for H being StableSubgroup of G
for N being normal StableSubgroup of H st H = IT . i & N = IT . (i + 1) holds
not H ./. N is trivial );
definition
let O be set ;
let G be GroupWithOperators of O;
let IT be CompositionSeries of G;
attrIT is jordan_holder means :Def31: :: GROUP_9:def 31
( IT is strictly_decreasing & ( for s being CompositionSeries of G holds
( not s <> IT or not s is strictly_decreasing or not s is_finer_than IT ) ) );
end;
:: deftheorem Def31 defines jordan_holder GROUP_9:def_31_:_
for O being set
for G being GroupWithOperators of O
for IT being CompositionSeries of G holds
( IT is jordan_holder iff ( IT is strictly_decreasing & ( for s being CompositionSeries of G holds
( not s <> IT or not s is strictly_decreasing or not s is_finer_than IT ) ) ) );
definition
let O be set ;
let G1, G2 be GroupWithOperators of O;
let s1 be CompositionSeries of G1;
let s2 be CompositionSeries of G2;
preds1 is_equivalent_with s2 means :Def32: :: GROUP_9:def 32
( len s1 = len s2 & ( for n being Nat st n + 1 = len s1 holds
ex p being Permutation of (Seg n) st
for H1 being StableSubgroup of G1
for H2 being StableSubgroup of G2
for N1 being normal StableSubgroup of H1
for N2 being normal StableSubgroup of H2
for i, j being Nat st 1 <= i & i <= n & j = p . i & H1 = s1 . i & H2 = s2 . j & N1 = s1 . (i + 1) & N2 = s2 . (j + 1) holds
H1 ./. N1,H2 ./. N2 are_isomorphic ) );
end;
:: deftheorem Def32 defines is_equivalent_with GROUP_9:def_32_:_
for O being set
for G1, G2 being GroupWithOperators of O
for s1 being CompositionSeries of G1
for s2 being CompositionSeries of G2 holds
( s1 is_equivalent_with s2 iff ( len s1 = len s2 & ( for n being Nat st n + 1 = len s1 holds
ex p being Permutation of (Seg n) st
for H1 being StableSubgroup of G1
for H2 being StableSubgroup of G2
for N1 being normal StableSubgroup of H1
for N2 being normal StableSubgroup of H2
for i, j being Nat st 1 <= i & i <= n & j = p . i & H1 = s1 . i & H2 = s2 . j & N1 = s1 . (i + 1) & N2 = s2 . (j + 1) holds
H1 ./. N1,H2 ./. N2 are_isomorphic ) ) );
definition
let O be set ;
let G be GroupWithOperators of O;
let s be CompositionSeries of G;
func the_series_of_quotients_of s -> FinSequence means :Def33: :: GROUP_9:def 33
( len s = (len it) + 1 & ( for i being Nat st i in dom it holds
for H being StableSubgroup of G
for N being normal StableSubgroup of H st H = s . i & N = s . (i + 1) holds
it . i = H ./. N ) ) if len s > 1
otherwise it = {} ;
existence
( ( len s > 1 implies ex b1 being FinSequence st
( len s = (len b1) + 1 & ( for i being Nat st i in dom b1 holds
for H being StableSubgroup of G
for N being normal StableSubgroup of H st H = s . i & N = s . (i + 1) holds
b1 . i = H ./. N ) ) ) & ( not len s > 1 implies ex b1 being FinSequence st b1 = {} ) )
proof
now__::_thesis:_(_len_s_>_1_implies_ex_f_being_FinSequence_st_
(_len_s_=_(len_f)_+_1_&_(_for_j_being_Nat_st_j_in_dom_f_holds_
for_H_being_StableSubgroup_of_G
for_N_being_normal_StableSubgroup_of_H_st_H_=_s_._j_&_N_=_s_._(j_+_1)_holds_
f_._j_=_H_./._N_)_)_)
set i = (len s) - 1;
assume len s > 1 ; ::_thesis: ex f being FinSequence st
( len s = (len f) + 1 & ( for j being Nat st j in dom f holds
for H being StableSubgroup of G
for N being normal StableSubgroup of H st H = s . j & N = s . (j + 1) holds
f . j = H ./. N ) )
then (len s) - 1 > 1 - 1 by XREAL_1:9;
then reconsider i = (len s) - 1 as Element of NAT by INT_1:3;
defpred S1[ set , set ] means for H being StableSubgroup of G
for N being normal StableSubgroup of H
for j being Nat st $1 in Seg i & j = $1 & H = s . j & N = s . (j + 1) holds
$2 = H ./. N;
A1: for k being Nat st k in Seg i holds
ex x being set st S1[k,x]
proof
let k be Nat; ::_thesis: ( k in Seg i implies ex x being set st S1[k,x] )
reconsider k1 = k as Element of NAT by ORDINAL1:def_12;
assume A2: k in Seg i ; ::_thesis: ex x being set st S1[k,x]
then A3: 1 <= k by FINSEQ_1:1;
k <= i by A2, FINSEQ_1:1;
then A4: k + 1 <= ((len s) - 1) + 1 by XREAL_1:6;
0 + k <= 1 + k by XREAL_1:6;
then k <= len s by A4, XXREAL_0:2;
then k1 in Seg (len s) by A3;
then A5: k in dom s by FINSEQ_1:def_3;
1 + 1 <= k + 1 by A3, XREAL_1:6;
then 1 <= k + 1 by XXREAL_0:2;
then k1 + 1 in Seg (len s) by A4;
then A6: k + 1 in dom s by FINSEQ_1:def_3;
then reconsider H = s . k, N = s . (k + 1) as Element of the_stable_subgroups_of G by A5, FINSEQ_2:11;
reconsider H = H, N = N as StableSubgroup of G by Def11;
reconsider N = N as normal StableSubgroup of H by A5, A6, Def28;
take H ./. N ; ::_thesis: S1[k,H ./. N]
thus S1[k,H ./. N] ; ::_thesis: verum
end;
consider f being FinSequence such that
A7: ( dom f = Seg i & ( for k being Nat st k in Seg i holds
S1[k,f . k] ) ) from FINSEQ_1:sch_1(A1);
take f = f; ::_thesis: ( len s = (len f) + 1 & ( for j being Nat st j in dom f holds
for H being StableSubgroup of G
for N being normal StableSubgroup of H st H = s . j & N = s . (j + 1) holds
f . j = H ./. N ) )
len f = i by A7, FINSEQ_1:def_3;
hence len s = (len f) + 1 ; ::_thesis: for j being Nat st j in dom f holds
for H being StableSubgroup of G
for N being normal StableSubgroup of H st H = s . j & N = s . (j + 1) holds
f . j = H ./. N
let j be Nat; ::_thesis: ( j in dom f implies for H being StableSubgroup of G
for N being normal StableSubgroup of H st H = s . j & N = s . (j + 1) holds
f . j = H ./. N )
assume A8: j in dom f ; ::_thesis: for H being StableSubgroup of G
for N being normal StableSubgroup of H st H = s . j & N = s . (j + 1) holds
f . j = H ./. N
let H be StableSubgroup of G; ::_thesis: for N being normal StableSubgroup of H st H = s . j & N = s . (j + 1) holds
f . j = H ./. N
let N be normal StableSubgroup of H; ::_thesis: ( H = s . j & N = s . (j + 1) implies f . j = H ./. N )
assume A9: H = s . j ; ::_thesis: ( N = s . (j + 1) implies f . j = H ./. N )
assume N = s . (j + 1) ; ::_thesis: f . j = H ./. N
hence f . j = H ./. N by A7, A8, A9; ::_thesis: verum
end;
hence ( ( len s > 1 implies ex b1 being FinSequence st
( len s = (len b1) + 1 & ( for i being Nat st i in dom b1 holds
for H being StableSubgroup of G
for N being normal StableSubgroup of H st H = s . i & N = s . (i + 1) holds
b1 . i = H ./. N ) ) ) & ( not len s > 1 implies ex b1 being FinSequence st b1 = {} ) ) ; ::_thesis: verum
end;
uniqueness
for b1, b2 being FinSequence holds
( ( len s > 1 & len s = (len b1) + 1 & ( for i being Nat st i in dom b1 holds
for H being StableSubgroup of G
for N being normal StableSubgroup of H st H = s . i & N = s . (i + 1) holds
b1 . i = H ./. N ) & len s = (len b2) + 1 & ( for i being Nat st i in dom b2 holds
for H being StableSubgroup of G
for N being normal StableSubgroup of H st H = s . i & N = s . (i + 1) holds
b2 . i = H ./. N ) implies b1 = b2 ) & ( not len s > 1 & b1 = {} & b2 = {} implies b1 = b2 ) )
proof
let f1, f2 be FinSequence; ::_thesis: ( ( len s > 1 & len s = (len f1) + 1 & ( for i being Nat st i in dom f1 holds
for H being StableSubgroup of G
for N being normal StableSubgroup of H st H = s . i & N = s . (i + 1) holds
f1 . i = H ./. N ) & len s = (len f2) + 1 & ( for i being Nat st i in dom f2 holds
for H being StableSubgroup of G
for N being normal StableSubgroup of H st H = s . i & N = s . (i + 1) holds
f2 . i = H ./. N ) implies f1 = f2 ) & ( not len s > 1 & f1 = {} & f2 = {} implies f1 = f2 ) )
now__::_thesis:_(_len_s_>_1_&_len_s_=_(len_f1)_+_1_&_(_for_i_being_Nat_st_i_in_dom_f1_holds_
for_H1_being_StableSubgroup_of_G
for_N1_being_normal_StableSubgroup_of_H1_st_H1_=_s_._i_&_N1_=_s_._(i_+_1)_holds_
f1_._i_=_H1_./._N1_)_&_len_s_=_(len_f2)_+_1_&_(_for_i_being_Nat_st_i_in_dom_f2_holds_
for_H1_being_StableSubgroup_of_G
for_N1_being_normal_StableSubgroup_of_H1_st_H1_=_s_._i_&_N1_=_s_._(i_+_1)_holds_
f2_._i_=_H1_./._N1_)_implies_f1_=_f2_)
assume len s > 1 ; ::_thesis: ( len s = (len f1) + 1 & ( for i being Nat st i in dom f1 holds
for H1 being StableSubgroup of G
for N1 being normal StableSubgroup of H1 st H1 = s . i & N1 = s . (i + 1) holds
f1 . i = H1 ./. N1 ) & len s = (len f2) + 1 & ( for i being Nat st i in dom f2 holds
for H1 being StableSubgroup of G
for N1 being normal StableSubgroup of H1 st H1 = s . i & N1 = s . (i + 1) holds
f2 . i = H1 ./. N1 ) implies f1 = f2 )
assume A10: len s = (len f1) + 1 ; ::_thesis: ( ( for i being Nat st i in dom f1 holds
for H1 being StableSubgroup of G
for N1 being normal StableSubgroup of H1 st H1 = s . i & N1 = s . (i + 1) holds
f1 . i = H1 ./. N1 ) & len s = (len f2) + 1 & ( for i being Nat st i in dom f2 holds
for H1 being StableSubgroup of G
for N1 being normal StableSubgroup of H1 st H1 = s . i & N1 = s . (i + 1) holds
f2 . i = H1 ./. N1 ) implies f1 = f2 )
assume A11: for i being Nat st i in dom f1 holds
for H1 being StableSubgroup of G
for N1 being normal StableSubgroup of H1 st H1 = s . i & N1 = s . (i + 1) holds
f1 . i = H1 ./. N1 ; ::_thesis: ( len s = (len f2) + 1 & ( for i being Nat st i in dom f2 holds
for H1 being StableSubgroup of G
for N1 being normal StableSubgroup of H1 st H1 = s . i & N1 = s . (i + 1) holds
f2 . i = H1 ./. N1 ) implies f1 = f2 )
assume A12: len s = (len f2) + 1 ; ::_thesis: ( ( for i being Nat st i in dom f2 holds
for H1 being StableSubgroup of G
for N1 being normal StableSubgroup of H1 st H1 = s . i & N1 = s . (i + 1) holds
f2 . i = H1 ./. N1 ) implies f1 = f2 )
assume A13: for i being Nat st i in dom f2 holds
for H1 being StableSubgroup of G
for N1 being normal StableSubgroup of H1 st H1 = s . i & N1 = s . (i + 1) holds
f2 . i = H1 ./. N1 ; ::_thesis: f1 = f2
for k being Nat st 1 <= k & k <= len f1 holds
f1 . k = f2 . k
proof
let k be Nat; ::_thesis: ( 1 <= k & k <= len f1 implies f1 . k = f2 . k )
reconsider k1 = k as Element of NAT by ORDINAL1:def_12;
assume that
A14: 1 <= k and
A15: k <= len f1 ; ::_thesis: f1 . k = f2 . k
A16: k + 1 <= ((len s) - 1) + 1 by A10, A15, XREAL_1:6;
0 + k <= 1 + k by XREAL_1:6;
then k <= len s by A16, XXREAL_0:2;
then k1 in Seg (len s) by A14;
then A17: k in dom s by FINSEQ_1:def_3;
1 + 1 <= k + 1 by A14, XREAL_1:6;
then 1 <= k + 1 by XXREAL_0:2;
then k1 + 1 in Seg (len s) by A16;
then A18: k + 1 in dom s by FINSEQ_1:def_3;
then reconsider H1 = s . k, N1 = s . (k + 1) as Element of the_stable_subgroups_of G by A17, FINSEQ_2:11;
reconsider H1 = H1, N1 = N1 as StableSubgroup of G by Def11;
reconsider N1 = N1 as normal StableSubgroup of H1 by A17, A18, Def28;
A19: k1 in Seg (len f1) by A14, A15;
then k in dom f1 by FINSEQ_1:def_3;
then A20: f1 . k = H1 ./. N1 by A11;
k in dom f2 by A10, A12, A19, FINSEQ_1:def_3;
hence f1 . k = f2 . k by A13, A20; ::_thesis: verum
end;
hence f1 = f2 by A10, A12, FINSEQ_1:14; ::_thesis: verum
end;
hence ( ( len s > 1 & len s = (len f1) + 1 & ( for i being Nat st i in dom f1 holds
for H being StableSubgroup of G
for N being normal StableSubgroup of H st H = s . i & N = s . (i + 1) holds
f1 . i = H ./. N ) & len s = (len f2) + 1 & ( for i being Nat st i in dom f2 holds
for H being StableSubgroup of G
for N being normal StableSubgroup of H st H = s . i & N = s . (i + 1) holds
f2 . i = H ./. N ) implies f1 = f2 ) & ( not len s > 1 & f1 = {} & f2 = {} implies f1 = f2 ) ) ; ::_thesis: verum
end;
consistency
for b1 being FinSequence holds verum ;
end;
:: deftheorem Def33 defines the_series_of_quotients_of GROUP_9:def_33_:_
for O being set
for G being GroupWithOperators of O
for s being CompositionSeries of G
for b4 being FinSequence holds
( ( len s > 1 implies ( b4 = the_series_of_quotients_of s iff ( len s = (len b4) + 1 & ( for i being Nat st i in dom b4 holds
for H being StableSubgroup of G
for N being normal StableSubgroup of H st H = s . i & N = s . (i + 1) holds
b4 . i = H ./. N ) ) ) ) & ( not len s > 1 implies ( b4 = the_series_of_quotients_of s iff b4 = {} ) ) );
definition
let O be set ;
let f1, f2 be FinSequence;
let p be Permutation of (dom f1);
predf1,f2 are_equivalent_under p,O means :Def34: :: GROUP_9:def 34
( len f1 = len f2 & ( for H1, H2 being GroupWithOperators of O
for i, j being Nat st i in dom f1 & j = (p ") . i & H1 = f1 . i & H2 = f2 . j holds
H1,H2 are_isomorphic ) );
end;
:: deftheorem Def34 defines are_equivalent_under GROUP_9:def_34_:_
for O being set
for f1, f2 being FinSequence
for p being Permutation of (dom f1) holds
( f1,f2 are_equivalent_under p,O iff ( len f1 = len f2 & ( for H1, H2 being GroupWithOperators of O
for i, j being Nat st i in dom f1 & j = (p ") . i & H1 = f1 . i & H2 = f2 . j holds
H1,H2 are_isomorphic ) ) );
theorem Th94: :: GROUP_9:94
for O being set
for G being GroupWithOperators of O
for s1 being CompositionSeries of G
for fs being FinSequence of the_stable_subgroups_of G
for i being Nat st i in dom s1 & i + 1 in dom s1 & s1 . i = s1 . (i + 1) & fs = Del (s1,i) holds
fs is composition_series
proof
let O be set ; ::_thesis: for G being GroupWithOperators of O
for s1 being CompositionSeries of G
for fs being FinSequence of the_stable_subgroups_of G
for i being Nat st i in dom s1 & i + 1 in dom s1 & s1 . i = s1 . (i + 1) & fs = Del (s1,i) holds
fs is composition_series
let G be GroupWithOperators of O; ::_thesis: for s1 being CompositionSeries of G
for fs being FinSequence of the_stable_subgroups_of G
for i being Nat st i in dom s1 & i + 1 in dom s1 & s1 . i = s1 . (i + 1) & fs = Del (s1,i) holds
fs is composition_series
let s1 be CompositionSeries of G; ::_thesis: for fs being FinSequence of the_stable_subgroups_of G
for i being Nat st i in dom s1 & i + 1 in dom s1 & s1 . i = s1 . (i + 1) & fs = Del (s1,i) holds
fs is composition_series
let fs be FinSequence of the_stable_subgroups_of G; ::_thesis: for i being Nat st i in dom s1 & i + 1 in dom s1 & s1 . i = s1 . (i + 1) & fs = Del (s1,i) holds
fs is composition_series
let i be Nat; ::_thesis: ( i in dom s1 & i + 1 in dom s1 & s1 . i = s1 . (i + 1) & fs = Del (s1,i) implies fs is composition_series )
assume A1: i in dom s1 ; ::_thesis: ( not i + 1 in dom s1 or not s1 . i = s1 . (i + 1) or not fs = Del (s1,i) or fs is composition_series )
then consider k being Nat such that
A2: len s1 = k + 1 and
A3: len (Del (s1,i)) = k by FINSEQ_3:104;
assume i + 1 in dom s1 ; ::_thesis: ( not s1 . i = s1 . (i + 1) or not fs = Del (s1,i) or fs is composition_series )
then i + 1 in Seg (len s1) by FINSEQ_1:def_3;
then A4: i + 1 <= len s1 by FINSEQ_1:1;
assume A5: s1 . i = s1 . (i + 1) ; ::_thesis: ( not fs = Del (s1,i) or fs is composition_series )
assume A6: fs = Del (s1,i) ; ::_thesis: fs is composition_series
A7: i in Seg (len s1) by A1, FINSEQ_1:def_3;
then A8: 1 <= i by FINSEQ_1:1;
then 1 + 1 <= i + 1 by XREAL_1:6;
then 1 + 1 <= (len fs) + 1 by A6, A4, A2, A3, XXREAL_0:2;
then A9: 1 <= len fs by XREAL_1:6;
percases ( len fs = 1 or len fs > 1 ) by A9, XXREAL_0:1;
supposeA10: len fs = 1 ; ::_thesis: fs is composition_series
A11: now__::_thesis:_for_n_being_Nat_st_n_in_dom_fs_&_n_+_1_in_dom_fs_holds_
for_H1,_H2_being_StableSubgroup_of_G_st_H1_=_fs_._n_&_H2_=_fs_._(n_+_1)_holds_
H2_is_normal_StableSubgroup_of_H1
let n be Nat; ::_thesis: ( n in dom fs & n + 1 in dom fs implies for H1, H2 being StableSubgroup of G st H1 = fs . n & H2 = fs . (n + 1) holds
H2 is normal StableSubgroup of H1 )
assume n in dom fs ; ::_thesis: ( n + 1 in dom fs implies for H1, H2 being StableSubgroup of G st H1 = fs . n & H2 = fs . (n + 1) holds
H2 is normal StableSubgroup of H1 )
then n in Seg 1 by A10, FINSEQ_1:def_3;
then A12: n = 1 by FINSEQ_1:2, TARSKI:def_1;
assume A13: n + 1 in dom fs ; ::_thesis: for H1, H2 being StableSubgroup of G st H1 = fs . n & H2 = fs . (n + 1) holds
H2 is normal StableSubgroup of H1
let H1, H2 be StableSubgroup of G; ::_thesis: ( H1 = fs . n & H2 = fs . (n + 1) implies H2 is normal StableSubgroup of H1 )
assume that
H1 = fs . n and
H2 = fs . (n + 1) ; ::_thesis: H2 is normal StableSubgroup of H1
2 in Seg 1 by A10, A12, A13, FINSEQ_1:def_3;
hence H2 is normal StableSubgroup of H1 by FINSEQ_1:2, TARSKI:def_1; ::_thesis: verum
end;
A14: s1 . 1 = (Omega). G by Def28;
A15: 1 <= i by A7, FINSEQ_1:1;
A16: i <= 1 by A6, A4, A2, A3, A10, XREAL_1:6;
then A17: i = 1 by A15, XXREAL_0:1;
dom s1 = Seg 2 by A6, A2, A3, A10, FINSEQ_1:def_3;
then 1 in dom s1 ;
then A18: i in dom s1 by A15, A16, XXREAL_0:1;
i <= 1 by A6, A4, A2, A3, A10, XREAL_1:6;
then A19: fs . (len fs) = s1 . (1 + 1) by A6, A2, A3, A10, A18, FINSEQ_3:111
.= (1). G by A6, A2, A3, A10, Def28 ;
s1 . 2 = (1). G by A6, A2, A3, A10, Def28;
hence fs is composition_series by A5, A10, A17, A14, A19, A11, Def28; ::_thesis: verum
end;
supposeA20: len fs > 1 ; ::_thesis: fs is composition_series
A21: fs . 1 = (Omega). G
proof
percases ( i = 1 or i > 1 ) by A8, XXREAL_0:1;
supposeA22: i = 1 ; ::_thesis: fs . 1 = (Omega). G
then fs . 1 = s1 . (1 + 1) by A1, A6, A2, A3, A20, FINSEQ_3:111;
hence fs . 1 = (Omega). G by A5, A22, Def28; ::_thesis: verum
end;
supposeA23: i > 1 ; ::_thesis: fs . 1 = (Omega). G
reconsider i = i as Element of NAT by INT_1:3;
fs . 1 = (Del (s1,i)) . 1 by A6
.= s1 . 1 by A23, FINSEQ_3:110 ;
hence fs . 1 = (Omega). G by Def28; ::_thesis: verum
end;
end;
end;
A24: now__::_thesis:_for_n_being_Nat_st_n_in_dom_fs_&_n_+_1_in_dom_fs_holds_
for_H1,_H2_being_StableSubgroup_of_G_st_H1_=_fs_._n_&_H2_=_fs_._(n_+_1)_holds_
H2_is_normal_StableSubgroup_of_H1
let n be Nat; ::_thesis: ( n in dom fs & n + 1 in dom fs implies for H1, H2 being StableSubgroup of G st H1 = fs . n & H2 = fs . (n + 1) holds
b5 is normal StableSubgroup of b4 )
assume that
A25: n in dom fs and
A26: n + 1 in dom fs ; ::_thesis: for H1, H2 being StableSubgroup of G st H1 = fs . n & H2 = fs . (n + 1) holds
b5 is normal StableSubgroup of b4
A27: n in Seg (len fs) by A25, FINSEQ_1:def_3;
then A28: n <= k by A6, A3, FINSEQ_1:1;
reconsider n1 = n + 1 as Nat ;
A29: n + 1 in Seg (len fs) by A26, FINSEQ_1:def_3;
then A30: n1 <= k by A6, A3, FINSEQ_1:1;
A31: 0 + (len fs) < 1 + (len fs) by XREAL_1:6;
then A32: Seg (len fs) c= Seg (len s1) by A6, A2, A3, FINSEQ_1:5;
then n in Seg (len s1) by A27;
then A33: n in dom s1 by FINSEQ_1:def_3;
n1 in Seg (len s1) by A29, A32;
then A34: n1 in dom s1 by FINSEQ_1:def_3;
n1 <= len fs by A29, FINSEQ_1:1;
then n1 < len s1 by A6, A2, A3, A31, XXREAL_0:2;
then n1 + 1 <= k + 1 by A2, NAT_1:13;
then Seg (n1 + 1) c= Seg (len s1) by A2, FINSEQ_1:5;
then A35: Seg (n1 + 1) c= dom s1 by FINSEQ_1:def_3;
A36: n1 + 1 in Seg (n1 + 1) by FINSEQ_1:4;
let H1, H2 be StableSubgroup of G; ::_thesis: ( H1 = fs . n & H2 = fs . (n + 1) implies b3 is normal StableSubgroup of b2 )
assume A37: H1 = fs . n ; ::_thesis: ( H2 = fs . (n + 1) implies b3 is normal StableSubgroup of b2 )
assume A38: H2 = fs . (n + 1) ; ::_thesis: b3 is normal StableSubgroup of b2
reconsider i = i, n = n as Nat ;
percases ( n < i or n >= i ) ;
supposeA39: n < i ; ::_thesis: b3 is normal StableSubgroup of b2
then A40: n1 <= i by NAT_1:13;
reconsider n9 = n, i = i as Element of NAT by INT_1:3;
A41: (Del (s1,i)) . n9 = s1 . n9 by A39, FINSEQ_3:110;
percases ( n1 < i or n1 = i ) by A40, XXREAL_0:1;
supposeA42: n1 < i ; ::_thesis: b3 is normal StableSubgroup of b2
reconsider n19 = n1, i = i as Element of NAT ;
(Del (s1,i)) . n19 = s1 . n19 by A42, FINSEQ_3:110;
hence H2 is normal StableSubgroup of H1 by A6, A37, A38, A33, A34, A41, Def28; ::_thesis: verum
end;
supposeA43: n1 = i ; ::_thesis: b3 is normal StableSubgroup of b2
then (Del (s1,i)) . n1 = s1 . (n1 + 1) by A1, A2, A30, FINSEQ_3:111;
hence H2 is normal StableSubgroup of H1 by A5, A6, A37, A38, A33, A34, A41, A43, Def28; ::_thesis: verum
end;
end;
end;
supposeA44: n >= i ; ::_thesis: b3 is normal StableSubgroup of b2
reconsider n9 = n, i = i as Element of NAT by INT_1:3;
A45: (Del (s1,i)) . n9 = s1 . (n9 + 1) by A1, A2, A28, A44, FINSEQ_3:111;
reconsider n19 = n1, i = i, k = k as Element of NAT by INT_1:3;
0 + n <= n + 1 by XREAL_1:6;
then A46: i <= n19 by A44, XXREAL_0:2;
n19 <= k by A6, A3, A29, FINSEQ_1:1;
then (Del (s1,i)) . n19 = s1 . (n19 + 1) by A1, A2, A46, FINSEQ_3:111;
hence H2 is normal StableSubgroup of H1 by A6, A37, A38, A34, A35, A36, A45, Def28; ::_thesis: verum
end;
end;
end;
i <= len fs by A6, A4, A2, A3, XREAL_1:6;
then fs . (len fs) = s1 . (len s1) by A1, A6, A2, A3, FINSEQ_3:111;
then fs . (len fs) = (1). G by Def28;
hence fs is composition_series by A21, A24, Def28; ::_thesis: verum
end;
end;
end;
theorem Th95: :: GROUP_9:95
for O being set
for G being GroupWithOperators of O
for s1, s2 being CompositionSeries of G st s1 is_finer_than s2 holds
ex n being Nat st len s1 = (len s2) + n
proof
let O be set ; ::_thesis: for G being GroupWithOperators of O
for s1, s2 being CompositionSeries of G st s1 is_finer_than s2 holds
ex n being Nat st len s1 = (len s2) + n
let G be GroupWithOperators of O; ::_thesis: for s1, s2 being CompositionSeries of G st s1 is_finer_than s2 holds
ex n being Nat st len s1 = (len s2) + n
let s1, s2 be CompositionSeries of G; ::_thesis: ( s1 is_finer_than s2 implies ex n being Nat st len s1 = (len s2) + n )
set n = (len s1) - (len s2);
assume s1 is_finer_than s2 ; ::_thesis: ex n being Nat st len s1 = (len s2) + n
then consider x being set such that
A1: x c= dom s1 and
A2: s2 = s1 * (Sgm x) by Def29;
A3: x c= Seg (len s1) by A1, FINSEQ_1:def_3;
reconsider x = x as finite set by A1;
now__::_thesis:_for_y1_being_set_st_y1_in_dom_s2_holds_
y1_in_dom_s1
let y1 be set ; ::_thesis: ( y1 in dom s2 implies y1 in dom s1 )
assume y1 in dom s2 ; ::_thesis: y1 in dom s1
then y1 in dom (Sgm x) by A2, FUNCT_1:11;
then A4: y1 in Seg (card x) by A3, FINSEQ_3:40;
card x <= card (dom s1) by A1, NAT_1:43;
then Seg (card x) c= Seg (card (dom s1)) by FINSEQ_1:5;
then y1 in Seg (card (dom s1)) by A4;
then y1 in Seg (card (Seg (len s1))) by FINSEQ_1:def_3;
then y1 in Seg (len s1) by FINSEQ_1:57;
hence y1 in dom s1 by FINSEQ_1:def_3; ::_thesis: verum
end;
then dom s2 c= dom s1 by TARSKI:def_3;
then Seg (len s2) c= dom s1 by FINSEQ_1:def_3;
then Seg (len s2) c= Seg (len s1) by FINSEQ_1:def_3;
then len s2 <= len s1 by FINSEQ_1:5;
then (len s2) - (len s2) <= (len s1) - (len s2) by XREAL_1:9;
then (len s1) - (len s2) in NAT by INT_1:3;
then reconsider n = (len s1) - (len s2) as Nat ;
take n ; ::_thesis: len s1 = (len s2) + n
thus len s1 = (len s2) + n ; ::_thesis: verum
end;
theorem Th96: :: GROUP_9:96
for O being set
for G being GroupWithOperators of O
for s2, s1 being CompositionSeries of G st len s2 = len s1 & s2 is_finer_than s1 holds
s1 = s2
proof
let O be set ; ::_thesis: for G being GroupWithOperators of O
for s2, s1 being CompositionSeries of G st len s2 = len s1 & s2 is_finer_than s1 holds
s1 = s2
let G be GroupWithOperators of O; ::_thesis: for s2, s1 being CompositionSeries of G st len s2 = len s1 & s2 is_finer_than s1 holds
s1 = s2
let s2, s1 be CompositionSeries of G; ::_thesis: ( len s2 = len s1 & s2 is_finer_than s1 implies s1 = s2 )
reconsider X = Seg (len s2) as finite set ;
assume len s2 = len s1 ; ::_thesis: ( not s2 is_finer_than s1 or s1 = s2 )
then A1: dom s1 = Seg (len s2) by FINSEQ_1:def_3
.= dom s2 by FINSEQ_1:def_3 ;
assume s2 is_finer_than s1 ; ::_thesis: s1 = s2
then consider x being set such that
A2: x c= dom s2 and
A3: s1 = s2 * (Sgm x) by Def29;
set y = X \ x;
A4: x c= Seg (len s2) by A2, FINSEQ_1:def_3;
then x = rng (Sgm x) by FINSEQ_1:def_13;
then A5: dom (s2 * (Sgm x)) = dom (Sgm x) by A2, RELAT_1:27;
reconsider x = x, y = X \ x as finite set by A2;
dom (Sgm x) = Seg (len s2) by A3, A1, A5, FINSEQ_1:def_3;
then len (Sgm x) = len s2 by FINSEQ_1:def_3;
then A6: card x = len s2 by A4, FINSEQ_3:39;
A7: X = X \/ x by A4, XBOOLE_1:12
.= x \/ y by XBOOLE_1:39 ;
card (x \/ y) = (card x) + (card y) by CARD_2:40, XBOOLE_1:79;
then len s2 = (card x) + (card y) by A7, FINSEQ_1:57;
then y = {} by A6;
then Sgm x = idseq (len s2) by A7, FINSEQ_3:48;
hence s1 = s2 by A3, FINSEQ_2:54; ::_thesis: verum
end;
theorem Th97: :: GROUP_9:97
for O being set
for G being GroupWithOperators of O
for s1, s2 being CompositionSeries of G st not s1 is empty & s2 is_finer_than s1 holds
not s2 is empty
proof
let O be set ; ::_thesis: for G being GroupWithOperators of O
for s1, s2 being CompositionSeries of G st not s1 is empty & s2 is_finer_than s1 holds
not s2 is empty
let G be GroupWithOperators of O; ::_thesis: for s1, s2 being CompositionSeries of G st not s1 is empty & s2 is_finer_than s1 holds
not s2 is empty
let s1, s2 be CompositionSeries of G; ::_thesis: ( not s1 is empty & s2 is_finer_than s1 implies not s2 is empty )
assume A1: not s1 is empty ; ::_thesis: ( not s2 is_finer_than s1 or not s2 is empty )
assume s2 is_finer_than s1 ; ::_thesis: not s2 is empty
then ex i being Nat st len s2 = (len s1) + i by Th95;
hence not s2 is empty by A1; ::_thesis: verum
end;
theorem Th98: :: GROUP_9:98
for O being set
for G being GroupWithOperators of O
for s1, s2 being CompositionSeries of G st s1 is_finer_than s2 & s1 is jordan_holder & s2 is jordan_holder holds
s1 = s2
proof
let O be set ; ::_thesis: for G being GroupWithOperators of O
for s1, s2 being CompositionSeries of G st s1 is_finer_than s2 & s1 is jordan_holder & s2 is jordan_holder holds
s1 = s2
let G be GroupWithOperators of O; ::_thesis: for s1, s2 being CompositionSeries of G st s1 is_finer_than s2 & s1 is jordan_holder & s2 is jordan_holder holds
s1 = s2
let s1, s2 be CompositionSeries of G; ::_thesis: ( s1 is_finer_than s2 & s1 is jordan_holder & s2 is jordan_holder implies s1 = s2 )
assume A1: s1 is_finer_than s2 ; ::_thesis: ( not s1 is jordan_holder or not s2 is jordan_holder or s1 = s2 )
assume s1 is jordan_holder ; ::_thesis: ( not s2 is jordan_holder or s1 = s2 )
then A2: s1 is strictly_decreasing by Def31;
assume ( s2 is jordan_holder & s1 <> s2 ) ; ::_thesis: contradiction
hence contradiction by A1, A2, Def31; ::_thesis: verum
end;
Lm36: for P, R being Relation holds
( P = (rng P) |` R iff P ~ = (R ~) | (dom (P ~)) )
proof
let P, R be Relation; ::_thesis: ( P = (rng P) |` R iff P ~ = (R ~) | (dom (P ~)) )
hereby ::_thesis: ( P ~ = (R ~) | (dom (P ~)) implies P = (rng P) |` R )
assume A1: P = (rng P) |` R ; ::_thesis: P ~ = (R ~) | (dom (P ~))
now__::_thesis:_for_x,_y_being_set_holds_
(_(_[x,y]_in_P_~_implies_[x,y]_in_(R_~)_|_(dom_(P_~))_)_&_(_[x,y]_in_(R_~)_|_(dom_(P_~))_implies_[x,y]_in_P_~_)_)
let x, y be set ; ::_thesis: ( ( [x,y] in P ~ implies [x,y] in (R ~) | (dom (P ~)) ) & ( [x,y] in (R ~) | (dom (P ~)) implies [x,y] in P ~ ) )
hereby ::_thesis: ( [x,y] in (R ~) | (dom (P ~)) implies [x,y] in P ~ )
assume A2: [x,y] in P ~ ; ::_thesis: [x,y] in (R ~) | (dom (P ~))
then [y,x] in P by RELAT_1:def_7;
then [y,x] in R by A1, RELAT_1:def_12;
then A3: [x,y] in R ~ by RELAT_1:def_7;
x in dom (P ~) by A2, XTUPLE_0:def_12;
hence [x,y] in (R ~) | (dom (P ~)) by A3, RELAT_1:def_11; ::_thesis: verum
end;
assume A4: [x,y] in (R ~) | (dom (P ~)) ; ::_thesis: [x,y] in P ~
then [x,y] in R ~ by RELAT_1:def_11;
then A5: [y,x] in R by RELAT_1:def_7;
x in dom (P ~) by A4, RELAT_1:def_11;
then x in rng P by RELAT_1:20;
then [y,x] in (rng P) |` R by A5, RELAT_1:def_12;
hence [x,y] in P ~ by A1, RELAT_1:def_7; ::_thesis: verum
end;
hence P ~ = (R ~) | (dom (P ~)) by RELAT_1:def_2; ::_thesis: verum
end;
assume A6: P ~ = (R ~) | (dom (P ~)) ; ::_thesis: P = (rng P) |` R
now__::_thesis:_for_x,_y_being_set_holds_
(_(_[x,y]_in_P_implies_[x,y]_in_(rng_P)_|`_R_)_&_(_[x,y]_in_(rng_P)_|`_R_implies_[x,y]_in_P_)_)
let x, y be set ; ::_thesis: ( ( [x,y] in P implies [x,y] in (rng P) |` R ) & ( [x,y] in (rng P) |` R implies [x,y] in P ) )
hereby ::_thesis: ( [x,y] in (rng P) |` R implies [x,y] in P )
assume [x,y] in P ; ::_thesis: [x,y] in (rng P) |` R
then A7: [y,x] in P ~ by RELAT_1:def_7;
then [y,x] in R ~ by A6, RELAT_1:def_11;
then A8: [x,y] in R by RELAT_1:def_7;
y in dom (P ~) by A6, A7, RELAT_1:def_11;
then y in rng P by RELAT_1:20;
hence [x,y] in (rng P) |` R by A8, RELAT_1:def_12; ::_thesis: verum
end;
assume A9: [x,y] in (rng P) |` R ; ::_thesis: [x,y] in P
then [x,y] in R by RELAT_1:def_12;
then A10: [y,x] in R ~ by RELAT_1:def_7;
y in rng P by A9, RELAT_1:def_12;
then y in dom (P ~) by RELAT_1:20;
then [y,x] in (R ~) | (dom (P ~)) by A10, RELAT_1:def_11;
hence [x,y] in P by A6, RELAT_1:def_7; ::_thesis: verum
end;
hence P = (rng P) |` R by RELAT_1:def_2; ::_thesis: verum
end;
Lm37: for X being set
for P, R being Relation holds P * (R | X) = (X |` P) * R
proof
let X be set ; ::_thesis: for P, R being Relation holds P * (R | X) = (X |` P) * R
let P, R be Relation; ::_thesis: P * (R | X) = (X |` P) * R
A1: now__::_thesis:_for_x_being_set_st_x_in_(X_|`_P)_*_R_holds_
x_in_P_*_(R_|_X)
let x be set ; ::_thesis: ( x in (X |` P) * R implies x in P * (R | X) )
assume A2: x in (X |` P) * R ; ::_thesis: x in P * (R | X)
then consider y, z being set such that
A3: x = [y,z] by RELAT_1:def_1;
consider w being set such that
A4: [y,w] in X |` P and
A5: [w,z] in R by A2, A3, RELAT_1:def_8;
w in X by A4, RELAT_1:def_12;
then A6: [w,z] in R | X by A5, RELAT_1:def_11;
[y,w] in P by A4, RELAT_1:def_12;
hence x in P * (R | X) by A3, A6, RELAT_1:def_8; ::_thesis: verum
end;
now__::_thesis:_for_x_being_set_st_x_in_P_*_(R_|_X)_holds_
x_in_(X_|`_P)_*_R
let x be set ; ::_thesis: ( x in P * (R | X) implies x in (X |` P) * R )
assume A7: x in P * (R | X) ; ::_thesis: x in (X |` P) * R
then consider y, z being set such that
A8: x = [y,z] by RELAT_1:def_1;
consider w being set such that
A9: [y,w] in P and
A10: [w,z] in R | X by A7, A8, RELAT_1:def_8;
w in X by A10, RELAT_1:def_11;
then A11: [y,w] in X |` P by A9, RELAT_1:def_12;
[w,z] in R by A10, RELAT_1:def_11;
hence x in (X |` P) * R by A8, A11, RELAT_1:def_8; ::_thesis: verum
end;
hence P * (R | X) = (X |` P) * R by A1, TARSKI:1; ::_thesis: verum
end;
Lm38: for n being Nat
for X being set
for f being PartFunc of REAL,REAL st X c= Seg n & X c= dom f & f | X is increasing & f .: X c= NAT \ {0} holds
Sgm (f .: X) = f * (Sgm X)
proof
let n be Nat; ::_thesis: for X being set
for f being PartFunc of REAL,REAL st X c= Seg n & X c= dom f & f | X is increasing & f .: X c= NAT \ {0} holds
Sgm (f .: X) = f * (Sgm X)
let X be set ; ::_thesis: for f being PartFunc of REAL,REAL st X c= Seg n & X c= dom f & f | X is increasing & f .: X c= NAT \ {0} holds
Sgm (f .: X) = f * (Sgm X)
let f be PartFunc of REAL,REAL; ::_thesis: ( X c= Seg n & X c= dom f & f | X is increasing & f .: X c= NAT \ {0} implies Sgm (f .: X) = f * (Sgm X) )
assume A1: X c= Seg n ; ::_thesis: ( not X c= dom f or not f | X is increasing or not f .: X c= NAT \ {0} or Sgm (f .: X) = f * (Sgm X) )
then A2: rng (Sgm X) = X by FINSEQ_1:def_13;
assume A3: X c= dom f ; ::_thesis: ( not f | X is increasing or not f .: X c= NAT \ {0} or Sgm (f .: X) = f * (Sgm X) )
assume A4: f | X is increasing ; ::_thesis: ( not f .: X c= NAT \ {0} or Sgm (f .: X) = f * (Sgm X) )
assume A5: f .: X c= NAT \ {0} ; ::_thesis: Sgm (f .: X) = f * (Sgm X)
percases ( X misses dom f or X meets dom f ) ;
supposeA6: X misses dom f ; ::_thesis: Sgm (f .: X) = f * (Sgm X)
then A7: f .: X = {} by RELAT_1:118;
then f .: X c= Seg 0 ;
then Sgm (f .: X) = {} by A7, FINSEQ_1:51;
hence Sgm (f .: X) = f * (Sgm X) by A2, A6, RELAT_1:44; ::_thesis: verum
end;
supposeA8: X meets dom f ; ::_thesis: Sgm (f .: X) = f * (Sgm X)
reconsider X9 = X as finite set by A1;
set fX = f .: X;
reconsider f9 = f as Function ;
f9 .: X9 is finite ;
then reconsider fX = f .: X as non empty finite real-membered set by A8, RELAT_1:118;
set k = max fX;
reconsider k = max fX as Nat by A5;
set fs = f * (Sgm X);
rng (Sgm X) c= dom f by A1, A3, FINSEQ_1:def_13;
then reconsider fs = f * (Sgm X) as FinSequence by FINSEQ_1:16;
f .: (rng (Sgm X)) c= NAT \ {0} by A1, A5, FINSEQ_1:def_13;
then A9: rng fs c= NAT \ {0} by RELAT_1:127;
NAT \ {0} c= NAT by XBOOLE_1:36;
then rng fs c= NAT by A9, XBOOLE_1:1;
then reconsider fs = fs as FinSequence of NAT by FINSEQ_1:def_4;
now__::_thesis:_for_x_being_set_st_x_in_f_.:_X_holds_
x_in_Seg_k
let x be set ; ::_thesis: ( x in f .: X implies x in Seg k )
assume A10: x in f .: X ; ::_thesis: x in Seg k
then reconsider k9 = x as Nat by A5;
not k9 in {0} by A5, A10, XBOOLE_0:def_5;
then k9 <> 0 by TARSKI:def_1;
then 0 + 1 < k9 + 1 by XREAL_1:6;
then A11: 1 <= k9 by NAT_1:13;
k9 <= k by A10, XXREAL_2:def_8;
hence x in Seg k by A11, FINSEQ_1:1; ::_thesis: verum
end;
then A12: f .: X c= Seg k by TARSKI:def_3;
A13: now__::_thesis:_for_l,_m,_k1,_k2_being_Nat_st_1_<=_l_&_l_<_m_&_m_<=_len_fs_&_k1_=_fs_._l_&_k2_=_fs_._m_holds_
k1_<_k2
A14: dom fs = Seg (len fs) by FINSEQ_1:def_3;
let l, m, k1, k2 be Nat; ::_thesis: ( 1 <= l & l < m & m <= len fs & k1 = fs . l & k2 = fs . m implies k1 < k2 )
assume that
A15: 1 <= l and
A16: l < m and
A17: m <= len fs ; ::_thesis: ( k1 = fs . l & k2 = fs . m implies k1 < k2 )
set k19 = (Sgm X) . l;
l <= len fs by A16, A17, XXREAL_0:2;
then A18: l in dom fs by A15, A14, FINSEQ_1:1;
then l in dom (Sgm X) by FUNCT_1:11;
then A19: (Sgm X) . l in X by A2, FUNCT_1:3;
set k29 = (Sgm X) . m;
1 <= m by A15, A16, XXREAL_0:2;
then A20: m in dom fs by A17, A14, FINSEQ_1:1;
then A21: m in dom (Sgm X) by FUNCT_1:11;
then A22: (Sgm X) . m in X by A2, FUNCT_1:3;
reconsider k19 = (Sgm X) . l, k29 = (Sgm X) . m as Nat ;
m in Seg (len (Sgm X)) by A21, FINSEQ_1:def_3;
then m <= len (Sgm X) by FINSEQ_1:1;
then A23: k19 < k29 by A1, A15, A16, FINSEQ_1:def_13;
reconsider k19 = k19, k29 = k29 as Element of NAT by ORDINAL1:def_12;
reconsider k19 = k19, k29 = k29 as Element of REAL ;
(Sgm X) . l in dom f by A18, FUNCT_1:11;
then A24: k19 in X /\ (dom f) by A19, XBOOLE_0:def_4;
assume that
A25: k1 = fs . l and
A26: k2 = fs . m ; ::_thesis: k1 < k2
A27: k2 = f . ((Sgm X) . m) by A26, A20, FUNCT_1:12;
(Sgm X) . m in dom f by A20, FUNCT_1:11;
then A28: k29 in X /\ (dom f) by A22, XBOOLE_0:def_4;
k1 = f . ((Sgm X) . l) by A25, A18, FUNCT_1:12;
hence k1 < k2 by A4, A27, A23, A24, A28, RFUNCT_2:20; ::_thesis: verum
end;
rng fs = f .: X by A2, RELAT_1:127;
hence Sgm (f .: X) = f * (Sgm X) by A12, A13, FINSEQ_1:def_13; ::_thesis: verum
end;
end;
end;
Lm39: for y being set
for n, i being Nat st y c= Seg (n + 1) & i in Seg (n + 1) & not i in y holds
ex x being set st
( Sgm x = ((Sgm ((Seg (n + 1)) \ {i})) ") * (Sgm y) & x c= Seg n )
proof
let y be set ; ::_thesis: for n, i being Nat st y c= Seg (n + 1) & i in Seg (n + 1) & not i in y holds
ex x being set st
( Sgm x = ((Sgm ((Seg (n + 1)) \ {i})) ") * (Sgm y) & x c= Seg n )
let n, i be Nat; ::_thesis: ( y c= Seg (n + 1) & i in Seg (n + 1) & not i in y implies ex x being set st
( Sgm x = ((Sgm ((Seg (n + 1)) \ {i})) ") * (Sgm y) & x c= Seg n ) )
set x1 = { k where k is Element of NAT : ( k in y & k < i ) } ;
set x2 = { (k - 1) where k is Element of NAT : ( k in y & k > i ) } ;
set x = { k where k is Element of NAT : ( k in y & k < i ) } \/ { (k - 1) where k is Element of NAT : ( k in y & k > i ) } ;
set f1 = id { k where k is Element of NAT : ( k in y & k < i ) } ;
assume A1: y c= Seg (n + 1) ; ::_thesis: ( not i in Seg (n + 1) or i in y or ex x being set st
( Sgm x = ((Sgm ((Seg (n + 1)) \ {i})) ") * (Sgm y) & x c= Seg n ) )
then A2: y = rng (Sgm y) by FINSEQ_1:def_13;
assume A3: i in Seg (n + 1) ; ::_thesis: ( i in y or ex x being set st
( Sgm x = ((Sgm ((Seg (n + 1)) \ {i})) ") * (Sgm y) & x c= Seg n ) )
then A4: 1 <= i by FINSEQ_1:1;
A5: i <= n + 1 by A3, FINSEQ_1:1;
A6: now__::_thesis:_for_z_being_set_st_z_in__{__k_where_k_is_Element_of_NAT_:_(_k_in_y_&_k_<_i_)__}__\/__{__(k_-_1)_where_k_is_Element_of_NAT_:_(_k_in_y_&_k_>_i_)__}__holds_
z_in_Seg_n
let z be set ; ::_thesis: ( z in { k where k is Element of NAT : ( k in y & k < i ) } \/ { (k - 1) where k is Element of NAT : ( k in y & k > i ) } implies b1 in Seg n )
assume A7: z in { k where k is Element of NAT : ( k in y & k < i ) } \/ { (k - 1) where k is Element of NAT : ( k in y & k > i ) } ; ::_thesis: b1 in Seg n
percases ( z in { k where k is Element of NAT : ( k in y & k < i ) } or z in { (k - 1) where k is Element of NAT : ( k in y & k > i ) } ) by A7, XBOOLE_0:def_3;
suppose z in { k where k is Element of NAT : ( k in y & k < i ) } ; ::_thesis: b1 in Seg n
then A8: ex k being Element of NAT st
( k = z & k in y & k < i ) ;
then reconsider z9 = z as Element of NAT ;
z9 < n + 1 by A5, A8, XXREAL_0:2;
then A9: z9 <= n by NAT_1:13;
1 <= z9 by A1, A8, FINSEQ_1:1;
hence z in Seg n by A9; ::_thesis: verum
end;
suppose z in { (k - 1) where k is Element of NAT : ( k in y & k > i ) } ; ::_thesis: b1 in Seg n
then consider k being Element of NAT such that
A10: k - 1 = z and
A11: k in y and
A12: k > i ;
reconsider z9 = z as integer number by A10;
1 < k by A4, A12, XXREAL_0:2;
then 1 + 1 < k + 1 by XREAL_1:6;
then 2 <= k by NAT_1:13;
then A13: 2 - 1 <= k - 1 by XREAL_1:9;
then reconsider z9 = z9 as Element of NAT by A10, INT_1:3;
k <= n + 1 by A1, A11, FINSEQ_1:1;
then k - 1 <= (n + 1) - 1 by XREAL_1:9;
then z9 <= n by A10;
hence z in Seg n by A10, A13; ::_thesis: verum
end;
end;
end;
then A14: { k where k is Element of NAT : ( k in y & k < i ) } \/ { (k - 1) where k is Element of NAT : ( k in y & k > i ) } c= Seg n by TARSKI:def_3;
then reconsider x9 = { k where k is Element of NAT : ( k in y & k < i ) } \/ { (k - 1) where k is Element of NAT : ( k in y & k > i ) } , y9 = y as finite set by A1;
set f2 = { [(k - 1),k] where k is Element of NAT : ( k in y9 & k > i ) } ;
now__::_thesis:_for_x_being_set_st_x_in__{__[(k_-_1),k]_where_k_is_Element_of_NAT_:_(_k_in_y9_&_k_>_i_)__}__holds_
ex_y,_z_being_set_st_x_=_[y,z]
let x be set ; ::_thesis: ( x in { [(k - 1),k] where k is Element of NAT : ( k in y9 & k > i ) } implies ex y, z being set st x = [y,z] )
assume x in { [(k - 1),k] where k is Element of NAT : ( k in y9 & k > i ) } ; ::_thesis: ex y, z being set st x = [y,z]
then consider k being Element of NAT such that
A15: [(k - 1),k] = x and
k in y9 and
k > i ;
reconsider y = k - 1, z = k as set ;
take y = y; ::_thesis: ex z being set st x = [y,z]
take z = z; ::_thesis: x = [y,z]
thus x = [y,z] by A15; ::_thesis: verum
end;
then reconsider f2 = { [(k - 1),k] where k is Element of NAT : ( k in y9 & k > i ) } as Relation by RELAT_1:def_1;
set f = (id { k where k is Element of NAT : ( k in y & k < i ) } ) \/ f2;
A16: now__::_thesis:_for_x_being_set_st_x_in__{__(k_-_1)_where_k_is_Element_of_NAT_:_(_k_in_y_&_k_>_i_)__}__holds_
x_in_dom_f2
let x be set ; ::_thesis: ( x in { (k - 1) where k is Element of NAT : ( k in y & k > i ) } implies x in dom f2 )
assume x in { (k - 1) where k is Element of NAT : ( k in y & k > i ) } ; ::_thesis: x in dom f2
then consider k being Element of NAT such that
A17: ( k - 1 = x & k in y9 & k > i ) ;
reconsider y = k as set ;
[x,y] in f2 by A17;
hence x in dom f2 by XTUPLE_0:def_12; ::_thesis: verum
end;
now__::_thesis:_for_x_being_set_st_x_in_dom_f2_holds_
x_in__{__(k_-_1)_where_k_is_Element_of_NAT_:_(_k_in_y_&_k_>_i_)__}_
let x be set ; ::_thesis: ( x in dom f2 implies x in { (k - 1) where k is Element of NAT : ( k in y & k > i ) } )
assume x in dom f2 ; ::_thesis: x in { (k - 1) where k is Element of NAT : ( k in y & k > i ) }
then consider y being set such that
A18: [x,y] in f2 by XTUPLE_0:def_12;
consider k being Element of NAT such that
A19: [(k - 1),k] = [x,y] and
A20: ( k in y9 & k > i ) by A18;
k - 1 = x by A19, XTUPLE_0:1;
hence x in { (k - 1) where k is Element of NAT : ( k in y & k > i ) } by A20; ::_thesis: verum
end;
then A21: dom f2 = { (k - 1) where k is Element of NAT : ( k in y & k > i ) } by A16, TARSKI:1;
A22: now__::_thesis:_for_x,_y1,_y2_being_set_st_[x,y1]_in_(id__{__k_where_k_is_Element_of_NAT_:_(_k_in_y_&_k_<_i_)__}__)_\/_f2_&_[x,y2]_in_(id__{__k_where_k_is_Element_of_NAT_:_(_k_in_y_&_k_<_i_)__}__)_\/_f2_holds_
y1_=_y2
let x, y1, y2 be set ; ::_thesis: ( [x,y1] in (id { k where k is Element of NAT : ( k in y & k < i ) } ) \/ f2 & [x,y2] in (id { k where k is Element of NAT : ( k in y & k < i ) } ) \/ f2 implies b2 = b3 )
assume A23: [x,y1] in (id { k where k is Element of NAT : ( k in y & k < i ) } ) \/ f2 ; ::_thesis: ( [x,y2] in (id { k where k is Element of NAT : ( k in y & k < i ) } ) \/ f2 implies b2 = b3 )
assume A24: [x,y2] in (id { k where k is Element of NAT : ( k in y & k < i ) } ) \/ f2 ; ::_thesis: b2 = b3
percases ( [x,y1] in id { k where k is Element of NAT : ( k in y & k < i ) } or [x,y1] in f2 ) by A23, XBOOLE_0:def_3;
supposeA25: [x,y1] in id { k where k is Element of NAT : ( k in y & k < i ) } ; ::_thesis: b2 = b3
then A26: x in dom (id { k where k is Element of NAT : ( k in y & k < i ) } ) by XTUPLE_0:def_12;
then (id { k where k is Element of NAT : ( k in y & k < i ) } ) . x = x by FUNCT_1:17;
then A27: y1 = x by A25, A26, FUNCT_1:def_2;
percases ( [x,y2] in id { k where k is Element of NAT : ( k in y & k < i ) } or [x,y2] in f2 ) by A24, XBOOLE_0:def_3;
supposeA28: [x,y2] in id { k where k is Element of NAT : ( k in y & k < i ) } ; ::_thesis: b2 = b3
then A29: x in dom (id { k where k is Element of NAT : ( k in y & k < i ) } ) by XTUPLE_0:def_12;
then (id { k where k is Element of NAT : ( k in y & k < i ) } ) . x = x by FUNCT_1:17;
hence y1 = y2 by A27, A28, A29, FUNCT_1:def_2; ::_thesis: verum
end;
supposeA30: [x,y2] in f2 ; ::_thesis: b2 = b3
x in { k where k is Element of NAT : ( k in y & k < i ) } by A26;
then consider k9 being Element of NAT such that
A31: k9 = x and
k9 in y and
A32: k9 < i ;
x in { (k - 1) where k is Element of NAT : ( k in y & k > i ) } by A21, A30, XTUPLE_0:def_12;
then ex k being Element of NAT st
( k - 1 = x & k in y & k > i ) ;
then k9 + 1 > i by A31;
hence y1 = y2 by A32, NAT_1:13; ::_thesis: verum
end;
end;
end;
suppose [x,y1] in f2 ; ::_thesis: b2 = b3
then consider k being Element of NAT such that
A33: [(k - 1),k] = [x,y1] and
k in y9 and
A34: k > i ;
A35: k - 1 = x by A33, XTUPLE_0:1;
percases ( [x,y2] in id { k where k is Element of NAT : ( k in y & k < i ) } or [x,y2] in f2 ) by A24, XBOOLE_0:def_3;
suppose [x,y2] in id { k where k is Element of NAT : ( k in y & k < i ) } ; ::_thesis: b2 = b3
then x in dom (id { k where k is Element of NAT : ( k in y & k < i ) } ) by XTUPLE_0:def_12;
then x in { k where k is Element of NAT : ( k in y & k < i ) } ;
then consider k9 being Element of NAT such that
A36: k9 = x and
k9 in y and
A37: k9 < i ;
k9 = k - 1 by A33, A36, XTUPLE_0:1;
then k9 + 1 > i by A34;
hence y1 = y2 by A37, NAT_1:13; ::_thesis: verum
end;
suppose [x,y2] in f2 ; ::_thesis: b2 = b3
then consider k9 being Element of NAT such that
A38: [(k9 - 1),k9] = [x,y2] and
k9 in y9 and
k9 > i ;
k9 - 1 = x by A38, XTUPLE_0:1;
hence y1 = y2 by A33, A35, A38, XTUPLE_0:1; ::_thesis: verum
end;
end;
end;
end;
end;
A39: now__::_thesis:_for_x,_y1,_y2_being_set_st_[x,y1]_in_f2_&_[x,y2]_in_f2_holds_
y1_=_y2
let x, y1, y2 be set ; ::_thesis: ( [x,y1] in f2 & [x,y2] in f2 implies y1 = y2 )
assume [x,y1] in f2 ; ::_thesis: ( [x,y2] in f2 implies y1 = y2 )
then consider k being Element of NAT such that
A40: [(k - 1),k] = [x,y1] and
k in y9 and
k > i ;
A41: k - 1 = x by A40, XTUPLE_0:1;
assume [x,y2] in f2 ; ::_thesis: y1 = y2
then consider k9 being Element of NAT such that
A42: [(k9 - 1),k9] = [x,y2] and
k9 in y9 and
k9 > i ;
k9 - 1 = x by A42, XTUPLE_0:1;
hence y1 = y2 by A40, A42, A41, XTUPLE_0:1; ::_thesis: verum
end;
reconsider f = (id { k where k is Element of NAT : ( k in y & k < i ) } ) \/ f2 as Function by A22, FUNCT_1:def_1;
A43: now__::_thesis:_for_x_being_set_st_x_in_dom_(id__{__k_where_k_is_Element_of_NAT_:_(_k_in_y_&_k_<_i_)__}__)_holds_
f_._x_=_(id__{__k_where_k_is_Element_of_NAT_:_(_k_in_y_&_k_<_i_)__}__)_._x
let x be set ; ::_thesis: ( x in dom (id { k where k is Element of NAT : ( k in y & k < i ) } ) implies f . x = (id { k where k is Element of NAT : ( k in y & k < i ) } ) . x )
A44: id { k where k is Element of NAT : ( k in y & k < i ) } c= f by XBOOLE_1:7;
dom f = (dom (id { k where k is Element of NAT : ( k in y & k < i ) } )) \/ (dom f2) by RELAT_1:1;
then A45: dom (id { k where k is Element of NAT : ( k in y & k < i ) } ) c= dom f by XBOOLE_1:7;
assume A46: x in dom (id { k where k is Element of NAT : ( k in y & k < i ) } ) ; ::_thesis: f . x = (id { k where k is Element of NAT : ( k in y & k < i ) } ) . x
then [x,((id { k where k is Element of NAT : ( k in y & k < i ) } ) . x)] in id { k where k is Element of NAT : ( k in y & k < i ) } by FUNCT_1:def_2;
hence f . x = (id { k where k is Element of NAT : ( k in y & k < i ) } ) . x by A46, A45, A44, FUNCT_1:def_2; ::_thesis: verum
end;
reconsider f2 = f2 as Function by A39, FUNCT_1:def_1;
assume A47: not i in y ; ::_thesis: ex x being set st
( Sgm x = ((Sgm ((Seg (n + 1)) \ {i})) ") * (Sgm y) & x c= Seg n )
A48: now__::_thesis:_for_z_being_set_st_z_in_y9_holds_
z_in_rng_f
let z be set ; ::_thesis: ( z in y9 implies b1 in rng f )
set k = z;
assume A49: z in y9 ; ::_thesis: b1 in rng f
then z in Seg (n + 1) by A1;
then reconsider k = z as Element of NAT ;
percases ( k <= i or k > i ) ;
suppose k <= i ; ::_thesis: b1 in rng f
then k < i by A47, A49, XXREAL_0:1;
then z in { k where k is Element of NAT : ( k in y & k < i ) } by A49;
then z in rng (id { k where k is Element of NAT : ( k in y & k < i ) } ) ;
then z in (rng (id { k where k is Element of NAT : ( k in y & k < i ) } )) \/ (rng f2) by XBOOLE_0:def_3;
hence z in rng f by RELAT_1:12; ::_thesis: verum
end;
supposeA50: k > i ; ::_thesis: b1 in rng f
set x99 = k - 1;
[(k - 1),z] in f2 by A49, A50;
then z in rng f2 by XTUPLE_0:def_13;
then z in (rng (id { k where k is Element of NAT : ( k in y & k < i ) } )) \/ (rng f2) by XBOOLE_0:def_3;
hence z in rng f by RELAT_1:12; ::_thesis: verum
end;
end;
end;
now__::_thesis:_for_z_being_set_st_z_in_rng_f_holds_
z_in_y9
let z be set ; ::_thesis: ( z in rng f implies b1 in y9 )
assume z in rng f ; ::_thesis: b1 in y9
then A51: z in (rng (id { k where k is Element of NAT : ( k in y & k < i ) } )) \/ (rng f2) by RELAT_1:12;
percases ( z in rng (id { k where k is Element of NAT : ( k in y & k < i ) } ) or z in rng f2 ) by A51, XBOOLE_0:def_3;
suppose z in rng (id { k where k is Element of NAT : ( k in y & k < i ) } ) ; ::_thesis: b1 in y9
then z in { k where k is Element of NAT : ( k in y & k < i ) } ;
then ex k being Element of NAT st
( k = z & k in y & k < i ) ;
hence z in y9 ; ::_thesis: verum
end;
suppose z in rng f2 ; ::_thesis: b1 in y9
then consider x99 being set such that
A52: [x99,z] in f2 by XTUPLE_0:def_13;
ex k being Element of NAT st
( [(k - 1),k] = [x99,z] & k in y9 & k > i ) by A52;
hence z in y9 by XTUPLE_0:1; ::_thesis: verum
end;
end;
end;
then A53: rng f = y9 by A48, TARSKI:1;
now__::_thesis:_for_a,_b_being_set_holds_
(_(_[a,b]_in_f_implies_[a,b]_in_(rng_f)_|`_(Sgm_((Seg_(n_+_1))_\_{i}))_)_&_(_[a,b]_in_(rng_f)_|`_(Sgm_((Seg_(n_+_1))_\_{i}))_implies_[a,b]_in_f_)_)
let a, b be set ; ::_thesis: ( ( [a,b] in f implies [a,b] in (rng f) |` (Sgm ((Seg (n + 1)) \ {i})) ) & ( [a,b] in (rng f) |` (Sgm ((Seg (n + 1)) \ {i})) implies [b1,b2] in f ) )
hereby ::_thesis: ( [a,b] in (rng f) |` (Sgm ((Seg (n + 1)) \ {i})) implies [b1,b2] in f )
assume A54: [a,b] in f ; ::_thesis: [a,b] in (rng f) |` (Sgm ((Seg (n + 1)) \ {i}))
percases ( [a,b] in id { k where k is Element of NAT : ( k in y & k < i ) } or [a,b] in f2 ) by A54, XBOOLE_0:def_3;
supposeA55: [a,b] in id { k where k is Element of NAT : ( k in y & k < i ) } ; ::_thesis: [a,b] in (rng f) |` (Sgm ((Seg (n + 1)) \ {i}))
reconsider i9 = i, n9 = n as Element of NAT by ORDINAL1:def_12;
A56: a = b by A55, RELAT_1:def_10;
a in { k where k is Element of NAT : ( k in y & k < i ) } by A55, RELAT_1:def_10;
then consider a9 being Element of NAT such that
A57: a9 = a and
A58: a9 in y and
A59: a9 < i ;
A60: 1 <= a9 by A1, A58, FINSEQ_1:1;
i <= n + 1 by A3, FINSEQ_1:1;
then a9 < n + 1 by A59, XXREAL_0:2;
then a9 <= n by NAT_1:13;
then A61: a in Seg n by A57, A60;
then a in Seg (len (Sgm ((Seg (n + 1)) \ {i}))) by A3, FINSEQ_3:107;
then A62: a in dom (Sgm ((Seg (n + 1)) \ {i})) by FINSEQ_1:def_3;
a9 = (Sgm ((Seg (n9 + 1)) \ {i9})) . a9 by A3, A57, A59, A60, A61, FINSEQ_3:108;
then [a,b] in Sgm ((Seg (n + 1)) \ {i}) by A56, A57, A62, FUNCT_1:1;
hence [a,b] in (rng f) |` (Sgm ((Seg (n + 1)) \ {i})) by A53, A56, A57, A58, RELAT_1:def_12; ::_thesis: verum
end;
supposeA63: [a,b] in f2 ; ::_thesis: [a,b] in (rng f) |` (Sgm ((Seg (n + 1)) \ {i}))
reconsider i9 = i, n9 = n as Element of NAT by ORDINAL1:def_12;
consider b9 being Element of NAT such that
A64: [a,b] = [(b9 - 1),b9] and
A65: b9 in y9 and
A66: b9 > i by A63;
A67: a = b9 - 1 by A64, XTUPLE_0:1;
reconsider a9 = b9 - 1 as integer number ;
i + 1 <= b9 by A66, NAT_1:13;
then A68: (i + 1) - 1 <= b9 - 1 by XREAL_1:9;
then A69: 1 <= a9 by A4, XXREAL_0:2;
reconsider a9 = a9 as Element of NAT by A68, INT_1:3;
b9 <= n + 1 by A1, A65, FINSEQ_1:1;
then A70: b9 - 1 <= (n + 1) - 1 by XREAL_1:9;
then A71: a9 in Seg n by A69;
then a in Seg n by A64, XTUPLE_0:1;
then a in Seg (len (Sgm ((Seg (n + 1)) \ {i}))) by A3, FINSEQ_3:107;
then A72: a in dom (Sgm ((Seg (n + 1)) \ {i})) by FINSEQ_1:def_3;
a9 + 1 = (Sgm ((Seg (n9 + 1)) \ {i9})) . a9 by A3, A70, A68, A71, FINSEQ_3:108;
then [a,b] in Sgm ((Seg (n + 1)) \ {i}) by A64, A67, A72, FUNCT_1:1;
hence [a,b] in (rng f) |` (Sgm ((Seg (n + 1)) \ {i})) by A53, A64, A65, RELAT_1:def_12; ::_thesis: verum
end;
end;
end;
assume A73: [a,b] in (rng f) |` (Sgm ((Seg (n + 1)) \ {i})) ; ::_thesis: [b1,b2] in f
then A74: [a,b] in Sgm ((Seg (n + 1)) \ {i}) by RELAT_1:def_12;
then A75: a in dom (Sgm ((Seg (n + 1)) \ {i})) by XTUPLE_0:def_12;
b in rng f by A73, RELAT_1:def_12;
then b in Seg (n + 1) by A1, A53;
then reconsider a9 = a, b9 = b as Element of NAT by A75;
A76: a in Seg (len (Sgm ((Seg (n + 1)) \ {i}))) by A75, FINSEQ_1:def_3;
then A77: 1 <= a9 by FINSEQ_1:1;
A78: b in y by A53, A73, RELAT_1:def_12;
A79: a in Seg n by A3, A76, FINSEQ_3:107;
reconsider i = i, n = n as Element of NAT by ORDINAL1:def_12;
A80: a9 <= n by A79, FINSEQ_1:1;
percases ( a9 < i or i <= a9 ) ;
supposeA81: a9 < i ; ::_thesis: [b1,b2] in f
then (Sgm ((Seg (n + 1)) \ {i})) . a9 = a9 by A3, A79, A77, FINSEQ_3:108;
then A82: b = a by A74, FUNCT_1:1;
then a9 in { k where k is Element of NAT : ( k in y & k < i ) } by A78, A81;
then [a,b] in id { k where k is Element of NAT : ( k in y & k < i ) } by A82, RELAT_1:def_10;
hence [a,b] in f by XBOOLE_0:def_3; ::_thesis: verum
end;
supposeA83: i <= a9 ; ::_thesis: [b1,b2] in f
then (Sgm ((Seg (n + 1)) \ {i})) . a9 = a9 + 1 by A3, A79, A80, FINSEQ_3:108;
then A84: b9 = a9 + 1 by A74, FUNCT_1:1;
then A85: b9 > i by A83, NAT_1:13;
A86: a = b9 - 1 by A84;
b9 in y9 by A53, A73, RELAT_1:def_12;
then [a,b] in f2 by A85, A86;
hence [a,b] in f by XBOOLE_0:def_3; ::_thesis: verum
end;
end;
end;
then A87: f = (rng f) |` (Sgm ((Seg (n + 1)) \ {i})) by RELAT_1:def_2;
reconsider g = f " as PartFunc of (dom (f ")),(rng (f ")) by RELSET_1:4;
A88: now__::_thesis:_for_x_being_set_st_x_in_dom_f2_holds_
f_._x_=_f2_._x
let x be set ; ::_thesis: ( x in dom f2 implies f . x = f2 . x )
A89: f2 c= f by XBOOLE_1:7;
dom f = (dom (id { k where k is Element of NAT : ( k in y & k < i ) } )) \/ (dom f2) by RELAT_1:1;
then A90: dom f2 c= dom f by XBOOLE_1:7;
assume A91: x in dom f2 ; ::_thesis: f . x = f2 . x
then [x,(f2 . x)] in f2 by FUNCT_1:def_2;
hence f . x = f2 . x by A91, A90, A89, FUNCT_1:def_2; ::_thesis: verum
end;
now__::_thesis:_for_y1,_y2_being_set_st_y1_in_dom_f_&_y2_in_dom_f_&_f_._y1_=_f_._y2_holds_
y1_=_y2
let y1, y2 be set ; ::_thesis: ( y1 in dom f & y2 in dom f & f . y1 = f . y2 implies b1 = b2 )
assume y1 in dom f ; ::_thesis: ( y2 in dom f & f . y1 = f . y2 implies b1 = b2 )
then A92: y1 in (dom (id { k where k is Element of NAT : ( k in y & k < i ) } )) \/ (dom f2) by RELAT_1:1;
assume y2 in dom f ; ::_thesis: ( f . y1 = f . y2 implies b1 = b2 )
then A93: y2 in (dom (id { k where k is Element of NAT : ( k in y & k < i ) } )) \/ (dom f2) by RELAT_1:1;
assume A94: f . y1 = f . y2 ; ::_thesis: b1 = b2
percases ( y1 in dom (id { k where k is Element of NAT : ( k in y & k < i ) } ) or y1 in dom f2 ) by A92, XBOOLE_0:def_3;
supposeA95: y1 in dom (id { k where k is Element of NAT : ( k in y & k < i ) } ) ; ::_thesis: b1 = b2
then A96: (id { k where k is Element of NAT : ( k in y & k < i ) } ) . y1 = y1 by FUNCT_1:17;
then A97: f . y1 = y1 by A43, A95;
percases ( y2 in dom (id { k where k is Element of NAT : ( k in y & k < i ) } ) or y2 in dom f2 ) by A93, XBOOLE_0:def_3;
supposeA98: y2 in dom (id { k where k is Element of NAT : ( k in y & k < i ) } ) ; ::_thesis: b1 = b2
then (id { k where k is Element of NAT : ( k in y & k < i ) } ) . y2 = y2 by FUNCT_1:17;
hence y1 = y2 by A43, A94, A97, A98; ::_thesis: verum
end;
supposeA99: y2 in dom f2 ; ::_thesis: b1 = b2
then f . y2 = f2 . y2 by A88;
then [y2,(f . y2)] in f2 by A99, FUNCT_1:def_2;
then A100: ex k being Element of NAT st
( [(k - 1),k] = [y2,(f . y2)] & k in y9 & k > i ) ;
f . y1 = (id { k where k is Element of NAT : ( k in y & k < i ) } ) . y1 by A43, A95;
then f . y1 in { k where k is Element of NAT : ( k in y & k < i ) } by A95, A96;
then ex k9 being Element of NAT st
( k9 = f . y1 & k9 in y & k9 < i ) ;
hence y1 = y2 by A94, A100, XTUPLE_0:1; ::_thesis: verum
end;
end;
end;
supposeA101: y1 in dom f2 ; ::_thesis: b1 = b2
then f . y1 = f2 . y1 by A88;
then [y1,(f . y1)] in f2 by A101, FUNCT_1:def_2;
then consider k being Element of NAT such that
A102: [(k - 1),k] = [y1,(f . y1)] and
k in y9 and
A103: k > i ;
A104: k = f . y1 by A102, XTUPLE_0:1;
percases ( y2 in dom (id { k where k is Element of NAT : ( k in y & k < i ) } ) or y2 in dom f2 ) by A93, XBOOLE_0:def_3;
supposeA105: y2 in dom (id { k where k is Element of NAT : ( k in y & k < i ) } ) ; ::_thesis: b1 = b2
then (id { k where k is Element of NAT : ( k in y & k < i ) } ) . y2 = y2 by FUNCT_1:17;
then f . y2 in dom (id { k where k is Element of NAT : ( k in y & k < i ) } ) by A43, A105;
then f . y2 in { k where k is Element of NAT : ( k in y & k < i ) } ;
then ex k9 being Element of NAT st
( k9 = f . y2 & k9 in y & k9 < i ) ;
hence y1 = y2 by A94, A102, A103, XTUPLE_0:1; ::_thesis: verum
end;
supposeA106: y2 in dom f2 ; ::_thesis: b1 = b2
then f . y2 = f2 . y2 by A88;
then [y2,(f . y2)] in f2 by A106, FUNCT_1:def_2;
then consider k being Element of NAT such that
A107: [(k - 1),k] = [y2,(f . y2)] and
k in y9 and
k > i ;
k = f . y2 by A107, XTUPLE_0:1;
hence y1 = y2 by A94, A102, A104, A107, XTUPLE_0:1; ::_thesis: verum
end;
end;
end;
end;
end;
then A108: f is one-to-one by FUNCT_1:def_4;
then f " = f ~ by FUNCT_1:def_5;
then A109: f " = ((Sgm ((Seg (n + 1)) \ {i})) ~) | (dom (f ")) by A87, Lm36;
dom (id { k where k is Element of NAT : ( k in y & k < i ) } ) = { k where k is Element of NAT : ( k in y & k < i ) } ;
then A110: dom f = x9 by A21, RELAT_1:1;
then dom f c= NAT by A14, XBOOLE_1:1;
then rng g c= NAT by A108, FUNCT_1:33;
then A111: rng g c= REAL by XBOOLE_1:1;
rng f c= NAT by A1, A53, XBOOLE_1:1;
then dom g c= NAT by A108, FUNCT_1:33;
then dom g c= REAL by XBOOLE_1:1;
then reconsider g = g as PartFunc of REAL,REAL by A111, RELSET_1:7;
A112: dom (f ") = y by A108, A53, FUNCT_1:33;
now__::_thesis:_for_r1,_r2_being_Element_of_REAL_st_r1_in_y_/\_(dom_g)_&_r2_in_y_/\_(dom_g)_&_r1_<_r2_holds_
g_._r1_<_g_._r2
let r1, r2 be Element of REAL ; ::_thesis: ( r1 in y /\ (dom g) & r2 in y /\ (dom g) & r1 < r2 implies g . b1 < g . b2 )
A113: g = ((id { k where k is Element of NAT : ( k in y & k < i ) } ) \/ f2) ~ by A108, FUNCT_1:def_5
.= ((id { k where k is Element of NAT : ( k in y & k < i ) } ) ~) \/ (f2 ~) by RELAT_1:23 ;
assume r1 in y /\ (dom g) ; ::_thesis: ( r2 in y /\ (dom g) & r1 < r2 implies g . b1 < g . b2 )
then A114: [r1,(g . r1)] in g by A112, FUNCT_1:1;
assume r2 in y /\ (dom g) ; ::_thesis: ( r1 < r2 implies g . b1 < g . b2 )
then A115: [r2,(g . r2)] in g by A112, FUNCT_1:1;
assume A116: r1 < r2 ; ::_thesis: g . b1 < g . b2
percases ( [r1,(g . r1)] in (id { k where k is Element of NAT : ( k in y & k < i ) } ) ~ or [r1,(g . r1)] in f2 ~ ) by A114, A113, XBOOLE_0:def_3;
suppose [r1,(g . r1)] in (id { k where k is Element of NAT : ( k in y & k < i ) } ) ~ ; ::_thesis: g . b1 < g . b2
then A117: [r1,(g . r1)] in id { k where k is Element of NAT : ( k in y & k < i ) } ;
then A118: r1 = g . r1 by RELAT_1:def_10;
r1 in { k where k is Element of NAT : ( k in y & k < i ) } by A117, RELAT_1:def_10;
then A119: ex k9 being Element of NAT st
( g . r1 = k9 & k9 in y & k9 < i ) by A118;
percases ( [r2,(g . r2)] in (id { k where k is Element of NAT : ( k in y & k < i ) } ) ~ or [r2,(g . r2)] in f2 ~ ) by A115, A113, XBOOLE_0:def_3;
suppose [r2,(g . r2)] in (id { k where k is Element of NAT : ( k in y & k < i ) } ) ~ ; ::_thesis: g . b1 < g . b2
then [r2,(g . r2)] in id { k where k is Element of NAT : ( k in y & k < i ) } ;
hence g . r1 < g . r2 by A116, A118, RELAT_1:def_10; ::_thesis: verum
end;
suppose [r2,(g . r2)] in f2 ~ ; ::_thesis: g . b1 < g . b2
then [(g . r2),r2] in f2 by RELAT_1:def_7;
then consider k99 being Element of NAT such that
A120: [(k99 - 1),k99] = [(g . r2),r2] and
k99 in y9 and
A121: k99 > i ;
reconsider k999 = g . r2, i9 = i - 1 as integer number by A120, XTUPLE_0:1;
k99 - 1 = g . r2 by A120, XTUPLE_0:1;
then i - 1 < g . r2 by A121, XREAL_1:9;
then i9 + 1 <= k999 by INT_1:7;
hence g . r1 < g . r2 by A119, XXREAL_0:2; ::_thesis: verum
end;
end;
end;
suppose [r1,(g . r1)] in f2 ~ ; ::_thesis: g . b1 < g . b2
then [(g . r1),r1] in f2 by RELAT_1:def_7;
then consider k9 being Element of NAT such that
A122: [(k9 - 1),k9] = [(g . r1),r1] and
k9 in y9 and
A123: k9 > i ;
A124: k9 - 1 = g . r1 by A122, XTUPLE_0:1;
A125: r1 = k9 by A122, XTUPLE_0:1;
percases ( [r2,(g . r2)] in (id { k where k is Element of NAT : ( k in y & k < i ) } ) ~ or [r2,(g . r2)] in f2 ~ ) by A115, A113, XBOOLE_0:def_3;
suppose [r2,(g . r2)] in (id { k where k is Element of NAT : ( k in y & k < i ) } ) ~ ; ::_thesis: g . b1 < g . b2
then [r2,(g . r2)] in id { k where k is Element of NAT : ( k in y & k < i ) } ;
then r2 in { k where k is Element of NAT : ( k in y & k < i ) } by RELAT_1:def_10;
then ex k99 being Element of NAT st
( r2 = k99 & k99 in y & k99 < i ) ;
hence g . r1 < g . r2 by A116, A123, A125, XXREAL_0:2; ::_thesis: verum
end;
suppose [r2,(g . r2)] in f2 ~ ; ::_thesis: g . b1 < g . b2
then [(g . r2),r2] in f2 by RELAT_1:def_7;
then consider k99 being Element of NAT such that
A126: [(k99 - 1),k99] = [(g . r2),r2] and
k99 in y9 and
k99 > i ;
( k99 - 1 = g . r2 & r2 = k99 ) by A126, XTUPLE_0:1;
hence g . r1 < g . r2 by A116, A124, A125, XREAL_1:9; ::_thesis: verum
end;
end;
end;
end;
end;
then A127: g | y is increasing by RFUNCT_2:20;
A128: rng (f ") = { k where k is Element of NAT : ( k in y & k < i ) } \/ { (k - 1) where k is Element of NAT : ( k in y & k > i ) } by A108, A110, FUNCT_1:33;
then A129: { k where k is Element of NAT : ( k in y & k < i ) } \/ { (k - 1) where k is Element of NAT : ( k in y & k > i ) } = (f ") .: y by A112, RELAT_1:113;
now__::_thesis:_for_x9_being_set_st_x9_in_g_.:_y_holds_
x9_in_NAT_\_{0}
let x9 be set ; ::_thesis: ( x9 in g .: y implies x9 in NAT \ {0} )
assume A130: x9 in g .: y ; ::_thesis: x9 in NAT \ {0}
then not x9 = 0 by A14, A129, FINSEQ_1:1;
then A131: not x9 in {0} by TARSKI:def_1;
x9 in Seg n by A6, A129, A130;
hence x9 in NAT \ {0} by A131, XBOOLE_0:def_5; ::_thesis: verum
end;
then A132: g .: y c= NAT \ {0} by TARSKI:def_3;
take { k where k is Element of NAT : ( k in y & k < i ) } \/ { (k - 1) where k is Element of NAT : ( k in y & k > i ) } ; ::_thesis: ( Sgm ( { k where k is Element of NAT : ( k in y & k < i ) } \/ { (k - 1) where k is Element of NAT : ( k in y & k > i ) } ) = ((Sgm ((Seg (n + 1)) \ {i})) ") * (Sgm y) & { k where k is Element of NAT : ( k in y & k < i ) } \/ { (k - 1) where k is Element of NAT : ( k in y & k > i ) } c= Seg n )
Sgm ((Seg (n + 1)) \ {i}) is one-to-one by FINSEQ_3:92, XBOOLE_1:36;
then A133: (Sgm ((Seg (n + 1)) \ {i})) " = (Sgm ((Seg (n + 1)) \ {i})) ~ by FUNCT_1:def_5;
Sgm ( { k where k is Element of NAT : ( k in y & k < i ) } \/ { (k - 1) where k is Element of NAT : ( k in y & k > i ) } ) = Sgm (g .: y) by A112, A128, RELAT_1:113
.= (((Sgm ((Seg (n + 1)) \ {i})) ") | y) * (Sgm y) by A1, A112, A127, A132, A133, A109, Lm38
.= ((Sgm ((Seg (n + 1)) \ {i})) ") * (y |` (Sgm y)) by Lm37
.= ((Sgm ((Seg (n + 1)) \ {i})) ") * (Sgm y) by A2, RELAT_1:95 ;
hence Sgm ( { k where k is Element of NAT : ( k in y & k < i ) } \/ { (k - 1) where k is Element of NAT : ( k in y & k > i ) } ) = ((Sgm ((Seg (n + 1)) \ {i})) ") * (Sgm y) ; ::_thesis: { k where k is Element of NAT : ( k in y & k < i ) } \/ { (k - 1) where k is Element of NAT : ( k in y & k > i ) } c= Seg n
thus { k where k is Element of NAT : ( k in y & k < i ) } \/ { (k - 1) where k is Element of NAT : ( k in y & k > i ) } c= Seg n by A6, TARSKI:def_3; ::_thesis: verum
end;
theorem Th99: :: GROUP_9:99
for O being set
for G being GroupWithOperators of O
for s1, s19, s2 being CompositionSeries of G
for i being Nat st i in dom s1 & i + 1 in dom s1 & s1 . i = s1 . (i + 1) & s19 = Del (s1,i) & s2 is jordan_holder & s1 is_finer_than s2 holds
s19 is_finer_than s2
proof
let O be set ; ::_thesis: for G being GroupWithOperators of O
for s1, s19, s2 being CompositionSeries of G
for i being Nat st i in dom s1 & i + 1 in dom s1 & s1 . i = s1 . (i + 1) & s19 = Del (s1,i) & s2 is jordan_holder & s1 is_finer_than s2 holds
s19 is_finer_than s2
let G be GroupWithOperators of O; ::_thesis: for s1, s19, s2 being CompositionSeries of G
for i being Nat st i in dom s1 & i + 1 in dom s1 & s1 . i = s1 . (i + 1) & s19 = Del (s1,i) & s2 is jordan_holder & s1 is_finer_than s2 holds
s19 is_finer_than s2
let s1, s19, s2 be CompositionSeries of G; ::_thesis: for i being Nat st i in dom s1 & i + 1 in dom s1 & s1 . i = s1 . (i + 1) & s19 = Del (s1,i) & s2 is jordan_holder & s1 is_finer_than s2 holds
s19 is_finer_than s2
let i be Nat; ::_thesis: ( i in dom s1 & i + 1 in dom s1 & s1 . i = s1 . (i + 1) & s19 = Del (s1,i) & s2 is jordan_holder & s1 is_finer_than s2 implies s19 is_finer_than s2 )
assume that
A1: i in dom s1 and
A2: i + 1 in dom s1 ; ::_thesis: ( not s1 . i = s1 . (i + 1) or not s19 = Del (s1,i) or not s2 is jordan_holder or not s1 is_finer_than s2 or s19 is_finer_than s2 )
A3: i in Seg (len s1) by A1, FINSEQ_1:def_3;
then A4: 1 <= i by FINSEQ_1:1;
set k = (len s1) - 1;
assume A5: s1 . i = s1 . (i + 1) ; ::_thesis: ( not s19 = Del (s1,i) or not s2 is jordan_holder or not s1 is_finer_than s2 or s19 is_finer_than s2 )
reconsider k = (len s1) - 1 as integer number ;
assume A6: s19 = Del (s1,i) ; ::_thesis: ( not s2 is jordan_holder or not s1 is_finer_than s2 or s19 is_finer_than s2 )
assume A7: s2 is jordan_holder ; ::_thesis: ( not s1 is_finer_than s2 or s19 is_finer_than s2 )
i <= len s1 by A3, FINSEQ_1:1;
then 1 <= len s1 by A4, XXREAL_0:2;
then 1 - 1 <= (len s1) - 1 by XREAL_1:9;
then reconsider k = k as Element of NAT by INT_1:3;
A8: dom s1 = Seg (k + 1) by FINSEQ_1:def_3;
assume s1 is_finer_than s2 ; ::_thesis: s19 is_finer_than s2
then consider z being set such that
A9: z c= dom s1 and
A10: s2 = s1 * (Sgm z) by Def29;
A11: i + 1 in Seg (len s1) by A2, FINSEQ_1:def_3;
now__::_thesis:_ex_y_being_set_st_
(_y_c=_Seg_(k_+_1)_&_not_i_in_y_&_s2_=_s1_*_(Sgm_y)_)
percases ( not i in z or i in z ) ;
supposeA12: not i in z ; ::_thesis: ex y being set st
( y c= Seg (k + 1) & not i in y & s2 = s1 * (Sgm y) )
set y = z;
take y = z; ::_thesis: ( y c= Seg (k + 1) & not i in y & s2 = s1 * (Sgm y) )
thus y c= Seg (k + 1) by A9, FINSEQ_1:def_3; ::_thesis: ( not i in y & s2 = s1 * (Sgm y) )
thus not i in y by A12; ::_thesis: s2 = s1 * (Sgm y)
thus s2 = s1 * (Sgm y) by A10; ::_thesis: verum
end;
supposeA13: i in z ; ::_thesis: ex y being set st
( y c= Seg (k + 1) & not i in y & s2 = s1 * (Sgm y) )
now__::_thesis:_for_x_being_set_holds_not_x_in_{(i_+_1)}_/\_{i}
let x be set ; ::_thesis: not x in {(i + 1)} /\ {i}
assume A14: x in {(i + 1)} /\ {i} ; ::_thesis: contradiction
then x in {i} by XBOOLE_0:def_4;
then A15: x = i by TARSKI:def_1;
x in {(i + 1)} by A14, XBOOLE_0:def_4;
then x = i + 1 by TARSKI:def_1;
hence contradiction by A15; ::_thesis: verum
end;
then {(i + 1)} /\ {i} = {} by XBOOLE_0:def_1;
then A16: {(i + 1)} misses {i} by XBOOLE_0:def_7;
reconsider y = (z \/ {(i + 1)}) \ {i} as set ;
take y = y; ::_thesis: ( y c= Seg (k + 1) & not i in y & s2 = s1 * (Sgm y) )
{(i + 1)} c= Seg (k + 1) by A11, ZFMISC_1:31;
then A17: z \/ {(i + 1)} c= Seg (k + 1) by A9, A8, XBOOLE_1:8;
hence A18: y c= Seg (k + 1) by XBOOLE_1:1; ::_thesis: ( not i in y & s2 = s1 * (Sgm y) )
then y c= dom s1 by FINSEQ_1:def_3;
then A19: rng (Sgm y) c= dom s1 by A18, FINSEQ_1:def_13;
reconsider y9 = y, z = z as finite set by A9;
A20: dom (Sgm y9) = Seg (card y9) by A17, FINSEQ_3:40, XBOOLE_1:1;
i in z \/ {(i + 1)} by A13, XBOOLE_0:def_3;
then {i} c= z \/ {(i + 1)} by ZFMISC_1:31;
then card ((z \/ {(i + 1)}) \ {i}) = (card (z \/ {(i + 1)})) - (card {i}) by CARD_2:44;
then A21: card y9 = (card (z \/ {(i + 1)})) - 1 by CARD_1:30;
A22: now__::_thesis:_not_i_+_1_in_z
A23: 0 + i < 1 + i by XREAL_1:6;
assume i + 1 in z ; ::_thesis: contradiction
then i + 1 in rng (Sgm z) by A9, A8, FINSEQ_1:def_13;
then consider x99 being set such that
A24: x99 in dom (Sgm z) and
A25: i + 1 = (Sgm z) . x99 by FUNCT_1:def_3;
i in rng (Sgm z) by A9, A8, A13, FINSEQ_1:def_13;
then consider x9 being set such that
A26: x9 in dom (Sgm z) and
A27: i = (Sgm z) . x9 by FUNCT_1:def_3;
reconsider x9 = x9, x99 = x99 as Element of NAT by A26, A24;
A28: dom (Sgm z) = Seg (len (Sgm z)) by FINSEQ_1:def_3;
then A29: x9 <= len (Sgm z) by A26, FINSEQ_1:1;
1 <= x99 by A24, A28, FINSEQ_1:1;
then x9 < x99 by A9, A8, A27, A25, A23, A29, FINSEQ_3:41;
then reconsider l = x99 - x9 as Element of NAT by INT_1:5;
percases ( l = 0 or 0 < l ) ;
suppose l = 0 ; ::_thesis: contradiction
hence contradiction by A27, A25, A23; ::_thesis: verum
end;
supposeA30: 0 < l ; ::_thesis: contradiction
set x999 = x9 + 1;
0 + 1 < l + 1 by A30, XREAL_1:6;
then (x9 + 1) - x9 <= x99 - x9 by NAT_1:13;
then A31: x9 + 1 <= x99 by XREAL_1:9;
x99 <= len (Sgm z) by A24, A28, FINSEQ_1:1;
then A32: x9 + 1 <= len (Sgm z) by A31, XXREAL_0:2;
A33: ( 1 + x9 > 0 + x9 & 1 <= x9 ) by A26, A28, FINSEQ_1:1, XREAL_1:6;
then 1 <= x9 + 1 by XXREAL_0:2;
then x9 + 1 in dom (Sgm z) by A28, A32;
then reconsider k3 = (Sgm z) . (x9 + 1) as Element of NAT by FINSEQ_2:11;
i < k3 by A9, A8, A27, A33, A32, FINSEQ_1:def_13;
then A34: i + 1 <= k3 by NAT_1:13;
A35: ( ( 1 <= x9 + 1 & x9 + 1 < x99 & x99 <= len (Sgm z) ) or x9 + 1 = x99 ) by A24, A28, A31, A33, FINSEQ_1:1, XXREAL_0:1, XXREAL_0:2;
then A36: x9 + 1 in dom s2 by A2, A9, A10, A8, A24, A25, A34, FINSEQ_1:def_13, FUNCT_1:11;
A37: s2 is strictly_decreasing by A7, Def31;
A38: x9 in dom s2 by A1, A10, A26, A27, FUNCT_1:11;
then reconsider H1 = s2 . x9, H2 = s2 . (x9 + 1) as Element of the_stable_subgroups_of G by A36, FINSEQ_2:11;
reconsider H1 = H1, H2 = H2 as StableSubgroup of G by Def11;
reconsider H1 = H1 as GroupWithOperators of O ;
reconsider H2 = H2 as normal StableSubgroup of H1 by A38, A36, Def28;
s2 . x9 = s1 . ((Sgm z) . x9) by A10, A26, FUNCT_1:13
.= s2 . (x9 + 1) by A5, A9, A10, A8, A27, A24, A25, A35, A34, FINSEQ_1:def_13, FUNCT_1:13 ;
then the carrier of H1 = the carrier of H2 ;
then H1 ./. H2 is trivial by Th77;
hence contradiction by A38, A36, A37, Def30; ::_thesis: verum
end;
end;
end;
then card (z \/ {(i + 1)}) = (card z) + 1 by CARD_2:41;
then A39: dom (Sgm y9) = dom (Sgm z) by A9, A8, A21, A20, FINSEQ_3:40;
set z2 = { x where x is Element of NAT : ( x in z & i + 1 < x ) } ;
set z1 = { x where x is Element of NAT : ( x in z & x < i ) } ;
A40: now__::_thesis:_for_x_being_set_st_x_in_(_{__x_where_x_is_Element_of_NAT_:_(_x_in_z_&_x_<_i_)__}__\/_{i})_\/__{__x_where_x_is_Element_of_NAT_:_(_x_in_z_&_i_+_1_<_x_)__}__holds_
x_in_z
let x be set ; ::_thesis: ( x in ( { x where x is Element of NAT : ( x in z & x < i ) } \/ {i}) \/ { x where x is Element of NAT : ( x in z & i + 1 < x ) } implies b1 in z )
assume x in ( { x where x is Element of NAT : ( x in z & x < i ) } \/ {i}) \/ { x where x is Element of NAT : ( x in z & i + 1 < x ) } ; ::_thesis: b1 in z
then ( x in { x where x is Element of NAT : ( x in z & x < i ) } \/ {i} or x in { x where x is Element of NAT : ( x in z & i + 1 < x ) } ) by XBOOLE_0:def_3;
then ( x in { x where x is Element of NAT : ( x in z & x < i ) } or x in {i} or x in { x where x is Element of NAT : ( x in z & i + 1 < x ) } ) by XBOOLE_0:def_3;
then consider x9, x99 being Element of NAT such that
A41: ( ( x = x9 & x9 in z & x9 < i ) or x in {i} or ( x = x99 & x99 in z & i + 1 < x99 ) ) ;
percases ( ( x = x9 & x9 in z & x9 < i ) or x in {i} or ( x = x99 & x99 in z & i + 1 < x99 ) ) by A41;
suppose ( x = x9 & x9 in z & x9 < i ) ; ::_thesis: b1 in z
hence x in z ; ::_thesis: verum
end;
suppose x in {i} ; ::_thesis: b1 in z
hence x in z by A13, TARSKI:def_1; ::_thesis: verum
end;
suppose ( x = x99 & x99 in z & i + 1 < x99 ) ; ::_thesis: b1 in z
hence x in z ; ::_thesis: verum
end;
end;
end;
A42: z c= Seg (k + 1) by A9, FINSEQ_1:def_3;
A43: now__::_thesis:_for_x_being_set_st_x_in_z_holds_
x_in_(_{__x_where_x_is_Element_of_NAT_:_(_x_in_z_&_x_<_i_)__}__\/_{i})_\/__{__x_where_x_is_Element_of_NAT_:_(_x_in_z_&_i_+_1_<_x_)__}_
let x be set ; ::_thesis: ( x in z implies x in ( { x where x is Element of NAT : ( x in z & x < i ) } \/ {i}) \/ { x where x is Element of NAT : ( x in z & i + 1 < x ) } )
assume A44: x in z ; ::_thesis: x in ( { x where x is Element of NAT : ( x in z & x < i ) } \/ {i}) \/ { x where x is Element of NAT : ( x in z & i + 1 < x ) }
then x in Seg (k + 1) by A42;
then reconsider x9 = x as Element of NAT ;
( x9 <= i or i + 1 <= x9 ) by NAT_1:13;
then ( x9 < i or x9 = i or i + 1 < x9 ) by A22, A44, XXREAL_0:1;
then ( x in { x where x is Element of NAT : ( x in z & x < i ) } or x in {i} or x in { x where x is Element of NAT : ( x in z & i + 1 < x ) } ) by A44, TARSKI:def_1;
then ( x in { x where x is Element of NAT : ( x in z & x < i ) } \/ {i} or x in { x where x is Element of NAT : ( x in z & i + 1 < x ) } ) by XBOOLE_0:def_3;
hence x in ( { x where x is Element of NAT : ( x in z & x < i ) } \/ {i}) \/ { x where x is Element of NAT : ( x in z & i + 1 < x ) } by XBOOLE_0:def_3; ::_thesis: verum
end;
then A45: z = ( { x where x is Element of NAT : ( x in z & x < i ) } \/ {i}) \/ { x where x is Element of NAT : ( x in z & i + 1 < x ) } by A40, TARSKI:1;
then { x where x is Element of NAT : ( x in z & i + 1 < x ) } c= z by XBOOLE_1:7;
then A46: { x where x is Element of NAT : ( x in z & i + 1 < x ) } c= Seg (k + 1) by A42, XBOOLE_1:1;
{ x where x is Element of NAT : ( x in z & x < i ) } c= z by A45, XBOOLE_1:7, XBOOLE_1:11;
then A47: { x where x is Element of NAT : ( x in z & x < i ) } c= Seg (k + 1) by A42, XBOOLE_1:1;
now__::_thesis:_for_x_being_set_holds_not_x_in__{__x_where_x_is_Element_of_NAT_:_(_x_in_z_&_i_+_1_<_x_)__}__/\_{i}
let x be set ; ::_thesis: not x in { x where x is Element of NAT : ( x in z & i + 1 < x ) } /\ {i}
assume A48: x in { x where x is Element of NAT : ( x in z & i + 1 < x ) } /\ {i} ; ::_thesis: contradiction
then x in { x where x is Element of NAT : ( x in z & i + 1 < x ) } by XBOOLE_0:def_4;
then consider x9 being Element of NAT such that
A49: x9 = x and
x9 in z and
A50: i + 1 < x9 ;
x in {i} by A48, XBOOLE_0:def_4;
then x9 = i by A49, TARSKI:def_1;
then (i + 1) - i < i - i by A50, XREAL_1:9;
then 1 < 0 ;
hence contradiction ; ::_thesis: verum
end;
then { x where x is Element of NAT : ( x in z & i + 1 < x ) } /\ {i} = {} by XBOOLE_0:def_1;
then A51: { x where x is Element of NAT : ( x in z & i + 1 < x ) } misses {i} by XBOOLE_0:def_7;
now__::_thesis:_for_x_being_set_holds_not_x_in__{__x_where_x_is_Element_of_NAT_:_(_x_in_z_&_x_<_i_)__}__/\_{i}
let x be set ; ::_thesis: not x in { x where x is Element of NAT : ( x in z & x < i ) } /\ {i}
assume A52: x in { x where x is Element of NAT : ( x in z & x < i ) } /\ {i} ; ::_thesis: contradiction
then x in { x where x is Element of NAT : ( x in z & x < i ) } by XBOOLE_0:def_4;
then A53: ex x9 being Element of NAT st
( x9 = x & x9 in z & x9 < i ) ;
x in {i} by A52, XBOOLE_0:def_4;
hence contradiction by A53, TARSKI:def_1; ::_thesis: verum
end;
then { x where x is Element of NAT : ( x in z & x < i ) } /\ {i} = {} by XBOOLE_0:def_1;
then A54: { x where x is Element of NAT : ( x in z & x < i ) } misses {i} by XBOOLE_0:def_7;
A55: y = ((( { x where x is Element of NAT : ( x in z & x < i ) } \/ {i}) \/ { x where x is Element of NAT : ( x in z & i + 1 < x ) } ) \/ {(i + 1)}) \ {i} by A43, A40, TARSKI:1
.= ((( { x where x is Element of NAT : ( x in z & x < i ) } \/ {i}) \/ { x where x is Element of NAT : ( x in z & i + 1 < x ) } ) \ {i}) \/ ({(i + 1)} \ {i}) by XBOOLE_1:42
.= ((( { x where x is Element of NAT : ( x in z & x < i ) } \/ {i}) \/ { x where x is Element of NAT : ( x in z & i + 1 < x ) } ) \ {i}) \/ {(i + 1)} by A16, XBOOLE_1:83
.= ((( { x where x is Element of NAT : ( x in z & x < i ) } \/ {i}) \ {i}) \/ ( { x where x is Element of NAT : ( x in z & i + 1 < x ) } \ {i})) \/ {(i + 1)} by XBOOLE_1:42
.= ((( { x where x is Element of NAT : ( x in z & x < i ) } \/ {i}) \ {i}) \/ { x where x is Element of NAT : ( x in z & i + 1 < x ) } ) \/ {(i + 1)} by A51, XBOOLE_1:83
.= ((( { x where x is Element of NAT : ( x in z & x < i ) } \ {i}) \/ ({i} \ {i})) \/ { x where x is Element of NAT : ( x in z & i + 1 < x ) } ) \/ {(i + 1)} by XBOOLE_1:42
.= ((( { x where x is Element of NAT : ( x in z & x < i ) } \ {i}) \/ {}) \/ { x where x is Element of NAT : ( x in z & i + 1 < x ) } ) \/ {(i + 1)} by XBOOLE_1:37
.= (( { x where x is Element of NAT : ( x in z & x < i ) } \/ {}) \/ { x where x is Element of NAT : ( x in z & i + 1 < x ) } ) \/ {(i + 1)} by A54, XBOOLE_1:83
.= ( { x where x is Element of NAT : ( x in z & x < i ) } \/ {(i + 1)}) \/ { x where x is Element of NAT : ( x in z & i + 1 < x ) } by XBOOLE_1:4 ;
then {(i + 1)} c= y by XBOOLE_1:7, XBOOLE_1:11;
then A56: {(i + 1)} c= Seg (k + 1) by A18, XBOOLE_1:1;
{ x where x is Element of NAT : ( x in z & i + 1 < x ) } c= y by A55, XBOOLE_1:7;
then A57: { x where x is Element of NAT : ( x in z & i + 1 < x ) } c= Seg (k + 1) by A18, XBOOLE_1:1;
now__::_thesis:_not_i_in_y
assume i in y ; ::_thesis: contradiction
then not i in {i} by XBOOLE_0:def_5;
hence contradiction by TARSKI:def_1; ::_thesis: verum
end;
hence not i in y ; ::_thesis: s2 = s1 * (Sgm y)
A58: now__::_thesis:_for_m,_n_being_Element_of_NAT_st_m_in__{__x_where_x_is_Element_of_NAT_:_(_x_in_z_&_x_<_i_)__}__\/_{i}_&_n_in__{__x_where_x_is_Element_of_NAT_:_(_x_in_z_&_i_+_1_<_x_)__}__holds_
m_<_n
let m, n be Element of NAT ; ::_thesis: ( m in { x where x is Element of NAT : ( x in z & x < i ) } \/ {i} & n in { x where x is Element of NAT : ( x in z & i + 1 < x ) } implies m < n )
assume m in { x where x is Element of NAT : ( x in z & x < i ) } \/ {i} ; ::_thesis: ( n in { x where x is Element of NAT : ( x in z & i + 1 < x ) } implies m < n )
then ( m in { x where x is Element of NAT : ( x in z & x < i ) } or m in {i} ) by XBOOLE_0:def_3;
then A59: ex x9 being Element of NAT st
( ( m = x9 & x9 in z & x9 < i ) or m in {i} ) ;
assume n in { x where x is Element of NAT : ( x in z & i + 1 < x ) } ; ::_thesis: m < n
then A60: ex x99 being Element of NAT st
( n = x99 & x99 in z & i + 1 < x99 ) ;
0 + i < 1 + i by XREAL_1:6;
then m < i + 1 by A59, TARSKI:def_1, XXREAL_0:2;
hence m < n by A60, XXREAL_0:2; ::_thesis: verum
end;
A61: now__::_thesis:_for_m,_n_being_Element_of_NAT_st_m_in__{__x_where_x_is_Element_of_NAT_:_(_x_in_z_&_x_<_i_)__}__&_n_in_{i}_holds_
m_<_n
let m, n be Element of NAT ; ::_thesis: ( m in { x where x is Element of NAT : ( x in z & x < i ) } & n in {i} implies m < n )
assume m in { x where x is Element of NAT : ( x in z & x < i ) } ; ::_thesis: ( n in {i} implies m < n )
then A62: ex x9 being Element of NAT st
( m = x9 & x9 in z & x9 < i ) ;
assume n in {i} ; ::_thesis: m < n
hence m < n by A62, TARSKI:def_1; ::_thesis: verum
end;
A63: now__::_thesis:_for_m,_n_being_Element_of_NAT_st_m_in__{__x_where_x_is_Element_of_NAT_:_(_x_in_z_&_x_<_i_)__}__\/_{(i_+_1)}_&_n_in__{__x_where_x_is_Element_of_NAT_:_(_x_in_z_&_i_+_1_<_x_)__}__holds_
m_<_n
let m, n be Element of NAT ; ::_thesis: ( m in { x where x is Element of NAT : ( x in z & x < i ) } \/ {(i + 1)} & n in { x where x is Element of NAT : ( x in z & i + 1 < x ) } implies m < n )
assume m in { x where x is Element of NAT : ( x in z & x < i ) } \/ {(i + 1)} ; ::_thesis: ( n in { x where x is Element of NAT : ( x in z & i + 1 < x ) } implies m < n )
then ( m in { x where x is Element of NAT : ( x in z & x < i ) } or m in {(i + 1)} ) by XBOOLE_0:def_3;
then A64: ex x9 being Element of NAT st
( ( m = x9 & x9 in z & x9 < i ) or m in {(i + 1)} ) ;
assume n in { x where x is Element of NAT : ( x in z & i + 1 < x ) } ; ::_thesis: m < n
then A65: ex x99 being Element of NAT st
( n = x99 & x99 in z & i + 1 < x99 ) ;
0 + i < 1 + i by XREAL_1:6;
then ( m < i + 1 or m = i + 1 ) by A64, TARSKI:def_1, XXREAL_0:2;
hence m < n by A65, XXREAL_0:2; ::_thesis: verum
end;
A66: now__::_thesis:_for_m,_n_being_Element_of_NAT_st_m_in__{__x_where_x_is_Element_of_NAT_:_(_x_in_z_&_x_<_i_)__}__&_n_in_{(i_+_1)}_holds_
m_<_n
let m, n be Element of NAT ; ::_thesis: ( m in { x where x is Element of NAT : ( x in z & x < i ) } & n in {(i + 1)} implies m < n )
A67: 0 + i < 1 + i by XREAL_1:6;
assume m in { x where x is Element of NAT : ( x in z & x < i ) } ; ::_thesis: ( n in {(i + 1)} implies m < n )
then A68: ex x9 being Element of NAT st
( m = x9 & x9 in z & x9 < i ) ;
assume n in {(i + 1)} ; ::_thesis: m < n
then n = i + 1 by TARSKI:def_1;
hence m < n by A68, A67, XXREAL_0:2; ::_thesis: verum
end;
{ x where x is Element of NAT : ( x in z & x < i ) } c= y by A55, XBOOLE_1:7, XBOOLE_1:11;
then { x where x is Element of NAT : ( x in z & x < i ) } c= Seg (k + 1) by A18, XBOOLE_1:1;
then A69: Sgm ( { x where x is Element of NAT : ( x in z & x < i ) } \/ {(i + 1)}) = (Sgm { x where x is Element of NAT : ( x in z & x < i ) } ) ^ (Sgm {(i + 1)}) by A56, A66, FINSEQ_3:42;
{ x where x is Element of NAT : ( x in z & x < i ) } \/ {i} c= z by A45, XBOOLE_1:7;
then { x where x is Element of NAT : ( x in z & x < i ) } \/ {i} c= Seg (k + 1) by A42, XBOOLE_1:1;
then A70: Sgm z = (Sgm ( { x where x is Element of NAT : ( x in z & x < i ) } \/ {i})) ^ (Sgm { x where x is Element of NAT : ( x in z & i + 1 < x ) } ) by A45, A46, A58, FINSEQ_3:42;
{i} c= z by A45, XBOOLE_1:7, XBOOLE_1:11;
then {i} c= Seg (k + 1) by A42, XBOOLE_1:1;
then A71: Sgm ( { x where x is Element of NAT : ( x in z & x < i ) } \/ {i}) = (Sgm { x where x is Element of NAT : ( x in z & x < i ) } ) ^ (Sgm {i}) by A47, A61, FINSEQ_3:42;
then A72: Sgm z = ((Sgm { x where x is Element of NAT : ( x in z & x < i ) } ) ^ <*i*>) ^ (Sgm { x where x is Element of NAT : ( x in z & i + 1 < x ) } ) by A4, A70, FINSEQ_3:44;
{ x where x is Element of NAT : ( x in z & x < i ) } \/ {(i + 1)} c= y by A55, XBOOLE_1:7;
then { x where x is Element of NAT : ( x in z & x < i ) } \/ {(i + 1)} c= Seg (k + 1) by A18, XBOOLE_1:1;
then A73: Sgm y = (Sgm ( { x where x is Element of NAT : ( x in z & x < i ) } \/ {(i + 1)})) ^ (Sgm { x where x is Element of NAT : ( x in z & i + 1 < x ) } ) by A55, A57, A63, FINSEQ_3:42;
then A74: Sgm y = ((Sgm { x where x is Element of NAT : ( x in z & x < i ) } ) ^ <*(i + 1)*>) ^ (Sgm { x where x is Element of NAT : ( x in z & i + 1 < x ) } ) by A69, FINSEQ_3:44;
A75: now__::_thesis:_for_x_being_set_st_x_in_dom_(Sgm_z)_holds_
(_(_(Sgm_z)_._x_<>_i_implies_(Sgm_y)_._x_=_(Sgm_z)_._x_)_&_(_(Sgm_z)_._x_=_i_implies_(Sgm_y)_._x_=_i_+_1_)_)
let x be set ; ::_thesis: ( x in dom (Sgm z) implies ( ( (Sgm z) . b1 <> i implies (Sgm y) . b1 = (Sgm z) . b1 ) & ( (Sgm z) . b1 = i implies (Sgm y) . b1 = i + 1 ) ) )
A76: len ((Sgm { x where x is Element of NAT : ( x in z & x < i ) } ) ^ <*i*>) = (len (Sgm { x where x is Element of NAT : ( x in z & x < i ) } )) + (len <*i*>) by FINSEQ_1:22
.= (len (Sgm { x where x is Element of NAT : ( x in z & x < i ) } )) + 1 by FINSEQ_1:40
.= (len (Sgm { x where x is Element of NAT : ( x in z & x < i ) } )) + (len <*(i + 1)*>) by FINSEQ_1:40
.= len ((Sgm { x where x is Element of NAT : ( x in z & x < i ) } ) ^ <*(i + 1)*>) by FINSEQ_1:22 ;
assume A77: x in dom (Sgm z) ; ::_thesis: ( ( (Sgm z) . b1 <> i implies (Sgm y) . b1 = (Sgm z) . b1 ) & ( (Sgm z) . b1 = i implies (Sgm y) . b1 = i + 1 ) )
then reconsider x9 = x as Element of NAT ;
A78: dom ((Sgm { x where x is Element of NAT : ( x in z & x < i ) } ) ^ <*i*>) = Seg (len ((Sgm { x where x is Element of NAT : ( x in z & x < i ) } ) ^ <*i*>)) by FINSEQ_1:def_3
.= dom ((Sgm { x where x is Element of NAT : ( x in z & x < i ) } ) ^ <*(i + 1)*>) by A76, FINSEQ_1:def_3 ;
percases ( x9 in dom ((Sgm { x where x is Element of NAT : ( x in z & x < i ) } ) ^ <*i*>) or ex x99 being Nat st
( x99 in dom (Sgm { x where x is Element of NAT : ( x in z & i + 1 < x ) } ) & x9 = (len ((Sgm { x where x is Element of NAT : ( x in z & x < i ) } ) ^ <*i*>)) + x99 ) ) by A72, A77, FINSEQ_1:25;
supposeA79: x9 in dom ((Sgm { x where x is Element of NAT : ( x in z & x < i ) } ) ^ <*i*>) ; ::_thesis: ( ( (Sgm z) . b1 <> i implies (Sgm y) . b1 = (Sgm z) . b1 ) & ( (Sgm z) . b1 = i implies (Sgm y) . b1 = i + 1 ) )
percases ( x9 in dom (Sgm { x where x is Element of NAT : ( x in z & x < i ) } ) or ex x99 being Nat st
( x99 in dom <*i*> & x9 = (len (Sgm { x where x is Element of NAT : ( x in z & x < i ) } )) + x99 ) ) by A79, FINSEQ_1:25;
supposeA80: x9 in dom (Sgm { x where x is Element of NAT : ( x in z & x < i ) } ) ; ::_thesis: ( ( (Sgm z) . b1 <> i implies (Sgm y) . b1 = (Sgm z) . b1 ) & ( (Sgm z) . b1 = i implies (Sgm y) . b1 = i + 1 ) )
then A81: (Sgm { x where x is Element of NAT : ( x in z & x < i ) } ) . x9 = ((Sgm { x where x is Element of NAT : ( x in z & x < i ) } ) ^ <*i*>) . x9 by FINSEQ_1:def_7
.= (((Sgm { x where x is Element of NAT : ( x in z & x < i ) } ) ^ <*i*>) ^ (Sgm { x where x is Element of NAT : ( x in z & i + 1 < x ) } )) . x9 by A79, FINSEQ_1:def_7
.= (Sgm z) . x9 by A4, A70, A71, FINSEQ_3:44 ;
(Sgm { x where x is Element of NAT : ( x in z & x < i ) } ) . x9 = ((Sgm { x where x is Element of NAT : ( x in z & x < i ) } ) ^ <*(i + 1)*>) . x9 by A80, FINSEQ_1:def_7
.= (((Sgm { x where x is Element of NAT : ( x in z & x < i ) } ) ^ <*(i + 1)*>) ^ (Sgm { x where x is Element of NAT : ( x in z & i + 1 < x ) } )) . x9 by A78, A79, FINSEQ_1:def_7
.= (Sgm y) . x9 by A73, A69, FINSEQ_3:44 ;
hence ( (Sgm z) . x <> i implies (Sgm y) . x = (Sgm z) . x ) by A81; ::_thesis: ( (Sgm z) . x = i implies (Sgm y) . x = i + 1 )
thus ( (Sgm z) . x = i implies (Sgm y) . x = i + 1 ) ::_thesis: verum
proof
assume (Sgm z) . x = i ; ::_thesis: (Sgm y) . x = i + 1
then i in rng (Sgm { x where x is Element of NAT : ( x in z & x < i ) } ) by A80, A81, FUNCT_1:3;
then i in { x where x is Element of NAT : ( x in z & x < i ) } by A47, FINSEQ_1:def_13;
then ex x999 being Element of NAT st
( x999 = i & x999 in z & x999 < i ) ;
hence (Sgm y) . x = i + 1 ; ::_thesis: verum
end;
end;
suppose ex x99 being Nat st
( x99 in dom <*i*> & x9 = (len (Sgm { x where x is Element of NAT : ( x in z & x < i ) } )) + x99 ) ; ::_thesis: ( ( (Sgm z) . b1 <> i implies (Sgm y) . b1 = (Sgm z) . b1 ) & ( (Sgm z) . b1 = i implies (Sgm y) . b1 = i + 1 ) )
then consider x99 being Nat such that
A82: x99 in dom <*i*> and
A83: x9 = (len (Sgm { x where x is Element of NAT : ( x in z & x < i ) } )) + x99 ;
A84: x99 in Seg 1 by A82, FINSEQ_1:38;
then A85: x99 = 1 by FINSEQ_1:2, TARSKI:def_1;
then i = <*i*> . x99 by FINSEQ_1:40
.= ((Sgm { x where x is Element of NAT : ( x in z & x < i ) } ) ^ <*i*>) . x9 by A82, A83, FINSEQ_1:def_7
.= (((Sgm { x where x is Element of NAT : ( x in z & x < i ) } ) ^ <*i*>) ^ (Sgm { x where x is Element of NAT : ( x in z & i + 1 < x ) } )) . x9 by A79, FINSEQ_1:def_7
.= (Sgm z) . x9 by A4, A70, A71, FINSEQ_3:44 ;
hence ( (Sgm z) . x <> i implies (Sgm y) . x = (Sgm z) . x ) ; ::_thesis: ( (Sgm z) . x = i implies (Sgm y) . x = i + 1 )
thus ( (Sgm z) . x = i implies (Sgm y) . x = i + 1 ) ::_thesis: verum
proof
assume (Sgm z) . x = i ; ::_thesis: (Sgm y) . x = i + 1
A86: x99 in dom <*(i + 1)*> by A84, FINSEQ_1:38;
i + 1 = <*(i + 1)*> . x99 by A85, FINSEQ_1:40
.= ((Sgm { x where x is Element of NAT : ( x in z & x < i ) } ) ^ <*(i + 1)*>) . x9 by A83, A86, FINSEQ_1:def_7
.= (((Sgm { x where x is Element of NAT : ( x in z & x < i ) } ) ^ <*(i + 1)*>) ^ (Sgm { x where x is Element of NAT : ( x in z & i + 1 < x ) } )) . x9 by A78, A79, FINSEQ_1:def_7 ;
hence (Sgm y) . x = i + 1 by A73, A69, FINSEQ_3:44; ::_thesis: verum
end;
end;
end;
end;
suppose ex x99 being Nat st
( x99 in dom (Sgm { x where x is Element of NAT : ( x in z & i + 1 < x ) } ) & x9 = (len ((Sgm { x where x is Element of NAT : ( x in z & x < i ) } ) ^ <*i*>)) + x99 ) ; ::_thesis: ( ( (Sgm z) . b1 <> i implies (Sgm y) . b1 = (Sgm z) . b1 ) & ( (Sgm z) . b1 = i implies (Sgm y) . b1 = i + 1 ) )
then consider x99 being Nat such that
A87: x99 in dom (Sgm { x where x is Element of NAT : ( x in z & i + 1 < x ) } ) and
A88: x9 = (len ((Sgm { x where x is Element of NAT : ( x in z & x < i ) } ) ^ <*i*>)) + x99 ;
(Sgm y) . x9 = (Sgm { x where x is Element of NAT : ( x in z & i + 1 < x ) } ) . x99 by A74, A76, A87, A88, FINSEQ_1:def_7;
hence ( (Sgm z) . x <> i implies (Sgm y) . x = (Sgm z) . x ) by A72, A87, A88, FINSEQ_1:def_7; ::_thesis: ( (Sgm z) . x = i implies (Sgm y) . x = i + 1 )
thus ( (Sgm z) . x = i implies (Sgm y) . x = i + 1 ) ::_thesis: verum
proof
assume (Sgm z) . x = i ; ::_thesis: (Sgm y) . x = i + 1
then (Sgm { x where x is Element of NAT : ( x in z & i + 1 < x ) } ) . x99 = i by A72, A87, A88, FINSEQ_1:def_7;
then i in rng (Sgm { x where x is Element of NAT : ( x in z & i + 1 < x ) } ) by A87, FUNCT_1:3;
then i in { x where x is Element of NAT : ( x in z & i + 1 < x ) } by A57, FINSEQ_1:def_13;
then ex x999 being Element of NAT st
( x999 = i & x999 in z & i + 1 < x999 ) ;
then (i + 1) - i < i - i by XREAL_1:9;
then 1 < 0 ;
hence (Sgm y) . x = i + 1 ; ::_thesis: verum
end;
end;
end;
end;
rng (Sgm z) c= dom s1 by A9, A8, FINSEQ_1:def_13;
then A89: dom (s1 * (Sgm z)) = dom (Sgm z) by RELAT_1:27;
then A90: dom (s1 * (Sgm y)) = dom (s1 * (Sgm z)) by A39, A19, RELAT_1:27;
now__::_thesis:_for_x_being_set_st_x_in_dom_(s1_*_(Sgm_y))_holds_
(s1_*_(Sgm_y))_._x_=_(s1_*_(Sgm_z))_._x
let x be set ; ::_thesis: ( x in dom (s1 * (Sgm y)) implies (s1 * (Sgm y)) . b1 = (s1 * (Sgm z)) . b1 )
assume A91: x in dom (s1 * (Sgm y)) ; ::_thesis: (s1 * (Sgm y)) . b1 = (s1 * (Sgm z)) . b1
then A92: x in dom (Sgm z) by A39, A19, RELAT_1:27;
A93: x in dom (s1 * (Sgm z)) by A39, A19, A89, A91, RELAT_1:27;
percases ( (Sgm z) . x = i or (Sgm z) . x <> i ) ;
supposeA94: (Sgm z) . x = i ; ::_thesis: (s1 * (Sgm y)) . b1 = (s1 * (Sgm z)) . b1
(s1 * (Sgm y)) . x = s1 . ((Sgm y) . x) by A91, FUNCT_1:12
.= s1 . ((Sgm z) . x) by A5, A75, A92, A94
.= (s1 * (Sgm z)) . x by A90, A91, FUNCT_1:12 ;
hence (s1 * (Sgm y)) . x = (s1 * (Sgm z)) . x ; ::_thesis: verum
end;
supposeA95: (Sgm z) . x <> i ; ::_thesis: (s1 * (Sgm y)) . b1 = (s1 * (Sgm z)) . b1
(s1 * (Sgm y)) . x = s1 . ((Sgm y) . x) by A91, FUNCT_1:12
.= s1 . ((Sgm z) . x) by A75, A92, A95
.= (s1 * (Sgm z)) . x by A93, FUNCT_1:12 ;
hence (s1 * (Sgm y)) . x = (s1 * (Sgm z)) . x ; ::_thesis: verum
end;
end;
end;
hence s2 = s1 * (Sgm y) by A10, A39, A19, A89, FUNCT_1:2, RELAT_1:27; ::_thesis: verum
end;
end;
end;
then consider y being set such that
A96: y c= Seg (k + 1) and
A97: not i in y and
A98: s2 = s1 * (Sgm y) ;
now__::_thesis:_ex_x_being_set_st_
(_x_c=_dom_s19_&_s2_=_s19_*_(Sgm_x)_)
consider x being set such that
A99: Sgm x = ((Sgm ((Seg (k + 1)) \ {i})) ") * (Sgm y) and
A100: x c= Seg k by A3, A96, A97, Lm39;
take x = x; ::_thesis: ( x c= dom s19 & s2 = s19 * (Sgm x) )
ex m being Nat st
( len s1 = m + 1 & len (Del (s1,i)) = m ) by A1, FINSEQ_3:104;
hence x c= dom s19 by A6, A100, FINSEQ_1:def_3; ::_thesis: s2 = s19 * (Sgm x)
set f = Sgm ((Seg (k + 1)) \ {i});
set X = dom (Sgm ((Seg (k + 1)) \ {i}));
set Y = rng (Sgm ((Seg (k + 1)) \ {i}));
reconsider f = Sgm ((Seg (k + 1)) \ {i}) as Function of (dom (Sgm ((Seg (k + 1)) \ {i}))),(rng (Sgm ((Seg (k + 1)) \ {i}))) by FUNCT_2:1;
A101: f is one-to-one by FINSEQ_3:92, XBOOLE_1:36;
(Seg (k + 1)) \ {i} c= Seg (k + 1) by XBOOLE_1:36;
then A102: rng f = (Seg (k + 1)) \ {i} by FINSEQ_1:def_13;
now__::_thesis:_for_x9_being_set_st_x9_in_y_holds_
x9_in_rng_f
let x9 be set ; ::_thesis: ( x9 in y implies x9 in rng f )
assume A103: x9 in y ; ::_thesis: x9 in rng f
then not x9 in {i} by A97, TARSKI:def_1;
hence x9 in rng f by A96, A102, A103, XBOOLE_0:def_5; ::_thesis: verum
end;
then y c= rng f by TARSKI:def_3;
then A104: rng (Sgm y) c= rng f by A96, FINSEQ_1:def_13;
A105: now__::_thesis:_not_rng_f_=_{}
1 <= i by A3, FINSEQ_1:1;
then A106: 1 + 1 <= i + 1 by XREAL_1:6;
i + 1 <= len s1 by A11, FINSEQ_1:1;
then 2 <= len s1 by A106, XXREAL_0:2;
then Seg 2 c= Seg (k + 1) by FINSEQ_1:5;
then A107: (Seg 2) \ {i} c= rng f by A102, XBOOLE_1:33;
assume A108: rng f = {} ; ::_thesis: contradiction
percases ( Seg 2 = {} or Seg 2 = {i} ) by A108, A107, XBOOLE_1:3, ZFMISC_1:58;
suppose Seg 2 = {} ; ::_thesis: contradiction
hence contradiction ; ::_thesis: verum
end;
suppose Seg 2 = {i} ; ::_thesis: contradiction
hence contradiction by FINSEQ_1:2, ZFMISC_1:5; ::_thesis: verum
end;
end;
end;
s19 * (Sgm x) = (s1 * f) * (Sgm x) by A6, FINSEQ_1:def_3
.= ((s1 * f) * (f ")) * (Sgm y) by A99, RELAT_1:36
.= (s1 * (f * (f "))) * (Sgm y) by RELAT_1:36
.= (s1 * (id (rng f))) * (Sgm y) by A101, A105, FUNCT_2:29
.= s1 * ((id (rng f)) * (Sgm y)) by RELAT_1:36
.= s1 * (Sgm y) by A104, RELAT_1:53 ;
hence s2 = s19 * (Sgm x) by A98; ::_thesis: verum
end;
hence s19 is_finer_than s2 by Def29; ::_thesis: verum
end;
theorem Th100: :: GROUP_9:100
for O being set
for G being GroupWithOperators of O
for s1, s2 being CompositionSeries of G st len s1 > 1 & s2 <> s1 & s2 is strictly_decreasing & s2 is_finer_than s1 holds
ex i, j being Nat st
( i in dom s1 & i in dom s2 & i + 1 in dom s1 & i + 1 in dom s2 & j in dom s2 & i + 1 < j & s1 . i = s2 . i & s1 . (i + 1) <> s2 . (i + 1) & s1 . (i + 1) = s2 . j )
proof
let O be set ; ::_thesis: for G being GroupWithOperators of O
for s1, s2 being CompositionSeries of G st len s1 > 1 & s2 <> s1 & s2 is strictly_decreasing & s2 is_finer_than s1 holds
ex i, j being Nat st
( i in dom s1 & i in dom s2 & i + 1 in dom s1 & i + 1 in dom s2 & j in dom s2 & i + 1 < j & s1 . i = s2 . i & s1 . (i + 1) <> s2 . (i + 1) & s1 . (i + 1) = s2 . j )
let G be GroupWithOperators of O; ::_thesis: for s1, s2 being CompositionSeries of G st len s1 > 1 & s2 <> s1 & s2 is strictly_decreasing & s2 is_finer_than s1 holds
ex i, j being Nat st
( i in dom s1 & i in dom s2 & i + 1 in dom s1 & i + 1 in dom s2 & j in dom s2 & i + 1 < j & s1 . i = s2 . i & s1 . (i + 1) <> s2 . (i + 1) & s1 . (i + 1) = s2 . j )
let s1, s2 be CompositionSeries of G; ::_thesis: ( len s1 > 1 & s2 <> s1 & s2 is strictly_decreasing & s2 is_finer_than s1 implies ex i, j being Nat st
( i in dom s1 & i in dom s2 & i + 1 in dom s1 & i + 1 in dom s2 & j in dom s2 & i + 1 < j & s1 . i = s2 . i & s1 . (i + 1) <> s2 . (i + 1) & s1 . (i + 1) = s2 . j ) )
assume len s1 > 1 ; ::_thesis: ( not s2 <> s1 or not s2 is strictly_decreasing or not s2 is_finer_than s1 or ex i, j being Nat st
( i in dom s1 & i in dom s2 & i + 1 in dom s1 & i + 1 in dom s2 & j in dom s2 & i + 1 < j & s1 . i = s2 . i & s1 . (i + 1) <> s2 . (i + 1) & s1 . (i + 1) = s2 . j ) )
then len s1 >= 1 + 1 by NAT_1:13;
then Seg 2 c= Seg (len s1) by FINSEQ_1:5;
then A1: Seg 2 c= dom s1 by FINSEQ_1:def_3;
assume A2: s2 <> s1 ; ::_thesis: ( not s2 is strictly_decreasing or not s2 is_finer_than s1 or ex i, j being Nat st
( i in dom s1 & i in dom s2 & i + 1 in dom s1 & i + 1 in dom s2 & j in dom s2 & i + 1 < j & s1 . i = s2 . i & s1 . (i + 1) <> s2 . (i + 1) & s1 . (i + 1) = s2 . j ) )
assume A3: s2 is strictly_decreasing ; ::_thesis: ( not s2 is_finer_than s1 or ex i, j being Nat st
( i in dom s1 & i in dom s2 & i + 1 in dom s1 & i + 1 in dom s2 & j in dom s2 & i + 1 < j & s1 . i = s2 . i & s1 . (i + 1) <> s2 . (i + 1) & s1 . (i + 1) = s2 . j ) )
assume A4: s2 is_finer_than s1 ; ::_thesis: ex i, j being Nat st
( i in dom s1 & i in dom s2 & i + 1 in dom s1 & i + 1 in dom s2 & j in dom s2 & i + 1 < j & s1 . i = s2 . i & s1 . (i + 1) <> s2 . (i + 1) & s1 . (i + 1) = s2 . j )
then consider n being Nat such that
A5: len s2 = (len s1) + n by Th95;
n <> 0 by A2, A4, A5, Th96;
then A6: 0 + (len s1) < n + (len s1) by XREAL_1:6;
then Seg (len s1) c= Seg (len s2) by A5, FINSEQ_1:5;
then Seg (len s1) c= dom s2 by FINSEQ_1:def_3;
then A7: dom s1 c= dom s2 by FINSEQ_1:def_3;
now__::_thesis:_ex_i_being_Element_of_NAT_st_
(_i_in_dom_s1_&_s1_._i_=_s2_._i_&_i_+_1_in_dom_s1_&_s1_._(i_+_1)_<>_s2_._(i_+_1)_)
set fX = { k where k is Element of NAT : ( k in dom s1 & s1 . k = s2 . k ) } ;
A8: 1 in Seg 2 ;
( s1 . 1 = (Omega). G & s2 . 1 = (Omega). G ) by Def28;
then A9: 1 in { k where k is Element of NAT : ( k in dom s1 & s1 . k = s2 . k ) } by A1, A8;
now__::_thesis:_for_x_being_set_st_x_in__{__k_where_k_is_Element_of_NAT_:_(_k_in_dom_s1_&_s1_._k_=_s2_._k_)__}__holds_
x_in_dom_s1
let x be set ; ::_thesis: ( x in { k where k is Element of NAT : ( k in dom s1 & s1 . k = s2 . k ) } implies x in dom s1 )
assume x in { k where k is Element of NAT : ( k in dom s1 & s1 . k = s2 . k ) } ; ::_thesis: x in dom s1
then ex k being Element of NAT st
( x = k & k in dom s1 & s1 . k = s2 . k ) ;
hence x in dom s1 ; ::_thesis: verum
end;
then { k where k is Element of NAT : ( k in dom s1 & s1 . k = s2 . k ) } c= dom s1 by TARSKI:def_3;
then reconsider fX = { k where k is Element of NAT : ( k in dom s1 & s1 . k = s2 . k ) } as non empty finite real-membered set by A9;
set i = max fX;
max fX in fX by XXREAL_2:def_8;
then A10: ex k being Element of NAT st
( max fX = k & k in dom s1 & s1 . k = s2 . k ) ;
then reconsider i = max fX as Element of NAT ;
take i = i; ::_thesis: ( i in dom s1 & s1 . i = s2 . i & i + 1 in dom s1 & s1 . (i + 1) <> s2 . (i + 1) )
thus ( i in dom s1 & s1 . i = s2 . i ) by A10; ::_thesis: ( i + 1 in dom s1 & s1 . (i + 1) <> s2 . (i + 1) )
A11: now__::_thesis:_i_+_1_in_dom_s1
assume not i + 1 in dom s1 ; ::_thesis: contradiction
then A12: not i + 1 in Seg (len s1) by FINSEQ_1:def_3;
percases ( 1 > i + 1 or i + 1 > len s1 ) by A12;
suppose 1 > i + 1 ; ::_thesis: contradiction
then 1 - 1 > (i + 1) - 1 by XREAL_1:9;
then 0 > i ;
hence contradiction ; ::_thesis: verum
end;
supposeA13: i + 1 > len s1 ; ::_thesis: contradiction
i in Seg (len s1) by A10, FINSEQ_1:def_3;
then A14: i <= len s1 by FINSEQ_1:1;
i >= len s1 by A13, NAT_1:13;
then A15: i = len s1 by A14, XXREAL_0:1;
then ( 0 + 1 <= i + 1 & i + 1 <= len s2 ) by A5, A6, NAT_1:13;
then i + 1 in Seg (len s2) ;
then A16: i + 1 in dom s2 by FINSEQ_1:def_3;
then reconsider H1 = s2 . i, H2 = s2 . (i + 1) as Element of the_stable_subgroups_of G by A10, FINSEQ_2:11;
reconsider H1 = H1, H2 = H2 as StableSubgroup of G by Def11;
A17: s2 . i = (1). G by A10, A15, Def28;
then A18: the carrier of H1 = {(1_ G)} by Def8;
reconsider H2 = H2 as normal StableSubgroup of H1 by A7, A10, A16, Def28;
1_ G in H2 by Lm18;
then 1_ G in the carrier of H2 by STRUCT_0:def_5;
then A19: {(1_ G)} c= the carrier of H2 by ZFMISC_1:31;
H2 is Subgroup of (1). G by A17, Def7;
then the carrier of H2 c= the carrier of ((1). G) by GROUP_2:def_5;
then the carrier of H2 c= {(1_ G)} by Def8;
then the carrier of H2 = {(1_ G)} by A19, XBOOLE_0:def_10;
then H1 ./. H2 is trivial by A18, Th77;
hence contradiction by A3, A7, A10, A16, Def30; ::_thesis: verum
end;
end;
end;
hence i + 1 in dom s1 ; ::_thesis: s1 . (i + 1) <> s2 . (i + 1)
now__::_thesis:_not_s1_._(i_+_1)_=_s2_._(i_+_1)
A20: 1 + i > 0 + i by XREAL_1:6;
assume s1 . (i + 1) = s2 . (i + 1) ; ::_thesis: contradiction
then consider k being Element of NAT such that
A21: k > i and
A22: ( k in dom s1 & s1 . k = s2 . k ) by A11, A20;
k in fX by A22;
hence contradiction by A21, XXREAL_2:def_8; ::_thesis: verum
end;
hence s1 . (i + 1) <> s2 . (i + 1) ; ::_thesis: verum
end;
then consider i being Nat such that
A23: i in dom s1 and
A24: i + 1 in dom s1 and
A25: s1 . i = s2 . i and
A26: s1 . (i + 1) <> s2 . (i + 1) ;
now__::_thesis:_ex_j_being_Element_of_NAT_st_
(_j_in_dom_s2_&_s1_._(i_+_1)_=_s2_._j_&_i_+_1_<_j_)
consider x being set such that
A27: x c= dom s2 and
A28: s1 = s2 * (Sgm x) by A4, Def29;
set j = (Sgm x) . (i + 1);
A29: x c= Seg (len s2) by A27, FINSEQ_1:def_3;
A30: i + 1 in dom (Sgm x) by A24, A28, FUNCT_1:11;
then (Sgm x) . (i + 1) in rng (Sgm x) by FUNCT_1:3;
then (Sgm x) . (i + 1) in x by A29, FINSEQ_1:def_13;
then A31: (Sgm x) . (i + 1) in Seg (len s2) by A29;
then reconsider j = (Sgm x) . (i + 1) as Element of NAT ;
A32: i + 1 <= j by A29, A30, FINSEQ_3:152;
take j = j; ::_thesis: ( j in dom s2 & s1 . (i + 1) = s2 . j & i + 1 < j )
thus j in dom s2 by A31, FINSEQ_1:def_3; ::_thesis: ( s1 . (i + 1) = s2 . j & i + 1 < j )
thus s1 . (i + 1) = s2 . j by A24, A28, FUNCT_1:12; ::_thesis: i + 1 < j
j <> i + 1 by A24, A26, A28, FUNCT_1:12;
hence i + 1 < j by A32, XXREAL_0:1; ::_thesis: verum
end;
then consider j being Nat such that
A33: ( j in dom s2 & i + 1 < j ) and
A34: s1 . (i + 1) = s2 . j ;
take i ; ::_thesis: ex j being Nat st
( i in dom s1 & i in dom s2 & i + 1 in dom s1 & i + 1 in dom s2 & j in dom s2 & i + 1 < j & s1 . i = s2 . i & s1 . (i + 1) <> s2 . (i + 1) & s1 . (i + 1) = s2 . j )
take j ; ::_thesis: ( i in dom s1 & i in dom s2 & i + 1 in dom s1 & i + 1 in dom s2 & j in dom s2 & i + 1 < j & s1 . i = s2 . i & s1 . (i + 1) <> s2 . (i + 1) & s1 . (i + 1) = s2 . j )
thus ( i in dom s1 & i in dom s2 ) by A7, A23; ::_thesis: ( i + 1 in dom s1 & i + 1 in dom s2 & j in dom s2 & i + 1 < j & s1 . i = s2 . i & s1 . (i + 1) <> s2 . (i + 1) & s1 . (i + 1) = s2 . j )
thus ( i + 1 in dom s1 & i + 1 in dom s2 ) by A7, A24; ::_thesis: ( j in dom s2 & i + 1 < j & s1 . i = s2 . i & s1 . (i + 1) <> s2 . (i + 1) & s1 . (i + 1) = s2 . j )
thus ( j in dom s2 & i + 1 < j ) by A33; ::_thesis: ( s1 . i = s2 . i & s1 . (i + 1) <> s2 . (i + 1) & s1 . (i + 1) = s2 . j )
thus ( s1 . i = s2 . i & s1 . (i + 1) <> s2 . (i + 1) ) by A25, A26; ::_thesis: s1 . (i + 1) = s2 . j
thus s1 . (i + 1) = s2 . j by A34; ::_thesis: verum
end;
theorem Th101: :: GROUP_9:101
for O being set
for G being GroupWithOperators of O
for H1, H2 being StableSubgroup of G
for s1 being CompositionSeries of G
for i, j being Nat st i in dom s1 & j in dom s1 & i <= j & H1 = s1 . i & H2 = s1 . j holds
H2 is StableSubgroup of H1
proof
let O be set ; ::_thesis: for G being GroupWithOperators of O
for H1, H2 being StableSubgroup of G
for s1 being CompositionSeries of G
for i, j being Nat st i in dom s1 & j in dom s1 & i <= j & H1 = s1 . i & H2 = s1 . j holds
H2 is StableSubgroup of H1
let G be GroupWithOperators of O; ::_thesis: for H1, H2 being StableSubgroup of G
for s1 being CompositionSeries of G
for i, j being Nat st i in dom s1 & j in dom s1 & i <= j & H1 = s1 . i & H2 = s1 . j holds
H2 is StableSubgroup of H1
let H1, H2 be StableSubgroup of G; ::_thesis: for s1 being CompositionSeries of G
for i, j being Nat st i in dom s1 & j in dom s1 & i <= j & H1 = s1 . i & H2 = s1 . j holds
H2 is StableSubgroup of H1
let s1 be CompositionSeries of G; ::_thesis: for i, j being Nat st i in dom s1 & j in dom s1 & i <= j & H1 = s1 . i & H2 = s1 . j holds
H2 is StableSubgroup of H1
let i, j be Nat; ::_thesis: ( i in dom s1 & j in dom s1 & i <= j & H1 = s1 . i & H2 = s1 . j implies H2 is StableSubgroup of H1 )
assume that
A1: i in dom s1 and
A2: j in dom s1 ; ::_thesis: ( not i <= j or not H1 = s1 . i or not H2 = s1 . j or H2 is StableSubgroup of H1 )
defpred S1[ Nat] means for n being Nat
for H2 being StableSubgroup of G st i + $1 in dom s1 & H2 = s1 . (i + $1) holds
H2 is StableSubgroup of H1;
assume A3: i <= j ; ::_thesis: ( not H1 = s1 . i or not H2 = s1 . j or H2 is StableSubgroup of H1 )
assume that
A4: H1 = s1 . i and
A5: H2 = s1 . j ; ::_thesis: H2 is StableSubgroup of H1
A6: for n being Nat st S1[n] holds
S1[n + 1]
proof
let n be Nat; ::_thesis: ( S1[n] implies S1[n + 1] )
assume A7: S1[n] ; ::_thesis: S1[n + 1]
set H2 = s1 . (i + n);
percases ( i + n in dom s1 or not i + n in dom s1 ) ;
supposeA8: i + n in dom s1 ; ::_thesis: S1[n + 1]
then reconsider H2 = s1 . (i + n) as Element of the_stable_subgroups_of G by FINSEQ_2:11;
reconsider H2 = H2 as StableSubgroup of G by Def11;
A9: H2 is StableSubgroup of H1 by A7, A8;
now__::_thesis:_for_k_being_Element_of_NAT_
for_H3_being_StableSubgroup_of_G_st_i_+_(n_+_1)_in_dom_s1_&_H3_=_s1_._(i_+_(n_+_1))_holds_
H3_is_StableSubgroup_of_H1
let k be Element of NAT ; ::_thesis: for H3 being StableSubgroup of G st i + (n + 1) in dom s1 & H3 = s1 . (i + (n + 1)) holds
H3 is StableSubgroup of H1
let H3 be StableSubgroup of G; ::_thesis: ( i + (n + 1) in dom s1 & H3 = s1 . (i + (n + 1)) implies H3 is StableSubgroup of H1 )
assume i + (n + 1) in dom s1 ; ::_thesis: ( H3 = s1 . (i + (n + 1)) implies H3 is StableSubgroup of H1 )
then A10: (i + n) + 1 in dom s1 ;
assume H3 = s1 . (i + (n + 1)) ; ::_thesis: H3 is StableSubgroup of H1
then H3 is StableSubgroup of H2 by A8, A10, Def28;
hence H3 is StableSubgroup of H1 by A9, Th11; ::_thesis: verum
end;
hence S1[n + 1] ; ::_thesis: verum
end;
suppose not i + n in dom s1 ; ::_thesis: S1[n + 1]
then A11: not i + n in Seg (len s1) by FINSEQ_1:def_3;
percases ( i + n < 0 + 1 or i + n > len s1 ) by A11, FINSEQ_1:1;
suppose i + n < 0 + 1 ; ::_thesis: S1[n + 1]
then n = 0 by NAT_1:13;
hence S1[n + 1] by A1, A4, Def28; ::_thesis: verum
end;
supposeA12: i + n > len s1 ; ::_thesis: S1[n + 1]
A13: 1 + (len s1) > 0 + (len s1) by XREAL_1:6;
(i + n) + 1 > (len s1) + 1 by A12, XREAL_1:6;
then (i + n) + 1 > len s1 by A13, XXREAL_0:2;
then not (i + n) + 1 in Seg (len s1) by FINSEQ_1:1;
hence S1[n + 1] by FINSEQ_1:def_3; ::_thesis: verum
end;
end;
end;
end;
end;
A14: S1[ 0 ] by A4, Th10;
A15: for n being Nat holds S1[n] from NAT_1:sch_2(A14, A6);
set n = j - i;
i - i <= j - i by A3, XREAL_1:9;
then reconsider n = j - i as Element of NAT by INT_1:3;
reconsider n = n as Nat ;
j = i + n ;
hence H2 is StableSubgroup of H1 by A2, A5, A15; ::_thesis: verum
end;
theorem Th102: :: GROUP_9:102
for O being set
for G being GroupWithOperators of O
for y being set
for s1 being CompositionSeries of G st y in rng (the_series_of_quotients_of s1) holds
y is strict GroupWithOperators of O
proof
let O be set ; ::_thesis: for G being GroupWithOperators of O
for y being set
for s1 being CompositionSeries of G st y in rng (the_series_of_quotients_of s1) holds
y is strict GroupWithOperators of O
let G be GroupWithOperators of O; ::_thesis: for y being set
for s1 being CompositionSeries of G st y in rng (the_series_of_quotients_of s1) holds
y is strict GroupWithOperators of O
let y be set ; ::_thesis: for s1 being CompositionSeries of G st y in rng (the_series_of_quotients_of s1) holds
y is strict GroupWithOperators of O
let s1 be CompositionSeries of G; ::_thesis: ( y in rng (the_series_of_quotients_of s1) implies y is strict GroupWithOperators of O )
assume A1: y in rng (the_series_of_quotients_of s1) ; ::_thesis: y is strict GroupWithOperators of O
set f1 = the_series_of_quotients_of s1;
A2: ( len (the_series_of_quotients_of s1) = 0 or len (the_series_of_quotients_of s1) >= 0 + 1 ) by NAT_1:13;
percases ( len (the_series_of_quotients_of s1) = 0 or len (the_series_of_quotients_of s1) >= 1 ) by A2;
suppose len (the_series_of_quotients_of s1) = 0 ; ::_thesis: y is strict GroupWithOperators of O
then the_series_of_quotients_of s1 = {} ;
hence y is strict GroupWithOperators of O by A1; ::_thesis: verum
end;
suppose len (the_series_of_quotients_of s1) >= 1 ; ::_thesis: y is strict GroupWithOperators of O
then A3: len s1 > 1 by Def33, CARD_1:27;
then A4: len s1 = (len (the_series_of_quotients_of s1)) + 1 by Def33;
consider i being set such that
A5: i in dom (the_series_of_quotients_of s1) and
A6: (the_series_of_quotients_of s1) . i = y by A1, FUNCT_1:def_3;
reconsider i = i as Nat by A5;
A7: i in Seg (len (the_series_of_quotients_of s1)) by A5, FINSEQ_1:def_3;
then A8: 1 <= i by FINSEQ_1:1;
1 <= i by A7, FINSEQ_1:1;
then 1 + 1 <= i + 1 by XREAL_1:6;
then A9: 1 <= i + 1 by XXREAL_0:2;
A10: i <= len (the_series_of_quotients_of s1) by A7, FINSEQ_1:1;
then ( 0 + i <= 1 + i & i + 1 <= (len (the_series_of_quotients_of s1)) + 1 ) by XREAL_1:6;
then i <= len s1 by A4, XXREAL_0:2;
then i in Seg (len s1) by A8, FINSEQ_1:1;
then A11: i in dom s1 by FINSEQ_1:def_3;
then s1 . i in the_stable_subgroups_of G by FINSEQ_2:11;
then reconsider H1 = s1 . i as strict StableSubgroup of G by Def11;
i + 1 <= (len (the_series_of_quotients_of s1)) + 1 by A10, XREAL_1:6;
then i + 1 <= len s1 by A3, Def33;
then i + 1 in Seg (len s1) by A9;
then A12: i + 1 in dom s1 by FINSEQ_1:def_3;
then s1 . (i + 1) in the_stable_subgroups_of G by FINSEQ_2:11;
then reconsider N1 = s1 . (i + 1) as strict StableSubgroup of G by Def11;
reconsider N1 = N1 as normal StableSubgroup of H1 by A11, A12, Def28;
y = H1 ./. N1 by A3, A5, A6, Def33;
hence y is strict GroupWithOperators of O ; ::_thesis: verum
end;
end;
end;
theorem Th103: :: GROUP_9:103
for O being set
for G being GroupWithOperators of O
for s1 being CompositionSeries of G
for i being Nat st i in dom (the_series_of_quotients_of s1) & ( for H being GroupWithOperators of O st H = (the_series_of_quotients_of s1) . i holds
H is trivial ) holds
( i in dom s1 & i + 1 in dom s1 & s1 . i = s1 . (i + 1) )
proof
let O be set ; ::_thesis: for G being GroupWithOperators of O
for s1 being CompositionSeries of G
for i being Nat st i in dom (the_series_of_quotients_of s1) & ( for H being GroupWithOperators of O st H = (the_series_of_quotients_of s1) . i holds
H is trivial ) holds
( i in dom s1 & i + 1 in dom s1 & s1 . i = s1 . (i + 1) )
let G be GroupWithOperators of O; ::_thesis: for s1 being CompositionSeries of G
for i being Nat st i in dom (the_series_of_quotients_of s1) & ( for H being GroupWithOperators of O st H = (the_series_of_quotients_of s1) . i holds
H is trivial ) holds
( i in dom s1 & i + 1 in dom s1 & s1 . i = s1 . (i + 1) )
let s1 be CompositionSeries of G; ::_thesis: for i being Nat st i in dom (the_series_of_quotients_of s1) & ( for H being GroupWithOperators of O st H = (the_series_of_quotients_of s1) . i holds
H is trivial ) holds
( i in dom s1 & i + 1 in dom s1 & s1 . i = s1 . (i + 1) )
let i be Nat; ::_thesis: ( i in dom (the_series_of_quotients_of s1) & ( for H being GroupWithOperators of O st H = (the_series_of_quotients_of s1) . i holds
H is trivial ) implies ( i in dom s1 & i + 1 in dom s1 & s1 . i = s1 . (i + 1) ) )
assume A1: i in dom (the_series_of_quotients_of s1) ; ::_thesis: ( ex H being GroupWithOperators of O st
( H = (the_series_of_quotients_of s1) . i & not H is trivial ) or ( i in dom s1 & i + 1 in dom s1 & s1 . i = s1 . (i + 1) ) )
set f1 = the_series_of_quotients_of s1;
assume A2: for H being GroupWithOperators of O st H = (the_series_of_quotients_of s1) . i holds
H is trivial ; ::_thesis: ( i in dom s1 & i + 1 in dom s1 & s1 . i = s1 . (i + 1) )
A3: ( len (the_series_of_quotients_of s1) = 0 or len (the_series_of_quotients_of s1) >= 0 + 1 ) by NAT_1:13;
percases ( len (the_series_of_quotients_of s1) = 0 or len (the_series_of_quotients_of s1) = 1 or len (the_series_of_quotients_of s1) > 1 ) by A3, XXREAL_0:1;
suppose len (the_series_of_quotients_of s1) = 0 ; ::_thesis: ( i in dom s1 & i + 1 in dom s1 & s1 . i = s1 . (i + 1) )
then the_series_of_quotients_of s1 = {} ;
hence ( i in dom s1 & i + 1 in dom s1 & s1 . i = s1 . (i + 1) ) by A1; ::_thesis: verum
end;
supposeA4: len (the_series_of_quotients_of s1) = 1 ; ::_thesis: ( i in dom s1 & i + 1 in dom s1 & s1 . i = s1 . (i + 1) )
(the_series_of_quotients_of s1) . i in rng (the_series_of_quotients_of s1) by A1, FUNCT_1:3;
then reconsider H = (the_series_of_quotients_of s1) . i as strict GroupWithOperators of O by Th102;
set H1 = (Omega). G;
A5: H is trivial by A2;
A6: len s1 > 1 by A4, Def33, CARD_1:27;
then A7: len s1 = (len (the_series_of_quotients_of s1)) + 1 by Def33;
then A8: s1 . 2 = (1). G by A4, Def28;
i in Seg 1 by A1, A4, FINSEQ_1:def_3;
then A9: i = 1 by FINSEQ_1:2, TARSKI:def_1;
then i in Seg 2 ;
hence i in dom s1 by A4, A7, FINSEQ_1:def_3; ::_thesis: ( i + 1 in dom s1 & s1 . i = s1 . (i + 1) )
reconsider N1 = (1). G as StableSubgroup of (Omega). G by Th16;
A10: s1 . 1 = (Omega). G by Def28;
A11: (1). G = (1). ((Omega). G) by Th15;
then reconsider N1 = N1 as normal StableSubgroup of (Omega). G ;
A12: (Omega). G,((Omega). G) ./. N1 are_isomorphic by A11, Th56;
i + 1 in Seg 2 by A9;
hence i + 1 in dom s1 by A4, A7, FINSEQ_1:def_3; ::_thesis: s1 . i = s1 . (i + 1)
for H1 being StableSubgroup of G
for N1 being normal StableSubgroup of H1 st H1 = s1 . i & N1 = s1 . (i + 1) holds
(the_series_of_quotients_of s1) . i = H1 ./. N1 by A1, A6, Def33;
then ((Omega). G) ./. N1 is trivial by A10, A8, A9, A5;
hence s1 . i = s1 . (i + 1) by A10, A8, A9, A11, A12, Th42, Th58; ::_thesis: verum
end;
supposeA13: len (the_series_of_quotients_of s1) > 1 ; ::_thesis: ( i in dom s1 & i + 1 in dom s1 & s1 . i = s1 . (i + 1) )
(the_series_of_quotients_of s1) . i in rng (the_series_of_quotients_of s1) by A1, FUNCT_1:3;
then reconsider H = (the_series_of_quotients_of s1) . i as strict GroupWithOperators of O by Th102;
A14: i in Seg (len (the_series_of_quotients_of s1)) by A1, FINSEQ_1:def_3;
then A15: 1 <= i by FINSEQ_1:1;
1 <= i by A14, FINSEQ_1:1;
then 1 + 1 <= i + 1 by XREAL_1:6;
then A16: 1 <= i + 1 by XXREAL_0:2;
A17: i <= len (the_series_of_quotients_of s1) by A14, FINSEQ_1:1;
then A18: ( 0 + i <= 1 + i & i + 1 <= (len (the_series_of_quotients_of s1)) + 1 ) by XREAL_1:6;
A19: len s1 > 1 by A13, Def33, CARD_1:27;
then len s1 = (len (the_series_of_quotients_of s1)) + 1 by Def33;
then i <= len s1 by A18, XXREAL_0:2;
then A20: i in Seg (len s1) by A15, FINSEQ_1:1;
hence i in dom s1 by FINSEQ_1:def_3; ::_thesis: ( i + 1 in dom s1 & s1 . i = s1 . (i + 1) )
i + 1 <= (len (the_series_of_quotients_of s1)) + 1 by A17, XREAL_1:6;
then i + 1 <= len s1 by A19, Def33;
then A21: i + 1 in Seg (len s1) by A16;
hence i + 1 in dom s1 by FINSEQ_1:def_3; ::_thesis: s1 . i = s1 . (i + 1)
A22: i + 1 in dom s1 by A21, FINSEQ_1:def_3;
then s1 . (i + 1) in the_stable_subgroups_of G by FINSEQ_2:11;
then reconsider N1 = s1 . (i + 1) as strict StableSubgroup of G by Def11;
A23: i in dom s1 by A20, FINSEQ_1:def_3;
then s1 . i in the_stable_subgroups_of G by FINSEQ_2:11;
then reconsider H1 = s1 . i as strict StableSubgroup of G by Def11;
reconsider N1 = N1 as normal StableSubgroup of H1 by A23, A22, Def28;
H is trivial by A2;
then H1 ./. N1 is trivial by A1, A19, Def33;
hence s1 . i = s1 . (i + 1) by Th76; ::_thesis: verum
end;
end;
end;
theorem Th104: :: GROUP_9:104
for O being set
for G being GroupWithOperators of O
for s1, s2 being CompositionSeries of G
for i being Nat st i in dom s1 & i + 1 in dom s1 & s1 . i = s1 . (i + 1) & s2 = Del (s1,i) holds
the_series_of_quotients_of s2 = Del ((the_series_of_quotients_of s1),i)
proof
let O be set ; ::_thesis: for G being GroupWithOperators of O
for s1, s2 being CompositionSeries of G
for i being Nat st i in dom s1 & i + 1 in dom s1 & s1 . i = s1 . (i + 1) & s2 = Del (s1,i) holds
the_series_of_quotients_of s2 = Del ((the_series_of_quotients_of s1),i)
let G be GroupWithOperators of O; ::_thesis: for s1, s2 being CompositionSeries of G
for i being Nat st i in dom s1 & i + 1 in dom s1 & s1 . i = s1 . (i + 1) & s2 = Del (s1,i) holds
the_series_of_quotients_of s2 = Del ((the_series_of_quotients_of s1),i)
let s1, s2 be CompositionSeries of G; ::_thesis: for i being Nat st i in dom s1 & i + 1 in dom s1 & s1 . i = s1 . (i + 1) & s2 = Del (s1,i) holds
the_series_of_quotients_of s2 = Del ((the_series_of_quotients_of s1),i)
let i be Nat; ::_thesis: ( i in dom s1 & i + 1 in dom s1 & s1 . i = s1 . (i + 1) & s2 = Del (s1,i) implies the_series_of_quotients_of s2 = Del ((the_series_of_quotients_of s1),i) )
set f1 = the_series_of_quotients_of s1;
assume A1: i in dom s1 ; ::_thesis: ( not i + 1 in dom s1 or not s1 . i = s1 . (i + 1) or not s2 = Del (s1,i) or the_series_of_quotients_of s2 = Del ((the_series_of_quotients_of s1),i) )
then consider k being Nat such that
A2: len s1 = k + 1 and
A3: len (Del (s1,i)) = k by FINSEQ_3:104;
assume i + 1 in dom s1 ; ::_thesis: ( not s1 . i = s1 . (i + 1) or not s2 = Del (s1,i) or the_series_of_quotients_of s2 = Del ((the_series_of_quotients_of s1),i) )
then i + 1 in Seg (len s1) by FINSEQ_1:def_3;
then A4: i + 1 <= len s1 by FINSEQ_1:1;
assume A5: s1 . i = s1 . (i + 1) ; ::_thesis: ( not s2 = Del (s1,i) or the_series_of_quotients_of s2 = Del ((the_series_of_quotients_of s1),i) )
A6: i in Seg (len s1) by A1, FINSEQ_1:def_3;
then 1 <= i by FINSEQ_1:1;
then A7: 1 + 1 <= i + 1 by XREAL_1:6;
then 2 <= len s1 by A4, XXREAL_0:2;
then A8: 1 < len s1 by XXREAL_0:2;
then A9: len s1 = (len (the_series_of_quotients_of s1)) + 1 by Def33;
assume A10: s2 = Del (s1,i) ; ::_thesis: the_series_of_quotients_of s2 = Del ((the_series_of_quotients_of s1),i)
then 1 + 1 <= (len s2) + 1 by A7, A4, A2, A3, XXREAL_0:2;
then A11: 1 <= len s2 by XREAL_1:6;
percases ( len s2 = 1 or len s2 > 1 ) by A11, XXREAL_0:1;
supposeA12: len s2 = 1 ; ::_thesis: the_series_of_quotients_of s2 = Del ((the_series_of_quotients_of s1),i)
then 1 in Seg (len (the_series_of_quotients_of s1)) by A10, A2, A3, A9;
then 1 in dom (the_series_of_quotients_of s1) by FINSEQ_1:def_3;
then A13: ex k1 being Nat st
( len (the_series_of_quotients_of s1) = k1 + 1 & len (Del ((the_series_of_quotients_of s1),1)) = k1 ) by FINSEQ_3:104;
A14: 1 <= i by A6, FINSEQ_1:1;
A15: the_series_of_quotients_of s2 = {} by A12, Def33;
i <= 1 by A10, A4, A2, A3, A12, XREAL_1:6;
then len (Del ((the_series_of_quotients_of s1),i)) = 0 by A10, A2, A3, A9, A12, A13, A14, XXREAL_0:1;
hence the_series_of_quotients_of s2 = Del ((the_series_of_quotients_of s1),i) by A15; ::_thesis: verum
end;
supposeA16: len s2 > 1 ; ::_thesis: the_series_of_quotients_of s2 = Del ((the_series_of_quotients_of s1),i)
( (i + 1) - 1 <= (len s1) - 1 & 1 <= i ) by A6, A4, FINSEQ_1:1, XREAL_1:9;
then i in Seg (len (the_series_of_quotients_of s1)) by A9, FINSEQ_1:1;
then A17: i in dom (the_series_of_quotients_of s1) by FINSEQ_1:def_3;
then consider k1 being Nat such that
A18: len (the_series_of_quotients_of s1) = k1 + 1 and
A19: len (Del ((the_series_of_quotients_of s1),i)) = k1 by FINSEQ_3:104;
now__::_thesis:_for_n_being_Nat_st_n_in_dom_(Del_((the_series_of_quotients_of_s1),i))_holds_
for_H1_being_StableSubgroup_of_G
for_N1_being_normal_StableSubgroup_of_H1_st_H1_=_s2_._n_&_N1_=_s2_._(n_+_1)_holds_
(Del_((the_series_of_quotients_of_s1),i))_._n_=_H1_./._N1
let n be Nat; ::_thesis: ( n in dom (Del ((the_series_of_quotients_of s1),i)) implies for H1 being StableSubgroup of G
for N1 being normal StableSubgroup of H1 st H1 = s2 . n & N1 = s2 . (n + 1) holds
(Del ((the_series_of_quotients_of s1),i)) . b3 = b4 ./. b5 )
set n1 = n + 1;
assume n in dom (Del ((the_series_of_quotients_of s1),i)) ; ::_thesis: for H1 being StableSubgroup of G
for N1 being normal StableSubgroup of H1 st H1 = s2 . n & N1 = s2 . (n + 1) holds
(Del ((the_series_of_quotients_of s1),i)) . b3 = b4 ./. b5
then A20: n in Seg (len (Del ((the_series_of_quotients_of s1),i))) by FINSEQ_1:def_3;
then A21: n <= k1 by A19, FINSEQ_1:1;
then A22: n + 1 <= k by A2, A9, A18, XREAL_1:6;
1 <= n by A20, FINSEQ_1:1;
then 1 + 1 <= n + 1 by XREAL_1:6;
then 1 <= n + 1 by XXREAL_0:2;
then n + 1 in Seg (len (the_series_of_quotients_of s1)) by A2, A9, A22;
then A23: n + 1 in dom (the_series_of_quotients_of s1) by FINSEQ_1:def_3;
reconsider n1 = n + 1 as Nat ;
let H1 be StableSubgroup of G; ::_thesis: for N1 being normal StableSubgroup of H1 st H1 = s2 . n & N1 = s2 . (n + 1) holds
(Del ((the_series_of_quotients_of s1),i)) . b2 = b3 ./. b4
let N1 be normal StableSubgroup of H1; ::_thesis: ( H1 = s2 . n & N1 = s2 . (n + 1) implies (Del ((the_series_of_quotients_of s1),i)) . b1 = b2 ./. b3 )
assume A24: H1 = s2 . n ; ::_thesis: ( N1 = s2 . (n + 1) implies (Del ((the_series_of_quotients_of s1),i)) . b1 = b2 ./. b3 )
0 + n < 1 + n by XREAL_1:6;
then A25: n <= k by A22, XXREAL_0:2;
((len (the_series_of_quotients_of s1)) - (len (Del ((the_series_of_quotients_of s1),i)))) + (len (Del ((the_series_of_quotients_of s1),i))) > 0 + (len (Del ((the_series_of_quotients_of s1),i))) by A18, A19, XREAL_1:6;
then Seg (len (Del ((the_series_of_quotients_of s1),i))) c= Seg (len (the_series_of_quotients_of s1)) by FINSEQ_1:5;
then n in Seg (len (the_series_of_quotients_of s1)) by A20;
then A26: n in dom (the_series_of_quotients_of s1) by FINSEQ_1:def_3;
assume A27: N1 = s2 . (n + 1) ; ::_thesis: (Del ((the_series_of_quotients_of s1),i)) . b1 = b2 ./. b3
percases ( n < i or n >= i ) ;
supposeA28: n < i ; ::_thesis: (Del ((the_series_of_quotients_of s1),i)) . b1 = b2 ./. b3
then A29: n1 <= i by NAT_1:13;
percases ( n1 < i or n1 = i ) by A29, XXREAL_0:1;
supposeA30: n1 < i ; ::_thesis: (Del ((the_series_of_quotients_of s1),i)) . b1 = b2 ./. b3
reconsider n9 = n as Element of NAT by INT_1:3;
A31: s1 . (n9 + 1) = N1 by A10, A27, A30, FINSEQ_3:110;
s1 . n9 = H1 by A10, A24, A28, FINSEQ_3:110;
then (the_series_of_quotients_of s1) . n = H1 ./. N1 by A8, A26, A31, Def33;
hence (Del ((the_series_of_quotients_of s1),i)) . n = H1 ./. N1 by A28, FINSEQ_3:110; ::_thesis: verum
end;
suppose n1 = i ; ::_thesis: (Del ((the_series_of_quotients_of s1),i)) . b1 = b2 ./. b3
then ( s1 . n = H1 & s1 . (n + 1) = N1 ) by A1, A5, A10, A2, A24, A27, A22, A28, FINSEQ_3:110, FINSEQ_3:111;
then (the_series_of_quotients_of s1) . n = H1 ./. N1 by A8, A26, Def33;
hence (Del ((the_series_of_quotients_of s1),i)) . n = H1 ./. N1 by A28, FINSEQ_3:110; ::_thesis: verum
end;
end;
end;
supposeA32: n >= i ; ::_thesis: (Del ((the_series_of_quotients_of s1),i)) . b1 = b2 ./. b3
reconsider n19 = n1 as Element of NAT ;
( 0 + i < 1 + i & n + 1 >= i + 1 ) by A32, XREAL_1:6;
then n1 >= i by XXREAL_0:2;
then A33: s1 . (n19 + 1) = N1 by A1, A10, A2, A27, A22, FINSEQ_3:111;
s1 . n19 = H1 by A1, A10, A2, A24, A25, A32, FINSEQ_3:111;
then (the_series_of_quotients_of s1) . n1 = H1 ./. N1 by A8, A23, A33, Def33;
hence (Del ((the_series_of_quotients_of s1),i)) . n = H1 ./. N1 by A17, A18, A21, A32, FINSEQ_3:111; ::_thesis: verum
end;
end;
end;
hence the_series_of_quotients_of s2 = Del ((the_series_of_quotients_of s1),i) by A10, A2, A3, A9, A16, A18, A19, Def33; ::_thesis: verum
end;
end;
end;
theorem :: GROUP_9:105
for O being set
for G being GroupWithOperators of O
for s1 being CompositionSeries of G
for f1 being FinSequence
for i being Nat st f1 = the_series_of_quotients_of s1 & i in dom f1 & ( for H being GroupWithOperators of O st H = f1 . i holds
H is trivial ) holds
( Del (s1,i) is CompositionSeries of G & ( for s2 being CompositionSeries of G st s2 = Del (s1,i) holds
the_series_of_quotients_of s2 = Del (f1,i) ) )
proof
let O be set ; ::_thesis: for G being GroupWithOperators of O
for s1 being CompositionSeries of G
for f1 being FinSequence
for i being Nat st f1 = the_series_of_quotients_of s1 & i in dom f1 & ( for H being GroupWithOperators of O st H = f1 . i holds
H is trivial ) holds
( Del (s1,i) is CompositionSeries of G & ( for s2 being CompositionSeries of G st s2 = Del (s1,i) holds
the_series_of_quotients_of s2 = Del (f1,i) ) )
let G be GroupWithOperators of O; ::_thesis: for s1 being CompositionSeries of G
for f1 being FinSequence
for i being Nat st f1 = the_series_of_quotients_of s1 & i in dom f1 & ( for H being GroupWithOperators of O st H = f1 . i holds
H is trivial ) holds
( Del (s1,i) is CompositionSeries of G & ( for s2 being CompositionSeries of G st s2 = Del (s1,i) holds
the_series_of_quotients_of s2 = Del (f1,i) ) )
let s1 be CompositionSeries of G; ::_thesis: for f1 being FinSequence
for i being Nat st f1 = the_series_of_quotients_of s1 & i in dom f1 & ( for H being GroupWithOperators of O st H = f1 . i holds
H is trivial ) holds
( Del (s1,i) is CompositionSeries of G & ( for s2 being CompositionSeries of G st s2 = Del (s1,i) holds
the_series_of_quotients_of s2 = Del (f1,i) ) )
let f1 be FinSequence; ::_thesis: for i being Nat st f1 = the_series_of_quotients_of s1 & i in dom f1 & ( for H being GroupWithOperators of O st H = f1 . i holds
H is trivial ) holds
( Del (s1,i) is CompositionSeries of G & ( for s2 being CompositionSeries of G st s2 = Del (s1,i) holds
the_series_of_quotients_of s2 = Del (f1,i) ) )
let i be Nat; ::_thesis: ( f1 = the_series_of_quotients_of s1 & i in dom f1 & ( for H being GroupWithOperators of O st H = f1 . i holds
H is trivial ) implies ( Del (s1,i) is CompositionSeries of G & ( for s2 being CompositionSeries of G st s2 = Del (s1,i) holds
the_series_of_quotients_of s2 = Del (f1,i) ) ) )
assume A1: f1 = the_series_of_quotients_of s1 ; ::_thesis: ( not i in dom f1 or ex H being GroupWithOperators of O st
( H = f1 . i & not H is trivial ) or ( Del (s1,i) is CompositionSeries of G & ( for s2 being CompositionSeries of G st s2 = Del (s1,i) holds
the_series_of_quotients_of s2 = Del (f1,i) ) ) )
assume A2: i in dom f1 ; ::_thesis: ( ex H being GroupWithOperators of O st
( H = f1 . i & not H is trivial ) or ( Del (s1,i) is CompositionSeries of G & ( for s2 being CompositionSeries of G st s2 = Del (s1,i) holds
the_series_of_quotients_of s2 = Del (f1,i) ) ) )
assume A3: for H being GroupWithOperators of O st H = f1 . i holds
H is trivial ; ::_thesis: ( Del (s1,i) is CompositionSeries of G & ( for s2 being CompositionSeries of G st s2 = Del (s1,i) holds
the_series_of_quotients_of s2 = Del (f1,i) ) )
then A4: s1 . i = s1 . (i + 1) by A1, A2, Th103;
A5: ( i in dom s1 & i + 1 in dom s1 ) by A1, A2, A3, Th103;
hence Del (s1,i) is CompositionSeries of G by A4, Th94, FINSEQ_3:105; ::_thesis: for s2 being CompositionSeries of G st s2 = Del (s1,i) holds
the_series_of_quotients_of s2 = Del (f1,i)
let s2 be CompositionSeries of G; ::_thesis: ( s2 = Del (s1,i) implies the_series_of_quotients_of s2 = Del (f1,i) )
assume s2 = Del (s1,i) ; ::_thesis: the_series_of_quotients_of s2 = Del (f1,i)
hence the_series_of_quotients_of s2 = Del (f1,i) by A1, A5, A4, Th104; ::_thesis: verum
end;
theorem Th106: :: GROUP_9:106
for O being set
for f1, f2 being FinSequence
for i, j being Nat st i in dom f1 & ex p being Permutation of (dom f1) st
( f1,f2 are_equivalent_under p,O & j = (p ") . i ) holds
ex p9 being Permutation of (dom (Del (f1,i))) st Del (f1,i), Del (f2,j) are_equivalent_under p9,O
proof
let O be set ; ::_thesis: for f1, f2 being FinSequence
for i, j being Nat st i in dom f1 & ex p being Permutation of (dom f1) st
( f1,f2 are_equivalent_under p,O & j = (p ") . i ) holds
ex p9 being Permutation of (dom (Del (f1,i))) st Del (f1,i), Del (f2,j) are_equivalent_under p9,O
let f1, f2 be FinSequence; ::_thesis: for i, j being Nat st i in dom f1 & ex p being Permutation of (dom f1) st
( f1,f2 are_equivalent_under p,O & j = (p ") . i ) holds
ex p9 being Permutation of (dom (Del (f1,i))) st Del (f1,i), Del (f2,j) are_equivalent_under p9,O
let i, j be Nat; ::_thesis: ( i in dom f1 & ex p being Permutation of (dom f1) st
( f1,f2 are_equivalent_under p,O & j = (p ") . i ) implies ex p9 being Permutation of (dom (Del (f1,i))) st Del (f1,i), Del (f2,j) are_equivalent_under p9,O )
A1: ( len f1 = 0 or len f1 >= 0 + 1 ) by NAT_1:13;
assume A2: i in dom f1 ; ::_thesis: ( for p being Permutation of (dom f1) holds
( not f1,f2 are_equivalent_under p,O or not j = (p ") . i ) or ex p9 being Permutation of (dom (Del (f1,i))) st Del (f1,i), Del (f2,j) are_equivalent_under p9,O )
given p being Permutation of (dom f1) such that A3: f1,f2 are_equivalent_under p,O and
A4: j = (p ") . i ; ::_thesis: ex p9 being Permutation of (dom (Del (f1,i))) st Del (f1,i), Del (f2,j) are_equivalent_under p9,O
A5: len f1 = len f2 by A3, Def34;
rng (p ") c= dom f1 ;
then A6: rng (p ") c= Seg (len f1) by FINSEQ_1:def_3;
(p ") . i in rng (p ") by A2, FUNCT_2:4;
then (p ") . i in Seg (len f1) by A6;
then A7: j in dom f2 by A4, A5, FINSEQ_1:def_3;
then A8: ex k2 being Nat st
( len f2 = k2 + 1 & len (Del (f2,j)) = k2 ) by FINSEQ_3:104;
consider k1 being Nat such that
A9: len f1 = k1 + 1 and
A10: len (Del (f1,i)) = k1 by A2, FINSEQ_3:104;
percases ( len f1 = 0 or len f1 = 1 or len f1 > 1 ) by A1, XXREAL_0:1;
supposeA11: len f1 = 0 ; ::_thesis: ex p9 being Permutation of (dom (Del (f1,i))) st Del (f1,i), Del (f2,j) are_equivalent_under p9,O
set p9 = the Permutation of (dom (Del (f1,i)));
take the Permutation of (dom (Del (f1,i))) ; ::_thesis: Del (f1,i), Del (f2,j) are_equivalent_under the Permutation of (dom (Del (f1,i))),O
thus Del (f1,i), Del (f2,j) are_equivalent_under the Permutation of (dom (Del (f1,i))),O by A9, A11; ::_thesis: verum
end;
supposeA12: len f1 = 1 ; ::_thesis: ex p9 being Permutation of (dom (Del (f1,i))) st Del (f1,i), Del (f2,j) are_equivalent_under p9,O
reconsider p9 = {} as Function of (dom {}),(rng {}) by FUNCT_2:1;
reconsider p9 = p9 as Function of {},{} ;
A13: p9 is onto by FUNCT_2:def_3;
Del (f1,i) = {} by A9, A10, A12;
then reconsider p9 = p9 as Permutation of (dom (Del (f1,i))) by A13;
take p9 ; ::_thesis: Del (f1,i), Del (f2,j) are_equivalent_under p9,O
for H1, H2 being GroupWithOperators of O
for l, n being Nat st l in dom (Del (f1,i)) & n = (p9 ") . l & H1 = (Del (f1,i)) . l & H2 = (Del (f2,j)) . n holds
H1,H2 are_isomorphic ;
hence Del (f1,i), Del (f2,j) are_equivalent_under p9,O by A5, A9, A10, A8, Def34; ::_thesis: verum
end;
supposeA14: len f1 > 1 ; ::_thesis: ex p9 being Permutation of (dom (Del (f1,i))) st Del (f1,i), Del (f2,j) are_equivalent_under p9,O
set Y = (dom f2) \ {j};
A15: now__::_thesis:_not_(dom_f2)_\_{j}_=_{}
assume (dom f2) \ {j} = {} ; ::_thesis: contradiction
then A16: dom f2 c= {j} by XBOOLE_1:37;
{j} c= dom f2 by A7, ZFMISC_1:31;
then A17: dom f2 = {j} by A16, XBOOLE_0:def_10;
consider k being Nat such that
A18: dom f2 = Seg k by FINSEQ_1:def_2;
k in NAT by ORDINAL1:def_12;
then k = len f2 by A18, FINSEQ_1:def_3;
then k >= 1 + 1 by A5, A14, NAT_1:13;
then Seg 2 c= Seg k by FINSEQ_1:5;
then {1,2} = {j} by A17, A18, FINSEQ_1:2, ZFMISC_1:21;
hence contradiction by ZFMISC_1:5; ::_thesis: verum
end;
set X = (dom f1) \ {i};
set p9 = (((Sgm ((dom f1) \ {i})) ") * p) * (Sgm ((dom f2) \ {j}));
(dom f2) \ {j} c= dom f2 by XBOOLE_1:36;
then A19: (dom f2) \ {j} c= Seg (len f2) by FINSEQ_1:def_3;
(dom f1) \ {i} c= dom f1 by XBOOLE_1:36;
then A20: (dom f1) \ {i} c= Seg (len f1) by FINSEQ_1:def_3;
then A21: rng (Sgm ((dom f1) \ {i})) = (dom f1) \ {i} by FINSEQ_1:def_13;
(dom f2) \ {j} c= dom f2 by XBOOLE_1:36;
then (dom f2) \ {j} c= Seg (len f2) by FINSEQ_1:def_3;
then A22: ( Sgm ((dom f2) \ {j}) is one-to-one & rng (Sgm ((dom f2) \ {j})) = (dom f2) \ {j} ) by FINSEQ_1:def_13, FINSEQ_3:92;
A23: dom f1 = Seg (len f1) by FINSEQ_1:def_3
.= Seg (len f2) by A3, Def34
.= dom f2 by FINSEQ_1:def_3 ;
A24: p . j = (p * (p ")) . i by A2, A4, FUNCT_2:15
.= (id (dom f1)) . i by FUNCT_2:61
.= i by A2, FUNCT_1:18 ;
A25: ( (((Sgm ((dom f1) \ {i})) ") * p) * (Sgm ((dom f2) \ {j})) is Permutation of (dom (Del (f1,i))) & ((((Sgm ((dom f1) \ {i})) ") * p) * (Sgm ((dom f2) \ {j}))) " = (((Sgm ((dom f2) \ {j})) ") * (p ")) * (Sgm ((dom f1) \ {i})) )
proof
set R6 = p;
set R5 = p " ;
set R4 = Sgm ((dom f1) \ {i});
set R3 = (Sgm ((dom f1) \ {i})) " ;
set R2 = Sgm ((dom f2) \ {j});
set R1 = (Sgm ((dom f2) \ {j})) " ;
set p99 = (((Sgm ((dom f2) \ {j})) ") * (p ")) * (Sgm ((dom f1) \ {i}));
A26: {i} c= dom f1 by A2, ZFMISC_1:31;
A27: ((dom f1) \ {i}) \/ {i} = (dom f1) \/ {i} by XBOOLE_1:39
.= dom f1 by A26, XBOOLE_1:12 ;
card (((dom f1) \ {i}) \/ {i}) = (card ((dom f1) \ {i})) + (card {i}) by CARD_2:40, XBOOLE_1:79;
then A28: (card ((dom f1) \ {i})) + 1 = card (((dom f1) \ {i}) \/ {i}) by CARD_1:30
.= card (Seg (len f1)) by A27, FINSEQ_1:def_3
.= k1 + 1 by A9, FINSEQ_1:57 ;
A29: {j} c= dom f2 by A7, ZFMISC_1:31;
A30: ((dom f2) \ {j}) \/ {j} = (dom f2) \/ {j} by XBOOLE_1:39
.= dom f2 by A29, XBOOLE_1:12 ;
A31: Sgm ((dom f1) \ {i}) is one-to-one by A20, FINSEQ_3:92;
then A32: dom ((Sgm ((dom f1) \ {i})) ") = (dom f1) \ {i} by A21, FUNCT_1:33;
then dom ((Sgm ((dom f1) \ {i})) ") c= dom f1 by XBOOLE_1:36;
then A33: dom ((Sgm ((dom f1) \ {i})) ") c= rng p by FUNCT_2:def_3;
A34: now__::_thesis:_for_x_being_set_st_x_in_(dom_f2)_\_{j}_holds_
x_in_dom_(((Sgm_((dom_f1)_\_{i}))_")_*_p)
let x be set ; ::_thesis: ( x in (dom f2) \ {j} implies x in dom (((Sgm ((dom f1) \ {i})) ") * p) )
assume A35: x in (dom f2) \ {j} ; ::_thesis: x in dom (((Sgm ((dom f1) \ {i})) ") * p)
dom f1 = dom p by A2, FUNCT_2:def_1;
then A36: x in dom p by A23, A35, XBOOLE_0:def_5;
not x in {j} by A35, XBOOLE_0:def_5;
then x <> j by TARSKI:def_1;
then p . x <> i by A7, A23, A24, A36, FUNCT_2:56;
then A37: not p . x in {i} by TARSKI:def_1;
dom f1 = rng p by FUNCT_2:def_3;
then p . x in dom f1 by A36, FUNCT_1:3;
then p . x in (dom f1) \ {i} by A37, XBOOLE_0:def_5;
hence x in dom (((Sgm ((dom f1) \ {i})) ") * p) by A32, A36, FUNCT_1:11; ::_thesis: verum
end;
now__::_thesis:_for_x_being_set_st_x_in_dom_(((Sgm_((dom_f1)_\_{i}))_")_*_p)_holds_
x_in_(dom_f2)_\_{j}
let x be set ; ::_thesis: ( x in dom (((Sgm ((dom f1) \ {i})) ") * p) implies x in (dom f2) \ {j} )
assume A38: x in dom (((Sgm ((dom f1) \ {i})) ") * p) ; ::_thesis: x in (dom f2) \ {j}
then p . x in dom ((Sgm ((dom f1) \ {i})) ") by FUNCT_1:11;
then p . x in (dom f1) \ {i} by A21, A31, FUNCT_1:33;
then not p . x in {i} by XBOOLE_0:def_5;
then p . x <> i by TARSKI:def_1;
then A39: not x in {j} by A24, TARSKI:def_1;
x in dom p by A38, FUNCT_1:11;
hence x in (dom f2) \ {j} by A23, A39, XBOOLE_0:def_5; ::_thesis: verum
end;
then dom (((Sgm ((dom f1) \ {i})) ") * p) = (dom f2) \ {j} by A34, TARSKI:1;
then A40: dom (((Sgm ((dom f1) \ {i})) ") * p) = rng (Sgm ((dom f2) \ {j})) by A19, FINSEQ_1:def_13;
then rng ((((Sgm ((dom f1) \ {i})) ") * p) * (Sgm ((dom f2) \ {j}))) = rng (((Sgm ((dom f1) \ {i})) ") * p) by RELAT_1:28
.= rng ((Sgm ((dom f1) \ {i})) ") by A33, RELAT_1:28
.= dom (Sgm ((dom f1) \ {i})) by A31, FUNCT_1:33 ;
then A41: rng ((((Sgm ((dom f1) \ {i})) ") * p) * (Sgm ((dom f2) \ {j}))) = Seg k1 by A20, A28, FINSEQ_3:40;
card (((dom f2) \ {j}) \/ {j}) = (card ((dom f2) \ {j})) + (card {j}) by CARD_2:40, XBOOLE_1:79;
then (card ((dom f2) \ {j})) + 1 = card (((dom f2) \ {j}) \/ {j}) by CARD_1:30
.= card (Seg (len f2)) by A30, FINSEQ_1:def_3
.= card (Seg (len f1)) by A3, Def34
.= k1 + 1 by A9, FINSEQ_1:57 ;
then dom (Sgm ((dom f2) \ {j})) = Seg k1 by A19, FINSEQ_3:40;
then A42: dom ((((Sgm ((dom f1) \ {i})) ") * p) * (Sgm ((dom f2) \ {j}))) = Seg k1 by A40, RELAT_1:27;
A43: dom (Del (f1,i)) = Seg k1 by A10, FINSEQ_1:def_3;
then reconsider p9 = (((Sgm ((dom f1) \ {i})) ") * p) * (Sgm ((dom f2) \ {j})) as Function of (dom (Del (f1,i))),(dom (Del (f1,i))) by A41, A42, FUNCT_2:1;
A44: p9 is onto by A43, A41, FUNCT_2:def_3;
Sgm ((dom f2) \ {j}) is one-to-one by A19, FINSEQ_3:92;
then reconsider p9 = p9 as Permutation of (dom (Del (f1,i))) by A31, A44;
set R7 = p9;
reconsider R1 = (Sgm ((dom f2) \ {j})) " , R2 = Sgm ((dom f2) \ {j}), R3 = (Sgm ((dom f1) \ {i})) " , R4 = Sgm ((dom f1) \ {i}), R5 = p " , R6 = p, R7 = p9, p9 = p9, p99 = (((Sgm ((dom f2) \ {j})) ") * (p ")) * (Sgm ((dom f1) \ {i})) as Function ;
A45: R3 = R4 ~ by A31, FUNCT_1:def_5;
A46: ( Sgm ((dom f2) \ {j}) is one-to-one & R5 = R6 ~ ) by A19, FINSEQ_3:92, FUNCT_1:def_5;
reconsider R1 = R1, R2 = R2, R3 = R3, R4 = R4, R5 = R5, R6 = R6, R7 = R7 as Relation ;
p9 " = R7 ~ by FUNCT_1:def_5
.= ((R6 * R3) ~) * (R2 ~) by RELAT_1:35
.= ((R3 ~) * (R6 ~)) * (R2 ~) by RELAT_1:35
.= (((R4 ~) ~) * R5) * R1 by A45, A46, FUNCT_1:def_5
.= p99 by RELAT_1:36 ;
hence ( (((Sgm ((dom f1) \ {i})) ") * p) * (Sgm ((dom f2) \ {j})) is Permutation of (dom (Del (f1,i))) & ((((Sgm ((dom f1) \ {i})) ") * p) * (Sgm ((dom f2) \ {j}))) " = (((Sgm ((dom f2) \ {j})) ") * (p ")) * (Sgm ((dom f1) \ {i})) ) ; ::_thesis: verum
end;
then reconsider p9 = (((Sgm ((dom f1) \ {i})) ") * p) * (Sgm ((dom f2) \ {j})) as Permutation of (dom (Del (f1,i))) ;
take p9 ; ::_thesis: Del (f1,i), Del (f2,j) are_equivalent_under p9,O
A47: Sgm ((dom f2) \ {j}) is Function of (dom (Sgm ((dom f2) \ {j}))),(rng (Sgm ((dom f2) \ {j}))) by FUNCT_2:1;
now__::_thesis:_for_H1,_H2_being_GroupWithOperators_of_O
for_l,_n_being_Nat_st_l_in_dom_(Del_(f1,i))_&_n_=_(p9_")_._l_&_H1_=_(Del_(f1,i))_._l_&_H2_=_(Del_(f2,j))_._n_holds_
H1,H2_are_isomorphic
let H1, H2 be GroupWithOperators of O; ::_thesis: for l, n being Nat st l in dom (Del (f1,i)) & n = (p9 ") . l & H1 = (Del (f1,i)) . l & H2 = (Del (f2,j)) . n holds
H1,H2 are_isomorphic
let l, n be Nat; ::_thesis: ( l in dom (Del (f1,i)) & n = (p9 ") . l & H1 = (Del (f1,i)) . l & H2 = (Del (f2,j)) . n implies H1,H2 are_isomorphic )
assume A48: l in dom (Del (f1,i)) ; ::_thesis: ( n = (p9 ") . l & H1 = (Del (f1,i)) . l & H2 = (Del (f2,j)) . n implies H1,H2 are_isomorphic )
set n1 = (Sgm ((dom f2) \ {j})) . n;
reconsider n1 = (Sgm ((dom f2) \ {j})) . n as Nat ;
A49: (Sgm ((dom f2) \ {j})) * (p9 ") = (Sgm ((dom f2) \ {j})) * (((Sgm ((dom f2) \ {j})) ") * ((p ") * (Sgm ((dom f1) \ {i})))) by A25, RELAT_1:36
.= ((Sgm ((dom f2) \ {j})) * ((Sgm ((dom f2) \ {j})) ")) * ((p ") * (Sgm ((dom f1) \ {i}))) by RELAT_1:36
.= (id ((dom f2) \ {j})) * ((p ") * (Sgm ((dom f1) \ {i}))) by A22, A15, A47, FUNCT_2:29
.= ((id ((dom f2) \ {j})) * (p ")) * (Sgm ((dom f1) \ {i})) by RELAT_1:36
.= (((dom f2) \ {j}) |` (p ")) * (Sgm ((dom f1) \ {i})) by RELAT_1:92 ;
assume A50: n = (p9 ") . l ; ::_thesis: ( H1 = (Del (f1,i)) . l & H2 = (Del (f2,j)) . n implies H1,H2 are_isomorphic )
A51: l in dom (p9 ") by A48, FUNCT_2:def_1;
then n in rng (p9 ") by A50, FUNCT_1:3;
then n in dom (Del (f1,i)) ;
then n in Seg (len (Del (f2,j))) by A5, A9, A10, A8, FINSEQ_1:def_3;
then A52: n in dom (Del (f2,j)) by FINSEQ_1:def_3;
set l1 = (Sgm ((dom f1) \ {i})) . l;
A53: dom (Del (f1,i)) c= dom (Sgm ((dom f1) \ {i})) by RELAT_1:25;
then (Sgm ((dom f1) \ {i})) . l in rng (Sgm ((dom f1) \ {i})) by A48, FUNCT_1:3;
then A54: (Sgm ((dom f1) \ {i})) . l in dom f1 by A21, XBOOLE_0:def_5;
assume that
A55: H1 = (Del (f1,i)) . l and
A56: H2 = (Del (f2,j)) . n ; ::_thesis: H1,H2 are_isomorphic
reconsider l1 = (Sgm ((dom f1) \ {i})) . l as Nat ;
A57: H1 = f1 . l1 by A48, A55, A53, FUNCT_1:13;
A58: dom f1 = rng p by FUNCT_2:def_3;
then A59: l1 in dom (p ") by A54, FUNCT_1:33;
A60: now__::_thesis:_not_(p_")_._l1_in_{j}
assume (p ") . l1 in {j} ; ::_thesis: contradiction
then A61: (p ") . l1 = (p ") . i by A4, TARSKI:def_1;
i in dom (p ") by A2, A58, FUNCT_1:33;
then l1 = i by A59, A61, FUNCT_1:def_4;
then i in rng (Sgm ((dom f1) \ {i})) by A48, A53, FUNCT_1:3;
then not i in {i} by A21, XBOOLE_0:def_5;
hence contradiction by TARSKI:def_1; ::_thesis: verum
end;
(p ") . l1 in rng (p ") by A59, FUNCT_1:3;
then A62: (p ") . l1 in (dom f2) \ {j} by A23, A60, XBOOLE_0:def_5;
dom (Del (f2,j)) c= dom (Sgm ((dom f2) \ {j})) by RELAT_1:25;
then A63: H2 = f2 . n1 by A56, A52, FUNCT_1:13;
n1 = ((Sgm ((dom f2) \ {j})) * (p9 ")) . l by A50, A51, FUNCT_1:13
.= (((dom f2) \ {j}) |` (p ")) . l1 by A48, A53, A49, FUNCT_1:13
.= (p ") . l1 by A54, A62, FUNCT_2:34 ;
hence H1,H2 are_isomorphic by A3, A54, A57, A63, Def34; ::_thesis: verum
end;
hence Del (f1,i), Del (f2,j) are_equivalent_under p9,O by A5, A9, A10, A8, Def34; ::_thesis: verum
end;
end;
end;
theorem Th107: :: GROUP_9:107
for O being set
for G1, G2 being GroupWithOperators of O
for s1 being CompositionSeries of G1
for s2 being CompositionSeries of G2 st s1 is empty & s2 is empty holds
s1 is_equivalent_with s2
proof
let O be set ; ::_thesis: for G1, G2 being GroupWithOperators of O
for s1 being CompositionSeries of G1
for s2 being CompositionSeries of G2 st s1 is empty & s2 is empty holds
s1 is_equivalent_with s2
let G1, G2 be GroupWithOperators of O; ::_thesis: for s1 being CompositionSeries of G1
for s2 being CompositionSeries of G2 st s1 is empty & s2 is empty holds
s1 is_equivalent_with s2
let s1 be CompositionSeries of G1; ::_thesis: for s2 being CompositionSeries of G2 st s1 is empty & s2 is empty holds
s1 is_equivalent_with s2
let s2 be CompositionSeries of G2; ::_thesis: ( s1 is empty & s2 is empty implies s1 is_equivalent_with s2 )
assume A1: s1 is empty ; ::_thesis: ( not s2 is empty or s1 is_equivalent_with s2 )
assume A2: s2 is empty ; ::_thesis: s1 is_equivalent_with s2
for n being Nat st n + 1 = len s1 holds
ex p being Permutation of (Seg n) st
for H1 being StableSubgroup of G1
for H2 being StableSubgroup of G2
for N1 being normal StableSubgroup of H1
for N2 being normal StableSubgroup of H2
for i, j being Nat st 1 <= i & i <= n & j = p . i & H1 = s1 . i & H2 = s2 . j & N1 = s1 . (i + 1) & N2 = s2 . (j + 1) holds
H1 ./. N1,H2 ./. N2 are_isomorphic by A1;
hence s1 is_equivalent_with s2 by A1, A2, Def32; ::_thesis: verum
end;
theorem Th108: :: GROUP_9:108
for O being set
for G1, G2 being GroupWithOperators of O
for s1 being CompositionSeries of G1
for s2 being CompositionSeries of G2 st not s1 is empty & not s2 is empty holds
( s1 is_equivalent_with s2 iff ex p being Permutation of (dom (the_series_of_quotients_of s1)) st the_series_of_quotients_of s1, the_series_of_quotients_of s2 are_equivalent_under p,O )
proof
let O be set ; ::_thesis: for G1, G2 being GroupWithOperators of O
for s1 being CompositionSeries of G1
for s2 being CompositionSeries of G2 st not s1 is empty & not s2 is empty holds
( s1 is_equivalent_with s2 iff ex p being Permutation of (dom (the_series_of_quotients_of s1)) st the_series_of_quotients_of s1, the_series_of_quotients_of s2 are_equivalent_under p,O )
let G1, G2 be GroupWithOperators of O; ::_thesis: for s1 being CompositionSeries of G1
for s2 being CompositionSeries of G2 st not s1 is empty & not s2 is empty holds
( s1 is_equivalent_with s2 iff ex p being Permutation of (dom (the_series_of_quotients_of s1)) st the_series_of_quotients_of s1, the_series_of_quotients_of s2 are_equivalent_under p,O )
let s1 be CompositionSeries of G1; ::_thesis: for s2 being CompositionSeries of G2 st not s1 is empty & not s2 is empty holds
( s1 is_equivalent_with s2 iff ex p being Permutation of (dom (the_series_of_quotients_of s1)) st the_series_of_quotients_of s1, the_series_of_quotients_of s2 are_equivalent_under p,O )
let s2 be CompositionSeries of G2; ::_thesis: ( not s1 is empty & not s2 is empty implies ( s1 is_equivalent_with s2 iff ex p being Permutation of (dom (the_series_of_quotients_of s1)) st the_series_of_quotients_of s1, the_series_of_quotients_of s2 are_equivalent_under p,O ) )
assume that
A1: not s1 is empty and
A2: not s2 is empty ; ::_thesis: ( s1 is_equivalent_with s2 iff ex p being Permutation of (dom (the_series_of_quotients_of s1)) st the_series_of_quotients_of s1, the_series_of_quotients_of s2 are_equivalent_under p,O )
set f2 = the_series_of_quotients_of s2;
set f1 = the_series_of_quotients_of s1;
hereby ::_thesis: ( ex p being Permutation of (dom (the_series_of_quotients_of s1)) st the_series_of_quotients_of s1, the_series_of_quotients_of s2 are_equivalent_under p,O implies s1 is_equivalent_with s2 )
assume A3: s1 is_equivalent_with s2 ; ::_thesis: ex p being Permutation of (dom (the_series_of_quotients_of s1)) st the_series_of_quotients_of s1, the_series_of_quotients_of s2 are_equivalent_under p,O
then A4: len s1 = len s2 by Def32;
percases ( len s1 <= 1 or len s1 > 1 ) ;
supposeA5: len s1 <= 1 ; ::_thesis: ex p being Permutation of (dom (the_series_of_quotients_of s1)) st the_series_of_quotients_of s1, the_series_of_quotients_of s2 are_equivalent_under p,O
reconsider fs1 = the_series_of_quotients_of s1, fs2 = the_series_of_quotients_of s2 as FinSequence ;
set p = the Permutation of (dom (the_series_of_quotients_of s1));
reconsider pf = the Permutation of (dom (the_series_of_quotients_of s1)) as Permutation of (dom fs1) ;
fs1 = {} by A5, Def33;
then A6: for H1, H2 being GroupWithOperators of O
for i, j being Nat st i in dom fs1 & j = (pf ") . i & H1 = fs1 . i & H2 = fs2 . j holds
H1,H2 are_isomorphic ;
take p = the Permutation of (dom (the_series_of_quotients_of s1)); ::_thesis: the_series_of_quotients_of s1, the_series_of_quotients_of s2 are_equivalent_under p,O
fs2 = {} by A4, A5, Def33;
then len (the_series_of_quotients_of s1) = len (the_series_of_quotients_of s2) by A5, Def33;
hence the_series_of_quotients_of s1, the_series_of_quotients_of s2 are_equivalent_under p,O by A6, Def34; ::_thesis: verum
end;
supposeA7: len s1 > 1 ; ::_thesis: ex p9 being Permutation of (dom (the_series_of_quotients_of s1)) st the_series_of_quotients_of s1, the_series_of_quotients_of s2 are_equivalent_under p9,O
set n = (len s1) - 1;
(len s1) - 1 > 1 - 1 by A7, XREAL_1:9;
then (len s1) - 1 in NAT by INT_1:3;
then reconsider n = (len s1) - 1 as Nat ;
n + 1 = len s1 ;
then consider p being Permutation of (Seg n) such that
A8: for H1 being StableSubgroup of G1
for H2 being StableSubgroup of G2
for N1 being normal StableSubgroup of H1
for N2 being normal StableSubgroup of H2
for i, j being Nat st 1 <= i & i <= n & j = p . i & H1 = s1 . i & H2 = s2 . j & N1 = s1 . (i + 1) & N2 = s2 . (j + 1) holds
H1 ./. N1,H2 ./. N2 are_isomorphic by A3, Def32;
A9: len s1 = (len (the_series_of_quotients_of s1)) + 1 by A7, Def33;
then dom (the_series_of_quotients_of s1) = Seg n by FINSEQ_1:def_3;
then reconsider p9 = p " as Permutation of (dom (the_series_of_quotients_of s1)) ;
reconsider fs1 = the_series_of_quotients_of s1, fs2 = the_series_of_quotients_of s2 as FinSequence ;
A10: len s2 = (len (the_series_of_quotients_of s2)) + 1 by A4, A7, Def33;
reconsider pf = p9 as Permutation of (dom fs1) ;
take p9 = p9; ::_thesis: the_series_of_quotients_of s1, the_series_of_quotients_of s2 are_equivalent_under p9,O
A11: pf " = p by FUNCT_1:43;
now__::_thesis:_for_H19,_H29_being_GroupWithOperators_of_O
for_i,_j_being_Nat_st_i_in_dom_fs1_&_j_=_(pf_")_._i_&_H19_=_fs1_._i_&_H29_=_fs2_._j_holds_
H19,H29_are_isomorphic
let H19, H29 be GroupWithOperators of O; ::_thesis: for i, j being Nat st i in dom fs1 & j = (pf ") . i & H19 = fs1 . i & H29 = fs2 . j holds
H19,H29 are_isomorphic
let i, j be Nat; ::_thesis: ( i in dom fs1 & j = (pf ") . i & H19 = fs1 . i & H29 = fs2 . j implies H19,H29 are_isomorphic )
set H1 = s1 . i;
set H2 = s2 . j;
set N1 = s1 . (i + 1);
set N2 = s2 . (j + 1);
assume A12: i in dom fs1 ; ::_thesis: ( j = (pf ") . i & H19 = fs1 . i & H29 = fs2 . j implies H19,H29 are_isomorphic )
then A13: i in Seg (len fs1) by FINSEQ_1:def_3;
then A14: 1 <= i by FINSEQ_1:1;
A15: i <= len fs1 by A13, FINSEQ_1:1;
then A16: i + 1 <= (len fs1) + 1 by XREAL_1:6;
0 + i < 1 + i by XREAL_1:6;
then 1 <= i + 1 by A14, XXREAL_0:2;
then i + 1 in Seg (len s1) by A9, A16;
then A17: i + 1 in dom s1 by FINSEQ_1:def_3;
assume A18: j = (pf ") . i ; ::_thesis: ( H19 = fs1 . i & H29 = fs2 . j implies H19,H29 are_isomorphic )
0 + (len fs1) < 1 + (len fs1) by XREAL_1:6;
then i <= len s1 by A9, A15, XXREAL_0:2;
then i in Seg (len s1) by A14, FINSEQ_1:1;
then A19: i in dom s1 by FINSEQ_1:def_3;
then reconsider H1 = s1 . i, N1 = s1 . (i + 1) as Element of the_stable_subgroups_of G1 by A17, FINSEQ_2:11;
reconsider H1 = H1, N1 = N1 as StableSubgroup of G1 by Def11;
reconsider N1 = N1 as normal StableSubgroup of H1 by A19, A17, Def28;
assume that
A20: H19 = fs1 . i and
A21: H29 = fs2 . j ; ::_thesis: H19,H29 are_isomorphic
i in dom p by A9, A13, FUNCT_2:def_1;
then A22: j in rng p by A11, A18, FUNCT_1:3;
then A23: 1 <= j by FINSEQ_1:1;
A24: j <= len fs2 by A4, A10, A22, FINSEQ_1:1;
then A25: j + 1 <= (len fs2) + 1 by XREAL_1:6;
0 + j < 1 + j by XREAL_1:6;
then 1 <= j + 1 by A23, XXREAL_0:2;
then j + 1 in Seg (len s2) by A10, A25;
then A26: j + 1 in dom s2 by FINSEQ_1:def_3;
0 + (len fs2) < 1 + (len fs2) by XREAL_1:6;
then j <= len s2 by A10, A24, XXREAL_0:2;
then j in Seg (len s2) by A23, FINSEQ_1:1;
then A27: j in dom s2 by FINSEQ_1:def_3;
then reconsider H2 = s2 . j, N2 = s2 . (j + 1) as Element of the_stable_subgroups_of G2 by A26, FINSEQ_2:11;
reconsider H2 = H2, N2 = N2 as StableSubgroup of G2 by Def11;
reconsider N2 = N2 as normal StableSubgroup of H2 by A27, A26, Def28;
dom fs1 = Seg n by A9, FINSEQ_1:def_3;
then ( 1 <= i & i <= n ) by A12, FINSEQ_1:1;
then A28: H1 ./. N1,H2 ./. N2 are_isomorphic by A8, A11, A18;
j in Seg (len (the_series_of_quotients_of s2)) by A4, A10, A22;
then j in dom fs2 by FINSEQ_1:def_3;
then H2 ./. N2 = H29 by A4, A7, A21, Def33;
hence H19,H29 are_isomorphic by A7, A12, A20, A28, Def33; ::_thesis: verum
end;
hence the_series_of_quotients_of s1, the_series_of_quotients_of s2 are_equivalent_under p9,O by A4, A9, A10, Def34; ::_thesis: verum
end;
end;
end;
given p being Permutation of (dom (the_series_of_quotients_of s1)) such that A29: the_series_of_quotients_of s1, the_series_of_quotients_of s2 are_equivalent_under p,O ; ::_thesis: s1 is_equivalent_with s2
A30: len (the_series_of_quotients_of s1) = len (the_series_of_quotients_of s2) by A29, Def34;
percases ( len s1 <= 1 or len s1 > 1 ) ;
supposeA31: len s1 <= 1 ; ::_thesis: s1 is_equivalent_with s2
A32: len s1 >= 0 + 1 by A1, NAT_1:13;
A33: now__::_thesis:_for_n_being_Nat_st_n_+_1_=_len_s1_holds_
ex_p_being_Permutation_of_(Seg_n)_st_
for_H1_being_StableSubgroup_of_G1
for_H2_being_StableSubgroup_of_G2
for_N1_being_normal_StableSubgroup_of_H1
for_N2_being_normal_StableSubgroup_of_H2
for_i,_j_being_Nat_st_1_<=_i_&_i_<=_n_&_j_=_p_._i_&_H1_=_s1_._i_&_H2_=_s2_._j_&_N1_=_s1_._(i_+_1)_&_N2_=_s2_._(j_+_1)_holds_
H1_./._N1,H2_./._N2_are_isomorphic
let n be Nat; ::_thesis: ( n + 1 = len s1 implies ex p being Permutation of (Seg n) st
for H1 being StableSubgroup of G1
for H2 being StableSubgroup of G2
for N1 being normal StableSubgroup of H1
for N2 being normal StableSubgroup of H2
for i, j being Nat st 1 <= i & i <= n & j = p . i & H1 = s1 . i & H2 = s2 . j & N1 = s1 . (i + 1) & N2 = s2 . (j + 1) holds
H1 ./. N1,H2 ./. N2 are_isomorphic )
set p = the Permutation of (Seg n);
assume n + 1 = len s1 ; ::_thesis: ex p being Permutation of (Seg n) st
for H1 being StableSubgroup of G1
for H2 being StableSubgroup of G2
for N1 being normal StableSubgroup of H1
for N2 being normal StableSubgroup of H2
for i, j being Nat st 1 <= i & i <= n & j = p . i & H1 = s1 . i & H2 = s2 . j & N1 = s1 . (i + 1) & N2 = s2 . (j + 1) holds
H1 ./. N1,H2 ./. N2 are_isomorphic
then n + 1 = 1 by A31, A32, XXREAL_0:1;
then A34: n = 0 ;
take p = the Permutation of (Seg n); ::_thesis: for H1 being StableSubgroup of G1
for H2 being StableSubgroup of G2
for N1 being normal StableSubgroup of H1
for N2 being normal StableSubgroup of H2
for i, j being Nat st 1 <= i & i <= n & j = p . i & H1 = s1 . i & H2 = s2 . j & N1 = s1 . (i + 1) & N2 = s2 . (j + 1) holds
H1 ./. N1,H2 ./. N2 are_isomorphic
let H1 be StableSubgroup of G1; ::_thesis: for H2 being StableSubgroup of G2
for N1 being normal StableSubgroup of H1
for N2 being normal StableSubgroup of H2
for i, j being Nat st 1 <= i & i <= n & j = p . i & H1 = s1 . i & H2 = s2 . j & N1 = s1 . (i + 1) & N2 = s2 . (j + 1) holds
H1 ./. N1,H2 ./. N2 are_isomorphic
let H2 be StableSubgroup of G2; ::_thesis: for N1 being normal StableSubgroup of H1
for N2 being normal StableSubgroup of H2
for i, j being Nat st 1 <= i & i <= n & j = p . i & H1 = s1 . i & H2 = s2 . j & N1 = s1 . (i + 1) & N2 = s2 . (j + 1) holds
H1 ./. N1,H2 ./. N2 are_isomorphic
let N1 be normal StableSubgroup of H1; ::_thesis: for N2 being normal StableSubgroup of H2
for i, j being Nat st 1 <= i & i <= n & j = p . i & H1 = s1 . i & H2 = s2 . j & N1 = s1 . (i + 1) & N2 = s2 . (j + 1) holds
H1 ./. N1,H2 ./. N2 are_isomorphic
let N2 be normal StableSubgroup of H2; ::_thesis: for i, j being Nat st 1 <= i & i <= n & j = p . i & H1 = s1 . i & H2 = s2 . j & N1 = s1 . (i + 1) & N2 = s2 . (j + 1) holds
H1 ./. N1,H2 ./. N2 are_isomorphic
let i, j be Nat; ::_thesis: ( 1 <= i & i <= n & j = p . i & H1 = s1 . i & H2 = s2 . j & N1 = s1 . (i + 1) & N2 = s2 . (j + 1) implies H1 ./. N1,H2 ./. N2 are_isomorphic )
assume that
A35: ( 1 <= i & i <= n ) and
j = p . i ; ::_thesis: ( H1 = s1 . i & H2 = s2 . j & N1 = s1 . (i + 1) & N2 = s2 . (j + 1) implies H1 ./. N1,H2 ./. N2 are_isomorphic )
assume that
H1 = s1 . i and
H2 = s2 . j ; ::_thesis: ( N1 = s1 . (i + 1) & N2 = s2 . (j + 1) implies H1 ./. N1,H2 ./. N2 are_isomorphic )
assume that
N1 = s1 . (i + 1) and
N2 = s2 . (j + 1) ; ::_thesis: H1 ./. N1,H2 ./. N2 are_isomorphic
thus H1 ./. N1,H2 ./. N2 are_isomorphic by A34, A35; ::_thesis: verum
end;
A36: the_series_of_quotients_of s1 = {} by A31, Def33;
now__::_thesis:_not_len_s2_<>_1
assume A37: len s2 <> 1 ; ::_thesis: contradiction
len s2 >= 0 + 1 by A2, NAT_1:13;
then len s2 > 1 by A37, XXREAL_0:1;
then (len (the_series_of_quotients_of s2)) + 1 > 0 + 1 by Def33;
hence contradiction by A30, A36; ::_thesis: verum
end;
then len s1 = len s2 by A31, A32, XXREAL_0:1;
hence s1 is_equivalent_with s2 by A33, Def32; ::_thesis: verum
end;
supposeA38: len s1 > 1 ; ::_thesis: s1 is_equivalent_with s2
then A39: len s1 = (len (the_series_of_quotients_of s1)) + 1 by Def33;
A40: now__::_thesis:_not_len_s2_<=_1
assume len s2 <= 1 ; ::_thesis: contradiction
then the_series_of_quotients_of s2 = {} by Def33;
then len (the_series_of_quotients_of s2) = 0 ;
hence contradiction by A30, A38, A39; ::_thesis: verum
end;
A41: now__::_thesis:_for_n_being_Nat_st_n_+_1_=_len_s1_holds_
ex_p9_being_Permutation_of_(Seg_n)_st_
for_H1_being_StableSubgroup_of_G1
for_H2_being_StableSubgroup_of_G2
for_N1_being_normal_StableSubgroup_of_H1
for_N2_being_normal_StableSubgroup_of_H2
for_i,_j_being_Nat_st_1_<=_i_&_i_<=_n_&_j_=_p9_._i_&_H1_=_s1_._i_&_H2_=_s2_._j_&_N1_=_s1_._(i_+_1)_&_N2_=_s2_._(j_+_1)_holds_
H1_./._N1,H2_./._N2_are_isomorphic
let n be Nat; ::_thesis: ( n + 1 = len s1 implies ex p9 being Permutation of (Seg n) st
for H1 being StableSubgroup of G1
for H2 being StableSubgroup of G2
for N1 being normal StableSubgroup of H1
for N2 being normal StableSubgroup of H2
for i, j being Nat st 1 <= i & i <= n & j = p9 . i & H1 = s1 . i & H2 = s2 . j & N1 = s1 . (i + 1) & N2 = s2 . (j + 1) holds
H1 ./. N1,H2 ./. N2 are_isomorphic )
assume A42: n + 1 = len s1 ; ::_thesis: ex p9 being Permutation of (Seg n) st
for H1 being StableSubgroup of G1
for H2 being StableSubgroup of G2
for N1 being normal StableSubgroup of H1
for N2 being normal StableSubgroup of H2
for i, j being Nat st 1 <= i & i <= n & j = p9 . i & H1 = s1 . i & H2 = s2 . j & N1 = s1 . (i + 1) & N2 = s2 . (j + 1) holds
H1 ./. N1,H2 ./. N2 are_isomorphic
then A43: dom (the_series_of_quotients_of s1) = Seg n by A39, FINSEQ_1:def_3;
then reconsider p9 = p " as Permutation of (Seg n) ;
take p9 = p9; ::_thesis: for H1 being StableSubgroup of G1
for H2 being StableSubgroup of G2
for N1 being normal StableSubgroup of H1
for N2 being normal StableSubgroup of H2
for i, j being Nat st 1 <= i & i <= n & j = p9 . i & H1 = s1 . i & H2 = s2 . j & N1 = s1 . (i + 1) & N2 = s2 . (j + 1) holds
H1 ./. N1,H2 ./. N2 are_isomorphic
let H1 be StableSubgroup of G1; ::_thesis: for H2 being StableSubgroup of G2
for N1 being normal StableSubgroup of H1
for N2 being normal StableSubgroup of H2
for i, j being Nat st 1 <= i & i <= n & j = p9 . i & H1 = s1 . i & H2 = s2 . j & N1 = s1 . (i + 1) & N2 = s2 . (j + 1) holds
H1 ./. N1,H2 ./. N2 are_isomorphic
let H2 be StableSubgroup of G2; ::_thesis: for N1 being normal StableSubgroup of H1
for N2 being normal StableSubgroup of H2
for i, j being Nat st 1 <= i & i <= n & j = p9 . i & H1 = s1 . i & H2 = s2 . j & N1 = s1 . (i + 1) & N2 = s2 . (j + 1) holds
H1 ./. N1,H2 ./. N2 are_isomorphic
let N1 be normal StableSubgroup of H1; ::_thesis: for N2 being normal StableSubgroup of H2
for i, j being Nat st 1 <= i & i <= n & j = p9 . i & H1 = s1 . i & H2 = s2 . j & N1 = s1 . (i + 1) & N2 = s2 . (j + 1) holds
H1 ./. N1,H2 ./. N2 are_isomorphic
let N2 be normal StableSubgroup of H2; ::_thesis: for i, j being Nat st 1 <= i & i <= n & j = p9 . i & H1 = s1 . i & H2 = s2 . j & N1 = s1 . (i + 1) & N2 = s2 . (j + 1) holds
H1 ./. N1,H2 ./. N2 are_isomorphic
let i, j be Nat; ::_thesis: ( 1 <= i & i <= n & j = p9 . i & H1 = s1 . i & H2 = s2 . j & N1 = s1 . (i + 1) & N2 = s2 . (j + 1) implies H1 ./. N1,H2 ./. N2 are_isomorphic )
assume ( 1 <= i & i <= n ) ; ::_thesis: ( j = p9 . i & H1 = s1 . i & H2 = s2 . j & N1 = s1 . (i + 1) & N2 = s2 . (j + 1) implies H1 ./. N1,H2 ./. N2 are_isomorphic )
then A44: i in dom (the_series_of_quotients_of s1) by A43, FINSEQ_1:1;
assume A45: j = p9 . i ; ::_thesis: ( H1 = s1 . i & H2 = s2 . j & N1 = s1 . (i + 1) & N2 = s2 . (j + 1) implies H1 ./. N1,H2 ./. N2 are_isomorphic )
assume that
A46: H1 = s1 . i and
A47: H2 = s2 . j ; ::_thesis: ( N1 = s1 . (i + 1) & N2 = s2 . (j + 1) implies H1 ./. N1,H2 ./. N2 are_isomorphic )
assume that
A48: N1 = s1 . (i + 1) and
A49: N2 = s2 . (j + 1) ; ::_thesis: H1 ./. N1,H2 ./. N2 are_isomorphic
i in dom p9 by A44, FUNCT_2:def_1;
then j in rng p9 by A45, FUNCT_1:3;
then j in Seg n ;
then j in dom (the_series_of_quotients_of s2) by A30, A39, A42, FINSEQ_1:def_3;
then A50: (the_series_of_quotients_of s2) . j = H2 ./. N2 by A40, A47, A49, Def33;
(the_series_of_quotients_of s1) . i = H1 ./. N1 by A38, A44, A46, A48, Def33;
hence H1 ./. N1,H2 ./. N2 are_isomorphic by A29, A44, A45, A50, Def34; ::_thesis: verum
end;
len s1 = len s2 by A30, A39, A40, Def33;
hence s1 is_equivalent_with s2 by A41, Def32; ::_thesis: verum
end;
end;
end;
theorem Th109: :: GROUP_9:109
for O being set
for G being GroupWithOperators of O
for s1, s2 being CompositionSeries of G st s1 is_finer_than s2 & s2 is jordan_holder & len s1 > len s2 holds
ex i being Nat st
( i in dom (the_series_of_quotients_of s1) & ( for H being GroupWithOperators of O st H = (the_series_of_quotients_of s1) . i holds
H is trivial ) )
proof
let O be set ; ::_thesis: for G being GroupWithOperators of O
for s1, s2 being CompositionSeries of G st s1 is_finer_than s2 & s2 is jordan_holder & len s1 > len s2 holds
ex i being Nat st
( i in dom (the_series_of_quotients_of s1) & ( for H being GroupWithOperators of O st H = (the_series_of_quotients_of s1) . i holds
H is trivial ) )
let G be GroupWithOperators of O; ::_thesis: for s1, s2 being CompositionSeries of G st s1 is_finer_than s2 & s2 is jordan_holder & len s1 > len s2 holds
ex i being Nat st
( i in dom (the_series_of_quotients_of s1) & ( for H being GroupWithOperators of O st H = (the_series_of_quotients_of s1) . i holds
H is trivial ) )
let s1, s2 be CompositionSeries of G; ::_thesis: ( s1 is_finer_than s2 & s2 is jordan_holder & len s1 > len s2 implies ex i being Nat st
( i in dom (the_series_of_quotients_of s1) & ( for H being GroupWithOperators of O st H = (the_series_of_quotients_of s1) . i holds
H is trivial ) ) )
assume A1: s1 is_finer_than s2 ; ::_thesis: ( not s2 is jordan_holder or not len s1 > len s2 or ex i being Nat st
( i in dom (the_series_of_quotients_of s1) & ( for H being GroupWithOperators of O st H = (the_series_of_quotients_of s1) . i holds
H is trivial ) ) )
assume A2: s2 is jordan_holder ; ::_thesis: ( not len s1 > len s2 or ex i being Nat st
( i in dom (the_series_of_quotients_of s1) & ( for H being GroupWithOperators of O st H = (the_series_of_quotients_of s1) . i holds
H is trivial ) ) )
assume A3: len s1 > len s2 ; ::_thesis: ex i being Nat st
( i in dom (the_series_of_quotients_of s1) & ( for H being GroupWithOperators of O st H = (the_series_of_quotients_of s1) . i holds
H is trivial ) )
then not s1 is strictly_decreasing by A1, A2, Def31;
then ex i being Nat st
( i in dom s1 & i + 1 in dom s1 & ex H1 being StableSubgroup of G ex N1 being normal StableSubgroup of H1 st
( H1 = s1 . i & N1 = s1 . (i + 1) & H1 ./. N1 is trivial ) ) by Def30;
then consider i being Nat, H1 being StableSubgroup of G, N1 being normal StableSubgroup of H1 such that
A4: i in dom s1 and
A5: i + 1 in dom s1 and
A6: ( H1 = s1 . i & N1 = s1 . (i + 1) & H1 ./. N1 is trivial ) ;
i + 1 in Seg (len s1) by A5, FINSEQ_1:def_3;
then A7: i + 1 <= len s1 by FINSEQ_1:1;
0 + 1 <= i + 1 by XREAL_1:6;
then A8: 1 <= len s1 by A7, XXREAL_0:2;
percases ( len s1 <= 1 or len s1 > 1 ) ;
suppose len s1 <= 1 ; ::_thesis: ex i being Nat st
( i in dom (the_series_of_quotients_of s1) & ( for H being GroupWithOperators of O st H = (the_series_of_quotients_of s1) . i holds
H is trivial ) )
then A9: len s1 = 1 by A8, XXREAL_0:1;
now__::_thesis:_for_i_being_Nat_st_i_in_dom_s1_&_i_+_1_in_dom_s1_holds_
for_H1_being_StableSubgroup_of_G
for_N1_being_normal_StableSubgroup_of_H1_st_H1_=_s1_._i_&_N1_=_s1_._(i_+_1)_holds_
not_H1_./._N1_is_trivial
let i be Nat; ::_thesis: ( i in dom s1 & i + 1 in dom s1 implies for H1 being StableSubgroup of G
for N1 being normal StableSubgroup of H1 st H1 = s1 . i & N1 = s1 . (i + 1) holds
not H1 ./. N1 is trivial )
assume i in dom s1 ; ::_thesis: ( i + 1 in dom s1 implies for H1 being StableSubgroup of G
for N1 being normal StableSubgroup of H1 st H1 = s1 . i & N1 = s1 . (i + 1) holds
not H1 ./. N1 is trivial )
then i in Seg 1 by A9, FINSEQ_1:def_3;
then A10: i = 1 by FINSEQ_1:2, TARSKI:def_1;
assume A11: i + 1 in dom s1 ; ::_thesis: for H1 being StableSubgroup of G
for N1 being normal StableSubgroup of H1 st H1 = s1 . i & N1 = s1 . (i + 1) holds
not H1 ./. N1 is trivial
let H1 be StableSubgroup of G; ::_thesis: for N1 being normal StableSubgroup of H1 st H1 = s1 . i & N1 = s1 . (i + 1) holds
not H1 ./. N1 is trivial
let N1 be normal StableSubgroup of H1; ::_thesis: ( H1 = s1 . i & N1 = s1 . (i + 1) implies not H1 ./. N1 is trivial )
assume H1 = s1 . i ; ::_thesis: ( N1 = s1 . (i + 1) implies not H1 ./. N1 is trivial )
assume N1 = s1 . (i + 1) ; ::_thesis: not H1 ./. N1 is trivial
assume H1 ./. N1 is trivial ; ::_thesis: contradiction
2 in Seg 1 by A9, A10, A11, FINSEQ_1:def_3;
hence contradiction by FINSEQ_1:2, TARSKI:def_1; ::_thesis: verum
end;
then s1 is strictly_decreasing by Def30;
hence ex i being Nat st
( i in dom (the_series_of_quotients_of s1) & ( for H being GroupWithOperators of O st H = (the_series_of_quotients_of s1) . i holds
H is trivial ) ) by A1, A2, A3, Def31; ::_thesis: verum
end;
supposeA12: len s1 > 1 ; ::_thesis: ex i being Nat st
( i in dom (the_series_of_quotients_of s1) & ( for H being GroupWithOperators of O st H = (the_series_of_quotients_of s1) . i holds
H is trivial ) )
take i ; ::_thesis: ( i in dom (the_series_of_quotients_of s1) & ( for H being GroupWithOperators of O st H = (the_series_of_quotients_of s1) . i holds
H is trivial ) )
A13: (i + 1) - 1 <= (len s1) - 1 by A7, XREAL_1:9;
i in Seg (len s1) by A4, FINSEQ_1:def_3;
then A14: 1 <= i by FINSEQ_1:1;
len s1 = (len (the_series_of_quotients_of s1)) + 1 by A12, Def33;
then i in Seg (len (the_series_of_quotients_of s1)) by A14, A13, FINSEQ_1:1;
hence A15: i in dom (the_series_of_quotients_of s1) by FINSEQ_1:def_3; ::_thesis: for H being GroupWithOperators of O st H = (the_series_of_quotients_of s1) . i holds
H is trivial
let H be GroupWithOperators of O; ::_thesis: ( H = (the_series_of_quotients_of s1) . i implies H is trivial )
assume H = (the_series_of_quotients_of s1) . i ; ::_thesis: H is trivial
hence H is trivial by A6, A12, A15, Def33; ::_thesis: verum
end;
end;
end;
Lm40: for k, m being Element of NAT holds
( k < m iff k <= m - 1 )
proof
let k, m be Element of NAT ; ::_thesis: ( k < m iff k <= m - 1 )
A1: now__::_thesis:_(_k_<=_m_-_1_implies_k_<_m_)
assume k <= m - 1 ; ::_thesis: k < m
then A2: k + 1 <= m by XREAL_1:19;
k < k + 1 by XREAL_1:29;
hence k < m by A2, XXREAL_0:2; ::_thesis: verum
end;
now__::_thesis:_(_k_<_m_implies_k_<=_m_-_1_)
assume k < m ; ::_thesis: k <= m - 1
then k + 1 <= m by INT_1:7;
hence k <= m - 1 by XREAL_1:19; ::_thesis: verum
end;
hence ( k < m iff k <= m - 1 ) by A1; ::_thesis: verum
end;
Lm41: for a being Element of NAT
for fs being FinSequence st a in dom fs holds
ex fs1, fs2 being FinSequence st
( fs = (fs1 ^ <*(fs . a)*>) ^ fs2 & len fs1 = a - 1 & len fs2 = (len fs) - a )
proof
let a be Element of NAT ; ::_thesis: for fs being FinSequence st a in dom fs holds
ex fs1, fs2 being FinSequence st
( fs = (fs1 ^ <*(fs . a)*>) ^ fs2 & len fs1 = a - 1 & len fs2 = (len fs) - a )
let fs be FinSequence; ::_thesis: ( a in dom fs implies ex fs1, fs2 being FinSequence st
( fs = (fs1 ^ <*(fs . a)*>) ^ fs2 & len fs1 = a - 1 & len fs2 = (len fs) - a ) )
assume A1: a in dom fs ; ::_thesis: ex fs1, fs2 being FinSequence st
( fs = (fs1 ^ <*(fs . a)*>) ^ fs2 & len fs1 = a - 1 & len fs2 = (len fs) - a )
then ( a >= 1 & a <= len fs ) by FINSEQ_3:25;
then reconsider b = (len fs) - a, d = a - 1 as Element of NAT by INT_1:5;
len fs = a + b ;
then consider fs3, fs2 being FinSequence such that
A2: len fs3 = a and
A3: len fs2 = b and
A4: fs = fs3 ^ fs2 by FINSEQ_2:22;
a = d + 1 ;
then consider fs1 being FinSequence, v being set such that
A5: fs3 = fs1 ^ <*v*> by A2, FINSEQ_2:18;
A6: (len fs1) + 1 = d + 1 by A2, A5, FINSEQ_2:16;
fs3 <> {} by A1, A2, FINSEQ_3:25;
then a in dom fs3 by A2, FINSEQ_5:6;
then fs3 . a = fs . a by A4, FINSEQ_1:def_7;
then fs . a = v by A5, A6, FINSEQ_1:42;
hence ex fs1, fs2 being FinSequence st
( fs = (fs1 ^ <*(fs . a)*>) ^ fs2 & len fs1 = a - 1 & len fs2 = (len fs) - a ) by A3, A4, A5, A6; ::_thesis: verum
end;
Lm42: for a being Element of NAT
for fs, fs1, fs2 being FinSequence
for v being set st a in dom fs & fs = (fs1 ^ <*v*>) ^ fs2 & len fs1 = a - 1 holds
Del (fs,a) = fs1 ^ fs2
proof
let a be Element of NAT ; ::_thesis: for fs, fs1, fs2 being FinSequence
for v being set st a in dom fs & fs = (fs1 ^ <*v*>) ^ fs2 & len fs1 = a - 1 holds
Del (fs,a) = fs1 ^ fs2
let fs, fs1, fs2 be FinSequence; ::_thesis: for v being set st a in dom fs & fs = (fs1 ^ <*v*>) ^ fs2 & len fs1 = a - 1 holds
Del (fs,a) = fs1 ^ fs2
let v be set ; ::_thesis: ( a in dom fs & fs = (fs1 ^ <*v*>) ^ fs2 & len fs1 = a - 1 implies Del (fs,a) = fs1 ^ fs2 )
assume that
A1: a in dom fs and
A2: fs = (fs1 ^ <*v*>) ^ fs2 and
A3: len fs1 = a - 1 ; ::_thesis: Del (fs,a) = fs1 ^ fs2
A4: (len (Del (fs,a))) + 1 = len fs by A1, WSIERP_1:def_1;
len fs = (len (fs1 ^ <*v*>)) + (len fs2) by A2, FINSEQ_1:22
.= ((len fs1) + 1) + (len fs2) by FINSEQ_2:16
.= a + (len fs2) by A3 ;
then len (Del (fs,a)) = (len fs2) + (len fs1) by A3, A4;
then A5: len (fs1 ^ fs2) = len (Del (fs,a)) by FINSEQ_1:22;
A6: len <*v*> = 1 by FINSEQ_1:39;
A7: fs = fs1 ^ (<*v*> ^ fs2) by A2, FINSEQ_1:32;
then len fs = (a - 1) + (len (<*v*> ^ fs2)) by A3, FINSEQ_1:22;
then A8: len (<*v*> ^ fs2) = (len fs) - (a - 1) ;
now__::_thesis:_for_e_being_Nat_st_1_<=_e_&_e_<=_len_(Del_(fs,a))_holds_
(fs1_^_fs2)_._e_=_(Del_(fs,a))_._e
let e be Nat; ::_thesis: ( 1 <= e & e <= len (Del (fs,a)) implies (fs1 ^ fs2) . e = (Del (fs,a)) . e )
assume that
A9: 1 <= e and
A10: e <= len (Del (fs,a)) ; ::_thesis: (fs1 ^ fs2) . e = (Del (fs,a)) . e
reconsider e1 = e as Element of NAT by ORDINAL1:def_12;
now__::_thesis:_(fs1_^_fs2)_._e_=_(Del_(fs,a))_._e
percases ( e < a or e >= a ) ;
supposeA11: e < a ; ::_thesis: (fs1 ^ fs2) . e = (Del (fs,a)) . e
then e1 <= a - 1 by Lm40;
then A12: e in dom fs1 by A3, A9, FINSEQ_3:25;
hence (fs1 ^ fs2) . e = fs1 . e by FINSEQ_1:def_7
.= fs . e1 by A7, A12, FINSEQ_1:def_7
.= (Del (fs,a)) . e by A1, A11, WSIERP_1:def_1 ;
::_thesis: verum
end;
supposeA13: e >= a ; ::_thesis: (fs1 ^ fs2) . e = (Del (fs,a)) . e
then A14: e1 > a - 1 by Lm40;
then A15: e + 1 > a by XREAL_1:19;
then (e + 1) - a > 0 by XREAL_1:50;
then A16: ((e + 1) - a) + 1 > 0 + 1 by XREAL_1:6;
A17: e + 1 > a - 1 by A15, XREAL_1:146, XXREAL_0:2;
then (e + 1) - (a - 1) > 0 by XREAL_1:50;
then reconsider f = (e + 1) - (a - 1) as Element of NAT by INT_1:3;
A18: e + 1 <= len fs by A4, A10, XREAL_1:6;
then A19: (e + 1) - (a - 1) <= len (<*v*> ^ fs2) by A8, XREAL_1:9;
thus (fs1 ^ fs2) . e = fs2 . (e - (len fs1)) by A3, A5, A10, A14, FINSEQ_1:24
.= fs2 . (f - 1) by A3
.= (<*v*> ^ fs2) . f by A6, A16, A19, FINSEQ_1:24
.= (fs1 ^ (<*v*> ^ fs2)) . (e1 + 1) by A3, A7, A17, A18, FINSEQ_1:24
.= (Del (fs,a)) . e by A1, A7, A13, WSIERP_1:def_1 ; ::_thesis: verum
end;
end;
end;
hence (fs1 ^ fs2) . e = (Del (fs,a)) . e ; ::_thesis: verum
end;
hence Del (fs,a) = fs1 ^ fs2 by A5, FINSEQ_1:14; ::_thesis: verum
end;
Lm43: for a being Element of NAT
for fs1, fs2 being FinSequence holds
( ( a <= len fs1 implies Del ((fs1 ^ fs2),a) = (Del (fs1,a)) ^ fs2 ) & ( a >= 1 implies Del ((fs1 ^ fs2),((len fs1) + a)) = fs1 ^ (Del (fs2,a)) ) )
proof
let a be Element of NAT ; ::_thesis: for fs1, fs2 being FinSequence holds
( ( a <= len fs1 implies Del ((fs1 ^ fs2),a) = (Del (fs1,a)) ^ fs2 ) & ( a >= 1 implies Del ((fs1 ^ fs2),((len fs1) + a)) = fs1 ^ (Del (fs2,a)) ) )
let fs1, fs2 be FinSequence; ::_thesis: ( ( a <= len fs1 implies Del ((fs1 ^ fs2),a) = (Del (fs1,a)) ^ fs2 ) & ( a >= 1 implies Del ((fs1 ^ fs2),((len fs1) + a)) = fs1 ^ (Del (fs2,a)) ) )
set f = fs1 ^ fs2;
A1: len (fs1 ^ fs2) = (len fs1) + (len fs2) by FINSEQ_1:22;
A2: now__::_thesis:_(_a_>=_1_implies_Del_((fs1_^_fs2),((len_fs1)_+_a))_=_fs1_^_(Del_(fs2,a))_)
set f2 = fs1 ^ (Del (fs2,a));
set f1 = Del ((fs1 ^ fs2),((len fs1) + a));
assume A3: a >= 1 ; ::_thesis: Del ((fs1 ^ fs2),((len fs1) + a)) = fs1 ^ (Del (fs2,a))
now__::_thesis:_Del_((fs1_^_fs2),((len_fs1)_+_a))_=_fs1_^_(Del_(fs2,a))
percases ( a > len fs2 or a <= len fs2 ) ;
supposeA4: a > len fs2 ; ::_thesis: Del ((fs1 ^ fs2),((len fs1) + a)) = fs1 ^ (Del (fs2,a))
then A5: not a in dom fs2 by FINSEQ_3:25;
(len fs1) + a > len (fs1 ^ fs2) by A1, A4, XREAL_1:6;
then not (len fs1) + a in dom (fs1 ^ fs2) by FINSEQ_3:25;
hence Del ((fs1 ^ fs2),((len fs1) + a)) = fs1 ^ fs2 by WSIERP_1:def_1
.= fs1 ^ (Del (fs2,a)) by A5, WSIERP_1:def_1 ;
::_thesis: verum
end;
supposeA6: a <= len fs2 ; ::_thesis: Del ((fs1 ^ fs2),((len fs1) + a)) = fs1 ^ (Del (fs2,a))
then A7: a in dom fs2 by A3, FINSEQ_3:25;
a - 1 >= 0 by A3, XREAL_1:48;
then A8: (a - 1) + (len fs1) >= 0 + (len fs1) by XREAL_1:6;
A9: (len fs1) + a >= 1 by A3, NAT_1:12;
(len fs1) + a <= len (fs1 ^ fs2) by A1, A6, XREAL_1:6;
then A10: (len fs1) + a in dom (fs1 ^ fs2) by A9, FINSEQ_3:25;
then consider g1, g2 being FinSequence such that
A11: fs1 ^ fs2 = (g1 ^ <*((fs1 ^ fs2) . ((len fs1) + a))*>) ^ g2 and
A12: len g1 = ((len fs1) + a) - 1 and
len g2 = (len (fs1 ^ fs2)) - ((len fs1) + a) by Lm41;
A13: Del ((fs1 ^ fs2),((len fs1) + a)) = g1 ^ g2 by A10, A11, A12, Lm42;
fs1 ^ fs2 = g1 ^ (<*((fs1 ^ fs2) . ((len fs1) + a))*> ^ g2) by A11, FINSEQ_1:32;
then consider t being FinSequence such that
A14: fs1 ^ t = g1 by A12, A8, FINSEQ_1:47;
fs1 ^ ((t ^ <*((fs1 ^ fs2) . ((len fs1) + a))*>) ^ g2) = (fs1 ^ (t ^ <*((fs1 ^ fs2) . ((len fs1) + a))*>)) ^ g2 by FINSEQ_1:32
.= fs1 ^ fs2 by A11, A14, FINSEQ_1:32 ;
then A15: fs2 = (t ^ <*((fs1 ^ fs2) . ((len fs1) + a))*>) ^ g2 by FINSEQ_1:33;
(len fs1) + (a - 1) = (len fs1) + (len t) by A12, A14, FINSEQ_1:22;
then Del (fs2,a) = t ^ g2 by A7, A15, Lm42;
hence Del ((fs1 ^ fs2),((len fs1) + a)) = fs1 ^ (Del (fs2,a)) by A13, A14, FINSEQ_1:32; ::_thesis: verum
end;
end;
end;
hence Del ((fs1 ^ fs2),((len fs1) + a)) = fs1 ^ (Del (fs2,a)) ; ::_thesis: verum
end;
now__::_thesis:_(_a_<=_len_fs1_implies_Del_((fs1_^_fs2),a)_=_(Del_(fs1,a))_^_fs2_)
set f3 = <*((fs1 ^ fs2) . a)*>;
set f2 = (Del (fs1,a)) ^ fs2;
set f1 = Del ((fs1 ^ fs2),a);
assume A16: a <= len fs1 ; ::_thesis: Del ((fs1 ^ fs2),a) = (Del (fs1,a)) ^ fs2
len fs1 <= len (fs1 ^ fs2) by A1, NAT_1:11;
then A17: a <= len (fs1 ^ fs2) by A16, XXREAL_0:2;
now__::_thesis:_Del_((fs1_^_fs2),a)_=_(Del_(fs1,a))_^_fs2
percases ( a < 1 or a >= 1 ) ;
supposeA18: a < 1 ; ::_thesis: Del ((fs1 ^ fs2),a) = (Del (fs1,a)) ^ fs2
then A19: not a in dom fs1 by FINSEQ_3:25;
not a in dom (fs1 ^ fs2) by A18, FINSEQ_3:25;
hence Del ((fs1 ^ fs2),a) = fs1 ^ fs2 by WSIERP_1:def_1
.= (Del (fs1,a)) ^ fs2 by A19, WSIERP_1:def_1 ;
::_thesis: verum
end;
supposeA20: a >= 1 ; ::_thesis: Del ((fs1 ^ fs2),a) = (Del (fs1,a)) ^ fs2
then A21: a in dom (fs1 ^ fs2) by A17, FINSEQ_3:25;
then consider g1, g2 being FinSequence such that
A22: fs1 ^ fs2 = (g1 ^ <*((fs1 ^ fs2) . a)*>) ^ g2 and
A23: len g1 = a - 1 and
len g2 = (len (fs1 ^ fs2)) - a by Lm41;
len (g1 ^ <*((fs1 ^ fs2) . a)*>) = (a - 1) + 1 by A23, FINSEQ_2:16
.= a ;
then consider t being FinSequence such that
A24: fs1 = (g1 ^ <*((fs1 ^ fs2) . a)*>) ^ t by A16, A22, FINSEQ_1:47;
(g1 ^ <*((fs1 ^ fs2) . a)*>) ^ g2 = (g1 ^ <*((fs1 ^ fs2) . a)*>) ^ (t ^ fs2) by A22, A24, FINSEQ_1:32;
then A25: g2 = t ^ fs2 by FINSEQ_1:33;
a in dom fs1 by A16, A20, FINSEQ_3:25;
then A26: Del (fs1,a) = g1 ^ t by A23, A24, Lm42;
thus Del ((fs1 ^ fs2),a) = g1 ^ g2 by A21, A22, A23, Lm42
.= (Del (fs1,a)) ^ fs2 by A26, A25, FINSEQ_1:32 ; ::_thesis: verum
end;
end;
end;
hence Del ((fs1 ^ fs2),a) = (Del (fs1,a)) ^ fs2 ; ::_thesis: verum
end;
hence ( ( a <= len fs1 implies Del ((fs1 ^ fs2),a) = (Del (fs1,a)) ^ fs2 ) & ( a >= 1 implies Del ((fs1 ^ fs2),((len fs1) + a)) = fs1 ^ (Del (fs2,a)) ) ) by A2; ::_thesis: verum
end;
Lm44: for D being non empty set
for f being FinSequence of D
for p being Element of D
for n being Nat st n in dom f holds
f = Del ((Ins (f,n,p)),(n + 1))
proof
let D be non empty set ; ::_thesis: for f being FinSequence of D
for p being Element of D
for n being Nat st n in dom f holds
f = Del ((Ins (f,n,p)),(n + 1))
let f be FinSequence of D; ::_thesis: for p being Element of D
for n being Nat st n in dom f holds
f = Del ((Ins (f,n,p)),(n + 1))
let p be Element of D; ::_thesis: for n being Nat st n in dom f holds
f = Del ((Ins (f,n,p)),(n + 1))
let n be Nat; ::_thesis: ( n in dom f implies f = Del ((Ins (f,n,p)),(n + 1)) )
set fs1 = (f | n) ^ <*p*>;
set fs2 = f /^ n;
assume n in dom f ; ::_thesis: f = Del ((Ins (f,n,p)),(n + 1))
then n in Seg (len f) by FINSEQ_1:def_3;
then n <= len f by FINSEQ_1:1;
then A1: len (f | n) = n by FINSEQ_1:59;
len ((f | n) ^ <*p*>) = (len (f | n)) + (len <*p*>) by FINSEQ_1:22
.= n + 1 by A1, FINSEQ_1:39 ;
then Del ((Ins (f,n,p)),(n + 1)) = (Del (((f | n) ^ <*p*>),(n + 1))) ^ (f /^ n) by Lm43
.= (f | n) ^ (f /^ n) by A1, WSIERP_1:40 ;
hence f = Del ((Ins (f,n,p)),(n + 1)) by RFINSEQ:8; ::_thesis: verum
end;
theorem Th110: :: GROUP_9:110
for O being set
for G being GroupWithOperators of O
for s1 being CompositionSeries of G st len s1 > 1 holds
( s1 is jordan_holder iff for i being Nat st i in dom (the_series_of_quotients_of s1) holds
(the_series_of_quotients_of s1) . i is strict simple GroupWithOperators of O )
proof
let O be set ; ::_thesis: for G being GroupWithOperators of O
for s1 being CompositionSeries of G st len s1 > 1 holds
( s1 is jordan_holder iff for i being Nat st i in dom (the_series_of_quotients_of s1) holds
(the_series_of_quotients_of s1) . i is strict simple GroupWithOperators of O )
let G be GroupWithOperators of O; ::_thesis: for s1 being CompositionSeries of G st len s1 > 1 holds
( s1 is jordan_holder iff for i being Nat st i in dom (the_series_of_quotients_of s1) holds
(the_series_of_quotients_of s1) . i is strict simple GroupWithOperators of O )
let s1 be CompositionSeries of G; ::_thesis: ( len s1 > 1 implies ( s1 is jordan_holder iff for i being Nat st i in dom (the_series_of_quotients_of s1) holds
(the_series_of_quotients_of s1) . i is strict simple GroupWithOperators of O ) )
assume A1: len s1 > 1 ; ::_thesis: ( s1 is jordan_holder iff for i being Nat st i in dom (the_series_of_quotients_of s1) holds
(the_series_of_quotients_of s1) . i is strict simple GroupWithOperators of O )
A2: now__::_thesis:_(_s1_is_jordan_holder_implies_for_i_being_Nat_st_i_in_dom_(the_series_of_quotients_of_s1)_holds_
(the_series_of_quotients_of_s1)_._i_is_strict_simple_GroupWithOperators_of_O_)
assume A3: s1 is jordan_holder ; ::_thesis: for i being Nat st i in dom (the_series_of_quotients_of s1) holds
(the_series_of_quotients_of s1) . i is strict simple GroupWithOperators of O
assume ex i being Nat st
( i in dom (the_series_of_quotients_of s1) & (the_series_of_quotients_of s1) . i is not strict simple GroupWithOperators of O ) ; ::_thesis: contradiction
then consider i being Nat such that
A4: i in dom (the_series_of_quotients_of s1) and
A5: (the_series_of_quotients_of s1) . i is not strict simple GroupWithOperators of O ;
A6: i in Seg (len (the_series_of_quotients_of s1)) by A4, FINSEQ_1:def_3;
then A7: i <= len (the_series_of_quotients_of s1) by FINSEQ_1:1;
len s1 = (len (the_series_of_quotients_of s1)) + 1 by A1, Def33;
then A8: i + 1 <= len s1 by A7, XREAL_1:6;
A9: 0 + 1 <= i + 1 by XREAL_1:6;
then i + 1 in Seg (len s1) by A8;
then A10: i + 1 in dom s1 by FINSEQ_1:def_3;
0 + (len (the_series_of_quotients_of s1)) < 1 + (len (the_series_of_quotients_of s1)) by XREAL_1:6;
then A11: len (the_series_of_quotients_of s1) < len s1 by A1, Def33;
then A12: i <= len s1 by A7, XXREAL_0:2;
1 <= i by A6, FINSEQ_1:1;
then i in Seg (len s1) by A12, FINSEQ_1:1;
then A13: i in dom s1 by FINSEQ_1:def_3;
then reconsider H1 = s1 . i, N1 = s1 . (i + 1) as Element of the_stable_subgroups_of G by A10, FINSEQ_2:11;
reconsider H1 = H1, N1 = N1 as strict StableSubgroup of G by Def11;
reconsider N1 = N1 as strict normal StableSubgroup of H1 by A13, A10, Def28;
A14: H1 ./. N1 is not strict simple GroupWithOperators of O by A1, A4, A5, Def33;
percases ( H1 ./. N1 is trivial or ex H being strict normal StableSubgroup of H1 ./. N1 st
( H <> (Omega). (H1 ./. N1) & H <> (1). (H1 ./. N1) ) ) by A14, Def13;
supposeA15: H1 ./. N1 is trivial ; ::_thesis: contradiction
s1 is strictly_decreasing by A3, Def31;
hence contradiction by A13, A10, A15, Def30; ::_thesis: verum
end;
suppose ex H being strict normal StableSubgroup of H1 ./. N1 st
( H <> (Omega). (H1 ./. N1) & H <> (1). (H1 ./. N1) ) ; ::_thesis: contradiction
then consider H being strict normal StableSubgroup of H1 ./. N1 such that
A16: H <> (Omega). (H1 ./. N1) and
A17: H <> (1). (H1 ./. N1) ;
N1 = Ker (nat_hom N1) by Th48;
then consider N2 being strict StableSubgroup of H1 such that
A18: the carrier of N2 = (nat_hom N1) " the carrier of H and
A19: ( H is normal implies ( N1 is normal StableSubgroup of N2 & N2 is normal ) ) by Th78;
A20: N2 is strict StableSubgroup of G by Th11;
reconsider i = i as Element of NAT by ORDINAL1:def_12;
A21: ( 1 <= i & not s1 is empty ) by A1, A6, FINSEQ_1:1;
reconsider N2 = N2 as Element of the_stable_subgroups_of G by A20, Def11;
set s2 = Ins (s1,i,N2);
A22: len (Ins (s1,i,N2)) = (len s1) + 1 by FINSEQ_5:69;
then A23: s1 <> Ins (s1,i,N2) ;
A24: now__::_thesis:_for_j_being_Nat_st_j_in_dom_(Ins_(s1,i,N2))_&_j_+_1_in_dom_(Ins_(s1,i,N2))_holds_
for_H19,_H29_being_StableSubgroup_of_G_st_H19_=_(Ins_(s1,i,N2))_._j_&_H29_=_(Ins_(s1,i,N2))_._(j_+_1)_holds_
H29_is_normal_StableSubgroup_of_H19
let j be Nat; ::_thesis: ( j in dom (Ins (s1,i,N2)) & j + 1 in dom (Ins (s1,i,N2)) implies for H19, H29 being StableSubgroup of G st H19 = (Ins (s1,i,N2)) . j & H29 = (Ins (s1,i,N2)) . (j + 1) holds
b5 is normal StableSubgroup of b4 )
assume A25: j in dom (Ins (s1,i,N2)) ; ::_thesis: ( j + 1 in dom (Ins (s1,i,N2)) implies for H19, H29 being StableSubgroup of G st H19 = (Ins (s1,i,N2)) . j & H29 = (Ins (s1,i,N2)) . (j + 1) holds
b5 is normal StableSubgroup of b4 )
then A26: j in Seg (len (Ins (s1,i,N2))) by FINSEQ_1:def_3;
then A27: 1 <= j by FINSEQ_1:1;
A28: j <= len (Ins (s1,i,N2)) by A26, FINSEQ_1:1;
( j < i or j = i or j > i ) by XXREAL_0:1;
then ( j + 1 <= i or j = i or j >= i + 1 ) by NAT_1:13;
then A29: ( (j + 1) - 1 <= i - 1 or j = i or j >= i + 1 ) by XREAL_1:9;
assume A30: j + 1 in dom (Ins (s1,i,N2)) ; ::_thesis: for H19, H29 being StableSubgroup of G st H19 = (Ins (s1,i,N2)) . j & H29 = (Ins (s1,i,N2)) . (j + 1) holds
b5 is normal StableSubgroup of b4
then A31: j + 1 in Seg (len (Ins (s1,i,N2))) by FINSEQ_1:def_3;
then A32: 1 <= j + 1 by FINSEQ_1:1;
A33: j + 1 <= len (Ins (s1,i,N2)) by A31, FINSEQ_1:1;
let H19, H29 be StableSubgroup of G; ::_thesis: ( H19 = (Ins (s1,i,N2)) . j & H29 = (Ins (s1,i,N2)) . (j + 1) implies b3 is normal StableSubgroup of b2 )
assume A34: H19 = (Ins (s1,i,N2)) . j ; ::_thesis: ( H29 = (Ins (s1,i,N2)) . (j + 1) implies b3 is normal StableSubgroup of b2 )
assume A35: H29 = (Ins (s1,i,N2)) . (j + 1) ; ::_thesis: b3 is normal StableSubgroup of b2
percases ( j <= i - 1 or j = i or j = i + 1 or i + 1 < j ) by A29, XXREAL_0:1;
supposeA36: j <= i - 1 ; ::_thesis: b3 is normal StableSubgroup of b2
A37: Seg (len (s1 | i)) = Seg i by A11, A7, FINSEQ_1:59, XXREAL_0:2;
A38: dom (s1 | i) c= dom s1 by RELAT_1:60;
(- 1) + i < 0 + i by XREAL_1:6;
then j <= i by A36, XXREAL_0:2;
then j in Seg (len (s1 | i)) by A27, A37, FINSEQ_1:1;
then A39: j in dom (s1 | i) by FINSEQ_1:def_3;
j + 1 <= (i - 1) + 1 by A36, XREAL_1:6;
then j + 1 in Seg (len (s1 | i)) by A32, A37;
then A40: j + 1 in dom (s1 | i) by FINSEQ_1:def_3;
A41: (Ins (s1,i,N2)) . (j + 1) = (Ins (s1,i,N2)) /. (j + 1) by A30, PARTFUN1:def_6
.= s1 /. (j + 1) by A40, FINSEQ_5:72
.= s1 . (j + 1) by A38, A40, PARTFUN1:def_6 ;
(Ins (s1,i,N2)) . j = (Ins (s1,i,N2)) /. j by A25, PARTFUN1:def_6
.= s1 /. j by A39, FINSEQ_5:72
.= s1 . j by A38, A39, PARTFUN1:def_6 ;
hence H29 is normal StableSubgroup of H19 by A34, A35, A38, A39, A40, A41, Def28; ::_thesis: verum
end;
supposeA42: j = i ; ::_thesis: b3 is normal StableSubgroup of b2
then A43: j in Seg i by A27;
Seg (len (s1 | i)) = Seg i by A11, A7, FINSEQ_1:59, XXREAL_0:2;
then A44: j in dom (s1 | i) by A43, FINSEQ_1:def_3;
A45: dom (s1 | i) c= dom s1 by RELAT_1:60;
A46: (Ins (s1,i,N2)) . j = (Ins (s1,i,N2)) /. j by A25, PARTFUN1:def_6
.= s1 /. j by A44, FINSEQ_5:72
.= s1 . j by A45, A44, PARTFUN1:def_6 ;
(Ins (s1,i,N2)) . (j + 1) = (Ins (s1,i,N2)) /. (i + 1) by A30, A42, PARTFUN1:def_6
.= N2 by A11, A7, FINSEQ_5:73, XXREAL_0:2 ;
hence H29 is normal StableSubgroup of H19 by A19, A34, A35, A42, A46; ::_thesis: verum
end;
supposeA47: j = i + 1 ; ::_thesis: b3 is normal StableSubgroup of b2
i + 1 <= (len s1) + 1 by A12, XREAL_1:6;
then i + 1 <= len (Ins (s1,i,N2)) by FINSEQ_5:69;
then i + 1 in Seg (len (Ins (s1,i,N2))) by A9;
then i + 1 in dom (Ins (s1,i,N2)) by FINSEQ_1:def_3;
then A48: H19 = (Ins (s1,i,N2)) /. (i + 1) by A34, A47, PARTFUN1:def_6
.= N2 by A11, A7, FINSEQ_5:73, XXREAL_0:2 ;
(i + 1) + 1 <= (len s1) + 1 by A8, XREAL_1:6;
then A49: (i + 1) + 1 <= len (Ins (s1,i,N2)) by FINSEQ_5:69;
1 <= (i + 1) + 1 by A9, XREAL_1:6;
then (i + 1) + 1 in Seg (len (Ins (s1,i,N2))) by A49;
then (i + 1) + 1 in dom (Ins (s1,i,N2)) by FINSEQ_1:def_3;
then H29 = (Ins (s1,i,N2)) /. ((i + 1) + 1) by A35, A47, PARTFUN1:def_6
.= s1 /. (i + 1) by A8, FINSEQ_5:74
.= N1 by A10, PARTFUN1:def_6 ;
hence H29 is normal StableSubgroup of H19 by A19, A48; ::_thesis: verum
end;
supposeA50: i + 1 < j ; ::_thesis: b3 is normal StableSubgroup of b2
set j9 = j - 1;
0 + 1 <= i + 1 by XREAL_1:6;
then A51: 0 + 1 < j by A50, XXREAL_0:2;
then A52: (0 + 1) - 1 < j - 1 by XREAL_1:9;
then reconsider j9 = j - 1 as Element of NAT by INT_1:3;
A53: j - 1 <= (len (Ins (s1,i,N2))) - 1 by A28, XREAL_1:9;
0 + 1 <= j9 by A52, NAT_1:13;
then j9 in Seg (len s1) by A22, A53;
then A54: j9 in dom s1 by FINSEQ_1:def_3;
(i + 1) + 1 <= j by A50, NAT_1:13;
then A55: ((i + 1) + 1) - 1 <= j - 1 by XREAL_1:9;
0 + j9 < 1 + j9 by XREAL_1:6;
then A56: i + 1 <= j9 + 1 by A55, XXREAL_0:2;
A57: (j + 1) - 1 <= (len (Ins (s1,i,N2))) - 1 by A33, XREAL_1:9;
then j9 + 1 in Seg (len s1) by A22, A51;
then A58: j9 + 1 in dom s1 by FINSEQ_1:def_3;
A59: (Ins (s1,i,N2)) . (j + 1) = (Ins (s1,i,N2)) /. ((j9 + 1) + 1) by A30, PARTFUN1:def_6
.= s1 /. (j9 + 1) by A22, A56, A57, FINSEQ_5:74
.= s1 . (j9 + 1) by A58, PARTFUN1:def_6 ;
(Ins (s1,i,N2)) . j = (Ins (s1,i,N2)) /. (j9 + 1) by A25, PARTFUN1:def_6
.= s1 /. j9 by A22, A55, A53, FINSEQ_5:74
.= s1 . j9 by A54, PARTFUN1:def_6 ;
hence H29 is normal StableSubgroup of H19 by A34, A35, A54, A58, A59, Def28; ::_thesis: verum
end;
end;
end;
len s1 in Seg (len s1) by A1;
then A60: len s1 in dom s1 by FINSEQ_1:def_3;
1 in Seg (len s1) by A1;
then A61: 1 in dom s1 by FINSEQ_1:def_3;
1 <= (len s1) + 1 by A1, NAT_1:13;
then A62: 1 <= len (Ins (s1,i,N2)) by FINSEQ_5:69;
then len (Ins (s1,i,N2)) in Seg (len (Ins (s1,i,N2))) ;
then len (Ins (s1,i,N2)) in dom (Ins (s1,i,N2)) by FINSEQ_1:def_3;
then A63: (Ins (s1,i,N2)) . (len (Ins (s1,i,N2))) = (Ins (s1,i,N2)) /. (len (Ins (s1,i,N2))) by PARTFUN1:def_6
.= (Ins (s1,i,N2)) /. ((len s1) + 1) by FINSEQ_5:69
.= s1 /. (len s1) by A8, FINSEQ_5:74
.= s1 . (len s1) by A60, PARTFUN1:def_6
.= (1). G by Def28 ;
1 in Seg (len (Ins (s1,i,N2))) by A62;
then 1 in dom (Ins (s1,i,N2)) by FINSEQ_1:def_3;
then (Ins (s1,i,N2)) . 1 = (Ins (s1,i,N2)) /. 1 by PARTFUN1:def_6
.= s1 /. 1 by A21, FINSEQ_5:75
.= s1 . 1 by A61, PARTFUN1:def_6
.= (Omega). G by Def28 ;
then reconsider s2 = Ins (s1,i,N2) as CompositionSeries of G by A63, A24, Def28;
now__::_thesis:_for_j_being_Nat_st_j_in_dom_s2_&_j_+_1_in_dom_s2_holds_
for_H19_being_StableSubgroup_of_G
for_N19_being_normal_StableSubgroup_of_H19_st_H19_=_s2_._j_&_N19_=_s2_._(j_+_1)_holds_
not_H19_./._N19_is_trivial
let j be Nat; ::_thesis: ( j in dom s2 & j + 1 in dom s2 implies for H19 being StableSubgroup of G
for N19 being normal StableSubgroup of H19 st H19 = s2 . j & N19 = s2 . (j + 1) holds
not b4 ./. b5 is trivial )
assume A64: j in dom s2 ; ::_thesis: ( j + 1 in dom s2 implies for H19 being StableSubgroup of G
for N19 being normal StableSubgroup of H19 st H19 = s2 . j & N19 = s2 . (j + 1) holds
not b4 ./. b5 is trivial )
then A65: j in Seg (len s2) by FINSEQ_1:def_3;
then A66: 1 <= j by FINSEQ_1:1;
( j < i or j = i or j > i ) by XXREAL_0:1;
then ( j + 1 <= i or j = i or j >= i + 1 ) by NAT_1:13;
then A67: ( (j + 1) - 1 <= i - 1 or j = i or j >= i + 1 ) by XREAL_1:9;
assume A68: j + 1 in dom s2 ; ::_thesis: for H19 being StableSubgroup of G
for N19 being normal StableSubgroup of H19 st H19 = s2 . j & N19 = s2 . (j + 1) holds
not b4 ./. b5 is trivial
then A69: j + 1 in Seg (len s2) by FINSEQ_1:def_3;
then A70: 1 <= j + 1 by FINSEQ_1:1;
A71: j + 1 <= len s2 by A69, FINSEQ_1:1;
let H19 be StableSubgroup of G; ::_thesis: for N19 being normal StableSubgroup of H19 st H19 = s2 . j & N19 = s2 . (j + 1) holds
not b3 ./. b4 is trivial
let N19 be normal StableSubgroup of H19; ::_thesis: ( H19 = s2 . j & N19 = s2 . (j + 1) implies not b2 ./. b3 is trivial )
assume A72: H19 = s2 . j ; ::_thesis: ( N19 = s2 . (j + 1) implies not b2 ./. b3 is trivial )
A73: j <= len s2 by A65, FINSEQ_1:1;
A74: s1 is strictly_decreasing by A3, Def31;
assume A75: N19 = s2 . (j + 1) ; ::_thesis: not b2 ./. b3 is trivial
percases ( j <= i - 1 or j = i or j = i + 1 or i + 1 < j ) by A67, XXREAL_0:1;
supposeA76: j <= i - 1 ; ::_thesis: not b2 ./. b3 is trivial
A77: Seg (len (s1 | i)) = Seg i by A11, A7, FINSEQ_1:59, XXREAL_0:2;
A78: dom (s1 | i) c= dom s1 by RELAT_1:60;
(- 1) + i < 0 + i by XREAL_1:6;
then j <= i by A76, XXREAL_0:2;
then j in Seg (len (s1 | i)) by A66, A77, FINSEQ_1:1;
then A79: j in dom (s1 | i) by FINSEQ_1:def_3;
j + 1 <= (i - 1) + 1 by A76, XREAL_1:6;
then j + 1 in Seg (len (s1 | i)) by A70, A77;
then A80: j + 1 in dom (s1 | i) by FINSEQ_1:def_3;
A81: s2 . (j + 1) = s2 /. (j + 1) by A68, PARTFUN1:def_6
.= s1 /. (j + 1) by A80, FINSEQ_5:72
.= s1 . (j + 1) by A78, A80, PARTFUN1:def_6 ;
s2 . j = s2 /. j by A64, PARTFUN1:def_6
.= s1 /. j by A79, FINSEQ_5:72
.= s1 . j by A78, A79, PARTFUN1:def_6 ;
hence not H19 ./. N19 is trivial by A72, A75, A74, A78, A79, A80, A81, Def30; ::_thesis: verum
end;
supposeA82: j = i ; ::_thesis: not b2 ./. b3 is trivial
then A83: j in Seg i by A66;
Seg (len (s1 | i)) = Seg i by A11, A7, FINSEQ_1:59, XXREAL_0:2;
then A84: j in dom (s1 | i) by A83, FINSEQ_1:def_3;
A85: s2 . (j + 1) = s2 /. (i + 1) by A68, A82, PARTFUN1:def_6
.= N2 by A11, A7, FINSEQ_5:73, XXREAL_0:2 ;
reconsider N2 = N2 as normal StableSubgroup of H1 by A19;
A86: dom (s1 | i) c= dom s1 by RELAT_1:60;
A87: s2 . j = s2 /. j by A64, PARTFUN1:def_6
.= s1 /. j by A84, FINSEQ_5:72
.= s1 . j by A86, A84, PARTFUN1:def_6 ;
now__::_thesis:_not_H19_./._N19_is_trivial
assume H19 ./. N19 is trivial ; ::_thesis: contradiction
then H1 = N2 by A72, A75, A82, A85, A87, Th76;
hence contradiction by A16, A18, Th80; ::_thesis: verum
end;
hence not H19 ./. N19 is trivial ; ::_thesis: verum
end;
supposeA88: j = i + 1 ; ::_thesis: not b2 ./. b3 is trivial
i + 1 <= (len s1) + 1 by A12, XREAL_1:6;
then i + 1 <= len s2 by FINSEQ_5:69;
then i + 1 in Seg (len s2) by A9;
then i + 1 in dom s2 by FINSEQ_1:def_3;
then A89: H19 = s2 /. (i + 1) by A72, A88, PARTFUN1:def_6
.= N2 by A11, A7, FINSEQ_5:73, XXREAL_0:2 ;
(i + 1) + 1 <= (len s1) + 1 by A8, XREAL_1:6;
then A90: (i + 1) + 1 <= len s2 by FINSEQ_5:69;
1 <= (i + 1) + 1 by A9, XREAL_1:6;
then (i + 1) + 1 in Seg (len s2) by A90;
then (i + 1) + 1 in dom s2 by FINSEQ_1:def_3;
then A91: N19 = s2 /. ((i + 1) + 1) by A75, A88, PARTFUN1:def_6
.= s1 /. (i + 1) by A8, FINSEQ_5:74
.= N1 by A10, PARTFUN1:def_6 ;
now__::_thesis:_not_H19_./._N19_is_trivial
assume H19 ./. N19 is trivial ; ::_thesis: contradiction
then the carrier of N1 = (nat_hom N1) " the carrier of H by A18, A89, A91, Th76;
hence contradiction by A17, Th81; ::_thesis: verum
end;
hence not H19 ./. N19 is trivial ; ::_thesis: verum
end;
supposeA92: i + 1 < j ; ::_thesis: not b2 ./. b3 is trivial
set j9 = j - 1;
A93: 0 + 1 <= i + 1 by XREAL_1:6;
then 0 + 1 < j by A92, XXREAL_0:2;
then A94: (0 + 1) - 1 < j - 1 by XREAL_1:9;
then reconsider j9 = j - 1 as Element of NAT by INT_1:3;
A95: (j + 1) - 1 <= (len s2) - 1 by A71, XREAL_1:9;
(i + 1) + 1 <= j by A92, NAT_1:13;
then A96: ((i + 1) + 1) - 1 <= j - 1 by XREAL_1:9;
1 <= j9 + 1 by A92, A93, XXREAL_0:2;
then j9 + 1 in Seg (len s1) by A22, A95;
then A97: j9 + 1 in dom s1 by FINSEQ_1:def_3;
0 + j9 < 1 + j9 by XREAL_1:6;
then A98: i + 1 <= j9 + 1 by A96, XXREAL_0:2;
A99: s2 . (j + 1) = s2 /. ((j9 + 1) + 1) by A68, PARTFUN1:def_6
.= s1 /. (j9 + 1) by A22, A98, A95, FINSEQ_5:74
.= s1 . (j9 + 1) by A97, PARTFUN1:def_6 ;
A100: j - 1 <= (len s2) - 1 by A73, XREAL_1:9;
0 + 1 <= j9 by A94, NAT_1:13;
then j9 in Seg (len s1) by A22, A100;
then A101: j9 in dom s1 by FINSEQ_1:def_3;
s2 . j = s2 /. (j9 + 1) by A64, PARTFUN1:def_6
.= s1 /. j9 by A22, A96, A100, FINSEQ_5:74
.= s1 . j9 by A101, PARTFUN1:def_6 ;
hence not H19 ./. N19 is trivial by A72, A75, A74, A101, A97, A99, Def30; ::_thesis: verum
end;
end;
end;
then A102: s2 is strictly_decreasing by Def30;
( (dom s2) \ {(i + 1)} c= dom s2 & s1 = Del (s2,(i + 1)) ) by A13, Lm44, XBOOLE_1:36;
then s2 is_finer_than s1 by Def29;
hence contradiction by A3, A23, A102, Def31; ::_thesis: verum
end;
end;
end;
now__::_thesis:_(_(_for_i_being_Nat_st_i_in_dom_(the_series_of_quotients_of_s1)_holds_
(the_series_of_quotients_of_s1)_._i_is_strict_simple_GroupWithOperators_of_O_)_implies_s1_is_jordan_holder_)
assume A103: for i being Nat st i in dom (the_series_of_quotients_of s1) holds
(the_series_of_quotients_of s1) . i is strict simple GroupWithOperators of O ; ::_thesis: s1 is jordan_holder
assume A104: not s1 is jordan_holder ; ::_thesis: contradiction
percases ( not s1 is strictly_decreasing or ex s2 being CompositionSeries of G st
( s2 <> s1 & s2 is strictly_decreasing & s2 is_finer_than s1 ) ) by A104, Def31;
suppose not s1 is strictly_decreasing ; ::_thesis: contradiction
then ex i being Nat st
( i in dom s1 & i + 1 in dom s1 & ex H1 being StableSubgroup of G ex N1 being normal StableSubgroup of H1 st
( H1 = s1 . i & N1 = s1 . (i + 1) & H1 ./. N1 is trivial ) ) by Def30;
then consider i being Nat, H1 being StableSubgroup of G, N1 being normal StableSubgroup of H1 such that
A105: i in dom s1 and
A106: i + 1 in dom s1 and
A107: ( H1 = s1 . i & N1 = s1 . (i + 1) ) and
A108: H1 ./. N1 is trivial ;
i + 1 in Seg (len s1) by A106, FINSEQ_1:def_3;
then A109: i + 1 <= len s1 by FINSEQ_1:1;
i in Seg (len s1) by A105, FINSEQ_1:def_3;
then A110: 1 <= i by FINSEQ_1:1;
then 1 + 1 <= i + 1 by XREAL_1:6;
then 1 + 1 <= len s1 by A109, XXREAL_0:2;
then A111: len s1 > 1 by NAT_1:13;
then (len (the_series_of_quotients_of s1)) + 1 = len s1 by Def33;
then len (the_series_of_quotients_of s1) = (len s1) - 1 ;
then (i + 1) - 1 <= len (the_series_of_quotients_of s1) by A109, XREAL_1:9;
then i in Seg (len (the_series_of_quotients_of s1)) by A110, FINSEQ_1:1;
then A112: i in dom (the_series_of_quotients_of s1) by FINSEQ_1:def_3;
then (the_series_of_quotients_of s1) . i = H1 ./. N1 by A107, A111, Def33;
then H1 ./. N1 is strict simple GroupWithOperators of O by A103, A112;
hence contradiction by A108, Def13; ::_thesis: verum
end;
suppose ex s2 being CompositionSeries of G st
( s2 <> s1 & s2 is strictly_decreasing & s2 is_finer_than s1 ) ; ::_thesis: contradiction
then consider s2 being CompositionSeries of G such that
A113: s2 <> s1 and
A114: s2 is strictly_decreasing and
A115: s2 is_finer_than s1 ;
consider i, j being Nat such that
A116: i in dom s1 and
A117: i in dom s2 and
A118: i + 1 in dom s1 and
A119: i + 1 in dom s2 and
A120: ( j in dom s2 & i + 1 < j ) and
A121: s1 . i = s2 . i and
A122: s1 . (i + 1) <> s2 . (i + 1) and
A123: s1 . (i + 1) = s2 . j by A1, A113, A114, A115, Th100;
reconsider H1 = s1 . i, H2 = s1 . (i + 1), H = s2 . (i + 1) as Element of the_stable_subgroups_of G by A116, A118, A119, FINSEQ_2:11;
reconsider H1 = H1, H2 = H2, H = H as strict StableSubgroup of G by Def11;
reconsider H2 = H2 as strict normal StableSubgroup of H1 by A116, A118, Def28;
reconsider H = H as strict normal StableSubgroup of H1 by A117, A119, A121, Def28;
reconsider H29 = H2 as normal StableSubgroup of H by A119, A120, A123, Th40, Th101;
reconsider J = H ./. H29 as strict normal StableSubgroup of H1 ./. H2 by Th44;
A124: now__::_thesis:_not_J_=_(Omega)._(H1_./._H2)
assume J = (Omega). (H1 ./. H2) ; ::_thesis: contradiction
then A125: the carrier of H = union (Cosets H2) by Th22;
then A126: H = H1 by Lm5, Th22;
then reconsider H1 = H1 as strict normal StableSubgroup of H ;
H1 = (Omega). H by A125, Lm5, Th22;
then H ./. H1 is trivial by Th57;
hence contradiction by A114, A117, A119, A121, A126, Def30; ::_thesis: verum
end;
reconsider H3 = HGrWOpStr(# the carrier of H2, the multF of H2, the action of H2 #) as strict normal StableSubgroup of H by A119, A120, A123, Th40, Th101;
now__::_thesis:_not_J_=_(1)._(H1_./._H2)
assume J = (1). (H1 ./. H2) ; ::_thesis: contradiction
then union (Cosets H3) = union {(1_ (H1 ./. H2))} by Def8;
then the carrier of H = union {(1_ (H1 ./. H2))} by Th22;
then the carrier of H = 1_ (H1 ./. H2) by ZFMISC_1:25;
then the carrier of H = carr H2 by Th43;
hence contradiction by A122, Lm5; ::_thesis: verum
end;
then A127: H1 ./. H2 is not simple GroupWithOperators of O by A124, Def13;
i + 1 in Seg (len s1) by A118, FINSEQ_1:def_3;
then A128: i + 1 <= len s1 by FINSEQ_1:1;
i in Seg (len s1) by A116, FINSEQ_1:def_3;
then A129: 1 <= i by FINSEQ_1:1;
then 1 + 1 <= i + 1 by XREAL_1:6;
then 1 + 1 <= len s1 by A128, XXREAL_0:2;
then A130: len s1 > 1 by NAT_1:13;
then (len (the_series_of_quotients_of s1)) + 1 = len s1 by Def33;
then len (the_series_of_quotients_of s1) = (len s1) - 1 ;
then (i + 1) - 1 <= len (the_series_of_quotients_of s1) by A128, XREAL_1:9;
then i in Seg (len (the_series_of_quotients_of s1)) by A129, FINSEQ_1:1;
then A131: i in dom (the_series_of_quotients_of s1) by FINSEQ_1:def_3;
then (the_series_of_quotients_of s1) . i = H1 ./. H2 by A130, Def33;
hence contradiction by A103, A127, A131; ::_thesis: verum
end;
end;
end;
hence ( s1 is jordan_holder iff for i being Nat st i in dom (the_series_of_quotients_of s1) holds
(the_series_of_quotients_of s1) . i is strict simple GroupWithOperators of O ) by A2; ::_thesis: verum
end;
theorem Th111: :: GROUP_9:111
for O being set
for G being GroupWithOperators of O
for s1 being CompositionSeries of G
for i being Nat st 1 <= i & i <= (len s1) - 1 holds
( s1 . i is strict StableSubgroup of G & s1 . (i + 1) is strict StableSubgroup of G )
proof
let O be set ; ::_thesis: for G being GroupWithOperators of O
for s1 being CompositionSeries of G
for i being Nat st 1 <= i & i <= (len s1) - 1 holds
( s1 . i is strict StableSubgroup of G & s1 . (i + 1) is strict StableSubgroup of G )
let G be GroupWithOperators of O; ::_thesis: for s1 being CompositionSeries of G
for i being Nat st 1 <= i & i <= (len s1) - 1 holds
( s1 . i is strict StableSubgroup of G & s1 . (i + 1) is strict StableSubgroup of G )
let s1 be CompositionSeries of G; ::_thesis: for i being Nat st 1 <= i & i <= (len s1) - 1 holds
( s1 . i is strict StableSubgroup of G & s1 . (i + 1) is strict StableSubgroup of G )
let i be Nat; ::_thesis: ( 1 <= i & i <= (len s1) - 1 implies ( s1 . i is strict StableSubgroup of G & s1 . (i + 1) is strict StableSubgroup of G ) )
assume that
A1: 1 <= i and
A2: i <= (len s1) - 1 ; ::_thesis: ( s1 . i is strict StableSubgroup of G & s1 . (i + 1) is strict StableSubgroup of G )
A3: 0 + i <= 1 + i by XREAL_1:6;
A4: i + 1 <= ((len s1) - 1) + 1 by A2, XREAL_1:6;
then i <= len s1 by A3, XXREAL_0:2;
then i in Seg (len s1) by A1, FINSEQ_1:1;
then i in dom s1 by FINSEQ_1:def_3;
then s1 . i is Element of the_stable_subgroups_of G by FINSEQ_2:11;
hence s1 . i is strict StableSubgroup of G by Def11; ::_thesis: s1 . (i + 1) is strict StableSubgroup of G
1 <= i + 1 by A1, A3, XXREAL_0:2;
then i + 1 in Seg (len s1) by A4;
then i + 1 in dom s1 by FINSEQ_1:def_3;
then s1 . (i + 1) is Element of the_stable_subgroups_of G by FINSEQ_2:11;
hence s1 . (i + 1) is strict StableSubgroup of G by Def11; ::_thesis: verum
end;
theorem Th112: :: GROUP_9:112
for O being set
for G being GroupWithOperators of O
for H1, H2 being StableSubgroup of G
for s1 being CompositionSeries of G
for i being Nat st 1 <= i & i <= (len s1) - 1 & H1 = s1 . i & H2 = s1 . (i + 1) holds
H2 is normal StableSubgroup of H1
proof
let O be set ; ::_thesis: for G being GroupWithOperators of O
for H1, H2 being StableSubgroup of G
for s1 being CompositionSeries of G
for i being Nat st 1 <= i & i <= (len s1) - 1 & H1 = s1 . i & H2 = s1 . (i + 1) holds
H2 is normal StableSubgroup of H1
let G be GroupWithOperators of O; ::_thesis: for H1, H2 being StableSubgroup of G
for s1 being CompositionSeries of G
for i being Nat st 1 <= i & i <= (len s1) - 1 & H1 = s1 . i & H2 = s1 . (i + 1) holds
H2 is normal StableSubgroup of H1
let H1, H2 be StableSubgroup of G; ::_thesis: for s1 being CompositionSeries of G
for i being Nat st 1 <= i & i <= (len s1) - 1 & H1 = s1 . i & H2 = s1 . (i + 1) holds
H2 is normal StableSubgroup of H1
let s1 be CompositionSeries of G; ::_thesis: for i being Nat st 1 <= i & i <= (len s1) - 1 & H1 = s1 . i & H2 = s1 . (i + 1) holds
H2 is normal StableSubgroup of H1
let i be Nat; ::_thesis: ( 1 <= i & i <= (len s1) - 1 & H1 = s1 . i & H2 = s1 . (i + 1) implies H2 is normal StableSubgroup of H1 )
assume that
A1: 1 <= i and
A2: i <= (len s1) - 1 ; ::_thesis: ( not H1 = s1 . i or not H2 = s1 . (i + 1) or H2 is normal StableSubgroup of H1 )
A3: i + 1 <= ((len s1) - 1) + 1 by A2, XREAL_1:6;
A4: 0 + i <= 1 + i by XREAL_1:6;
then 1 <= i + 1 by A1, XXREAL_0:2;
then i + 1 in Seg (len s1) by A3;
then A5: i + 1 in dom s1 by FINSEQ_1:def_3;
i <= len s1 by A4, A3, XXREAL_0:2;
then i in Seg (len s1) by A1, FINSEQ_1:1;
then A6: i in dom s1 by FINSEQ_1:def_3;
assume ( H1 = s1 . i & H2 = s1 . (i + 1) ) ; ::_thesis: H2 is normal StableSubgroup of H1
hence H2 is normal StableSubgroup of H1 by A5, A6, Def28; ::_thesis: verum
end;
theorem Th113: :: GROUP_9:113
for O being set
for G being GroupWithOperators of O
for s1 being CompositionSeries of G holds s1 is_equivalent_with s1
proof
let O be set ; ::_thesis: for G being GroupWithOperators of O
for s1 being CompositionSeries of G holds s1 is_equivalent_with s1
let G be GroupWithOperators of O; ::_thesis: for s1 being CompositionSeries of G holds s1 is_equivalent_with s1
let s1 be CompositionSeries of G; ::_thesis: s1 is_equivalent_with s1
percases ( s1 is empty or not s1 is empty ) ;
suppose s1 is empty ; ::_thesis: s1 is_equivalent_with s1
hence s1 is_equivalent_with s1 by Th107; ::_thesis: verum
end;
supposeA1: not s1 is empty ; ::_thesis: s1 is_equivalent_with s1
set f1 = the_series_of_quotients_of s1;
now__::_thesis:_ex_p_being_Permutation_of_(dom_(the_series_of_quotients_of_s1))_st_the_series_of_quotients_of_s1,_the_series_of_quotients_of_s1_are_equivalent_under_p,O
set p = id (dom (the_series_of_quotients_of s1));
reconsider p = id (dom (the_series_of_quotients_of s1)) as Function of (dom (the_series_of_quotients_of s1)),(dom (the_series_of_quotients_of s1)) ;
rng p = dom (the_series_of_quotients_of s1) by RELAT_1:45;
then p is onto by FUNCT_2:def_3;
then reconsider p = p as Permutation of (dom (the_series_of_quotients_of s1)) ;
take p = p; ::_thesis: the_series_of_quotients_of s1, the_series_of_quotients_of s1 are_equivalent_under p,O
A2: now__::_thesis:_for_H1,_H2_being_GroupWithOperators_of_O
for_i,_j_being_Nat_st_i_in_dom_(the_series_of_quotients_of_s1)_&_j_=_(p_")_._i_&_H1_=_(the_series_of_quotients_of_s1)_._i_&_H2_=_(the_series_of_quotients_of_s1)_._j_holds_
H1,H2_are_isomorphic
let H1, H2 be GroupWithOperators of O; ::_thesis: for i, j being Nat st i in dom (the_series_of_quotients_of s1) & j = (p ") . i & H1 = (the_series_of_quotients_of s1) . i & H2 = (the_series_of_quotients_of s1) . j holds
H1,H2 are_isomorphic
let i, j be Nat; ::_thesis: ( i in dom (the_series_of_quotients_of s1) & j = (p ") . i & H1 = (the_series_of_quotients_of s1) . i & H2 = (the_series_of_quotients_of s1) . j implies H1,H2 are_isomorphic )
assume A3: ( i in dom (the_series_of_quotients_of s1) & j = (p ") . i ) ; ::_thesis: ( H1 = (the_series_of_quotients_of s1) . i & H2 = (the_series_of_quotients_of s1) . j implies H1,H2 are_isomorphic )
A4: p " = p by FUNCT_1:45;
assume ( H1 = (the_series_of_quotients_of s1) . i & H2 = (the_series_of_quotients_of s1) . j ) ; ::_thesis: H1,H2 are_isomorphic
hence H1,H2 are_isomorphic by A3, A4, FUNCT_1:18; ::_thesis: verum
end;
len (the_series_of_quotients_of s1) = len (the_series_of_quotients_of s1) ;
hence the_series_of_quotients_of s1, the_series_of_quotients_of s1 are_equivalent_under p,O by A2, Def34; ::_thesis: verum
end;
hence s1 is_equivalent_with s1 by A1, Th108; ::_thesis: verum
end;
end;
end;
theorem Th114: :: GROUP_9:114
for O being set
for G being GroupWithOperators of O
for s1, s2 being CompositionSeries of G st ( len s1 <= 1 or len s2 <= 1 ) & len s1 <= len s2 holds
s2 is_finer_than s1
proof
let O be set ; ::_thesis: for G being GroupWithOperators of O
for s1, s2 being CompositionSeries of G st ( len s1 <= 1 or len s2 <= 1 ) & len s1 <= len s2 holds
s2 is_finer_than s1
let G be GroupWithOperators of O; ::_thesis: for s1, s2 being CompositionSeries of G st ( len s1 <= 1 or len s2 <= 1 ) & len s1 <= len s2 holds
s2 is_finer_than s1
let s1, s2 be CompositionSeries of G; ::_thesis: ( ( len s1 <= 1 or len s2 <= 1 ) & len s1 <= len s2 implies s2 is_finer_than s1 )
assume A1: ( len s1 <= 1 or len s2 <= 1 ) ; ::_thesis: ( not len s1 <= len s2 or s2 is_finer_than s1 )
assume A2: len s1 <= len s2 ; ::_thesis: s2 is_finer_than s1
then A3: len s1 <= 1 by A1, XXREAL_0:2;
percases ( len s1 = 1 or len s1 <> 1 ) ;
supposeA4: len s1 = 1 ; ::_thesis: s2 is_finer_than s1
then A5: s1 = <*(s1 . 1)*> by FINSEQ_1:40;
now__::_thesis:_ex_x_being_set_st_
(_x_c=_dom_s2_&_s1_=_s2_*_(Sgm_x)_)
reconsider D = Seg (len s2) as non empty set by A2, A4;
set x = {1};
take x = {1}; ::_thesis: ( x c= dom s2 & s1 = s2 * (Sgm x) )
set f = s2;
set p = <*1*>;
( dom s2 = Seg (len s2) & rng s2 c= the_stable_subgroups_of G ) by FINSEQ_1:def_3;
then reconsider f = s2 as Function of D,(the_stable_subgroups_of G) by FUNCT_2:2;
A6: 1 in Seg (len s2) by A2, A4;
then 1 in dom s2 by FINSEQ_1:def_3;
hence x c= dom s2 by ZFMISC_1:31; ::_thesis: s1 = s2 * (Sgm x)
{1} c= D by A6, ZFMISC_1:31;
then rng <*1*> c= D by FINSEQ_1:38;
then reconsider p = <*1*> as FinSequence of D by FINSEQ_1:def_4;
( Sgm x = p & f * p = <*(f . 1)*> ) by FINSEQ_2:35, FINSEQ_3:44;
then s2 * (Sgm x) = <*((Omega). G)*> by Def28;
hence s1 = s2 * (Sgm x) by A5, Def28; ::_thesis: verum
end;
hence s2 is_finer_than s1 by Def29; ::_thesis: verum
end;
suppose len s1 <> 1 ; ::_thesis: s2 is_finer_than s1
then len s1 < 0 + 1 by A3, XXREAL_0:1;
then A7: s1 = {} by NAT_1:13;
now__::_thesis:_ex_x_being_set_st_
(_x_c=_dom_s2_&_s1_=_s2_*_(Sgm_x)_)
set x = {} ;
take x = {} ; ::_thesis: ( x c= dom s2 & s1 = s2 * (Sgm x) )
thus x c= dom s2 by XBOOLE_1:2; ::_thesis: s1 = s2 * (Sgm x)
thus s1 = s2 * (Sgm x) by A7, FINSEQ_3:43; ::_thesis: verum
end;
hence s2 is_finer_than s1 by Def29; ::_thesis: verum
end;
end;
end;
theorem Th115: :: GROUP_9:115
for O being set
for G being GroupWithOperators of O
for s1, s2 being CompositionSeries of G st s1 is_equivalent_with s2 & s1 is jordan_holder holds
s2 is jordan_holder
proof
let O be set ; ::_thesis: for G being GroupWithOperators of O
for s1, s2 being CompositionSeries of G st s1 is_equivalent_with s2 & s1 is jordan_holder holds
s2 is jordan_holder
let G be GroupWithOperators of O; ::_thesis: for s1, s2 being CompositionSeries of G st s1 is_equivalent_with s2 & s1 is jordan_holder holds
s2 is jordan_holder
let s1, s2 be CompositionSeries of G; ::_thesis: ( s1 is_equivalent_with s2 & s1 is jordan_holder implies s2 is jordan_holder )
assume A1: s1 is_equivalent_with s2 ; ::_thesis: ( not s1 is jordan_holder or s2 is jordan_holder )
assume A2: s1 is jordan_holder ; ::_thesis: s2 is jordan_holder
percases ( len s1 <= 0 + 1 or len s1 > 1 ) ;
supposeA3: len s1 <= 0 + 1 ; ::_thesis: s2 is jordan_holder
percases ( len s1 = 0 or len s1 = 1 ) by A3, NAT_1:25;
supposeA4: len s1 = 0 ; ::_thesis: s2 is jordan_holder
then len s2 = 0 by A1, Def32;
then A5: s2 = {} ;
s1 = {} by A4;
hence s2 is jordan_holder by A2, A5; ::_thesis: verum
end;
supposeA6: len s1 = 1 ; ::_thesis: s2 is jordan_holder
then A7: s1 . 1 = (1). G by Def28;
A8: len s2 = 1 by A1, A6, Def32;
s1 = <*(s1 . 1)*> by A6, FINSEQ_1:40
.= <*(s2 . 1)*> by A7, A8, Def28
.= s2 by A8, FINSEQ_1:40 ;
hence s2 is jordan_holder by A2; ::_thesis: verum
end;
end;
end;
supposeA9: len s1 > 1 ; ::_thesis: s2 is jordan_holder
set f2 = the_series_of_quotients_of s2;
set f1 = the_series_of_quotients_of s1;
A10: not s1 is empty by A9;
A11: len s2 > 1 by A1, A9, Def32;
then not s2 is empty ;
then consider p being Permutation of (dom (the_series_of_quotients_of s1)) such that
A12: the_series_of_quotients_of s1, the_series_of_quotients_of s2 are_equivalent_under p,O by A1, A10, Th108;
A13: len (the_series_of_quotients_of s1) = len (the_series_of_quotients_of s2) by A12, Def34;
now__::_thesis:_for_j_being_Nat_st_j_in_dom_(the_series_of_quotients_of_s2)_holds_
(the_series_of_quotients_of_s2)_._j_is_strict_simple_GroupWithOperators_of_O
let j be Nat; ::_thesis: ( j in dom (the_series_of_quotients_of s2) implies (the_series_of_quotients_of s2) . j is strict simple GroupWithOperators of O )
set i = p . j;
set H1 = (the_series_of_quotients_of s1) . (p . j);
set H2 = (the_series_of_quotients_of s2) . j;
assume A14: j in dom (the_series_of_quotients_of s2) ; ::_thesis: (the_series_of_quotients_of s2) . j is strict simple GroupWithOperators of O
then A15: (the_series_of_quotients_of s2) . j in rng (the_series_of_quotients_of s2) by FUNCT_1:3;
A16: dom (the_series_of_quotients_of s1) = Seg (len (the_series_of_quotients_of s1)) by FINSEQ_1:def_3
.= Seg (len (the_series_of_quotients_of s2)) by A12, Def34
.= dom (the_series_of_quotients_of s2) by FINSEQ_1:def_3 ;
then A17: p . j in dom (the_series_of_quotients_of s2) by A14, FUNCT_2:5;
then reconsider i = p . j as Element of NAT ;
p . j in Seg (len (the_series_of_quotients_of s2)) by A17, FINSEQ_1:def_3;
then A18: i in dom (the_series_of_quotients_of s1) by A13, FINSEQ_1:def_3;
then (the_series_of_quotients_of s1) . i in rng (the_series_of_quotients_of s1) by FUNCT_1:3;
then reconsider H1 = (the_series_of_quotients_of s1) . (p . j), H2 = (the_series_of_quotients_of s2) . j as strict GroupWithOperators of O by A15, Th102;
( i in dom (the_series_of_quotients_of s1) & j = (p ") . i ) by A14, A16, FUNCT_2:5, FUNCT_2:26;
then A19: H1,H2 are_isomorphic by A12, Def34;
H1 is strict simple GroupWithOperators of O by A2, A9, A18, Th110;
hence (the_series_of_quotients_of s2) . j is strict simple GroupWithOperators of O by A19, Th82; ::_thesis: verum
end;
hence s2 is jordan_holder by A11, Th110; ::_thesis: verum
end;
end;
end;
Lm45: for O being set
for G being GroupWithOperators of O
for s1, s2 being CompositionSeries of G
for k, l being Nat st k in Seg l & len s1 > 1 & len s2 > 1 & l = (((len s1) - 1) * ((len s2) - 1)) + 1 & not k = (((len s1) - 1) * ((len s2) - 1)) + 1 holds
ex i, j being Nat st
( k = ((i - 1) * ((len s2) - 1)) + j & 1 <= i & i <= (len s1) - 1 & 1 <= j & j <= (len s2) - 1 )
proof
let O be set ; ::_thesis: for G being GroupWithOperators of O
for s1, s2 being CompositionSeries of G
for k, l being Nat st k in Seg l & len s1 > 1 & len s2 > 1 & l = (((len s1) - 1) * ((len s2) - 1)) + 1 & not k = (((len s1) - 1) * ((len s2) - 1)) + 1 holds
ex i, j being Nat st
( k = ((i - 1) * ((len s2) - 1)) + j & 1 <= i & i <= (len s1) - 1 & 1 <= j & j <= (len s2) - 1 )
let G be GroupWithOperators of O; ::_thesis: for s1, s2 being CompositionSeries of G
for k, l being Nat st k in Seg l & len s1 > 1 & len s2 > 1 & l = (((len s1) - 1) * ((len s2) - 1)) + 1 & not k = (((len s1) - 1) * ((len s2) - 1)) + 1 holds
ex i, j being Nat st
( k = ((i - 1) * ((len s2) - 1)) + j & 1 <= i & i <= (len s1) - 1 & 1 <= j & j <= (len s2) - 1 )
let s1, s2 be CompositionSeries of G; ::_thesis: for k, l being Nat st k in Seg l & len s1 > 1 & len s2 > 1 & l = (((len s1) - 1) * ((len s2) - 1)) + 1 & not k = (((len s1) - 1) * ((len s2) - 1)) + 1 holds
ex i, j being Nat st
( k = ((i - 1) * ((len s2) - 1)) + j & 1 <= i & i <= (len s1) - 1 & 1 <= j & j <= (len s2) - 1 )
let k, l be Nat; ::_thesis: ( k in Seg l & len s1 > 1 & len s2 > 1 & l = (((len s1) - 1) * ((len s2) - 1)) + 1 & not k = (((len s1) - 1) * ((len s2) - 1)) + 1 implies ex i, j being Nat st
( k = ((i - 1) * ((len s2) - 1)) + j & 1 <= i & i <= (len s1) - 1 & 1 <= j & j <= (len s2) - 1 ) )
set l9 = (len s1) - 1;
set l99 = (len s2) - 1;
assume A1: k in Seg l ; ::_thesis: ( not len s1 > 1 or not len s2 > 1 or not l = (((len s1) - 1) * ((len s2) - 1)) + 1 or k = (((len s1) - 1) * ((len s2) - 1)) + 1 or ex i, j being Nat st
( k = ((i - 1) * ((len s2) - 1)) + j & 1 <= i & i <= (len s1) - 1 & 1 <= j & j <= (len s2) - 1 ) )
then A2: k <= l by FINSEQ_1:1;
assume that
A3: len s1 > 1 and
A4: len s2 > 1 and
A5: l = (((len s1) - 1) * ((len s2) - 1)) + 1 ; ::_thesis: ( k = (((len s1) - 1) * ((len s2) - 1)) + 1 or ex i, j being Nat st
( k = ((i - 1) * ((len s2) - 1)) + j & 1 <= i & i <= (len s1) - 1 & 1 <= j & j <= (len s2) - 1 ) )
assume not k = (((len s1) - 1) * ((len s2) - 1)) + 1 ; ::_thesis: ex i, j being Nat st
( k = ((i - 1) * ((len s2) - 1)) + j & 1 <= i & i <= (len s1) - 1 & 1 <= j & j <= (len s2) - 1 )
then A6: k < l by A2, A5, XXREAL_0:1;
(len s2) + 1 > 1 + 1 by A4, XREAL_1:6;
then len s2 >= 2 by NAT_1:13;
then A7: (len s2) - 1 >= 2 - 1 by XREAL_1:9;
(len s1) - 1 > 1 - 1 by A3, XREAL_1:9;
then reconsider l9 = (len s1) - 1 as Element of NAT by INT_1:3;
A8: (len s2) - 1 > 1 - 1 by A4, XREAL_1:9;
then reconsider l99 = (len s2) - 1 as Element of NAT by INT_1:3;
A9: k = ((k div l99) * l99) + (k mod l99) by A8, NAT_D:2;
percases ( k mod l99 = 0 or k mod l99 <> 0 ) ;
supposeA10: k mod l99 = 0 ; ::_thesis: ex i, j being Nat st
( k = ((i - 1) * ((len s2) - 1)) + j & 1 <= i & i <= (len s1) - 1 & 1 <= j & j <= (len s2) - 1 )
set i = k div l99;
set j = l99;
take k div l99 ; ::_thesis: ex j being Nat st
( k = (((k div l99) - 1) * ((len s2) - 1)) + j & 1 <= k div l99 & k div l99 <= (len s1) - 1 & 1 <= j & j <= (len s2) - 1 )
take l99 ; ::_thesis: ( k = (((k div l99) - 1) * ((len s2) - 1)) + l99 & 1 <= k div l99 & k div l99 <= (len s1) - 1 & 1 <= l99 & l99 <= (len s2) - 1 )
thus k = (((k div l99) - 1) * ((len s2) - 1)) + l99 by A9, A10; ::_thesis: ( 1 <= k div l99 & k div l99 <= (len s1) - 1 & 1 <= l99 & l99 <= (len s2) - 1 )
k div l99 > 0 by A1, A9, A10, FINSEQ_1:1;
then (k div l99) + 1 > 0 + 1 by XREAL_1:6;
hence 1 <= k div l99 by NAT_1:13; ::_thesis: ( k div l99 <= (len s1) - 1 & 1 <= l99 & l99 <= (len s2) - 1 )
(k div l99) * l99 <= ((len s1) - 1) * l99 by A5, A6, A9, A10, INT_1:7;
then ((k div l99) * l99) / l99 <= (((len s1) - 1) * l99) / l99 by XREAL_1:72;
then k div l99 <= (((len s1) - 1) * l99) / l99 by A8, XCMPLX_1:89;
hence k div l99 <= (len s1) - 1 by A8, XCMPLX_1:89; ::_thesis: ( 1 <= l99 & l99 <= (len s2) - 1 )
thus ( 1 <= l99 & l99 <= (len s2) - 1 ) by A7; ::_thesis: verum
end;
supposeA11: k mod l99 <> 0 ; ::_thesis: ex i, j being Nat st
( k = ((i - 1) * ((len s2) - 1)) + j & 1 <= i & i <= (len s1) - 1 & 1 <= j & j <= (len s2) - 1 )
set i = (k div l99) + 1;
set j = k mod l99;
take (k div l99) + 1 ; ::_thesis: ex j being Nat st
( k = ((((k div l99) + 1) - 1) * ((len s2) - 1)) + j & 1 <= (k div l99) + 1 & (k div l99) + 1 <= (len s1) - 1 & 1 <= j & j <= (len s2) - 1 )
take k mod l99 ; ::_thesis: ( k = ((((k div l99) + 1) - 1) * ((len s2) - 1)) + (k mod l99) & 1 <= (k div l99) + 1 & (k div l99) + 1 <= (len s1) - 1 & 1 <= k mod l99 & k mod l99 <= (len s2) - 1 )
thus k = ((((k div l99) + 1) - 1) * ((len s2) - 1)) + (k mod l99) by A8, NAT_D:2; ::_thesis: ( 1 <= (k div l99) + 1 & (k div l99) + 1 <= (len s1) - 1 & 1 <= k mod l99 & k mod l99 <= (len s2) - 1 )
0 + 1 <= (k div l99) + 1 by XREAL_1:6;
hence 1 <= (k div l99) + 1 ; ::_thesis: ( (k div l99) + 1 <= (len s1) - 1 & 1 <= k mod l99 & k mod l99 <= (len s2) - 1 )
k + 1 <= l by A6, INT_1:7;
then A12: (k + 1) - 1 <= l - 1 by XREAL_1:9;
(k mod l99) + (l99 * (k div l99)) >= 0 + (l99 * (k div l99)) by XREAL_1:6;
then A13: (k div l99) * l99 <= k by A8, NAT_D:2;
k <> l9 * l99 by A11, NAT_D:13;
then k < ((len s1) - 1) * l99 by A5, A12, XXREAL_0:1;
then (k div l99) * l99 < ((len s1) - 1) * l99 by A13, XXREAL_0:2;
then ((k div l99) * l99) / l99 < (((len s1) - 1) * l99) / l99 by A8, XREAL_1:74;
then k div l99 < (((len s1) - 1) * l99) / l99 by A8, XCMPLX_1:89;
then k div l99 < (len s1) - 1 by A8, XCMPLX_1:89;
hence (k div l99) + 1 <= (len s1) - 1 by INT_1:7; ::_thesis: ( 1 <= k mod l99 & k mod l99 <= (len s2) - 1 )
(k mod l99) + 1 > 0 + 1 by A11, XREAL_1:6;
hence 1 <= k mod l99 by NAT_1:13; ::_thesis: k mod l99 <= (len s2) - 1
thus k mod l99 <= (len s2) - 1 by A8, NAT_D:1; ::_thesis: verum
end;
end;
end;
Lm46: for O being set
for G being GroupWithOperators of O
for i1, j1, i2, j2 being Nat
for s1, s2 being CompositionSeries of G st len s2 > 1 & ((i1 - 1) * ((len s2) - 1)) + j1 = ((i2 - 1) * ((len s2) - 1)) + j2 & 1 <= i1 & 1 <= j1 & j1 <= (len s2) - 1 & 1 <= i2 & 1 <= j2 & j2 <= (len s2) - 1 holds
( j1 = j2 & i1 = i2 )
proof
let O be set ; ::_thesis: for G being GroupWithOperators of O
for i1, j1, i2, j2 being Nat
for s1, s2 being CompositionSeries of G st len s2 > 1 & ((i1 - 1) * ((len s2) - 1)) + j1 = ((i2 - 1) * ((len s2) - 1)) + j2 & 1 <= i1 & 1 <= j1 & j1 <= (len s2) - 1 & 1 <= i2 & 1 <= j2 & j2 <= (len s2) - 1 holds
( j1 = j2 & i1 = i2 )
let G be GroupWithOperators of O; ::_thesis: for i1, j1, i2, j2 being Nat
for s1, s2 being CompositionSeries of G st len s2 > 1 & ((i1 - 1) * ((len s2) - 1)) + j1 = ((i2 - 1) * ((len s2) - 1)) + j2 & 1 <= i1 & 1 <= j1 & j1 <= (len s2) - 1 & 1 <= i2 & 1 <= j2 & j2 <= (len s2) - 1 holds
( j1 = j2 & i1 = i2 )
let i1, j1, i2, j2 be Nat; ::_thesis: for s1, s2 being CompositionSeries of G st len s2 > 1 & ((i1 - 1) * ((len s2) - 1)) + j1 = ((i2 - 1) * ((len s2) - 1)) + j2 & 1 <= i1 & 1 <= j1 & j1 <= (len s2) - 1 & 1 <= i2 & 1 <= j2 & j2 <= (len s2) - 1 holds
( j1 = j2 & i1 = i2 )
let s1, s2 be CompositionSeries of G; ::_thesis: ( len s2 > 1 & ((i1 - 1) * ((len s2) - 1)) + j1 = ((i2 - 1) * ((len s2) - 1)) + j2 & 1 <= i1 & 1 <= j1 & j1 <= (len s2) - 1 & 1 <= i2 & 1 <= j2 & j2 <= (len s2) - 1 implies ( j1 = j2 & i1 = i2 ) )
set l99 = (len s2) - 1;
set i19 = i1 - 1;
set i29 = i2 - 1;
assume len s2 > 1 ; ::_thesis: ( not ((i1 - 1) * ((len s2) - 1)) + j1 = ((i2 - 1) * ((len s2) - 1)) + j2 or not 1 <= i1 or not 1 <= j1 or not j1 <= (len s2) - 1 or not 1 <= i2 or not 1 <= j2 or not j2 <= (len s2) - 1 or ( j1 = j2 & i1 = i2 ) )
then A1: (len s2) - 1 > 1 - 1 by XREAL_1:9;
then reconsider l99 = (len s2) - 1 as Element of NAT by INT_1:3;
A2: l99 / l99 = 1 by A1, XCMPLX_1:60;
assume A3: ((i1 - 1) * ((len s2) - 1)) + j1 = ((i2 - 1) * ((len s2) - 1)) + j2 ; ::_thesis: ( not 1 <= i1 or not 1 <= j1 or not j1 <= (len s2) - 1 or not 1 <= i2 or not 1 <= j2 or not j2 <= (len s2) - 1 or ( j1 = j2 & i1 = i2 ) )
assume that
A4: 1 <= i1 and
A5: 1 <= j1 and
A6: j1 <= (len s2) - 1 ; ::_thesis: ( not 1 <= i2 or not 1 <= j2 or not j2 <= (len s2) - 1 or ( j1 = j2 & i1 = i2 ) )
i1 - 1 >= 1 - 1 by A4, XREAL_1:9;
then reconsider i19 = i1 - 1 as Element of NAT by INT_1:3;
assume that
A7: 1 <= i2 and
A8: 1 <= j2 and
A9: j2 <= (len s2) - 1 ; ::_thesis: ( j1 = j2 & i1 = i2 )
i2 - 1 >= 1 - 1 by A7, XREAL_1:9;
then reconsider i29 = i2 - 1 as Element of NAT by INT_1:3;
A10: j1 mod l99 = ((i19 * l99) + j1) mod l99 by NAT_D:21
.= ((i29 * l99) + j2) mod l99 by A3
.= j2 mod l99 by NAT_D:21 ;
A11: j1 = j2
proof
percases ( j1 = l99 or j1 <> l99 ) ;
supposeA12: j1 = l99 ; ::_thesis: j1 = j2
assume j2 <> j1 ; ::_thesis: contradiction
then j2 < l99 by A9, A12, XXREAL_0:1;
then j2 = j1 mod l99 by A10, NAT_D:24;
hence contradiction by A8, A12, NAT_D:25; ::_thesis: verum
end;
suppose j1 <> l99 ; ::_thesis: j1 = j2
then j1 < l99 by A6, XXREAL_0:1;
then A13: j1 = j2 mod l99 by A10, NAT_D:24;
percases ( j2 = l99 or j2 <> l99 ) ;
suppose j2 = l99 ; ::_thesis: j1 = j2
hence j1 = j2 by A5, A13, NAT_D:25; ::_thesis: verum
end;
suppose j2 <> l99 ; ::_thesis: j1 = j2
then j2 < l99 by A9, XXREAL_0:1;
hence j1 = j2 by A13, NAT_D:24; ::_thesis: verum
end;
end;
end;
end;
end;
hence j1 = j2 ; ::_thesis: i1 = i2
i19 * (l99 / l99) = (i29 * l99) / l99 by A3, A11, XCMPLX_1:74;
then i19 * 1 = i29 * 1 by A2, XCMPLX_1:74;
hence i1 = i2 ; ::_thesis: verum
end;
Lm47: for O being set
for G being GroupWithOperators of O
for k being integer number
for i, j being Nat
for s1, s2 being CompositionSeries of G st len s2 > 1 & k = ((i - 1) * ((len s2) - 1)) + j & 1 <= i & i <= (len s1) - 1 & 1 <= j & j <= (len s2) - 1 holds
( 1 <= k & k <= ((len s1) - 1) * ((len s2) - 1) )
proof
let O be set ; ::_thesis: for G being GroupWithOperators of O
for k being integer number
for i, j being Nat
for s1, s2 being CompositionSeries of G st len s2 > 1 & k = ((i - 1) * ((len s2) - 1)) + j & 1 <= i & i <= (len s1) - 1 & 1 <= j & j <= (len s2) - 1 holds
( 1 <= k & k <= ((len s1) - 1) * ((len s2) - 1) )
let G be GroupWithOperators of O; ::_thesis: for k being integer number
for i, j being Nat
for s1, s2 being CompositionSeries of G st len s2 > 1 & k = ((i - 1) * ((len s2) - 1)) + j & 1 <= i & i <= (len s1) - 1 & 1 <= j & j <= (len s2) - 1 holds
( 1 <= k & k <= ((len s1) - 1) * ((len s2) - 1) )
let k be integer number ; ::_thesis: for i, j being Nat
for s1, s2 being CompositionSeries of G st len s2 > 1 & k = ((i - 1) * ((len s2) - 1)) + j & 1 <= i & i <= (len s1) - 1 & 1 <= j & j <= (len s2) - 1 holds
( 1 <= k & k <= ((len s1) - 1) * ((len s2) - 1) )
let i, j be Nat; ::_thesis: for s1, s2 being CompositionSeries of G st len s2 > 1 & k = ((i - 1) * ((len s2) - 1)) + j & 1 <= i & i <= (len s1) - 1 & 1 <= j & j <= (len s2) - 1 holds
( 1 <= k & k <= ((len s1) - 1) * ((len s2) - 1) )
let s1, s2 be CompositionSeries of G; ::_thesis: ( len s2 > 1 & k = ((i - 1) * ((len s2) - 1)) + j & 1 <= i & i <= (len s1) - 1 & 1 <= j & j <= (len s2) - 1 implies ( 1 <= k & k <= ((len s1) - 1) * ((len s2) - 1) ) )
set l9 = (len s1) - 1;
set l99 = (len s2) - 1;
assume len s2 > 1 ; ::_thesis: ( not k = ((i - 1) * ((len s2) - 1)) + j or not 1 <= i or not i <= (len s1) - 1 or not 1 <= j or not j <= (len s2) - 1 or ( 1 <= k & k <= ((len s1) - 1) * ((len s2) - 1) ) )
then A1: (len s2) - 1 > 1 - 1 by XREAL_1:9;
assume A2: k = ((i - 1) * ((len s2) - 1)) + j ; ::_thesis: ( not 1 <= i or not i <= (len s1) - 1 or not 1 <= j or not j <= (len s2) - 1 or ( 1 <= k & k <= ((len s1) - 1) * ((len s2) - 1) ) )
assume that
A3: 1 <= i and
A4: i <= (len s1) - 1 ; ::_thesis: ( not 1 <= j or not j <= (len s2) - 1 or ( 1 <= k & k <= ((len s1) - 1) * ((len s2) - 1) ) )
assume that
A5: 1 <= j and
A6: j <= (len s2) - 1 ; ::_thesis: ( 1 <= k & k <= ((len s1) - 1) * ((len s2) - 1) )
i - 1 <= ((len s1) - 1) - 1 by A4, XREAL_1:9;
then (i - 1) * ((len s2) - 1) <= (((len s1) - 1) - 1) * ((len s2) - 1) by A1, XREAL_1:64;
then A7: k <= ((((len s1) - 1) * ((len s2) - 1)) - (1 * ((len s2) - 1))) + ((len s2) - 1) by A2, A6, XREAL_1:7;
1 - 1 <= i - 1 by A3, XREAL_1:9;
then 0 + 1 <= ((i - 1) * ((len s2) - 1)) + j by A5, A1, XREAL_1:7;
hence ( 1 <= k & k <= ((len s1) - 1) * ((len s2) - 1) ) by A2, A7; ::_thesis: verum
end;
begin
definition
let O be set ;
let G be GroupWithOperators of O;
let s1, s2 be CompositionSeries of G;
assume that
A1: len s1 > 1 and
A2: len s2 > 1 ;
func the_schreier_series_of (s1,s2) -> CompositionSeries of G means :Def35: :: GROUP_9:def 35
for k, i, j being Nat
for H1, H2, H3 being StableSubgroup of G holds
( ( k = ((i - 1) * ((len s2) - 1)) + j & 1 <= i & i <= (len s1) - 1 & 1 <= j & j <= (len s2) - 1 & H1 = s1 . (i + 1) & H2 = s1 . i & H3 = s2 . j implies it . k = H1 "\/" (H2 /\ H3) ) & ( k = (((len s1) - 1) * ((len s2) - 1)) + 1 implies it . k = (1). G ) & len it = (((len s1) - 1) * ((len s2) - 1)) + 1 );
existence
ex b1 being CompositionSeries of G st
for k, i, j being Nat
for H1, H2, H3 being StableSubgroup of G holds
( ( k = ((i - 1) * ((len s2) - 1)) + j & 1 <= i & i <= (len s1) - 1 & 1 <= j & j <= (len s2) - 1 & H1 = s1 . (i + 1) & H2 = s1 . i & H3 = s2 . j implies b1 . k = H1 "\/" (H2 /\ H3) ) & ( k = (((len s1) - 1) * ((len s2) - 1)) + 1 implies b1 . k = (1). G ) & len b1 = (((len s1) - 1) * ((len s2) - 1)) + 1 )
proof
(len s2) - 1 > 1 - 1 by A2, XREAL_1:9;
then reconsider l99 = (len s2) - 1 as Element of NAT by INT_1:3;
(len s2) + 1 > 1 + 1 by A2, XREAL_1:6;
then len s2 >= 2 by NAT_1:13;
then A3: (len s2) - 1 >= 2 - 1 by XREAL_1:9;
(len s1) - 1 > 1 - 1 by A1, XREAL_1:9;
then reconsider l9 = (len s1) - 1 as Element of NAT by INT_1:3;
defpred S1[ set , set ] means for i, j being Nat
for H1, H2, H3 being StableSubgroup of G holds
( ( $1 = ((i - 1) * ((len s2) - 1)) + j & 1 <= i & i <= (len s1) - 1 & 1 <= j & j <= (len s2) - 1 & H1 = s1 . (i + 1) & H2 = s1 . i & H3 = s2 . j implies $2 = H1 "\/" (H2 /\ H3) ) & ( $1 = (((len s1) - 1) * ((len s2) - 1)) + 1 implies $2 = (1). G ) );
(len s2) - 1 > 1 - 1 by A2, XREAL_1:9;
then A4: l99 / l99 = 1 by XCMPLX_1:60;
(len s1) + 1 > 1 + 1 by A1, XREAL_1:6;
then len s1 >= 2 by NAT_1:13;
then A5: (len s1) - 1 >= 2 - 1 by XREAL_1:9;
then A6: (((len s1) - 1) * ((len s2) - 1)) + 1 >= 0 + 1 by A3, XREAL_1:6;
reconsider l = (((len s1) - 1) * ((len s2) - 1)) + 1 as Element of NAT by A5, A3, INT_1:3;
A7: 1 in Seg l by A6;
A8: for k being Nat st k = (((len s1) - 1) * ((len s2) - 1)) + 1 holds
for i, j being Nat holds
( not k = ((i - 1) * ((len s2) - 1)) + j or not 1 <= i or not i <= (len s1) - 1 or not 1 <= j or not j <= (len s2) - 1 )
proof
let k be Nat; ::_thesis: ( k = (((len s1) - 1) * ((len s2) - 1)) + 1 implies for i, j being Nat holds
( not k = ((i - 1) * ((len s2) - 1)) + j or not 1 <= i or not i <= (len s1) - 1 or not 1 <= j or not j <= (len s2) - 1 ) )
assume A9: k = (((len s1) - 1) * ((len s2) - 1)) + 1 ; ::_thesis: for i, j being Nat holds
( not k = ((i - 1) * ((len s2) - 1)) + j or not 1 <= i or not i <= (len s1) - 1 or not 1 <= j or not j <= (len s2) - 1 )
assume ex i, j being Nat st
( k = ((i - 1) * ((len s2) - 1)) + j & 1 <= i & i <= (len s1) - 1 & 1 <= j & j <= (len s2) - 1 ) ; ::_thesis: contradiction
then consider i, j being Nat such that
A10: k = ((i - 1) * ((len s2) - 1)) + j and
A11: 1 <= i and
A12: i <= (len s1) - 1 and
A13: 1 <= j and
A14: j <= (len s2) - 1 ;
set i9 = i - 1;
i - 1 >= 1 - 1 by A11, XREAL_1:9;
then reconsider i9 = i - 1 as Element of NAT by INT_1:3;
A15: 1 mod l99 = ((l9 * l99) + 1) mod l99 by NAT_D:21
.= ((i9 * l99) + j) mod l99 by A9, A10
.= j mod l99 by NAT_D:21 ;
j = 1
proof
percases ( j = l99 or j <> l99 ) ;
supposeA16: j = l99 ; ::_thesis: j = 1
assume j <> 1 ; ::_thesis: contradiction
then 1 < l99 by A13, A16, XXREAL_0:1;
then 1 = j mod l99 by A15, NAT_D:24;
hence contradiction by A16, NAT_D:25; ::_thesis: verum
end;
suppose j <> l99 ; ::_thesis: j = 1
then j < l99 by A14, XXREAL_0:1;
then A17: j mod l99 = j by NAT_D:24;
percases ( 1 = l99 or 1 <> l99 ) ;
suppose 1 = l99 ; ::_thesis: j = 1
hence j = 1 by A13, A14, XXREAL_0:1; ::_thesis: verum
end;
supposeA18: 1 <> l99 ; ::_thesis: j = 1
1 <= l99 by A13, A14, XXREAL_0:2;
then 1 < l99 by A18, XXREAL_0:1;
hence j = 1 by A15, A17, NAT_D:24; ::_thesis: verum
end;
end;
end;
end;
end;
then A19: l9 * (l99 / l99) = (i9 * l99) / l99 by A9, A10, XCMPLX_1:74;
l99 / l99 = 1 by A13, A14, XCMPLX_1:60;
then A20: l9 * 1 = i9 * 1 by A19, XCMPLX_1:74;
(- 1) + i < 0 + i by XREAL_1:6;
hence contradiction by A12, A20; ::_thesis: verum
end;
A21: for k being Nat st k in Seg l holds
ex x being set st S1[k,x]
proof
let k be Nat; ::_thesis: ( k in Seg l implies ex x being set st S1[k,x] )
assume A22: k in Seg l ; ::_thesis: ex x being set st S1[k,x]
percases ( ex i, j being Nat st
( k = ((i - 1) * ((len s2) - 1)) + j & 1 <= i & i <= (len s1) - 1 & 1 <= j & j <= (len s2) - 1 ) or k = (((len s1) - 1) * ((len s2) - 1)) + 1 ) by A1, A2, A22, Lm45;
supposeA23: ex i, j being Nat st
( k = ((i - 1) * ((len s2) - 1)) + j & 1 <= i & i <= (len s1) - 1 & 1 <= j & j <= (len s2) - 1 ) ; ::_thesis: ex x being set st S1[k,x]
then consider i, j being Nat such that
A24: k = ((i - 1) * ((len s2) - 1)) + j and
A25: 1 <= i and
A26: i <= (len s1) - 1 and
A27: ( 1 <= j & j <= (len s2) - 1 ) ;
reconsider H1 = s1 . (i + 1), H2 = s1 . i, H3 = s2 . j as StableSubgroup of G by A25, A26, A27, Th111;
take x = H1 "\/" (H2 /\ H3); ::_thesis: S1[k,x]
now__::_thesis:_for_i1,_j1_being_Nat
for_H1,_H2,_H3_being_StableSubgroup_of_G_holds_
(_(_k_=_((i1_-_1)_*_((len_s2)_-_1))_+_j1_&_1_<=_i1_&_i1_<=_(len_s1)_-_1_&_1_<=_j1_&_j1_<=_(len_s2)_-_1_&_H1_=_s1_._(i1_+_1)_&_H2_=_s1_._i1_&_H3_=_s2_._j1_implies_x_=_H1_"\/"_(H2_/\_H3)_)_&_(_k_=_(((len_s1)_-_1)_*_((len_s2)_-_1))_+_1_implies_x_=_(1)._G_)_)
let i1, j1 be Nat; ::_thesis: for H1, H2, H3 being StableSubgroup of G holds
( ( k = ((i1 - 1) * ((len s2) - 1)) + j1 & 1 <= i1 & i1 <= (len s1) - 1 & 1 <= j1 & j1 <= (len s2) - 1 & H1 = s1 . (i1 + 1) & H2 = s1 . i1 & H3 = s2 . j1 implies x = H1 "\/" (H2 /\ H3) ) & ( k = (((len s1) - 1) * ((len s2) - 1)) + 1 implies x = (1). G ) )
let H1, H2, H3 be StableSubgroup of G; ::_thesis: ( ( k = ((i1 - 1) * ((len s2) - 1)) + j1 & 1 <= i1 & i1 <= (len s1) - 1 & 1 <= j1 & j1 <= (len s2) - 1 & H1 = s1 . (i1 + 1) & H2 = s1 . i1 & H3 = s2 . j1 implies x = H1 "\/" (H2 /\ H3) ) & ( k = (((len s1) - 1) * ((len s2) - 1)) + 1 implies x = (1). G ) )
thus ( k = ((i1 - 1) * ((len s2) - 1)) + j1 & 1 <= i1 & i1 <= (len s1) - 1 & 1 <= j1 & j1 <= (len s2) - 1 & H1 = s1 . (i1 + 1) & H2 = s1 . i1 & H3 = s2 . j1 implies x = H1 "\/" (H2 /\ H3) ) ::_thesis: ( k = (((len s1) - 1) * ((len s2) - 1)) + 1 implies x = (1). G )
proof
assume that
A28: k = ((i1 - 1) * ((len s2) - 1)) + j1 and
A29: 1 <= i1 and
i1 <= (len s1) - 1 and
A30: ( 1 <= j1 & j1 <= (len s2) - 1 ) ; ::_thesis: ( not H1 = s1 . (i1 + 1) or not H2 = s1 . i1 or not H3 = s2 . j1 or x = H1 "\/" (H2 /\ H3) )
assume A31: ( H1 = s1 . (i1 + 1) & H2 = s1 . i1 & H3 = s2 . j1 ) ; ::_thesis: x = H1 "\/" (H2 /\ H3)
i = i1 by A2, A24, A25, A27, A28, A29, A30, Lm46;
hence x = H1 "\/" (H2 /\ H3) by A24, A28, A31; ::_thesis: verum
end;
assume k = (((len s1) - 1) * ((len s2) - 1)) + 1 ; ::_thesis: x = (1). G
hence x = (1). G by A8, A23; ::_thesis: verum
end;
hence S1[k,x] ; ::_thesis: verum
end;
supposeA32: k = (((len s1) - 1) * ((len s2) - 1)) + 1 ; ::_thesis: ex x being set st S1[k,x]
take (1). G ; ::_thesis: S1[k, (1). G]
thus S1[k, (1). G] by A8, A32; ::_thesis: verum
end;
end;
end;
consider f being FinSequence such that
A33: ( dom f = Seg l & ( for k being Nat st k in Seg l holds
S1[k,f . k] ) ) from FINSEQ_1:sch_1(A21);
for k being Nat st k in dom f holds
f . k in the_stable_subgroups_of G
proof
let k be Nat; ::_thesis: ( k in dom f implies f . k in the_stable_subgroups_of G )
assume A34: k in dom f ; ::_thesis: f . k in the_stable_subgroups_of G
then A35: for i, j being Nat
for H1, H2, H3 being StableSubgroup of G holds
( ( k = ((i - 1) * ((len s2) - 1)) + j & 1 <= i & i <= (len s1) - 1 & 1 <= j & j <= (len s2) - 1 & H1 = s1 . (i + 1) & H2 = s1 . i & H3 = s2 . j implies f . k = H1 "\/" (H2 /\ H3) ) & ( k = (((len s1) - 1) * ((len s2) - 1)) + 1 implies f . k = (1). G ) ) by A33;
percases ( ex i, j being Nat st
( k = ((i - 1) * ((len s2) - 1)) + j & 1 <= i & i <= (len s1) - 1 & 1 <= j & j <= (len s2) - 1 ) or k = (((len s1) - 1) * ((len s2) - 1)) + 1 ) by A1, A2, A33, A34, Lm45;
suppose ex i, j being Nat st
( k = ((i - 1) * ((len s2) - 1)) + j & 1 <= i & i <= (len s1) - 1 & 1 <= j & j <= (len s2) - 1 ) ; ::_thesis: f . k in the_stable_subgroups_of G
then consider i, j being Nat such that
A36: k = ((i - 1) * ((len s2) - 1)) + j and
A37: ( 1 <= i & i <= (len s1) - 1 & 1 <= j & j <= (len s2) - 1 ) ;
reconsider H1 = s1 . (i + 1), H2 = s1 . i, H3 = s2 . j as StableSubgroup of G by A37, Th111;
f . k = H1 "\/" (H2 /\ H3) by A33, A34, A36, A37;
hence f . k in the_stable_subgroups_of G by Def11; ::_thesis: verum
end;
suppose k = (((len s1) - 1) * ((len s2) - 1)) + 1 ; ::_thesis: f . k in the_stable_subgroups_of G
hence f . k in the_stable_subgroups_of G by A35, Def11; ::_thesis: verum
end;
end;
end;
then reconsider f = f as FinSequence of the_stable_subgroups_of G by FINSEQ_2:12;
l in Seg l by A6;
then S1[l,f . l] by A33;
then A38: f . (len f) = (1). G by A33, FINSEQ_1:def_3;
A39: for i being Nat
for s1 being CompositionSeries of G
for H being GroupWithOperators of O st 1 <= i & i <= (len s1) - 1 & H = s1 . i holds
s1 . (i + 1) is normal StableSubgroup of H
proof
let i be Nat; ::_thesis: for s1 being CompositionSeries of G
for H being GroupWithOperators of O st 1 <= i & i <= (len s1) - 1 & H = s1 . i holds
s1 . (i + 1) is normal StableSubgroup of H
let s1 be CompositionSeries of G; ::_thesis: for H being GroupWithOperators of O st 1 <= i & i <= (len s1) - 1 & H = s1 . i holds
s1 . (i + 1) is normal StableSubgroup of H
let H be GroupWithOperators of O; ::_thesis: ( 1 <= i & i <= (len s1) - 1 & H = s1 . i implies s1 . (i + 1) is normal StableSubgroup of H )
assume that
A40: 1 <= i and
A41: i <= (len s1) - 1 ; ::_thesis: ( not H = s1 . i or s1 . (i + 1) is normal StableSubgroup of H )
A42: 0 + i <= 1 + i by XREAL_1:6;
A43: i + 1 <= ((len s1) - 1) + 1 by A41, XREAL_1:6;
then i <= len s1 by A42, XXREAL_0:2;
then i in Seg (len s1) by A40, FINSEQ_1:1;
then A44: i in dom s1 by FINSEQ_1:def_3;
reconsider H1 = s1 . i, H2 = s1 . (i + 1) as StableSubgroup of G by A40, A41, Th111;
assume A45: H = s1 . i ; ::_thesis: s1 . (i + 1) is normal StableSubgroup of H
1 <= i + 1 by A40, A42, XXREAL_0:2;
then i + 1 in Seg (len s1) by A43;
then i + 1 in dom s1 by FINSEQ_1:def_3;
then H2 is normal StableSubgroup of H1 by A44, Def28;
hence s1 . (i + 1) is normal StableSubgroup of H by A45; ::_thesis: verum
end;
A46: for k being Nat st k in dom f & k + 1 in dom f holds
for H1, H2 being StableSubgroup of G st H1 = f . k & H2 = f . (k + 1) holds
H2 is normal StableSubgroup of H1
proof
let k be Nat; ::_thesis: ( k in dom f & k + 1 in dom f implies for H1, H2 being StableSubgroup of G st H1 = f . k & H2 = f . (k + 1) holds
H2 is normal StableSubgroup of H1 )
assume A47: k in dom f ; ::_thesis: ( not k + 1 in dom f or for H1, H2 being StableSubgroup of G st H1 = f . k & H2 = f . (k + 1) holds
H2 is normal StableSubgroup of H1 )
set k9 = k + 1;
assume A48: k + 1 in dom f ; ::_thesis: for H1, H2 being StableSubgroup of G st H1 = f . k & H2 = f . (k + 1) holds
H2 is normal StableSubgroup of H1
then k + 1 <= l by A33, FINSEQ_1:1;
then k <> l by NAT_1:13;
then consider i, j being Nat such that
A49: k = ((i - 1) * ((len s2) - 1)) + j and
A50: 1 <= i and
A51: i <= (len s1) - 1 and
A52: 1 <= j and
A53: j <= (len s2) - 1 by A1, A2, A33, A47, Lm45;
reconsider H19 = s1 . (i + 1), H29 = s1 . i, H39 = s2 . j as strict StableSubgroup of G by A50, A51, A52, A53, Th111;
A54: f . k = H19 "\/" (H29 /\ H39) by A33, A47, A49, A50, A51, A52, A53;
let H1, H2 be StableSubgroup of G; ::_thesis: ( H1 = f . k & H2 = f . (k + 1) implies H2 is normal StableSubgroup of H1 )
assume A55: H1 = f . k ; ::_thesis: ( not H2 = f . (k + 1) or H2 is normal StableSubgroup of H1 )
A56: H19 is normal StableSubgroup of H29 by A39, A50, A51;
assume A57: H2 = f . (k + 1) ; ::_thesis: H2 is normal StableSubgroup of H1
percases ( j <> (len s2) - 1 or j = (len s2) - 1 ) ;
supposeA58: j <> (len s2) - 1 ; ::_thesis: H2 is normal StableSubgroup of H1
reconsider j9 = j + 1 as Nat ;
j < (len s2) - 1 by A53, A58, XXREAL_0:1;
then A59: j9 <= (len s2) - 1 by INT_1:7;
reconsider H399 = s2 . j9 as strict StableSubgroup of G by A52, A53, Th111;
0 + j <= 1 + j by XREAL_1:6;
then A60: 1 <= j9 by A52, XXREAL_0:2;
A61: H399 is normal StableSubgroup of H39 by A39, A52, A53;
k + 1 = ((i - 1) * ((len s2) - 1)) + j9 by A49;
then H2 = H19 "\/" (H29 /\ H399) by A33, A48, A50, A51, A57, A59, A60;
hence H2 is normal StableSubgroup of H1 by A55, A56, A54, A61, Th92; ::_thesis: verum
end;
supposeA62: j = (len s2) - 1 ; ::_thesis: H2 is normal StableSubgroup of H1
percases ( i <> (len s1) - 1 or i = (len s1) - 1 ) ;
supposeA63: i <> (len s1) - 1 ; ::_thesis: H2 is normal StableSubgroup of H1
set i9 = i + 1;
A64: 0 + (i + 1) <= 1 + (i + 1) by XREAL_1:6;
set j9 = 1;
H19 is StableSubgroup of H1 by A55, A54, Th35;
then H19 is Subgroup of H1 by Def7;
then A65: the carrier of H19 c= the carrier of H1 by GROUP_2:def_5;
1 + 1 <= i + 1 by A50, XREAL_1:6;
then A66: 1 <= i + 1 by XXREAL_0:2;
i < l9 by A51, A63, XXREAL_0:1;
then A67: i + 1 <= l9 by NAT_1:13;
then A68: (i + 1) + 1 <= ((len s1) - 1) + 1 by XREAL_1:6;
then i + 1 <= len s1 by A64, XXREAL_0:2;
then i + 1 in Seg (len s1) by A66;
then A69: i + 1 in dom s1 by FINSEQ_1:def_3;
(len s2) - 1 > 1 - 1 by A2, XREAL_1:9;
then A70: l99 >= 0 + 1 by NAT_1:13;
then reconsider H199 = s1 . ((i + 1) + 1), H299 = s1 . (i + 1), H399 = s2 . 1 as strict StableSubgroup of G by A67, A66, Th111;
1 <= (i + 1) + 1 by A66, A64, XXREAL_0:2;
then (i + 1) + 1 in Seg (len s1) by A68;
then (i + 1) + 1 in dom s1 by FINSEQ_1:def_3;
then A71: H199 is normal StableSubgroup of H299 by A69, Def28;
now__::_thesis:_for_x_being_set_st_x_in_the_carrier_of_H299_holds_
x_in_the_carrier_of_((Omega)._G)
let x be set ; ::_thesis: ( x in the carrier of H299 implies x in the carrier of ((Omega). G) )
H299 is Subgroup of G by Def7;
then A72: the carrier of H299 c= the carrier of G by GROUP_2:def_5;
assume x in the carrier of H299 ; ::_thesis: x in the carrier of ((Omega). G)
hence x in the carrier of ((Omega). G) by A72; ::_thesis: verum
end;
then the carrier of H299 c= the carrier of ((Omega). G) by TARSKI:def_3;
then A73: the carrier of H299 = the carrier of H299 /\ the carrier of ((Omega). G) by XBOOLE_1:28;
A74: H399 = (Omega). G by Def28;
k + 1 = (((i + 1) - 1) * ((len s2) - 1)) + 1 by A49, A62;
then H2 = H199 "\/" (H299 /\ H399) by A33, A48, A57, A67, A66, A70;
then H2 = H199 "\/" H299 by A74, A73, Th18;
then A75: H2 = H19 by A71, Th36;
H29 /\ H39 is StableSubgroup of H29 by Lm34;
then A76: H1 is StableSubgroup of H29 by A55, A56, A54, Th37;
then A77: H1 is Subgroup of H29 by Def7;
now__::_thesis:_for_H9_being_strict_Subgroup_of_H1_st_H9_=_multMagma(#_the_carrier_of_H2,_the_multF_of_H2_#)_holds_
H9_is_normal
let H9 be strict Subgroup of H1; ::_thesis: ( H9 = multMagma(# the carrier of H2, the multF of H2 #) implies H9 is normal )
assume A78: H9 = multMagma(# the carrier of H2, the multF of H2 #) ; ::_thesis: H9 is normal
now__::_thesis:_for_a_being_Element_of_H1_holds_a_*_H9_c=_H9_*_a
let a be Element of H1; ::_thesis: a * H9 c= H9 * a
reconsider a9 = a as Element of H29 by A76, Th2;
now__::_thesis:_for_x_being_set_st_x_in_a_*_H9_holds_
x_in_H9_*_a
reconsider H1s9 = multMagma(# the carrier of H19, the multF of H19 #) as normal Subgroup of H29 by A56, Lm7;
let x be set ; ::_thesis: ( x in a * H9 implies x in H9 * a )
assume x in a * H9 ; ::_thesis: x in H9 * a
then consider b being Element of H1 such that
A79: x = a * b and
A80: b in H9 by GROUP_2:103;
set b9 = b;
A81: H1 is Subgroup of H29 by A76, Def7;
then reconsider b9 = b as Element of H29 by GROUP_2:42;
x = a9 * b9 by A79, A81, GROUP_2:43;
then ( a9 * H1s9 c= H1s9 * a9 & x in a9 * H1s9 ) by A75, A78, A80, GROUP_2:103, GROUP_3:118;
then consider b99 being Element of H29 such that
A82: x = b99 * a9 and
A83: b99 in H1s9 by GROUP_2:104;
b99 in the carrier of H19 by A83, STRUCT_0:def_5;
then reconsider b99 = b99 as Element of H1 by A65;
x = b99 * a by A77, A82, GROUP_2:43;
hence x in H9 * a by A75, A78, A83, GROUP_2:104; ::_thesis: verum
end;
hence a * H9 c= H9 * a by TARSKI:def_3; ::_thesis: verum
end;
hence H9 is normal by GROUP_3:118; ::_thesis: verum
end;
hence H2 is normal StableSubgroup of H1 by A55, A54, A75, Def10, Th35; ::_thesis: verum
end;
suppose i = (len s1) - 1 ; ::_thesis: H2 is normal StableSubgroup of H1
then H2 = (1). G by A33, A48, A49, A57, A62
.= (1). H1 by Th15 ;
hence H2 is normal StableSubgroup of H1 ; ::_thesis: verum
end;
end;
end;
end;
end;
( (len s1) - 1 > 1 - 1 & (len s2) - 1 > 1 - 1 ) by A1, A2, XREAL_1:9;
then ((len s1) - 1) * ((len s2) - 1) > 0 * ((len s2) - 1) by XREAL_1:68;
then 1 <> l ;
then consider i, j being Nat such that
A84: 1 = ((i - 1) * ((len s2) - 1)) + j and
A85: 1 <= i and
A86: i <= (len s1) - 1 and
A87: 1 <= j and
A88: j <= (len s2) - 1 by A1, A2, A7, Lm45;
set i9 = i - 1;
i - 1 >= 1 - 1 by A85, XREAL_1:9;
then reconsider i9 = i - 1 as Element of NAT by INT_1:3;
reconsider H1 = s1 . (i + 1), H2 = s1 . i, H3 = s2 . j as StableSubgroup of G by A85, A86, A87, A88, Th111;
1 mod l99 = ((i9 * l99) + j) mod l99 by A84;
then A89: 1 mod l99 = j mod l99 by NAT_D:21;
A90: j = 1
proof
percases ( l99 = 1 or l99 <> 1 ) ;
suppose l99 = 1 ; ::_thesis: j = 1
hence j = 1 by A87, A88, XXREAL_0:1; ::_thesis: verum
end;
suppose l99 <> 1 ; ::_thesis: j = 1
then 1 < l99 by A3, XXREAL_0:1;
then A91: 1 = j mod l99 by A89, NAT_D:14;
then j <> l99 by NAT_D:25;
then l99 > j by A88, XXREAL_0:1;
hence j = 1 by A91, NAT_D:24; ::_thesis: verum
end;
end;
end;
then A92: H3 = (Omega). G by Def28;
(i9 * l99) / l99 = 0 / l99 by A84, A90;
then i9 * 1 = 0 by A4, XCMPLX_1:74;
then A93: H2 = (Omega). G by Def28;
f . 1 = H1 "\/" (H2 /\ H3) by A33, A7, A84, A85, A86, A87, A88;
then f . 1 = H1 "\/" ((Omega). G) by A93, A92, Th19;
then f . 1 = (Omega). G by Th34;
then reconsider f = f as CompositionSeries of G by A38, A46, Def28;
take f ; ::_thesis: for k, i, j being Nat
for H1, H2, H3 being StableSubgroup of G holds
( ( k = ((i - 1) * ((len s2) - 1)) + j & 1 <= i & i <= (len s1) - 1 & 1 <= j & j <= (len s2) - 1 & H1 = s1 . (i + 1) & H2 = s1 . i & H3 = s2 . j implies f . k = H1 "\/" (H2 /\ H3) ) & ( k = (((len s1) - 1) * ((len s2) - 1)) + 1 implies f . k = (1). G ) & len f = (((len s1) - 1) * ((len s2) - 1)) + 1 )
let k, i, j be Nat; ::_thesis: for H1, H2, H3 being StableSubgroup of G holds
( ( k = ((i - 1) * ((len s2) - 1)) + j & 1 <= i & i <= (len s1) - 1 & 1 <= j & j <= (len s2) - 1 & H1 = s1 . (i + 1) & H2 = s1 . i & H3 = s2 . j implies f . k = H1 "\/" (H2 /\ H3) ) & ( k = (((len s1) - 1) * ((len s2) - 1)) + 1 implies f . k = (1). G ) & len f = (((len s1) - 1) * ((len s2) - 1)) + 1 )
let H1, H2, H3 be StableSubgroup of G; ::_thesis: ( ( k = ((i - 1) * ((len s2) - 1)) + j & 1 <= i & i <= (len s1) - 1 & 1 <= j & j <= (len s2) - 1 & H1 = s1 . (i + 1) & H2 = s1 . i & H3 = s2 . j implies f . k = H1 "\/" (H2 /\ H3) ) & ( k = (((len s1) - 1) * ((len s2) - 1)) + 1 implies f . k = (1). G ) & len f = (((len s1) - 1) * ((len s2) - 1)) + 1 )
A94: for k, i, j being Nat st k = ((i - 1) * ((len s2) - 1)) + j & 1 <= i & i <= (len s1) - 1 & 1 <= j & j <= (len s2) - 1 holds
k in Seg l
proof
let k, i, j be Nat; ::_thesis: ( k = ((i - 1) * ((len s2) - 1)) + j & 1 <= i & i <= (len s1) - 1 & 1 <= j & j <= (len s2) - 1 implies k in Seg l )
assume A95: k = ((i - 1) * ((len s2) - 1)) + j ; ::_thesis: ( not 1 <= i or not i <= (len s1) - 1 or not 1 <= j or not j <= (len s2) - 1 or k in Seg l )
assume that
A96: 1 <= i and
A97: i <= (len s1) - 1 ; ::_thesis: ( not 1 <= j or not j <= (len s2) - 1 or k in Seg l )
assume that
A98: 1 <= j and
A99: j <= (len s2) - 1 ; ::_thesis: k in Seg l
i - 1 <= l9 - 1 by A97, XREAL_1:9;
then (i - 1) * l99 <= (l9 - 1) * l99 by XREAL_1:64;
then ( 0 + (l9 * l99) <= 1 + (l9 * l99) & k <= ((l9 * l99) - (1 * l99)) + l99 ) by A95, A99, XREAL_1:7;
then A100: k <= (((len s1) - 1) * ((len s2) - 1)) + 1 by XXREAL_0:2;
1 - 1 <= i - 1 by A96, XREAL_1:9;
then 0 + 1 <= ((i - 1) * ((len s2) - 1)) + j by A3, A98, XREAL_1:7;
hence k in Seg l by A95, A100, FINSEQ_1:1; ::_thesis: verum
end;
now__::_thesis:_(_k_=_((i_-_1)_*_((len_s2)_-_1))_+_j_&_1_<=_i_&_i_<=_(len_s1)_-_1_&_1_<=_j_&_j_<=_(len_s2)_-_1_&_H1_=_s1_._(i_+_1)_&_H2_=_s1_._i_&_H3_=_s2_._j_implies_f_._k_=_H1_"\/"_(H2_/\_H3)_)
assume that
A101: ( k = ((i - 1) * ((len s2) - 1)) + j & 1 <= i & i <= (len s1) - 1 & 1 <= j & j <= (len s2) - 1 ) and
A102: ( H1 = s1 . (i + 1) & H2 = s1 . i & H3 = s2 . j ) ; ::_thesis: f . k = H1 "\/" (H2 /\ H3)
k in Seg l by A94, A101;
hence f . k = H1 "\/" (H2 /\ H3) by A33, A101, A102; ::_thesis: verum
end;
hence ( k = ((i - 1) * ((len s2) - 1)) + j & 1 <= i & i <= (len s1) - 1 & 1 <= j & j <= (len s2) - 1 & H1 = s1 . (i + 1) & H2 = s1 . i & H3 = s2 . j implies f . k = H1 "\/" (H2 /\ H3) ) ; ::_thesis: ( ( k = (((len s1) - 1) * ((len s2) - 1)) + 1 implies f . k = (1). G ) & len f = (((len s1) - 1) * ((len s2) - 1)) + 1 )
now__::_thesis:_(_k_=_(((len_s1)_-_1)_*_((len_s2)_-_1))_+_1_implies_f_._k_=_(1)._G_)
assume A103: k = (((len s1) - 1) * ((len s2) - 1)) + 1 ; ::_thesis: f . k = (1). G
then k in Seg l by A6;
hence f . k = (1). G by A33, A103; ::_thesis: verum
end;
hence ( ( k = (((len s1) - 1) * ((len s2) - 1)) + 1 implies f . k = (1). G ) & len f = (((len s1) - 1) * ((len s2) - 1)) + 1 ) by A33, FINSEQ_1:def_3; ::_thesis: verum
end;
uniqueness
for b1, b2 being CompositionSeries of G st ( for k, i, j being Nat
for H1, H2, H3 being StableSubgroup of G holds
( ( k = ((i - 1) * ((len s2) - 1)) + j & 1 <= i & i <= (len s1) - 1 & 1 <= j & j <= (len s2) - 1 & H1 = s1 . (i + 1) & H2 = s1 . i & H3 = s2 . j implies b1 . k = H1 "\/" (H2 /\ H3) ) & ( k = (((len s1) - 1) * ((len s2) - 1)) + 1 implies b1 . k = (1). G ) & len b1 = (((len s1) - 1) * ((len s2) - 1)) + 1 ) ) & ( for k, i, j being Nat
for H1, H2, H3 being StableSubgroup of G holds
( ( k = ((i - 1) * ((len s2) - 1)) + j & 1 <= i & i <= (len s1) - 1 & 1 <= j & j <= (len s2) - 1 & H1 = s1 . (i + 1) & H2 = s1 . i & H3 = s2 . j implies b2 . k = H1 "\/" (H2 /\ H3) ) & ( k = (((len s1) - 1) * ((len s2) - 1)) + 1 implies b2 . k = (1). G ) & len b2 = (((len s1) - 1) * ((len s2) - 1)) + 1 ) ) holds
b1 = b2
proof
let f1, f2 be CompositionSeries of G; ::_thesis: ( ( for k, i, j being Nat
for H1, H2, H3 being StableSubgroup of G holds
( ( k = ((i - 1) * ((len s2) - 1)) + j & 1 <= i & i <= (len s1) - 1 & 1 <= j & j <= (len s2) - 1 & H1 = s1 . (i + 1) & H2 = s1 . i & H3 = s2 . j implies f1 . k = H1 "\/" (H2 /\ H3) ) & ( k = (((len s1) - 1) * ((len s2) - 1)) + 1 implies f1 . k = (1). G ) & len f1 = (((len s1) - 1) * ((len s2) - 1)) + 1 ) ) & ( for k, i, j being Nat
for H1, H2, H3 being StableSubgroup of G holds
( ( k = ((i - 1) * ((len s2) - 1)) + j & 1 <= i & i <= (len s1) - 1 & 1 <= j & j <= (len s2) - 1 & H1 = s1 . (i + 1) & H2 = s1 . i & H3 = s2 . j implies f2 . k = H1 "\/" (H2 /\ H3) ) & ( k = (((len s1) - 1) * ((len s2) - 1)) + 1 implies f2 . k = (1). G ) & len f2 = (((len s1) - 1) * ((len s2) - 1)) + 1 ) ) implies f1 = f2 )
assume A104: for k, i, j being Nat
for H1, H2, H3 being StableSubgroup of G holds
( ( k = ((i - 1) * ((len s2) - 1)) + j & 1 <= i & i <= (len s1) - 1 & 1 <= j & j <= (len s2) - 1 & H1 = s1 . (i + 1) & H2 = s1 . i & H3 = s2 . j implies f1 . k = H1 "\/" (H2 /\ H3) ) & ( k = (((len s1) - 1) * ((len s2) - 1)) + 1 implies f1 . k = (1). G ) & len f1 = (((len s1) - 1) * ((len s2) - 1)) + 1 ) ; ::_thesis: ( ex k, i, j being Nat ex H1, H2, H3 being StableSubgroup of G st
( ( k = ((i - 1) * ((len s2) - 1)) + j & 1 <= i & i <= (len s1) - 1 & 1 <= j & j <= (len s2) - 1 & H1 = s1 . (i + 1) & H2 = s1 . i & H3 = s2 . j implies f2 . k = H1 "\/" (H2 /\ H3) ) & ( k = (((len s1) - 1) * ((len s2) - 1)) + 1 implies f2 . k = (1). G ) implies not len f2 = (((len s1) - 1) * ((len s2) - 1)) + 1 ) or f1 = f2 )
assume A105: for k, i, j being Nat
for H1, H2, H3 being StableSubgroup of G holds
( ( k = ((i - 1) * ((len s2) - 1)) + j & 1 <= i & i <= (len s1) - 1 & 1 <= j & j <= (len s2) - 1 & H1 = s1 . (i + 1) & H2 = s1 . i & H3 = s2 . j implies f2 . k = H1 "\/" (H2 /\ H3) ) & ( k = (((len s1) - 1) * ((len s2) - 1)) + 1 implies f2 . k = (1). G ) & len f2 = (((len s1) - 1) * ((len s2) - 1)) + 1 ) ; ::_thesis: f1 = f2
A106: now__::_thesis:_for_k_being_Nat_st_k_in_dom_f1_holds_
f1_._k_=_f2_._k
set l = len f1;
let k be Nat; ::_thesis: ( k in dom f1 implies f1 . b1 = f2 . b1 )
assume k in dom f1 ; ::_thesis: f1 . b1 = f2 . b1
then A107: k in Seg (len f1) by FINSEQ_1:def_3;
percases ( ex i, j being Nat st
( k = ((i - 1) * ((len s2) - 1)) + j & 1 <= i & i <= (len s1) - 1 & 1 <= j & j <= (len s2) - 1 ) or k = (((len s1) - 1) * ((len s2) - 1)) + 1 ) by A1, A2, A104, A107, Lm45;
suppose ex i, j being Nat st
( k = ((i - 1) * ((len s2) - 1)) + j & 1 <= i & i <= (len s1) - 1 & 1 <= j & j <= (len s2) - 1 ) ; ::_thesis: f1 . b1 = f2 . b1
then consider i, j being Nat such that
A108: k = ((i - 1) * ((len s2) - 1)) + j and
A109: ( 1 <= i & i <= (len s1) - 1 & 1 <= j & j <= (len s2) - 1 ) ;
reconsider H1 = s1 . (i + 1), H2 = s1 . i, H3 = s2 . j as StableSubgroup of G by A109, Th111;
f1 . k = H1 "\/" (H2 /\ H3) by A104, A108, A109;
hence f1 . k = f2 . k by A105, A108, A109; ::_thesis: verum
end;
supposeA110: k = (((len s1) - 1) * ((len s2) - 1)) + 1 ; ::_thesis: f1 . b1 = f2 . b1
then f1 . k = (1). G by A104;
hence f1 . k = f2 . k by A105, A110; ::_thesis: verum
end;
end;
end;
dom f1 = Seg (len f2) by A104, A105, FINSEQ_1:def_3
.= dom f2 by FINSEQ_1:def_3 ;
hence f1 = f2 by A106, FINSEQ_1:13; ::_thesis: verum
end;
end;
:: deftheorem Def35 defines the_schreier_series_of GROUP_9:def_35_:_
for O being set
for G being GroupWithOperators of O
for s1, s2 being CompositionSeries of G st len s1 > 1 & len s2 > 1 holds
for b5 being CompositionSeries of G holds
( b5 = the_schreier_series_of (s1,s2) iff for k, i, j being Nat
for H1, H2, H3 being StableSubgroup of G holds
( ( k = ((i - 1) * ((len s2) - 1)) + j & 1 <= i & i <= (len s1) - 1 & 1 <= j & j <= (len s2) - 1 & H1 = s1 . (i + 1) & H2 = s1 . i & H3 = s2 . j implies b5 . k = H1 "\/" (H2 /\ H3) ) & ( k = (((len s1) - 1) * ((len s2) - 1)) + 1 implies b5 . k = (1). G ) & len b5 = (((len s1) - 1) * ((len s2) - 1)) + 1 ) );
theorem Th116: :: GROUP_9:116
for O being set
for G being GroupWithOperators of O
for s1, s2 being CompositionSeries of G st len s1 > 1 & len s2 > 1 holds
the_schreier_series_of (s1,s2) is_finer_than s1
proof
let O be set ; ::_thesis: for G being GroupWithOperators of O
for s1, s2 being CompositionSeries of G st len s1 > 1 & len s2 > 1 holds
the_schreier_series_of (s1,s2) is_finer_than s1
let G be GroupWithOperators of O; ::_thesis: for s1, s2 being CompositionSeries of G st len s1 > 1 & len s2 > 1 holds
the_schreier_series_of (s1,s2) is_finer_than s1
let s1, s2 be CompositionSeries of G; ::_thesis: ( len s1 > 1 & len s2 > 1 implies the_schreier_series_of (s1,s2) is_finer_than s1 )
assume that
A1: len s1 > 1 and
A2: len s2 > 1 ; ::_thesis: the_schreier_series_of (s1,s2) is_finer_than s1
now__::_thesis:_ex_fX_being_Element_of_bool_REAL_st_
(_fX_c=_dom_(the_schreier_series_of_(s1,s2))_&_s1_=_(the_schreier_series_of_(s1,s2))_*_(Sgm_fX)_)
set rR = rng s1;
set R = s1;
set l = (((len s1) - 1) * ((len s2) - 1)) + 1;
set X = Seg (len s1);
set g = { [k,(((k - 1) * ((len s2) - 1)) + 1)] where k is Element of NAT : ( 1 <= k & k <= len s1 ) } ;
now__::_thesis:_for_x_being_set_st_x_in__{__[k,(((k_-_1)_*_((len_s2)_-_1))_+_1)]_where_k_is_Element_of_NAT_:_(_1_<=_k_&_k_<=_len_s1_)__}__holds_
ex_y,_z_being_set_st_x_=_[y,z]
let x be set ; ::_thesis: ( x in { [k,(((k - 1) * ((len s2) - 1)) + 1)] where k is Element of NAT : ( 1 <= k & k <= len s1 ) } implies ex y, z being set st x = [y,z] )
assume x in { [k,(((k - 1) * ((len s2) - 1)) + 1)] where k is Element of NAT : ( 1 <= k & k <= len s1 ) } ; ::_thesis: ex y, z being set st x = [y,z]
then consider k being Element of NAT such that
A3: [k,(((k - 1) * ((len s2) - 1)) + 1)] = x and
1 <= k and
k <= len s1 ;
set z = ((k - 1) * ((len s2) - 1)) + 1;
set y = k;
reconsider y = k, z = ((k - 1) * ((len s2) - 1)) + 1 as set ;
take y = y; ::_thesis: ex z being set st x = [y,z]
take z = z; ::_thesis: x = [y,z]
thus x = [y,z] by A3; ::_thesis: verum
end;
then reconsider g = { [k,(((k - 1) * ((len s2) - 1)) + 1)] where k is Element of NAT : ( 1 <= k & k <= len s1 ) } as Relation by RELAT_1:def_1;
A4: now__::_thesis:_for_y_being_set_st_y_in_rng_g_holds_
y_in_REAL
let y be set ; ::_thesis: ( y in rng g implies y in REAL )
assume y in rng g ; ::_thesis: y in REAL
then consider x being set such that
A5: [x,y] in g by XTUPLE_0:def_13;
consider k being Element of NAT such that
A6: [k,(((k - 1) * ((len s2) - 1)) + 1)] = [x,y] and
1 <= k and
k <= len s1 by A5;
((k - 1) * ((len s2) - 1)) + 1 = y by A6, XTUPLE_0:1;
hence y in REAL by XREAL_0:def_1; ::_thesis: verum
end;
A7: now__::_thesis:_for_x,_y1,_y2_being_set_st_[x,y1]_in_g_&_[x,y2]_in_g_holds_
y1_=_y2
let x, y1, y2 be set ; ::_thesis: ( [x,y1] in g & [x,y2] in g implies y1 = y2 )
assume [x,y1] in g ; ::_thesis: ( [x,y2] in g implies y1 = y2 )
then consider k being Element of NAT such that
A8: [k,(((k - 1) * ((len s2) - 1)) + 1)] = [x,y1] and
1 <= k and
k <= len s1 ;
A9: k = x by A8, XTUPLE_0:1;
assume [x,y2] in g ; ::_thesis: y1 = y2
then consider k9 being Element of NAT such that
A10: [k9,(((k9 - 1) * ((len s2) - 1)) + 1)] = [x,y2] and
1 <= k9 and
k9 <= len s1 ;
k9 = x by A10, XTUPLE_0:1;
hence y1 = y2 by A8, A10, A9, XTUPLE_0:1; ::_thesis: verum
end;
now__::_thesis:_for_x_being_set_st_x_in_dom_g_holds_
x_in_NAT
let x be set ; ::_thesis: ( x in dom g implies x in NAT )
assume x in dom g ; ::_thesis: x in NAT
then consider y being set such that
A11: [x,y] in g by XTUPLE_0:def_12;
consider k being Element of NAT such that
A12: [k,(((k - 1) * ((len s2) - 1)) + 1)] = [x,y] and
1 <= k and
k <= len s1 by A11;
k = x by A12, XTUPLE_0:1;
hence x in NAT ; ::_thesis: verum
end;
then A13: dom g c= NAT by TARSKI:def_3;
reconsider g = g as Function by A7, FUNCT_1:def_1;
A14: rng g c= REAL by A4, TARSKI:def_3;
reconsider f = g as PartFunc of (dom g),(rng g) by RELSET_1:4;
dom g c= REAL by A13, XBOOLE_1:1;
then reconsider f = f as PartFunc of REAL,REAL by A14, RELSET_1:7;
set dR = dom s1;
set t = the_schreier_series_of (s1,s2);
set fX = f .: (Seg (len s1));
take fX = f .: (Seg (len s1)); ::_thesis: ( fX c= dom (the_schreier_series_of (s1,s2)) & s1 = (the_schreier_series_of (s1,s2)) * (Sgm fX) )
reconsider R = s1 as Relation of (dom s1),(rng s1) by FUNCT_2:1;
A15: (id (dom s1)) * R = R by FUNCT_2:17;
(len s2) + 1 > 1 + 1 by A2, XREAL_1:6;
then len s2 >= 2 by NAT_1:13;
then A16: (len s2) - 1 >= 2 - 1 by XREAL_1:9;
(len s1) + 1 > 1 + 1 by A1, XREAL_1:6;
then len s1 >= 2 by NAT_1:13;
then (len s1) - 1 >= 2 - 1 by XREAL_1:9;
then reconsider l = (((len s1) - 1) * ((len s2) - 1)) + 1 as Element of NAT by A16, INT_1:3;
A17: len (the_schreier_series_of (s1,s2)) = l by A1, A2, Def35;
then A18: dom (the_schreier_series_of (s1,s2)) = Seg l by FINSEQ_1:def_3;
(len s2) + 1 > 1 + 1 by A2, XREAL_1:6;
then len s2 >= 2 by NAT_1:13;
then A19: (len s2) - 1 >= 2 - 1 by XREAL_1:9;
now__::_thesis:_for_y_being_set_st_y_in_fX_holds_
y_in_Seg_l
let y be set ; ::_thesis: ( y in fX implies y in Seg l )
assume y in fX ; ::_thesis: y in Seg l
then consider x being set such that
A20: [x,y] in g and
x in Seg (len s1) by RELAT_1:def_13;
consider k being Element of NAT such that
A21: [k,(((k - 1) * ((len s2) - 1)) + 1)] = [x,y] and
A22: 1 <= k and
A23: k <= len s1 by A20;
reconsider y9 = y as integer number by A21, XTUPLE_0:1;
A24: k - 1 >= 1 - 1 by A22, XREAL_1:9;
then A25: y9 > 0 by A19, A21, XTUPLE_0:1;
k - 1 <= (len s1) - 1 by A23, XREAL_1:9;
then A26: (k - 1) * ((len s2) - 1) <= ((len s1) - 1) * ((len s2) - 1) by A19, XREAL_1:64;
((k - 1) * ((len s2) - 1)) + 1 >= 0 + 1 by A19, A24, XREAL_1:6;
then A27: y9 >= 1 by A21, XTUPLE_0:1;
reconsider y9 = y9 as Element of NAT by A25, INT_1:3;
((k - 1) * ((len s2) - 1)) + 1 = y by A21, XTUPLE_0:1;
then y9 <= l by A26, XREAL_1:6;
hence y in Seg l by A27; ::_thesis: verum
end;
then A28: fX c= Seg l by TARSKI:def_3;
hence fX c= dom (the_schreier_series_of (s1,s2)) by A17, FINSEQ_1:def_3; ::_thesis: s1 = (the_schreier_series_of (s1,s2)) * (Sgm fX)
now__::_thesis:_for_x_being_set_st_x_in_Seg_(len_s1)_holds_
x_in_dom_f
let x be set ; ::_thesis: ( x in Seg (len s1) implies x in dom f )
assume A29: x in Seg (len s1) ; ::_thesis: x in dom f
then reconsider k = x as Element of NAT ;
set y = ((k - 1) * ((len s2) - 1)) + 1;
( 1 <= k & k <= len s1 ) by A29, FINSEQ_1:1;
then [x,(((k - 1) * ((len s2) - 1)) + 1)] in f ;
hence x in dom f by XTUPLE_0:def_12; ::_thesis: verum
end;
then A30: Seg (len s1) c= dom f by TARSKI:def_3;
then A31: dom s1 c= dom f by FINSEQ_1:def_3;
now__::_thesis:_for_x_being_set_st_x_in_dom_f_holds_
x_in_dom_s1
let x be set ; ::_thesis: ( x in dom f implies x in dom s1 )
assume x in dom f ; ::_thesis: x in dom s1
then consider y being set such that
A32: [x,y] in f by XTUPLE_0:def_12;
consider k being Element of NAT such that
A33: ( [k,(((k - 1) * ((len s2) - 1)) + 1)] = [x,y] & 1 <= k & k <= len s1 ) by A32;
( k in Seg (len s1) & k = x ) by A33, XTUPLE_0:1;
hence x in dom s1 by FINSEQ_1:def_3; ::_thesis: verum
end;
then dom f c= dom s1 by TARSKI:def_3;
then A34: dom s1 = dom f by A31, XBOOLE_0:def_10;
then Seg (len s1) = dom f by FINSEQ_1:def_3;
then A35: rng f c= Seg l by A28, RELAT_1:113;
then A36: dom s1 = dom ((the_schreier_series_of (s1,s2)) * f) by A18, A34, RELAT_1:27;
A37: now__::_thesis:_for_x_being_set_st_x_in_dom_s1_holds_
s1_._x_=_((the_schreier_series_of_(s1,s2))_*_f)_._x
let x be set ; ::_thesis: ( x in dom s1 implies s1 . b1 = ((the_schreier_series_of (s1,s2)) * f) . b1 )
assume A38: x in dom s1 ; ::_thesis: s1 . b1 = ((the_schreier_series_of (s1,s2)) * f) . b1
then [x,(f . x)] in f by A31, FUNCT_1:def_2;
then consider i being Element of NAT such that
A39: [i,(((i - 1) * ((len s2) - 1)) + 1)] = [x,(f . x)] and
A40: 1 <= i and
A41: i <= len s1 ;
set k = ((i - 1) * ((len s2) - 1)) + 1;
((i - 1) * ((len s2) - 1)) + 1 = f . x by A39, XTUPLE_0:1;
then ((i - 1) * ((len s2) - 1)) + 1 in rng f by A31, A38, FUNCT_1:3;
then ((i - 1) * ((len s2) - 1)) + 1 in Seg l by A35;
then reconsider k = ((i - 1) * ((len s2) - 1)) + 1 as Element of NAT ;
A42: x in dom ((the_schreier_series_of (s1,s2)) * f) by A18, A34, A35, A38, RELAT_1:27;
percases ( i = len s1 or i <> len s1 ) ;
supposeA43: i = len s1 ; ::_thesis: s1 . b1 = ((the_schreier_series_of (s1,s2)) * f) . b1
((the_schreier_series_of (s1,s2)) * f) . x = (the_schreier_series_of (s1,s2)) . (f . x) by A42, FUNCT_1:12
.= (the_schreier_series_of (s1,s2)) . k by A39, XTUPLE_0:1
.= (1). G by A1, A2, A43, Def35
.= s1 . (len s1) by Def28 ;
hence s1 . x = ((the_schreier_series_of (s1,s2)) * f) . x by A39, A43, XTUPLE_0:1; ::_thesis: verum
end;
suppose i <> len s1 ; ::_thesis: s1 . b1 = ((the_schreier_series_of (s1,s2)) * f) . b1
then i < len s1 by A41, XXREAL_0:1;
then A44: i + 1 <= len s1 by NAT_1:13;
then A45: (i + 1) - 1 <= (len s1) - 1 by XREAL_1:9;
A46: s2 . 1 = (Omega). G by Def28;
then reconsider H1 = s1 . (i + 1), H2 = s1 . i, H3 = s2 . 1 as strict StableSubgroup of G by A40, A45, Th111;
now__::_thesis:_for_x_being_set_st_x_in_the_carrier_of_H2_holds_
x_in_the_carrier_of_((Omega)._G)
let x be set ; ::_thesis: ( x in the carrier of H2 implies x in the carrier of ((Omega). G) )
H2 is Subgroup of G by Def7;
then A47: the carrier of H2 c= the carrier of G by GROUP_2:def_5;
assume x in the carrier of H2 ; ::_thesis: x in the carrier of ((Omega). G)
hence x in the carrier of ((Omega). G) by A47; ::_thesis: verum
end;
then the carrier of H2 c= the carrier of ((Omega). G) by TARSKI:def_3;
then A48: the carrier of H2 = the carrier of H2 /\ the carrier of H3 by A46, XBOOLE_1:28;
(len s2) - 1 > 1 - 1 by A2, XREAL_1:9;
then A49: (len s2) - 1 >= 0 + 1 by INT_1:7;
0 + i <= 1 + i by XREAL_1:6;
then 1 <= i + 1 by A40, XXREAL_0:2;
then i + 1 in Seg (len s1) by A44;
then A50: i + 1 in dom s1 by FINSEQ_1:def_3;
i in Seg (len s1) by A40, A41;
then i in dom s1 by FINSEQ_1:def_3;
then A51: H1 is normal StableSubgroup of H2 by A50, Def28;
((the_schreier_series_of (s1,s2)) * f) . x = (the_schreier_series_of (s1,s2)) . (f . x) by A42, FUNCT_1:12
.= (the_schreier_series_of (s1,s2)) . k by A39, XTUPLE_0:1
.= H1 "\/" (H2 /\ H3) by A1, A2, A40, A45, A49, Def35
.= H1 "\/" H2 by A48, Th18
.= H2 by A51, Th36 ;
hence s1 . x = ((the_schreier_series_of (s1,s2)) * f) . x by A39, XTUPLE_0:1; ::_thesis: verum
end;
end;
end;
now__::_thesis:_for_r1,_r2_being_Element_of_REAL_st_r1_in_(Seg_(len_s1))_/\_(dom_f)_&_r2_in_(Seg_(len_s1))_/\_(dom_f)_&_r1_<_r2_holds_
f_._r1_<_f_._r2
let r1, r2 be Element of REAL ; ::_thesis: ( r1 in (Seg (len s1)) /\ (dom f) & r2 in (Seg (len s1)) /\ (dom f) & r1 < r2 implies f . r1 < f . r2 )
assume r1 in (Seg (len s1)) /\ (dom f) ; ::_thesis: ( r2 in (Seg (len s1)) /\ (dom f) & r1 < r2 implies f . r1 < f . r2 )
then r1 in dom f by XBOOLE_0:def_4;
then [r1,(f . r1)] in f by FUNCT_1:1;
then consider k9 being Element of NAT such that
A52: [k9,(((k9 - 1) * ((len s2) - 1)) + 1)] = [r1,(f . r1)] and
1 <= k9 and
k9 <= len s1 ;
assume r2 in (Seg (len s1)) /\ (dom f) ; ::_thesis: ( r1 < r2 implies f . r1 < f . r2 )
then r2 in dom f by XBOOLE_0:def_4;
then [r2,(f . r2)] in f by FUNCT_1:1;
then consider k99 being Element of NAT such that
A53: [k99,(((k99 - 1) * ((len s2) - 1)) + 1)] = [r2,(f . r2)] and
1 <= k99 and
k99 <= len s1 ;
A54: k99 = r2 by A53, XTUPLE_0:1;
assume A55: r1 < r2 ; ::_thesis: f . r1 < f . r2
k9 = r1 by A52, XTUPLE_0:1;
then k9 - 1 < k99 - 1 by A55, A54, XREAL_1:9;
then A56: (k9 - 1) * ((len s2) - 1) < (k99 - 1) * ((len s2) - 1) by A19, XREAL_1:68;
A57: ((k99 - 1) * ((len s2) - 1)) + 1 = f . r2 by A53, XTUPLE_0:1;
((k9 - 1) * ((len s2) - 1)) + 1 = f . r1 by A52, XTUPLE_0:1;
hence f . r1 < f . r2 by A57, A56, XREAL_1:6; ::_thesis: verum
end;
then A58: f | (Seg (len s1)) is increasing by RFUNCT_2:20;
now__::_thesis:_for_y_being_set_st_y_in_f_.:_(Seg_(len_s1))_holds_
y_in_NAT_\_{0}
let y be set ; ::_thesis: ( y in f .: (Seg (len s1)) implies y in NAT \ {0} )
assume y in f .: (Seg (len s1)) ; ::_thesis: y in NAT \ {0}
then consider x being set such that
A59: [x,y] in g and
x in Seg (len s1) by RELAT_1:def_13;
consider k being Element of NAT such that
A60: [k,(((k - 1) * ((len s2) - 1)) + 1)] = [x,y] and
A61: 1 <= k and
k <= len s1 by A59;
reconsider y9 = y as integer number by A60, XTUPLE_0:1;
( ((k - 1) * ((len s2) - 1)) + 1 = y & k - 1 >= 1 - 1 ) by A60, A61, XREAL_1:9, XTUPLE_0:1;
then ( y9 in NAT & not y in {0} ) by A19, INT_1:3, TARSKI:def_1;
hence y in NAT \ {0} by XBOOLE_0:def_5; ::_thesis: verum
end;
then f .: (Seg (len s1)) c= NAT \ {0} by TARSKI:def_3;
then (the_schreier_series_of (s1,s2)) * (Sgm fX) = (the_schreier_series_of (s1,s2)) * (f * (Sgm (Seg (len s1)))) by A30, A58, Lm38
.= ((the_schreier_series_of (s1,s2)) * f) * (Sgm (Seg (len s1))) by RELAT_1:36
.= s1 * (Sgm (Seg (len s1))) by A36, A37, FUNCT_1:2
.= s1 * (idseq (len s1)) by FINSEQ_3:48
.= s1 * (id (Seg (len s1))) by FINSEQ_2:def_1
.= s1 * (id (dom s1)) by FINSEQ_1:def_3 ;
hence s1 = (the_schreier_series_of (s1,s2)) * (Sgm fX) by A15; ::_thesis: verum
end;
hence the_schreier_series_of (s1,s2) is_finer_than s1 by Def29; ::_thesis: verum
end;
theorem Th117: :: GROUP_9:117
for O being set
for G being GroupWithOperators of O
for s1, s2 being CompositionSeries of G st len s1 > 1 & len s2 > 1 holds
the_schreier_series_of (s1,s2) is_equivalent_with the_schreier_series_of (s2,s1)
proof
let O be set ; ::_thesis: for G being GroupWithOperators of O
for s1, s2 being CompositionSeries of G st len s1 > 1 & len s2 > 1 holds
the_schreier_series_of (s1,s2) is_equivalent_with the_schreier_series_of (s2,s1)
let G be GroupWithOperators of O; ::_thesis: for s1, s2 being CompositionSeries of G st len s1 > 1 & len s2 > 1 holds
the_schreier_series_of (s1,s2) is_equivalent_with the_schreier_series_of (s2,s1)
let s1, s2 be CompositionSeries of G; ::_thesis: ( len s1 > 1 & len s2 > 1 implies the_schreier_series_of (s1,s2) is_equivalent_with the_schreier_series_of (s2,s1) )
assume that
A1: len s1 > 1 and
A2: len s2 > 1 ; ::_thesis: the_schreier_series_of (s1,s2) is_equivalent_with the_schreier_series_of (s2,s1)
set s21 = the_schreier_series_of (s2,s1);
A3: ( (len s1) - 1 > 1 - 1 & (len s2) - 1 > 1 - 1 ) by A1, A2, XREAL_1:9;
set s12 = the_schreier_series_of (s1,s2);
A4: len (the_schreier_series_of (s1,s2)) = (((len s1) - 1) * ((len s2) - 1)) + 1 by A1, A2, Def35;
A5: len (the_schreier_series_of (s2,s1)) = (((len s1) - 1) * ((len s2) - 1)) + 1 by A1, A2, Def35;
then A6: not the_schreier_series_of (s2,s1) is empty by A3;
((len s1) - 1) * ((len s2) - 1) > 0 * ((len s2) - 1) by A3, XREAL_1:68;
then A7: (((len s1) - 1) * ((len s2) - 1)) + 1 > 0 + 1 by XREAL_1:6;
A8: now__::_thesis:_ex_p_being_Permutation_of_(dom_(the_series_of_quotients_of_(the_schreier_series_of_(s1,s2))))_st_the_series_of_quotients_of_(the_schreier_series_of_(s1,s2)),_the_series_of_quotients_of_(the_schreier_series_of_(s2,s1))_are_equivalent_under_p,O
set p = { [(((j - 1) * ((len s1) - 1)) + i),(((i - 1) * ((len s2) - 1)) + j)] where i, j is Element of NAT : ( 1 <= i & i <= (len s1) - 1 & 1 <= j & j <= (len s2) - 1 ) } ;
now__::_thesis:_for_x_being_set_st_x_in__{__[(((j_-_1)_*_((len_s1)_-_1))_+_i),(((i_-_1)_*_((len_s2)_-_1))_+_j)]_where_i,_j_is_Element_of_NAT_:_(_1_<=_i_&_i_<=_(len_s1)_-_1_&_1_<=_j_&_j_<=_(len_s2)_-_1_)__}__holds_
ex_y,_z_being_set_st_x_=_[y,z]
let x be set ; ::_thesis: ( x in { [(((j - 1) * ((len s1) - 1)) + i),(((i - 1) * ((len s2) - 1)) + j)] where i, j is Element of NAT : ( 1 <= i & i <= (len s1) - 1 & 1 <= j & j <= (len s2) - 1 ) } implies ex y, z being set st x = [y,z] )
assume x in { [(((j - 1) * ((len s1) - 1)) + i),(((i - 1) * ((len s2) - 1)) + j)] where i, j is Element of NAT : ( 1 <= i & i <= (len s1) - 1 & 1 <= j & j <= (len s2) - 1 ) } ; ::_thesis: ex y, z being set st x = [y,z]
then consider i, j being Element of NAT such that
A9: [(((j - 1) * ((len s1) - 1)) + i),(((i - 1) * ((len s2) - 1)) + j)] = x and
1 <= i and
i <= (len s1) - 1 and
1 <= j and
j <= (len s2) - 1 ;
set z = ((i - 1) * ((len s2) - 1)) + j;
set y = ((j - 1) * ((len s1) - 1)) + i;
reconsider y = ((j - 1) * ((len s1) - 1)) + i, z = ((i - 1) * ((len s2) - 1)) + j as set ;
take y = y; ::_thesis: ex z being set st x = [y,z]
take z = z; ::_thesis: x = [y,z]
thus x = [y,z] by A9; ::_thesis: verum
end;
then reconsider p = { [(((j - 1) * ((len s1) - 1)) + i),(((i - 1) * ((len s2) - 1)) + j)] where i, j is Element of NAT : ( 1 <= i & i <= (len s1) - 1 & 1 <= j & j <= (len s2) - 1 ) } as Relation by RELAT_1:def_1;
set X = dom (the_series_of_quotients_of (the_schreier_series_of (s1,s2)));
set f1 = the_series_of_quotients_of (the_schreier_series_of (s1,s2));
set f2 = the_series_of_quotients_of (the_schreier_series_of (s2,s1));
now__::_thesis:_for_x,_y1,_y2_being_set_st_[x,y1]_in_p_&_[x,y2]_in_p_holds_
y1_=_y2
let x, y1, y2 be set ; ::_thesis: ( [x,y1] in p & [x,y2] in p implies y1 = y2 )
assume [x,y1] in p ; ::_thesis: ( [x,y2] in p implies y1 = y2 )
then consider i1, j1 being Element of NAT such that
A10: [(((j1 - 1) * ((len s1) - 1)) + i1),(((i1 - 1) * ((len s2) - 1)) + j1)] = [x,y1] and
A11: ( 1 <= i1 & i1 <= (len s1) - 1 & 1 <= j1 ) and
j1 <= (len s2) - 1 ;
A12: ((j1 - 1) * ((len s1) - 1)) + i1 = x by A10, XTUPLE_0:1;
assume [x,y2] in p ; ::_thesis: y1 = y2
then consider i2, j2 being Element of NAT such that
A13: [(((j2 - 1) * ((len s1) - 1)) + i2),(((i2 - 1) * ((len s2) - 1)) + j2)] = [x,y2] and
A14: ( 1 <= i2 & i2 <= (len s1) - 1 & 1 <= j2 ) and
j2 <= (len s2) - 1 ;
A15: ((j2 - 1) * ((len s1) - 1)) + i2 = x by A13, XTUPLE_0:1;
then j1 = j2 by A1, A11, A14, A12, Lm46;
hence y1 = y2 by A10, A13, A12, A15, XTUPLE_0:1; ::_thesis: verum
end;
then reconsider p = p as Function by FUNCT_1:def_1;
A16: len (the_schreier_series_of (s1,s2)) > 1 by A1, A2, A7, Def35;
then A17: (len (the_series_of_quotients_of (the_schreier_series_of (s1,s2)))) + 1 = len (the_schreier_series_of (s1,s2)) by Def33;
A18: len (the_schreier_series_of (s1,s2)) = (((len s1) - 1) * ((len s2) - 1)) + 1 by A1, A2, Def35;
now__::_thesis:_for_y_being_set_st_y_in_dom_(the_series_of_quotients_of_(the_schreier_series_of_(s1,s2)))_holds_
y_in_rng_p
set l9 = ((len s1) - 1) * ((len s2) - 1);
reconsider l9 = ((len s1) - 1) * ((len s2) - 1) as Element of NAT by A3, INT_1:3;
let y be set ; ::_thesis: ( y in dom (the_series_of_quotients_of (the_schreier_series_of (s1,s2))) implies y in rng p )
assume A19: y in dom (the_series_of_quotients_of (the_schreier_series_of (s1,s2))) ; ::_thesis: y in rng p
then reconsider k = y as Element of NAT ;
A20: y in Seg (len (the_series_of_quotients_of (the_schreier_series_of (s1,s2)))) by A19, FINSEQ_1:def_3;
then A21: 1 <= k by FINSEQ_1:1;
A22: k <= ((len s1) - 1) * ((len s2) - 1) by A17, A18, A20, FINSEQ_1:1;
0 + (((len s1) - 1) * ((len s2) - 1)) <= 1 + (((len s1) - 1) * ((len s2) - 1)) by XREAL_1:6;
then k <= l9 + 1 by A22, XXREAL_0:2;
then A23: k in Seg (l9 + 1) by A21;
k <> l9 + 1 by A22, NAT_1:13;
then consider i, j being Nat such that
A24: ( k = ((i - 1) * ((len s2) - 1)) + j & 1 <= i & i <= (len s1) - 1 & 1 <= j & j <= (len s2) - 1 ) by A1, A2, A23, Lm45;
reconsider j = j, i = i as Element of NAT by INT_1:3;
set x = ((j - 1) * ((len s1) - 1)) + i;
reconsider x = ((j - 1) * ((len s1) - 1)) + i as set ;
[x,y] in p by A24;
hence y in rng p by XTUPLE_0:def_13; ::_thesis: verum
end;
then A25: dom (the_series_of_quotients_of (the_schreier_series_of (s1,s2))) c= rng p by TARSKI:def_3;
A26: dom (the_series_of_quotients_of (the_schreier_series_of (s1,s2))) = Seg (len (the_series_of_quotients_of (the_schreier_series_of (s1,s2)))) by FINSEQ_1:def_3;
now__::_thesis:_for_x_being_set_st_x_in_dom_(the_series_of_quotients_of_(the_schreier_series_of_(s1,s2)))_holds_
x_in_dom_p
set l9 = ((len s1) - 1) * ((len s2) - 1);
reconsider l9 = ((len s1) - 1) * ((len s2) - 1) as Element of NAT by A3, INT_1:3;
let x be set ; ::_thesis: ( x in dom (the_series_of_quotients_of (the_schreier_series_of (s1,s2))) implies x in dom p )
assume A27: x in dom (the_series_of_quotients_of (the_schreier_series_of (s1,s2))) ; ::_thesis: x in dom p
then reconsider k = x as Element of NAT ;
A28: k <= ((len s1) - 1) * ((len s2) - 1) by A17, A18, A26, A27, FINSEQ_1:1;
0 + (((len s1) - 1) * ((len s2) - 1)) <= 1 + (((len s1) - 1) * ((len s2) - 1)) by XREAL_1:6;
then A29: k <= l9 + 1 by A28, XXREAL_0:2;
1 <= k by A26, A27, FINSEQ_1:1;
then A30: k in Seg (l9 + 1) by A29;
k <> l9 + 1 by A28, NAT_1:13;
then consider j, i being Nat such that
A31: ( k = ((j - 1) * ((len s1) - 1)) + i & 1 <= j & j <= (len s2) - 1 & 1 <= i & i <= (len s1) - 1 ) by A1, A2, A30, Lm45;
reconsider j = j, i = i as Element of NAT by INT_1:3;
set y = ((i - 1) * ((len s2) - 1)) + j;
reconsider y = ((i - 1) * ((len s2) - 1)) + j as set ;
[x,y] in p by A31;
hence x in dom p by XTUPLE_0:def_12; ::_thesis: verum
end;
then A32: dom (the_series_of_quotients_of (the_schreier_series_of (s1,s2))) c= dom p by TARSKI:def_3;
now__::_thesis:_for_y_being_set_st_y_in_rng_p_holds_
y_in_dom_(the_series_of_quotients_of_(the_schreier_series_of_(s1,s2)))
let y be set ; ::_thesis: ( y in rng p implies y in dom (the_series_of_quotients_of (the_schreier_series_of (s1,s2))) )
set k = y;
assume y in rng p ; ::_thesis: y in dom (the_series_of_quotients_of (the_schreier_series_of (s1,s2)))
then consider x being set such that
A33: [x,y] in p by XTUPLE_0:def_13;
consider i, j being Element of NAT such that
A34: [(((j - 1) * ((len s1) - 1)) + i),(((i - 1) * ((len s2) - 1)) + j)] = [x,y] and
A35: ( 1 <= i & i <= (len s1) - 1 & 1 <= j & j <= (len s2) - 1 ) by A33;
A36: y = ((i - 1) * ((len s2) - 1)) + j by A34, XTUPLE_0:1;
reconsider k = y as integer number by A34, XTUPLE_0:1;
1 <= k by A2, A35, A36, Lm47;
then reconsider k = k as Element of NAT by INT_1:3;
( 1 <= k & k <= len (the_series_of_quotients_of (the_schreier_series_of (s1,s2))) ) by A2, A17, A18, A35, A36, Lm47;
hence y in dom (the_series_of_quotients_of (the_schreier_series_of (s1,s2))) by A26; ::_thesis: verum
end;
then rng p c= dom (the_series_of_quotients_of (the_schreier_series_of (s1,s2))) by TARSKI:def_3;
then A37: rng p = dom (the_series_of_quotients_of (the_schreier_series_of (s1,s2))) by A25, XBOOLE_0:def_10;
now__::_thesis:_for_x_being_set_st_x_in_dom_p_holds_
x_in_dom_(the_series_of_quotients_of_(the_schreier_series_of_(s1,s2)))
let x be set ; ::_thesis: ( x in dom p implies x in dom (the_series_of_quotients_of (the_schreier_series_of (s1,s2))) )
set k = x;
assume x in dom p ; ::_thesis: x in dom (the_series_of_quotients_of (the_schreier_series_of (s1,s2)))
then consider y being set such that
A38: [x,y] in p by XTUPLE_0:def_12;
consider i, j being Element of NAT such that
A39: [(((j - 1) * ((len s1) - 1)) + i),(((i - 1) * ((len s2) - 1)) + j)] = [x,y] and
A40: ( 1 <= i & i <= (len s1) - 1 & 1 <= j & j <= (len s2) - 1 ) by A38;
A41: x = ((j - 1) * ((len s1) - 1)) + i by A39, XTUPLE_0:1;
reconsider k = x as integer number by A39, XTUPLE_0:1;
1 <= k by A1, A40, A41, Lm47;
then reconsider k = k as Element of NAT by INT_1:3;
( 1 <= k & k <= len (the_series_of_quotients_of (the_schreier_series_of (s1,s2))) ) by A1, A17, A18, A40, A41, Lm47;
hence x in dom (the_series_of_quotients_of (the_schreier_series_of (s1,s2))) by A26; ::_thesis: verum
end;
then dom p c= dom (the_series_of_quotients_of (the_schreier_series_of (s1,s2))) by TARSKI:def_3;
then A42: dom p = dom (the_series_of_quotients_of (the_schreier_series_of (s1,s2))) by A32, XBOOLE_0:def_10;
then reconsider p = p as Function of (dom (the_series_of_quotients_of (the_schreier_series_of (s1,s2)))),(dom (the_series_of_quotients_of (the_schreier_series_of (s1,s2)))) by A37, FUNCT_2:1;
A43: p is onto by A37, FUNCT_2:def_3;
now__::_thesis:_for_x1,_x2_being_set_st_x1_in_dom_(the_series_of_quotients_of_(the_schreier_series_of_(s1,s2)))_&_x2_in_dom_(the_series_of_quotients_of_(the_schreier_series_of_(s1,s2)))_&_p_._x1_=_p_._x2_holds_
x1_=_x2
let x1, x2 be set ; ::_thesis: ( x1 in dom (the_series_of_quotients_of (the_schreier_series_of (s1,s2))) & x2 in dom (the_series_of_quotients_of (the_schreier_series_of (s1,s2))) & p . x1 = p . x2 implies x1 = x2 )
assume that
A44: x1 in dom (the_series_of_quotients_of (the_schreier_series_of (s1,s2))) and
A45: x2 in dom (the_series_of_quotients_of (the_schreier_series_of (s1,s2))) ; ::_thesis: ( p . x1 = p . x2 implies x1 = x2 )
assume A46: p . x1 = p . x2 ; ::_thesis: x1 = x2
[x1,(p . x1)] in p by A32, A44, FUNCT_1:def_2;
then consider i1, j1 being Element of NAT such that
A47: [(((j1 - 1) * ((len s1) - 1)) + i1),(((i1 - 1) * ((len s2) - 1)) + j1)] = [x1,(p . x1)] and
A48: 1 <= i1 and
i1 <= (len s1) - 1 and
A49: ( 1 <= j1 & j1 <= (len s2) - 1 ) ;
[x2,(p . x2)] in p by A32, A45, FUNCT_1:def_2;
then consider i2, j2 being Element of NAT such that
A50: [(((j2 - 1) * ((len s1) - 1)) + i2),(((i2 - 1) * ((len s2) - 1)) + j2)] = [x2,(p . x2)] and
A51: 1 <= i2 and
i2 <= (len s1) - 1 and
A52: ( 1 <= j2 & j2 <= (len s2) - 1 ) ;
A53: ((i2 - 1) * ((len s2) - 1)) + j2 = p . x2 by A50, XTUPLE_0:1;
A54: ((i1 - 1) * ((len s2) - 1)) + j1 = p . x1 by A47, XTUPLE_0:1;
then i1 = i2 by A2, A46, A48, A49, A51, A52, A53, Lm46;
hence x1 = x2 by A46, A47, A50, A54, A53, XTUPLE_0:1; ::_thesis: verum
end;
then p is one-to-one by FUNCT_2:56;
then reconsider p = p as Permutation of (dom (the_series_of_quotients_of (the_schreier_series_of (s1,s2)))) by A43;
take p = p; ::_thesis: the_series_of_quotients_of (the_schreier_series_of (s1,s2)), the_series_of_quotients_of (the_schreier_series_of (s2,s1)) are_equivalent_under p,O
A55: len (the_schreier_series_of (s2,s1)) > 1 by A1, A2, A7, Def35;
then A56: (len (the_series_of_quotients_of (the_schreier_series_of (s2,s1)))) + 1 = len (the_schreier_series_of (s2,s1)) by Def33;
now__::_thesis:_for_H1,_H2_being_GroupWithOperators_of_O
for_k1,_k2_being_Nat_st_k1_in_dom_(the_series_of_quotients_of_(the_schreier_series_of_(s1,s2)))_&_k2_=_(p_")_._k1_&_H1_=_(the_series_of_quotients_of_(the_schreier_series_of_(s1,s2)))_._k1_&_H2_=_(the_series_of_quotients_of_(the_schreier_series_of_(s2,s1)))_._k2_holds_
H1,H2_are_isomorphic
(len s2) + 1 > 1 + 1 by A2, XREAL_1:6;
then len s2 >= 2 by NAT_1:13;
then A57: (len s2) - 1 >= 2 - 1 by XREAL_1:9;
set l = (((len s1) - 1) * ((len s2) - 1)) + 1;
let H1, H2 be GroupWithOperators of O; ::_thesis: for k1, k2 being Nat st k1 in dom (the_series_of_quotients_of (the_schreier_series_of (s1,s2))) & k2 = (p ") . k1 & H1 = (the_series_of_quotients_of (the_schreier_series_of (s1,s2))) . k1 & H2 = (the_series_of_quotients_of (the_schreier_series_of (s2,s1))) . k2 holds
H1,H2 are_isomorphic
let k1, k2 be Nat; ::_thesis: ( k1 in dom (the_series_of_quotients_of (the_schreier_series_of (s1,s2))) & k2 = (p ") . k1 & H1 = (the_series_of_quotients_of (the_schreier_series_of (s1,s2))) . k1 & H2 = (the_series_of_quotients_of (the_schreier_series_of (s2,s1))) . k2 implies H1,H2 are_isomorphic )
assume that
A58: k1 in dom (the_series_of_quotients_of (the_schreier_series_of (s1,s2))) and
A59: k2 = (p ") . k1 ; ::_thesis: ( H1 = (the_series_of_quotients_of (the_schreier_series_of (s1,s2))) . k1 & H2 = (the_series_of_quotients_of (the_schreier_series_of (s2,s1))) . k2 implies H1,H2 are_isomorphic )
(len s1) + 1 > 1 + 1 by A1, XREAL_1:6;
then len s1 >= 2 by NAT_1:13;
then (len s1) - 1 >= 2 - 1 by XREAL_1:9;
then reconsider l = (((len s1) - 1) * ((len s2) - 1)) + 1 as Element of NAT by A57, INT_1:3;
assume that
A60: H1 = (the_series_of_quotients_of (the_schreier_series_of (s1,s2))) . k1 and
A61: H2 = (the_series_of_quotients_of (the_schreier_series_of (s2,s1))) . k2 ; ::_thesis: H1,H2 are_isomorphic
A62: len (the_schreier_series_of (s1,s2)) = (((len s1) - 1) * ((len s2) - 1)) + 1 by A1, A2, Def35;
0 + (((len s1) - 1) * ((len s2) - 1)) <= 1 + (((len s1) - 1) * ((len s2) - 1)) by XREAL_1:6;
then A63: Seg (len (the_series_of_quotients_of (the_schreier_series_of (s1,s2)))) c= Seg l by A17, A62, FINSEQ_1:5;
A64: k1 in Seg (len (the_series_of_quotients_of (the_schreier_series_of (s1,s2)))) by A58, FINSEQ_1:def_3;
then k1 <= len (the_series_of_quotients_of (the_schreier_series_of (s1,s2))) by FINSEQ_1:1;
then k1 <> (((len s1) - 1) * ((len s2) - 1)) + 1 by A17, A62, NAT_1:13;
then consider i, j being Nat such that
A65: k1 = ((i - 1) * ((len s2) - 1)) + j and
A66: 1 <= i and
A67: i <= (len s1) - 1 and
A68: 1 <= j and
A69: j <= (len s2) - 1 by A1, A2, A64, A63, Lm45;
reconsider H = s1 . i, K = s2 . j, H9 = s1 . (i + 1), K9 = s2 . (j + 1) as strict StableSubgroup of G by A66, A67, A68, A69, Th111;
A70: (p ") . k1 in rng (p ") by A58, FUNCT_2:4;
p . k2 = k1 by A25, A58, A59, FUNCT_1:35;
then [k2,k1] in p by A42, A59, A70, FUNCT_1:1;
then consider i9, j9 being Element of NAT such that
A71: [k2,k1] = [(((j9 - 1) * ((len s1) - 1)) + i9),(((i9 - 1) * ((len s2) - 1)) + j9)] and
A72: 1 <= i9 and
i9 <= (len s1) - 1 and
A73: ( 1 <= j9 & j9 <= (len s2) - 1 ) ;
set JK = K9 "\/" (K /\ H9);
A74: ((i - 1) * ((len s2) - 1)) + j = ((i9 - 1) * ((len s2) - 1)) + j9 by A65, A71, XTUPLE_0:1;
then A75: i = i9 by A2, A66, A68, A69, A72, A73, Lm46;
A76: now__::_thesis:_(the_schreier_series_of_(s2,s1))_._(k2_+_1)_=_K9_"\/"_(K_/\_H9)
percases ( i = (len s1) - 1 or i <> (len s1) - 1 ) ;
supposeA77: i = (len s1) - 1 ; ::_thesis: (the_schreier_series_of (s2,s1)) . (k2 + 1) = K9 "\/" (K /\ H9)
percases ( j <> (len s2) - 1 or j = (len s2) - 1 ) ;
supposeA78: j <> (len s2) - 1 ; ::_thesis: (the_schreier_series_of (s2,s1)) . (k2 + 1) = K9 "\/" (K /\ H9)
set j9 = j + 1;
A79: 0 + (j + 1) <= 1 + (j + 1) by XREAL_1:6;
set i9 = 1;
set H3 = s1 . 1;
H9 = (1). G by A77, Def28;
then A80: K9 "\/" (K /\ H9) = K9 "\/" ((1). G) by Th21
.= K9 by Th33 ;
set H2 = s2 . (j + 1);
set H1 = s2 . ((j + 1) + 1);
1 + 1 <= j + 1 by A68, XREAL_1:6;
then A81: 1 <= j + 1 by XXREAL_0:2;
j < (len s2) - 1 by A69, A78, XXREAL_0:1;
then A82: j + 1 <= (len s2) - 1 by INT_1:7;
then A83: (j + 1) + 1 <= ((len s2) - 1) + 1 by XREAL_1:6;
then j + 1 <= len s2 by A79, XXREAL_0:2;
then j + 1 in Seg (len s2) by A81;
then A84: j + 1 in dom s2 by FINSEQ_1:def_3;
(len s1) - 1 > 1 - 1 by A1, XREAL_1:9;
then A85: (len s1) - 1 >= 0 + 1 by INT_1:7;
then reconsider H1 = s2 . ((j + 1) + 1), H2 = s2 . (j + 1), H3 = s1 . 1 as strict StableSubgroup of G by A82, A81, Th111;
A86: H3 = (Omega). G by Def28;
now__::_thesis:_for_x_being_set_st_x_in_the_carrier_of_H2_holds_
x_in_the_carrier_of_((Omega)._G)
let x be set ; ::_thesis: ( x in the carrier of H2 implies x in the carrier of ((Omega). G) )
H2 is Subgroup of G by Def7;
then A87: the carrier of H2 c= the carrier of G by GROUP_2:def_5;
assume x in the carrier of H2 ; ::_thesis: x in the carrier of ((Omega). G)
hence x in the carrier of ((Omega). G) by A87; ::_thesis: verum
end;
then the carrier of H2 c= the carrier of ((Omega). G) by TARSKI:def_3;
then A88: the carrier of H2 = the carrier of H2 /\ the carrier of ((Omega). G) by XBOOLE_1:28;
k2 + 1 = (((j + 1) - 1) * ((len s1) - 1)) + 1 by A71, A74, A75, A77, XTUPLE_0:1;
then (the_schreier_series_of (s2,s1)) . (k2 + 1) = H1 "\/" (H2 /\ H3) by A1, A2, A82, A81, A85, Def35;
then A89: (the_schreier_series_of (s2,s1)) . (k2 + 1) = H1 "\/" H2 by A86, A88, Th18;
1 <= (j + 1) + 1 by A81, A79, XXREAL_0:2;
then (j + 1) + 1 in Seg (len s2) by A83;
then (j + 1) + 1 in dom s2 by FINSEQ_1:def_3;
then H1 is normal StableSubgroup of H2 by A84, Def28;
hence (the_schreier_series_of (s2,s1)) . (k2 + 1) = K9 "\/" (K /\ H9) by A89, A80, Th36; ::_thesis: verum
end;
supposeA90: j = (len s2) - 1 ; ::_thesis: (the_schreier_series_of (s2,s1)) . (k2 + 1) = K9 "\/" (K /\ H9)
then A91: K9 = (1). G by Def28;
H9 = (1). G by A77, Def28;
then A92: K9 "\/" (K /\ H9) = ((1). G) "\/" ((1). G) by A91, Th21
.= (1). G by Th33 ;
k2 = ((len s1) - 1) * ((len s2) - 1) by A71, A74, A75, A77, A90, XTUPLE_0:1;
hence (the_schreier_series_of (s2,s1)) . (k2 + 1) = K9 "\/" (K /\ H9) by A1, A2, A92, Def35; ::_thesis: verum
end;
end;
end;
suppose i <> (len s1) - 1 ; ::_thesis: (the_schreier_series_of (s2,s1)) . (k2 + 1) = K9 "\/" (K /\ H9)
then i < (len s1) - 1 by A67, XXREAL_0:1;
then A93: i + 1 <= (len s1) - 1 by INT_1:7;
set i9 = i + 1;
set k29 = k2 + 1;
1 + 1 <= i + 1 by A66, XREAL_1:6;
then A94: 1 <= i + 1 by XXREAL_0:2;
k2 + 1 = (((j - 1) * ((len s1) - 1)) + i) + 1 by A71, A74, A75, XTUPLE_0:1;
then k2 + 1 = ((j - 1) * ((len s1) - 1)) + (i + 1) ;
hence (the_schreier_series_of (s2,s1)) . (k2 + 1) = K9 "\/" (K /\ H9) by A1, A2, A68, A69, A93, A94, Def35; ::_thesis: verum
end;
end;
end;
rng (p ") c= dom (the_series_of_quotients_of (the_schreier_series_of (s1,s2))) ;
then rng (p ") c= Seg (len (the_series_of_quotients_of (the_schreier_series_of (s1,s2)))) by FINSEQ_1:def_3;
then (p ") . k1 in Seg (len (the_series_of_quotients_of (the_schreier_series_of (s1,s2)))) by A70;
then A95: k2 in dom (the_series_of_quotients_of (the_schreier_series_of (s2,s1))) by A4, A5, A17, A56, A59, FINSEQ_1:def_3;
A96: ( H9 is normal StableSubgroup of H & K9 is normal StableSubgroup of K ) by A66, A67, A68, A69, Th112;
then reconsider JK = K9 "\/" (K /\ H9) as normal StableSubgroup of K9 "\/" (K /\ H) by Th92;
k2 = ((j - 1) * ((len s1) - 1)) + i by A71, A74, A75, XTUPLE_0:1;
then (the_schreier_series_of (s2,s1)) . k2 = K9 "\/" (K /\ H) by A1, A2, A66, A67, A68, A69, Def35;
then A97: H2 = (K9 "\/" (K /\ H)) ./. JK by A55, A61, A95, A76, Def33;
set JH = H9 "\/" (H /\ K9);
A98: now__::_thesis:_(the_schreier_series_of_(s1,s2))_._(k1_+_1)_=_H9_"\/"_(H_/\_K9)
percases ( j = (len s2) - 1 or j <> (len s2) - 1 ) ;
supposeA99: j = (len s2) - 1 ; ::_thesis: (the_schreier_series_of (s1,s2)) . (k1 + 1) = H9 "\/" (H /\ K9)
percases ( i <> (len s1) - 1 or i = (len s1) - 1 ) ;
supposeA100: i <> (len s1) - 1 ; ::_thesis: (the_schreier_series_of (s1,s2)) . (k1 + 1) = H9 "\/" (H /\ K9)
set j9 = 1;
set H3 = s2 . 1;
set i9 = i + 1;
A101: 0 + (i + 1) <= 1 + (i + 1) by XREAL_1:6;
set H2 = s1 . (i + 1);
set H1 = s1 . ((i + 1) + 1);
1 + 1 <= i + 1 by A66, XREAL_1:6;
then A102: 1 <= i + 1 by XXREAL_0:2;
i < (len s1) - 1 by A67, A100, XXREAL_0:1;
then A103: i + 1 <= (len s1) - 1 by INT_1:7;
then A104: (i + 1) + 1 <= ((len s1) - 1) + 1 by XREAL_1:6;
then i + 1 <= len s1 by A101, XXREAL_0:2;
then i + 1 in Seg (len s1) by A102;
then A105: i + 1 in dom s1 by FINSEQ_1:def_3;
(len s2) - 1 > 1 - 1 by A2, XREAL_1:9;
then A106: (len s2) - 1 >= 0 + 1 by INT_1:7;
then reconsider H1 = s1 . ((i + 1) + 1), H2 = s1 . (i + 1), H3 = s2 . 1 as strict StableSubgroup of G by A103, A102, Th111;
A107: H3 = (Omega). G by Def28;
now__::_thesis:_for_x_being_set_st_x_in_the_carrier_of_H2_holds_
x_in_the_carrier_of_((Omega)._G)
let x be set ; ::_thesis: ( x in the carrier of H2 implies x in the carrier of ((Omega). G) )
H2 is Subgroup of G by Def7;
then A108: the carrier of H2 c= the carrier of G by GROUP_2:def_5;
assume x in the carrier of H2 ; ::_thesis: x in the carrier of ((Omega). G)
hence x in the carrier of ((Omega). G) by A108; ::_thesis: verum
end;
then the carrier of H2 c= the carrier of ((Omega). G) by TARSKI:def_3;
then A109: the carrier of H2 = the carrier of H2 /\ the carrier of ((Omega). G) by XBOOLE_1:28;
k1 + 1 = (((i + 1) - 1) * ((len s2) - 1)) + 1 by A65, A99;
then (the_schreier_series_of (s1,s2)) . (k1 + 1) = H1 "\/" (H2 /\ H3) by A1, A2, A103, A102, A106, Def35;
then A110: (the_schreier_series_of (s1,s2)) . (k1 + 1) = H1 "\/" H2 by A107, A109, Th18;
1 <= (i + 1) + 1 by A102, A101, XXREAL_0:2;
then (i + 1) + 1 in Seg (len s1) by A104;
then (i + 1) + 1 in dom s1 by FINSEQ_1:def_3;
then A111: H1 is normal StableSubgroup of H2 by A105, Def28;
H9 "\/" (H /\ K9) = H9 "\/" (H /\ ((1). G)) by A99, Def28
.= H9 "\/" ((1). G) by Th21
.= H9 by Th33 ;
hence (the_schreier_series_of (s1,s2)) . (k1 + 1) = H9 "\/" (H /\ K9) by A110, A111, Th36; ::_thesis: verum
end;
supposeA112: i = (len s1) - 1 ; ::_thesis: (the_schreier_series_of (s1,s2)) . (k1 + 1) = H9 "\/" (H /\ K9)
then A113: k1 = ((len s1) - 1) * ((len s2) - 1) by A65, A99;
A114: K9 = (1). G by A99, Def28;
H9 = (1). G by A112, Def28;
then H9 "\/" (H /\ K9) = ((1). G) "\/" ((1). G) by A114, Th21
.= (1). G by Th33 ;
hence (the_schreier_series_of (s1,s2)) . (k1 + 1) = H9 "\/" (H /\ K9) by A1, A2, A113, Def35; ::_thesis: verum
end;
end;
end;
suppose j <> (len s2) - 1 ; ::_thesis: (the_schreier_series_of (s1,s2)) . (k1 + 1) = H9 "\/" (H /\ K9)
then j < (len s2) - 1 by A69, XXREAL_0:1;
then A115: j + 1 <= (len s2) - 1 by INT_1:7;
set j9 = j + 1;
set k19 = k1 + 1;
1 + 1 <= j + 1 by A68, XREAL_1:6;
then A116: 1 <= j + 1 by XXREAL_0:2;
k1 + 1 = ((i - 1) * ((len s2) - 1)) + (j + 1) by A65;
hence (the_schreier_series_of (s1,s2)) . (k1 + 1) = H9 "\/" (H /\ K9) by A1, A2, A66, A67, A115, A116, Def35; ::_thesis: verum
end;
end;
end;
reconsider JH = H9 "\/" (H /\ K9) as normal StableSubgroup of H9 "\/" (H /\ K) by A96, Th92;
(the_schreier_series_of (s1,s2)) . k1 = H9 "\/" (H /\ K) by A1, A2, A65, A66, A67, A68, A69, Def35;
then H1 = (H9 "\/" (H /\ K)) ./. JH by A16, A58, A60, A98, Def33;
hence H1,H2 are_isomorphic by A96, A97, Th93; ::_thesis: verum
end;
hence the_series_of_quotients_of (the_schreier_series_of (s1,s2)), the_series_of_quotients_of (the_schreier_series_of (s2,s1)) are_equivalent_under p,O by A4, A5, A17, A56, Def34; ::_thesis: verum
end;
not the_schreier_series_of (s1,s2) is empty by A3, A4;
hence the_schreier_series_of (s1,s2) is_equivalent_with the_schreier_series_of (s2,s1) by A6, A8, Th108; ::_thesis: verum
end;
theorem Th118: :: GROUP_9:118
for O being set
for G being GroupWithOperators of O
for s1, s2 being CompositionSeries of G ex s19, s29 being CompositionSeries of G st
( s19 is_finer_than s1 & s29 is_finer_than s2 & s19 is_equivalent_with s29 )
proof
let O be set ; ::_thesis: for G being GroupWithOperators of O
for s1, s2 being CompositionSeries of G ex s19, s29 being CompositionSeries of G st
( s19 is_finer_than s1 & s29 is_finer_than s2 & s19 is_equivalent_with s29 )
let G be GroupWithOperators of O; ::_thesis: for s1, s2 being CompositionSeries of G ex s19, s29 being CompositionSeries of G st
( s19 is_finer_than s1 & s29 is_finer_than s2 & s19 is_equivalent_with s29 )
let s1, s2 be CompositionSeries of G; ::_thesis: ex s19, s29 being CompositionSeries of G st
( s19 is_finer_than s1 & s29 is_finer_than s2 & s19 is_equivalent_with s29 )
percases ( ( len s1 > 1 & len s2 > 1 ) or len s1 <= 1 or len s2 <= 1 ) ;
supposeA1: ( len s1 > 1 & len s2 > 1 ) ; ::_thesis: ex s19, s29 being CompositionSeries of G st
( s19 is_finer_than s1 & s29 is_finer_than s2 & s19 is_equivalent_with s29 )
set s29 = the_schreier_series_of (s2,s1);
set s19 = the_schreier_series_of (s1,s2);
take the_schreier_series_of (s1,s2) ; ::_thesis: ex s29 being CompositionSeries of G st
( the_schreier_series_of (s1,s2) is_finer_than s1 & s29 is_finer_than s2 & the_schreier_series_of (s1,s2) is_equivalent_with s29 )
take the_schreier_series_of (s2,s1) ; ::_thesis: ( the_schreier_series_of (s1,s2) is_finer_than s1 & the_schreier_series_of (s2,s1) is_finer_than s2 & the_schreier_series_of (s1,s2) is_equivalent_with the_schreier_series_of (s2,s1) )
thus ( the_schreier_series_of (s1,s2) is_finer_than s1 & the_schreier_series_of (s2,s1) is_finer_than s2 ) by A1, Th116; ::_thesis: the_schreier_series_of (s1,s2) is_equivalent_with the_schreier_series_of (s2,s1)
thus the_schreier_series_of (s1,s2) is_equivalent_with the_schreier_series_of (s2,s1) by A1, Th117; ::_thesis: verum
end;
supposeA2: ( len s1 <= 1 or len s2 <= 1 ) ; ::_thesis: ex s19, s29 being CompositionSeries of G st
( s19 is_finer_than s1 & s29 is_finer_than s2 & s19 is_equivalent_with s29 )
percases ( len s1 <= len s2 or len s1 > len s2 ) ;
supposeA3: len s1 <= len s2 ; ::_thesis: ex s19, s29 being CompositionSeries of G st
( s19 is_finer_than s1 & s29 is_finer_than s2 & s19 is_equivalent_with s29 )
set s29 = s2;
set s19 = s2;
take s2 ; ::_thesis: ex s29 being CompositionSeries of G st
( s2 is_finer_than s1 & s29 is_finer_than s2 & s2 is_equivalent_with s29 )
take s2 ; ::_thesis: ( s2 is_finer_than s1 & s2 is_finer_than s2 & s2 is_equivalent_with s2 )
thus ( s2 is_finer_than s1 & s2 is_finer_than s2 ) by A2, A3, Th114; ::_thesis: s2 is_equivalent_with s2
thus s2 is_equivalent_with s2 by Th113; ::_thesis: verum
end;
supposeA4: len s1 > len s2 ; ::_thesis: ex s19, s29 being CompositionSeries of G st
( s19 is_finer_than s1 & s29 is_finer_than s2 & s19 is_equivalent_with s29 )
set s29 = s1;
set s19 = s1;
take s1 ; ::_thesis: ex s29 being CompositionSeries of G st
( s1 is_finer_than s1 & s29 is_finer_than s2 & s1 is_equivalent_with s29 )
take s1 ; ::_thesis: ( s1 is_finer_than s1 & s1 is_finer_than s2 & s1 is_equivalent_with s1 )
thus ( s1 is_finer_than s1 & s1 is_finer_than s2 ) by A2, A4, Th114; ::_thesis: s1 is_equivalent_with s1
thus s1 is_equivalent_with s1 by Th113; ::_thesis: verum
end;
end;
end;
end;
end;
begin
theorem :: GROUP_9:119
for O being set
for G being GroupWithOperators of O
for s1, s2 being CompositionSeries of G st s1 is jordan_holder & s2 is jordan_holder holds
s1 is_equivalent_with s2
proof
let O be set ; ::_thesis: for G being GroupWithOperators of O
for s1, s2 being CompositionSeries of G st s1 is jordan_holder & s2 is jordan_holder holds
s1 is_equivalent_with s2
let G be GroupWithOperators of O; ::_thesis: for s1, s2 being CompositionSeries of G st s1 is jordan_holder & s2 is jordan_holder holds
s1 is_equivalent_with s2
let s1, s2 be CompositionSeries of G; ::_thesis: ( s1 is jordan_holder & s2 is jordan_holder implies s1 is_equivalent_with s2 )
assume A1: s1 is jordan_holder ; ::_thesis: ( not s2 is jordan_holder or s1 is_equivalent_with s2 )
assume A2: s2 is jordan_holder ; ::_thesis: s1 is_equivalent_with s2
percases ( s1 is empty or not s1 is empty ) ;
supposeA3: s1 is empty ; ::_thesis: s1 is_equivalent_with s2
now__::_thesis:_s2_is_empty
now__::_thesis:_ex_x_being_set_st_
(_x_c=_dom_s2_&_s1_=_s2_*_(Sgm_x)_)
set x = {} ;
take x = {} ; ::_thesis: ( x c= dom s2 & s1 = s2 * (Sgm x) )
thus x c= dom s2 by XBOOLE_1:2; ::_thesis: s1 = s2 * (Sgm x)
thus s1 = s2 * (Sgm x) by A3, FINSEQ_3:43; ::_thesis: verum
end;
then A4: s2 is_finer_than s1 by Def29;
assume A5: not s2 is empty ; ::_thesis: contradiction
s2 is strictly_decreasing by A2, Def31;
hence contradiction by A1, A3, A5, A4, Def31; ::_thesis: verum
end;
hence s1 is_equivalent_with s2 by A3, Th107; ::_thesis: verum
end;
supposeA6: not s1 is empty ; ::_thesis: s1 is_equivalent_with s2
defpred S1[ Nat] means for s19, s29 being CompositionSeries of G st not s19 is empty & not s29 is empty & len s19 = (len s1) + $1 & s19 is_finer_than s1 & s29 is_finer_than s2 & ex p being Permutation of (dom (the_series_of_quotients_of s19)) st the_series_of_quotients_of s19, the_series_of_quotients_of s29 are_equivalent_under p,O holds
ex p being Permutation of (dom (the_series_of_quotients_of s1)) st the_series_of_quotients_of s1, the_series_of_quotients_of s2 are_equivalent_under p,O;
A7: now__::_thesis:_not_s2_is_empty
assume A8: s2 is empty ; ::_thesis: contradiction
now__::_thesis:_ex_x_being_set_st_
(_x_c=_dom_s1_&_s2_=_s1_*_(Sgm_x)_)
set x = {} ;
take x = {} ; ::_thesis: ( x c= dom s1 & s2 = s1 * (Sgm x) )
thus x c= dom s1 by XBOOLE_1:2; ::_thesis: s2 = s1 * (Sgm x)
thus s2 = s1 * (Sgm x) by A8, FINSEQ_3:43; ::_thesis: verum
end;
then A9: s1 is_finer_than s2 by Def29;
s1 is strictly_decreasing by A1, Def31;
hence contradiction by A2, A6, A8, A9, Def31; ::_thesis: verum
end;
A10: for n being Nat st S1[n] holds
S1[n + 1]
proof
let n be Nat; ::_thesis: ( S1[n] implies S1[n + 1] )
assume A11: S1[n] ; ::_thesis: S1[n + 1]
now__::_thesis:_for_s19,_s29_being_CompositionSeries_of_G_st_not_s19_is_empty_&_not_s29_is_empty_&_len_s19_=_((len_s1)_+_n)_+_1_&_s19_is_finer_than_s1_&_s29_is_finer_than_s2_&_ex_p_being_Permutation_of_(dom_(the_series_of_quotients_of_s19))_st_the_series_of_quotients_of_s19,_the_series_of_quotients_of_s29_are_equivalent_under_p,O_holds_
S1[n_+_1]
let s19, s29 be CompositionSeries of G; ::_thesis: ( not s19 is empty & not s29 is empty & len s19 = ((len s1) + n) + 1 & s19 is_finer_than s1 & s29 is_finer_than s2 & ex p being Permutation of (dom (the_series_of_quotients_of s19)) st the_series_of_quotients_of s19, the_series_of_quotients_of s29 are_equivalent_under p,O implies S1[n + 1] )
assume that
not s19 is empty and
not s29 is empty ; ::_thesis: ( len s19 = ((len s1) + n) + 1 & s19 is_finer_than s1 & s29 is_finer_than s2 & ex p being Permutation of (dom (the_series_of_quotients_of s19)) st the_series_of_quotients_of s19, the_series_of_quotients_of s29 are_equivalent_under p,O implies S1[n + 1] )
assume A12: len s19 = ((len s1) + n) + 1 ; ::_thesis: ( s19 is_finer_than s1 & s29 is_finer_than s2 & ex p being Permutation of (dom (the_series_of_quotients_of s19)) st the_series_of_quotients_of s19, the_series_of_quotients_of s29 are_equivalent_under p,O implies S1[n + 1] )
set f1 = the_series_of_quotients_of s19;
assume A13: s19 is_finer_than s1 ; ::_thesis: ( s29 is_finer_than s2 & ex p being Permutation of (dom (the_series_of_quotients_of s19)) st the_series_of_quotients_of s19, the_series_of_quotients_of s29 are_equivalent_under p,O implies S1[n + 1] )
(n + 1) + (len s1) > 0 + (len s1) by XREAL_1:6;
then consider i being Nat such that
A14: i in dom (the_series_of_quotients_of s19) and
A15: for H being GroupWithOperators of O st H = (the_series_of_quotients_of s19) . i holds
H is trivial by A1, A12, A13, Th109;
reconsider s199 = Del (s19,i) as FinSequence of the_stable_subgroups_of G by FINSEQ_3:105;
A16: i in dom s19 by A14, A15, Th103;
A17: ( i + 1 in dom s19 & s19 . i = s19 . (i + 1) ) by A14, A15, Th103;
then reconsider s199 = s199 as CompositionSeries of G by A16, Th94;
A18: the_series_of_quotients_of s199 = Del ((the_series_of_quotients_of s19),i) by A16, A17, Th104;
set f2 = the_series_of_quotients_of s29;
assume A19: s29 is_finer_than s2 ; ::_thesis: ( ex p being Permutation of (dom (the_series_of_quotients_of s19)) st the_series_of_quotients_of s19, the_series_of_quotients_of s29 are_equivalent_under p,O implies S1[n + 1] )
given p being Permutation of (dom (the_series_of_quotients_of s19)) such that A20: the_series_of_quotients_of s19, the_series_of_quotients_of s29 are_equivalent_under p,O ; ::_thesis: S1[n + 1]
set H1 = (the_series_of_quotients_of s19) . i;
A21: (the_series_of_quotients_of s19) . i in rng (the_series_of_quotients_of s19) by A14, FUNCT_1:3;
set j = (p ") . i;
reconsider j = (p ") . i as Nat ;
set H2 = (the_series_of_quotients_of s29) . j;
reconsider s299 = Del (s29,j) as FinSequence of the_stable_subgroups_of G by FINSEQ_3:105;
rng (p ") c= dom (the_series_of_quotients_of s19) ;
then A22: rng (p ") c= Seg (len (the_series_of_quotients_of s19)) by FINSEQ_1:def_3;
A23: len (the_series_of_quotients_of s19) = len (the_series_of_quotients_of s29) by A20, Def34;
(p ") . i in rng (p ") by A14, FUNCT_2:4;
then (p ") . i in Seg (len (the_series_of_quotients_of s19)) by A22;
then A24: j in dom (the_series_of_quotients_of s29) by A23, FINSEQ_1:def_3;
then (the_series_of_quotients_of s29) . j in rng (the_series_of_quotients_of s29) by FUNCT_1:3;
then reconsider H1 = (the_series_of_quotients_of s19) . i, H2 = (the_series_of_quotients_of s29) . j as strict GroupWithOperators of O by A21, Th102;
A25: H1 is trivial by A15;
H1,H2 are_isomorphic by A20, A14, Def34;
then A26: for H being GroupWithOperators of O st H = (the_series_of_quotients_of s29) . j holds
H is trivial by A25, Th58;
then A27: ( j in dom s29 & j + 1 in dom s29 ) by A24, Th103;
A28: s29 . j = s29 . (j + 1) by A24, A26, Th103;
then reconsider s299 = s299 as CompositionSeries of G by A27, Th94;
A29: ( s299 is_finer_than s2 & not s299 is empty ) by A2, A7, A19, A27, A28, Th97, Th99;
A30: len s199 = (len s1) + n by A12, A16, FINSEQ_3:109;
the_series_of_quotients_of s299 = Del ((the_series_of_quotients_of s29),j) by A27, A28, Th104;
then A31: ex p being Permutation of (dom (the_series_of_quotients_of s199)) st the_series_of_quotients_of s199, the_series_of_quotients_of s299 are_equivalent_under p,O by A20, A14, A18, Th106;
( s199 is_finer_than s1 & not s199 is empty ) by A1, A6, A13, A16, A17, Th97, Th99;
hence S1[n + 1] by A11, A30, A29, A31; ::_thesis: verum
end;
hence S1[n + 1] ; ::_thesis: verum
end;
A32: S1[ 0 ]
proof
let s19, s29 be CompositionSeries of G; ::_thesis: ( not s19 is empty & not s29 is empty & len s19 = (len s1) + 0 & s19 is_finer_than s1 & s29 is_finer_than s2 & ex p being Permutation of (dom (the_series_of_quotients_of s19)) st the_series_of_quotients_of s19, the_series_of_quotients_of s29 are_equivalent_under p,O implies ex p being Permutation of (dom (the_series_of_quotients_of s1)) st the_series_of_quotients_of s1, the_series_of_quotients_of s2 are_equivalent_under p,O )
assume A33: ( not s19 is empty & not s29 is empty ) ; ::_thesis: ( not len s19 = (len s1) + 0 or not s19 is_finer_than s1 or not s29 is_finer_than s2 or for p being Permutation of (dom (the_series_of_quotients_of s19)) holds not the_series_of_quotients_of s19, the_series_of_quotients_of s29 are_equivalent_under p,O or ex p being Permutation of (dom (the_series_of_quotients_of s1)) st the_series_of_quotients_of s1, the_series_of_quotients_of s2 are_equivalent_under p,O )
assume A34: ( len s19 = (len s1) + 0 & s19 is_finer_than s1 ) ; ::_thesis: ( not s29 is_finer_than s2 or for p being Permutation of (dom (the_series_of_quotients_of s19)) holds not the_series_of_quotients_of s19, the_series_of_quotients_of s29 are_equivalent_under p,O or ex p being Permutation of (dom (the_series_of_quotients_of s1)) st the_series_of_quotients_of s1, the_series_of_quotients_of s2 are_equivalent_under p,O )
assume A35: s29 is_finer_than s2 ; ::_thesis: ( for p being Permutation of (dom (the_series_of_quotients_of s19)) holds not the_series_of_quotients_of s19, the_series_of_quotients_of s29 are_equivalent_under p,O or ex p being Permutation of (dom (the_series_of_quotients_of s1)) st the_series_of_quotients_of s1, the_series_of_quotients_of s2 are_equivalent_under p,O )
given p being Permutation of (dom (the_series_of_quotients_of s19)) such that A36: the_series_of_quotients_of s19, the_series_of_quotients_of s29 are_equivalent_under p,O ; ::_thesis: ex p being Permutation of (dom (the_series_of_quotients_of s1)) st the_series_of_quotients_of s1, the_series_of_quotients_of s2 are_equivalent_under p,O
A37: s19 is_equivalent_with s29 by A33, A36, Th108;
s19 = s1 by A34, Th96;
then s29 is jordan_holder by A1, A37, Th115;
then s29 = s2 by A2, A35, Th98;
then s1 is_equivalent_with s2 by A34, A37, Th96;
hence ex p being Permutation of (dom (the_series_of_quotients_of s1)) st the_series_of_quotients_of s1, the_series_of_quotients_of s2 are_equivalent_under p,O by A6, A7, Th108; ::_thesis: verum
end;
A38: for n being Nat holds S1[n] from NAT_1:sch_2(A32, A10);
consider s19, s29 being CompositionSeries of G such that
A39: s19 is_finer_than s1 and
A40: s29 is_finer_than s2 and
A41: s19 is_equivalent_with s29 by Th118;
A42: not s19 is empty by A6, A39, Th97;
A43: ex n being Nat st len s19 = (len s1) + n by A39, Th95;
A44: not s29 is empty by A7, A40, Th97;
then ex p9 being Permutation of (dom (the_series_of_quotients_of s19)) st the_series_of_quotients_of s19, the_series_of_quotients_of s29 are_equivalent_under p9,O by A41, A42, Th108;
then ex p being Permutation of (dom (the_series_of_quotients_of s1)) st the_series_of_quotients_of s1, the_series_of_quotients_of s2 are_equivalent_under p,O by A39, A40, A42, A44, A38, A43;
hence s1 is_equivalent_with s2 by A6, A7, Th108; ::_thesis: verum
end;
end;
end;
begin
theorem :: GROUP_9:120
for P, R being Relation holds
( P = (rng P) |` R iff P ~ = (R ~) | (dom (P ~)) ) by Lm36;
theorem :: GROUP_9:121
for X being set
for P, R being Relation holds P * (R | X) = (X |` P) * R by Lm37;
theorem :: GROUP_9:122
for n being Nat
for X being set
for f being PartFunc of REAL,REAL st X c= Seg n & X c= dom f & f | X is increasing & f .: X c= NAT \ {0} holds
Sgm (f .: X) = f * (Sgm X) by Lm38;
theorem :: GROUP_9:123
for y being set
for i, n being Nat st y c= Seg (n + 1) & i in Seg (n + 1) & not i in y holds
ex x being set st
( Sgm x = ((Sgm ((Seg (n + 1)) \ {i})) ") * (Sgm y) & x c= Seg n ) by Lm39;
theorem :: GROUP_9:124
for D being non empty set
for f being FinSequence of D
for p being Element of D
for n being Nat st n in dom f holds
f = Del ((Ins (f,n,p)),(n + 1)) by Lm44;
theorem :: GROUP_9:125
for G, H being Group
for F1 being FinSequence of the carrier of G
for F2 being FinSequence of the carrier of H
for I being FinSequence of INT
for f being Homomorphism of G,H st ( for k being Nat st k in dom F1 holds
F2 . k = f . (F1 . k) ) & len F1 = len I & len F2 = len I holds
f . (Product (F1 |^ I)) = Product (F2 |^ I) by Lm24;