:: ZMODUL01 semantic presentation
begin
definition
attrc1 is strict ;
struct Z_ModuleStruct -> addLoopStr ;
aggrZ_ModuleStruct(# carrier, ZeroF, addF, Mult #) -> Z_ModuleStruct ;
sel Mult c1 -> Function of [:INT, the carrier of c1:], the carrier of c1;
end;
registration
cluster non empty for Z_ModuleStruct ;
existence
not for b1 being Z_ModuleStruct holds b1 is empty
proof
set ZS = the non empty set ;
set O = the Element of the non empty set ;
set F = the BinOp of the non empty set ;
set G = the Function of [:INT, the non empty set :], the non empty set ;
take Z_ModuleStruct(# the non empty set , the Element of the non empty set , the BinOp of the non empty set , the Function of [:INT, the non empty set :], the non empty set #) ; ::_thesis: not Z_ModuleStruct(# the non empty set , the Element of the non empty set , the BinOp of the non empty set , the Function of [:INT, the non empty set :], the non empty set #) is empty
thus not the carrier of Z_ModuleStruct(# the non empty set , the Element of the non empty set , the BinOp of the non empty set , the Function of [:INT, the non empty set :], the non empty set #) is empty ; :: according to STRUCT_0:def_1 ::_thesis: verum
end;
end;
definition
let V be Z_ModuleStruct ;
mode VECTOR of V is Element of V;
end;
definition
let V be non empty Z_ModuleStruct ;
let v be VECTOR of V;
let a be integer number ;
funca * v -> Element of V equals :: ZMODUL01:def 1
the Mult of V . (a,v);
coherence
the Mult of V . (a,v) is Element of V
proof
reconsider a = a as Element of INT by INT_1:def_2;
the Mult of V . (a,v) is Element of V ;
hence the Mult of V . (a,v) is Element of V ; ::_thesis: verum
end;
end;
:: deftheorem defines * ZMODUL01:def_1_:_
for V being non empty Z_ModuleStruct
for v being VECTOR of V
for a being integer number holds a * v = the Mult of V . (a,v);
registration
let ZS be non empty set ;
let O be Element of ZS;
let F be BinOp of ZS;
let G be Function of [:INT,ZS:],ZS;
cluster Z_ModuleStruct(# ZS,O,F,G #) -> non empty ;
coherence
not Z_ModuleStruct(# ZS,O,F,G #) is empty ;
end;
definition
let IT be non empty Z_ModuleStruct ;
attrIT is vector-distributive means :Def2: :: ZMODUL01:def 2
for a being integer number
for v, w being VECTOR of IT holds a * (v + w) = (a * v) + (a * w);
attrIT is scalar-distributive means :Def3: :: ZMODUL01:def 3
for a, b being integer number
for v being VECTOR of IT holds (a + b) * v = (a * v) + (b * v);
attrIT is scalar-associative means :Def4: :: ZMODUL01:def 4
for a, b being integer number
for v being VECTOR of IT holds (a * b) * v = a * (b * v);
attrIT is scalar-unital means :Def5: :: ZMODUL01:def 5
for v being VECTOR of IT holds 1 * v = v;
end;
:: deftheorem Def2 defines vector-distributive ZMODUL01:def_2_:_
for IT being non empty Z_ModuleStruct holds
( IT is vector-distributive iff for a being integer number
for v, w being VECTOR of IT holds a * (v + w) = (a * v) + (a * w) );
:: deftheorem Def3 defines scalar-distributive ZMODUL01:def_3_:_
for IT being non empty Z_ModuleStruct holds
( IT is scalar-distributive iff for a, b being integer number
for v being VECTOR of IT holds (a + b) * v = (a * v) + (b * v) );
:: deftheorem Def4 defines scalar-associative ZMODUL01:def_4_:_
for IT being non empty Z_ModuleStruct holds
( IT is scalar-associative iff for a, b being integer number
for v being VECTOR of IT holds (a * b) * v = a * (b * v) );
:: deftheorem Def5 defines scalar-unital ZMODUL01:def_5_:_
for IT being non empty Z_ModuleStruct holds
( IT is scalar-unital iff for v being VECTOR of IT holds 1 * v = v );
definition
func Trivial-Z_ModuleStruct -> strict Z_ModuleStruct equals :: ZMODUL01:def 6
Z_ModuleStruct(# 1,op0,op2,(pr2 (INT,1)) #);
coherence
Z_ModuleStruct(# 1,op0,op2,(pr2 (INT,1)) #) is strict Z_ModuleStruct ;
end;
:: deftheorem defines Trivial-Z_ModuleStruct ZMODUL01:def_6_:_
Trivial-Z_ModuleStruct = Z_ModuleStruct(# 1,op0,op2,(pr2 (INT,1)) #);
registration
cluster Trivial-Z_ModuleStruct -> non empty trivial strict ;
coherence
( Trivial-Z_ModuleStruct is trivial & not Trivial-Z_ModuleStruct is empty ) by CARD_1:49;
end;
registration
cluster non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital for Z_ModuleStruct ;
existence
ex b1 being non empty Z_ModuleStruct st
( b1 is strict & b1 is Abelian & b1 is add-associative & b1 is right_zeroed & b1 is right_complementable & b1 is scalar-distributive & b1 is vector-distributive & b1 is scalar-associative & b1 is scalar-unital )
proof
take S = Trivial-Z_ModuleStruct ; ::_thesis: ( S is strict & S is Abelian & S is add-associative & S is right_zeroed & S is right_complementable & S is scalar-distributive & S is vector-distributive & S is scalar-associative & S is scalar-unital )
thus S is strict ; ::_thesis: ( S is Abelian & S is add-associative & S is right_zeroed & S is right_complementable & S is scalar-distributive & S is vector-distributive & S is scalar-associative & S is scalar-unital )
thus S is Abelian ; ::_thesis: ( S is add-associative & S is right_zeroed & S is right_complementable & S is scalar-distributive & S is vector-distributive & S is scalar-associative & S is scalar-unital )
thus S is add-associative ; ::_thesis: ( S is right_zeroed & S is right_complementable & S is scalar-distributive & S is vector-distributive & S is scalar-associative & S is scalar-unital )
thus S is right_zeroed ; ::_thesis: ( S is right_complementable & S is scalar-distributive & S is vector-distributive & S is scalar-associative & S is scalar-unital )
thus S is right_complementable ; ::_thesis: ( S is scalar-distributive & S is vector-distributive & S is scalar-associative & S is scalar-unital )
thus for a, b being integer number
for v being VECTOR of S holds (a + b) * v = (a * v) + (b * v) by STRUCT_0:def_10; :: according to ZMODUL01:def_3 ::_thesis: ( S is vector-distributive & S is scalar-associative & S is scalar-unital )
thus for a being integer number
for v, w being VECTOR of S holds a * (v + w) = (a * v) + (a * w) by STRUCT_0:def_10; :: according to ZMODUL01:def_2 ::_thesis: ( S is scalar-associative & S is scalar-unital )
thus for a, b being integer number
for v being VECTOR of S holds (a * b) * v = a * (b * v) by STRUCT_0:def_10; :: according to ZMODUL01:def_4 ::_thesis: S is scalar-unital
thus for v being VECTOR of S holds 1 * v = v by STRUCT_0:def_10; :: according to ZMODUL01:def_5 ::_thesis: verum
end;
end;
definition
mode Z_Module is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital Z_ModuleStruct ;
end;
definition
let IT be non empty Z_ModuleStruct ;
attrIT is Mult-cancelable means :Def7: :: ZMODUL01:def 7
for a being integer number
for v being VECTOR of IT holds
( not a * v = 0. IT or a = 0 or v = 0. IT );
end;
:: deftheorem Def7 defines Mult-cancelable ZMODUL01:def_7_:_
for IT being non empty Z_ModuleStruct holds
( IT is Mult-cancelable iff for a being integer number
for v being VECTOR of IT holds
( not a * v = 0. IT or a = 0 or v = 0. IT ) );
theorem Th1: :: ZMODUL01:1
for a being integer number
for V being Z_Module
for v being VECTOR of V st ( a = 0 or v = 0. V ) holds
a * v = 0. V
proof
let a be integer number ; ::_thesis: for V being Z_Module
for v being VECTOR of V st ( a = 0 or v = 0. V ) holds
a * v = 0. V
let V be Z_Module; ::_thesis: for v being VECTOR of V st ( a = 0 or v = 0. V ) holds
a * v = 0. V
let v be VECTOR of V; ::_thesis: ( ( a = 0 or v = 0. V ) implies a * v = 0. V )
reconsider N1 = 1, N0 = 0 as Integer ;
assume A1: ( a = 0 or v = 0. V ) ; ::_thesis: a * v = 0. V
now__::_thesis:_a_*_v_=_0._V
percases ( a = 0 or v = 0. V ) by A1;
supposeA2: a = 0 ; ::_thesis: a * v = 0. V
v + (N0 * v) = (N1 * v) + (N0 * v) by Def5
.= (N1 + N0) * v by Def3
.= v by Def5
.= v + (0. V) by RLVECT_1:4 ;
hence a * v = 0. V by A2, RLVECT_1:8; ::_thesis: verum
end;
supposeA3: v = 0. V ; ::_thesis: a * v = 0. V
(a * (0. V)) + (a * (0. V)) = a * ((0. V) + (0. V)) by Def2
.= a * (0. V) by RLVECT_1:4
.= (a * (0. V)) + (0. V) by RLVECT_1:4 ;
hence a * v = 0. V by A3, RLVECT_1:8; ::_thesis: verum
end;
end;
end;
hence a * v = 0. V ; ::_thesis: verum
end;
theorem Th2: :: ZMODUL01:2
for V being Z_Module
for v being VECTOR of V holds - v = (- 1) * v
proof
let V be Z_Module; ::_thesis: for v being VECTOR of V holds - v = (- 1) * v
let v be VECTOR of V; ::_thesis: - v = (- 1) * v
reconsider N1 = 1, N0 = 0 , M1 = - 1 as Integer ;
v + ((- 1) * v) = (1 * v) + ((- 1) * v) by Def5
.= (1 + (- 1)) * v by Def3
.= 0. V by Th1 ;
hence - v = (- v) + (v + ((- 1) * v)) by RLVECT_1:4
.= ((- v) + v) + ((- 1) * v) by RLVECT_1:def_3
.= (0. V) + ((- 1) * v) by RLVECT_1:def_10
.= (- 1) * v by RLVECT_1:4 ;
::_thesis: verum
end;
theorem Th3: :: ZMODUL01:3
for V being Z_Module
for v being VECTOR of V st V is Mult-cancelable & v = - v holds
v = 0. V
proof
let V be Z_Module; ::_thesis: for v being VECTOR of V st V is Mult-cancelable & v = - v holds
v = 0. V
let v be VECTOR of V; ::_thesis: ( V is Mult-cancelable & v = - v implies v = 0. V )
assume A1: V is Mult-cancelable ; ::_thesis: ( not v = - v or v = 0. V )
assume v = - v ; ::_thesis: v = 0. V
then 0. V = v + v by RLVECT_1:def_10
.= (1 * v) + v by Def5
.= (1 * v) + (1 * v) by Def5
.= (1 + 1) * v by Def3
.= 2 * v ;
hence v = 0. V by A1, Def7; ::_thesis: verum
end;
theorem :: ZMODUL01:4
for V being Z_Module
for v being VECTOR of V st V is Mult-cancelable & v + v = 0. V holds
v = 0. V
proof
let V be Z_Module; ::_thesis: for v being VECTOR of V st V is Mult-cancelable & v + v = 0. V holds
v = 0. V
let v be VECTOR of V; ::_thesis: ( V is Mult-cancelable & v + v = 0. V implies v = 0. V )
assume A1: V is Mult-cancelable ; ::_thesis: ( not v + v = 0. V or v = 0. V )
assume v + v = 0. V ; ::_thesis: v = 0. V
then v = - v by RLVECT_1:def_10;
hence v = 0. V by A1, Th3; ::_thesis: verum
end;
theorem Th5: :: ZMODUL01:5
for a being integer number
for V being Z_Module
for v being VECTOR of V holds a * (- v) = (- a) * v
proof
let a be integer number ; ::_thesis: for V being Z_Module
for v being VECTOR of V holds a * (- v) = (- a) * v
let V be Z_Module; ::_thesis: for v being VECTOR of V holds a * (- v) = (- a) * v
let v be VECTOR of V; ::_thesis: a * (- v) = (- a) * v
thus a * (- v) = a * ((- 1) * v) by Th2
.= (a * (- 1)) * v by Def4
.= (- a) * v ; ::_thesis: verum
end;
theorem Th6: :: ZMODUL01:6
for a being integer number
for V being Z_Module
for v being VECTOR of V holds a * (- v) = - (a * v)
proof
let a be integer number ; ::_thesis: for V being Z_Module
for v being VECTOR of V holds a * (- v) = - (a * v)
let V be Z_Module; ::_thesis: for v being VECTOR of V holds a * (- v) = - (a * v)
let v be VECTOR of V; ::_thesis: a * (- v) = - (a * v)
thus a * (- v) = (- (1 * a)) * v by Th5
.= ((- 1) * a) * v
.= (- 1) * (a * v) by Def4
.= - (a * v) by Th2 ; ::_thesis: verum
end;
theorem :: ZMODUL01:7
for a being integer number
for V being Z_Module
for v being VECTOR of V holds (- a) * (- v) = a * v
proof
let a be integer number ; ::_thesis: for V being Z_Module
for v being VECTOR of V holds (- a) * (- v) = a * v
let V be Z_Module; ::_thesis: for v being VECTOR of V holds (- a) * (- v) = a * v
let v be VECTOR of V; ::_thesis: (- a) * (- v) = a * v
thus (- a) * (- v) = (- (- a)) * v by Th5
.= a * v ; ::_thesis: verum
end;
theorem Th8: :: ZMODUL01:8
for a being integer number
for V being Z_Module
for v, w being VECTOR of V holds a * (v - w) = (a * v) - (a * w)
proof
let a be integer number ; ::_thesis: for V being Z_Module
for v, w being VECTOR of V holds a * (v - w) = (a * v) - (a * w)
let V be Z_Module; ::_thesis: for v, w being VECTOR of V holds a * (v - w) = (a * v) - (a * w)
let v, w be VECTOR of V; ::_thesis: a * (v - w) = (a * v) - (a * w)
thus a * (v - w) = (a * v) + (a * (- w)) by Def2
.= (a * v) - (a * w) by Th6 ; ::_thesis: verum
end;
theorem Th9: :: ZMODUL01:9
for a, b being integer number
for V being Z_Module
for v being VECTOR of V holds (a - b) * v = (a * v) - (b * v)
proof
let a, b be integer number ; ::_thesis: for V being Z_Module
for v being VECTOR of V holds (a - b) * v = (a * v) - (b * v)
let V be Z_Module; ::_thesis: for v being VECTOR of V holds (a - b) * v = (a * v) - (b * v)
let v be VECTOR of V; ::_thesis: (a - b) * v = (a * v) - (b * v)
thus (a - b) * v = (a + (- b)) * v
.= (a * v) + ((- b) * v) by Def3
.= (a * v) + (b * (- v)) by Th5
.= (a * v) - (b * v) by Th6 ; ::_thesis: verum
end;
theorem :: ZMODUL01:10
for a being integer number
for V being Z_Module
for v, w being VECTOR of V st V is Mult-cancelable & a <> 0 & a * v = a * w holds
v = w
proof
let a be integer number ; ::_thesis: for V being Z_Module
for v, w being VECTOR of V st V is Mult-cancelable & a <> 0 & a * v = a * w holds
v = w
let V be Z_Module; ::_thesis: for v, w being VECTOR of V st V is Mult-cancelable & a <> 0 & a * v = a * w holds
v = w
let v, w be VECTOR of V; ::_thesis: ( V is Mult-cancelable & a <> 0 & a * v = a * w implies v = w )
assume A1: V is Mult-cancelable ; ::_thesis: ( not a <> 0 or not a * v = a * w or v = w )
assume that
A2: a <> 0 and
A3: a * v = a * w ; ::_thesis: v = w
0. V = (a * v) - (a * w) by A3, RLVECT_1:15
.= a * (v - w) by Th8 ;
then v - w = 0. V by A2, A1, Def7;
hence v = w by RLVECT_1:21; ::_thesis: verum
end;
theorem :: ZMODUL01:11
for a, b being integer number
for V being Z_Module
for v being VECTOR of V st V is Mult-cancelable & v <> 0. V & a * v = b * v holds
a = b
proof
let a, b be integer number ; ::_thesis: for V being Z_Module
for v being VECTOR of V st V is Mult-cancelable & v <> 0. V & a * v = b * v holds
a = b
let V be Z_Module; ::_thesis: for v being VECTOR of V st V is Mult-cancelable & v <> 0. V & a * v = b * v holds
a = b
let v be VECTOR of V; ::_thesis: ( V is Mult-cancelable & v <> 0. V & a * v = b * v implies a = b )
assume A1: V is Mult-cancelable ; ::_thesis: ( not v <> 0. V or not a * v = b * v or a = b )
assume that
A2: v <> 0. V and
A3: a * v = b * v ; ::_thesis: a = b
0. V = (a * v) - (b * v) by A3, RLVECT_1:15
.= (a - b) * v by Th9 ;
then (- b) + a = 0 by A2, A1, Def7;
hence a = b ; ::_thesis: verum
end;
Lm1: for V being Z_Module holds Sum (<*> the carrier of V) = 0. V
proof
let V be Z_Module; ::_thesis: Sum (<*> the carrier of V) = 0. V
set S = <*> the carrier of V;
ex f being Function of NAT, the carrier of V st
( Sum (<*> the carrier of V) = f . (len (<*> the carrier of V)) & f . 0 = 0. V & ( for j being Element of NAT
for v being VECTOR of V st j < len (<*> the carrier of V) & v = (<*> the carrier of V) . (j + 1) holds
f . (j + 1) = (f . j) + v ) ) by RLVECT_1:def_12;
hence Sum (<*> the carrier of V) = 0. V ; ::_thesis: verum
end;
Lm2: for V being Z_Module
for F being FinSequence of V st len F = 0 holds
Sum F = 0. V
proof
let V be Z_Module; ::_thesis: for F being FinSequence of V st len F = 0 holds
Sum F = 0. V
let F be FinSequence of V; ::_thesis: ( len F = 0 implies Sum F = 0. V )
assume len F = 0 ; ::_thesis: Sum F = 0. V
then F = <*> the carrier of V ;
hence Sum F = 0. V by Lm1; ::_thesis: verum
end;
theorem Th12: :: ZMODUL01:12
for a being integer number
for V being Z_Module
for F, G being FinSequence of V st len F = len G & ( for k being Element of NAT
for v being VECTOR of V st k in dom F & v = G . k holds
F . k = a * v ) holds
Sum F = a * (Sum G)
proof
let a be integer number ; ::_thesis: for V being Z_Module
for F, G being FinSequence of V st len F = len G & ( for k being Element of NAT
for v being VECTOR of V st k in dom F & v = G . k holds
F . k = a * v ) holds
Sum F = a * (Sum G)
let V be Z_Module; ::_thesis: for F, G being FinSequence of V st len F = len G & ( for k being Element of NAT
for v being VECTOR of V st k in dom F & v = G . k holds
F . k = a * v ) holds
Sum F = a * (Sum G)
let F, G be FinSequence of V; ::_thesis: ( len F = len G & ( for k being Element of NAT
for v being VECTOR of V st k in dom F & v = G . k holds
F . k = a * v ) implies Sum F = a * (Sum G) )
defpred S1[ set ] means for H, I being FinSequence of V st len H = len I & len H = $1 & ( for k being Element of NAT
for v being VECTOR of V st k in Seg (len H) & v = I . k holds
H . k = a * v ) holds
Sum H = a * (Sum I);
A1: dom F = Seg (len F) by FINSEQ_1:def_3;
now__::_thesis:_for_n_being_Element_of_NAT_st_(_for_H,_I_being_FinSequence_of_V_st_len_H_=_len_I_&_len_H_=_n_&_(_for_k_being_Element_of_NAT_
for_v_being_VECTOR_of_V_st_k_in_Seg_(len_H)_&_v_=_I_._k_holds_
H_._k_=_a_*_v_)_holds_
Sum_H_=_a_*_(Sum_I)_)_holds_
for_H,_I_being_FinSequence_of_V_st_len_H_=_len_I_&_len_H_=_n_+_1_&_(_for_k_being_Element_of_NAT_
for_v_being_VECTOR_of_V_st_k_in_Seg_(len_H)_&_v_=_I_._k_holds_
H_._k_=_a_*_v_)_holds_
Sum_H_=_a_*_(Sum_I)
let n be Element of NAT ; ::_thesis: ( ( for H, I being FinSequence of V st len H = len I & len H = n & ( for k being Element of NAT
for v being VECTOR of V st k in Seg (len H) & v = I . k holds
H . k = a * v ) holds
Sum H = a * (Sum I) ) implies for H, I being FinSequence of V st len H = len I & len H = n + 1 & ( for k being Element of NAT
for v being VECTOR of V st k in Seg (len H) & v = I . k holds
H . k = a * v ) holds
Sum H = a * (Sum I) )
assume A2: for H, I being FinSequence of V st len H = len I & len H = n & ( for k being Element of NAT
for v being VECTOR of V st k in Seg (len H) & v = I . k holds
H . k = a * v ) holds
Sum H = a * (Sum I) ; ::_thesis: for H, I being FinSequence of V st len H = len I & len H = n + 1 & ( for k being Element of NAT
for v being VECTOR of V st k in Seg (len H) & v = I . k holds
H . k = a * v ) holds
Sum H = a * (Sum I)
let H, I be FinSequence of V; ::_thesis: ( len H = len I & len H = n + 1 & ( for k being Element of NAT
for v being VECTOR of V st k in Seg (len H) & v = I . k holds
H . k = a * v ) implies Sum H = a * (Sum I) )
assume that
A3: len H = len I and
A4: len H = n + 1 and
A5: for k being Element of NAT
for v being VECTOR of V st k in Seg (len H) & v = I . k holds
H . k = a * v ; ::_thesis: Sum H = a * (Sum I)
reconsider p = H | (Seg n), q = I | (Seg n) as FinSequence of the carrier of V by FINSEQ_1:18;
A6: n <= n + 1 by NAT_1:12;
then A7: len q = n by A3, A4, FINSEQ_1:17;
A8: len p = n by A4, A6, FINSEQ_1:17;
A9: now__::_thesis:_for_k_being_Element_of_NAT_
for_v_being_VECTOR_of_V_st_k_in_Seg_(len_p)_&_v_=_q_._k_holds_
p_._k_=_a_*_v
len p <= len H by A4, A6, FINSEQ_1:17;
then A10: Seg (len p) c= Seg (len H) by FINSEQ_1:5;
A11: dom p = Seg n by A4, A6, FINSEQ_1:17;
let k be Element of NAT ; ::_thesis: for v being VECTOR of V st k in Seg (len p) & v = q . k holds
p . k = a * v
let v be VECTOR of V; ::_thesis: ( k in Seg (len p) & v = q . k implies p . k = a * v )
assume that
A12: k in Seg (len p) and
A13: v = q . k ; ::_thesis: p . k = a * v
dom q = Seg n by A3, A4, A6, FINSEQ_1:17;
then I . k = q . k by A8, A12, FUNCT_1:47;
then H . k = a * v by A5, A12, A13, A10;
hence p . k = a * v by A8, A12, A11, FUNCT_1:47; ::_thesis: verum
end;
1 <= n + 1 by NAT_1:11;
then ( n + 1 in dom H & n + 1 in dom I ) by A3, A4, FINSEQ_3:25;
then reconsider v1 = H . (n + 1), v2 = I . (n + 1) as VECTOR of V by FUNCT_1:102;
A14: v1 = a * v2 by A4, A5, FINSEQ_1:4;
A15: dom q = Seg (len q) by FINSEQ_1:def_3;
dom p = Seg (len p) by FINSEQ_1:def_3;
hence Sum H = (Sum p) + v1 by A4, A8, RLVECT_1:38
.= (a * (Sum q)) + (a * v2) by A2, A8, A7, A9, A14
.= a * ((Sum q) + v2) by Def2
.= a * (Sum I) by A3, A4, A7, A15, RLVECT_1:38 ;
::_thesis: verum
end;
then A16: for n being Element of NAT st S1[n] holds
S1[n + 1] ;
now__::_thesis:_for_H,_I_being_FinSequence_of_V_st_len_H_=_len_I_&_len_H_=_0_&_(_for_k_being_Element_of_NAT_
for_v_being_VECTOR_of_V_st_k_in_Seg_(len_H)_&_v_=_I_._k_holds_
H_._k_=_a_*_v_)_holds_
Sum_H_=_a_*_(Sum_I)
let H, I be FinSequence of V; ::_thesis: ( len H = len I & len H = 0 & ( for k being Element of NAT
for v being VECTOR of V st k in Seg (len H) & v = I . k holds
H . k = a * v ) implies Sum H = a * (Sum I) )
assume that
A17: len H = len I and
A18: len H = 0 and
for k being Element of NAT
for v being VECTOR of V st k in Seg (len H) & v = I . k holds
H . k = a * v ; ::_thesis: Sum H = a * (Sum I)
Sum H = 0. V by A18, Lm2;
hence Sum H = a * (Sum I) by A17, A18, Lm2, Th1; ::_thesis: verum
end;
then A19: S1[ 0 ] ;
for n being Element of NAT holds S1[n] from NAT_1:sch_1(A19, A16);
hence ( len F = len G & ( for k being Element of NAT
for v being VECTOR of V st k in dom F & v = G . k holds
F . k = a * v ) implies Sum F = a * (Sum G) ) by A1; ::_thesis: verum
end;
theorem :: ZMODUL01:13
for V being Z_Module
for a being Integer holds a * (Sum (<*> the carrier of V)) = 0. V by Lm1, Th1;
theorem :: ZMODUL01:14
for V being Z_Module
for a being Integer
for v, u being VECTOR of V holds a * (Sum <*v,u*>) = (a * v) + (a * u)
proof
let V be Z_Module; ::_thesis: for a being Integer
for v, u being VECTOR of V holds a * (Sum <*v,u*>) = (a * v) + (a * u)
let a be Integer; ::_thesis: for v, u being VECTOR of V holds a * (Sum <*v,u*>) = (a * v) + (a * u)
let v, u be VECTOR of V; ::_thesis: a * (Sum <*v,u*>) = (a * v) + (a * u)
thus a * (Sum <*v,u*>) = a * (v + u) by RLVECT_1:45
.= (a * v) + (a * u) by Def2 ; ::_thesis: verum
end;
theorem :: ZMODUL01:15
for V being Z_Module
for a being Integer
for v, u, w being VECTOR of V holds a * (Sum <*v,u,w*>) = ((a * v) + (a * u)) + (a * w)
proof
let V be Z_Module; ::_thesis: for a being Integer
for v, u, w being VECTOR of V holds a * (Sum <*v,u,w*>) = ((a * v) + (a * u)) + (a * w)
let a be Integer; ::_thesis: for v, u, w being VECTOR of V holds a * (Sum <*v,u,w*>) = ((a * v) + (a * u)) + (a * w)
let v, u, w be VECTOR of V; ::_thesis: a * (Sum <*v,u,w*>) = ((a * v) + (a * u)) + (a * w)
thus a * (Sum <*v,u,w*>) = a * ((v + u) + w) by RLVECT_1:46
.= (a * (v + u)) + (a * w) by Def2
.= ((a * v) + (a * u)) + (a * w) by Def2 ; ::_thesis: verum
end;
theorem :: ZMODUL01:16
for a being integer number
for V being Z_Module
for v being VECTOR of V holds (- a) * v = - (a * v)
proof
let a be integer number ; ::_thesis: for V being Z_Module
for v being VECTOR of V holds (- a) * v = - (a * v)
let V be Z_Module; ::_thesis: for v being VECTOR of V holds (- a) * v = - (a * v)
let v be VECTOR of V; ::_thesis: (- a) * v = - (a * v)
thus (- a) * v = a * (- v) by Th5
.= - (a * v) by Th6 ; ::_thesis: verum
end;
theorem :: ZMODUL01:17
for a being integer number
for V being Z_Module
for F, G being FinSequence of V st len F = len G & ( for k being Element of NAT st k in dom F holds
G . k = a * (F /. k) ) holds
Sum G = a * (Sum F)
proof
let a be integer number ; ::_thesis: for V being Z_Module
for F, G being FinSequence of V st len F = len G & ( for k being Element of NAT st k in dom F holds
G . k = a * (F /. k) ) holds
Sum G = a * (Sum F)
let V be Z_Module; ::_thesis: for F, G being FinSequence of V st len F = len G & ( for k being Element of NAT st k in dom F holds
G . k = a * (F /. k) ) holds
Sum G = a * (Sum F)
let F, G be FinSequence of V; ::_thesis: ( len F = len G & ( for k being Element of NAT st k in dom F holds
G . k = a * (F /. k) ) implies Sum G = a * (Sum F) )
assume that
A1: len F = len G and
A2: for k being Element of NAT st k in dom F holds
G . k = a * (F /. k) ; ::_thesis: Sum G = a * (Sum F)
A3: ( dom F = Seg (len F) & dom G = Seg (len G) ) by FINSEQ_1:def_3;
now__::_thesis:_for_k_being_Element_of_NAT_
for_v_being_VECTOR_of_V_st_k_in_dom_G_&_v_=_F_._k_holds_
G_._k_=_a_*_v
let k be Element of NAT ; ::_thesis: for v being VECTOR of V st k in dom G & v = F . k holds
G . k = a * v
let v be VECTOR of V; ::_thesis: ( k in dom G & v = F . k implies G . k = a * v )
assume that
A4: k in dom G and
A5: v = F . k ; ::_thesis: G . k = a * v
v = F /. k by A1, A3, A4, A5, PARTFUN1:def_6;
hence G . k = a * v by A1, A2, A3, A4; ::_thesis: verum
end;
hence Sum G = a * (Sum F) by A1, Th12; ::_thesis: verum
end;
begin
definition
let V be Z_Module;
let V1 be Subset of V;
attrV1 is linearly-closed means :Def8: :: ZMODUL01:def 8
( ( for v, u being VECTOR of V st v in V1 & u in V1 holds
v + u in V1 ) & ( for a being integer number
for v being VECTOR of V st v in V1 holds
a * v in V1 ) );
end;
:: deftheorem Def8 defines linearly-closed ZMODUL01:def_8_:_
for V being Z_Module
for V1 being Subset of V holds
( V1 is linearly-closed iff ( ( for v, u being VECTOR of V st v in V1 & u in V1 holds
v + u in V1 ) & ( for a being integer number
for v being VECTOR of V st v in V1 holds
a * v in V1 ) ) );
theorem Th18: :: ZMODUL01:18
for V being Z_Module
for V1 being Subset of V st V1 <> {} & V1 is linearly-closed holds
0. V in V1
proof
let V be Z_Module; ::_thesis: for V1 being Subset of V st V1 <> {} & V1 is linearly-closed holds
0. V in V1
let V1 be Subset of V; ::_thesis: ( V1 <> {} & V1 is linearly-closed implies 0. V in V1 )
assume that
A1: V1 <> {} and
A2: V1 is linearly-closed ; ::_thesis: 0. V in V1
set x = the Element of V1;
reconsider x = the Element of V1 as Element of V by A1, TARSKI:def_3;
0 * x in V1 by A1, A2, Def8;
hence 0. V in V1 by Th1; ::_thesis: verum
end;
theorem Th19: :: ZMODUL01:19
for V being Z_Module
for V1 being Subset of V st V1 is linearly-closed holds
for v being VECTOR of V st v in V1 holds
- v in V1
proof
let V be Z_Module; ::_thesis: for V1 being Subset of V st V1 is linearly-closed holds
for v being VECTOR of V st v in V1 holds
- v in V1
let V1 be Subset of V; ::_thesis: ( V1 is linearly-closed implies for v being VECTOR of V st v in V1 holds
- v in V1 )
assume A1: V1 is linearly-closed ; ::_thesis: for v being VECTOR of V st v in V1 holds
- v in V1
let v be VECTOR of V; ::_thesis: ( v in V1 implies - v in V1 )
assume v in V1 ; ::_thesis: - v in V1
then (- 1) * v in V1 by A1, Def8;
hence - v in V1 by Th2; ::_thesis: verum
end;
theorem Th20: :: ZMODUL01:20
for V being Z_Module
for V1 being Subset of V st V1 is linearly-closed holds
for v, u being VECTOR of V st v in V1 & u in V1 holds
v - u in V1
proof
let V be Z_Module; ::_thesis: for V1 being Subset of V st V1 is linearly-closed holds
for v, u being VECTOR of V st v in V1 & u in V1 holds
v - u in V1
let V1 be Subset of V; ::_thesis: ( V1 is linearly-closed implies for v, u being VECTOR of V st v in V1 & u in V1 holds
v - u in V1 )
assume A1: V1 is linearly-closed ; ::_thesis: for v, u being VECTOR of V st v in V1 & u in V1 holds
v - u in V1
let v, u be VECTOR of V; ::_thesis: ( v in V1 & u in V1 implies v - u in V1 )
assume that
A2: v in V1 and
A3: u in V1 ; ::_thesis: v - u in V1
- u in V1 by A1, A3, Th19;
hence v - u in V1 by A1, A2, Def8; ::_thesis: verum
end;
theorem :: ZMODUL01:21
for V being Z_Module
for V1 being Subset of V st the carrier of V = V1 holds
V1 is linearly-closed
proof
let V be Z_Module; ::_thesis: for V1 being Subset of V st the carrier of V = V1 holds
V1 is linearly-closed
let V1 be Subset of V; ::_thesis: ( the carrier of V = V1 implies V1 is linearly-closed )
assume A1: the carrier of V = V1 ; ::_thesis: V1 is linearly-closed
hence for v, u being VECTOR of V st v in V1 & u in V1 holds
v + u in V1 ; :: according to ZMODUL01:def_8 ::_thesis: for a being integer number
for v being VECTOR of V st v in V1 holds
a * v in V1
let a be integer number ; ::_thesis: for v being VECTOR of V st v in V1 holds
a * v in V1
let v be VECTOR of V; ::_thesis: ( v in V1 implies a * v in V1 )
thus ( v in V1 implies a * v in V1 ) by A1; ::_thesis: verum
end;
theorem Th22: :: ZMODUL01:22
for V being Z_Module
for V1, V2, V3 being Subset of V st V1 is linearly-closed & V2 is linearly-closed & V3 = { (v + u) where v, u is VECTOR of V : ( v in V1 & u in V2 ) } holds
V3 is linearly-closed
proof
let V be Z_Module; ::_thesis: for V1, V2, V3 being Subset of V st V1 is linearly-closed & V2 is linearly-closed & V3 = { (v + u) where v, u is VECTOR of V : ( v in V1 & u in V2 ) } holds
V3 is linearly-closed
let V1, V2, V3 be Subset of V; ::_thesis: ( V1 is linearly-closed & V2 is linearly-closed & V3 = { (v + u) where v, u is VECTOR of V : ( v in V1 & u in V2 ) } implies V3 is linearly-closed )
assume that
A1: ( V1 is linearly-closed & V2 is linearly-closed ) and
A2: V3 = { (v + u) where v, u is VECTOR of V : ( v in V1 & u in V2 ) } ; ::_thesis: V3 is linearly-closed
thus for v, u being VECTOR of V st v in V3 & u in V3 holds
v + u in V3 :: according to ZMODUL01:def_8 ::_thesis: for a being integer number
for v being VECTOR of V st v in V3 holds
a * v in V3
proof
let v, u be VECTOR of V; ::_thesis: ( v in V3 & u in V3 implies v + u in V3 )
assume that
A3: v in V3 and
A4: u in V3 ; ::_thesis: v + u in V3
consider v1, v2 being VECTOR of V such that
A5: v = v1 + v2 and
A6: ( v1 in V1 & v2 in V2 ) by A2, A3;
consider u1, u2 being VECTOR of V such that
A7: u = u1 + u2 and
A8: ( u1 in V1 & u2 in V2 ) by A2, A4;
A9: v + u = ((v1 + v2) + u1) + u2 by A5, A7, RLVECT_1:def_3
.= ((v1 + u1) + v2) + u2 by RLVECT_1:def_3
.= (v1 + u1) + (v2 + u2) by RLVECT_1:def_3 ;
( v1 + u1 in V1 & v2 + u2 in V2 ) by A1, A6, A8, Def8;
hence v + u in V3 by A2, A9; ::_thesis: verum
end;
let a be integer number ; ::_thesis: for v being VECTOR of V st v in V3 holds
a * v in V3
let v be VECTOR of V; ::_thesis: ( v in V3 implies a * v in V3 )
assume v in V3 ; ::_thesis: a * v in V3
then consider v1, v2 being VECTOR of V such that
A10: v = v1 + v2 and
A11: ( v1 in V1 & v2 in V2 ) by A2;
A12: a * v = (a * v1) + (a * v2) by A10, Def2;
( a * v1 in V1 & a * v2 in V2 ) by A1, A11, Def8;
hence a * v in V3 by A2, A12; ::_thesis: verum
end;
registration
let V be Z_Module;
cluster{(0. V)} -> linearly-closed for Subset of V;
coherence
for b1 being Subset of V st b1 = {(0. V)} holds
b1 is linearly-closed
proof
let S be Subset of V; ::_thesis: ( S = {(0. V)} implies S is linearly-closed )
assume A1: S = {(0. V)} ; ::_thesis: S is linearly-closed
thus for v, u being VECTOR of V st v in S & u in S holds
v + u in S :: according to ZMODUL01:def_8 ::_thesis: for a being integer number
for v being VECTOR of V st v in S holds
a * v in S
proof
let v, u be VECTOR of V; ::_thesis: ( v in S & u in S implies v + u in S )
assume ( v in S & u in S ) ; ::_thesis: v + u in S
then ( v = 0. V & u = 0. V ) by A1, TARSKI:def_1;
then v + u = 0. V by RLVECT_1:4;
hence v + u in S by A1, TARSKI:def_1; ::_thesis: verum
end;
let a be integer number ; ::_thesis: for v being VECTOR of V st v in S holds
a * v in S
let v be VECTOR of V; ::_thesis: ( v in S implies a * v in S )
assume A2: v in S ; ::_thesis: a * v in S
then v = 0. V by A1, TARSKI:def_1;
hence a * v in S by A2, Th1; ::_thesis: verum
end;
end;
registration
let V be Z_Module;
cluster linearly-closed for Element of bool the carrier of V;
existence
ex b1 being Subset of V st b1 is linearly-closed
proof
take {(0. V)} ; ::_thesis: {(0. V)} is linearly-closed
thus {(0. V)} is linearly-closed ; ::_thesis: verum
end;
end;
registration
let V be Z_Module;
let V1, V2 be linearly-closed Subset of V;
clusterV1 /\ V2 -> linearly-closed for Subset of V;
coherence
for b1 being Subset of V st b1 = V1 /\ V2 holds
b1 is linearly-closed
proof
let S be Subset of V; ::_thesis: ( S = V1 /\ V2 implies S is linearly-closed )
assume A1: S = V1 /\ V2 ; ::_thesis: S is linearly-closed
thus for v, u being VECTOR of V st v in S & u in S holds
v + u in S :: according to ZMODUL01:def_8 ::_thesis: for a being integer number
for v being VECTOR of V st v in S holds
a * v in S
proof
let v, u be VECTOR of V; ::_thesis: ( v in S & u in S implies v + u in S )
assume A2: ( v in S & u in S ) ; ::_thesis: v + u in S
then ( v in V2 & u in V2 ) by A1, XBOOLE_0:def_4;
then A3: v + u in V2 by Def8;
( v in V1 & u in V1 ) by A2, A1, XBOOLE_0:def_4;
then v + u in V1 by Def8;
hence v + u in S by A3, A1, XBOOLE_0:def_4; ::_thesis: verum
end;
let a be integer number ; ::_thesis: for v being VECTOR of V st v in S holds
a * v in S
let v be VECTOR of V; ::_thesis: ( v in S implies a * v in S )
assume A4: v in S ; ::_thesis: a * v in S
then v in V2 by A1, XBOOLE_0:def_4;
then A5: a * v in V2 by Def8;
v in V1 by A4, A1, XBOOLE_0:def_4;
then a * v in V1 by Def8;
hence a * v in S by A5, A1, XBOOLE_0:def_4; ::_thesis: verum
end;
end;
definition
let V be Z_Module;
mode Submodule of V -> Z_Module means :Def9: :: ZMODUL01:def 9
( the carrier of it c= the carrier of V & 0. it = 0. V & the addF of it = the addF of V || the carrier of it & the Mult of it = the Mult of V | [:INT, the carrier of it:] );
existence
ex b1 being Z_Module st
( the carrier of b1 c= the carrier of V & 0. b1 = 0. V & the addF of b1 = the addF of V || the carrier of b1 & the Mult of b1 = the Mult of V | [:INT, the carrier of b1:] )
proof
( the addF of V = the addF of V || the carrier of V & the Mult of V = the Mult of V | [:INT, the carrier of V:] ) by RELSET_1:19;
hence ex b1 being Z_Module st
( the carrier of b1 c= the carrier of V & 0. b1 = 0. V & the addF of b1 = the addF of V || the carrier of b1 & the Mult of b1 = the Mult of V | [:INT, the carrier of b1:] ) ; ::_thesis: verum
end;
end;
:: deftheorem Def9 defines Submodule ZMODUL01:def_9_:_
for V, b2 being Z_Module holds
( b2 is Submodule of V iff ( the carrier of b2 c= the carrier of V & 0. b2 = 0. V & the addF of b2 = the addF of V || the carrier of b2 & the Mult of b2 = the Mult of V | [:INT, the carrier of b2:] ) );
theorem Th23: :: ZMODUL01:23
for V being Z_Module
for x being set
for W1, W2 being Submodule of V st x in W1 & W1 is Submodule of W2 holds
x in W2
proof
let V be Z_Module; ::_thesis: for x being set
for W1, W2 being Submodule of V st x in W1 & W1 is Submodule of W2 holds
x in W2
let x be set ; ::_thesis: for W1, W2 being Submodule of V st x in W1 & W1 is Submodule of W2 holds
x in W2
let W1, W2 be Submodule of V; ::_thesis: ( x in W1 & W1 is Submodule of W2 implies x in W2 )
assume ( x in W1 & W1 is Submodule of W2 ) ; ::_thesis: x in W2
then ( x in the carrier of W1 & the carrier of W1 c= the carrier of W2 ) by Def9, STRUCT_0:def_5;
hence x in W2 by STRUCT_0:def_5; ::_thesis: verum
end;
theorem Th24: :: ZMODUL01:24
for V being Z_Module
for x being set
for W being Submodule of V st x in W holds
x in V
proof
let V be Z_Module; ::_thesis: for x being set
for W being Submodule of V st x in W holds
x in V
let x be set ; ::_thesis: for W being Submodule of V st x in W holds
x in V
let W be Submodule of V; ::_thesis: ( x in W implies x in V )
assume x in W ; ::_thesis: x in V
then A1: x in the carrier of W by STRUCT_0:def_5;
the carrier of W c= the carrier of V by Def9;
hence x in V by A1, STRUCT_0:def_5; ::_thesis: verum
end;
theorem Th25: :: ZMODUL01:25
for V being Z_Module
for W being Submodule of V
for w being VECTOR of W holds w is VECTOR of V
proof
let V be Z_Module; ::_thesis: for W being Submodule of V
for w being VECTOR of W holds w is VECTOR of V
let W be Submodule of V; ::_thesis: for w being VECTOR of W holds w is VECTOR of V
let w be VECTOR of W; ::_thesis: w is VECTOR of V
w in V by Th24, RLVECT_1:1;
hence w is VECTOR of V by STRUCT_0:def_5; ::_thesis: verum
end;
theorem :: ZMODUL01:26
for V being Z_Module
for W being Submodule of V holds 0. W = 0. V by Def9;
theorem :: ZMODUL01:27
for V being Z_Module
for W1, W2 being Submodule of V holds 0. W1 = 0. W2
proof
let V be Z_Module; ::_thesis: for W1, W2 being Submodule of V holds 0. W1 = 0. W2
let W1, W2 be Submodule of V; ::_thesis: 0. W1 = 0. W2
thus 0. W1 = 0. V by Def9
.= 0. W2 by Def9 ; ::_thesis: verum
end;
theorem Th28: :: ZMODUL01:28
for V being Z_Module
for v, u being VECTOR of V
for W being Submodule of V
for w1, w2 being VECTOR of W st w1 = v & w2 = u holds
w1 + w2 = v + u
proof
let V be Z_Module; ::_thesis: for v, u being VECTOR of V
for W being Submodule of V
for w1, w2 being VECTOR of W st w1 = v & w2 = u holds
w1 + w2 = v + u
let v, u be VECTOR of V; ::_thesis: for W being Submodule of V
for w1, w2 being VECTOR of W st w1 = v & w2 = u holds
w1 + w2 = v + u
let W be Submodule of V; ::_thesis: for w1, w2 being VECTOR of W st w1 = v & w2 = u holds
w1 + w2 = v + u
let w1, w2 be VECTOR of W; ::_thesis: ( w1 = v & w2 = u implies w1 + w2 = v + u )
assume A1: ( v = w1 & u = w2 ) ; ::_thesis: w1 + w2 = v + u
w1 + w2 = ( the addF of V || the carrier of W) . [w1,w2] by Def9;
hence w1 + w2 = v + u by A1, FUNCT_1:49; ::_thesis: verum
end;
theorem Th29: :: ZMODUL01:29
for V being Z_Module
for v being VECTOR of V
for a being integer number
for W being Submodule of V
for w being VECTOR of W st w = v holds
a * w = a * v
proof
let V be Z_Module; ::_thesis: for v being VECTOR of V
for a being integer number
for W being Submodule of V
for w being VECTOR of W st w = v holds
a * w = a * v
let v be VECTOR of V; ::_thesis: for a being integer number
for W being Submodule of V
for w being VECTOR of W st w = v holds
a * w = a * v
let a be integer number ; ::_thesis: for W being Submodule of V
for w being VECTOR of W st w = v holds
a * w = a * v
let W be Submodule of V; ::_thesis: for w being VECTOR of W st w = v holds
a * w = a * v
let w be VECTOR of W; ::_thesis: ( w = v implies a * w = a * v )
reconsider a = a as Element of INT by INT_1:def_2;
assume A1: w = v ; ::_thesis: a * w = a * v
a * w = ( the Mult of V | [:INT, the carrier of W:]) . [a,w] by Def9;
hence a * w = a * v by A1, FUNCT_1:49; ::_thesis: verum
end;
theorem Th30: :: ZMODUL01:30
for V being Z_Module
for v being VECTOR of V
for W being Submodule of V
for w being VECTOR of W st w = v holds
- v = - w
proof
let V be Z_Module; ::_thesis: for v being VECTOR of V
for W being Submodule of V
for w being VECTOR of W st w = v holds
- v = - w
let v be VECTOR of V; ::_thesis: for W being Submodule of V
for w being VECTOR of W st w = v holds
- v = - w
let W be Submodule of V; ::_thesis: for w being VECTOR of W st w = v holds
- v = - w
let w be VECTOR of W; ::_thesis: ( w = v implies - v = - w )
A1: ( - v = (- 1) * v & - w = (- 1) * w ) by Th2;
assume w = v ; ::_thesis: - v = - w
hence - v = - w by A1, Th29; ::_thesis: verum
end;
theorem Th31: :: ZMODUL01:31
for V being Z_Module
for v, u being VECTOR of V
for W being Submodule of V
for w1, w2 being VECTOR of W st w1 = v & w2 = u holds
w1 - w2 = v - u
proof
let V be Z_Module; ::_thesis: for v, u being VECTOR of V
for W being Submodule of V
for w1, w2 being VECTOR of W st w1 = v & w2 = u holds
w1 - w2 = v - u
let v, u be VECTOR of V; ::_thesis: for W being Submodule of V
for w1, w2 being VECTOR of W st w1 = v & w2 = u holds
w1 - w2 = v - u
let W be Submodule of V; ::_thesis: for w1, w2 being VECTOR of W st w1 = v & w2 = u holds
w1 - w2 = v - u
let w1, w2 be VECTOR of W; ::_thesis: ( w1 = v & w2 = u implies w1 - w2 = v - u )
assume that
A1: w1 = v and
A2: w2 = u ; ::_thesis: w1 - w2 = v - u
- w2 = - u by A2, Th30;
hence w1 - w2 = v - u by A1, Th28; ::_thesis: verum
end;
Lm3: for V being Z_Module
for V1 being Subset of V
for W being Submodule of V st the carrier of W = V1 holds
V1 is linearly-closed
proof
let V be Z_Module; ::_thesis: for V1 being Subset of V
for W being Submodule of V st the carrier of W = V1 holds
V1 is linearly-closed
let V1 be Subset of V; ::_thesis: for W being Submodule of V st the carrier of W = V1 holds
V1 is linearly-closed
let W be Submodule of V; ::_thesis: ( the carrier of W = V1 implies V1 is linearly-closed )
set VW = the carrier of W;
reconsider WW = W as Z_Module ;
assume A1: the carrier of W = V1 ; ::_thesis: V1 is linearly-closed
thus for v, u being VECTOR of V st v in V1 & u in V1 holds
v + u in V1 :: according to ZMODUL01:def_8 ::_thesis: for a being integer number
for v being VECTOR of V st v in V1 holds
a * v in V1
proof
let v, u be VECTOR of V; ::_thesis: ( v in V1 & u in V1 implies v + u in V1 )
assume ( v in V1 & u in V1 ) ; ::_thesis: v + u in V1
then reconsider vv = v, uu = u as VECTOR of WW by A1;
reconsider vw = vv + uu as Element of the carrier of W ;
vw in V1 by A1;
hence v + u in V1 by Th28; ::_thesis: verum
end;
let a be integer number ; ::_thesis: for v being VECTOR of V st v in V1 holds
a * v in V1
let v be VECTOR of V; ::_thesis: ( v in V1 implies a * v in V1 )
assume v in V1 ; ::_thesis: a * v in V1
then reconsider vv = v as VECTOR of WW by A1;
reconsider vw = a * vv as Element of the carrier of W ;
vw in V1 by A1;
hence a * v in V1 by Th29; ::_thesis: verum
end;
theorem Th32: :: ZMODUL01:32
for V being Z_Module holds V is Submodule of V
proof
let V be Z_Module; ::_thesis: V is Submodule of V
thus ( the carrier of V c= the carrier of V & 0. V = 0. V ) ; :: according to ZMODUL01:def_9 ::_thesis: ( the addF of V = the addF of V || the carrier of V & the Mult of V = the Mult of V | [:INT, the carrier of V:] )
thus ( the addF of V = the addF of V || the carrier of V & the Mult of V = the Mult of V | [:INT, the carrier of V:] ) by RELSET_1:19; ::_thesis: verum
end;
theorem Th33: :: ZMODUL01:33
for V being Z_Module
for W being Submodule of V holds 0. V in W
proof
let V be Z_Module; ::_thesis: for W being Submodule of V holds 0. V in W
let W be Submodule of V; ::_thesis: 0. V in W
0. W in W by RLVECT_1:1;
hence 0. V in W by Def9; ::_thesis: verum
end;
theorem :: ZMODUL01:34
for V being Z_Module
for W1, W2 being Submodule of V holds 0. W1 in W2
proof
let V be Z_Module; ::_thesis: for W1, W2 being Submodule of V holds 0. W1 in W2
let W1, W2 be Submodule of V; ::_thesis: 0. W1 in W2
0. W1 = 0. V by Def9;
hence 0. W1 in W2 by Th33; ::_thesis: verum
end;
theorem :: ZMODUL01:35
for V being Z_Module
for W being Submodule of V holds 0. W in V by Th24, RLVECT_1:1;
theorem Th36: :: ZMODUL01:36
for V being Z_Module
for u, v being VECTOR of V
for W being Submodule of V st u in W & v in W holds
u + v in W
proof
let V be Z_Module; ::_thesis: for u, v being VECTOR of V
for W being Submodule of V st u in W & v in W holds
u + v in W
let u, v be VECTOR of V; ::_thesis: for W being Submodule of V st u in W & v in W holds
u + v in W
let W be Submodule of V; ::_thesis: ( u in W & v in W implies u + v in W )
reconsider VW = the carrier of W as Subset of V by Def9;
assume ( u in W & v in W ) ; ::_thesis: u + v in W
then A1: ( u in the carrier of W & v in the carrier of W ) by STRUCT_0:def_5;
VW is linearly-closed by Lm3;
then u + v in the carrier of W by A1, Def8;
hence u + v in W by STRUCT_0:def_5; ::_thesis: verum
end;
theorem Th37: :: ZMODUL01:37
for V being Z_Module
for v being VECTOR of V
for a being integer number
for W being Submodule of V st v in W holds
a * v in W
proof
let V be Z_Module; ::_thesis: for v being VECTOR of V
for a being integer number
for W being Submodule of V st v in W holds
a * v in W
let v be VECTOR of V; ::_thesis: for a being integer number
for W being Submodule of V st v in W holds
a * v in W
let a be integer number ; ::_thesis: for W being Submodule of V st v in W holds
a * v in W
let W be Submodule of V; ::_thesis: ( v in W implies a * v in W )
reconsider VW = the carrier of W as Subset of V by Def9;
assume v in W ; ::_thesis: a * v in W
then A1: v in the carrier of W by STRUCT_0:def_5;
VW is linearly-closed by Lm3;
then a * v in the carrier of W by A1, Def8;
hence a * v in W by STRUCT_0:def_5; ::_thesis: verum
end;
theorem Th38: :: ZMODUL01:38
for V being Z_Module
for v being VECTOR of V
for W being Submodule of V st v in W holds
- v in W
proof
let V be Z_Module; ::_thesis: for v being VECTOR of V
for W being Submodule of V st v in W holds
- v in W
let v be VECTOR of V; ::_thesis: for W being Submodule of V st v in W holds
- v in W
let W be Submodule of V; ::_thesis: ( v in W implies - v in W )
assume v in W ; ::_thesis: - v in W
then (- 1) * v in W by Th37;
hence - v in W by Th2; ::_thesis: verum
end;
theorem Th39: :: ZMODUL01:39
for V being Z_Module
for u, v being VECTOR of V
for W being Submodule of V st u in W & v in W holds
u - v in W
proof
let V be Z_Module; ::_thesis: for u, v being VECTOR of V
for W being Submodule of V st u in W & v in W holds
u - v in W
let u, v be VECTOR of V; ::_thesis: for W being Submodule of V st u in W & v in W holds
u - v in W
let W be Submodule of V; ::_thesis: ( u in W & v in W implies u - v in W )
assume that
A1: u in W and
A2: v in W ; ::_thesis: u - v in W
- v in W by A2, Th38;
hence u - v in W by A1, Th36; ::_thesis: verum
end;
theorem Th40: :: ZMODUL01:40
for V being Z_Module
for V1 being Subset of V
for D being non empty set
for d1 being Element of D
for A being BinOp of D
for M being Function of [:INT,D:],D st V1 = D & d1 = 0. V & A = the addF of V || V1 & M = the Mult of V | [:INT,V1:] holds
Z_ModuleStruct(# D,d1,A,M #) is Submodule of V
proof
let V be Z_Module; ::_thesis: for V1 being Subset of V
for D being non empty set
for d1 being Element of D
for A being BinOp of D
for M being Function of [:INT,D:],D st V1 = D & d1 = 0. V & A = the addF of V || V1 & M = the Mult of V | [:INT,V1:] holds
Z_ModuleStruct(# D,d1,A,M #) is Submodule of V
let V1 be Subset of V; ::_thesis: for D being non empty set
for d1 being Element of D
for A being BinOp of D
for M being Function of [:INT,D:],D st V1 = D & d1 = 0. V & A = the addF of V || V1 & M = the Mult of V | [:INT,V1:] holds
Z_ModuleStruct(# D,d1,A,M #) is Submodule of V
let D be non empty set ; ::_thesis: for d1 being Element of D
for A being BinOp of D
for M being Function of [:INT,D:],D st V1 = D & d1 = 0. V & A = the addF of V || V1 & M = the Mult of V | [:INT,V1:] holds
Z_ModuleStruct(# D,d1,A,M #) is Submodule of V
let d1 be Element of D; ::_thesis: for A being BinOp of D
for M being Function of [:INT,D:],D st V1 = D & d1 = 0. V & A = the addF of V || V1 & M = the Mult of V | [:INT,V1:] holds
Z_ModuleStruct(# D,d1,A,M #) is Submodule of V
let A be BinOp of D; ::_thesis: for M being Function of [:INT,D:],D st V1 = D & d1 = 0. V & A = the addF of V || V1 & M = the Mult of V | [:INT,V1:] holds
Z_ModuleStruct(# D,d1,A,M #) is Submodule of V
let M be Function of [:INT,D:],D; ::_thesis: ( V1 = D & d1 = 0. V & A = the addF of V || V1 & M = the Mult of V | [:INT,V1:] implies Z_ModuleStruct(# D,d1,A,M #) is Submodule of V )
assume that
A1: V1 = D and
A2: d1 = 0. V and
A3: A = the addF of V || V1 and
A4: M = the Mult of V | [:INT,V1:] ; ::_thesis: Z_ModuleStruct(# D,d1,A,M #) is Submodule of V
set W = Z_ModuleStruct(# D,d1,A,M #);
A5: for a being integer number
for x being VECTOR of Z_ModuleStruct(# D,d1,A,M #) holds a * x = the Mult of V . (a,x)
proof
let a be integer number ; ::_thesis: for x being VECTOR of Z_ModuleStruct(# D,d1,A,M #) holds a * x = the Mult of V . (a,x)
let x be VECTOR of Z_ModuleStruct(# D,d1,A,M #); ::_thesis: a * x = the Mult of V . (a,x)
reconsider a1 = a as Element of INT by INT_1:def_2;
thus a * x = the Mult of V . [a1,x] by A1, A4, FUNCT_1:49
.= the Mult of V . (a,x) ; ::_thesis: verum
end;
A6: for x, y being VECTOR of Z_ModuleStruct(# D,d1,A,M #) holds x + y = the addF of V . (x,y)
proof
let x, y be VECTOR of Z_ModuleStruct(# D,d1,A,M #); ::_thesis: x + y = the addF of V . (x,y)
thus x + y = the addF of V . [x,y] by A1, A3, FUNCT_1:49
.= the addF of V . (x,y) ; ::_thesis: verum
end;
A7: ( Z_ModuleStruct(# D,d1,A,M #) is Abelian & Z_ModuleStruct(# D,d1,A,M #) is add-associative & Z_ModuleStruct(# D,d1,A,M #) is right_zeroed & Z_ModuleStruct(# D,d1,A,M #) is right_complementable & Z_ModuleStruct(# D,d1,A,M #) is vector-distributive & Z_ModuleStruct(# D,d1,A,M #) is scalar-distributive & Z_ModuleStruct(# D,d1,A,M #) is scalar-associative & Z_ModuleStruct(# D,d1,A,M #) is scalar-unital )
proof
set MV = the Mult of V;
set AV = the addF of V;
thus Z_ModuleStruct(# D,d1,A,M #) is Abelian ::_thesis: ( Z_ModuleStruct(# D,d1,A,M #) is add-associative & Z_ModuleStruct(# D,d1,A,M #) is right_zeroed & Z_ModuleStruct(# D,d1,A,M #) is right_complementable & Z_ModuleStruct(# D,d1,A,M #) is vector-distributive & Z_ModuleStruct(# D,d1,A,M #) is scalar-distributive & Z_ModuleStruct(# D,d1,A,M #) is scalar-associative & Z_ModuleStruct(# D,d1,A,M #) is scalar-unital )
proof
let x, y be VECTOR of Z_ModuleStruct(# D,d1,A,M #); :: according to RLVECT_1:def_2 ::_thesis: x + y = y + x
reconsider x1 = x, y1 = y as VECTOR of V by A1, TARSKI:def_3;
thus x + y = x1 + y1 by A6
.= y1 + x1
.= y + x by A6 ; ::_thesis: verum
end;
thus Z_ModuleStruct(# D,d1,A,M #) is add-associative ::_thesis: ( Z_ModuleStruct(# D,d1,A,M #) is right_zeroed & Z_ModuleStruct(# D,d1,A,M #) is right_complementable & Z_ModuleStruct(# D,d1,A,M #) is vector-distributive & Z_ModuleStruct(# D,d1,A,M #) is scalar-distributive & Z_ModuleStruct(# D,d1,A,M #) is scalar-associative & Z_ModuleStruct(# D,d1,A,M #) is scalar-unital )
proof
let x, y, z be VECTOR of Z_ModuleStruct(# D,d1,A,M #); :: according to RLVECT_1:def_3 ::_thesis: (x + y) + z = x + (y + z)
reconsider x1 = x, y1 = y, z1 = z as VECTOR of V by A1, TARSKI:def_3;
thus (x + y) + z = the addF of V . ((x + y),z1) by A6
.= (x1 + y1) + z1 by A6
.= x1 + (y1 + z1) by RLVECT_1:def_3
.= the addF of V . (x1,(y + z)) by A6
.= x + (y + z) by A6 ; ::_thesis: verum
end;
thus Z_ModuleStruct(# D,d1,A,M #) is right_zeroed ::_thesis: ( Z_ModuleStruct(# D,d1,A,M #) is right_complementable & Z_ModuleStruct(# D,d1,A,M #) is vector-distributive & Z_ModuleStruct(# D,d1,A,M #) is scalar-distributive & Z_ModuleStruct(# D,d1,A,M #) is scalar-associative & Z_ModuleStruct(# D,d1,A,M #) is scalar-unital )
proof
let x be VECTOR of Z_ModuleStruct(# D,d1,A,M #); :: according to RLVECT_1:def_4 ::_thesis: x + (0. Z_ModuleStruct(# D,d1,A,M #)) = x
reconsider y = x as VECTOR of V by A1, TARSKI:def_3;
thus x + (0. Z_ModuleStruct(# D,d1,A,M #)) = y + (0. V) by A2, A6
.= x by RLVECT_1:4 ; ::_thesis: verum
end;
thus Z_ModuleStruct(# D,d1,A,M #) is right_complementable ::_thesis: ( Z_ModuleStruct(# D,d1,A,M #) is vector-distributive & Z_ModuleStruct(# D,d1,A,M #) is scalar-distributive & Z_ModuleStruct(# D,d1,A,M #) is scalar-associative & Z_ModuleStruct(# D,d1,A,M #) is scalar-unital )
proof
let x be VECTOR of Z_ModuleStruct(# D,d1,A,M #); :: according to ALGSTR_0:def_16 ::_thesis: x is right_complementable
reconsider x1 = x as VECTOR of V by A1, TARSKI:def_3;
consider v being VECTOR of V such that
A8: x1 + v = 0. V by ALGSTR_0:def_11;
v = - x1 by A8, RLVECT_1:def_10
.= (- 1) * x1 by Th2
.= (- 1) * x by A5 ;
then reconsider y = v as VECTOR of Z_ModuleStruct(# D,d1,A,M #) ;
take y ; :: according to ALGSTR_0:def_11 ::_thesis: x + y = 0. Z_ModuleStruct(# D,d1,A,M #)
thus x + y = 0. Z_ModuleStruct(# D,d1,A,M #) by A2, A6, A8; ::_thesis: verum
end;
thus for a being integer number
for x, y being VECTOR of Z_ModuleStruct(# D,d1,A,M #) holds a * (x + y) = (a * x) + (a * y) :: according to ZMODUL01:def_2 ::_thesis: ( Z_ModuleStruct(# D,d1,A,M #) is scalar-distributive & Z_ModuleStruct(# D,d1,A,M #) is scalar-associative & Z_ModuleStruct(# D,d1,A,M #) is scalar-unital )
proof
let a be integer number ; ::_thesis: for x, y being VECTOR of Z_ModuleStruct(# D,d1,A,M #) holds a * (x + y) = (a * x) + (a * y)
let x, y be VECTOR of Z_ModuleStruct(# D,d1,A,M #); ::_thesis: a * (x + y) = (a * x) + (a * y)
reconsider x1 = x, y1 = y as VECTOR of V by A1, TARSKI:def_3;
reconsider a = a as Integer ;
a * (x + y) = the Mult of V . (a,(x + y)) by A5
.= a * (x1 + y1) by A6
.= (a * x1) + (a * y1) by Def2
.= the addF of V . (( the Mult of V . (a,x1)),(a * y)) by A5
.= the addF of V . ((a * x),(a * y)) by A5
.= (a * x) + (a * y) by A6 ;
hence a * (x + y) = (a * x) + (a * y) ; ::_thesis: verum
end;
thus for a, b being integer number
for x being VECTOR of Z_ModuleStruct(# D,d1,A,M #) holds (a + b) * x = (a * x) + (b * x) :: according to ZMODUL01:def_3 ::_thesis: ( Z_ModuleStruct(# D,d1,A,M #) is scalar-associative & Z_ModuleStruct(# D,d1,A,M #) is scalar-unital )
proof
let a, b be integer number ; ::_thesis: for x being VECTOR of Z_ModuleStruct(# D,d1,A,M #) holds (a + b) * x = (a * x) + (b * x)
let x be VECTOR of Z_ModuleStruct(# D,d1,A,M #); ::_thesis: (a + b) * x = (a * x) + (b * x)
reconsider y = x as VECTOR of V by A1, TARSKI:def_3;
reconsider a = a, b = b as Integer ;
(a + b) * x = (a + b) * y by A5
.= (a * y) + (b * y) by Def3
.= the addF of V . (( the Mult of V . (a,y)),(b * x)) by A5
.= the addF of V . ((a * x),(b * x)) by A5
.= (a * x) + (b * x) by A6 ;
hence (a + b) * x = (a * x) + (b * x) ; ::_thesis: verum
end;
thus for a, b being integer number
for x being VECTOR of Z_ModuleStruct(# D,d1,A,M #) holds (a * b) * x = a * (b * x) :: according to ZMODUL01:def_4 ::_thesis: Z_ModuleStruct(# D,d1,A,M #) is scalar-unital
proof
let a, b be integer number ; ::_thesis: for x being VECTOR of Z_ModuleStruct(# D,d1,A,M #) holds (a * b) * x = a * (b * x)
let x be VECTOR of Z_ModuleStruct(# D,d1,A,M #); ::_thesis: (a * b) * x = a * (b * x)
reconsider y = x as VECTOR of V by A1, TARSKI:def_3;
reconsider a = a, b = b as Integer ;
(a * b) * x = (a * b) * y by A5
.= a * (b * y) by Def4
.= the Mult of V . (a,(b * x)) by A5
.= a * (b * x) by A5 ;
hence (a * b) * x = a * (b * x) ; ::_thesis: verum
end;
let x be VECTOR of Z_ModuleStruct(# D,d1,A,M #); :: according to ZMODUL01:def_5 ::_thesis: 1 * x = x
reconsider y = x as VECTOR of V by A1, TARSKI:def_3;
thus 1 * x = 1 * y by A5
.= x by Def5 ; ::_thesis: verum
end;
0. Z_ModuleStruct(# D,d1,A,M #) = 0. V by A2;
hence Z_ModuleStruct(# D,d1,A,M #) is Submodule of V by A1, A3, A4, A7, Def9; ::_thesis: verum
end;
theorem Th41: :: ZMODUL01:41
for V, X being strict Z_Module st V is Submodule of X & X is Submodule of V holds
V = X
proof
let V, X be strict Z_Module; ::_thesis: ( V is Submodule of X & X is Submodule of V implies V = X )
assume that
A1: V is Submodule of X and
A2: X is Submodule of V ; ::_thesis: V = X
set VX = the carrier of X;
set VV = the carrier of V;
( the carrier of V c= the carrier of X & the carrier of X c= the carrier of V ) by A1, A2, Def9;
then A3: the carrier of V = the carrier of X by XBOOLE_0:def_10;
set AX = the addF of X;
set AV = the addF of V;
( the addF of V = the addF of X || the carrier of V & the addF of X = the addF of V || the carrier of X ) by A1, A2, Def9;
then A4: the addF of V = the addF of X by A3, RELAT_1:72;
set MX = the Mult of X;
set MV = the Mult of V;
A5: the Mult of X = the Mult of V | [:INT, the carrier of X:] by A2, Def9;
( 0. V = 0. X & the Mult of V = the Mult of X | [:INT, the carrier of V:] ) by A1, Def9;
hence V = X by A3, A4, A5, RELAT_1:72; ::_thesis: verum
end;
theorem Th42: :: ZMODUL01:42
for V, X, Y being Z_Module st V is Submodule of X & X is Submodule of Y holds
V is Submodule of Y
proof
let V, X, Y be Z_Module; ::_thesis: ( V is Submodule of X & X is Submodule of Y implies V is Submodule of Y )
assume that
A1: V is Submodule of X and
A2: X is Submodule of Y ; ::_thesis: V is Submodule of Y
( the carrier of V c= the carrier of X & the carrier of X c= the carrier of Y ) by A1, A2, Def9;
hence the carrier of V c= the carrier of Y by XBOOLE_1:1; :: according to ZMODUL01:def_9 ::_thesis: ( 0. V = 0. Y & the addF of V = the addF of Y || the carrier of V & the Mult of V = the Mult of Y | [:INT, the carrier of V:] )
0. V = 0. X by A1, Def9;
hence 0. V = 0. Y by A2, Def9; ::_thesis: ( the addF of V = the addF of Y || the carrier of V & the Mult of V = the Mult of Y | [:INT, the carrier of V:] )
thus the addF of V = the addF of Y || the carrier of V ::_thesis: the Mult of V = the Mult of Y | [:INT, the carrier of V:]
proof
set AY = the addF of Y;
set VX = the carrier of X;
set AX = the addF of X;
set VV = the carrier of V;
set AV = the addF of V;
the carrier of V c= the carrier of X by A1, Def9;
then A3: [: the carrier of V, the carrier of V:] c= [: the carrier of X, the carrier of X:] by ZFMISC_1:96;
the addF of V = the addF of X || the carrier of V by A1, Def9;
then the addF of V = ( the addF of Y || the carrier of X) || the carrier of V by A2, Def9;
hence the addF of V = the addF of Y || the carrier of V by A3, FUNCT_1:51; ::_thesis: verum
end;
set MY = the Mult of Y;
set MX = the Mult of X;
set MV = the Mult of V;
set VX = the carrier of X;
set VV = the carrier of V;
the carrier of V c= the carrier of X by A1, Def9;
then A4: [:INT, the carrier of V:] c= [:INT, the carrier of X:] by ZFMISC_1:95;
the Mult of V = the Mult of X | [:INT, the carrier of V:] by A1, Def9;
then the Mult of V = ( the Mult of Y | [:INT, the carrier of X:]) | [:INT, the carrier of V:] by A2, Def9;
hence the Mult of V = the Mult of Y | [:INT, the carrier of V:] by A4, FUNCT_1:51; ::_thesis: verum
end;
theorem Th43: :: ZMODUL01:43
for V being Z_Module
for W1, W2 being Submodule of V st the carrier of W1 c= the carrier of W2 holds
W1 is Submodule of W2
proof
let V be Z_Module; ::_thesis: for W1, W2 being Submodule of V st the carrier of W1 c= the carrier of W2 holds
W1 is Submodule of W2
let W1, W2 be Submodule of V; ::_thesis: ( the carrier of W1 c= the carrier of W2 implies W1 is Submodule of W2 )
set VW1 = the carrier of W1;
set VW2 = the carrier of W2;
set AV = the addF of V;
set MV = the Mult of V;
assume A1: the carrier of W1 c= the carrier of W2 ; ::_thesis: W1 is Submodule of W2
then A2: [: the carrier of W1, the carrier of W1:] c= [: the carrier of W2, the carrier of W2:] by ZFMISC_1:96;
0. W1 = 0. V by Def9;
hence ( the carrier of W1 c= the carrier of W2 & 0. W1 = 0. W2 ) by A1, Def9; :: according to ZMODUL01:def_9 ::_thesis: ( the addF of W1 = the addF of W2 || the carrier of W1 & the Mult of W1 = the Mult of W2 | [:INT, the carrier of W1:] )
( the addF of W1 = the addF of V || the carrier of W1 & the addF of W2 = the addF of V || the carrier of W2 ) by Def9;
hence the addF of W1 = the addF of W2 || the carrier of W1 by A2, FUNCT_1:51; ::_thesis: the Mult of W1 = the Mult of W2 | [:INT, the carrier of W1:]
A3: [:INT, the carrier of W1:] c= [:INT, the carrier of W2:] by A1, ZFMISC_1:95;
( the Mult of W1 = the Mult of V | [:INT, the carrier of W1:] & the Mult of W2 = the Mult of V | [:INT, the carrier of W2:] ) by Def9;
hence the Mult of W1 = the Mult of W2 | [:INT, the carrier of W1:] by A3, FUNCT_1:51; ::_thesis: verum
end;
theorem :: ZMODUL01:44
for V being Z_Module
for W1, W2 being Submodule of V st ( for v being VECTOR of V st v in W1 holds
v in W2 ) holds
W1 is Submodule of W2
proof
let V be Z_Module; ::_thesis: for W1, W2 being Submodule of V st ( for v being VECTOR of V st v in W1 holds
v in W2 ) holds
W1 is Submodule of W2
let W1, W2 be Submodule of V; ::_thesis: ( ( for v being VECTOR of V st v in W1 holds
v in W2 ) implies W1 is Submodule of W2 )
assume A1: for v being VECTOR of V st v in W1 holds
v in W2 ; ::_thesis: W1 is Submodule of W2
the carrier of W1 c= the carrier of W2
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in the carrier of W1 or x in the carrier of W2 )
assume A2: x in the carrier of W1 ; ::_thesis: x in the carrier of W2
the carrier of W1 c= the carrier of V by Def9;
then reconsider v = x as VECTOR of V by A2;
v in W1 by A2, STRUCT_0:def_5;
then v in W2 by A1;
hence x in the carrier of W2 by STRUCT_0:def_5; ::_thesis: verum
end;
hence W1 is Submodule of W2 by Th43; ::_thesis: verum
end;
registration
let V be Z_Module;
cluster non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable Abelian add-associative right_zeroed V129() V130() V131() V132() strict vector-distributive scalar-distributive scalar-associative scalar-unital for Submodule of V;
existence
ex b1 being Submodule of V st b1 is strict
proof
( the carrier of V is Subset of V iff the carrier of V c= the carrier of V ) ;
then reconsider V1 = the carrier of V as Subset of V ;
( the addF of V = the addF of V || V1 & the Mult of V = the Mult of V | [:INT,V1:] ) by RELSET_1:19;
then Z_ModuleStruct(# the carrier of V,(0. V), the addF of V, the Mult of V #) is Submodule of V by Th40;
hence ex b1 being Submodule of V st b1 is strict ; ::_thesis: verum
end;
end;
theorem Th45: :: ZMODUL01:45
for V being Z_Module
for W1, W2 being strict Submodule of V st the carrier of W1 = the carrier of W2 holds
W1 = W2
proof
let V be Z_Module; ::_thesis: for W1, W2 being strict Submodule of V st the carrier of W1 = the carrier of W2 holds
W1 = W2
let W1, W2 be strict Submodule of V; ::_thesis: ( the carrier of W1 = the carrier of W2 implies W1 = W2 )
assume the carrier of W1 = the carrier of W2 ; ::_thesis: W1 = W2
then ( W1 is Submodule of W2 & W2 is Submodule of W1 ) by Th43;
hence W1 = W2 by Th41; ::_thesis: verum
end;
theorem Th46: :: ZMODUL01:46
for V being Z_Module
for W1, W2 being strict Submodule of V st ( for v being VECTOR of V holds
( v in W1 iff v in W2 ) ) holds
W1 = W2
proof
let V be Z_Module; ::_thesis: for W1, W2 being strict Submodule of V st ( for v being VECTOR of V holds
( v in W1 iff v in W2 ) ) holds
W1 = W2
let W1, W2 be strict Submodule of V; ::_thesis: ( ( for v being VECTOR of V holds
( v in W1 iff v in W2 ) ) implies W1 = W2 )
assume A1: for v being VECTOR of V holds
( v in W1 iff v in W2 ) ; ::_thesis: W1 = W2
for x being set holds
( x in the carrier of W1 iff x in the carrier of W2 )
proof
let x be set ; ::_thesis: ( x in the carrier of W1 iff x in the carrier of W2 )
thus ( x in the carrier of W1 implies x in the carrier of W2 ) ::_thesis: ( x in the carrier of W2 implies x in the carrier of W1 )
proof
assume A2: x in the carrier of W1 ; ::_thesis: x in the carrier of W2
the carrier of W1 c= the carrier of V by Def9;
then reconsider v = x as VECTOR of V by A2;
v in W1 by A2, STRUCT_0:def_5;
then v in W2 by A1;
hence x in the carrier of W2 by STRUCT_0:def_5; ::_thesis: verum
end;
assume A3: x in the carrier of W2 ; ::_thesis: x in the carrier of W1
the carrier of W2 c= the carrier of V by Def9;
then reconsider v = x as VECTOR of V by A3;
v in W2 by A3, STRUCT_0:def_5;
then v in W1 by A1;
hence x in the carrier of W1 by STRUCT_0:def_5; ::_thesis: verum
end;
then the carrier of W1 = the carrier of W2 by TARSKI:1;
hence W1 = W2 by Th45; ::_thesis: verum
end;
theorem :: ZMODUL01:47
for V being strict Z_Module
for W being strict Submodule of V st the carrier of W = the carrier of V holds
W = V
proof
let V be strict Z_Module; ::_thesis: for W being strict Submodule of V st the carrier of W = the carrier of V holds
W = V
let W be strict Submodule of V; ::_thesis: ( the carrier of W = the carrier of V implies W = V )
assume A1: the carrier of W = the carrier of V ; ::_thesis: W = V
V is Submodule of V by Th32;
hence W = V by A1, Th45; ::_thesis: verum
end;
theorem :: ZMODUL01:48
for V being strict Z_Module
for W being strict Submodule of V st ( for v being VECTOR of V holds
( v in W iff v in V ) ) holds
W = V
proof
let V be strict Z_Module; ::_thesis: for W being strict Submodule of V st ( for v being VECTOR of V holds
( v in W iff v in V ) ) holds
W = V
let W be strict Submodule of V; ::_thesis: ( ( for v being VECTOR of V holds
( v in W iff v in V ) ) implies W = V )
assume A1: for v being VECTOR of V holds
( v in W iff v in V ) ; ::_thesis: W = V
V is Submodule of V by Th32;
hence W = V by A1, Th46; ::_thesis: verum
end;
theorem :: ZMODUL01:49
for V being Z_Module
for V1 being Subset of V
for W being Submodule of V st the carrier of W = V1 holds
V1 is linearly-closed by Lm3;
theorem Th50: :: ZMODUL01:50
for V being Z_Module
for V1 being Subset of V st V1 <> {} & V1 is linearly-closed holds
ex W being strict Submodule of V st V1 = the carrier of W
proof
let V be Z_Module; ::_thesis: for V1 being Subset of V st V1 <> {} & V1 is linearly-closed holds
ex W being strict Submodule of V st V1 = the carrier of W
let V1 be Subset of V; ::_thesis: ( V1 <> {} & V1 is linearly-closed implies ex W being strict Submodule of V st V1 = the carrier of W )
assume that
A1: V1 <> {} and
A2: V1 is linearly-closed ; ::_thesis: ex W being strict Submodule of V st V1 = the carrier of W
reconsider D = V1 as non empty set by A1;
set M = the Mult of V | [:INT,V1:];
set VV = the carrier of V;
dom the Mult of V = [:INT, the carrier of V:] by FUNCT_2:def_1;
then A3: dom ( the Mult of V | [:INT,V1:]) = [:INT, the carrier of V:] /\ [:INT,V1:] by RELAT_1:61;
[:INT,V1:] c= [:INT, the carrier of V:] by ZFMISC_1:95;
then A4: dom ( the Mult of V | [:INT,V1:]) = [:INT,D:] by A3, XBOOLE_1:28;
now__::_thesis:_for_y_being_set_holds_
(_(_y_in_D_implies_ex_x_being_set_st_
(_x_in_dom_(_the_Mult_of_V_|_[:INT,V1:])_&_y_=_(_the_Mult_of_V_|_[:INT,V1:])_._x_)_)_&_(_ex_x_being_set_st_
(_x_in_dom_(_the_Mult_of_V_|_[:INT,V1:])_&_y_=_(_the_Mult_of_V_|_[:INT,V1:])_._x_)_implies_y_in_D_)_)
let y be set ; ::_thesis: ( ( y in D implies ex x being set st
( x in dom ( the Mult of V | [:INT,V1:]) & y = ( the Mult of V | [:INT,V1:]) . x ) ) & ( ex x being set st
( x in dom ( the Mult of V | [:INT,V1:]) & y = ( the Mult of V | [:INT,V1:]) . x ) implies y in D ) )
thus ( y in D implies ex x being set st
( x in dom ( the Mult of V | [:INT,V1:]) & y = ( the Mult of V | [:INT,V1:]) . x ) ) ::_thesis: ( ex x being set st
( x in dom ( the Mult of V | [:INT,V1:]) & y = ( the Mult of V | [:INT,V1:]) . x ) implies y in D )
proof
assume A5: y in D ; ::_thesis: ex x being set st
( x in dom ( the Mult of V | [:INT,V1:]) & y = ( the Mult of V | [:INT,V1:]) . x )
then reconsider v1 = y as Element of the carrier of V ;
reconsider N1 = 1 as Element of INT by INT_1:def_2;
A6: [N1,y] in [:INT,D:] by A5, ZFMISC_1:87;
then ( the Mult of V | [:INT,V1:]) . [1,y] = 1 * v1 by FUNCT_1:49
.= y by Def5 ;
hence ex x being set st
( x in dom ( the Mult of V | [:INT,V1:]) & y = ( the Mult of V | [:INT,V1:]) . x ) by A4, A6; ::_thesis: verum
end;
given x being set such that A7: x in dom ( the Mult of V | [:INT,V1:]) and
A8: y = ( the Mult of V | [:INT,V1:]) . x ; ::_thesis: y in D
consider x1, x2 being set such that
A9: x1 in INT and
A10: x2 in D and
A11: x = [x1,x2] by A4, A7, ZFMISC_1:def_2;
reconsider xx1 = x1 as Integer by A9;
reconsider v2 = x2 as Element of the carrier of V by A10;
[x1,x2] in [:INT,V1:] by A9, A10, ZFMISC_1:87;
then y = xx1 * v2 by A8, A11, FUNCT_1:49;
hence y in D by A2, A10, Def8; ::_thesis: verum
end;
then D = rng ( the Mult of V | [:INT,V1:]) by FUNCT_1:def_3;
then reconsider M = the Mult of V | [:INT,V1:] as Function of [:INT,D:],D by A4, FUNCT_2:def_1, RELSET_1:4;
set A = the addF of V || V1;
reconsider d1 = 0. V as Element of D by A2, Th18;
dom the addF of V = [: the carrier of V, the carrier of V:] by FUNCT_2:def_1;
then dom ( the addF of V || V1) = [: the carrier of V, the carrier of V:] /\ [:V1,V1:] by RELAT_1:61;
then A12: dom ( the addF of V || V1) = [:D,D:] by XBOOLE_1:28;
now__::_thesis:_for_y_being_set_holds_
(_(_y_in_D_implies_ex_x_being_set_st_
(_x_in_dom_(_the_addF_of_V_||_V1)_&_y_=_(_the_addF_of_V_||_V1)_._x_)_)_&_(_ex_x_being_set_st_
(_x_in_dom_(_the_addF_of_V_||_V1)_&_y_=_(_the_addF_of_V_||_V1)_._x_)_implies_y_in_D_)_)
let y be set ; ::_thesis: ( ( y in D implies ex x being set st
( x in dom ( the addF of V || V1) & y = ( the addF of V || V1) . x ) ) & ( ex x being set st
( x in dom ( the addF of V || V1) & y = ( the addF of V || V1) . x ) implies y in D ) )
thus ( y in D implies ex x being set st
( x in dom ( the addF of V || V1) & y = ( the addF of V || V1) . x ) ) ::_thesis: ( ex x being set st
( x in dom ( the addF of V || V1) & y = ( the addF of V || V1) . x ) implies y in D )
proof
assume A13: y in D ; ::_thesis: ex x being set st
( x in dom ( the addF of V || V1) & y = ( the addF of V || V1) . x )
then reconsider v1 = y, v0 = d1 as Element of the carrier of V ;
A14: [d1,y] in [:D,D:] by A13, ZFMISC_1:87;
then ( the addF of V || V1) . [d1,y] = v0 + v1 by FUNCT_1:49
.= y by RLVECT_1:4 ;
hence ex x being set st
( x in dom ( the addF of V || V1) & y = ( the addF of V || V1) . x ) by A12, A14; ::_thesis: verum
end;
given x being set such that A15: x in dom ( the addF of V || V1) and
A16: y = ( the addF of V || V1) . x ; ::_thesis: y in D
consider x1, x2 being set such that
A17: ( x1 in D & x2 in D ) and
A18: x = [x1,x2] by A12, A15, ZFMISC_1:def_2;
reconsider v1 = x1, v2 = x2 as Element of the carrier of V by A17;
[x1,x2] in [:V1,V1:] by A17, ZFMISC_1:87;
then y = v1 + v2 by A16, A18, FUNCT_1:49;
hence y in D by A2, A17, Def8; ::_thesis: verum
end;
then D = rng ( the addF of V || V1) by FUNCT_1:def_3;
then reconsider A = the addF of V || V1 as Function of [:D,D:],D by A12, FUNCT_2:def_1, RELSET_1:4;
set W = Z_ModuleStruct(# D,d1,A,M #);
Z_ModuleStruct(# D,d1,A,M #) is Submodule of V by Th40;
hence ex W being strict Submodule of V st V1 = the carrier of W ; ::_thesis: verum
end;
definition
let V be Z_Module;
func (0). V -> strict Submodule of V means :Def10: :: ZMODUL01:def 10
the carrier of it = {(0. V)};
correctness
existence
ex b1 being strict Submodule of V st the carrier of b1 = {(0. V)};
uniqueness
for b1, b2 being strict Submodule of V st the carrier of b1 = {(0. V)} & the carrier of b2 = {(0. V)} holds
b1 = b2;
by Th45, Th50;
end;
:: deftheorem Def10 defines (0). ZMODUL01:def_10_:_
for V being Z_Module
for b2 being strict Submodule of V holds
( b2 = (0). V iff the carrier of b2 = {(0. V)} );
definition
let V be Z_Module;
func (Omega). V -> strict Submodule of V equals :: ZMODUL01:def 11
Z_ModuleStruct(# the carrier of V, the ZeroF of V, the addF of V, the Mult of V #);
coherence
Z_ModuleStruct(# the carrier of V, the ZeroF of V, the addF of V, the Mult of V #) is strict Submodule of V
proof
set W = Z_ModuleStruct(# the carrier of V, the ZeroF of V, the addF of V, the Mult of V #);
A1: for u, v, w being VECTOR of Z_ModuleStruct(# the carrier of V, the ZeroF of V, the addF of V, the Mult of V #) holds (u + v) + w = u + (v + w)
proof
let u, v, w be VECTOR of Z_ModuleStruct(# the carrier of V, the ZeroF of V, the addF of V, the Mult of V #); ::_thesis: (u + v) + w = u + (v + w)
reconsider u9 = u, v9 = v, w9 = w as VECTOR of V ;
thus (u + v) + w = (u9 + v9) + w9
.= u9 + (v9 + w9) by RLVECT_1:def_3
.= u + (v + w) ; ::_thesis: verum
end;
A2: for v being VECTOR of Z_ModuleStruct(# the carrier of V, the ZeroF of V, the addF of V, the Mult of V #) holds v + (0. Z_ModuleStruct(# the carrier of V, the ZeroF of V, the addF of V, the Mult of V #)) = v
proof
let v be VECTOR of Z_ModuleStruct(# the carrier of V, the ZeroF of V, the addF of V, the Mult of V #); ::_thesis: v + (0. Z_ModuleStruct(# the carrier of V, the ZeroF of V, the addF of V, the Mult of V #)) = v
reconsider v9 = v as VECTOR of V ;
thus v + (0. Z_ModuleStruct(# the carrier of V, the ZeroF of V, the addF of V, the Mult of V #)) = v9 + (0. V)
.= v by RLVECT_1:4 ; ::_thesis: verum
end;
A3: Z_ModuleStruct(# the carrier of V, the ZeroF of V, the addF of V, the Mult of V #) is right_complementable
proof
let v be VECTOR of Z_ModuleStruct(# the carrier of V, the ZeroF of V, the addF of V, the Mult of V #); :: according to ALGSTR_0:def_16 ::_thesis: v is right_complementable
reconsider v9 = v as VECTOR of V ;
consider w9 being VECTOR of V such that
A4: v9 + w9 = 0. V by ALGSTR_0:def_11;
reconsider w = w9 as VECTOR of Z_ModuleStruct(# the carrier of V, the ZeroF of V, the addF of V, the Mult of V #) ;
take w ; :: according to ALGSTR_0:def_11 ::_thesis: v + w = 0. Z_ModuleStruct(# the carrier of V, the ZeroF of V, the addF of V, the Mult of V #)
thus v + w = 0. Z_ModuleStruct(# the carrier of V, the ZeroF of V, the addF of V, the Mult of V #) by A4; ::_thesis: verum
end;
A5: for a being integer number
for v, w being VECTOR of Z_ModuleStruct(# the carrier of V, the ZeroF of V, the addF of V, the Mult of V #) holds a * (v + w) = (a * v) + (a * w)
proof
let a be integer number ; ::_thesis: for v, w being VECTOR of Z_ModuleStruct(# the carrier of V, the ZeroF of V, the addF of V, the Mult of V #) holds a * (v + w) = (a * v) + (a * w)
let v, w be VECTOR of Z_ModuleStruct(# the carrier of V, the ZeroF of V, the addF of V, the Mult of V #); ::_thesis: a * (v + w) = (a * v) + (a * w)
reconsider v9 = v, w9 = w as VECTOR of V ;
thus a * (v + w) = a * (v9 + w9)
.= (a * v9) + (a * w9) by Def2
.= (a * v) + (a * w) ; ::_thesis: verum
end;
A6: for a, b being integer number
for v being VECTOR of Z_ModuleStruct(# the carrier of V, the ZeroF of V, the addF of V, the Mult of V #) holds (a * b) * v = a * (b * v)
proof
let a, b be integer number ; ::_thesis: for v being VECTOR of Z_ModuleStruct(# the carrier of V, the ZeroF of V, the addF of V, the Mult of V #) holds (a * b) * v = a * (b * v)
let v be VECTOR of Z_ModuleStruct(# the carrier of V, the ZeroF of V, the addF of V, the Mult of V #); ::_thesis: (a * b) * v = a * (b * v)
reconsider v9 = v as VECTOR of V ;
thus (a * b) * v = (a * b) * v9
.= a * (b * v9) by Def4
.= a * (b * v) ; ::_thesis: verum
end;
A7: for a, b being integer number
for v being VECTOR of Z_ModuleStruct(# the carrier of V, the ZeroF of V, the addF of V, the Mult of V #) holds (a + b) * v = (a * v) + (b * v)
proof
let a, b be integer number ; ::_thesis: for v being VECTOR of Z_ModuleStruct(# the carrier of V, the ZeroF of V, the addF of V, the Mult of V #) holds (a + b) * v = (a * v) + (b * v)
let v be VECTOR of Z_ModuleStruct(# the carrier of V, the ZeroF of V, the addF of V, the Mult of V #); ::_thesis: (a + b) * v = (a * v) + (b * v)
reconsider v9 = v as VECTOR of V ;
thus (a + b) * v = (a + b) * v9
.= (a * v9) + (b * v9) by Def3
.= (a * v) + (b * v) ; ::_thesis: verum
end;
A8: for a being integer number
for v, w being VECTOR of Z_ModuleStruct(# the carrier of V, the ZeroF of V, the addF of V, the Mult of V #)
for v9, w9 being VECTOR of V st v = v9 & w = w9 holds
( v + w = v9 + w9 & a * v = a * v9 ) ;
A9: for v, w being VECTOR of Z_ModuleStruct(# the carrier of V, the ZeroF of V, the addF of V, the Mult of V #) holds v + w = w + v
proof
let v, w be VECTOR of Z_ModuleStruct(# the carrier of V, the ZeroF of V, the addF of V, the Mult of V #); ::_thesis: v + w = w + v
reconsider v9 = v, w9 = w as VECTOR of V ;
thus v + w = w9 + v9 by A8
.= w + v ; ::_thesis: verum
end;
for v being VECTOR of Z_ModuleStruct(# the carrier of V, the ZeroF of V, the addF of V, the Mult of V #) holds 1 * v = v
proof
let v be VECTOR of Z_ModuleStruct(# the carrier of V, the ZeroF of V, the addF of V, the Mult of V #); ::_thesis: 1 * v = v
reconsider v9 = v as VECTOR of V ;
thus 1 * v = 1 * v9
.= v by Def5 ; ::_thesis: verum
end;
then reconsider W = Z_ModuleStruct(# the carrier of V, the ZeroF of V, the addF of V, the Mult of V #) as Z_Module by A9, A1, A2, A3, A5, A7, A6, Def2, Def3, Def4, Def5, RLVECT_1:def_2, RLVECT_1:def_3, RLVECT_1:def_4;
A10: the Mult of W = the Mult of V | [:INT, the carrier of W:] by RELSET_1:19;
( 0. W = 0. V & the addF of W = the addF of V || the carrier of W ) by RELSET_1:19;
hence Z_ModuleStruct(# the carrier of V, the ZeroF of V, the addF of V, the Mult of V #) is strict Submodule of V by A10, Def9; ::_thesis: verum
end;
end;
:: deftheorem defines (Omega). ZMODUL01:def_11_:_
for V being Z_Module holds (Omega). V = Z_ModuleStruct(# the carrier of V, the ZeroF of V, the addF of V, the Mult of V #);
theorem Th51: :: ZMODUL01:51
for V being Z_Module
for W being Submodule of V holds (0). W = (0). V
proof
let V be Z_Module; ::_thesis: for W being Submodule of V holds (0). W = (0). V
let W be Submodule of V; ::_thesis: (0). W = (0). V
( the carrier of ((0). W) = {(0. W)} & the carrier of ((0). V) = {(0. V)} ) by Def10;
then A1: the carrier of ((0). W) = the carrier of ((0). V) by Def9;
(0). W is Submodule of V by Th42;
hence (0). W = (0). V by A1, Th45; ::_thesis: verum
end;
theorem Th52: :: ZMODUL01:52
for V being Z_Module
for W1, W2 being Submodule of V holds (0). W1 = (0). W2
proof
let V be Z_Module; ::_thesis: for W1, W2 being Submodule of V holds (0). W1 = (0). W2
let W1, W2 be Submodule of V; ::_thesis: (0). W1 = (0). W2
(0). W1 = (0). V by Th51;
hence (0). W1 = (0). W2 by Th51; ::_thesis: verum
end;
theorem :: ZMODUL01:53
for V being Z_Module
for W being Submodule of V holds (0). W is Submodule of V by Th42;
theorem Th54: :: ZMODUL01:54
for V being Z_Module
for W being Submodule of V holds (0). V is Submodule of W
proof
let V be Z_Module; ::_thesis: for W being Submodule of V holds (0). V is Submodule of W
let W be Submodule of V; ::_thesis: (0). V is Submodule of W
the carrier of ((0). V) = {(0. V)} by Def10
.= {(0. W)} by Def9 ;
hence (0). V is Submodule of W by Th43; ::_thesis: verum
end;
theorem :: ZMODUL01:55
for V being Z_Module
for W1, W2 being Submodule of V holds (0). W1 is Submodule of W2
proof
let V be Z_Module; ::_thesis: for W1, W2 being Submodule of V holds (0). W1 is Submodule of W2
let W1, W2 be Submodule of V; ::_thesis: (0). W1 is Submodule of W2
(0). W1 = (0). W2 by Th52;
hence (0). W1 is Submodule of W2 ; ::_thesis: verum
end;
theorem :: ZMODUL01:56
for V being strict Z_Module holds V is Submodule of (Omega). V ;
definition
let V be Z_Module;
let v be VECTOR of V;
let W be Submodule of V;
funcv + W -> Subset of V equals :: ZMODUL01:def 12
{ (v + u) where u is VECTOR of V : u in W } ;
coherence
{ (v + u) where u is VECTOR of V : u in W } is Subset of V
proof
set Y = { (v + u) where u is VECTOR of V : u in W } ;
defpred S1[ set ] means ex u being VECTOR of V st
( $1 = v + u & u in W );
consider X being set such that
A1: for x being set holds
( x in X iff ( x in the carrier of V & S1[x] ) ) from XBOOLE_0:sch_1();
X c= the carrier of V
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in X or x in the carrier of V )
assume x in X ; ::_thesis: x in the carrier of V
hence x in the carrier of V by A1; ::_thesis: verum
end;
then reconsider X = X as Subset of V ;
A2: { (v + u) where u is VECTOR of V : u in W } c= X
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in { (v + u) where u is VECTOR of V : u in W } or x in X )
assume x in { (v + u) where u is VECTOR of V : u in W } ; ::_thesis: x in X
then ex u being VECTOR of V st
( x = v + u & u in W ) ;
hence x in X by A1; ::_thesis: verum
end;
X c= { (v + u) where u is VECTOR of V : u in W }
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in X or x in { (v + u) where u is VECTOR of V : u in W } )
assume x in X ; ::_thesis: x in { (v + u) where u is VECTOR of V : u in W }
then ex u being VECTOR of V st
( x = v + u & u in W ) by A1;
hence x in { (v + u) where u is VECTOR of V : u in W } ; ::_thesis: verum
end;
hence { (v + u) where u is VECTOR of V : u in W } is Subset of V by A2, XBOOLE_0:def_10; ::_thesis: verum
end;
end;
:: deftheorem defines + ZMODUL01:def_12_:_
for V being Z_Module
for v being VECTOR of V
for W being Submodule of V holds v + W = { (v + u) where u is VECTOR of V : u in W } ;
Lm4: for V being Z_Module
for W being Submodule of V holds (0. V) + W = the carrier of W
proof
let V be Z_Module; ::_thesis: for W being Submodule of V holds (0. V) + W = the carrier of W
let W be Submodule of V; ::_thesis: (0. V) + W = the carrier of W
set A = { ((0. V) + u) where u is VECTOR of V : u in W } ;
A1: the carrier of W c= { ((0. V) + u) where u is VECTOR of V : u in W }
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in the carrier of W or x in { ((0. V) + u) where u is VECTOR of V : u in W } )
assume x in the carrier of W ; ::_thesis: x in { ((0. V) + u) where u is VECTOR of V : u in W }
then A2: x in W by STRUCT_0:def_5;
then x in V by Th24;
then reconsider y = x as Element of V by STRUCT_0:def_5;
(0. V) + y = x by RLVECT_1:4;
hence x in { ((0. V) + u) where u is VECTOR of V : u in W } by A2; ::_thesis: verum
end;
{ ((0. V) + u) where u is VECTOR of V : u in W } c= the carrier of W
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in { ((0. V) + u) where u is VECTOR of V : u in W } or x in the carrier of W )
assume x in { ((0. V) + u) where u is VECTOR of V : u in W } ; ::_thesis: x in the carrier of W
then consider u being VECTOR of V such that
A3: x = (0. V) + u and
A4: u in W ;
x = u by A3, RLVECT_1:4;
hence x in the carrier of W by A4, STRUCT_0:def_5; ::_thesis: verum
end;
hence (0. V) + W = the carrier of W by A1, XBOOLE_0:def_10; ::_thesis: verum
end;
definition
let V be Z_Module;
let W be Submodule of V;
mode Coset of W -> Subset of V means :Def13: :: ZMODUL01:def 13
ex v being VECTOR of V st it = v + W;
existence
ex b1 being Subset of V ex v being VECTOR of V st b1 = v + W
proof
reconsider VW = the carrier of W as Subset of V by Def9;
take VW ; ::_thesis: ex v being VECTOR of V st VW = v + W
take 0. V ; ::_thesis: VW = (0. V) + W
thus VW = (0. V) + W by Lm4; ::_thesis: verum
end;
end;
:: deftheorem Def13 defines Coset ZMODUL01:def_13_:_
for V being Z_Module
for W being Submodule of V
for b3 being Subset of V holds
( b3 is Coset of W iff ex v being VECTOR of V st b3 = v + W );
theorem Th57: :: ZMODUL01:57
for V being Z_Module
for v being VECTOR of V
for W being Submodule of V holds
( 0. V in v + W iff v in W )
proof
let V be Z_Module; ::_thesis: for v being VECTOR of V
for W being Submodule of V holds
( 0. V in v + W iff v in W )
let v be VECTOR of V; ::_thesis: for W being Submodule of V holds
( 0. V in v + W iff v in W )
let W be Submodule of V; ::_thesis: ( 0. V in v + W iff v in W )
thus ( 0. V in v + W implies v in W ) ::_thesis: ( v in W implies 0. V in v + W )
proof
assume 0. V in v + W ; ::_thesis: v in W
then consider u being VECTOR of V such that
A1: 0. V = v + u and
A2: u in W ;
v = - u by A1, RLVECT_1:def_10;
hence v in W by A2, Th38; ::_thesis: verum
end;
assume v in W ; ::_thesis: 0. V in v + W
then A3: - v in W by Th38;
0. V = v - v by RLVECT_1:15
.= v + (- v) ;
hence 0. V in v + W by A3; ::_thesis: verum
end;
theorem Th58: :: ZMODUL01:58
for V being Z_Module
for v being VECTOR of V
for W being Submodule of V holds v in v + W
proof
let V be Z_Module; ::_thesis: for v being VECTOR of V
for W being Submodule of V holds v in v + W
let v be VECTOR of V; ::_thesis: for W being Submodule of V holds v in v + W
let W be Submodule of V; ::_thesis: v in v + W
( v + (0. V) = v & 0. V in W ) by Th33, RLVECT_1:4;
hence v in v + W ; ::_thesis: verum
end;
theorem :: ZMODUL01:59
for V being Z_Module
for W being Submodule of V holds (0. V) + W = the carrier of W by Lm4;
theorem Th60: :: ZMODUL01:60
for V being Z_Module
for v being VECTOR of V holds v + ((0). V) = {v}
proof
let V be Z_Module; ::_thesis: for v being VECTOR of V holds v + ((0). V) = {v}
let v be VECTOR of V; ::_thesis: v + ((0). V) = {v}
thus v + ((0). V) c= {v} :: according to XBOOLE_0:def_10 ::_thesis: {v} c= v + ((0). V)
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in v + ((0). V) or x in {v} )
assume x in v + ((0). V) ; ::_thesis: x in {v}
then consider u being VECTOR of V such that
A1: x = v + u and
A2: u in (0). V ;
A3: the carrier of ((0). V) = {(0. V)} by Def10;
u in the carrier of ((0). V) by A2, STRUCT_0:def_5;
then u = 0. V by A3, TARSKI:def_1;
then x = v by A1, RLVECT_1:4;
hence x in {v} by TARSKI:def_1; ::_thesis: verum
end;
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in {v} or x in v + ((0). V) )
assume x in {v} ; ::_thesis: x in v + ((0). V)
then A4: x = v by TARSKI:def_1;
( 0. V in (0). V & v = v + (0. V) ) by Th33, RLVECT_1:4;
hence x in v + ((0). V) by A4; ::_thesis: verum
end;
Lm5: for V being Z_Module
for v being VECTOR of V
for W being Submodule of V holds
( v in W iff v + W = the carrier of W )
proof
let V be Z_Module; ::_thesis: for v being VECTOR of V
for W being Submodule of V holds
( v in W iff v + W = the carrier of W )
let v be VECTOR of V; ::_thesis: for W being Submodule of V holds
( v in W iff v + W = the carrier of W )
let W be Submodule of V; ::_thesis: ( v in W iff v + W = the carrier of W )
( 0. V in W & v + (0. V) = v ) by Th33, RLVECT_1:4;
then A1: v in { (v + u) where u is VECTOR of V : u in W } ;
thus ( v in W implies v + W = the carrier of W ) ::_thesis: ( v + W = the carrier of W implies v in W )
proof
assume A2: v in W ; ::_thesis: v + W = the carrier of W
thus v + W c= the carrier of W :: according to XBOOLE_0:def_10 ::_thesis: the carrier of W c= v + W
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in v + W or x in the carrier of W )
assume x in v + W ; ::_thesis: x in the carrier of W
then consider u being VECTOR of V such that
A3: x = v + u and
A4: u in W ;
v + u in W by A2, A4, Th36;
hence x in the carrier of W by A3, STRUCT_0:def_5; ::_thesis: verum
end;
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in the carrier of W or x in v + W )
assume x in the carrier of W ; ::_thesis: x in v + W
then reconsider y = x, z = v as Element of W by A2, STRUCT_0:def_5;
reconsider y1 = y, z1 = z as VECTOR of V by Th25;
A5: z + (y - z) = (y + z) - z by RLVECT_1:def_3
.= y + (z - z) by RLVECT_1:def_3
.= y + (0. W) by RLVECT_1:15
.= x by RLVECT_1:4 ;
y - z in W by STRUCT_0:def_5;
then A6: y1 - z1 in W by Th31;
y - z = y1 - z1 by Th31;
then z1 + (y1 - z1) = x by A5, Th28;
hence x in v + W by A6; ::_thesis: verum
end;
thus ( v + W = the carrier of W implies v in W ) by A1, STRUCT_0:def_5; ::_thesis: verum
end;
theorem Th61: :: ZMODUL01:61
for V being Z_Module
for v being VECTOR of V holds v + ((Omega). V) = the carrier of V
proof
let V be Z_Module; ::_thesis: for v being VECTOR of V holds v + ((Omega). V) = the carrier of V
let v be VECTOR of V; ::_thesis: v + ((Omega). V) = the carrier of V
v in (Omega). V by STRUCT_0:def_5;
hence v + ((Omega). V) = the carrier of V by Lm5; ::_thesis: verum
end;
theorem Th62: :: ZMODUL01:62
for V being Z_Module
for v being VECTOR of V
for W being Submodule of V holds
( 0. V in v + W iff v + W = the carrier of W )
proof
let V be Z_Module; ::_thesis: for v being VECTOR of V
for W being Submodule of V holds
( 0. V in v + W iff v + W = the carrier of W )
let v be VECTOR of V; ::_thesis: for W being Submodule of V holds
( 0. V in v + W iff v + W = the carrier of W )
let W be Submodule of V; ::_thesis: ( 0. V in v + W iff v + W = the carrier of W )
( 0. V in v + W iff v in W ) by Th57;
hence ( 0. V in v + W iff v + W = the carrier of W ) by Lm5; ::_thesis: verum
end;
theorem :: ZMODUL01:63
for V being Z_Module
for v being VECTOR of V
for W being Submodule of V holds
( v in W iff v + W = the carrier of W ) by Lm5;
theorem :: ZMODUL01:64
for V being Z_Module
for v being VECTOR of V
for a being integer number
for W being Submodule of V st v in W holds
(a * v) + W = the carrier of W
proof
let V be Z_Module; ::_thesis: for v being VECTOR of V
for a being integer number
for W being Submodule of V st v in W holds
(a * v) + W = the carrier of W
let v be VECTOR of V; ::_thesis: for a being integer number
for W being Submodule of V st v in W holds
(a * v) + W = the carrier of W
let a be integer number ; ::_thesis: for W being Submodule of V st v in W holds
(a * v) + W = the carrier of W
let W be Submodule of V; ::_thesis: ( v in W implies (a * v) + W = the carrier of W )
assume A1: v in W ; ::_thesis: (a * v) + W = the carrier of W
thus (a * v) + W c= the carrier of W :: according to XBOOLE_0:def_10 ::_thesis: the carrier of W c= (a * v) + W
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in (a * v) + W or x in the carrier of W )
assume x in (a * v) + W ; ::_thesis: x in the carrier of W
then consider u being VECTOR of V such that
A2: x = (a * v) + u and
A3: u in W ;
a * v in W by A1, Th37;
then (a * v) + u in W by A3, Th36;
hence x in the carrier of W by A2, STRUCT_0:def_5; ::_thesis: verum
end;
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in the carrier of W or x in (a * v) + W )
assume A4: x in the carrier of W ; ::_thesis: x in (a * v) + W
then A5: x in W by STRUCT_0:def_5;
the carrier of W c= the carrier of V by Def9;
then reconsider y = x as Element of V by A4;
A6: (a * v) + (y - (a * v)) = (y + (a * v)) - (a * v) by RLVECT_1:def_3
.= y + ((a * v) - (a * v)) by RLVECT_1:def_3
.= y + (0. V) by RLVECT_1:15
.= x by RLVECT_1:4 ;
a * v in W by A1, Th37;
then y - (a * v) in W by A5, Th39;
hence x in (a * v) + W by A6; ::_thesis: verum
end;
theorem Th65: :: ZMODUL01:65
for V being Z_Module
for u, v being VECTOR of V
for W being Submodule of V holds
( u in W iff v + W = (v + u) + W )
proof
let V be Z_Module; ::_thesis: for u, v being VECTOR of V
for W being Submodule of V holds
( u in W iff v + W = (v + u) + W )
let u, v be VECTOR of V; ::_thesis: for W being Submodule of V holds
( u in W iff v + W = (v + u) + W )
let W be Submodule of V; ::_thesis: ( u in W iff v + W = (v + u) + W )
thus ( u in W implies v + W = (v + u) + W ) ::_thesis: ( v + W = (v + u) + W implies u in W )
proof
assume A1: u in W ; ::_thesis: v + W = (v + u) + W
thus v + W c= (v + u) + W :: according to XBOOLE_0:def_10 ::_thesis: (v + u) + W c= v + W
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in v + W or x in (v + u) + W )
assume x in v + W ; ::_thesis: x in (v + u) + W
then consider v1 being VECTOR of V such that
A2: x = v + v1 and
A3: v1 in W ;
A4: (v + u) + (v1 - u) = v + (u + (v1 - u)) by RLVECT_1:def_3
.= v + ((v1 + u) - u) by RLVECT_1:def_3
.= v + (v1 + (u - u)) by RLVECT_1:def_3
.= v + (v1 + (0. V)) by RLVECT_1:15
.= x by A2, RLVECT_1:4 ;
v1 - u in W by A1, A3, Th39;
hence x in (v + u) + W by A4; ::_thesis: verum
end;
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in (v + u) + W or x in v + W )
assume x in (v + u) + W ; ::_thesis: x in v + W
then consider v2 being VECTOR of V such that
A5: x = (v + u) + v2 and
A6: v2 in W ;
A7: x = v + (u + v2) by A5, RLVECT_1:def_3;
u + v2 in W by A1, A6, Th36;
hence x in v + W by A7; ::_thesis: verum
end;
assume A8: v + W = (v + u) + W ; ::_thesis: u in W
( 0. V in W & v + (0. V) = v ) by Th33, RLVECT_1:4;
then v in (v + u) + W by A8;
then consider u1 being VECTOR of V such that
A9: v = (v + u) + u1 and
A10: u1 in W ;
( v = v + (0. V) & v = v + (u + u1) ) by A9, RLVECT_1:4, RLVECT_1:def_3;
then u + u1 = 0. V by RLVECT_1:8;
then u = - u1 by RLVECT_1:def_10;
hence u in W by A10, Th38; ::_thesis: verum
end;
theorem :: ZMODUL01:66
for V being Z_Module
for u, v being VECTOR of V
for W being Submodule of V holds
( u in W iff v + W = (v - u) + W )
proof
let V be Z_Module; ::_thesis: for u, v being VECTOR of V
for W being Submodule of V holds
( u in W iff v + W = (v - u) + W )
let u, v be VECTOR of V; ::_thesis: for W being Submodule of V holds
( u in W iff v + W = (v - u) + W )
let W be Submodule of V; ::_thesis: ( u in W iff v + W = (v - u) + W )
A1: ( - u in W implies u in W )
proof
assume - u in W ; ::_thesis: u in W
then - (- u) in W by Th38;
hence u in W by RLVECT_1:17; ::_thesis: verum
end;
( - u in W iff v + W = (v + (- u)) + W ) by Th65;
hence ( u in W iff v + W = (v - u) + W ) by A1, Th38; ::_thesis: verum
end;
theorem Th67: :: ZMODUL01:67
for V being Z_Module
for v, u being VECTOR of V
for W being Submodule of V holds
( v in u + W iff u + W = v + W )
proof
let V be Z_Module; ::_thesis: for v, u being VECTOR of V
for W being Submodule of V holds
( v in u + W iff u + W = v + W )
let v, u be VECTOR of V; ::_thesis: for W being Submodule of V holds
( v in u + W iff u + W = v + W )
let W be Submodule of V; ::_thesis: ( v in u + W iff u + W = v + W )
thus ( v in u + W implies u + W = v + W ) ::_thesis: ( u + W = v + W implies v in u + W )
proof
assume v in u + W ; ::_thesis: u + W = v + W
then consider z being VECTOR of V such that
A1: v = u + z and
A2: z in W ;
thus u + W c= v + W :: according to XBOOLE_0:def_10 ::_thesis: v + W c= u + W
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in u + W or x in v + W )
assume x in u + W ; ::_thesis: x in v + W
then consider v1 being VECTOR of V such that
A3: x = u + v1 and
A4: v1 in W ;
v - z = u + (z - z) by A1, RLVECT_1:def_3
.= u + (0. V) by RLVECT_1:15
.= u by RLVECT_1:4 ;
then A5: x = v + (v1 + (- z)) by A3, RLVECT_1:def_3
.= v + (v1 - z) ;
v1 - z in W by A2, A4, Th39;
hence x in v + W by A5; ::_thesis: verum
end;
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in v + W or x in u + W )
assume x in v + W ; ::_thesis: x in u + W
then consider v2 being VECTOR of V such that
A6: ( x = v + v2 & v2 in W ) ;
( z + v2 in W & x = u + (z + v2) ) by A1, A2, A6, Th36, RLVECT_1:def_3;
hence x in u + W ; ::_thesis: verum
end;
thus ( u + W = v + W implies v in u + W ) by Th58; ::_thesis: verum
end;
theorem Th68: :: ZMODUL01:68
for V being Z_Module
for u, v1, v2 being VECTOR of V
for W being Submodule of V st u in v1 + W & u in v2 + W holds
v1 + W = v2 + W
proof
let V be Z_Module; ::_thesis: for u, v1, v2 being VECTOR of V
for W being Submodule of V st u in v1 + W & u in v2 + W holds
v1 + W = v2 + W
let u, v1, v2 be VECTOR of V; ::_thesis: for W being Submodule of V st u in v1 + W & u in v2 + W holds
v1 + W = v2 + W
let W be Submodule of V; ::_thesis: ( u in v1 + W & u in v2 + W implies v1 + W = v2 + W )
assume that
A1: u in v1 + W and
A2: u in v2 + W ; ::_thesis: v1 + W = v2 + W
consider x1 being VECTOR of V such that
A3: u = v1 + x1 and
A4: x1 in W by A1;
consider x2 being VECTOR of V such that
A5: u = v2 + x2 and
A6: x2 in W by A2;
thus v1 + W c= v2 + W :: according to XBOOLE_0:def_10 ::_thesis: v2 + W c= v1 + W
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in v1 + W or x in v2 + W )
assume x in v1 + W ; ::_thesis: x in v2 + W
then consider u1 being VECTOR of V such that
A7: x = v1 + u1 and
A8: u1 in W ;
x2 - x1 in W by A4, A6, Th39;
then A9: (x2 - x1) + u1 in W by A8, Th36;
u - x1 = v1 + (x1 - x1) by A3, RLVECT_1:def_3
.= v1 + (0. V) by RLVECT_1:15
.= v1 by RLVECT_1:4 ;
then x = (v2 + (x2 - x1)) + u1 by A5, A7, RLVECT_1:def_3
.= v2 + ((x2 - x1) + u1) by RLVECT_1:def_3 ;
hence x in v2 + W by A9; ::_thesis: verum
end;
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in v2 + W or x in v1 + W )
assume x in v2 + W ; ::_thesis: x in v1 + W
then consider u1 being VECTOR of V such that
A10: x = v2 + u1 and
A11: u1 in W ;
x1 - x2 in W by A4, A6, Th39;
then A12: (x1 - x2) + u1 in W by A11, Th36;
u - x2 = v2 + (x2 - x2) by A5, RLVECT_1:def_3
.= v2 + (0. V) by RLVECT_1:15
.= v2 by RLVECT_1:4 ;
then x = (v1 + (x1 - x2)) + u1 by A3, A10, RLVECT_1:def_3
.= v1 + ((x1 - x2) + u1) by RLVECT_1:def_3 ;
hence x in v1 + W by A12; ::_thesis: verum
end;
theorem :: ZMODUL01:69
for V being Z_Module
for v being VECTOR of V
for a being integer number
for W being Submodule of V st v in W holds
a * v in v + W
proof
let V be Z_Module; ::_thesis: for v being VECTOR of V
for a being integer number
for W being Submodule of V st v in W holds
a * v in v + W
let v be VECTOR of V; ::_thesis: for a being integer number
for W being Submodule of V st v in W holds
a * v in v + W
let a be integer number ; ::_thesis: for W being Submodule of V st v in W holds
a * v in v + W
let W be Submodule of V; ::_thesis: ( v in W implies a * v in v + W )
assume v in W ; ::_thesis: a * v in v + W
then A1: (a - 1) * v in W by Th37;
a * v = ((a - 1) + 1) * v
.= ((a - 1) * v) + (1 * v) by Def3
.= v + ((a - 1) * v) by Def5 ;
hence a * v in v + W by A1; ::_thesis: verum
end;
theorem Th70: :: ZMODUL01:70
for V being Z_Module
for u, v being VECTOR of V
for W being Submodule of V holds
( u + v in v + W iff u in W )
proof
let V be Z_Module; ::_thesis: for u, v being VECTOR of V
for W being Submodule of V holds
( u + v in v + W iff u in W )
let u, v be VECTOR of V; ::_thesis: for W being Submodule of V holds
( u + v in v + W iff u in W )
let W be Submodule of V; ::_thesis: ( u + v in v + W iff u in W )
thus ( u + v in v + W implies u in W ) ::_thesis: ( u in W implies u + v in v + W )
proof
assume u + v in v + W ; ::_thesis: u in W
then ex v1 being VECTOR of V st
( u + v = v + v1 & v1 in W ) ;
hence u in W by RLVECT_1:8; ::_thesis: verum
end;
assume u in W ; ::_thesis: u + v in v + W
hence u + v in v + W ; ::_thesis: verum
end;
theorem :: ZMODUL01:71
for V being Z_Module
for v, u being VECTOR of V
for W being Submodule of V holds
( v - u in v + W iff u in W )
proof
let V be Z_Module; ::_thesis: for v, u being VECTOR of V
for W being Submodule of V holds
( v - u in v + W iff u in W )
let v, u be VECTOR of V; ::_thesis: for W being Submodule of V holds
( v - u in v + W iff u in W )
let W be Submodule of V; ::_thesis: ( v - u in v + W iff u in W )
A1: v - u = (- u) + v ;
A2: ( - u in W implies - (- u) in W ) by Th38;
( u in W implies - u in W ) by Th38;
hence ( v - u in v + W iff u in W ) by A1, A2, Th70, RLVECT_1:17; ::_thesis: verum
end;
theorem Th72: :: ZMODUL01:72
for V being Z_Module
for u, v being VECTOR of V
for W being Submodule of V holds
( u in v + W iff ex v1 being VECTOR of V st
( v1 in W & u = v + v1 ) )
proof
let V be Z_Module; ::_thesis: for u, v being VECTOR of V
for W being Submodule of V holds
( u in v + W iff ex v1 being VECTOR of V st
( v1 in W & u = v + v1 ) )
let u, v be VECTOR of V; ::_thesis: for W being Submodule of V holds
( u in v + W iff ex v1 being VECTOR of V st
( v1 in W & u = v + v1 ) )
let W be Submodule of V; ::_thesis: ( u in v + W iff ex v1 being VECTOR of V st
( v1 in W & u = v + v1 ) )
thus ( u in v + W implies ex v1 being VECTOR of V st
( v1 in W & u = v + v1 ) ) ::_thesis: ( ex v1 being VECTOR of V st
( v1 in W & u = v + v1 ) implies u in v + W )
proof
assume u in v + W ; ::_thesis: ex v1 being VECTOR of V st
( v1 in W & u = v + v1 )
then ex v1 being VECTOR of V st
( u = v + v1 & v1 in W ) ;
hence ex v1 being VECTOR of V st
( v1 in W & u = v + v1 ) ; ::_thesis: verum
end;
given v1 being VECTOR of V such that A1: ( v1 in W & u = v + v1 ) ; ::_thesis: u in v + W
thus u in v + W by A1; ::_thesis: verum
end;
theorem Th73: :: ZMODUL01:73
for V being Z_Module
for u, v being VECTOR of V
for W being Submodule of V holds
( u in v + W iff ex v1 being VECTOR of V st
( v1 in W & u = v - v1 ) )
proof
let V be Z_Module; ::_thesis: for u, v being VECTOR of V
for W being Submodule of V holds
( u in v + W iff ex v1 being VECTOR of V st
( v1 in W & u = v - v1 ) )
let u, v be VECTOR of V; ::_thesis: for W being Submodule of V holds
( u in v + W iff ex v1 being VECTOR of V st
( v1 in W & u = v - v1 ) )
let W be Submodule of V; ::_thesis: ( u in v + W iff ex v1 being VECTOR of V st
( v1 in W & u = v - v1 ) )
thus ( u in v + W implies ex v1 being VECTOR of V st
( v1 in W & u = v - v1 ) ) ::_thesis: ( ex v1 being VECTOR of V st
( v1 in W & u = v - v1 ) implies u in v + W )
proof
assume u in v + W ; ::_thesis: ex v1 being VECTOR of V st
( v1 in W & u = v - v1 )
then consider v1 being VECTOR of V such that
A1: u = v + v1 and
A2: v1 in W ;
take x = - v1; ::_thesis: ( x in W & u = v - x )
thus x in W by A2, Th38; ::_thesis: u = v - x
thus u = v - x by A1, RLVECT_1:17; ::_thesis: verum
end;
given v1 being VECTOR of V such that A3: v1 in W and
A4: u = v - v1 ; ::_thesis: u in v + W
- v1 in W by A3, Th38;
hence u in v + W by A4; ::_thesis: verum
end;
theorem Th74: :: ZMODUL01:74
for V being Z_Module
for v1, v2 being VECTOR of V
for W being Submodule of V holds
( ex v being VECTOR of V st
( v1 in v + W & v2 in v + W ) iff v1 - v2 in W )
proof
let V be Z_Module; ::_thesis: for v1, v2 being VECTOR of V
for W being Submodule of V holds
( ex v being VECTOR of V st
( v1 in v + W & v2 in v + W ) iff v1 - v2 in W )
let v1, v2 be VECTOR of V; ::_thesis: for W being Submodule of V holds
( ex v being VECTOR of V st
( v1 in v + W & v2 in v + W ) iff v1 - v2 in W )
let W be Submodule of V; ::_thesis: ( ex v being VECTOR of V st
( v1 in v + W & v2 in v + W ) iff v1 - v2 in W )
thus ( ex v being VECTOR of V st
( v1 in v + W & v2 in v + W ) implies v1 - v2 in W ) ::_thesis: ( v1 - v2 in W implies ex v being VECTOR of V st
( v1 in v + W & v2 in v + W ) )
proof
given v being VECTOR of V such that A1: v1 in v + W and
A2: v2 in v + W ; ::_thesis: v1 - v2 in W
consider u2 being VECTOR of V such that
A3: u2 in W and
A4: v2 = v + u2 by A2, Th72;
consider u1 being VECTOR of V such that
A5: u1 in W and
A6: v1 = v + u1 by A1, Th72;
v1 - v2 = (u1 + v) + ((- v) - u2) by A6, A4, RLVECT_1:30
.= ((u1 + v) + (- v)) - u2 by RLVECT_1:def_3
.= (u1 + (v + (- v))) - u2 by RLVECT_1:def_3
.= (u1 + (0. V)) - u2 by RLVECT_1:5
.= u1 - u2 by RLVECT_1:4 ;
hence v1 - v2 in W by A5, A3, Th39; ::_thesis: verum
end;
assume v1 - v2 in W ; ::_thesis: ex v being VECTOR of V st
( v1 in v + W & v2 in v + W )
then A7: - (v1 - v2) in W by Th38;
take v1 ; ::_thesis: ( v1 in v1 + W & v2 in v1 + W )
thus v1 in v1 + W by Th58; ::_thesis: v2 in v1 + W
v1 + (- (v1 - v2)) = v1 + ((- v1) + v2) by RLVECT_1:33
.= (v1 + (- v1)) + v2 by RLVECT_1:def_3
.= (0. V) + v2 by RLVECT_1:5
.= v2 by RLVECT_1:4 ;
hence v2 in v1 + W by A7; ::_thesis: verum
end;
theorem Th75: :: ZMODUL01:75
for V being Z_Module
for v, u being VECTOR of V
for W being Submodule of V st v + W = u + W holds
ex v1 being VECTOR of V st
( v1 in W & v + v1 = u )
proof
let V be Z_Module; ::_thesis: for v, u being VECTOR of V
for W being Submodule of V st v + W = u + W holds
ex v1 being VECTOR of V st
( v1 in W & v + v1 = u )
let v, u be VECTOR of V; ::_thesis: for W being Submodule of V st v + W = u + W holds
ex v1 being VECTOR of V st
( v1 in W & v + v1 = u )
let W be Submodule of V; ::_thesis: ( v + W = u + W implies ex v1 being VECTOR of V st
( v1 in W & v + v1 = u ) )
assume v + W = u + W ; ::_thesis: ex v1 being VECTOR of V st
( v1 in W & v + v1 = u )
then v in u + W by Th58;
then consider u1 being VECTOR of V such that
A1: v = u + u1 and
A2: u1 in W ;
take v1 = u - v; ::_thesis: ( v1 in W & v + v1 = u )
0. V = (u + u1) - v by A1, RLVECT_1:15
.= u1 + (u - v) by RLVECT_1:def_3 ;
then v1 = - u1 by RLVECT_1:def_10;
hence v1 in W by A2, Th38; ::_thesis: v + v1 = u
thus v + v1 = (u + v) - v by RLVECT_1:def_3
.= u + (v - v) by RLVECT_1:def_3
.= u + (0. V) by RLVECT_1:15
.= u by RLVECT_1:4 ; ::_thesis: verum
end;
theorem Th76: :: ZMODUL01:76
for V being Z_Module
for v, u being VECTOR of V
for W being Submodule of V st v + W = u + W holds
ex v1 being VECTOR of V st
( v1 in W & v - v1 = u )
proof
let V be Z_Module; ::_thesis: for v, u being VECTOR of V
for W being Submodule of V st v + W = u + W holds
ex v1 being VECTOR of V st
( v1 in W & v - v1 = u )
let v, u be VECTOR of V; ::_thesis: for W being Submodule of V st v + W = u + W holds
ex v1 being VECTOR of V st
( v1 in W & v - v1 = u )
let W be Submodule of V; ::_thesis: ( v + W = u + W implies ex v1 being VECTOR of V st
( v1 in W & v - v1 = u ) )
assume v + W = u + W ; ::_thesis: ex v1 being VECTOR of V st
( v1 in W & v - v1 = u )
then u in v + W by Th58;
then consider u1 being VECTOR of V such that
A1: u = v + u1 and
A2: u1 in W ;
take v1 = v - u; ::_thesis: ( v1 in W & v - v1 = u )
0. V = (v + u1) - u by A1, RLVECT_1:15
.= u1 + (v - u) by RLVECT_1:def_3 ;
then v1 = - u1 by RLVECT_1:def_10;
hence v1 in W by A2, Th38; ::_thesis: v - v1 = u
thus v - v1 = (v - v) + u by RLVECT_1:29
.= (0. V) + u by RLVECT_1:15
.= u by RLVECT_1:4 ; ::_thesis: verum
end;
theorem Th77: :: ZMODUL01:77
for V being Z_Module
for v being VECTOR of V
for W1, W2 being strict Submodule of V st v + W1 = v + W2 holds
W1 = W2
proof
let V be Z_Module; ::_thesis: for v being VECTOR of V
for W1, W2 being strict Submodule of V st v + W1 = v + W2 holds
W1 = W2
let v be VECTOR of V; ::_thesis: for W1, W2 being strict Submodule of V st v + W1 = v + W2 holds
W1 = W2
let W1, W2 be strict Submodule of V; ::_thesis: ( v + W1 = v + W2 implies W1 = W2 )
assume A1: v + W1 = v + W2 ; ::_thesis: W1 = W2
the carrier of W1 = the carrier of W2
proof
A2: the carrier of W1 c= the carrier of V by Def9;
thus the carrier of W1 c= the carrier of W2 :: according to XBOOLE_0:def_10 ::_thesis: the carrier of W2 c= the carrier of W1
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in the carrier of W1 or x in the carrier of W2 )
assume A3: x in the carrier of W1 ; ::_thesis: x in the carrier of W2
then reconsider y = x as Element of V by A2;
set z = v + y;
x in W1 by A3, STRUCT_0:def_5;
then v + y in v + W2 by A1;
then consider u being VECTOR of V such that
A4: v + y = v + u and
A5: u in W2 ;
y = u by A4, RLVECT_1:8;
hence x in the carrier of W2 by A5, STRUCT_0:def_5; ::_thesis: verum
end;
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in the carrier of W2 or x in the carrier of W1 )
assume A6: x in the carrier of W2 ; ::_thesis: x in the carrier of W1
the carrier of W2 c= the carrier of V by Def9;
then reconsider y = x as Element of V by A6;
set z = v + y;
x in W2 by A6, STRUCT_0:def_5;
then v + y in v + W1 by A1;
then consider u being VECTOR of V such that
A7: v + y = v + u and
A8: u in W1 ;
y = u by A7, RLVECT_1:8;
hence x in the carrier of W1 by A8, STRUCT_0:def_5; ::_thesis: verum
end;
hence W1 = W2 by Th45; ::_thesis: verum
end;
theorem Th78: :: ZMODUL01:78
for V being Z_Module
for v, u being VECTOR of V
for W1, W2 being strict Submodule of V st v + W1 = u + W2 holds
W1 = W2
proof
let V be Z_Module; ::_thesis: for v, u being VECTOR of V
for W1, W2 being strict Submodule of V st v + W1 = u + W2 holds
W1 = W2
let v, u be VECTOR of V; ::_thesis: for W1, W2 being strict Submodule of V st v + W1 = u + W2 holds
W1 = W2
let W1, W2 be strict Submodule of V; ::_thesis: ( v + W1 = u + W2 implies W1 = W2 )
assume A1: v + W1 = u + W2 ; ::_thesis: W1 = W2
set V2 = the carrier of W2;
set V1 = the carrier of W1;
assume A2: W1 <> W2 ; ::_thesis: contradiction
A3: now__::_thesis:_not_the_carrier_of_W1_\_the_carrier_of_W2_<>_{}
set x = the Element of the carrier of W1 \ the carrier of W2;
assume the carrier of W1 \ the carrier of W2 <> {} ; ::_thesis: contradiction
then the Element of the carrier of W1 \ the carrier of W2 in the carrier of W1 by XBOOLE_0:def_5;
then A4: the Element of the carrier of W1 \ the carrier of W2 in W1 by STRUCT_0:def_5;
then the Element of the carrier of W1 \ the carrier of W2 in V by Th24;
then reconsider x = the Element of the carrier of W1 \ the carrier of W2 as Element of V by STRUCT_0:def_5;
set z = v + x;
v + x in u + W2 by A1, A4;
then consider u1 being VECTOR of V such that
A5: v + x = u + u1 and
A6: u1 in W2 ;
x = (0. V) + x by RLVECT_1:4
.= (v - v) + x by RLVECT_1:15
.= (- v) + (u + u1) by A5, RLVECT_1:def_3 ;
then A7: (v + ((- v) + (u + u1))) + W1 = v + W1 by A4, Th65;
v + ((- v) + (u + u1)) = (v - v) + (u + u1) by RLVECT_1:def_3
.= (0. V) + (u + u1) by RLVECT_1:15
.= u + u1 by RLVECT_1:4 ;
then (u + u1) + W2 = (u + u1) + W1 by A1, A6, A7, Th65;
hence contradiction by A2, Th77; ::_thesis: verum
end;
A8: now__::_thesis:_not_the_carrier_of_W2_\_the_carrier_of_W1_<>_{}
set x = the Element of the carrier of W2 \ the carrier of W1;
assume the carrier of W2 \ the carrier of W1 <> {} ; ::_thesis: contradiction
then the Element of the carrier of W2 \ the carrier of W1 in the carrier of W2 by XBOOLE_0:def_5;
then A9: the Element of the carrier of W2 \ the carrier of W1 in W2 by STRUCT_0:def_5;
then the Element of the carrier of W2 \ the carrier of W1 in V by Th24;
then reconsider x = the Element of the carrier of W2 \ the carrier of W1 as Element of V by STRUCT_0:def_5;
set z = u + x;
u + x in v + W1 by A1, A9;
then consider u1 being VECTOR of V such that
A10: u + x = v + u1 and
A11: u1 in W1 ;
x = (0. V) + x by RLVECT_1:4
.= (u - u) + x by RLVECT_1:15
.= (- u) + (v + u1) by A10, RLVECT_1:def_3 ;
then A12: (u + ((- u) + (v + u1))) + W2 = u + W2 by A9, Th65;
u + ((- u) + (v + u1)) = (u - u) + (v + u1) by RLVECT_1:def_3
.= (0. V) + (v + u1) by RLVECT_1:15
.= v + u1 by RLVECT_1:4 ;
then (v + u1) + W1 = (v + u1) + W2 by A1, A11, A12, Th65;
hence contradiction by A2, Th77; ::_thesis: verum
end;
the carrier of W1 <> the carrier of W2 by A2, Th45;
then ( not the carrier of W1 c= the carrier of W2 or not the carrier of W2 c= the carrier of W1 ) by XBOOLE_0:def_10;
hence contradiction by A3, A8, XBOOLE_1:37; ::_thesis: verum
end;
theorem :: ZMODUL01:79
for V being Z_Module
for W being Submodule of V
for C being Coset of W holds
( C is linearly-closed iff C = the carrier of W )
proof
let V be Z_Module; ::_thesis: for W being Submodule of V
for C being Coset of W holds
( C is linearly-closed iff C = the carrier of W )
let W be Submodule of V; ::_thesis: for C being Coset of W holds
( C is linearly-closed iff C = the carrier of W )
let C be Coset of W; ::_thesis: ( C is linearly-closed iff C = the carrier of W )
thus ( C is linearly-closed implies C = the carrier of W ) ::_thesis: ( C = the carrier of W implies C is linearly-closed )
proof
assume A1: C is linearly-closed ; ::_thesis: C = the carrier of W
consider v being VECTOR of V such that
A2: C = v + W by Def13;
C <> {} by A2, Th58;
then 0. V in v + W by A1, A2, Th18;
hence C = the carrier of W by A2, Th62; ::_thesis: verum
end;
thus ( C = the carrier of W implies C is linearly-closed ) by Lm3; ::_thesis: verum
end;
theorem :: ZMODUL01:80
for V being Z_Module
for W1, W2 being strict Submodule of V
for C1 being Coset of W1
for C2 being Coset of W2 st C1 = C2 holds
W1 = W2
proof
let V be Z_Module; ::_thesis: for W1, W2 being strict Submodule of V
for C1 being Coset of W1
for C2 being Coset of W2 st C1 = C2 holds
W1 = W2
let W1, W2 be strict Submodule of V; ::_thesis: for C1 being Coset of W1
for C2 being Coset of W2 st C1 = C2 holds
W1 = W2
let C1 be Coset of W1; ::_thesis: for C2 being Coset of W2 st C1 = C2 holds
W1 = W2
let C2 be Coset of W2; ::_thesis: ( C1 = C2 implies W1 = W2 )
( ex v1 being VECTOR of V st C1 = v1 + W1 & ex v2 being VECTOR of V st C2 = v2 + W2 ) by Def13;
hence ( C1 = C2 implies W1 = W2 ) by Th78; ::_thesis: verum
end;
theorem :: ZMODUL01:81
for V being Z_Module
for v being VECTOR of V holds {v} is Coset of (0). V
proof
let V be Z_Module; ::_thesis: for v being VECTOR of V holds {v} is Coset of (0). V
let v be VECTOR of V; ::_thesis: {v} is Coset of (0). V
v + ((0). V) = {v} by Th60;
hence {v} is Coset of (0). V by Def13; ::_thesis: verum
end;
theorem Th82: :: ZMODUL01:82
for V being Z_Module
for V1 being Subset of V st V1 is Coset of (0). V holds
ex v being VECTOR of V st V1 = {v}
proof
let V be Z_Module; ::_thesis: for V1 being Subset of V st V1 is Coset of (0). V holds
ex v being VECTOR of V st V1 = {v}
let V1 be Subset of V; ::_thesis: ( V1 is Coset of (0). V implies ex v being VECTOR of V st V1 = {v} )
assume V1 is Coset of (0). V ; ::_thesis: ex v being VECTOR of V st V1 = {v}
then consider v being VECTOR of V such that
A1: V1 = v + ((0). V) by Def13;
take v ; ::_thesis: V1 = {v}
thus V1 = {v} by A1, Th60; ::_thesis: verum
end;
theorem Th83: :: ZMODUL01:83
for V being Z_Module
for W being Submodule of V holds the carrier of W is Coset of W
proof
let V be Z_Module; ::_thesis: for W being Submodule of V holds the carrier of W is Coset of W
let W be Submodule of V; ::_thesis: the carrier of W is Coset of W
the carrier of W = (0. V) + W by Lm4;
hence the carrier of W is Coset of W by Def13; ::_thesis: verum
end;
theorem :: ZMODUL01:84
for V being Z_Module holds the carrier of V is Coset of (Omega). V
proof
let V be Z_Module; ::_thesis: the carrier of V is Coset of (Omega). V
set v = the VECTOR of V;
( the carrier of V is Subset of V iff the carrier of V c= the carrier of V ) ;
then reconsider A = the carrier of V as Subset of V ;
A = the VECTOR of V + ((Omega). V) by Th61;
hence the carrier of V is Coset of (Omega). V by Def13; ::_thesis: verum
end;
theorem :: ZMODUL01:85
for V being Z_Module
for V1 being Subset of V st V1 is Coset of (Omega). V holds
V1 = the carrier of V
proof
let V be Z_Module; ::_thesis: for V1 being Subset of V st V1 is Coset of (Omega). V holds
V1 = the carrier of V
let V1 be Subset of V; ::_thesis: ( V1 is Coset of (Omega). V implies V1 = the carrier of V )
assume V1 is Coset of (Omega). V ; ::_thesis: V1 = the carrier of V
then ex v being VECTOR of V st V1 = v + ((Omega). V) by Def13;
hence V1 = the carrier of V by Th61; ::_thesis: verum
end;
theorem :: ZMODUL01:86
for V being Z_Module
for W being Submodule of V
for C being Coset of W holds
( 0. V in C iff C = the carrier of W )
proof
let V be Z_Module; ::_thesis: for W being Submodule of V
for C being Coset of W holds
( 0. V in C iff C = the carrier of W )
let W be Submodule of V; ::_thesis: for C being Coset of W holds
( 0. V in C iff C = the carrier of W )
let C be Coset of W; ::_thesis: ( 0. V in C iff C = the carrier of W )
ex v being VECTOR of V st C = v + W by Def13;
hence ( 0. V in C iff C = the carrier of W ) by Th62; ::_thesis: verum
end;
theorem Th87: :: ZMODUL01:87
for V being Z_Module
for u being VECTOR of V
for W being Submodule of V
for C being Coset of W holds
( u in C iff C = u + W )
proof
let V be Z_Module; ::_thesis: for u being VECTOR of V
for W being Submodule of V
for C being Coset of W holds
( u in C iff C = u + W )
let u be VECTOR of V; ::_thesis: for W being Submodule of V
for C being Coset of W holds
( u in C iff C = u + W )
let W be Submodule of V; ::_thesis: for C being Coset of W holds
( u in C iff C = u + W )
let C be Coset of W; ::_thesis: ( u in C iff C = u + W )
thus ( u in C implies C = u + W ) ::_thesis: ( C = u + W implies u in C )
proof
assume A1: u in C ; ::_thesis: C = u + W
ex v being VECTOR of V st C = v + W by Def13;
hence C = u + W by A1, Th67; ::_thesis: verum
end;
thus ( C = u + W implies u in C ) by Th58; ::_thesis: verum
end;
theorem :: ZMODUL01:88
for V being Z_Module
for u, v being VECTOR of V
for W being Submodule of V
for C being Coset of W st u in C & v in C holds
ex v1 being VECTOR of V st
( v1 in W & u + v1 = v )
proof
let V be Z_Module; ::_thesis: for u, v being VECTOR of V
for W being Submodule of V
for C being Coset of W st u in C & v in C holds
ex v1 being VECTOR of V st
( v1 in W & u + v1 = v )
let u, v be VECTOR of V; ::_thesis: for W being Submodule of V
for C being Coset of W st u in C & v in C holds
ex v1 being VECTOR of V st
( v1 in W & u + v1 = v )
let W be Submodule of V; ::_thesis: for C being Coset of W st u in C & v in C holds
ex v1 being VECTOR of V st
( v1 in W & u + v1 = v )
let C be Coset of W; ::_thesis: ( u in C & v in C implies ex v1 being VECTOR of V st
( v1 in W & u + v1 = v ) )
assume ( u in C & v in C ) ; ::_thesis: ex v1 being VECTOR of V st
( v1 in W & u + v1 = v )
then ( C = u + W & C = v + W ) by Th87;
hence ex v1 being VECTOR of V st
( v1 in W & u + v1 = v ) by Th75; ::_thesis: verum
end;
theorem Th89: :: ZMODUL01:89
for V being Z_Module
for u, v being VECTOR of V
for W being Submodule of V
for C being Coset of W st u in C & v in C holds
ex v1 being VECTOR of V st
( v1 in W & u - v1 = v )
proof
let V be Z_Module; ::_thesis: for u, v being VECTOR of V
for W being Submodule of V
for C being Coset of W st u in C & v in C holds
ex v1 being VECTOR of V st
( v1 in W & u - v1 = v )
let u, v be VECTOR of V; ::_thesis: for W being Submodule of V
for C being Coset of W st u in C & v in C holds
ex v1 being VECTOR of V st
( v1 in W & u - v1 = v )
let W be Submodule of V; ::_thesis: for C being Coset of W st u in C & v in C holds
ex v1 being VECTOR of V st
( v1 in W & u - v1 = v )
let C be Coset of W; ::_thesis: ( u in C & v in C implies ex v1 being VECTOR of V st
( v1 in W & u - v1 = v ) )
assume ( u in C & v in C ) ; ::_thesis: ex v1 being VECTOR of V st
( v1 in W & u - v1 = v )
then ( C = u + W & C = v + W ) by Th87;
hence ex v1 being VECTOR of V st
( v1 in W & u - v1 = v ) by Th76; ::_thesis: verum
end;
theorem :: ZMODUL01:90
for V being Z_Module
for v1, v2 being VECTOR of V
for W being Submodule of V holds
( ex C being Coset of W st
( v1 in C & v2 in C ) iff v1 - v2 in W )
proof
let V be Z_Module; ::_thesis: for v1, v2 being VECTOR of V
for W being Submodule of V holds
( ex C being Coset of W st
( v1 in C & v2 in C ) iff v1 - v2 in W )
let v1, v2 be VECTOR of V; ::_thesis: for W being Submodule of V holds
( ex C being Coset of W st
( v1 in C & v2 in C ) iff v1 - v2 in W )
let W be Submodule of V; ::_thesis: ( ex C being Coset of W st
( v1 in C & v2 in C ) iff v1 - v2 in W )
thus ( ex C being Coset of W st
( v1 in C & v2 in C ) implies v1 - v2 in W ) ::_thesis: ( v1 - v2 in W implies ex C being Coset of W st
( v1 in C & v2 in C ) )
proof
given C being Coset of W such that A1: ( v1 in C & v2 in C ) ; ::_thesis: v1 - v2 in W
ex v being VECTOR of V st C = v + W by Def13;
hence v1 - v2 in W by A1, Th74; ::_thesis: verum
end;
assume v1 - v2 in W ; ::_thesis: ex C being Coset of W st
( v1 in C & v2 in C )
then consider v being VECTOR of V such that
A2: ( v1 in v + W & v2 in v + W ) by Th74;
reconsider C = v + W as Coset of W by Def13;
take C ; ::_thesis: ( v1 in C & v2 in C )
thus ( v1 in C & v2 in C ) by A2; ::_thesis: verum
end;
theorem :: ZMODUL01:91
for V being Z_Module
for u being VECTOR of V
for W being Submodule of V
for B, C being Coset of W st u in B & u in C holds
B = C
proof
let V be Z_Module; ::_thesis: for u being VECTOR of V
for W being Submodule of V
for B, C being Coset of W st u in B & u in C holds
B = C
let u be VECTOR of V; ::_thesis: for W being Submodule of V
for B, C being Coset of W st u in B & u in C holds
B = C
let W be Submodule of V; ::_thesis: for B, C being Coset of W st u in B & u in C holds
B = C
let B, C be Coset of W; ::_thesis: ( u in B & u in C implies B = C )
assume A1: ( u in B & u in C ) ; ::_thesis: B = C
( ex v1 being VECTOR of V st B = v1 + W & ex v2 being VECTOR of V st C = v2 + W ) by Def13;
hence B = C by A1, Th68; ::_thesis: verum
end;
begin
definition
let V be Z_Module;
let W1, W2 be Submodule of V;
funcW1 + W2 -> strict Submodule of V means :Def14: :: ZMODUL01:def 14
the carrier of it = { (v + u) where v, u is VECTOR of V : ( v in W1 & u in W2 ) } ;
existence
ex b1 being strict Submodule of V st the carrier of b1 = { (v + u) where v, u is VECTOR of V : ( v in W1 & u in W2 ) }
proof
reconsider V1 = the carrier of W1, V2 = the carrier of W2 as Subset of V by Def9;
set VS = { (v + u) where v, u is VECTOR of V : ( v in W1 & u in W2 ) } ;
{ (v + u) where v, u is VECTOR of V : ( v in W1 & u in W2 ) } c= the carrier of V
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in { (v + u) where v, u is VECTOR of V : ( v in W1 & u in W2 ) } or x in the carrier of V )
assume x in { (v + u) where v, u is VECTOR of V : ( v in W1 & u in W2 ) } ; ::_thesis: x in the carrier of V
then ex v1, v2 being VECTOR of V st
( x = v1 + v2 & v1 in W1 & v2 in W2 ) ;
hence x in the carrier of V ; ::_thesis: verum
end;
then reconsider VS = { (v + u) where v, u is VECTOR of V : ( v in W1 & u in W2 ) } as Subset of V ;
A1: 0. V = (0. V) + (0. V) by RLVECT_1:4;
( 0. V in W1 & 0. V in W2 ) by Th33;
then A2: 0. V in VS by A1;
A3: VS = { (v + u) where v, u is VECTOR of V : ( v in V1 & u in V2 ) }
proof
thus VS c= { (v + u) where v, u is VECTOR of V : ( v in V1 & u in V2 ) } :: according to XBOOLE_0:def_10 ::_thesis: { (v + u) where v, u is VECTOR of V : ( v in V1 & u in V2 ) } c= VS
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in VS or x in { (v + u) where v, u is VECTOR of V : ( v in V1 & u in V2 ) } )
assume x in VS ; ::_thesis: x in { (v + u) where v, u is VECTOR of V : ( v in V1 & u in V2 ) }
then consider v, u being VECTOR of V such that
A4: x = v + u and
A5: ( v in W1 & u in W2 ) ;
( v in V1 & u in V2 ) by A5, STRUCT_0:def_5;
hence x in { (v + u) where v, u is VECTOR of V : ( v in V1 & u in V2 ) } by A4; ::_thesis: verum
end;
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in { (v + u) where v, u is VECTOR of V : ( v in V1 & u in V2 ) } or x in VS )
assume x in { (v + u) where v, u is VECTOR of V : ( v in V1 & u in V2 ) } ; ::_thesis: x in VS
then consider v, u being VECTOR of V such that
A6: x = v + u and
A7: ( v in V1 & u in V2 ) ;
( v in W1 & u in W2 ) by A7, STRUCT_0:def_5;
hence x in VS by A6; ::_thesis: verum
end;
( V1 is linearly-closed & V2 is linearly-closed ) by Lm3;
hence ex b1 being strict Submodule of V st the carrier of b1 = { (v + u) where v, u is VECTOR of V : ( v in W1 & u in W2 ) } by A2, A3, Th22, Th50; ::_thesis: verum
end;
uniqueness
for b1, b2 being strict Submodule of V st the carrier of b1 = { (v + u) where v, u is VECTOR of V : ( v in W1 & u in W2 ) } & the carrier of b2 = { (v + u) where v, u is VECTOR of V : ( v in W1 & u in W2 ) } holds
b1 = b2 by Th45;
commutativity
for b1 being strict Submodule of V
for W1, W2 being Submodule of V st the carrier of b1 = { (v + u) where v, u is VECTOR of V : ( v in W1 & u in W2 ) } holds
the carrier of b1 = { (v + u) where v, u is VECTOR of V : ( v in W2 & u in W1 ) }
proof
let W be strict Submodule of V; ::_thesis: for W1, W2 being Submodule of V st the carrier of W = { (v + u) where v, u is VECTOR of V : ( v in W1 & u in W2 ) } holds
the carrier of W = { (v + u) where v, u is VECTOR of V : ( v in W2 & u in W1 ) }
let W1, W2 be Submodule of V; ::_thesis: ( the carrier of W = { (v + u) where v, u is VECTOR of V : ( v in W1 & u in W2 ) } implies the carrier of W = { (v + u) where v, u is VECTOR of V : ( v in W2 & u in W1 ) } )
set A = { (v + u) where v, u is VECTOR of V : ( v in W1 & u in W2 ) } ;
set B = { (v + u) where v, u is VECTOR of V : ( v in W2 & u in W1 ) } ;
A8: { (v + u) where v, u is VECTOR of V : ( v in W2 & u in W1 ) } c= { (v + u) where v, u is VECTOR of V : ( v in W1 & u in W2 ) }
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in { (v + u) where v, u is VECTOR of V : ( v in W2 & u in W1 ) } or x in { (v + u) where v, u is VECTOR of V : ( v in W1 & u in W2 ) } )
assume x in { (v + u) where v, u is VECTOR of V : ( v in W2 & u in W1 ) } ; ::_thesis: x in { (v + u) where v, u is VECTOR of V : ( v in W1 & u in W2 ) }
then ex v, u being VECTOR of V st
( x = v + u & v in W2 & u in W1 ) ;
hence x in { (v + u) where v, u is VECTOR of V : ( v in W1 & u in W2 ) } ; ::_thesis: verum
end;
{ (v + u) where v, u is VECTOR of V : ( v in W1 & u in W2 ) } c= { (v + u) where v, u is VECTOR of V : ( v in W2 & u in W1 ) }
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in { (v + u) where v, u is VECTOR of V : ( v in W1 & u in W2 ) } or x in { (v + u) where v, u is VECTOR of V : ( v in W2 & u in W1 ) } )
assume x in { (v + u) where v, u is VECTOR of V : ( v in W1 & u in W2 ) } ; ::_thesis: x in { (v + u) where v, u is VECTOR of V : ( v in W2 & u in W1 ) }
then ex v, u being VECTOR of V st
( x = v + u & v in W1 & u in W2 ) ;
hence x in { (v + u) where v, u is VECTOR of V : ( v in W2 & u in W1 ) } ; ::_thesis: verum
end;
hence ( the carrier of W = { (v + u) where v, u is VECTOR of V : ( v in W1 & u in W2 ) } implies the carrier of W = { (v + u) where v, u is VECTOR of V : ( v in W2 & u in W1 ) } ) by A8, XBOOLE_0:def_10; ::_thesis: verum
end;
end;
:: deftheorem Def14 defines + ZMODUL01:def_14_:_
for V being Z_Module
for W1, W2 being Submodule of V
for b4 being strict Submodule of V holds
( b4 = W1 + W2 iff the carrier of b4 = { (v + u) where v, u is VECTOR of V : ( v in W1 & u in W2 ) } );
definition
let V be Z_Module;
let W1, W2 be Submodule of V;
funcW1 /\ W2 -> strict Submodule of V means :Def15: :: ZMODUL01:def 15
the carrier of it = the carrier of W1 /\ the carrier of W2;
existence
ex b1 being strict Submodule of V st the carrier of b1 = the carrier of W1 /\ the carrier of W2
proof
set VW2 = the carrier of W2;
set VW1 = the carrier of W1;
set VV = the carrier of V;
0. V in W2 by Th33;
then A1: 0. V in the carrier of W2 by STRUCT_0:def_5;
( the carrier of W1 c= the carrier of V & the carrier of W2 c= the carrier of V ) by Def9;
then the carrier of W1 /\ the carrier of W2 c= the carrier of V /\ the carrier of V by XBOOLE_1:27;
then reconsider V1 = the carrier of W1, V2 = the carrier of W2, V3 = the carrier of W1 /\ the carrier of W2 as Subset of V by Def9;
( V1 is linearly-closed & V2 is linearly-closed ) by Lm3;
then A2: V3 is linearly-closed ;
0. V in W1 by Th33;
then 0. V in the carrier of W1 by STRUCT_0:def_5;
then the carrier of W1 /\ the carrier of W2 <> {} by A1, XBOOLE_0:def_4;
hence ex b1 being strict Submodule of V st the carrier of b1 = the carrier of W1 /\ the carrier of W2 by A2, Th50; ::_thesis: verum
end;
uniqueness
for b1, b2 being strict Submodule of V st the carrier of b1 = the carrier of W1 /\ the carrier of W2 & the carrier of b2 = the carrier of W1 /\ the carrier of W2 holds
b1 = b2 by Th45;
commutativity
for b1 being strict Submodule of V
for W1, W2 being Submodule of V st the carrier of b1 = the carrier of W1 /\ the carrier of W2 holds
the carrier of b1 = the carrier of W2 /\ the carrier of W1 ;
end;
:: deftheorem Def15 defines /\ ZMODUL01:def_15_:_
for V being Z_Module
for W1, W2 being Submodule of V
for b4 being strict Submodule of V holds
( b4 = W1 /\ W2 iff the carrier of b4 = the carrier of W1 /\ the carrier of W2 );
theorem Th92: :: ZMODUL01:92
for x being set
for V being Z_Module
for W1, W2 being Submodule of V holds
( x in W1 + W2 iff ex v1, v2 being VECTOR of V st
( v1 in W1 & v2 in W2 & x = v1 + v2 ) )
proof
let x be set ; ::_thesis: for V being Z_Module
for W1, W2 being Submodule of V holds
( x in W1 + W2 iff ex v1, v2 being VECTOR of V st
( v1 in W1 & v2 in W2 & x = v1 + v2 ) )
let V be Z_Module; ::_thesis: for W1, W2 being Submodule of V holds
( x in W1 + W2 iff ex v1, v2 being VECTOR of V st
( v1 in W1 & v2 in W2 & x = v1 + v2 ) )
let W1, W2 be Submodule of V; ::_thesis: ( x in W1 + W2 iff ex v1, v2 being VECTOR of V st
( v1 in W1 & v2 in W2 & x = v1 + v2 ) )
thus ( x in W1 + W2 implies ex v1, v2 being VECTOR of V st
( v1 in W1 & v2 in W2 & x = v1 + v2 ) ) ::_thesis: ( ex v1, v2 being VECTOR of V st
( v1 in W1 & v2 in W2 & x = v1 + v2 ) implies x in W1 + W2 )
proof
assume x in W1 + W2 ; ::_thesis: ex v1, v2 being VECTOR of V st
( v1 in W1 & v2 in W2 & x = v1 + v2 )
then x in the carrier of (W1 + W2) by STRUCT_0:def_5;
then x in { (v + u) where v, u is VECTOR of V : ( v in W1 & u in W2 ) } by Def14;
then consider v1, v2 being VECTOR of V such that
A1: ( x = v1 + v2 & v1 in W1 & v2 in W2 ) ;
take v1 ; ::_thesis: ex v2 being VECTOR of V st
( v1 in W1 & v2 in W2 & x = v1 + v2 )
take v2 ; ::_thesis: ( v1 in W1 & v2 in W2 & x = v1 + v2 )
thus ( v1 in W1 & v2 in W2 & x = v1 + v2 ) by A1; ::_thesis: verum
end;
given v1, v2 being VECTOR of V such that A2: ( v1 in W1 & v2 in W2 & x = v1 + v2 ) ; ::_thesis: x in W1 + W2
x in { (v + u) where v, u is VECTOR of V : ( v in W1 & u in W2 ) } by A2;
then x in the carrier of (W1 + W2) by Def14;
hence x in W1 + W2 by STRUCT_0:def_5; ::_thesis: verum
end;
theorem :: ZMODUL01:93
for V being Z_Module
for W1, W2 being Submodule of V
for v being VECTOR of V st ( v in W1 or v in W2 ) holds
v in W1 + W2
proof
let V be Z_Module; ::_thesis: for W1, W2 being Submodule of V
for v being VECTOR of V st ( v in W1 or v in W2 ) holds
v in W1 + W2
let W1, W2 be Submodule of V; ::_thesis: for v being VECTOR of V st ( v in W1 or v in W2 ) holds
v in W1 + W2
let v be VECTOR of V; ::_thesis: ( ( v in W1 or v in W2 ) implies v in W1 + W2 )
assume A1: ( v in W1 or v in W2 ) ; ::_thesis: v in W1 + W2
now__::_thesis:_v_in_W1_+_W2
percases ( v in W1 or v in W2 ) by A1;
supposeA2: v in W1 ; ::_thesis: v in W1 + W2
( v = v + (0. V) & 0. V in W2 ) by Th33, RLVECT_1:4;
hence v in W1 + W2 by A2, Th92; ::_thesis: verum
end;
supposeA3: v in W2 ; ::_thesis: v in W1 + W2
( v = (0. V) + v & 0. V in W1 ) by Th33, RLVECT_1:4;
hence v in W1 + W2 by A3, Th92; ::_thesis: verum
end;
end;
end;
hence v in W1 + W2 ; ::_thesis: verum
end;
theorem Th94: :: ZMODUL01:94
for x being set
for V being Z_Module
for W1, W2 being Submodule of V holds
( x in W1 /\ W2 iff ( x in W1 & x in W2 ) )
proof
let x be set ; ::_thesis: for V being Z_Module
for W1, W2 being Submodule of V holds
( x in W1 /\ W2 iff ( x in W1 & x in W2 ) )
let V be Z_Module; ::_thesis: for W1, W2 being Submodule of V holds
( x in W1 /\ W2 iff ( x in W1 & x in W2 ) )
let W1, W2 be Submodule of V; ::_thesis: ( x in W1 /\ W2 iff ( x in W1 & x in W2 ) )
( x in W1 /\ W2 iff x in the carrier of (W1 /\ W2) ) by STRUCT_0:def_5;
then ( x in W1 /\ W2 iff x in the carrier of W1 /\ the carrier of W2 ) by Def15;
then ( x in W1 /\ W2 iff ( x in the carrier of W1 & x in the carrier of W2 ) ) by XBOOLE_0:def_4;
hence ( x in W1 /\ W2 iff ( x in W1 & x in W2 ) ) by STRUCT_0:def_5; ::_thesis: verum
end;
Lm6: for V being Z_Module
for W1, W2 being Submodule of V holds the carrier of W1 c= the carrier of (W1 + W2)
proof
let V be Z_Module; ::_thesis: for W1, W2 being Submodule of V holds the carrier of W1 c= the carrier of (W1 + W2)
let W1, W2 be Submodule of V; ::_thesis: the carrier of W1 c= the carrier of (W1 + W2)
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in the carrier of W1 or x in the carrier of (W1 + W2) )
set A = { (v + u) where v, u is VECTOR of V : ( v in W1 & u in W2 ) } ;
assume x in the carrier of W1 ; ::_thesis: x in the carrier of (W1 + W2)
then reconsider v = x as Element of W1 ;
reconsider v = v as VECTOR of V by Th25;
A1: v = v + (0. V) by RLVECT_1:4;
( v in W1 & 0. V in W2 ) by Th33, STRUCT_0:def_5;
then x in { (v + u) where v, u is VECTOR of V : ( v in W1 & u in W2 ) } by A1;
hence x in the carrier of (W1 + W2) by Def14; ::_thesis: verum
end;
Lm7: for V being Z_Module
for W1 being Submodule of V
for W2 being strict Submodule of V st the carrier of W1 c= the carrier of W2 holds
W1 + W2 = W2
proof
let V be Z_Module; ::_thesis: for W1 being Submodule of V
for W2 being strict Submodule of V st the carrier of W1 c= the carrier of W2 holds
W1 + W2 = W2
let W1 be Submodule of V; ::_thesis: for W2 being strict Submodule of V st the carrier of W1 c= the carrier of W2 holds
W1 + W2 = W2
let W2 be strict Submodule of V; ::_thesis: ( the carrier of W1 c= the carrier of W2 implies W1 + W2 = W2 )
assume A1: the carrier of W1 c= the carrier of W2 ; ::_thesis: W1 + W2 = W2
A2: the carrier of (W1 + W2) c= the carrier of W2
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in the carrier of (W1 + W2) or x in the carrier of W2 )
assume x in the carrier of (W1 + W2) ; ::_thesis: x in the carrier of W2
then x in { (v + u) where v, u is VECTOR of V : ( v in W1 & u in W2 ) } by Def14;
then consider v, u being VECTOR of V such that
A3: x = v + u and
A4: v in W1 and
A5: u in W2 ;
W1 is Submodule of W2 by A1, Th43;
then v in W2 by A4, Th23;
then v + u in W2 by A5, Th36;
hence x in the carrier of W2 by A3, STRUCT_0:def_5; ::_thesis: verum
end;
the carrier of W2 c= the carrier of (W1 + W2) by Lm6;
then the carrier of (W1 + W2) = the carrier of W2 by A2, XBOOLE_0:def_10;
hence W1 + W2 = W2 by Th45; ::_thesis: verum
end;
theorem :: ZMODUL01:95
for V being Z_Module
for W being strict Submodule of V holds W + W = W by Lm7;
theorem Th96: :: ZMODUL01:96
for V being Z_Module
for W1, W2, W3 being Submodule of V holds W1 + (W2 + W3) = (W1 + W2) + W3
proof
let V be Z_Module; ::_thesis: for W1, W2, W3 being Submodule of V holds W1 + (W2 + W3) = (W1 + W2) + W3
let W1, W2, W3 be Submodule of V; ::_thesis: W1 + (W2 + W3) = (W1 + W2) + W3
set A = { (v + u) where v, u is VECTOR of V : ( v in W1 & u in W2 ) } ;
set B = { (v + u) where v, u is VECTOR of V : ( v in W2 & u in W3 ) } ;
set C = { (v + u) where v, u is VECTOR of V : ( v in W1 + W2 & u in W3 ) } ;
set D = { (v + u) where v, u is VECTOR of V : ( v in W1 & u in W2 + W3 ) } ;
A1: the carrier of (W1 + (W2 + W3)) = { (v + u) where v, u is VECTOR of V : ( v in W1 & u in W2 + W3 ) } by Def14;
A2: { (v + u) where v, u is VECTOR of V : ( v in W1 + W2 & u in W3 ) } c= { (v + u) where v, u is VECTOR of V : ( v in W1 & u in W2 + W3 ) }
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in { (v + u) where v, u is VECTOR of V : ( v in W1 + W2 & u in W3 ) } or x in { (v + u) where v, u is VECTOR of V : ( v in W1 & u in W2 + W3 ) } )
assume x in { (v + u) where v, u is VECTOR of V : ( v in W1 + W2 & u in W3 ) } ; ::_thesis: x in { (v + u) where v, u is VECTOR of V : ( v in W1 & u in W2 + W3 ) }
then consider v, u being VECTOR of V such that
A3: x = v + u and
A4: v in W1 + W2 and
A5: u in W3 ;
v in the carrier of (W1 + W2) by A4, STRUCT_0:def_5;
then v in { (v + u) where v, u is VECTOR of V : ( v in W1 & u in W2 ) } by Def14;
then consider u1, u2 being VECTOR of V such that
A6: v = u1 + u2 and
A7: u1 in W1 and
A8: u2 in W2 ;
u2 + u in { (v + u) where v, u is VECTOR of V : ( v in W2 & u in W3 ) } by A5, A8;
then u2 + u in the carrier of (W2 + W3) by Def14;
then A9: u2 + u in W2 + W3 by STRUCT_0:def_5;
v + u = u1 + (u2 + u) by A6, RLVECT_1:def_3;
hence x in { (v + u) where v, u is VECTOR of V : ( v in W1 & u in W2 + W3 ) } by A3, A7, A9; ::_thesis: verum
end;
{ (v + u) where v, u is VECTOR of V : ( v in W1 & u in W2 + W3 ) } c= { (v + u) where v, u is VECTOR of V : ( v in W1 + W2 & u in W3 ) }
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in { (v + u) where v, u is VECTOR of V : ( v in W1 & u in W2 + W3 ) } or x in { (v + u) where v, u is VECTOR of V : ( v in W1 + W2 & u in W3 ) } )
assume x in { (v + u) where v, u is VECTOR of V : ( v in W1 & u in W2 + W3 ) } ; ::_thesis: x in { (v + u) where v, u is VECTOR of V : ( v in W1 + W2 & u in W3 ) }
then consider v, u being VECTOR of V such that
A10: x = v + u and
A11: v in W1 and
A12: u in W2 + W3 ;
u in the carrier of (W2 + W3) by A12, STRUCT_0:def_5;
then u in { (v + u) where v, u is VECTOR of V : ( v in W2 & u in W3 ) } by Def14;
then consider u1, u2 being VECTOR of V such that
A13: u = u1 + u2 and
A14: u1 in W2 and
A15: u2 in W3 ;
v + u1 in { (v + u) where v, u is VECTOR of V : ( v in W1 & u in W2 ) } by A11, A14;
then v + u1 in the carrier of (W1 + W2) by Def14;
then A16: v + u1 in W1 + W2 by STRUCT_0:def_5;
v + u = (v + u1) + u2 by A13, RLVECT_1:def_3;
hence x in { (v + u) where v, u is VECTOR of V : ( v in W1 + W2 & u in W3 ) } by A10, A15, A16; ::_thesis: verum
end;
then { (v + u) where v, u is VECTOR of V : ( v in W1 & u in W2 + W3 ) } = { (v + u) where v, u is VECTOR of V : ( v in W1 + W2 & u in W3 ) } by A2, XBOOLE_0:def_10;
hence W1 + (W2 + W3) = (W1 + W2) + W3 by A1, Def14; ::_thesis: verum
end;
theorem Th97: :: ZMODUL01:97
for V being Z_Module
for W1, W2 being Submodule of V holds W1 is Submodule of W1 + W2
proof
let V be Z_Module; ::_thesis: for W1, W2 being Submodule of V holds W1 is Submodule of W1 + W2
let W1, W2 be Submodule of V; ::_thesis: W1 is Submodule of W1 + W2
the carrier of W1 c= the carrier of (W1 + W2) by Lm6;
hence W1 is Submodule of W1 + W2 by Th43; ::_thesis: verum
end;
theorem Th98: :: ZMODUL01:98
for V being Z_Module
for W1 being Submodule of V
for W2 being strict Submodule of V holds
( W1 is Submodule of W2 iff W1 + W2 = W2 )
proof
let V be Z_Module; ::_thesis: for W1 being Submodule of V
for W2 being strict Submodule of V holds
( W1 is Submodule of W2 iff W1 + W2 = W2 )
let W1 be Submodule of V; ::_thesis: for W2 being strict Submodule of V holds
( W1 is Submodule of W2 iff W1 + W2 = W2 )
let W2 be strict Submodule of V; ::_thesis: ( W1 is Submodule of W2 iff W1 + W2 = W2 )
thus ( W1 is Submodule of W2 implies W1 + W2 = W2 ) ::_thesis: ( W1 + W2 = W2 implies W1 is Submodule of W2 )
proof
assume W1 is Submodule of W2 ; ::_thesis: W1 + W2 = W2
then the carrier of W1 c= the carrier of W2 by Def9;
hence W1 + W2 = W2 by Lm7; ::_thesis: verum
end;
thus ( W1 + W2 = W2 implies W1 is Submodule of W2 ) by Th97; ::_thesis: verum
end;
theorem Th99: :: ZMODUL01:99
for V being Z_Module
for W being strict Submodule of V holds ((0). V) + W = W
proof
let V be Z_Module; ::_thesis: for W being strict Submodule of V holds ((0). V) + W = W
let W be strict Submodule of V; ::_thesis: ((0). V) + W = W
(0). V is Submodule of W by Th54;
then the carrier of ((0). V) c= the carrier of W by Def9;
hence ((0). V) + W = W by Lm7; ::_thesis: verum
end;
theorem :: ZMODUL01:100
for V being Z_Module holds ((0). V) + ((Omega). V) = Z_ModuleStruct(# the carrier of V, the ZeroF of V, the addF of V, the Mult of V #) by Th99;
theorem Th101: :: ZMODUL01:101
for V being Z_Module
for W being Submodule of V holds ((Omega). V) + W = Z_ModuleStruct(# the carrier of V, the ZeroF of V, the addF of V, the Mult of V #)
proof
let V be Z_Module; ::_thesis: for W being Submodule of V holds ((Omega). V) + W = Z_ModuleStruct(# the carrier of V, the ZeroF of V, the addF of V, the Mult of V #)
let W be Submodule of V; ::_thesis: ((Omega). V) + W = Z_ModuleStruct(# the carrier of V, the ZeroF of V, the addF of V, the Mult of V #)
the carrier of W c= the carrier of V by Def9;
hence ((Omega). V) + W = Z_ModuleStruct(# the carrier of V, the ZeroF of V, the addF of V, the Mult of V #) by Lm7; ::_thesis: verum
end;
theorem :: ZMODUL01:102
for V being strict Z_Module holds ((Omega). V) + ((Omega). V) = V by Th101;
theorem :: ZMODUL01:103
for V being Z_Module
for W being strict Submodule of V holds W /\ W = W
proof
let V be Z_Module; ::_thesis: for W being strict Submodule of V holds W /\ W = W
let W be strict Submodule of V; ::_thesis: W /\ W = W
the carrier of W = the carrier of W /\ the carrier of W ;
hence W /\ W = W by Def15; ::_thesis: verum
end;
theorem Th104: :: ZMODUL01:104
for V being Z_Module
for W1, W2, W3 being Submodule of V holds W1 /\ (W2 /\ W3) = (W1 /\ W2) /\ W3
proof
let V be Z_Module; ::_thesis: for W1, W2, W3 being Submodule of V holds W1 /\ (W2 /\ W3) = (W1 /\ W2) /\ W3
let W1, W2, W3 be Submodule of V; ::_thesis: W1 /\ (W2 /\ W3) = (W1 /\ W2) /\ W3
set V1 = the carrier of W1;
set V2 = the carrier of W2;
set V3 = the carrier of W3;
the carrier of (W1 /\ (W2 /\ W3)) = the carrier of W1 /\ the carrier of (W2 /\ W3) by Def15
.= the carrier of W1 /\ ( the carrier of W2 /\ the carrier of W3) by Def15
.= ( the carrier of W1 /\ the carrier of W2) /\ the carrier of W3 by XBOOLE_1:16
.= the carrier of (W1 /\ W2) /\ the carrier of W3 by Def15 ;
hence W1 /\ (W2 /\ W3) = (W1 /\ W2) /\ W3 by Def15; ::_thesis: verum
end;
Lm8: for V being Z_Module
for W1, W2 being Submodule of V holds the carrier of (W1 /\ W2) c= the carrier of W1
proof
let V be Z_Module; ::_thesis: for W1, W2 being Submodule of V holds the carrier of (W1 /\ W2) c= the carrier of W1
let W1, W2 be Submodule of V; ::_thesis: the carrier of (W1 /\ W2) c= the carrier of W1
the carrier of (W1 /\ W2) = the carrier of W1 /\ the carrier of W2 by Def15;
hence the carrier of (W1 /\ W2) c= the carrier of W1 by XBOOLE_1:17; ::_thesis: verum
end;
theorem Th105: :: ZMODUL01:105
for V being Z_Module
for W1, W2 being Submodule of V holds W1 /\ W2 is Submodule of W1
proof
let V be Z_Module; ::_thesis: for W1, W2 being Submodule of V holds W1 /\ W2 is Submodule of W1
let W1, W2 be Submodule of V; ::_thesis: W1 /\ W2 is Submodule of W1
the carrier of (W1 /\ W2) c= the carrier of W1 by Lm8;
hence W1 /\ W2 is Submodule of W1 by Th43; ::_thesis: verum
end;
theorem Th106: :: ZMODUL01:106
for V being Z_Module
for W2 being Submodule of V
for W1 being strict Submodule of V holds
( W1 is Submodule of W2 iff W1 /\ W2 = W1 )
proof
let V be Z_Module; ::_thesis: for W2 being Submodule of V
for W1 being strict Submodule of V holds
( W1 is Submodule of W2 iff W1 /\ W2 = W1 )
let W2 be Submodule of V; ::_thesis: for W1 being strict Submodule of V holds
( W1 is Submodule of W2 iff W1 /\ W2 = W1 )
let W1 be strict Submodule of V; ::_thesis: ( W1 is Submodule of W2 iff W1 /\ W2 = W1 )
thus ( W1 is Submodule of W2 implies W1 /\ W2 = W1 ) ::_thesis: ( W1 /\ W2 = W1 implies W1 is Submodule of W2 )
proof
assume W1 is Submodule of W2 ; ::_thesis: W1 /\ W2 = W1
then A1: the carrier of W1 c= the carrier of W2 by Def9;
the carrier of (W1 /\ W2) = the carrier of W1 /\ the carrier of W2 by Def15;
hence W1 /\ W2 = W1 by A1, Th45, XBOOLE_1:28; ::_thesis: verum
end;
thus ( W1 /\ W2 = W1 implies W1 is Submodule of W2 ) by Th105; ::_thesis: verum
end;
theorem Th107: :: ZMODUL01:107
for V being Z_Module
for W being Submodule of V holds ((0). V) /\ W = (0). V
proof
let V be Z_Module; ::_thesis: for W being Submodule of V holds ((0). V) /\ W = (0). V
let W be Submodule of V; ::_thesis: ((0). V) /\ W = (0). V
0. V in W by Th33;
then 0. V in the carrier of W by STRUCT_0:def_5;
then {(0. V)} c= the carrier of W by ZFMISC_1:31;
then A1: {(0. V)} /\ the carrier of W = {(0. V)} by XBOOLE_1:28;
the carrier of (((0). V) /\ W) = the carrier of ((0). V) /\ the carrier of W by Def15
.= {(0. V)} /\ the carrier of W by Def10 ;
hence ((0). V) /\ W = (0). V by A1, Def10; ::_thesis: verum
end;
theorem :: ZMODUL01:108
for V being Z_Module holds ((0). V) /\ ((Omega). V) = (0). V by Th107;
theorem Th109: :: ZMODUL01:109
for V being Z_Module
for W being strict Submodule of V holds ((Omega). V) /\ W = W
proof
let V be Z_Module; ::_thesis: for W being strict Submodule of V holds ((Omega). V) /\ W = W
let W be strict Submodule of V; ::_thesis: ((Omega). V) /\ W = W
( the carrier of (((Omega). V) /\ W) = the carrier of V /\ the carrier of W & the carrier of W c= the carrier of V ) by Def15, Def9;
hence ((Omega). V) /\ W = W by Th45, XBOOLE_1:28; ::_thesis: verum
end;
theorem :: ZMODUL01:110
for V being strict Z_Module holds ((Omega). V) /\ ((Omega). V) = V by Th109;
Lm9: for V being Z_Module
for W1, W2 being Submodule of V holds the carrier of (W1 /\ W2) c= the carrier of (W1 + W2)
proof
let V be Z_Module; ::_thesis: for W1, W2 being Submodule of V holds the carrier of (W1 /\ W2) c= the carrier of (W1 + W2)
let W1, W2 be Submodule of V; ::_thesis: the carrier of (W1 /\ W2) c= the carrier of (W1 + W2)
( the carrier of (W1 /\ W2) c= the carrier of W1 & the carrier of W1 c= the carrier of (W1 + W2) ) by Lm6, Lm8;
hence the carrier of (W1 /\ W2) c= the carrier of (W1 + W2) by XBOOLE_1:1; ::_thesis: verum
end;
theorem :: ZMODUL01:111
for V being Z_Module
for W1, W2 being Submodule of V holds W1 /\ W2 is Submodule of W1 + W2 by Lm9, Th43;
Lm10: for V being Z_Module
for W1, W2 being Submodule of V holds the carrier of ((W1 /\ W2) + W2) = the carrier of W2
proof
let V be Z_Module; ::_thesis: for W1, W2 being Submodule of V holds the carrier of ((W1 /\ W2) + W2) = the carrier of W2
let W1, W2 be Submodule of V; ::_thesis: the carrier of ((W1 /\ W2) + W2) = the carrier of W2
thus the carrier of ((W1 /\ W2) + W2) c= the carrier of W2 :: according to XBOOLE_0:def_10 ::_thesis: the carrier of W2 c= the carrier of ((W1 /\ W2) + W2)
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in the carrier of ((W1 /\ W2) + W2) or x in the carrier of W2 )
assume x in the carrier of ((W1 /\ W2) + W2) ; ::_thesis: x in the carrier of W2
then x in { (u + v) where u, v is VECTOR of V : ( u in W1 /\ W2 & v in W2 ) } by Def14;
then consider u, v being VECTOR of V such that
A1: x = u + v and
A2: u in W1 /\ W2 and
A3: v in W2 ;
u in W2 by A2, Th94;
then u + v in W2 by A3, Th36;
hence x in the carrier of W2 by A1, STRUCT_0:def_5; ::_thesis: verum
end;
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in the carrier of W2 or x in the carrier of ((W1 /\ W2) + W2) )
the carrier of W2 c= the carrier of ((W1 /\ W2) + W2) by Lm6;
hence ( not x in the carrier of W2 or x in the carrier of ((W1 /\ W2) + W2) ) ; ::_thesis: verum
end;
theorem :: ZMODUL01:112
for V being Z_Module
for W1 being Submodule of V
for W2 being strict Submodule of V holds (W1 /\ W2) + W2 = W2 by Lm10, Th45;
Lm11: for V being Z_Module
for W1, W2 being Submodule of V holds the carrier of (W1 /\ (W1 + W2)) = the carrier of W1
proof
let V be Z_Module; ::_thesis: for W1, W2 being Submodule of V holds the carrier of (W1 /\ (W1 + W2)) = the carrier of W1
let W1, W2 be Submodule of V; ::_thesis: the carrier of (W1 /\ (W1 + W2)) = the carrier of W1
thus the carrier of (W1 /\ (W1 + W2)) c= the carrier of W1 :: according to XBOOLE_0:def_10 ::_thesis: the carrier of W1 c= the carrier of (W1 /\ (W1 + W2))
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in the carrier of (W1 /\ (W1 + W2)) or x in the carrier of W1 )
assume A1: x in the carrier of (W1 /\ (W1 + W2)) ; ::_thesis: x in the carrier of W1
the carrier of (W1 /\ (W1 + W2)) = the carrier of W1 /\ the carrier of (W1 + W2) by Def15;
hence x in the carrier of W1 by A1, XBOOLE_0:def_4; ::_thesis: verum
end;
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in the carrier of W1 or x in the carrier of (W1 /\ (W1 + W2)) )
assume A2: x in the carrier of W1 ; ::_thesis: x in the carrier of (W1 /\ (W1 + W2))
the carrier of W1 c= the carrier of V by Def9;
then reconsider x1 = x as Element of V by A2;
A3: ( x1 + (0. V) = x1 & 0. V in W2 ) by Th33, RLVECT_1:4;
x in W1 by A2, STRUCT_0:def_5;
then x in { (u + v) where u, v is VECTOR of V : ( u in W1 & v in W2 ) } by A3;
then x in the carrier of (W1 + W2) by Def14;
then x in the carrier of W1 /\ the carrier of (W1 + W2) by A2, XBOOLE_0:def_4;
hence x in the carrier of (W1 /\ (W1 + W2)) by Def15; ::_thesis: verum
end;
theorem :: ZMODUL01:113
for V being Z_Module
for W2 being Submodule of V
for W1 being strict Submodule of V holds W1 /\ (W1 + W2) = W1 by Lm11, Th45;
Lm12: for V being Z_Module
for W1, W2, W3 being Submodule of V holds the carrier of ((W1 /\ W2) + (W2 /\ W3)) c= the carrier of (W2 /\ (W1 + W3))
proof
let V be Z_Module; ::_thesis: for W1, W2, W3 being Submodule of V holds the carrier of ((W1 /\ W2) + (W2 /\ W3)) c= the carrier of (W2 /\ (W1 + W3))
let W1, W2, W3 be Submodule of V; ::_thesis: the carrier of ((W1 /\ W2) + (W2 /\ W3)) c= the carrier of (W2 /\ (W1 + W3))
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in the carrier of ((W1 /\ W2) + (W2 /\ W3)) or x in the carrier of (W2 /\ (W1 + W3)) )
assume x in the carrier of ((W1 /\ W2) + (W2 /\ W3)) ; ::_thesis: x in the carrier of (W2 /\ (W1 + W3))
then x in { (u + v) where u, v is VECTOR of V : ( u in W1 /\ W2 & v in W2 /\ W3 ) } by Def14;
then consider u, v being VECTOR of V such that
A1: x = u + v and
A2: ( u in W1 /\ W2 & v in W2 /\ W3 ) ;
( u in W2 & v in W2 ) by A2, Th94;
then A3: x in W2 by A1, Th36;
( u in W1 & v in W3 ) by A2, Th94;
then x in W1 + W3 by A1, Th92;
then x in W2 /\ (W1 + W3) by A3, Th94;
hence x in the carrier of (W2 /\ (W1 + W3)) by STRUCT_0:def_5; ::_thesis: verum
end;
theorem :: ZMODUL01:114
for V being Z_Module
for W1, W2, W3 being Submodule of V holds (W1 /\ W2) + (W2 /\ W3) is Submodule of W2 /\ (W1 + W3) by Lm12, Th43;
Lm13: for V being Z_Module
for W1, W2, W3 being Submodule of V st W1 is Submodule of W2 holds
the carrier of (W2 /\ (W1 + W3)) = the carrier of ((W1 /\ W2) + (W2 /\ W3))
proof
let V be Z_Module; ::_thesis: for W1, W2, W3 being Submodule of V st W1 is Submodule of W2 holds
the carrier of (W2 /\ (W1 + W3)) = the carrier of ((W1 /\ W2) + (W2 /\ W3))
let W1, W2, W3 be Submodule of V; ::_thesis: ( W1 is Submodule of W2 implies the carrier of (W2 /\ (W1 + W3)) = the carrier of ((W1 /\ W2) + (W2 /\ W3)) )
assume A1: W1 is Submodule of W2 ; ::_thesis: the carrier of (W2 /\ (W1 + W3)) = the carrier of ((W1 /\ W2) + (W2 /\ W3))
thus the carrier of (W2 /\ (W1 + W3)) c= the carrier of ((W1 /\ W2) + (W2 /\ W3)) :: according to XBOOLE_0:def_10 ::_thesis: the carrier of ((W1 /\ W2) + (W2 /\ W3)) c= the carrier of (W2 /\ (W1 + W3))
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in the carrier of (W2 /\ (W1 + W3)) or x in the carrier of ((W1 /\ W2) + (W2 /\ W3)) )
assume x in the carrier of (W2 /\ (W1 + W3)) ; ::_thesis: x in the carrier of ((W1 /\ W2) + (W2 /\ W3))
then A2: x in the carrier of W2 /\ the carrier of (W1 + W3) by Def15;
then x in the carrier of (W1 + W3) by XBOOLE_0:def_4;
then x in { (u + v) where u, v is VECTOR of V : ( u in W1 & v in W3 ) } by Def14;
then consider u1, v1 being VECTOR of V such that
A3: x = u1 + v1 and
A4: u1 in W1 and
A5: v1 in W3 ;
A6: u1 in W2 by A1, A4, Th23;
x in the carrier of W2 by A2, XBOOLE_0:def_4;
then u1 + v1 in W2 by A3, STRUCT_0:def_5;
then (v1 + u1) - u1 in W2 by A6, Th39;
then v1 + (u1 - u1) in W2 by RLVECT_1:def_3;
then v1 + (0. V) in W2 by RLVECT_1:15;
then v1 in W2 by RLVECT_1:4;
then A7: v1 in W2 /\ W3 by A5, Th94;
u1 in W1 /\ W2 by A4, A6, Th94;
then x in (W1 /\ W2) + (W2 /\ W3) by A3, A7, Th92;
hence x in the carrier of ((W1 /\ W2) + (W2 /\ W3)) by STRUCT_0:def_5; ::_thesis: verum
end;
thus the carrier of ((W1 /\ W2) + (W2 /\ W3)) c= the carrier of (W2 /\ (W1 + W3)) by Lm12; ::_thesis: verum
end;
theorem :: ZMODUL01:115
for V being Z_Module
for W1, W2, W3 being Submodule of V st W1 is Submodule of W2 holds
W2 /\ (W1 + W3) = (W1 /\ W2) + (W2 /\ W3) by Lm13, Th45;
Lm14: for V being Z_Module
for W2, W1, W3 being Submodule of V holds the carrier of (W2 + (W1 /\ W3)) c= the carrier of ((W1 + W2) /\ (W2 + W3))
proof
let V be Z_Module; ::_thesis: for W2, W1, W3 being Submodule of V holds the carrier of (W2 + (W1 /\ W3)) c= the carrier of ((W1 + W2) /\ (W2 + W3))
let W2, W1, W3 be Submodule of V; ::_thesis: the carrier of (W2 + (W1 /\ W3)) c= the carrier of ((W1 + W2) /\ (W2 + W3))
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in the carrier of (W2 + (W1 /\ W3)) or x in the carrier of ((W1 + W2) /\ (W2 + W3)) )
assume x in the carrier of (W2 + (W1 /\ W3)) ; ::_thesis: x in the carrier of ((W1 + W2) /\ (W2 + W3))
then x in { (u + v) where u, v is VECTOR of V : ( u in W2 & v in W1 /\ W3 ) } by Def14;
then consider u, v being VECTOR of V such that
A1: ( x = u + v & u in W2 ) and
A2: v in W1 /\ W3 ;
v in W3 by A2, Th94;
then x in { (u1 + u2) where u1, u2 is VECTOR of V : ( u1 in W2 & u2 in W3 ) } by A1;
then A3: x in the carrier of (W2 + W3) by Def14;
v in W1 by A2, Th94;
then x in { (v1 + v2) where v1, v2 is VECTOR of V : ( v1 in W1 & v2 in W2 ) } by A1;
then x in the carrier of (W1 + W2) by Def14;
then x in the carrier of (W1 + W2) /\ the carrier of (W2 + W3) by A3, XBOOLE_0:def_4;
hence x in the carrier of ((W1 + W2) /\ (W2 + W3)) by Def15; ::_thesis: verum
end;
theorem :: ZMODUL01:116
for V being Z_Module
for W2, W1, W3 being Submodule of V holds W2 + (W1 /\ W3) is Submodule of (W1 + W2) /\ (W2 + W3) by Lm14, Th43;
Lm15: for V being Z_Module
for W1, W2, W3 being Submodule of V st W1 is Submodule of W2 holds
the carrier of (W2 + (W1 /\ W3)) = the carrier of ((W1 + W2) /\ (W2 + W3))
proof
let V be Z_Module; ::_thesis: for W1, W2, W3 being Submodule of V st W1 is Submodule of W2 holds
the carrier of (W2 + (W1 /\ W3)) = the carrier of ((W1 + W2) /\ (W2 + W3))
let W1, W2, W3 be Submodule of V; ::_thesis: ( W1 is Submodule of W2 implies the carrier of (W2 + (W1 /\ W3)) = the carrier of ((W1 + W2) /\ (W2 + W3)) )
reconsider V2 = the carrier of W2 as Subset of V by Def9;
A1: V2 is linearly-closed by Lm3;
assume W1 is Submodule of W2 ; ::_thesis: the carrier of (W2 + (W1 /\ W3)) = the carrier of ((W1 + W2) /\ (W2 + W3))
then A2: the carrier of W1 c= the carrier of W2 by Def9;
thus the carrier of (W2 + (W1 /\ W3)) c= the carrier of ((W1 + W2) /\ (W2 + W3)) by Lm14; :: according to XBOOLE_0:def_10 ::_thesis: the carrier of ((W1 + W2) /\ (W2 + W3)) c= the carrier of (W2 + (W1 /\ W3))
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in the carrier of ((W1 + W2) /\ (W2 + W3)) or x in the carrier of (W2 + (W1 /\ W3)) )
assume x in the carrier of ((W1 + W2) /\ (W2 + W3)) ; ::_thesis: x in the carrier of (W2 + (W1 /\ W3))
then x in the carrier of (W1 + W2) /\ the carrier of (W2 + W3) by Def15;
then x in the carrier of (W1 + W2) by XBOOLE_0:def_4;
then x in { (u1 + u2) where u1, u2 is VECTOR of V : ( u1 in W1 & u2 in W2 ) } by Def14;
then consider u1, u2 being VECTOR of V such that
A3: x = u1 + u2 and
A4: ( u1 in W1 & u2 in W2 ) ;
( u1 in the carrier of W1 & u2 in the carrier of W2 ) by A4, STRUCT_0:def_5;
then u1 + u2 in V2 by A2, A1, Def8;
then A5: u1 + u2 in W2 by STRUCT_0:def_5;
( 0. V in W1 /\ W3 & (u1 + u2) + (0. V) = u1 + u2 ) by Th33, RLVECT_1:4;
then x in { (u + v) where u, v is VECTOR of V : ( u in W2 & v in W1 /\ W3 ) } by A3, A5;
hence x in the carrier of (W2 + (W1 /\ W3)) by Def14; ::_thesis: verum
end;
theorem :: ZMODUL01:117
for V being Z_Module
for W1, W2, W3 being Submodule of V st W1 is Submodule of W2 holds
W2 + (W1 /\ W3) = (W1 + W2) /\ (W2 + W3) by Lm15, Th45;
theorem Th118: :: ZMODUL01:118
for V being Z_Module
for W1, W3, W2 being Submodule of V st W1 is strict Submodule of W3 holds
W1 + (W2 /\ W3) = (W1 + W2) /\ W3
proof
let V be Z_Module; ::_thesis: for W1, W3, W2 being Submodule of V st W1 is strict Submodule of W3 holds
W1 + (W2 /\ W3) = (W1 + W2) /\ W3
let W1, W3, W2 be Submodule of V; ::_thesis: ( W1 is strict Submodule of W3 implies W1 + (W2 /\ W3) = (W1 + W2) /\ W3 )
assume A1: W1 is strict Submodule of W3 ; ::_thesis: W1 + (W2 /\ W3) = (W1 + W2) /\ W3
thus (W1 + W2) /\ W3 = (W1 /\ W3) + (W3 /\ W2) by A1, Lm13, Th45
.= W1 + (W2 /\ W3) by A1, Th106 ; ::_thesis: verum
end;
theorem :: ZMODUL01:119
for V being Z_Module
for W1, W2 being strict Submodule of V holds
( W1 + W2 = W2 iff W1 /\ W2 = W1 )
proof
let V be Z_Module; ::_thesis: for W1, W2 being strict Submodule of V holds
( W1 + W2 = W2 iff W1 /\ W2 = W1 )
let W1, W2 be strict Submodule of V; ::_thesis: ( W1 + W2 = W2 iff W1 /\ W2 = W1 )
( W1 + W2 = W2 iff W1 is Submodule of W2 ) by Th98;
hence ( W1 + W2 = W2 iff W1 /\ W2 = W1 ) by Th106; ::_thesis: verum
end;
theorem :: ZMODUL01:120
for V being Z_Module
for W1 being Submodule of V
for W2, W3 being strict Submodule of V st W1 is Submodule of W2 holds
W1 + W3 is Submodule of W2 + W3
proof
let V be Z_Module; ::_thesis: for W1 being Submodule of V
for W2, W3 being strict Submodule of V st W1 is Submodule of W2 holds
W1 + W3 is Submodule of W2 + W3
let W1 be Submodule of V; ::_thesis: for W2, W3 being strict Submodule of V st W1 is Submodule of W2 holds
W1 + W3 is Submodule of W2 + W3
let W2, W3 be strict Submodule of V; ::_thesis: ( W1 is Submodule of W2 implies W1 + W3 is Submodule of W2 + W3 )
assume A1: W1 is Submodule of W2 ; ::_thesis: W1 + W3 is Submodule of W2 + W3
(W1 + W3) + (W2 + W3) = ((W1 + W3) + W3) + W2 by Th96
.= (W1 + (W3 + W3)) + W2 by Th96
.= (W1 + W3) + W2 by Lm7
.= (W1 + W2) + W3 by Th96
.= W2 + W3 by A1, Th98 ;
hence W1 + W3 is Submodule of W2 + W3 by Th98; ::_thesis: verum
end;
theorem :: ZMODUL01:121
for V being Z_Module
for W1, W2 being Submodule of V holds
( ( W1 is Submodule of W2 or W2 is Submodule of W1 ) iff ex W being Submodule of V st the carrier of W = the carrier of W1 \/ the carrier of W2 )
proof
let V be Z_Module; ::_thesis: for W1, W2 being Submodule of V holds
( ( W1 is Submodule of W2 or W2 is Submodule of W1 ) iff ex W being Submodule of V st the carrier of W = the carrier of W1 \/ the carrier of W2 )
let W1, W2 be Submodule of V; ::_thesis: ( ( W1 is Submodule of W2 or W2 is Submodule of W1 ) iff ex W being Submodule of V st the carrier of W = the carrier of W1 \/ the carrier of W2 )
set VW1 = the carrier of W1;
set VW2 = the carrier of W2;
thus ( for W being Submodule of V holds not the carrier of W = the carrier of W1 \/ the carrier of W2 or W1 is Submodule of W2 or W2 is Submodule of W1 ) ::_thesis: ( ( W1 is Submodule of W2 or W2 is Submodule of W1 ) implies ex W being Submodule of V st the carrier of W = the carrier of W1 \/ the carrier of W2 )
proof
given W being Submodule of V such that A1: the carrier of W = the carrier of W1 \/ the carrier of W2 ; ::_thesis: ( W1 is Submodule of W2 or W2 is Submodule of W1 )
set VW = the carrier of W;
assume that
A2: W1 is not Submodule of W2 and
A3: W2 is not Submodule of W1 ; ::_thesis: contradiction
not the carrier of W2 c= the carrier of W1 by A3, Th43;
then consider y being set such that
A4: y in the carrier of W2 and
A5: not y in the carrier of W1 by TARSKI:def_3;
reconsider y = y as Element of the carrier of W2 by A4;
reconsider y = y as VECTOR of V by Th25;
reconsider A1 = the carrier of W as Subset of V by Def9;
A6: A1 is linearly-closed by Lm3;
not the carrier of W1 c= the carrier of W2 by A2, Th43;
then consider x being set such that
A7: x in the carrier of W1 and
A8: not x in the carrier of W2 by TARSKI:def_3;
reconsider x = x as Element of the carrier of W1 by A7;
reconsider x = x as VECTOR of V by Th25;
A9: now__::_thesis:_not_x_+_y_in_the_carrier_of_W2
reconsider A2 = the carrier of W2 as Subset of V by Def9;
A10: A2 is linearly-closed by Lm3;
assume x + y in the carrier of W2 ; ::_thesis: contradiction
then (x + y) - y in the carrier of W2 by A10, Th20;
then x + (y - y) in the carrier of W2 by RLVECT_1:def_3;
then x + (0. V) in the carrier of W2 by RLVECT_1:15;
hence contradiction by A8, RLVECT_1:4; ::_thesis: verum
end;
A11: now__::_thesis:_not_x_+_y_in_the_carrier_of_W1
reconsider A2 = the carrier of W1 as Subset of V by Def9;
A12: A2 is linearly-closed by Lm3;
assume x + y in the carrier of W1 ; ::_thesis: contradiction
then (y + x) - x in the carrier of W1 by A12, Th20;
then y + (x - x) in the carrier of W1 by RLVECT_1:def_3;
then y + (0. V) in the carrier of W1 by RLVECT_1:15;
hence contradiction by A5, RLVECT_1:4; ::_thesis: verum
end;
( x in the carrier of W & y in the carrier of W ) by A1, XBOOLE_0:def_3;
then x + y in the carrier of W by A6, Def8;
hence contradiction by A1, A11, A9, XBOOLE_0:def_3; ::_thesis: verum
end;
A13: now__::_thesis:_(_W1_is_Submodule_of_W2_&_(_W1_is_Submodule_of_W2_or_W2_is_Submodule_of_W1_)_implies_ex_W_being_Submodule_of_V_st_the_carrier_of_W_=_the_carrier_of_W1_\/_the_carrier_of_W2_)
assume W1 is Submodule of W2 ; ::_thesis: ( ( W1 is Submodule of W2 or W2 is Submodule of W1 ) implies ex W being Submodule of V st the carrier of W = the carrier of W1 \/ the carrier of W2 )
then the carrier of W1 c= the carrier of W2 by Def9;
then the carrier of W1 \/ the carrier of W2 = the carrier of W2 by XBOOLE_1:12;
hence ( ( W1 is Submodule of W2 or W2 is Submodule of W1 ) implies ex W being Submodule of V st the carrier of W = the carrier of W1 \/ the carrier of W2 ) ; ::_thesis: verum
end;
A14: now__::_thesis:_(_W2_is_Submodule_of_W1_&_(_W1_is_Submodule_of_W2_or_W2_is_Submodule_of_W1_)_implies_ex_W_being_Submodule_of_V_st_the_carrier_of_W_=_the_carrier_of_W1_\/_the_carrier_of_W2_)
assume W2 is Submodule of W1 ; ::_thesis: ( ( W1 is Submodule of W2 or W2 is Submodule of W1 ) implies ex W being Submodule of V st the carrier of W = the carrier of W1 \/ the carrier of W2 )
then the carrier of W2 c= the carrier of W1 by Def9;
then the carrier of W1 \/ the carrier of W2 = the carrier of W1 by XBOOLE_1:12;
hence ( ( W1 is Submodule of W2 or W2 is Submodule of W1 ) implies ex W being Submodule of V st the carrier of W = the carrier of W1 \/ the carrier of W2 ) ; ::_thesis: verum
end;
assume ( W1 is Submodule of W2 or W2 is Submodule of W1 ) ; ::_thesis: ex W being Submodule of V st the carrier of W = the carrier of W1 \/ the carrier of W2
hence ex W being Submodule of V st the carrier of W = the carrier of W1 \/ the carrier of W2 by A13, A14; ::_thesis: verum
end;
definition
let V be Z_Module;
func Submodules V -> set means :Def16: :: ZMODUL01:def 16
for x being set holds
( x in it iff x is strict Submodule of V );
existence
ex b1 being set st
for x being set holds
( x in b1 iff x is strict Submodule of V )
proof
defpred S1[ set , set ] means ex W being strict Submodule of V st
( $2 = W & $1 = the carrier of W );
defpred S2[ set ] means ex W being strict Submodule of V st $1 = the carrier of W;
consider B being set such that
A1: for x being set holds
( x in B iff ( x in bool the carrier of V & 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 Th45;
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 W being strict Submodule of V such that
A5: x = the carrier of W by A1;
reconsider y = W 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 Submodule of V )
proof
let y be set ; ::_thesis: ( y in rng f iff y is strict Submodule of V )
thus ( y in rng f implies y is strict Submodule of V ) ::_thesis: ( y is strict Submodule of V implies y in rng f )
proof
assume y in rng f ; ::_thesis: y is strict Submodule of V
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 W being strict Submodule of V st
( y = W & x = the carrier of W ) by A3;
hence y is strict Submodule of V ; ::_thesis: verum
end;
assume y is strict Submodule of V ; ::_thesis: y in rng f
then reconsider W = y as strict Submodule of V ;
reconsider x = the carrier of W as set ;
the carrier of W c= the carrier of V by Def9;
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 Submodule of V ) ; ::_thesis: verum
end;
uniqueness
for b1, b2 being set st ( for x being set holds
( x in b1 iff x is strict Submodule of V ) ) & ( for x being set holds
( x in b2 iff x is strict Submodule of V ) ) holds
b1 = b2
proof
let D1, D2 be set ; ::_thesis: ( ( for x being set holds
( x in D1 iff x is strict Submodule of V ) ) & ( for x being set holds
( x in D2 iff x is strict Submodule of V ) ) implies D1 = D2 )
assume A10: for x being set holds
( x in D1 iff x is strict Submodule of V ) ; ::_thesis: ( ex x being set st
( ( x in D2 implies x is strict Submodule of V ) implies ( x is strict Submodule of V & not x in D2 ) ) or D1 = D2 )
assume A11: for x being set holds
( x in D2 iff x is strict Submodule of V ) ; ::_thesis: D1 = D2
now__::_thesis:_for_x_being_set_holds_
(_(_x_in_D1_implies_x_in_D2_)_&_(_x_in_D2_implies_x_in_D1_)_)
let x be set ; ::_thesis: ( ( x in D1 implies x in D2 ) & ( x in D2 implies x in D1 ) )
thus ( x in D1 implies x in D2 ) ::_thesis: ( x in D2 implies x in D1 )
proof
assume x in D1 ; ::_thesis: x in D2
then x is strict Submodule of V by A10;
hence x in D2 by A11; ::_thesis: verum
end;
assume x in D2 ; ::_thesis: x in D1
then x is strict Submodule of V by A11;
hence x in D1 by A10; ::_thesis: verum
end;
hence D1 = D2 by TARSKI:1; ::_thesis: verum
end;
end;
:: deftheorem Def16 defines Submodules ZMODUL01:def_16_:_
for V being Z_Module
for b2 being set holds
( b2 = Submodules V iff for x being set holds
( x in b2 iff x is strict Submodule of V ) );
registration
let V be Z_Module;
cluster Submodules V -> non empty ;
coherence
not Submodules V is empty
proof
set x = the strict Submodule of V;
the strict Submodule of V in Submodules V by Def16;
hence not Submodules V is empty ; ::_thesis: verum
end;
end;
theorem :: ZMODUL01:122
for V being strict Z_Module holds V in Submodules V
proof
let V be strict Z_Module; ::_thesis: V in Submodules V
(Omega). V in Submodules V by Def16;
hence V in Submodules V ; ::_thesis: verum
end;
definition
let V be Z_Module;
let W1, W2 be Submodule of V;
predV is_the_direct_sum_of W1,W2 means :Def17: :: ZMODUL01:def 17
( Z_ModuleStruct(# the carrier of V, the ZeroF of V, the addF of V, the Mult of V #) = W1 + W2 & W1 /\ W2 = (0). V );
end;
:: deftheorem Def17 defines is_the_direct_sum_of ZMODUL01:def_17_:_
for V being Z_Module
for W1, W2 being Submodule of V holds
( V is_the_direct_sum_of W1,W2 iff ( Z_ModuleStruct(# the carrier of V, the ZeroF of V, the addF of V, the Mult of V #) = W1 + W2 & W1 /\ W2 = (0). V ) );
Lm16: for V being Z_Module
for W being strict Submodule of V st ( for v being VECTOR of V holds v in W ) holds
W = Z_ModuleStruct(# the carrier of V, the ZeroF of V, the addF of V, the Mult of V #)
proof
let V be Z_Module; ::_thesis: for W being strict Submodule of V st ( for v being VECTOR of V holds v in W ) holds
W = Z_ModuleStruct(# the carrier of V, the ZeroF of V, the addF of V, the Mult of V #)
let W be strict Submodule of V; ::_thesis: ( ( for v being VECTOR of V holds v in W ) implies W = Z_ModuleStruct(# the carrier of V, the ZeroF of V, the addF of V, the Mult of V #) )
assume for v being VECTOR of V holds v in W ; ::_thesis: W = Z_ModuleStruct(# the carrier of V, the ZeroF of V, the addF of V, the Mult of V #)
then for v being VECTOR of V holds
( v in W iff v in (Omega). V ) by RLVECT_1:1;
hence W = Z_ModuleStruct(# the carrier of V, the ZeroF of V, the addF of V, the Mult of V #) by Th46; ::_thesis: verum
end;
Lm17: for V being Z_Module
for W1, W2 being Submodule of V holds
( W1 + W2 = Z_ModuleStruct(# the carrier of V, the ZeroF of V, the addF of V, the Mult of V #) iff for v being VECTOR of V ex v1, v2 being VECTOR of V st
( v1 in W1 & v2 in W2 & v = v1 + v2 ) )
proof
let V be Z_Module; ::_thesis: for W1, W2 being Submodule of V holds
( W1 + W2 = Z_ModuleStruct(# the carrier of V, the ZeroF of V, the addF of V, the Mult of V #) iff for v being VECTOR of V ex v1, v2 being VECTOR of V st
( v1 in W1 & v2 in W2 & v = v1 + v2 ) )
let W1, W2 be Submodule of V; ::_thesis: ( W1 + W2 = Z_ModuleStruct(# the carrier of V, the ZeroF of V, the addF of V, the Mult of V #) iff for v being VECTOR of V ex v1, v2 being VECTOR of V st
( v1 in W1 & v2 in W2 & v = v1 + v2 ) )
thus ( W1 + W2 = Z_ModuleStruct(# the carrier of V, the ZeroF of V, the addF of V, the Mult of V #) implies for v being VECTOR of V ex v1, v2 being VECTOR of V st
( v1 in W1 & v2 in W2 & v = v1 + v2 ) ) ::_thesis: ( ( for v being VECTOR of V ex v1, v2 being VECTOR of V st
( v1 in W1 & v2 in W2 & v = v1 + v2 ) ) implies W1 + W2 = Z_ModuleStruct(# the carrier of V, the ZeroF of V, the addF of V, the Mult of V #) )
proof
assume A1: W1 + W2 = Z_ModuleStruct(# the carrier of V, the ZeroF of V, the addF of V, the Mult of V #) ; ::_thesis: for v being VECTOR of V ex v1, v2 being VECTOR of V st
( v1 in W1 & v2 in W2 & v = v1 + v2 )
let v be VECTOR of V; ::_thesis: ex v1, v2 being VECTOR of V st
( v1 in W1 & v2 in W2 & v = v1 + v2 )
v in Z_ModuleStruct(# the carrier of V, the ZeroF of V, the addF of V, the Mult of V #) by RLVECT_1:1;
hence ex v1, v2 being VECTOR of V st
( v1 in W1 & v2 in W2 & v = v1 + v2 ) by A1, Th92; ::_thesis: verum
end;
assume A2: for v being VECTOR of V ex v1, v2 being VECTOR of V st
( v1 in W1 & v2 in W2 & v = v1 + v2 ) ; ::_thesis: W1 + W2 = Z_ModuleStruct(# the carrier of V, the ZeroF of V, the addF of V, the Mult of V #)
now__::_thesis:_for_u_being_VECTOR_of_V_holds_u_in_W1_+_W2
let u be VECTOR of V; ::_thesis: u in W1 + W2
ex v1, v2 being VECTOR of V st
( v1 in W1 & v2 in W2 & u = v1 + v2 ) by A2;
hence u in W1 + W2 by Th92; ::_thesis: verum
end;
hence W1 + W2 = Z_ModuleStruct(# the carrier of V, the ZeroF of V, the addF of V, the Mult of V #) by Lm16; ::_thesis: verum
end;
definition
let V be Z_Module;
let W be Submodule of V;
attrW is with_Linear_Compl means :Def18: :: ZMODUL01:def 18
ex C being Submodule of V st V is_the_direct_sum_of C,W;
end;
:: deftheorem Def18 defines with_Linear_Compl ZMODUL01:def_18_:_
for V being Z_Module
for W being Submodule of V holds
( W is with_Linear_Compl iff ex C being Submodule of V st V is_the_direct_sum_of C,W );
registration
let V be Z_Module;
cluster non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable Abelian add-associative right_zeroed V129() V130() V131() V132() vector-distributive scalar-distributive scalar-associative scalar-unital with_Linear_Compl for Submodule of V;
correctness
existence
ex b1 being Submodule of V st b1 is with_Linear_Compl ;
proof
( ((0). V) + ((Omega). V) = Z_ModuleStruct(# the carrier of V, the ZeroF of V, the addF of V, the Mult of V #) & (0). V = ((0). V) /\ ((Omega). V) ) by Th99, Th107;
then V is_the_direct_sum_of (0). V, (Omega). V by Def17;
then (Omega). V is with_Linear_Compl by Def18;
hence ex b1 being Submodule of V st b1 is with_Linear_Compl ; ::_thesis: verum
end;
end;
definition
let V be Z_Module;
let W be Submodule of V;
assume A1: W is with_Linear_Compl ;
mode Linear_Compl of W -> Submodule of V means :Def19: :: ZMODUL01:def 19
V is_the_direct_sum_of it,W;
existence
ex b1 being Submodule of V st V is_the_direct_sum_of b1,W by A1, Def18;
end;
:: deftheorem Def19 defines Linear_Compl ZMODUL01:def_19_:_
for V being Z_Module
for W being Submodule of V st W is with_Linear_Compl holds
for b3 being Submodule of V holds
( b3 is Linear_Compl of W iff V is_the_direct_sum_of b3,W );
Lm18: for V being Z_Module
for W1, W2 being Submodule of V st V is_the_direct_sum_of W1,W2 holds
V is_the_direct_sum_of W2,W1
proof
let V be Z_Module; ::_thesis: for W1, W2 being Submodule of V st V is_the_direct_sum_of W1,W2 holds
V is_the_direct_sum_of W2,W1
let W1, W2 be Submodule of V; ::_thesis: ( V is_the_direct_sum_of W1,W2 implies V is_the_direct_sum_of W2,W1 )
assume A1: V is_the_direct_sum_of W1,W2 ; ::_thesis: V is_the_direct_sum_of W2,W1
then A2: W2 /\ W1 = (0). V by Def17;
Z_ModuleStruct(# the carrier of V, the ZeroF of V, the addF of V, the Mult of V #) = W1 + W2 by A1, Def17;
hence V is_the_direct_sum_of W2,W1 by A2, Def17; ::_thesis: verum
end;
theorem :: ZMODUL01:123
for V being Z_Module
for W1, W2 being Submodule of V st V is_the_direct_sum_of W1,W2 holds
W2 is Linear_Compl of W1
proof
let V be Z_Module; ::_thesis: for W1, W2 being Submodule of V st V is_the_direct_sum_of W1,W2 holds
W2 is Linear_Compl of W1
let W1, W2 be Submodule of V; ::_thesis: ( V is_the_direct_sum_of W1,W2 implies W2 is Linear_Compl of W1 )
assume V is_the_direct_sum_of W1,W2 ; ::_thesis: W2 is Linear_Compl of W1
then A1: V is_the_direct_sum_of W2,W1 by Lm18;
then W1 is with_Linear_Compl by Def18;
hence W2 is Linear_Compl of W1 by Def19, A1; ::_thesis: verum
end;
theorem Th124: :: ZMODUL01:124
for V being Z_Module
for W being with_Linear_Compl Submodule of V
for L being Linear_Compl of W holds
( V is_the_direct_sum_of L,W & V is_the_direct_sum_of W,L )
proof
let V be Z_Module; ::_thesis: for W being with_Linear_Compl Submodule of V
for L being Linear_Compl of W holds
( V is_the_direct_sum_of L,W & V is_the_direct_sum_of W,L )
let W be with_Linear_Compl Submodule of V; ::_thesis: for L being Linear_Compl of W holds
( V is_the_direct_sum_of L,W & V is_the_direct_sum_of W,L )
let L be Linear_Compl of W; ::_thesis: ( V is_the_direct_sum_of L,W & V is_the_direct_sum_of W,L )
thus V is_the_direct_sum_of L,W by Def19; ::_thesis: V is_the_direct_sum_of W,L
hence V is_the_direct_sum_of W,L by Lm18; ::_thesis: verum
end;
theorem :: ZMODUL01:125
for V being Z_Module
for W being with_Linear_Compl Submodule of V
for L being Linear_Compl of W holds W + L = Z_ModuleStruct(# the carrier of V, the ZeroF of V, the addF of V, the Mult of V #)
proof
let V be Z_Module; ::_thesis: for W being with_Linear_Compl Submodule of V
for L being Linear_Compl of W holds W + L = Z_ModuleStruct(# the carrier of V, the ZeroF of V, the addF of V, the Mult of V #)
let W be with_Linear_Compl Submodule of V; ::_thesis: for L being Linear_Compl of W holds W + L = Z_ModuleStruct(# the carrier of V, the ZeroF of V, the addF of V, the Mult of V #)
let L be Linear_Compl of W; ::_thesis: W + L = Z_ModuleStruct(# the carrier of V, the ZeroF of V, the addF of V, the Mult of V #)
V is_the_direct_sum_of W,L by Th124;
hence W + L = Z_ModuleStruct(# the carrier of V, the ZeroF of V, the addF of V, the Mult of V #) by Def17; ::_thesis: verum
end;
theorem :: ZMODUL01:126
for V being Z_Module
for W being with_Linear_Compl Submodule of V
for L being Linear_Compl of W holds W /\ L = (0). V
proof
let V be Z_Module; ::_thesis: for W being with_Linear_Compl Submodule of V
for L being Linear_Compl of W holds W /\ L = (0). V
let W be with_Linear_Compl Submodule of V; ::_thesis: for L being Linear_Compl of W holds W /\ L = (0). V
let L be Linear_Compl of W; ::_thesis: W /\ L = (0). V
V is_the_direct_sum_of W,L by Th124;
hence W /\ L = (0). V by Def17; ::_thesis: verum
end;
theorem :: ZMODUL01:127
for V being Z_Module
for W1, W2 being Submodule of V st V is_the_direct_sum_of W1,W2 holds
V is_the_direct_sum_of W2,W1 by Lm18;
theorem :: ZMODUL01:128
for V being Z_Module
for W being with_Linear_Compl Submodule of V
for L being Linear_Compl of W holds W is Linear_Compl of L
proof
let V be Z_Module; ::_thesis: for W being with_Linear_Compl Submodule of V
for L being Linear_Compl of W holds W is Linear_Compl of L
let W be with_Linear_Compl Submodule of V; ::_thesis: for L being Linear_Compl of W holds W is Linear_Compl of L
let L be Linear_Compl of W; ::_thesis: W is Linear_Compl of L
V is_the_direct_sum_of L,W by Def19;
then A1: V is_the_direct_sum_of W,L by Lm18;
then L is with_Linear_Compl by Def18;
hence W is Linear_Compl of L by Def19, A1; ::_thesis: verum
end;
theorem Th129: :: ZMODUL01:129
for V being Z_Module holds
( V is_the_direct_sum_of (0). V, (Omega). V & V is_the_direct_sum_of (Omega). V, (0). V )
proof
let V be Z_Module; ::_thesis: ( V is_the_direct_sum_of (0). V, (Omega). V & V is_the_direct_sum_of (Omega). V, (0). V )
A1: ( ((0). V) + ((Omega). V) = Z_ModuleStruct(# the carrier of V, the ZeroF of V, the addF of V, the Mult of V #) & (0). V = ((0). V) /\ ((Omega). V) ) by Th99, Th107;
hence V is_the_direct_sum_of (0). V, (Omega). V by Def17; ::_thesis: V is_the_direct_sum_of (Omega). V, (0). V
thus V is_the_direct_sum_of (Omega). V, (0). V by A1, Def17; ::_thesis: verum
end;
theorem :: ZMODUL01:130
for V being Z_Module holds
( (0). V is Linear_Compl of (Omega). V & (Omega). V is Linear_Compl of (0). V )
proof
let V be Z_Module; ::_thesis: ( (0). V is Linear_Compl of (Omega). V & (Omega). V is Linear_Compl of (0). V )
A1: ( V is_the_direct_sum_of (0). V, (Omega). V & V is_the_direct_sum_of (Omega). V, (0). V ) by Th129;
then A2: (Omega). V is with_Linear_Compl by Def18;
(0). V is with_Linear_Compl by A1, Def18;
hence ( (0). V is Linear_Compl of (Omega). V & (Omega). V is Linear_Compl of (0). V ) by Def19, A1, A2; ::_thesis: verum
end;
theorem Th131: :: ZMODUL01:131
for V being Z_Module
for W1, W2 being Submodule of V
for C1 being Coset of W1
for C2 being Coset of W2 st C1 meets C2 holds
C1 /\ C2 is Coset of W1 /\ W2
proof
let V be Z_Module; ::_thesis: for W1, W2 being Submodule of V
for C1 being Coset of W1
for C2 being Coset of W2 st C1 meets C2 holds
C1 /\ C2 is Coset of W1 /\ W2
let W1, W2 be Submodule of V; ::_thesis: for C1 being Coset of W1
for C2 being Coset of W2 st C1 meets C2 holds
C1 /\ C2 is Coset of W1 /\ W2
let C1 be Coset of W1; ::_thesis: for C2 being Coset of W2 st C1 meets C2 holds
C1 /\ C2 is Coset of W1 /\ W2
let C2 be Coset of W2; ::_thesis: ( C1 meets C2 implies C1 /\ C2 is Coset of W1 /\ W2 )
set v = the Element of C1 /\ C2;
set C = C1 /\ C2;
assume A1: C1 /\ C2 <> {} ; :: according to XBOOLE_0:def_7 ::_thesis: C1 /\ C2 is Coset of W1 /\ W2
then reconsider v = the Element of C1 /\ C2 as Element of V by TARSKI:def_3;
v in C2 by A1, XBOOLE_0:def_4;
then A2: C2 = v + W2 by Th87;
v in C1 by A1, XBOOLE_0:def_4;
then A3: C1 = v + W1 by Th87;
C1 /\ C2 is Coset of W1 /\ W2
proof
take v ; :: according to ZMODUL01:def_13 ::_thesis: C1 /\ C2 = v + (W1 /\ W2)
thus C1 /\ C2 c= v + (W1 /\ W2) :: according to XBOOLE_0:def_10 ::_thesis: v + (W1 /\ W2) c= C1 /\ C2
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in C1 /\ C2 or x in v + (W1 /\ W2) )
assume A4: x in C1 /\ C2 ; ::_thesis: x in v + (W1 /\ W2)
then x in C1 by XBOOLE_0:def_4;
then consider u1 being VECTOR of V such that
A5: u1 in W1 and
A6: x = v + u1 by A3, Th72;
x in C2 by A4, XBOOLE_0:def_4;
then consider u2 being VECTOR of V such that
A7: u2 in W2 and
A8: x = v + u2 by A2, Th72;
u1 = u2 by A6, A8, RLVECT_1:8;
then u1 in W1 /\ W2 by A5, A7, Th94;
hence x in v + (W1 /\ W2) by A6; ::_thesis: verum
end;
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in v + (W1 /\ W2) or x in C1 /\ C2 )
assume x in v + (W1 /\ W2) ; ::_thesis: x in C1 /\ C2
then consider u being VECTOR of V such that
A9: u in W1 /\ W2 and
A10: x = v + u by Th72;
u in W2 by A9, Th94;
then A11: x in { (v + u2) where u2 is VECTOR of V : u2 in W2 } by A10;
u in W1 by A9, Th94;
then x in { (v + u1) where u1 is VECTOR of V : u1 in W1 } by A10;
hence x in C1 /\ C2 by A3, A2, A11, XBOOLE_0:def_4; ::_thesis: verum
end;
hence C1 /\ C2 is Coset of W1 /\ W2 ; ::_thesis: verum
end;
Lm19: for V being Z_Module
for W being Submodule of V
for v being VECTOR of V ex C being Coset of W st v in C
proof
let V be Z_Module; ::_thesis: for W being Submodule of V
for v being VECTOR of V ex C being Coset of W st v in C
let W be Submodule of V; ::_thesis: for v being VECTOR of V ex C being Coset of W st v in C
let v be VECTOR of V; ::_thesis: ex C being Coset of W st v in C
reconsider C = v + W as Coset of W by Def13;
take C ; ::_thesis: v in C
thus v in C by Th58; ::_thesis: verum
end;
theorem Th132: :: ZMODUL01:132
for V being Z_Module
for W1, W2 being Submodule of V holds
( V is_the_direct_sum_of W1,W2 iff for C1 being Coset of W1
for C2 being Coset of W2 ex v being VECTOR of V st C1 /\ C2 = {v} )
proof
let V be Z_Module; ::_thesis: for W1, W2 being Submodule of V holds
( V is_the_direct_sum_of W1,W2 iff for C1 being Coset of W1
for C2 being Coset of W2 ex v being VECTOR of V st C1 /\ C2 = {v} )
let W1, W2 be Submodule of V; ::_thesis: ( V is_the_direct_sum_of W1,W2 iff for C1 being Coset of W1
for C2 being Coset of W2 ex v being VECTOR of V st C1 /\ C2 = {v} )
set VW1 = the carrier of W1;
set VW2 = the carrier of W2;
0. V in W2 by Th33;
then A1: 0. V in the carrier of W2 by STRUCT_0:def_5;
thus ( V is_the_direct_sum_of W1,W2 implies for C1 being Coset of W1
for C2 being Coset of W2 ex v being VECTOR of V st C1 /\ C2 = {v} ) ::_thesis: ( ( for C1 being Coset of W1
for C2 being Coset of W2 ex v being VECTOR of V st C1 /\ C2 = {v} ) implies V is_the_direct_sum_of W1,W2 )
proof
assume A2: V is_the_direct_sum_of W1,W2 ; ::_thesis: for C1 being Coset of W1
for C2 being Coset of W2 ex v being VECTOR of V st C1 /\ C2 = {v}
then A3: Z_ModuleStruct(# the carrier of V, the ZeroF of V, the addF of V, the Mult of V #) = W1 + W2 by Def17;
let C1 be Coset of W1; ::_thesis: for C2 being Coset of W2 ex v being VECTOR of V st C1 /\ C2 = {v}
let C2 be Coset of W2; ::_thesis: ex v being VECTOR of V st C1 /\ C2 = {v}
consider v1 being VECTOR of V such that
A4: C1 = v1 + W1 by Def13;
v1 in Z_ModuleStruct(# the carrier of V, the ZeroF of V, the addF of V, the Mult of V #) by RLVECT_1:1;
then consider v11, v12 being VECTOR of V such that
A5: v11 in W1 and
A6: v12 in W2 and
A7: v1 = v11 + v12 by A3, Th92;
consider v2 being VECTOR of V such that
A8: C2 = v2 + W2 by Def13;
v2 in Z_ModuleStruct(# the carrier of V, the ZeroF of V, the addF of V, the Mult of V #) by RLVECT_1:1;
then consider v21, v22 being VECTOR of V such that
A9: v21 in W1 and
A10: v22 in W2 and
A11: v2 = v21 + v22 by A3, Th92;
take v = v12 + v21; ::_thesis: C1 /\ C2 = {v}
{v} = C1 /\ C2
proof
thus A12: {v} c= C1 /\ C2 :: according to XBOOLE_0:def_10 ::_thesis: C1 /\ C2 c= {v}
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in {v} or x in C1 /\ C2 )
assume x in {v} ; ::_thesis: x in C1 /\ C2
then A13: x = v by TARSKI:def_1;
v21 = v2 - v22 by A11, RLSUB_2:61;
then v21 in C2 by A8, A10, Th73;
then C2 = v21 + W2 by Th87;
then A14: x in C2 by A6, A13;
v12 = v1 - v11 by A7, RLSUB_2:61;
then v12 in C1 by A4, A5, Th73;
then C1 = v12 + W1 by Th87;
then x in C1 by A9, A13;
hence x in C1 /\ C2 by A14, XBOOLE_0:def_4; ::_thesis: verum
end;
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in C1 /\ C2 or x in {v} )
assume A15: x in C1 /\ C2 ; ::_thesis: x in {v}
then C1 meets C2 by XBOOLE_0:4;
then reconsider C = C1 /\ C2 as Coset of W1 /\ W2 by Th131;
A16: v in {v} by TARSKI:def_1;
W1 /\ W2 = (0). V by A2, Def17;
then ex u being VECTOR of V st C = {u} by Th82;
hence x in {v} by A12, A15, A16, TARSKI:def_1; ::_thesis: verum
end;
hence C1 /\ C2 = {v} ; ::_thesis: verum
end;
assume A17: for C1 being Coset of W1
for C2 being Coset of W2 ex v being VECTOR of V st C1 /\ C2 = {v} ; ::_thesis: V is_the_direct_sum_of W1,W2
A18: the carrier of W2 is Coset of W2 by Th83;
now__::_thesis:_for_u_being_VECTOR_of_V_holds_u_in_W1_+_W2
let u be VECTOR of V; ::_thesis: u in W1 + W2
consider C1 being Coset of W1 such that
A19: u in C1 by Lm19;
consider v being VECTOR of V such that
A20: C1 /\ the carrier of W2 = {v} by A18, A17;
A21: v in {v} by TARSKI:def_1;
then v in C1 by A20, XBOOLE_0:def_4;
then consider v1 being VECTOR of V such that
A22: v1 in W1 and
A23: u - v1 = v by A19, Th89;
v in the carrier of W2 by A20, A21, XBOOLE_0:def_4;
then A24: v in W2 by STRUCT_0:def_5;
u = v1 + v by A23, RLSUB_2:61;
hence u in W1 + W2 by A24, A22, Th92; ::_thesis: verum
end;
hence Z_ModuleStruct(# the carrier of V, the ZeroF of V, the addF of V, the Mult of V #) = W1 + W2 by Lm16; :: according to ZMODUL01:def_17 ::_thesis: W1 /\ W2 = (0). V
the carrier of W1 is Coset of W1 by Th83;
then consider v being VECTOR of V such that
A25: the carrier of W1 /\ the carrier of W2 = {v} by A18, A17;
0. V in W1 by Th33;
then 0. V in the carrier of W1 by STRUCT_0:def_5;
then A26: 0. V in {v} by A25, A1, XBOOLE_0:def_4;
the carrier of ((0). V) = {(0. V)} by Def10
.= the carrier of W1 /\ the carrier of W2 by A25, A26, TARSKI:def_1
.= the carrier of (W1 /\ W2) by Def15 ;
hence W1 /\ W2 = (0). V by Th45; ::_thesis: verum
end;
theorem :: ZMODUL01:133
for V being Z_Module
for W1, W2 being Submodule of V holds
( W1 + W2 = Z_ModuleStruct(# the carrier of V, the ZeroF of V, the addF of V, the Mult of V #) iff for v being VECTOR of V ex v1, v2 being VECTOR of V st
( v1 in W1 & v2 in W2 & v = v1 + v2 ) ) by Lm17;
theorem Th134: :: ZMODUL01:134
for V being Z_Module
for W1, W2 being Submodule of V
for v1, v2, u1, u2 being VECTOR of V st V is_the_direct_sum_of W1,W2 & v1 + v2 = u1 + u2 & v1 in W1 & u1 in W1 & v2 in W2 & u2 in W2 holds
( v1 = u1 & v2 = u2 )
proof
let V be Z_Module; ::_thesis: for W1, W2 being Submodule of V
for v1, v2, u1, u2 being VECTOR of V st V is_the_direct_sum_of W1,W2 & v1 + v2 = u1 + u2 & v1 in W1 & u1 in W1 & v2 in W2 & u2 in W2 holds
( v1 = u1 & v2 = u2 )
let W1, W2 be Submodule of V; ::_thesis: for v1, v2, u1, u2 being VECTOR of V st V is_the_direct_sum_of W1,W2 & v1 + v2 = u1 + u2 & v1 in W1 & u1 in W1 & v2 in W2 & u2 in W2 holds
( v1 = u1 & v2 = u2 )
let v1, v2, u1, u2 be VECTOR of V; ::_thesis: ( V is_the_direct_sum_of W1,W2 & v1 + v2 = u1 + u2 & v1 in W1 & u1 in W1 & v2 in W2 & u2 in W2 implies ( v1 = u1 & v2 = u2 ) )
reconsider C2 = v1 + W2 as Coset of W2 by Def13;
reconsider C1 = the carrier of W1 as Coset of W1 by Th83;
A1: v1 in C2 by Th58;
assume V is_the_direct_sum_of W1,W2 ; ::_thesis: ( not v1 + v2 = u1 + u2 or not v1 in W1 or not u1 in W1 or not v2 in W2 or not u2 in W2 or ( v1 = u1 & v2 = u2 ) )
then consider u being VECTOR of V such that
A2: C1 /\ C2 = {u} by Th132;
assume that
A3: v1 + v2 = u1 + u2 and
A4: v1 in W1 and
A5: u1 in W1 and
A6: ( v2 in W2 & u2 in W2 ) ; ::_thesis: ( v1 = u1 & v2 = u2 )
A7: v2 - u2 in W2 by A6, Th39;
v1 in C1 by A4, STRUCT_0:def_5;
then v1 in C1 /\ C2 by A1, XBOOLE_0:def_4;
then A8: v1 = u by A2, TARSKI:def_1;
u1 = (v1 + v2) - u2 by A3, RLSUB_2:61
.= v1 + (v2 - u2) by RLVECT_1:def_3 ;
then A9: u1 in C2 by A7;
u1 in C1 by A5, STRUCT_0:def_5;
then A10: u1 in C1 /\ C2 by A9, XBOOLE_0:def_4;
hence v1 = u1 by A2, A8, TARSKI:def_1; ::_thesis: v2 = u2
u1 = u by A10, A2, TARSKI:def_1;
hence v2 = u2 by A3, A8, RLVECT_1:8; ::_thesis: verum
end;
theorem :: ZMODUL01:135
for V being Z_Module
for W1, W2 being Submodule of V st V = W1 + W2 & ex v being VECTOR of V st
for v1, v2, u1, u2 being VECTOR of V st v1 + v2 = u1 + u2 & v1 in W1 & u1 in W1 & v2 in W2 & u2 in W2 holds
( v1 = u1 & v2 = u2 ) holds
V is_the_direct_sum_of W1,W2
proof
let V be Z_Module; ::_thesis: for W1, W2 being Submodule of V st V = W1 + W2 & ex v being VECTOR of V st
for v1, v2, u1, u2 being VECTOR of V st v1 + v2 = u1 + u2 & v1 in W1 & u1 in W1 & v2 in W2 & u2 in W2 holds
( v1 = u1 & v2 = u2 ) holds
V is_the_direct_sum_of W1,W2
let W1, W2 be Submodule of V; ::_thesis: ( V = W1 + W2 & ex v being VECTOR of V st
for v1, v2, u1, u2 being VECTOR of V st v1 + v2 = u1 + u2 & v1 in W1 & u1 in W1 & v2 in W2 & u2 in W2 holds
( v1 = u1 & v2 = u2 ) implies V is_the_direct_sum_of W1,W2 )
assume A1: V = W1 + W2 ; ::_thesis: ( for v being VECTOR of V ex v1, v2, u1, u2 being VECTOR of V st
( v1 + v2 = u1 + u2 & v1 in W1 & u1 in W1 & v2 in W2 & u2 in W2 & not ( v1 = u1 & v2 = u2 ) ) or V is_the_direct_sum_of W1,W2 )
( the carrier of ((0). V) = {(0. V)} & (0). V is Submodule of W1 /\ W2 ) by Th54, Def10;
then A2: {(0. V)} c= the carrier of (W1 /\ W2) by Def9;
given v being VECTOR of V such that A3: for v1, v2, u1, u2 being VECTOR of V st v1 + v2 = u1 + u2 & v1 in W1 & u1 in W1 & v2 in W2 & u2 in W2 holds
( v1 = u1 & v2 = u2 ) ; ::_thesis: V is_the_direct_sum_of W1,W2
assume not V is_the_direct_sum_of W1,W2 ; ::_thesis: contradiction
then W1 /\ W2 <> (0). V by A1, Def17;
then the carrier of (W1 /\ W2) <> {(0. V)} by Def10;
then {(0. V)} c< the carrier of (W1 /\ W2) by A2, XBOOLE_0:def_8;
then consider x being set such that
A4: x in the carrier of (W1 /\ W2) and
A5: not x in {(0. V)} by XBOOLE_0:6;
A6: x <> 0. V by A5, TARSKI:def_1;
A7: x in W1 /\ W2 by A4, STRUCT_0:def_5;
then x in V by Th24;
then reconsider u = x as VECTOR of V by STRUCT_0:def_5;
consider v1, v2 being VECTOR of V such that
A8: v1 in W1 and
A9: v2 in W2 and
A10: v = v1 + v2 by A1, Lm17;
A11: v = (v1 + v2) + (0. V) by A10, RLVECT_1:4
.= (v1 + v2) + (u - u) by RLVECT_1:15
.= ((v1 + v2) + u) - u by RLVECT_1:def_3
.= ((v1 + u) + v2) - u by RLVECT_1:def_3
.= (v1 + u) + (v2 - u) by RLVECT_1:def_3 ;
x in W2 by A7, Th94;
then A12: v2 - u in W2 by A9, Th39;
x in W1 by A7, Th94;
then v1 + u in W1 by A8, Th36;
then v2 - u = v2 by A3, A8, A9, A10, A11, A12
.= v2 - (0. V) by RLVECT_1:13 ;
hence contradiction by A6, RLVECT_1:23; ::_thesis: verum
end;
definition
let V be Z_Module;
let v be VECTOR of V;
let W1, W2 be Submodule of V;
assume A1: V is_the_direct_sum_of W1,W2 ;
funcv |-- (W1,W2) -> Element of [: the carrier of V, the carrier of V:] means :Def20: :: ZMODUL01:def 20
( v = (it `1) + (it `2) & it `1 in W1 & it `2 in W2 );
existence
ex b1 being Element of [: the carrier of V, the carrier of V:] st
( v = (b1 `1) + (b1 `2) & b1 `1 in W1 & b1 `2 in W2 )
proof
W1 + W2 = Z_ModuleStruct(# the carrier of V, the ZeroF of V, the addF of V, the Mult of V #) by A1, Def17;
then consider v1, v2 being VECTOR of V such that
A2: ( v1 in W1 & v2 in W2 & v = v1 + v2 ) by Lm17;
take [v1,v2] ; ::_thesis: ( v = ([v1,v2] `1) + ([v1,v2] `2) & [v1,v2] `1 in W1 & [v1,v2] `2 in W2 )
[v1,v2] `1 = v1 by MCART_1:7;
hence ( v = ([v1,v2] `1) + ([v1,v2] `2) & [v1,v2] `1 in W1 & [v1,v2] `2 in W2 ) by A2, MCART_1:7; ::_thesis: verum
end;
uniqueness
for b1, b2 being Element of [: the carrier of V, the carrier of V:] st v = (b1 `1) + (b1 `2) & b1 `1 in W1 & b1 `2 in W2 & v = (b2 `1) + (b2 `2) & b2 `1 in W1 & b2 `2 in W2 holds
b1 = b2
proof
let t1, t2 be Element of [: the carrier of V, the carrier of V:]; ::_thesis: ( v = (t1 `1) + (t1 `2) & t1 `1 in W1 & t1 `2 in W2 & v = (t2 `1) + (t2 `2) & t2 `1 in W1 & t2 `2 in W2 implies t1 = t2 )
assume ( v = (t1 `1) + (t1 `2) & t1 `1 in W1 & t1 `2 in W2 & v = (t2 `1) + (t2 `2) & t2 `1 in W1 & t2 `2 in W2 ) ; ::_thesis: t1 = t2
then A3: ( t1 `1 = t2 `1 & t1 `2 = t2 `2 ) by A1, Th134;
t1 = [(t1 `1),(t1 `2)] by MCART_1:21;
hence t1 = t2 by A3, MCART_1:21; ::_thesis: verum
end;
end;
:: deftheorem Def20 defines |-- ZMODUL01:def_20_:_
for V being Z_Module
for v being VECTOR of V
for W1, W2 being Submodule of V st V is_the_direct_sum_of W1,W2 holds
for b5 being Element of [: the carrier of V, the carrier of V:] holds
( b5 = v |-- (W1,W2) iff ( v = (b5 `1) + (b5 `2) & b5 `1 in W1 & b5 `2 in W2 ) );
theorem Th136: :: ZMODUL01:136
for V being Z_Module
for W1, W2 being Submodule of V
for v being VECTOR of V st V is_the_direct_sum_of W1,W2 holds
(v |-- (W1,W2)) `1 = (v |-- (W2,W1)) `2
proof
let V be Z_Module; ::_thesis: for W1, W2 being Submodule of V
for v being VECTOR of V st V is_the_direct_sum_of W1,W2 holds
(v |-- (W1,W2)) `1 = (v |-- (W2,W1)) `2
let W1, W2 be Submodule of V; ::_thesis: for v being VECTOR of V st V is_the_direct_sum_of W1,W2 holds
(v |-- (W1,W2)) `1 = (v |-- (W2,W1)) `2
let v be VECTOR of V; ::_thesis: ( V is_the_direct_sum_of W1,W2 implies (v |-- (W1,W2)) `1 = (v |-- (W2,W1)) `2 )
assume A1: V is_the_direct_sum_of W1,W2 ; ::_thesis: (v |-- (W1,W2)) `1 = (v |-- (W2,W1)) `2
then A2: (v |-- (W1,W2)) `2 in W2 by Def20;
A3: V is_the_direct_sum_of W2,W1 by A1, Lm18;
then A4: ( v = ((v |-- (W2,W1)) `2) + ((v |-- (W2,W1)) `1) & (v |-- (W2,W1)) `1 in W2 ) by Def20;
A5: (v |-- (W2,W1)) `2 in W1 by A3, Def20;
( v = ((v |-- (W1,W2)) `1) + ((v |-- (W1,W2)) `2) & (v |-- (W1,W2)) `1 in W1 ) by A1, Def20;
hence (v |-- (W1,W2)) `1 = (v |-- (W2,W1)) `2 by A1, A2, A4, A5, Th134; ::_thesis: verum
end;
theorem Th137: :: ZMODUL01:137
for V being Z_Module
for W1, W2 being Submodule of V
for v being VECTOR of V st V is_the_direct_sum_of W1,W2 holds
(v |-- (W1,W2)) `2 = (v |-- (W2,W1)) `1
proof
let V be Z_Module; ::_thesis: for W1, W2 being Submodule of V
for v being VECTOR of V st V is_the_direct_sum_of W1,W2 holds
(v |-- (W1,W2)) `2 = (v |-- (W2,W1)) `1
let W1, W2 be Submodule of V; ::_thesis: for v being VECTOR of V st V is_the_direct_sum_of W1,W2 holds
(v |-- (W1,W2)) `2 = (v |-- (W2,W1)) `1
let v be VECTOR of V; ::_thesis: ( V is_the_direct_sum_of W1,W2 implies (v |-- (W1,W2)) `2 = (v |-- (W2,W1)) `1 )
assume A1: V is_the_direct_sum_of W1,W2 ; ::_thesis: (v |-- (W1,W2)) `2 = (v |-- (W2,W1)) `1
then A2: (v |-- (W1,W2)) `2 in W2 by Def20;
A3: V is_the_direct_sum_of W2,W1 by A1, Lm18;
then A4: ( v = ((v |-- (W2,W1)) `2) + ((v |-- (W2,W1)) `1) & (v |-- (W2,W1)) `1 in W2 ) by Def20;
A5: (v |-- (W2,W1)) `2 in W1 by A3, Def20;
( v = ((v |-- (W1,W2)) `1) + ((v |-- (W1,W2)) `2) & (v |-- (W1,W2)) `1 in W1 ) by A1, Def20;
hence (v |-- (W1,W2)) `2 = (v |-- (W2,W1)) `1 by A1, A2, A4, A5, Th134; ::_thesis: verum
end;
theorem :: ZMODUL01:138
for V being Z_Module
for W being with_Linear_Compl Submodule of V
for L being Linear_Compl of W
for v being VECTOR of V
for t being Element of [: the carrier of V, the carrier of V:] st (t `1) + (t `2) = v & t `1 in W & t `2 in L holds
t = v |-- (W,L)
proof
let V be Z_Module; ::_thesis: for W being with_Linear_Compl Submodule of V
for L being Linear_Compl of W
for v being VECTOR of V
for t being Element of [: the carrier of V, the carrier of V:] st (t `1) + (t `2) = v & t `1 in W & t `2 in L holds
t = v |-- (W,L)
let W be with_Linear_Compl Submodule of V; ::_thesis: for L being Linear_Compl of W
for v being VECTOR of V
for t being Element of [: the carrier of V, the carrier of V:] st (t `1) + (t `2) = v & t `1 in W & t `2 in L holds
t = v |-- (W,L)
let L be Linear_Compl of W; ::_thesis: for v being VECTOR of V
for t being Element of [: the carrier of V, the carrier of V:] st (t `1) + (t `2) = v & t `1 in W & t `2 in L holds
t = v |-- (W,L)
V is_the_direct_sum_of W,L by Th124;
hence for v being VECTOR of V
for t being Element of [: the carrier of V, the carrier of V:] st (t `1) + (t `2) = v & t `1 in W & t `2 in L holds
t = v |-- (W,L) by Def20; ::_thesis: verum
end;
theorem :: ZMODUL01:139
for V being Z_Module
for W being with_Linear_Compl Submodule of V
for L being Linear_Compl of W
for v being VECTOR of V holds ((v |-- (W,L)) `1) + ((v |-- (W,L)) `2) = v
proof
let V be Z_Module; ::_thesis: for W being with_Linear_Compl Submodule of V
for L being Linear_Compl of W
for v being VECTOR of V holds ((v |-- (W,L)) `1) + ((v |-- (W,L)) `2) = v
let W be with_Linear_Compl Submodule of V; ::_thesis: for L being Linear_Compl of W
for v being VECTOR of V holds ((v |-- (W,L)) `1) + ((v |-- (W,L)) `2) = v
let L be Linear_Compl of W; ::_thesis: for v being VECTOR of V holds ((v |-- (W,L)) `1) + ((v |-- (W,L)) `2) = v
V is_the_direct_sum_of W,L by Th124;
hence for v being VECTOR of V holds ((v |-- (W,L)) `1) + ((v |-- (W,L)) `2) = v by Def20; ::_thesis: verum
end;
theorem :: ZMODUL01:140
for V being Z_Module
for W being with_Linear_Compl Submodule of V
for L being Linear_Compl of W
for v being VECTOR of V holds
( (v |-- (W,L)) `1 in W & (v |-- (W,L)) `2 in L )
proof
let V be Z_Module; ::_thesis: for W being with_Linear_Compl Submodule of V
for L being Linear_Compl of W
for v being VECTOR of V holds
( (v |-- (W,L)) `1 in W & (v |-- (W,L)) `2 in L )
let W be with_Linear_Compl Submodule of V; ::_thesis: for L being Linear_Compl of W
for v being VECTOR of V holds
( (v |-- (W,L)) `1 in W & (v |-- (W,L)) `2 in L )
let L be Linear_Compl of W; ::_thesis: for v being VECTOR of V holds
( (v |-- (W,L)) `1 in W & (v |-- (W,L)) `2 in L )
V is_the_direct_sum_of W,L by Th124;
hence for v being VECTOR of V holds
( (v |-- (W,L)) `1 in W & (v |-- (W,L)) `2 in L ) by Def20; ::_thesis: verum
end;
theorem :: ZMODUL01:141
for V being Z_Module
for W being with_Linear_Compl Submodule of V
for L being Linear_Compl of W
for v being VECTOR of V holds (v |-- (W,L)) `1 = (v |-- (L,W)) `2
proof
let V be Z_Module; ::_thesis: for W being with_Linear_Compl Submodule of V
for L being Linear_Compl of W
for v being VECTOR of V holds (v |-- (W,L)) `1 = (v |-- (L,W)) `2
let W be with_Linear_Compl Submodule of V; ::_thesis: for L being Linear_Compl of W
for v being VECTOR of V holds (v |-- (W,L)) `1 = (v |-- (L,W)) `2
let L be Linear_Compl of W; ::_thesis: for v being VECTOR of V holds (v |-- (W,L)) `1 = (v |-- (L,W)) `2
V is_the_direct_sum_of W,L by Th124;
hence for v being VECTOR of V holds (v |-- (W,L)) `1 = (v |-- (L,W)) `2 by Th136; ::_thesis: verum
end;
theorem :: ZMODUL01:142
for V being Z_Module
for W being with_Linear_Compl Submodule of V
for L being Linear_Compl of W
for v being VECTOR of V holds (v |-- (W,L)) `2 = (v |-- (L,W)) `1
proof
let V be Z_Module; ::_thesis: for W being with_Linear_Compl Submodule of V
for L being Linear_Compl of W
for v being VECTOR of V holds (v |-- (W,L)) `2 = (v |-- (L,W)) `1
let W be with_Linear_Compl Submodule of V; ::_thesis: for L being Linear_Compl of W
for v being VECTOR of V holds (v |-- (W,L)) `2 = (v |-- (L,W)) `1
let L be Linear_Compl of W; ::_thesis: for v being VECTOR of V holds (v |-- (W,L)) `2 = (v |-- (L,W)) `1
V is_the_direct_sum_of W,L by Th124;
hence for v being VECTOR of V holds (v |-- (W,L)) `2 = (v |-- (L,W)) `1 by Th137; ::_thesis: verum
end;
definition
let V be Z_Module;
func SubJoin V -> BinOp of (Submodules V) means :Def21: :: ZMODUL01:def 21
for A1, A2 being Element of Submodules V
for W1, W2 being Submodule of V st A1 = W1 & A2 = W2 holds
it . (A1,A2) = W1 + W2;
existence
ex b1 being BinOp of (Submodules V) st
for A1, A2 being Element of Submodules V
for W1, W2 being Submodule of V st A1 = W1 & A2 = W2 holds
b1 . (A1,A2) = W1 + W2
proof
defpred S1[ Element of Submodules V, Element of Submodules V, Element of Submodules V] means for W1, W2 being Submodule of V st $1 = W1 & $2 = W2 holds
$3 = W1 + W2;
A1: for A1, A2 being Element of Submodules V ex B being Element of Submodules V st S1[A1,A2,B]
proof
let A1, A2 be Element of Submodules V; ::_thesis: ex B being Element of Submodules V st S1[A1,A2,B]
reconsider W1 = A1, W2 = A2 as Submodule of V by Def16;
reconsider C = W1 + W2 as Element of Submodules V by Def16;
take C ; ::_thesis: S1[A1,A2,C]
thus S1[A1,A2,C] ; ::_thesis: verum
end;
ex o being BinOp of (Submodules V) st
for a, b being Element of Submodules V holds S1[a,b,o . (a,b)] from BINOP_1:sch_3(A1);
hence ex b1 being BinOp of (Submodules V) st
for A1, A2 being Element of Submodules V
for W1, W2 being Submodule of V st A1 = W1 & A2 = W2 holds
b1 . (A1,A2) = W1 + W2 ; ::_thesis: verum
end;
uniqueness
for b1, b2 being BinOp of (Submodules V) st ( for A1, A2 being Element of Submodules V
for W1, W2 being Submodule of V st A1 = W1 & A2 = W2 holds
b1 . (A1,A2) = W1 + W2 ) & ( for A1, A2 being Element of Submodules V
for W1, W2 being Submodule of V st A1 = W1 & A2 = W2 holds
b2 . (A1,A2) = W1 + W2 ) holds
b1 = b2
proof
let o1, o2 be BinOp of (Submodules V); ::_thesis: ( ( for A1, A2 being Element of Submodules V
for W1, W2 being Submodule of V st A1 = W1 & A2 = W2 holds
o1 . (A1,A2) = W1 + W2 ) & ( for A1, A2 being Element of Submodules V
for W1, W2 being Submodule of V st A1 = W1 & A2 = W2 holds
o2 . (A1,A2) = W1 + W2 ) implies o1 = o2 )
assume A2: for A1, A2 being Element of Submodules V
for W1, W2 being Submodule of V st A1 = W1 & A2 = W2 holds
o1 . (A1,A2) = W1 + W2 ; ::_thesis: ( ex A1, A2 being Element of Submodules V ex W1, W2 being Submodule of V st
( A1 = W1 & A2 = W2 & not o2 . (A1,A2) = W1 + W2 ) or o1 = o2 )
assume A3: for A1, A2 being Element of Submodules V
for W1, W2 being Submodule of V st A1 = W1 & A2 = W2 holds
o2 . (A1,A2) = W1 + W2 ; ::_thesis: o1 = o2
now__::_thesis:_for_x,_y_being_set_st_x_in_Submodules_V_&_y_in_Submodules_V_holds_
o1_._(x,y)_=_o2_._(x,y)
let x, y be set ; ::_thesis: ( x in Submodules V & y in Submodules V implies o1 . (x,y) = o2 . (x,y) )
assume A4: ( x in Submodules V & y in Submodules V ) ; ::_thesis: o1 . (x,y) = o2 . (x,y)
then reconsider A = x, B = y as Element of Submodules V ;
reconsider W1 = x, W2 = y as Submodule of V by A4, Def16;
o1 . (A,B) = W1 + W2 by A2;
hence o1 . (x,y) = o2 . (x,y) by A3; ::_thesis: verum
end;
hence o1 = o2 by BINOP_1:1; ::_thesis: verum
end;
end;
:: deftheorem Def21 defines SubJoin ZMODUL01:def_21_:_
for V being Z_Module
for b2 being BinOp of (Submodules V) holds
( b2 = SubJoin V iff for A1, A2 being Element of Submodules V
for W1, W2 being Submodule of V st A1 = W1 & A2 = W2 holds
b2 . (A1,A2) = W1 + W2 );
definition
let V be Z_Module;
func SubMeet V -> BinOp of (Submodules V) means :Def22: :: ZMODUL01:def 22
for A1, A2 being Element of Submodules V
for W1, W2 being Submodule of V st A1 = W1 & A2 = W2 holds
it . (A1,A2) = W1 /\ W2;
existence
ex b1 being BinOp of (Submodules V) st
for A1, A2 being Element of Submodules V
for W1, W2 being Submodule of V st A1 = W1 & A2 = W2 holds
b1 . (A1,A2) = W1 /\ W2
proof
defpred S1[ Element of Submodules V, Element of Submodules V, Element of Submodules V] means for W1, W2 being Submodule of V st $1 = W1 & $2 = W2 holds
$3 = W1 /\ W2;
A1: for A1, A2 being Element of Submodules V ex B being Element of Submodules V st S1[A1,A2,B]
proof
let A1, A2 be Element of Submodules V; ::_thesis: ex B being Element of Submodules V st S1[A1,A2,B]
reconsider W1 = A1, W2 = A2 as Submodule of V by Def16;
reconsider C = W1 /\ W2 as Element of Submodules V by Def16;
take C ; ::_thesis: S1[A1,A2,C]
thus S1[A1,A2,C] ; ::_thesis: verum
end;
ex o being BinOp of (Submodules V) st
for a, b being Element of Submodules V holds S1[a,b,o . (a,b)] from BINOP_1:sch_3(A1);
hence ex b1 being BinOp of (Submodules V) st
for A1, A2 being Element of Submodules V
for W1, W2 being Submodule of V st A1 = W1 & A2 = W2 holds
b1 . (A1,A2) = W1 /\ W2 ; ::_thesis: verum
end;
uniqueness
for b1, b2 being BinOp of (Submodules V) st ( for A1, A2 being Element of Submodules V
for W1, W2 being Submodule of V st A1 = W1 & A2 = W2 holds
b1 . (A1,A2) = W1 /\ W2 ) & ( for A1, A2 being Element of Submodules V
for W1, W2 being Submodule of V st A1 = W1 & A2 = W2 holds
b2 . (A1,A2) = W1 /\ W2 ) holds
b1 = b2
proof
let o1, o2 be BinOp of (Submodules V); ::_thesis: ( ( for A1, A2 being Element of Submodules V
for W1, W2 being Submodule of V st A1 = W1 & A2 = W2 holds
o1 . (A1,A2) = W1 /\ W2 ) & ( for A1, A2 being Element of Submodules V
for W1, W2 being Submodule of V st A1 = W1 & A2 = W2 holds
o2 . (A1,A2) = W1 /\ W2 ) implies o1 = o2 )
assume A2: for A1, A2 being Element of Submodules V
for W1, W2 being Submodule of V st A1 = W1 & A2 = W2 holds
o1 . (A1,A2) = W1 /\ W2 ; ::_thesis: ( ex A1, A2 being Element of Submodules V ex W1, W2 being Submodule of V st
( A1 = W1 & A2 = W2 & not o2 . (A1,A2) = W1 /\ W2 ) or o1 = o2 )
assume A3: for A1, A2 being Element of Submodules V
for W1, W2 being Submodule of V st A1 = W1 & A2 = W2 holds
o2 . (A1,A2) = W1 /\ W2 ; ::_thesis: o1 = o2
now__::_thesis:_for_x,_y_being_set_st_x_in_Submodules_V_&_y_in_Submodules_V_holds_
o1_._(x,y)_=_o2_._(x,y)
let x, y be set ; ::_thesis: ( x in Submodules V & y in Submodules V implies o1 . (x,y) = o2 . (x,y) )
assume A4: ( x in Submodules V & y in Submodules V ) ; ::_thesis: o1 . (x,y) = o2 . (x,y)
then reconsider A = x, B = y as Element of Submodules V ;
reconsider W1 = x, W2 = y as Submodule of V by A4, Def16;
o1 . (A,B) = W1 /\ W2 by A2;
hence o1 . (x,y) = o2 . (x,y) by A3; ::_thesis: verum
end;
hence o1 = o2 by BINOP_1:1; ::_thesis: verum
end;
end;
:: deftheorem Def22 defines SubMeet ZMODUL01:def_22_:_
for V being Z_Module
for b2 being BinOp of (Submodules V) holds
( b2 = SubMeet V iff for A1, A2 being Element of Submodules V
for W1, W2 being Submodule of V st A1 = W1 & A2 = W2 holds
b2 . (A1,A2) = W1 /\ W2 );
theorem Th143: :: ZMODUL01:143
for V being Z_Module holds LattStr(# (Submodules V),(SubJoin V),(SubMeet V) #) is Lattice
proof
let V be Z_Module; ::_thesis: LattStr(# (Submodules V),(SubJoin V),(SubMeet V) #) is Lattice
set S = LattStr(# (Submodules V),(SubJoin V),(SubMeet V) #);
A1: for A, B being Element of LattStr(# (Submodules V),(SubJoin V),(SubMeet V) #) holds A "/\" B = B "/\" A
proof
let A, B be Element of LattStr(# (Submodules V),(SubJoin V),(SubMeet V) #); ::_thesis: A "/\" B = B "/\" A
reconsider W1 = A, W2 = B as Submodule of V by Def16;
thus A "/\" B = W1 /\ W2 by Def22
.= B "/\" A by Def22 ; ::_thesis: verum
end;
A2: for A, B being Element of LattStr(# (Submodules V),(SubJoin V),(SubMeet V) #) holds (A "/\" B) "\/" B = B
proof
let A, B be Element of LattStr(# (Submodules V),(SubJoin V),(SubMeet V) #); ::_thesis: (A "/\" B) "\/" B = B
reconsider W1 = A, W2 = B as strict Submodule of V by Def16;
reconsider AB = W1 /\ W2 as Element of LattStr(# (Submodules V),(SubJoin V),(SubMeet V) #) by Def16;
thus (A "/\" B) "\/" B = (SubJoin V) . (AB,B) by Def22
.= (W1 /\ W2) + W2 by Def21
.= B by Lm10, Th45 ; ::_thesis: verum
end;
A3: for A, B, C being Element of LattStr(# (Submodules V),(SubJoin V),(SubMeet V) #) holds A "\/" (B "\/" C) = (A "\/" B) "\/" C
proof
let A, B, C be Element of LattStr(# (Submodules V),(SubJoin V),(SubMeet V) #); ::_thesis: A "\/" (B "\/" C) = (A "\/" B) "\/" C
reconsider W1 = A, W2 = B, W3 = C as Submodule of V by Def16;
reconsider AB = W1 + W2, BC = W2 + W3 as Element of LattStr(# (Submodules V),(SubJoin V),(SubMeet V) #) by Def16;
thus A "\/" (B "\/" C) = (SubJoin V) . (A,BC) by Def21
.= W1 + (W2 + W3) by Def21
.= (W1 + W2) + W3 by Th96
.= (SubJoin V) . (AB,C) by Def21
.= (A "\/" B) "\/" C by Def21 ; ::_thesis: verum
end;
A4: for A, B being Element of LattStr(# (Submodules V),(SubJoin V),(SubMeet V) #) holds A "/\" (A "\/" B) = A
proof
let A, B be Element of LattStr(# (Submodules V),(SubJoin V),(SubMeet V) #); ::_thesis: A "/\" (A "\/" B) = A
reconsider W1 = A, W2 = B as strict Submodule of V by Def16;
reconsider AB = W1 + W2 as Element of LattStr(# (Submodules V),(SubJoin V),(SubMeet V) #) by Def16;
thus A "/\" (A "\/" B) = (SubMeet V) . (A,AB) by Def21
.= W1 /\ (W1 + W2) by Def22
.= A by Lm11, Th45 ; ::_thesis: verum
end;
A5: for A, B, C being Element of LattStr(# (Submodules V),(SubJoin V),(SubMeet V) #) holds A "/\" (B "/\" C) = (A "/\" B) "/\" C
proof
let A, B, C be Element of LattStr(# (Submodules V),(SubJoin V),(SubMeet V) #); ::_thesis: A "/\" (B "/\" C) = (A "/\" B) "/\" C
reconsider W1 = A, W2 = B, W3 = C as Submodule of V by Def16;
reconsider AB = W1 /\ W2, BC = W2 /\ W3 as Element of LattStr(# (Submodules V),(SubJoin V),(SubMeet V) #) by Def16;
thus A "/\" (B "/\" C) = (SubMeet V) . (A,BC) by Def22
.= W1 /\ (W2 /\ W3) by Def22
.= (W1 /\ W2) /\ W3 by Th104
.= (SubMeet V) . (AB,C) by Def22
.= (A "/\" B) "/\" C by Def22 ; ::_thesis: verum
end;
for A, B being Element of LattStr(# (Submodules V),(SubJoin V),(SubMeet V) #) holds A "\/" B = B "\/" A
proof
let A, B be Element of LattStr(# (Submodules V),(SubJoin V),(SubMeet V) #); ::_thesis: A "\/" B = B "\/" A
reconsider W1 = A, W2 = B as Submodule of V by Def16;
thus A "\/" B = W1 + W2 by Def21
.= B "\/" A by Def21 ; ::_thesis: verum
end;
then ( LattStr(# (Submodules V),(SubJoin V),(SubMeet V) #) is join-commutative & LattStr(# (Submodules V),(SubJoin V),(SubMeet V) #) is join-associative & LattStr(# (Submodules V),(SubJoin V),(SubMeet V) #) is meet-absorbing & LattStr(# (Submodules V),(SubJoin V),(SubMeet V) #) is meet-commutative & LattStr(# (Submodules V),(SubJoin V),(SubMeet V) #) is meet-associative & LattStr(# (Submodules V),(SubJoin V),(SubMeet V) #) is join-absorbing ) by A3, A2, A1, A5, A4, LATTICES:def_4, LATTICES:def_5, LATTICES:def_6, LATTICES:def_7, LATTICES:def_8, LATTICES:def_9;
hence LattStr(# (Submodules V),(SubJoin V),(SubMeet V) #) is Lattice ; ::_thesis: verum
end;
registration
let V be Z_Module;
cluster LattStr(# (Submodules V),(SubJoin V),(SubMeet V) #) -> Lattice-like ;
coherence
LattStr(# (Submodules V),(SubJoin V),(SubMeet V) #) is Lattice-like by Th143;
end;
theorem Th144: :: ZMODUL01:144
for V being Z_Module holds LattStr(# (Submodules V),(SubJoin V),(SubMeet V) #) is lower-bounded
proof
let V be Z_Module; ::_thesis: LattStr(# (Submodules V),(SubJoin V),(SubMeet V) #) is lower-bounded
set S = LattStr(# (Submodules V),(SubJoin V),(SubMeet V) #);
ex C being Element of LattStr(# (Submodules V),(SubJoin V),(SubMeet V) #) st
for A being Element of LattStr(# (Submodules V),(SubJoin V),(SubMeet V) #) holds
( C "/\" A = C & A "/\" C = C )
proof
reconsider C = (0). V as Element of LattStr(# (Submodules V),(SubJoin V),(SubMeet V) #) by Def16;
take C ; ::_thesis: for A being Element of LattStr(# (Submodules V),(SubJoin V),(SubMeet V) #) holds
( C "/\" A = C & A "/\" C = C )
let A be Element of LattStr(# (Submodules V),(SubJoin V),(SubMeet V) #); ::_thesis: ( C "/\" A = C & A "/\" C = C )
reconsider W = A as Submodule of V by Def16;
thus C "/\" A = ((0). V) /\ W by Def22
.= C by Th107 ; ::_thesis: A "/\" C = C
hence A "/\" C = C ; ::_thesis: verum
end;
hence LattStr(# (Submodules V),(SubJoin V),(SubMeet V) #) is lower-bounded by LATTICES:def_13; ::_thesis: verum
end;
theorem Th145: :: ZMODUL01:145
for V being Z_Module holds LattStr(# (Submodules V),(SubJoin V),(SubMeet V) #) is upper-bounded
proof
let V be Z_Module; ::_thesis: LattStr(# (Submodules V),(SubJoin V),(SubMeet V) #) is upper-bounded
set S = LattStr(# (Submodules V),(SubJoin V),(SubMeet V) #);
ex C being Element of LattStr(# (Submodules V),(SubJoin V),(SubMeet V) #) st
for A being Element of LattStr(# (Submodules V),(SubJoin V),(SubMeet V) #) holds
( C "\/" A = C & A "\/" C = C )
proof
reconsider C = (Omega). V as Element of LattStr(# (Submodules V),(SubJoin V),(SubMeet V) #) by Def16;
take C ; ::_thesis: for A being Element of LattStr(# (Submodules V),(SubJoin V),(SubMeet V) #) holds
( C "\/" A = C & A "\/" C = C )
let A be Element of LattStr(# (Submodules V),(SubJoin V),(SubMeet V) #); ::_thesis: ( C "\/" A = C & A "\/" C = C )
reconsider W = A as Submodule of V by Def16;
thus C "\/" A = ((Omega). V) + W by Def21
.= C by Th101 ; ::_thesis: A "\/" C = C
hence A "\/" C = C ; ::_thesis: verum
end;
hence LattStr(# (Submodules V),(SubJoin V),(SubMeet V) #) is upper-bounded by LATTICES:def_14; ::_thesis: verum
end;
theorem :: ZMODUL01:146
for V being Z_Module holds LattStr(# (Submodules V),(SubJoin V),(SubMeet V) #) is 01_Lattice
proof
let V be Z_Module; ::_thesis: LattStr(# (Submodules V),(SubJoin V),(SubMeet V) #) is 01_Lattice
LattStr(# (Submodules V),(SubJoin V),(SubMeet V) #) is lower-bounded upper-bounded Lattice by Th144, Th145;
hence LattStr(# (Submodules V),(SubJoin V),(SubMeet V) #) is 01_Lattice ; ::_thesis: verum
end;
theorem :: ZMODUL01:147
for V being Z_Module holds LattStr(# (Submodules V),(SubJoin V),(SubMeet V) #) is modular
proof
let V be Z_Module; ::_thesis: LattStr(# (Submodules V),(SubJoin V),(SubMeet V) #) is modular
set S = LattStr(# (Submodules V),(SubJoin V),(SubMeet V) #);
for A, B, C being Element of LattStr(# (Submodules V),(SubJoin V),(SubMeet V) #) st A [= C holds
A "\/" (B "/\" C) = (A "\/" B) "/\" C
proof
let A, B, C be Element of LattStr(# (Submodules V),(SubJoin V),(SubMeet V) #); ::_thesis: ( A [= C implies A "\/" (B "/\" C) = (A "\/" B) "/\" C )
reconsider W1 = A, W2 = B, W3 = C as strict Submodule of V by Def16;
assume A1: A [= C ; ::_thesis: A "\/" (B "/\" C) = (A "\/" B) "/\" C
reconsider AB = W1 + W2 as Element of LattStr(# (Submodules V),(SubJoin V),(SubMeet V) #) by Def16;
reconsider BC = W2 /\ W3 as Element of LattStr(# (Submodules V),(SubJoin V),(SubMeet V) #) by Def16;
W1 + W3 = A "\/" C by Def21
.= W3 by A1, LATTICES:def_3 ;
then A2: W1 is Submodule of W3 by Th98;
thus A "\/" (B "/\" C) = (SubJoin V) . (A,BC) by Def22
.= W1 + (W2 /\ W3) by Def21
.= (W1 + W2) /\ W3 by A2, Th118
.= (SubMeet V) . (AB,C) by Def22
.= (A "\/" B) "/\" C by Def21 ; ::_thesis: verum
end;
hence LattStr(# (Submodules V),(SubJoin V),(SubMeet V) #) is modular by LATTICES:def_12; ::_thesis: verum
end;
theorem :: ZMODUL01:148
for V being Z_Module
for W1, W2, W3 being strict Submodule of V st W1 is Submodule of W2 holds
W1 /\ W3 is Submodule of W2 /\ W3
proof
let V be Z_Module; ::_thesis: for W1, W2, W3 being strict Submodule of V st W1 is Submodule of W2 holds
W1 /\ W3 is Submodule of W2 /\ W3
let W1, W2, W3 be strict Submodule of V; ::_thesis: ( W1 is Submodule of W2 implies W1 /\ W3 is Submodule of W2 /\ W3 )
set S = LattStr(# (Submodules V),(SubJoin V),(SubMeet V) #);
reconsider A = W1, B = W2, C = W3, AC = W1 /\ W3, BC = W2 /\ W3 as Element of LattStr(# (Submodules V),(SubJoin V),(SubMeet V) #) by Def16;
assume A1: W1 is Submodule of W2 ; ::_thesis: W1 /\ W3 is Submodule of W2 /\ W3
A "\/" B = W1 + W2 by Def21
.= B by A1, Th98 ;
then A [= B by LATTICES:def_3;
then A "/\" C [= B "/\" C by LATTICES:9;
then A2: (A "/\" C) "\/" (B "/\" C) = B "/\" C by LATTICES:def_3;
A3: B "/\" C = W2 /\ W3 by Def22;
(A "/\" C) "\/" (B "/\" C) = (SubJoin V) . (((SubMeet V) . (A,C)),BC) by Def22
.= (SubJoin V) . (AC,BC) by Def22
.= (W1 /\ W3) + (W2 /\ W3) by Def21 ;
hence W1 /\ W3 is Submodule of W2 /\ W3 by A2, A3, Th98; ::_thesis: verum
end;
theorem :: ZMODUL01:149
for V being Z_Module
for W being strict Submodule of V st ( for v being VECTOR of V holds v in W ) holds
W = Z_ModuleStruct(# the carrier of V, the ZeroF of V, the addF of V, the Mult of V #) by Lm16;
theorem :: ZMODUL01:150
for V being Z_Module
for W being Submodule of V
for v being VECTOR of V ex C being Coset of W st v in C by Lm19;
begin
definition
let AG be non empty addLoopStr ;
func Int-mult-left AG -> Function of [:INT, the carrier of AG:], the carrier of AG means :Def23: :: ZMODUL01:def 23
for i being Element of INT
for a being Element of AG holds
( ( i >= 0 implies it . (i,a) = (Nat-mult-left AG) . (i,a) ) & ( i < 0 implies it . (i,a) = (Nat-mult-left AG) . ((- i),(- a)) ) );
existence
ex b1 being Function of [:INT, the carrier of AG:], the carrier of AG st
for i being Element of INT
for a being Element of AG holds
( ( i >= 0 implies b1 . (i,a) = (Nat-mult-left AG) . (i,a) ) & ( i < 0 implies b1 . (i,a) = (Nat-mult-left AG) . ((- i),(- a)) ) )
proof
defpred S1[ Element of INT , Element of AG, Element of AG] means ( ( $1 >= 0 implies $3 = (Nat-mult-left AG) . ($1,$2) ) & ( $1 < 0 implies $3 = (Nat-mult-left AG) . ((- $1),(- $2)) ) );
A1: for x being Element of INT
for y being Element of AG ex z being Element of AG st S1[x,y,z]
proof
let x be Element of INT ; ::_thesis: for y being Element of AG ex z being Element of AG st S1[x,y,z]
let y be Element of AG; ::_thesis: ex z being Element of AG st S1[x,y,z]
percases ( x >= 0 or x < 0 ) ;
suppose x >= 0 ; ::_thesis: ex z being Element of AG st S1[x,y,z]
then reconsider x0 = x as Element of NAT by INT_1:3;
reconsider z = (Nat-mult-left AG) . (x0,y) as Element of AG ;
S1[x,y,z] ;
hence ex z being Element of AG st S1[x,y,z] ; ::_thesis: verum
end;
supposeA2: x < 0 ; ::_thesis: ex z being Element of AG st S1[x,y,z]
then reconsider x0 = - x as Element of NAT by INT_1:3;
reconsider z = (Nat-mult-left AG) . (x0,(- y)) as Element of AG ;
S1[x,y,z] by A2;
hence ex z being Element of AG st S1[x,y,z] ; ::_thesis: verum
end;
end;
end;
consider f being Function of [:INT, the carrier of AG:], the carrier of AG such that
A3: for x being Element of INT
for y being Element of the carrier of AG holds S1[x,y,f . (x,y)] from BINOP_1:sch_3(A1);
take f ; ::_thesis: for i being Element of INT
for a being Element of AG holds
( ( i >= 0 implies f . (i,a) = (Nat-mult-left AG) . (i,a) ) & ( i < 0 implies f . (i,a) = (Nat-mult-left AG) . ((- i),(- a)) ) )
thus for i being Element of INT
for a being Element of AG holds
( ( i >= 0 implies f . (i,a) = (Nat-mult-left AG) . (i,a) ) & ( i < 0 implies f . (i,a) = (Nat-mult-left AG) . ((- i),(- a)) ) ) by A3; ::_thesis: verum
end;
uniqueness
for b1, b2 being Function of [:INT, the carrier of AG:], the carrier of AG st ( for i being Element of INT
for a being Element of AG holds
( ( i >= 0 implies b1 . (i,a) = (Nat-mult-left AG) . (i,a) ) & ( i < 0 implies b1 . (i,a) = (Nat-mult-left AG) . ((- i),(- a)) ) ) ) & ( for i being Element of INT
for a being Element of AG holds
( ( i >= 0 implies b2 . (i,a) = (Nat-mult-left AG) . (i,a) ) & ( i < 0 implies b2 . (i,a) = (Nat-mult-left AG) . ((- i),(- a)) ) ) ) holds
b1 = b2
proof
let f1, f2 be Function of [:INT, the carrier of AG:], the carrier of AG; ::_thesis: ( ( for i being Element of INT
for a being Element of AG holds
( ( i >= 0 implies f1 . (i,a) = (Nat-mult-left AG) . (i,a) ) & ( i < 0 implies f1 . (i,a) = (Nat-mult-left AG) . ((- i),(- a)) ) ) ) & ( for i being Element of INT
for a being Element of AG holds
( ( i >= 0 implies f2 . (i,a) = (Nat-mult-left AG) . (i,a) ) & ( i < 0 implies f2 . (i,a) = (Nat-mult-left AG) . ((- i),(- a)) ) ) ) implies f1 = f2 )
assume A4: for i being Element of INT
for a being Element of AG holds
( ( i >= 0 implies f1 . (i,a) = (Nat-mult-left AG) . (i,a) ) & ( i < 0 implies f1 . (i,a) = (Nat-mult-left AG) . ((- i),(- a)) ) ) ; ::_thesis: ( ex i being Element of INT ex a being Element of AG st
( ( i >= 0 implies f2 . (i,a) = (Nat-mult-left AG) . (i,a) ) implies ( i < 0 & not f2 . (i,a) = (Nat-mult-left AG) . ((- i),(- a)) ) ) or f1 = f2 )
assume A5: for i being Element of INT
for a being Element of AG holds
( ( i >= 0 implies f2 . (i,a) = (Nat-mult-left AG) . (i,a) ) & ( i < 0 implies f2 . (i,a) = (Nat-mult-left AG) . ((- i),(- a)) ) ) ; ::_thesis: f1 = f2
for x, y being set st x in INT & y in the carrier of AG holds
f1 . (x,y) = f2 . (x,y)
proof
let x, y be set ; ::_thesis: ( x in INT & y in the carrier of AG implies f1 . (x,y) = f2 . (x,y) )
assume A6: ( x in INT & y in the carrier of AG ) ; ::_thesis: f1 . (x,y) = f2 . (x,y)
then reconsider x0 = x as Element of INT ;
reconsider y0 = y as Element of AG by A6;
percases ( 0 <= x0 or 0 > x0 ) ;
supposeA7: 0 <= x0 ; ::_thesis: f1 . (x,y) = f2 . (x,y)
hence f1 . (x,y) = (Nat-mult-left AG) . (x0,y0) by A4
.= f2 . (x,y) by A5, A7 ;
::_thesis: verum
end;
supposeA8: 0 > x0 ; ::_thesis: f1 . (x,y) = f2 . (x,y)
hence f1 . (x,y) = (Nat-mult-left AG) . ((- x0),(- y0)) by A4
.= f2 . (x,y) by A5, A8 ;
::_thesis: verum
end;
end;
end;
hence f1 = f2 by BINOP_1:def_21; ::_thesis: verum
end;
end;
:: deftheorem Def23 defines Int-mult-left ZMODUL01:def_23_:_
for AG being non empty addLoopStr
for b2 being Function of [:INT, the carrier of AG:], the carrier of AG holds
( b2 = Int-mult-left AG iff for i being Element of INT
for a being Element of AG holds
( ( i >= 0 implies b2 . (i,a) = (Nat-mult-left AG) . (i,a) ) & ( i < 0 implies b2 . (i,a) = (Nat-mult-left AG) . ((- i),(- a)) ) ) );
theorem :: ZMODUL01:151
for R being non empty addLoopStr
for a being Element of R
for i being Element of INT
for i1 being Element of NAT st i = i1 holds
(Int-mult-left R) . (i,a) = i1 * a by Def23;
theorem Th152: :: ZMODUL01:152
for R being non empty addLoopStr
for a being Element of R
for i being Element of INT st i = 0 holds
(Int-mult-left R) . (i,a) = 0. R
proof
let R be non empty addLoopStr ; ::_thesis: for a being Element of R
for i being Element of INT st i = 0 holds
(Int-mult-left R) . (i,a) = 0. R
let a be Element of R; ::_thesis: for i being Element of INT st i = 0 holds
(Int-mult-left R) . (i,a) = 0. R
let i be Element of INT ; ::_thesis: ( i = 0 implies (Int-mult-left R) . (i,a) = 0. R )
assume i = 0 ; ::_thesis: (Int-mult-left R) . (i,a) = 0. R
hence (Int-mult-left R) . (i,a) = 0 * a by Def23
.= 0. R by BINOM:12 ;
::_thesis: verum
end;
theorem Th153: :: ZMODUL01:153
for R being non empty right_complementable add-associative right_zeroed addLoopStr
for i being Element of NAT holds (Nat-mult-left R) . (i,(0. R)) = 0. R
proof
let R be non empty right_complementable add-associative right_zeroed addLoopStr ; ::_thesis: for i being Element of NAT holds (Nat-mult-left R) . (i,(0. R)) = 0. R
let i be Element of NAT ; ::_thesis: (Nat-mult-left R) . (i,(0. R)) = 0. R
defpred S1[ Element of NAT ] means (Nat-mult-left R) . ($1,(0. R)) = 0. R;
A1: S1[ 0 ] by BINOM:def_3;
A2: for n being Element of NAT st S1[n] holds
S1[n + 1]
proof
let n be Element of NAT ; ::_thesis: ( S1[n] implies S1[n + 1] )
assume A3: S1[n] ; ::_thesis: S1[n + 1]
(Nat-mult-left R) . ((n + 1),(0. R)) = (0. R) + (0. R) by A3, BINOM:def_3
.= 0. R by RLVECT_1:4 ;
hence S1[n + 1] ; ::_thesis: verum
end;
for n being Element of NAT holds S1[n] from NAT_1:sch_1(A1, A2);
hence (Nat-mult-left R) . (i,(0. R)) = 0. R ; ::_thesis: verum
end;
theorem Th154: :: ZMODUL01:154
for R being non empty right_complementable add-associative right_zeroed addLoopStr
for i being Element of INT holds (Int-mult-left R) . (i,(0. R)) = 0. R
proof
let R be non empty right_complementable add-associative right_zeroed addLoopStr ; ::_thesis: for i being Element of INT holds (Int-mult-left R) . (i,(0. R)) = 0. R
let i be Element of INT ; ::_thesis: (Int-mult-left R) . (i,(0. R)) = 0. R
percases ( 0 <= i or 0 > i ) ;
suppose 0 <= i ; ::_thesis: (Int-mult-left R) . (i,(0. R)) = 0. R
then reconsider i1 = i as Element of NAT by INT_1:3;
thus (Int-mult-left R) . (i,(0. R)) = (Nat-mult-left R) . (i1,(0. R)) by Def23
.= 0. R by Th153 ; ::_thesis: verum
end;
supposeA1: 0 > i ; ::_thesis: (Int-mult-left R) . (i,(0. R)) = 0. R
then reconsider i1 = - i as Element of NAT by INT_1:3;
thus (Int-mult-left R) . (i,(0. R)) = (Nat-mult-left R) . (i1,(- (0. R))) by Def23, A1
.= (Nat-mult-left R) . (i1,(0. R)) by RLVECT_1:12
.= 0. R by Th153 ; ::_thesis: verum
end;
end;
end;
theorem Th155: :: ZMODUL01:155
for R being non empty right_zeroed addLoopStr
for a being Element of R
for i being Element of INT st i = 1 holds
(Int-mult-left R) . (i,a) = a
proof
let R be non empty right_zeroed addLoopStr ; ::_thesis: for a being Element of R
for i being Element of INT st i = 1 holds
(Int-mult-left R) . (i,a) = a
let a be Element of R; ::_thesis: for i being Element of INT st i = 1 holds
(Int-mult-left R) . (i,a) = a
let i be Element of INT ; ::_thesis: ( i = 1 implies (Int-mult-left R) . (i,a) = a )
assume i = 1 ; ::_thesis: (Int-mult-left R) . (i,a) = a
hence (Int-mult-left R) . (i,a) = 1 * a by Def23
.= a by BINOM:13 ;
::_thesis: verum
end;
theorem Th156: :: ZMODUL01:156
for R being non empty right_complementable Abelian add-associative right_zeroed addLoopStr
for a being Element of R
for i, j, k being Element of NAT st i <= j & k = j - i holds
(Nat-mult-left R) . (k,a) = ((Nat-mult-left R) . (j,a)) - ((Nat-mult-left R) . (i,a))
proof
let R be non empty right_complementable Abelian add-associative right_zeroed addLoopStr ; ::_thesis: for a being Element of R
for i, j, k being Element of NAT st i <= j & k = j - i holds
(Nat-mult-left R) . (k,a) = ((Nat-mult-left R) . (j,a)) - ((Nat-mult-left R) . (i,a))
let a be Element of R; ::_thesis: for i, j, k being Element of NAT st i <= j & k = j - i holds
(Nat-mult-left R) . (k,a) = ((Nat-mult-left R) . (j,a)) - ((Nat-mult-left R) . (i,a))
let i, j, k be Element of NAT ; ::_thesis: ( i <= j & k = j - i implies (Nat-mult-left R) . (k,a) = ((Nat-mult-left R) . (j,a)) - ((Nat-mult-left R) . (i,a)) )
assume A1: ( i <= j & k = j - i ) ; ::_thesis: (Nat-mult-left R) . (k,a) = ((Nat-mult-left R) . (j,a)) - ((Nat-mult-left R) . (i,a))
A2: j * a = (k + i) * a by A1
.= (k * a) + (i * a) by BINOM:15 ;
thus ((Nat-mult-left R) . (j,a)) - ((Nat-mult-left R) . (i,a)) = (k * a) + ((i * a) - (i * a)) by A2, RLVECT_1:28
.= (k * a) + (0. R) by RLVECT_1:15
.= (Nat-mult-left R) . (k,a) by RLVECT_1:4 ; ::_thesis: verum
end;
theorem Th157: :: ZMODUL01:157
for R being non empty right_complementable Abelian add-associative right_zeroed addLoopStr
for a being Element of R
for i being Element of NAT holds - ((Nat-mult-left R) . (i,a)) = (Nat-mult-left R) . (i,(- a))
proof
let R be non empty right_complementable Abelian add-associative right_zeroed addLoopStr ; ::_thesis: for a being Element of R
for i being Element of NAT holds - ((Nat-mult-left R) . (i,a)) = (Nat-mult-left R) . (i,(- a))
let a be Element of R; ::_thesis: for i being Element of NAT holds - ((Nat-mult-left R) . (i,a)) = (Nat-mult-left R) . (i,(- a))
let i be Element of NAT ; ::_thesis: - ((Nat-mult-left R) . (i,a)) = (Nat-mult-left R) . (i,(- a))
defpred S1[ Element of NAT ] means ((Nat-mult-left R) . ($1,a)) + ((Nat-mult-left R) . ($1,(- a))) = 0. R;
A1: S1[ 0 ]
proof
((Nat-mult-left R) . (0,a)) + ((Nat-mult-left R) . (0,(- a))) = (0. R) + ((Nat-mult-left R) . (0,(- a))) by BINOM:def_3
.= (0. R) + (0. R) by BINOM:def_3
.= 0. R by RLVECT_1:4 ;
hence S1[ 0 ] ; ::_thesis: verum
end;
A2: for n being Element of NAT st S1[n] holds
S1[n + 1]
proof
let n be Element of NAT ; ::_thesis: ( S1[n] implies S1[n + 1] )
assume A3: S1[n] ; ::_thesis: S1[n + 1]
((Nat-mult-left R) . ((n + 1),a)) + ((Nat-mult-left R) . ((n + 1),(- a))) = (a + ((Nat-mult-left R) . (n,a))) + ((Nat-mult-left R) . ((n + 1),(- a))) by BINOM:def_3
.= (a + ((Nat-mult-left R) . (n,a))) + ((- a) + ((Nat-mult-left R) . (n,(- a)))) by BINOM:def_3
.= ((a + ((Nat-mult-left R) . (n,a))) + ((Nat-mult-left R) . (n,(- a)))) + (- a) by RLVECT_1:def_3
.= (a + (((Nat-mult-left R) . (n,a)) + ((Nat-mult-left R) . (n,(- a))))) + (- a) by RLVECT_1:def_3
.= a + (- a) by A3, RLVECT_1:4
.= 0. R by RLVECT_1:5 ;
hence S1[n + 1] ; ::_thesis: verum
end;
for n being Element of NAT holds S1[n] from NAT_1:sch_1(A1, A2);
then ((Nat-mult-left R) . (i,a)) + ((Nat-mult-left R) . (i,(- a))) = 0. R ;
hence - ((Nat-mult-left R) . (i,a)) = (Nat-mult-left R) . (i,(- a)) by RLVECT_1:6; ::_thesis: verum
end;
theorem Th158: :: ZMODUL01:158
for R being non empty right_complementable Abelian add-associative right_zeroed addLoopStr
for a being Element of R
for i, j being Element of INT st i in NAT & not j in NAT holds
(Int-mult-left R) . ((i + j),a) = ((Int-mult-left R) . (i,a)) + ((Int-mult-left R) . (j,a))
proof
let R be non empty right_complementable Abelian add-associative right_zeroed addLoopStr ; ::_thesis: for a being Element of R
for i, j being Element of INT st i in NAT & not j in NAT holds
(Int-mult-left R) . ((i + j),a) = ((Int-mult-left R) . (i,a)) + ((Int-mult-left R) . (j,a))
let a be Element of R; ::_thesis: for i, j being Element of INT st i in NAT & not j in NAT holds
(Int-mult-left R) . ((i + j),a) = ((Int-mult-left R) . (i,a)) + ((Int-mult-left R) . (j,a))
let i, j be Element of INT ; ::_thesis: ( i in NAT & not j in NAT implies (Int-mult-left R) . ((i + j),a) = ((Int-mult-left R) . (i,a)) + ((Int-mult-left R) . (j,a)) )
assume A1: ( i in NAT & not j in NAT ) ; ::_thesis: (Int-mult-left R) . ((i + j),a) = ((Int-mult-left R) . (i,a)) + ((Int-mult-left R) . (j,a))
then reconsider i1 = i as Element of NAT ;
A2: j < 0 by A1, INT_1:3;
then reconsider j1 = - j as Element of NAT by INT_1:3;
A3: i + j is Element of INT by INT_1:def_2;
percases ( j1 <= i1 or j1 > i1 ) ;
supposeA4: j1 <= i1 ; ::_thesis: (Int-mult-left R) . ((i + j),a) = ((Int-mult-left R) . (i,a)) + ((Int-mult-left R) . (j,a))
reconsider k1 = i1 - j1 as Element of NAT by A4, INT_1:5;
thus (Int-mult-left R) . ((i + j),a) = (Nat-mult-left R) . (k1,a) by A3, Def23
.= ((Nat-mult-left R) . (i1,a)) - ((Nat-mult-left R) . (j1,a)) by Th156, A4
.= ((Nat-mult-left R) . (i1,a)) + ((Nat-mult-left R) . (j1,(- a))) by Th157
.= ((Int-mult-left R) . (i,a)) + ((Nat-mult-left R) . (j1,(- a))) by Def23
.= ((Int-mult-left R) . (i,a)) + ((Int-mult-left R) . (j,a)) by A2, Def23 ; ::_thesis: verum
end;
supposeA5: j1 > i1 ; ::_thesis: (Int-mult-left R) . ((i + j),a) = ((Int-mult-left R) . (i,a)) + ((Int-mult-left R) . (j,a))
then reconsider k1 = j1 - i1 as Element of NAT by INT_1:5;
A6: i1 - j1 < 0 by A5, XREAL_1:49;
thus (Int-mult-left R) . ((i + j),a) = (Nat-mult-left R) . ((- (i1 - j1)),(- a)) by A3, A6, Def23
.= (Nat-mult-left R) . (k1,(- a))
.= ((Nat-mult-left R) . (j1,(- a))) - ((Nat-mult-left R) . (i1,(- a))) by Th156, A5
.= ((Nat-mult-left R) . (j1,(- a))) + ((Nat-mult-left R) . (i1,(- (- a)))) by Th157
.= ((Nat-mult-left R) . (j1,(- a))) + ((Nat-mult-left R) . (i1,a)) by RLVECT_1:17
.= ((Int-mult-left R) . (j,a)) + ((Nat-mult-left R) . (i1,a)) by A2, Def23
.= ((Int-mult-left R) . (i,a)) + ((Int-mult-left R) . (j,a)) by Def23 ; ::_thesis: verum
end;
end;
end;
theorem Th159: :: ZMODUL01:159
for R being non empty right_complementable Abelian add-associative right_zeroed addLoopStr
for a being Element of R
for i, j being Element of INT holds (Int-mult-left R) . ((i + j),a) = ((Int-mult-left R) . (i,a)) + ((Int-mult-left R) . (j,a))
proof
let R be non empty right_complementable Abelian add-associative right_zeroed addLoopStr ; ::_thesis: for a being Element of R
for i, j being Element of INT holds (Int-mult-left R) . ((i + j),a) = ((Int-mult-left R) . (i,a)) + ((Int-mult-left R) . (j,a))
let a be Element of R; ::_thesis: for i, j being Element of INT holds (Int-mult-left R) . ((i + j),a) = ((Int-mult-left R) . (i,a)) + ((Int-mult-left R) . (j,a))
let i, j be Element of INT ; ::_thesis: (Int-mult-left R) . ((i + j),a) = ((Int-mult-left R) . (i,a)) + ((Int-mult-left R) . (j,a))
percases ( ( i in NAT & j in NAT ) or ( i in NAT & not j in NAT ) or ( not i in NAT & j in NAT ) or ( not i in NAT & not j in NAT ) ) ;
supposeA1: ( i in NAT & j in NAT ) ; ::_thesis: (Int-mult-left R) . ((i + j),a) = ((Int-mult-left R) . (i,a)) + ((Int-mult-left R) . (j,a))
then reconsider i1 = i as Element of NAT ;
reconsider j1 = j as Element of NAT by A1;
A2: i + j is Element of INT by INT_1:def_2;
thus (Int-mult-left R) . ((i + j),a) = (i1 + j1) * a by A2, Def23
.= (i1 * a) + (j1 * a) by BINOM:15
.= ((Int-mult-left R) . (i,a)) + ((Nat-mult-left R) . (j1,a)) by Def23
.= ((Int-mult-left R) . (i,a)) + ((Int-mult-left R) . (j,a)) by Def23 ; ::_thesis: verum
end;
suppose ( i in NAT & not j in NAT ) ; ::_thesis: (Int-mult-left R) . ((i + j),a) = ((Int-mult-left R) . (i,a)) + ((Int-mult-left R) . (j,a))
hence (Int-mult-left R) . ((i + j),a) = ((Int-mult-left R) . (i,a)) + ((Int-mult-left R) . (j,a)) by Th158; ::_thesis: verum
end;
suppose ( not i in NAT & j in NAT ) ; ::_thesis: (Int-mult-left R) . ((i + j),a) = ((Int-mult-left R) . (i,a)) + ((Int-mult-left R) . (j,a))
hence (Int-mult-left R) . ((i + j),a) = ((Int-mult-left R) . (i,a)) + ((Int-mult-left R) . (j,a)) by Th158; ::_thesis: verum
end;
suppose ( not i in NAT & not j in NAT ) ; ::_thesis: (Int-mult-left R) . ((i + j),a) = ((Int-mult-left R) . (i,a)) + ((Int-mult-left R) . (j,a))
then A3: ( i < 0 & j < 0 ) by INT_1:3;
then reconsider i1 = - i as Element of NAT by INT_1:3;
reconsider j1 = - j as Element of NAT by A3, INT_1:3;
A4: - (i + j) = i1 + j1 ;
A5: i + j is Element of INT by INT_1:def_2;
thus (Int-mult-left R) . ((i + j),a) = (i1 + j1) * (- a) by A3, A4, A5, Def23
.= (i1 * (- a)) + (j1 * (- a)) by BINOM:15
.= ((Int-mult-left R) . (i,a)) + ((Nat-mult-left R) . (j1,(- a))) by A3, Def23
.= ((Int-mult-left R) . (i,a)) + ((Int-mult-left R) . (j,a)) by A3, Def23 ; ::_thesis: verum
end;
end;
end;
theorem Th160: :: ZMODUL01:160
for R being non empty right_complementable Abelian add-associative right_zeroed addLoopStr
for a, b being Element of R
for i being Element of NAT holds (Nat-mult-left R) . (i,(a + b)) = ((Nat-mult-left R) . (i,a)) + ((Nat-mult-left R) . (i,b))
proof
let R be non empty right_complementable Abelian add-associative right_zeroed addLoopStr ; ::_thesis: for a, b being Element of R
for i being Element of NAT holds (Nat-mult-left R) . (i,(a + b)) = ((Nat-mult-left R) . (i,a)) + ((Nat-mult-left R) . (i,b))
let a, b be Element of R; ::_thesis: for i being Element of NAT holds (Nat-mult-left R) . (i,(a + b)) = ((Nat-mult-left R) . (i,a)) + ((Nat-mult-left R) . (i,b))
let i be Element of NAT ; ::_thesis: (Nat-mult-left R) . (i,(a + b)) = ((Nat-mult-left R) . (i,a)) + ((Nat-mult-left R) . (i,b))
defpred S1[ Element of NAT ] means (Nat-mult-left R) . ($1,(a + b)) = ((Nat-mult-left R) . ($1,a)) + ((Nat-mult-left R) . ($1,b));
A1: S1[ 0 ]
proof
(Nat-mult-left R) . (0,(a + b)) = 0. R by BINOM:def_3
.= (0. R) + (0. R) by RLVECT_1:4
.= ((Nat-mult-left R) . (0,a)) + (0. R) by BINOM:def_3
.= ((Nat-mult-left R) . (0,a)) + ((Nat-mult-left R) . (0,b)) by BINOM:def_3 ;
hence S1[ 0 ] ; ::_thesis: verum
end;
A2: for n being Element of NAT st S1[n] holds
S1[n + 1]
proof
let n be Element of NAT ; ::_thesis: ( S1[n] implies S1[n + 1] )
assume A3: S1[n] ; ::_thesis: S1[n + 1]
(Nat-mult-left R) . ((n + 1),(a + b)) = (a + b) + ((Nat-mult-left R) . (n,(a + b))) by BINOM:def_3
.= ((a + b) + ((Nat-mult-left R) . (n,a))) + ((Nat-mult-left R) . (n,b)) by A3, RLVECT_1:def_3
.= ((a + ((Nat-mult-left R) . (n,a))) + b) + ((Nat-mult-left R) . (n,b)) by RLVECT_1:def_3
.= (((Nat-mult-left R) . ((n + 1),a)) + b) + ((Nat-mult-left R) . (n,b)) by BINOM:def_3
.= ((Nat-mult-left R) . ((n + 1),a)) + (b + ((Nat-mult-left R) . (n,b))) by RLVECT_1:def_3
.= ((Nat-mult-left R) . ((n + 1),a)) + ((Nat-mult-left R) . ((n + 1),b)) by BINOM:def_3 ;
hence S1[n + 1] ; ::_thesis: verum
end;
for n being Element of NAT holds S1[n] from NAT_1:sch_1(A1, A2);
hence (Nat-mult-left R) . (i,(a + b)) = ((Nat-mult-left R) . (i,a)) + ((Nat-mult-left R) . (i,b)) ; ::_thesis: verum
end;
theorem Th161: :: ZMODUL01:161
for R being non empty right_complementable Abelian add-associative right_zeroed addLoopStr
for a, b being Element of R
for i being Element of INT holds (Int-mult-left R) . (i,(a + b)) = ((Int-mult-left R) . (i,a)) + ((Int-mult-left R) . (i,b))
proof
let R be non empty right_complementable Abelian add-associative right_zeroed addLoopStr ; ::_thesis: for a, b being Element of R
for i being Element of INT holds (Int-mult-left R) . (i,(a + b)) = ((Int-mult-left R) . (i,a)) + ((Int-mult-left R) . (i,b))
let a, b be Element of R; ::_thesis: for i being Element of INT holds (Int-mult-left R) . (i,(a + b)) = ((Int-mult-left R) . (i,a)) + ((Int-mult-left R) . (i,b))
let i be Element of INT ; ::_thesis: (Int-mult-left R) . (i,(a + b)) = ((Int-mult-left R) . (i,a)) + ((Int-mult-left R) . (i,b))
percases ( 0 <= i or 0 > i ) ;
suppose 0 <= i ; ::_thesis: (Int-mult-left R) . (i,(a + b)) = ((Int-mult-left R) . (i,a)) + ((Int-mult-left R) . (i,b))
then reconsider i1 = i as Element of NAT by INT_1:3;
thus (Int-mult-left R) . (i,(a + b)) = (Nat-mult-left R) . (i1,(a + b)) by Def23
.= (i1 * a) + (i1 * b) by Th160
.= ((Int-mult-left R) . (i,a)) + ((Nat-mult-left R) . (i1,b)) by Def23
.= ((Int-mult-left R) . (i,a)) + ((Int-mult-left R) . (i,b)) by Def23 ; ::_thesis: verum
end;
supposeA1: 0 > i ; ::_thesis: (Int-mult-left R) . (i,(a + b)) = ((Int-mult-left R) . (i,a)) + ((Int-mult-left R) . (i,b))
then reconsider i1 = - i as Element of NAT by INT_1:3;
(a + b) + ((- a) + (- b)) = ((b + a) + (- a)) + (- b) by RLVECT_1:def_3
.= (b + (a + (- a))) + (- b) by RLVECT_1:def_3
.= (b + (0. R)) + (- b) by RLVECT_1:5
.= b + (- b) by RLVECT_1:4
.= 0. R by RLVECT_1:5 ;
then A2: - (a + b) = (- a) + (- b) by RLVECT_1:6;
thus (Int-mult-left R) . (i,(a + b)) = (Nat-mult-left R) . (i1,(- (a + b))) by Def23, A1
.= (i1 * (- a)) + (i1 * (- b)) by A2, Th160
.= ((Int-mult-left R) . (i,a)) + ((Nat-mult-left R) . (i1,(- b))) by A1, Def23
.= ((Int-mult-left R) . (i,a)) + ((Int-mult-left R) . (i,b)) by A1, Def23 ; ::_thesis: verum
end;
end;
end;
theorem Th162: :: ZMODUL01:162
for R being non empty right_complementable Abelian add-associative right_zeroed addLoopStr
for a being Element of R
for i, j being Element of NAT holds (Nat-mult-left R) . ((i * j),a) = (Nat-mult-left R) . (i,((Nat-mult-left R) . (j,a)))
proof
let R be non empty right_complementable Abelian add-associative right_zeroed addLoopStr ; ::_thesis: for a being Element of R
for i, j being Element of NAT holds (Nat-mult-left R) . ((i * j),a) = (Nat-mult-left R) . (i,((Nat-mult-left R) . (j,a)))
let a be Element of R; ::_thesis: for i, j being Element of NAT holds (Nat-mult-left R) . ((i * j),a) = (Nat-mult-left R) . (i,((Nat-mult-left R) . (j,a)))
let i, j be Element of NAT ; ::_thesis: (Nat-mult-left R) . ((i * j),a) = (Nat-mult-left R) . (i,((Nat-mult-left R) . (j,a)))
defpred S1[ Element of NAT ] means (Nat-mult-left R) . (($1 * j),a) = (Nat-mult-left R) . ($1,((Nat-mult-left R) . (j,a)));
A1: S1[ 0 ]
proof
(Nat-mult-left R) . ((0 * j),a) = 0. R by BINOM:def_3
.= (Nat-mult-left R) . (0,((Nat-mult-left R) . (j,a))) by BINOM:def_3 ;
hence S1[ 0 ] ; ::_thesis: verum
end;
A2: for n being Element of NAT st S1[n] holds
S1[n + 1]
proof
let n be Element of NAT ; ::_thesis: ( S1[n] implies S1[n + 1] )
assume A3: S1[n] ; ::_thesis: S1[n + 1]
(Nat-mult-left R) . (((n + 1) * j),a) = (j + (n * j)) * a
.= (j * a) + ((n * j) * a) by BINOM:15
.= (Nat-mult-left R) . ((n + 1),(j * a)) by A3, BINOM:def_3 ;
hence S1[n + 1] ; ::_thesis: verum
end;
for n being Element of NAT holds S1[n] from NAT_1:sch_1(A1, A2);
hence (Nat-mult-left R) . ((i * j),a) = (Nat-mult-left R) . (i,((Nat-mult-left R) . (j,a))) ; ::_thesis: verum
end;
Lm20: for R being non empty right_complementable Abelian add-associative right_zeroed addLoopStr
for a being Element of R
for i, j being Element of INT st i <> 0 & j <> 0 holds
(Int-mult-left R) . ((i * j),a) = (Int-mult-left R) . (i,((Int-mult-left R) . (j,a)))
proof
let R be non empty right_complementable Abelian add-associative right_zeroed addLoopStr ; ::_thesis: for a being Element of R
for i, j being Element of INT st i <> 0 & j <> 0 holds
(Int-mult-left R) . ((i * j),a) = (Int-mult-left R) . (i,((Int-mult-left R) . (j,a)))
let a be Element of R; ::_thesis: for i, j being Element of INT st i <> 0 & j <> 0 holds
(Int-mult-left R) . ((i * j),a) = (Int-mult-left R) . (i,((Int-mult-left R) . (j,a)))
let i, j be Element of INT ; ::_thesis: ( i <> 0 & j <> 0 implies (Int-mult-left R) . ((i * j),a) = (Int-mult-left R) . (i,((Int-mult-left R) . (j,a))) )
assume A1: ( i <> 0 & j <> 0 ) ; ::_thesis: (Int-mult-left R) . ((i * j),a) = (Int-mult-left R) . (i,((Int-mult-left R) . (j,a)))
percases ( ( i in NAT & j in NAT ) or ( i in NAT & not j in NAT ) or ( not i in NAT & j in NAT ) or ( not i in NAT & not j in NAT ) ) ;
supposeA2: ( i in NAT & j in NAT ) ; ::_thesis: (Int-mult-left R) . ((i * j),a) = (Int-mult-left R) . (i,((Int-mult-left R) . (j,a)))
then reconsider i1 = i as Element of NAT ;
reconsider j1 = j as Element of NAT by A2;
A3: i * j is Element of INT by INT_1:def_2;
thus (Int-mult-left R) . ((i * j),a) = (Nat-mult-left R) . ((i1 * j1),a) by A3, Def23
.= (Nat-mult-left R) . (i1,((Nat-mult-left R) . (j1,a))) by Th162
.= (Nat-mult-left R) . (i1,((Int-mult-left R) . (j,a))) by Def23
.= (Int-mult-left R) . (i,((Int-mult-left R) . (j,a))) by Def23 ; ::_thesis: verum
end;
supposeA4: ( i in NAT & not j in NAT ) ; ::_thesis: (Int-mult-left R) . ((i * j),a) = (Int-mult-left R) . (i,((Int-mult-left R) . (j,a)))
then A5: ( 0 < i & j < 0 ) by A1, INT_1:3;
reconsider i1 = i as Element of NAT by A4;
reconsider j1 = - j as Element of NAT by A5, INT_1:3;
A6: - (i * j) = i1 * j1 ;
A7: j * i < 0 * i by A5, XREAL_1:68;
i * j is Element of INT by INT_1:def_2;
hence (Int-mult-left R) . ((i * j),a) = (Nat-mult-left R) . ((i1 * j1),(- a)) by A7, A6, Def23
.= (Nat-mult-left R) . (i1,((Nat-mult-left R) . (j1,(- a)))) by Th162
.= (Nat-mult-left R) . (i1,((Int-mult-left R) . (j,a))) by Def23, A5
.= (Int-mult-left R) . (i,((Int-mult-left R) . (j,a))) by Def23 ;
::_thesis: verum
end;
supposeA8: ( not i in NAT & j in NAT ) ; ::_thesis: (Int-mult-left R) . ((i * j),a) = (Int-mult-left R) . (i,((Int-mult-left R) . (j,a)))
then A9: ( 0 < j & i < 0 ) by A1, INT_1:3;
then reconsider i1 = - i as Element of NAT by INT_1:3;
reconsider j1 = j as Element of NAT by A8;
A10: - (i * j) = i1 * j1 ;
A11: i * j < 0 * j by A9, XREAL_1:68;
A12: i * j is Element of INT by INT_1:def_2;
thus (Int-mult-left R) . ((i * j),a) = (Nat-mult-left R) . ((i1 * j1),(- a)) by A11, A10, A12, Def23
.= (Nat-mult-left R) . (i1,((Nat-mult-left R) . (j1,(- a)))) by Th162
.= (Nat-mult-left R) . (i1,(- ((Nat-mult-left R) . (j1,a)))) by Th157
.= (Nat-mult-left R) . (i1,(- ((Int-mult-left R) . (j,a)))) by Def23
.= (Int-mult-left R) . (i,((Int-mult-left R) . (j,a))) by A9, Def23 ; ::_thesis: verum
end;
suppose ( not i in NAT & not j in NAT ) ; ::_thesis: (Int-mult-left R) . ((i * j),a) = (Int-mult-left R) . (i,((Int-mult-left R) . (j,a)))
then A13: ( i < 0 & j < 0 ) by INT_1:3;
then reconsider i1 = - i as Element of NAT by INT_1:3;
reconsider j1 = - j as Element of NAT by A13, INT_1:3;
A14: i * j is Element of INT by INT_1:def_2;
- ((Nat-mult-left R) . (j1,a)) = (Nat-mult-left R) . (j1,(- a)) by Th157;
then ((Nat-mult-left R) . (j1,(- a))) + ((Nat-mult-left R) . (j1,a)) = 0. R by RLVECT_1:def_10;
then A15: (Nat-mult-left R) . (j1,a) = - ((Nat-mult-left R) . (j1,(- a))) by RLVECT_1:def_10;
thus (Int-mult-left R) . ((i * j),a) = (Nat-mult-left R) . ((i1 * j1),a) by A14, Def23
.= (Nat-mult-left R) . (i1,((Nat-mult-left R) . (j1,a))) by Th162
.= (Nat-mult-left R) . (i1,(- ((Int-mult-left R) . (j,a)))) by A13, A15, Def23
.= (Int-mult-left R) . (i,((Int-mult-left R) . (j,a))) by A13, Def23 ; ::_thesis: verum
end;
end;
end;
Lm21: for R being non empty right_complementable Abelian add-associative right_zeroed addLoopStr
for a being Element of R
for i, j being Element of INT st ( i = 0 or j = 0 ) holds
(Int-mult-left R) . ((i * j),a) = (Int-mult-left R) . (i,((Int-mult-left R) . (j,a)))
proof
let R be non empty right_complementable Abelian add-associative right_zeroed addLoopStr ; ::_thesis: for a being Element of R
for i, j being Element of INT st ( i = 0 or j = 0 ) holds
(Int-mult-left R) . ((i * j),a) = (Int-mult-left R) . (i,((Int-mult-left R) . (j,a)))
let a be Element of R; ::_thesis: for i, j being Element of INT st ( i = 0 or j = 0 ) holds
(Int-mult-left R) . ((i * j),a) = (Int-mult-left R) . (i,((Int-mult-left R) . (j,a)))
let i, j be Element of INT ; ::_thesis: ( ( i = 0 or j = 0 ) implies (Int-mult-left R) . ((i * j),a) = (Int-mult-left R) . (i,((Int-mult-left R) . (j,a))) )
assume A1: ( i = 0 or j = 0 ) ; ::_thesis: (Int-mult-left R) . ((i * j),a) = (Int-mult-left R) . (i,((Int-mult-left R) . (j,a)))
percases ( i = 0 or j = 0 ) by A1;
supposeA2: i = 0 ; ::_thesis: (Int-mult-left R) . ((i * j),a) = (Int-mult-left R) . (i,((Int-mult-left R) . (j,a)))
hence (Int-mult-left R) . ((i * j),a) = 0. R by Th152
.= (Int-mult-left R) . (i,((Int-mult-left R) . (j,a))) by A2, Th152 ;
::_thesis: verum
end;
supposeA3: j = 0 ; ::_thesis: (Int-mult-left R) . ((i * j),a) = (Int-mult-left R) . (i,((Int-mult-left R) . (j,a)))
hence (Int-mult-left R) . ((i * j),a) = 0. R by Th152
.= (Int-mult-left R) . (i,(0. R)) by Th154
.= (Int-mult-left R) . (i,((Int-mult-left R) . (j,a))) by Th152, A3 ;
::_thesis: verum
end;
end;
end;
theorem Th163: :: ZMODUL01:163
for R being non empty right_complementable Abelian add-associative right_zeroed addLoopStr
for a being Element of R
for i, j being Element of INT holds (Int-mult-left R) . ((i * j),a) = (Int-mult-left R) . (i,((Int-mult-left R) . (j,a)))
proof
let R be non empty right_complementable Abelian add-associative right_zeroed addLoopStr ; ::_thesis: for a being Element of R
for i, j being Element of INT holds (Int-mult-left R) . ((i * j),a) = (Int-mult-left R) . (i,((Int-mult-left R) . (j,a)))
let a be Element of R; ::_thesis: for i, j being Element of INT holds (Int-mult-left R) . ((i * j),a) = (Int-mult-left R) . (i,((Int-mult-left R) . (j,a)))
let i, j be Element of INT ; ::_thesis: (Int-mult-left R) . ((i * j),a) = (Int-mult-left R) . (i,((Int-mult-left R) . (j,a)))
percases ( i = 0 or j = 0 or ( not i = 0 & not j = 0 ) ) ;
suppose ( i = 0 or j = 0 ) ; ::_thesis: (Int-mult-left R) . ((i * j),a) = (Int-mult-left R) . (i,((Int-mult-left R) . (j,a)))
hence (Int-mult-left R) . ((i * j),a) = (Int-mult-left R) . (i,((Int-mult-left R) . (j,a))) by Lm21; ::_thesis: verum
end;
suppose ( not i = 0 & not j = 0 ) ; ::_thesis: (Int-mult-left R) . ((i * j),a) = (Int-mult-left R) . (i,((Int-mult-left R) . (j,a)))
hence (Int-mult-left R) . ((i * j),a) = (Int-mult-left R) . (i,((Int-mult-left R) . (j,a))) by Lm20; ::_thesis: verum
end;
end;
end;
theorem :: ZMODUL01:164
for AG being non empty right_complementable Abelian add-associative right_zeroed addLoopStr holds Z_ModuleStruct(# the carrier of AG, the ZeroF of AG, the addF of AG,(Int-mult-left AG) #) is Z_Module
proof
let AG be non empty right_complementable Abelian add-associative right_zeroed addLoopStr ; ::_thesis: Z_ModuleStruct(# the carrier of AG, the ZeroF of AG, the addF of AG,(Int-mult-left AG) #) is Z_Module
reconsider ZS = Z_ModuleStruct(# the carrier of AG, the ZeroF of AG, the addF of AG,(Int-mult-left AG) #) as non empty Z_ModuleStruct ;
set ML = the Mult of ZS;
set AD = the addF of ZS;
set CA = the carrier of ZS;
set Z0 = the ZeroF of ZS;
set MLI = Int-mult-left AG;
A1: for v, w being Element of ZS holds v + w = w + v
proof
let v, w be Element of ZS; ::_thesis: v + w = w + v
reconsider v1 = v, w1 = w as Element of AG ;
thus v + w = v1 + w1
.= w1 + v1
.= w + v ; ::_thesis: verum
end;
A2: for u, v, w being Element of ZS holds (u + v) + w = u + (v + w)
proof
let u, v, w be Element of ZS; ::_thesis: (u + v) + w = u + (v + w)
reconsider u1 = u, v1 = v, w1 = w as Element of AG ;
thus (u + v) + w = (u1 + v1) + w1
.= u1 + (v1 + w1) by RLVECT_1:def_3
.= u + (v + w) ; ::_thesis: verum
end;
A3: for v being Element of ZS holds v + (0. ZS) = v
proof
let v be VECTOR of ZS; ::_thesis: v + (0. ZS) = v
reconsider v1 = v as Element of AG ;
thus v + (0. ZS) = v1 + (0. AG)
.= v by RLVECT_1:def_4 ; ::_thesis: verum
end;
A4: now__::_thesis:_for_v_being_VECTOR_of_ZS_holds_v_is_right_complementable
let v be VECTOR of ZS; ::_thesis: v is right_complementable
reconsider v1 = v as Element of AG ;
consider w1 being Element of AG such that
A5: v1 + w1 = 0. AG by ALGSTR_0:def_11;
reconsider w = w1 as Element of ZS ;
v + w = 0. ZS by A5;
hence v is right_complementable by ALGSTR_0:def_11; ::_thesis: verum
end;
A6: for a, b being integer number
for v being VECTOR of ZS holds (a + b) * v = (a * v) + (b * v)
proof
let a, b be integer number ; ::_thesis: for v being VECTOR of ZS holds (a + b) * v = (a * v) + (b * v)
let v be VECTOR of ZS; ::_thesis: (a + b) * v = (a * v) + (b * v)
reconsider a1 = a, b1 = b as Element of INT by INT_1:def_2;
reconsider v1 = v as Element of AG ;
thus (a + b) * v = ((Int-mult-left AG) . (a1,v1)) + ((Int-mult-left AG) . (b1,v1)) by Th159
.= (a * v) + (b * v) ; ::_thesis: verum
end;
A7: for a being integer number
for v, w being VECTOR of ZS holds a * (v + w) = (a * v) + (a * w)
proof
let a be integer number ; ::_thesis: for v, w being VECTOR of ZS holds a * (v + w) = (a * v) + (a * w)
let v, w be VECTOR of ZS; ::_thesis: a * (v + w) = (a * v) + (a * w)
reconsider a1 = a as Element of INT by INT_1:def_2;
reconsider v1 = v, w1 = w as Element of AG ;
thus a * (v + w) = (Int-mult-left AG) . (a1,(v1 + w1))
.= ((Int-mult-left AG) . (a1,v1)) + ((Int-mult-left AG) . (a1,w1)) by Th161
.= (a * v) + (a * w) ; ::_thesis: verum
end;
A8: for a, b being integer number
for v being VECTOR of ZS holds (a * b) * v = a * (b * v)
proof
let a, b be integer number ; ::_thesis: for v being VECTOR of ZS holds (a * b) * v = a * (b * v)
let v be VECTOR of ZS; ::_thesis: (a * b) * v = a * (b * v)
reconsider a1 = a, b1 = b as Element of INT by INT_1:def_2;
reconsider v1 = v as Element of AG ;
thus (a * b) * v = (Int-mult-left AG) . (a1,((Int-mult-left AG) . (b1,v1))) by Th163
.= a * (b * v) ; ::_thesis: verum
end;
for v being VECTOR of ZS holds 1 * v = v
proof
let v be VECTOR of ZS; ::_thesis: 1 * v = v
reconsider i = 1 as Element of INT by INT_1:def_2;
reconsider v1 = v as Element of AG ;
thus 1 * v = (Int-mult-left AG) . (i,v1)
.= v by Th155 ; ::_thesis: verum
end;
hence Z_ModuleStruct(# the carrier of AG, the ZeroF of AG, the addF of AG,(Int-mult-left AG) #) is Z_Module by A1, A2, A3, A4, A6, A7, A8, Def2, Def3, Def4, Def5, ALGSTR_0:def_16, RLVECT_1:def_2, RLVECT_1:def_3, RLVECT_1:def_4; ::_thesis: verum
end;