:: IDEAL_1 semantic presentation
begin
registration
cluster non empty add-associative right_zeroed left_zeroed for addLoopStr ;
existence
ex b1 being non empty addLoopStr st
( b1 is add-associative & b1 is left_zeroed & b1 is right_zeroed )
proof
set R = the non degenerated comRing;
take the non degenerated comRing ; ::_thesis: ( the non degenerated comRing is add-associative & the non degenerated comRing is left_zeroed & the non degenerated comRing is right_zeroed )
thus ( the non degenerated comRing is add-associative & the non degenerated comRing is left_zeroed & the non degenerated comRing is right_zeroed ) ; ::_thesis: verum
end;
end;
registration
cluster non empty non trivial add-cancelable Abelian add-associative right_zeroed associative commutative well-unital distributive left_zeroed for doubleLoopStr ;
existence
ex b1 being non trivial doubleLoopStr st
( b1 is Abelian & b1 is left_zeroed & b1 is right_zeroed & b1 is add-cancelable & b1 is well-unital & b1 is add-associative & b1 is associative & b1 is commutative & b1 is distributive )
proof
set R = the non degenerated comRing;
take the non degenerated comRing ; ::_thesis: ( the non degenerated comRing is Abelian & the non degenerated comRing is left_zeroed & the non degenerated comRing is right_zeroed & the non degenerated comRing is add-cancelable & the non degenerated comRing is well-unital & the non degenerated comRing is add-associative & the non degenerated comRing is associative & the non degenerated comRing is commutative & the non degenerated comRing is distributive )
thus ( the non degenerated comRing is Abelian & the non degenerated comRing is left_zeroed & the non degenerated comRing is right_zeroed & the non degenerated comRing is add-cancelable & the non degenerated comRing is well-unital & the non degenerated comRing is add-associative & the non degenerated comRing is associative & the non degenerated comRing is commutative & the non degenerated comRing is distributive ) ; ::_thesis: verum
end;
end;
theorem Th1: :: IDEAL_1:1
for V being non empty add-associative right_zeroed left_zeroed addLoopStr
for v, u being Element of V holds Sum <*v,u*> = v + u
proof
let V be non empty add-associative right_zeroed left_zeroed addLoopStr ; ::_thesis: for v, u being Element of V holds Sum <*v,u*> = v + u
let v, u be Element of V; ::_thesis: Sum <*v,u*> = v + u
<*v,u*> = <*v*> ^ <*u*> by FINSEQ_1:def_9;
then Sum <*v,u*> = (Sum <*v*>) + (Sum <*u*>) by RLVECT_1:41
.= v + (Sum <*u*>) by BINOM:3
.= v + u by BINOM:3 ;
hence Sum <*v,u*> = v + u ; ::_thesis: verum
end;
begin
definition
let L be non empty addLoopStr ;
let F be Subset of L;
attrF is add-closed means :Def1: :: IDEAL_1:def 1
for x, y being Element of L st x in F & y in F holds
x + y in F;
end;
:: deftheorem Def1 defines add-closed IDEAL_1:def_1_:_
for L being non empty addLoopStr
for F being Subset of L holds
( F is add-closed iff for x, y being Element of L st x in F & y in F holds
x + y in F );
definition
let L be non empty multMagma ;
let F be Subset of L;
attrF is left-ideal means :Def2: :: IDEAL_1:def 2
for p, x being Element of L st x in F holds
p * x in F;
attrF is right-ideal means :Def3: :: IDEAL_1:def 3
for p, x being Element of L st x in F holds
x * p in F;
end;
:: deftheorem Def2 defines left-ideal IDEAL_1:def_2_:_
for L being non empty multMagma
for F being Subset of L holds
( F is left-ideal iff for p, x being Element of L st x in F holds
p * x in F );
:: deftheorem Def3 defines right-ideal IDEAL_1:def_3_:_
for L being non empty multMagma
for F being Subset of L holds
( F is right-ideal iff for p, x being Element of L st x in F holds
x * p in F );
registration
let L be non empty addLoopStr ;
cluster non empty add-closed for Element of bool the carrier of L;
existence
ex b1 being non empty Subset of L st b1 is add-closed
proof
set M = the carrier of L;
for u being set st u in the carrier of L holds
u in the carrier of L ;
then reconsider M = the carrier of L as Subset of L by TARSKI:def_3;
reconsider M = M as non empty Subset of L ;
take M ; ::_thesis: M is add-closed
for x, y being Element of L st x in M & y in M holds
x + y in M ;
hence M is add-closed by Def1; ::_thesis: verum
end;
end;
registration
let L be non empty multMagma ;
cluster non empty left-ideal for Element of bool the carrier of L;
existence
ex b1 being non empty Subset of L st b1 is left-ideal
proof
set M = the carrier of L;
for u being set st u in the carrier of L holds
u in the carrier of L ;
then reconsider M = the carrier of L as Subset of L by TARSKI:def_3;
reconsider M = M as non empty Subset of L ;
take M ; ::_thesis: M is left-ideal
for p, x being Element of L st x in M holds
p * x in M ;
hence M is left-ideal by Def2; ::_thesis: verum
end;
cluster non empty right-ideal for Element of bool the carrier of L;
existence
ex b1 being non empty Subset of L st b1 is right-ideal
proof
set M = the carrier of L;
for u being set st u in the carrier of L holds
u in the carrier of L ;
then reconsider M = the carrier of L as Subset of L by TARSKI:def_3;
reconsider M = M as non empty Subset of L ;
take M ; ::_thesis: M is right-ideal
for p, x being Element of L st x in M holds
x * p in M ;
hence M is right-ideal by Def3; ::_thesis: verum
end;
end;
registration
let L be non empty doubleLoopStr ;
cluster non empty add-closed left-ideal right-ideal for Element of bool the carrier of L;
existence
ex b1 being non empty Subset of L st
( b1 is add-closed & b1 is left-ideal & b1 is right-ideal )
proof
set M = the carrier of L;
for u being set st u in the carrier of L holds
u in the carrier of L ;
then reconsider M = the carrier of L as Subset of L by TARSKI:def_3;
reconsider M = M as non empty Subset of L ;
take M ; ::_thesis: ( M is add-closed & M is left-ideal & M is right-ideal )
A1: for p, x being Element of L st x in M holds
x * p in M ;
( ( for x, y being Element of L st x in M & y in M holds
x + y in M ) & ( for p, x being Element of L st x in M holds
p * x in M ) ) ;
hence ( M is add-closed & M is left-ideal & M is right-ideal ) by A1, Def1, Def2, Def3; ::_thesis: verum
end;
cluster non empty add-closed right-ideal for Element of bool the carrier of L;
existence
ex b1 being non empty Subset of L st
( b1 is add-closed & b1 is right-ideal )
proof
set M = the carrier of L;
for u being set st u in the carrier of L holds
u in the carrier of L ;
then reconsider M = the carrier of L as Subset of L by TARSKI:def_3;
reconsider M = M as non empty Subset of L ;
take M ; ::_thesis: ( M is add-closed & M is right-ideal )
( ( for x, y being Element of L st x in M & y in M holds
x + y in M ) & ( for p, x being Element of L st x in M holds
x * p in M ) ) ;
hence ( M is add-closed & M is right-ideal ) by Def1, Def3; ::_thesis: verum
end;
cluster non empty add-closed left-ideal for Element of bool the carrier of L;
existence
ex b1 being non empty Subset of L st
( b1 is add-closed & b1 is left-ideal )
proof
set M = the carrier of L;
for u being set st u in the carrier of L holds
u in the carrier of L ;
then reconsider M = the carrier of L as Subset of L by TARSKI:def_3;
reconsider M = M as non empty Subset of L ;
take M ; ::_thesis: ( M is add-closed & M is left-ideal )
( ( for x, y being Element of L st x in M & y in M holds
x + y in M ) & ( for p, x being Element of L st x in M holds
p * x in M ) ) ;
hence ( M is add-closed & M is left-ideal ) by Def1, Def2; ::_thesis: verum
end;
end;
registration
let R be non empty commutative multMagma ;
cluster non empty left-ideal -> non empty right-ideal for Element of bool the carrier of R;
coherence
for b1 being non empty Subset of R st b1 is left-ideal holds
b1 is right-ideal
proof
let I be non empty Subset of R; ::_thesis: ( I is left-ideal implies I is right-ideal )
assume I is left-ideal ; ::_thesis: I is right-ideal
then for p, x being Element of R st x in I holds
x * p in I by Def2;
hence I is right-ideal by Def3; ::_thesis: verum
end;
cluster non empty right-ideal -> non empty left-ideal for Element of bool the carrier of R;
coherence
for b1 being non empty Subset of R st b1 is right-ideal holds
b1 is left-ideal
proof
let I be non empty Subset of R; ::_thesis: ( I is right-ideal implies I is left-ideal )
assume I is right-ideal ; ::_thesis: I is left-ideal
then for p, x being Element of R st x in I holds
p * x in I by Def3;
hence I is left-ideal by Def2; ::_thesis: verum
end;
end;
definition
let L be non empty doubleLoopStr ;
mode Ideal of L is non empty add-closed left-ideal right-ideal Subset of L;
end;
definition
let L be non empty doubleLoopStr ;
mode RightIdeal of L is non empty add-closed right-ideal Subset of L;
end;
definition
let L be non empty doubleLoopStr ;
mode LeftIdeal of L is non empty add-closed left-ideal Subset of L;
end;
theorem Th2: :: IDEAL_1:2
for R being non empty left_add-cancelable right_zeroed left-distributive doubleLoopStr
for I being non empty left-ideal Subset of R holds 0. R in I
proof
let R be non empty left_add-cancelable right_zeroed left-distributive doubleLoopStr ; ::_thesis: for I being non empty left-ideal Subset of R holds 0. R in I
let I be non empty left-ideal Subset of R; ::_thesis: 0. R in I
set a = the Element of I;
(0. R) * the Element of I in I by Def2;
hence 0. R in I by BINOM:1; ::_thesis: verum
end;
theorem Th3: :: IDEAL_1:3
for R being non empty right_add-cancelable right-distributive left_zeroed doubleLoopStr
for I being non empty right-ideal Subset of R holds 0. R in I
proof
let R be non empty right_add-cancelable right-distributive left_zeroed doubleLoopStr ; ::_thesis: for I being non empty right-ideal Subset of R holds 0. R in I
let I be non empty right-ideal Subset of R; ::_thesis: 0. R in I
set a = the Element of I;
the Element of I * (0. R) in I by Def3;
hence 0. R in I by BINOM:2; ::_thesis: verum
end;
theorem Th4: :: IDEAL_1:4
for L being non empty right_zeroed addLoopStr holds {(0. L)} is add-closed
proof
let L be non empty right_zeroed addLoopStr ; ::_thesis: {(0. L)} is add-closed
let x, y be Element of L; :: according to IDEAL_1:def_1 ::_thesis: ( x in {(0. L)} & y in {(0. L)} implies x + y in {(0. L)} )
assume ( x in {(0. L)} & y in {(0. L)} ) ; ::_thesis: x + y in {(0. L)}
then ( x = 0. L & y = 0. L ) by TARSKI:def_1;
then x + y = 0. L by RLVECT_1:def_4;
hence x + y in {(0. L)} by TARSKI:def_1; ::_thesis: verum
end;
theorem Th5: :: IDEAL_1:5
for L being non empty right_add-cancelable right-distributive left_zeroed doubleLoopStr holds {(0. L)} is left-ideal
proof
let L be non empty right_add-cancelable right-distributive left_zeroed doubleLoopStr ; ::_thesis: {(0. L)} is left-ideal
let p, x be Element of L; :: according to IDEAL_1:def_2 ::_thesis: ( x in {(0. L)} implies p * x in {(0. L)} )
reconsider p9 = p as Element of L ;
assume x in {(0. L)} ; ::_thesis: p * x in {(0. L)}
then x = 0. L by TARSKI:def_1;
then p9 * x = 0. L by BINOM:2;
hence p * x in {(0. L)} by TARSKI:def_1; ::_thesis: verum
end;
theorem Th6: :: IDEAL_1:6
for L being non empty left_add-cancelable right_zeroed left-distributive doubleLoopStr holds {(0. L)} is right-ideal
proof
let L be non empty left_add-cancelable right_zeroed left-distributive doubleLoopStr ; ::_thesis: {(0. L)} is right-ideal
let p, x be Element of L; :: according to IDEAL_1:def_3 ::_thesis: ( x in {(0. L)} implies x * p in {(0. L)} )
reconsider p9 = p as Element of L ;
assume x in {(0. L)} ; ::_thesis: x * p in {(0. L)}
then x = 0. L by TARSKI:def_1;
then x * p9 = 0. L by BINOM:1;
hence x * p in {(0. L)} by TARSKI:def_1; ::_thesis: verum
end;
registration
let L be non empty right_zeroed addLoopStr ;
cluster{(0. L)} -> add-closed for Subset of L;
coherence
for b1 being Subset of L st b1 = {(0. L)} holds
b1 is add-closed by Th4;
end;
registration
let L be non empty right_add-cancelable right-distributive left_zeroed doubleLoopStr ;
cluster{(0. L)} -> left-ideal for Subset of L;
coherence
for b1 being Subset of L st b1 = {(0. L)} holds
b1 is left-ideal by Th5;
end;
registration
let L be non empty left_add-cancelable right_zeroed left-distributive doubleLoopStr ;
cluster{(0. L)} -> right-ideal for Subset of L;
coherence
for b1 being Subset of L st b1 = {(0. L)} holds
b1 is right-ideal by Th6;
end;
theorem :: IDEAL_1:7
for L being non empty right_complementable add-associative right_zeroed distributive doubleLoopStr holds {(0. L)} is Ideal of L ;
theorem :: IDEAL_1:8
for L being non empty right_complementable add-associative right_zeroed right-distributive doubleLoopStr holds {(0. L)} is LeftIdeal of L ;
theorem :: IDEAL_1:9
for L being non empty right_complementable add-associative right_zeroed left-distributive doubleLoopStr holds {(0. L)} is RightIdeal of L ;
theorem Th10: :: IDEAL_1:10
for L being non empty doubleLoopStr holds the carrier of L is Ideal of L
proof
let L be non empty doubleLoopStr ; ::_thesis: the carrier of L is Ideal of L
the carrier of L c= the carrier of L ;
then reconsider cL = the carrier of L as Subset of L ;
A1: cL is left-ideal
proof
let x, y be Element of L; :: according to IDEAL_1:def_2 ::_thesis: ( y in cL implies x * y in cL )
thus ( y in cL implies x * y in cL ) ; ::_thesis: verum
end;
A2: cL is right-ideal
proof
let x, y be Element of L; :: according to IDEAL_1:def_3 ::_thesis: ( y in cL implies y * x in cL )
thus ( y in cL implies y * x in cL ) ; ::_thesis: verum
end;
cL is add-closed
proof
let x, y be Element of L; :: according to IDEAL_1:def_1 ::_thesis: ( x in cL & y in cL implies x + y in cL )
thus ( x in cL & y in cL implies x + y in cL ) ; ::_thesis: verum
end;
hence the carrier of L is Ideal of L by A1, A2; ::_thesis: verum
end;
theorem Th11: :: IDEAL_1:11
for L being non empty doubleLoopStr holds the carrier of L is LeftIdeal of L
proof
let L be non empty doubleLoopStr ; ::_thesis: the carrier of L is LeftIdeal of L
the carrier of L c= the carrier of L ;
then reconsider cL = the carrier of L as Subset of L ;
A1: cL is left-ideal
proof
let x, y be Element of L; :: according to IDEAL_1:def_2 ::_thesis: ( y in cL implies x * y in cL )
thus ( y in cL implies x * y in cL ) ; ::_thesis: verum
end;
cL is add-closed
proof
let x, y be Element of L; :: according to IDEAL_1:def_1 ::_thesis: ( x in cL & y in cL implies x + y in cL )
thus ( x in cL & y in cL implies x + y in cL ) ; ::_thesis: verum
end;
hence the carrier of L is LeftIdeal of L by A1; ::_thesis: verum
end;
theorem Th12: :: IDEAL_1:12
for L being non empty doubleLoopStr holds the carrier of L is RightIdeal of L
proof
let L be non empty doubleLoopStr ; ::_thesis: the carrier of L is RightIdeal of L
the carrier of L c= the carrier of L ;
then reconsider cL = the carrier of L as Subset of L ;
A1: cL is right-ideal
proof
let x, y be Element of L; :: according to IDEAL_1:def_3 ::_thesis: ( y in cL implies y * x in cL )
thus ( y in cL implies y * x in cL ) ; ::_thesis: verum
end;
cL is add-closed
proof
let x, y be Element of L; :: according to IDEAL_1:def_1 ::_thesis: ( x in cL & y in cL implies x + y in cL )
thus ( x in cL & y in cL implies x + y in cL ) ; ::_thesis: verum
end;
hence the carrier of L is RightIdeal of L by A1; ::_thesis: verum
end;
definition
let R be non empty add-cancelable right_zeroed distributive left_zeroed doubleLoopStr ;
let I be Ideal of R;
:: original: trivial
redefine attrI is trivial means :: IDEAL_1:def 4
I = {(0. R)};
compatibility
( I is trivial iff I = {(0. R)} )
proof
now__::_thesis:_(_I_is_trivial_implies_I_=_{(0._R)}_)
assume I is trivial ; ::_thesis: I = {(0. R)}
then consider x being set such that
A1: I = {x} by ZFMISC_1:131;
0. R in {x} by A1, Th3;
hence I = {(0. R)} by A1, TARSKI:def_1; ::_thesis: verum
end;
hence ( I is trivial iff I = {(0. R)} ) ; ::_thesis: verum
end;
end;
:: deftheorem defines trivial IDEAL_1:def_4_:_
for R being non empty add-cancelable right_zeroed distributive left_zeroed doubleLoopStr
for I being Ideal of R holds
( I is trivial iff I = {(0. R)} );
registration
let R be non empty non trivial add-cancelable right_zeroed distributive left_zeroed doubleLoopStr ;
cluster non empty proper add-closed left-ideal right-ideal for Element of bool the carrier of R;
existence
ex b1 being Ideal of R st b1 is proper
proof
reconsider M = {(0. R)} as Ideal of R ;
M is proper by SUBSET_1:def_6;
hence ex b1 being Ideal of R st b1 is proper ; ::_thesis: verum
end;
end;
theorem Th13: :: IDEAL_1:13
for L being non empty right_complementable add-associative right_zeroed left-distributive left_unital doubleLoopStr
for I being non empty left-ideal Subset of L
for x being Element of L st x in I holds
- x in I
proof
let L be non empty right_complementable add-associative right_zeroed left-distributive left_unital doubleLoopStr ; ::_thesis: for I being non empty left-ideal Subset of L
for x being Element of L st x in I holds
- x in I
let I be non empty left-ideal Subset of L; ::_thesis: for x being Element of L st x in I holds
- x in I
let x be Element of L; ::_thesis: ( x in I implies - x in I )
assume x in I ; ::_thesis: - x in I
then A1: (- (1. L)) * x in I by Def2;
0. L = (0. L) * x by VECTSP_1:7
.= ((1. L) + (- (1. L))) * x by RLVECT_1:def_10
.= ((1. L) * x) + ((- (1. L)) * x) by VECTSP_1:def_3
.= x + ((- (1. L)) * x) by VECTSP_1:def_8 ;
hence - x in I by A1, RLVECT_1:def_10; ::_thesis: verum
end;
theorem Th14: :: IDEAL_1:14
for L being non empty right_complementable add-associative right_zeroed right-distributive right_unital doubleLoopStr
for I being non empty right-ideal Subset of L
for x being Element of L st x in I holds
- x in I
proof
let L be non empty right_complementable add-associative right_zeroed right-distributive right_unital doubleLoopStr ; ::_thesis: for I being non empty right-ideal Subset of L
for x being Element of L st x in I holds
- x in I
let I be non empty right-ideal Subset of L; ::_thesis: for x being Element of L st x in I holds
- x in I
let x be Element of L; ::_thesis: ( x in I implies - x in I )
assume x in I ; ::_thesis: - x in I
then A1: x * (- (1. L)) in I by Def3;
0. L = x * (0. L) by VECTSP_1:6
.= x * ((1. L) + (- (1. L))) by RLVECT_1:def_10
.= (x * (1. L)) + (x * (- (1. L))) by VECTSP_1:def_2
.= x + (x * (- (1. L))) by VECTSP_1:def_4 ;
hence - x in I by A1, RLVECT_1:def_10; ::_thesis: verum
end;
theorem :: IDEAL_1:15
for L being non empty right_complementable add-associative right_zeroed left-distributive left_unital doubleLoopStr
for I being LeftIdeal of L
for x, y being Element of L st x in I & y in I holds
x - y in I
proof
let L be non empty right_complementable add-associative right_zeroed left-distributive left_unital doubleLoopStr ; ::_thesis: for I being LeftIdeal of L
for x, y being Element of L st x in I & y in I holds
x - y in I
let I be LeftIdeal of L; ::_thesis: for x, y being Element of L st x in I & y in I holds
x - y in I
let x, y be Element of L; ::_thesis: ( x in I & y in I implies x - y in I )
assume that
A1: x in I and
A2: y in I ; ::_thesis: x - y in I
- y in I by A2, Th13;
hence x - y in I by A1, Def1; ::_thesis: verum
end;
theorem :: IDEAL_1:16
for L being non empty right_complementable add-associative right_zeroed right-distributive right_unital doubleLoopStr
for I being RightIdeal of L
for x, y being Element of L st x in I & y in I holds
x - y in I
proof
let L be non empty right_complementable add-associative right_zeroed right-distributive right_unital doubleLoopStr ; ::_thesis: for I being RightIdeal of L
for x, y being Element of L st x in I & y in I holds
x - y in I
let I be RightIdeal of L; ::_thesis: for x, y being Element of L st x in I & y in I holds
x - y in I
let x, y be Element of L; ::_thesis: ( x in I & y in I implies x - y in I )
assume that
A1: x in I and
A2: y in I ; ::_thesis: x - y in I
- y in I by A2, Th14;
hence x - y in I by A1, Def1; ::_thesis: verum
end;
theorem Th17: :: IDEAL_1:17
for R being non empty add-cancelable add-associative right_zeroed distributive left_zeroed doubleLoopStr
for I being non empty add-closed right-ideal Subset of R
for a being Element of I
for n being Element of NAT holds n * a in I
proof
let R be non empty add-cancelable add-associative right_zeroed distributive left_zeroed doubleLoopStr ; ::_thesis: for I being non empty add-closed right-ideal Subset of R
for a being Element of I
for n being Element of NAT holds n * a in I
let I be non empty add-closed right-ideal Subset of R; ::_thesis: for a being Element of I
for n being Element of NAT holds n * a in I
let a be Element of I; ::_thesis: for n being Element of NAT holds n * a in I
let n be Element of NAT ; ::_thesis: n * a in I
defpred S1[ Element of NAT ] means $1 * a in I;
A1: 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] )
A2: (n + 1) * a = (1 * a) + (n * a) by BINOM:15
.= a + (n * a) by BINOM:13 ;
assume n * a in I ; ::_thesis: S1[n + 1]
hence S1[n + 1] by A2, Def1; ::_thesis: verum
end;
0 * a = 0. R by BINOM:12;
then A3: S1[ 0 ] by Th3;
for n being Element of NAT holds S1[n] from NAT_1:sch_1(A3, A1);
hence n * a in I ; ::_thesis: verum
end;
theorem :: IDEAL_1:18
for R being non empty add-cancelable right_zeroed associative well-unital distributive left_zeroed doubleLoopStr
for I being non empty right-ideal Subset of R
for a being Element of I
for n being Element of NAT st n <> 0 holds
a |^ n in I
proof
let R be non empty add-cancelable right_zeroed associative well-unital distributive left_zeroed doubleLoopStr ; ::_thesis: for I being non empty right-ideal Subset of R
for a being Element of I
for n being Element of NAT st n <> 0 holds
a |^ n in I
let I be non empty right-ideal Subset of R; ::_thesis: for a being Element of I
for n being Element of NAT st n <> 0 holds
a |^ n in I
let a be Element of I; ::_thesis: for n being Element of NAT st n <> 0 holds
a |^ n in I
let n be Element of NAT ; ::_thesis: ( n <> 0 implies a |^ n in I )
defpred S1[ Nat] means a |^ $1 in I;
assume A1: n <> 0 ; ::_thesis: a |^ n in I
A2: for n being Nat st 1 <= n & S1[n] holds
S1[n + 1]
proof
let n be Nat; ::_thesis: ( 1 <= n & S1[n] implies S1[n + 1] )
assume 1 <= n ; ::_thesis: ( not S1[n] or S1[n + 1] )
A3: a |^ (n + 1) = (a |^ n) * (a |^ 1) by BINOM:10;
assume a |^ n in I ; ::_thesis: S1[n + 1]
hence S1[n + 1] by A3, Def3; ::_thesis: verum
end;
a |^ 1 = a by BINOM:8;
then A4: S1[1] ;
for n being Nat st 1 <= n holds
S1[n] from NAT_1:sch_8(A4, A2);
hence a |^ n in I by A1, NAT_1:14; ::_thesis: verum
end;
definition
let R be non empty addLoopStr ;
let I be non empty add-closed Subset of R;
func add| (I,R) -> BinOp of I equals :: IDEAL_1:def 5
the addF of R || I;
coherence
the addF of R || I is BinOp of I
proof
reconsider f = the addF of R || I as Function of [:I,I:], the carrier of R by FUNCT_2:32;
A1: dom f = [:I,I:] by FUNCT_2:def_1;
for x being set st x in [:I,I:] holds
f . x in I
proof
let x be set ; ::_thesis: ( x in [:I,I:] implies f . x in I )
assume A2: x in [:I,I:] ; ::_thesis: f . x in I
then consider u, v being set such that
A3: ( u in I & v in I ) and
A4: x = [u,v] by ZFMISC_1:def_2;
reconsider u = u, v = v as Element of R by A3;
reconsider u = u, v = v as Element of R ;
f . x = u + v by A1, A2, A4, FUNCT_1:47;
hence f . x in I by A3, Def1; ::_thesis: verum
end;
hence the addF of R || I is BinOp of I by A1, FUNCT_2:3; ::_thesis: verum
end;
end;
:: deftheorem defines add| IDEAL_1:def_5_:_
for R being non empty addLoopStr
for I being non empty add-closed Subset of R holds add| (I,R) = the addF of R || I;
definition
let R be non empty multMagma ;
let I be non empty right-ideal Subset of R;
func mult| (I,R) -> BinOp of I equals :: IDEAL_1:def 6
the multF of R || I;
coherence
the multF of R || I is BinOp of I
proof
reconsider f = the multF of R || I as Function of [:I,I:], the carrier of R by FUNCT_2:32;
A1: dom f = [:I,I:] by FUNCT_2:def_1;
for x being set st x in [:I,I:] holds
f . x in I
proof
let x be set ; ::_thesis: ( x in [:I,I:] implies f . x in I )
assume x in [:I,I:] ; ::_thesis: f . x in I
then consider u, v being set such that
A2: ( u in I & v in I ) and
A3: x = [u,v] by ZFMISC_1:def_2;
reconsider u = u, v = v as Element of I by A2;
f . x = the multF of R . [u,v] by A1, A3, FUNCT_1:47
.= u * v ;
hence f . x in I by Def3; ::_thesis: verum
end;
hence the multF of R || I is BinOp of I by A1, FUNCT_2:3; ::_thesis: verum
end;
end;
:: deftheorem defines mult| IDEAL_1:def_6_:_
for R being non empty multMagma
for I being non empty right-ideal Subset of R holds mult| (I,R) = the multF of R || I;
definition
let R be non empty addLoopStr ;
let I be non empty add-closed Subset of R;
func Gr (I,R) -> non empty addLoopStr equals :: IDEAL_1:def 7
addLoopStr(# I,(add| (I,R)),(In ((0. R),I)) #);
coherence
addLoopStr(# I,(add| (I,R)),(In ((0. R),I)) #) is non empty addLoopStr ;
end;
:: deftheorem defines Gr IDEAL_1:def_7_:_
for R being non empty addLoopStr
for I being non empty add-closed Subset of R holds Gr (I,R) = addLoopStr(# I,(add| (I,R)),(In ((0. R),I)) #);
registration
let R be non empty add-cancelable add-associative right_zeroed distributive left_zeroed doubleLoopStr ;
let I be non empty add-closed Subset of R;
cluster Gr (I,R) -> non empty add-associative ;
coherence
Gr (I,R) is add-associative
proof
set M = addLoopStr(# I,(add| (I,R)),(In ((0. R),I)) #);
reconsider M = addLoopStr(# I,(add| (I,R)),(In ((0. R),I)) #) as non empty addLoopStr ;
now__::_thesis:_for_u_being_set_st_u_in_[:I,I:]_holds_
u_in_dom_the_addF_of_R
let u be set ; ::_thesis: ( u in [:I,I:] implies u in dom the addF of R )
A1: dom the addF of R = [: the carrier of R, the carrier of R:] by FUNCT_2:def_1;
assume u in [:I,I:] ; ::_thesis: u in dom the addF of R
hence u in dom the addF of R by A1; ::_thesis: verum
end;
then [:I,I:] c= dom the addF of R by TARSKI:def_3;
then A2: dom ( the addF of R || I) = [:I,I:] by RELAT_1:62;
A3: for a, b being Element of M
for a9, b9 being Element of I st a9 = a & b9 = b holds
a + b = a9 + b9
proof
let a, b be Element of M; ::_thesis: for a9, b9 being Element of I st a9 = a & b9 = b holds
a + b = a9 + b9
let a9, b9 be Element of I; ::_thesis: ( a9 = a & b9 = b implies a + b = a9 + b9 )
assume A4: ( a9 = a & b9 = b ) ; ::_thesis: a + b = a9 + b9
[a9,b9] in dom ( the addF of R || I) by A2;
hence a + b = a9 + b9 by A4, FUNCT_1:47; ::_thesis: verum
end;
now__::_thesis:_for_a,_b,_c_being_Element_of_M_holds_(a_+_b)_+_c_=_a_+_(b_+_c)
let a, b, c be Element of M; ::_thesis: (a + b) + c = a + (b + c)
reconsider a9 = a, b9 = b, c9 = c as Element of I ;
a9 + b9 in I by Def1;
then A5: [(a9 + b9),c9] in dom ( the addF of R || I) by A2, ZFMISC_1:def_2;
b9 + c9 in I by Def1;
then A6: [a9,(b9 + c9)] in dom ( the addF of R || I) by A2, ZFMISC_1:def_2;
thus (a + b) + c = ( the addF of R || I) . [(a9 + b9),c9] by A3
.= (a9 + b9) + c9 by A5, FUNCT_1:47
.= a9 + (b9 + c9) by RLVECT_1:def_3
.= (add| (I,R)) . [a9,(b9 + c9)] by A6, FUNCT_1:47
.= a + (b + c) by A3 ; ::_thesis: verum
end;
hence Gr (I,R) is add-associative by RLVECT_1:def_3; ::_thesis: verum
end;
end;
registration
let R be non empty add-cancelable add-associative right_zeroed distributive left_zeroed doubleLoopStr ;
let I be non empty add-closed right-ideal Subset of R;
cluster Gr (I,R) -> non empty right_zeroed ;
coherence
Gr (I,R) is right_zeroed
proof
set M = addLoopStr(# I,(add| (I,R)),(In ((0. R),I)) #);
reconsider M = addLoopStr(# I,(add| (I,R)),(In ((0. R),I)) #) as non empty addLoopStr ;
now__::_thesis:_for_u_being_set_st_u_in_[:I,I:]_holds_
u_in_dom_the_addF_of_R
let u be set ; ::_thesis: ( u in [:I,I:] implies u in dom the addF of R )
A1: dom the addF of R = [: the carrier of R, the carrier of R:] by FUNCT_2:def_1;
assume u in [:I,I:] ; ::_thesis: u in dom the addF of R
hence u in dom the addF of R by A1; ::_thesis: verum
end;
then [:I,I:] c= dom the addF of R by TARSKI:def_3;
then A2: dom ( the addF of R || I) = [:I,I:] by RELAT_1:62;
now__::_thesis:_for_a_being_Element_of_M_holds_a_+_(0._M)_=_a
let a be Element of M; ::_thesis: a + (0. M) = a
reconsider a9 = a as Element of I ;
0. R in I by Th3;
then A3: [a9,(0. R)] in dom ( the addF of R || I) by A2, ZFMISC_1:def_2;
thus a + (0. M) = ( the addF of R || I) . [a9,(0. R)] by Th3, FUNCT_7:def_1
.= a9 + (0. R) by A3, FUNCT_1:47
.= a by RLVECT_1:def_4 ; ::_thesis: verum
end;
hence Gr (I,R) is right_zeroed by RLVECT_1:def_4; ::_thesis: verum
end;
end;
registration
let R be non empty Abelian doubleLoopStr ;
let I be non empty add-closed Subset of R;
cluster Gr (I,R) -> non empty Abelian ;
coherence
Gr (I,R) is Abelian
proof
set M = addLoopStr(# I,(add| (I,R)),(In ((0. R),I)) #);
reconsider M = addLoopStr(# I,(add| (I,R)),(In ((0. R),I)) #) as non empty addLoopStr ;
now__::_thesis:_for_u_being_set_st_u_in_[:I,I:]_holds_
u_in_dom_the_addF_of_R
let u be set ; ::_thesis: ( u in [:I,I:] implies u in dom the addF of R )
A1: dom the addF of R = [: the carrier of R, the carrier of R:] by FUNCT_2:def_1;
assume u in [:I,I:] ; ::_thesis: u in dom the addF of R
hence u in dom the addF of R by A1; ::_thesis: verum
end;
then [:I,I:] c= dom the addF of R by TARSKI:def_3;
then A2: dom ( the addF of R || I) = [:I,I:] by RELAT_1:62;
A3: for a, b being Element of M
for a9, b9 being Element of I st a9 = a & b9 = b holds
a + b = a9 + b9
proof
let a, b be Element of M; ::_thesis: for a9, b9 being Element of I st a9 = a & b9 = b holds
a + b = a9 + b9
let a9, b9 be Element of I; ::_thesis: ( a9 = a & b9 = b implies a + b = a9 + b9 )
assume A4: ( a9 = a & b9 = b ) ; ::_thesis: a + b = a9 + b9
[a9,b9] in dom ( the addF of R || I) by A2;
hence a + b = a9 + b9 by A4, FUNCT_1:47; ::_thesis: verum
end;
now__::_thesis:_for_a,_b_being_Element_of_M_holds_a_+_b_=_b_+_a
let a, b be Element of M; ::_thesis: a + b = b + a
reconsider a9 = a, b9 = b as Element of I ;
thus a + b = a9 + b9 by A3
.= b + a by A3 ; ::_thesis: verum
end;
hence Gr (I,R) is Abelian by RLVECT_1:def_2; ::_thesis: verum
end;
end;
registration
let R be non empty right_complementable Abelian add-associative right_zeroed right_unital distributive left_zeroed doubleLoopStr ;
let I be non empty add-closed right-ideal Subset of R;
cluster Gr (I,R) -> non empty right_complementable ;
coherence
Gr (I,R) is right_complementable
proof
set M = addLoopStr(# I,(add| (I,R)),(In ((0. R),I)) #);
reconsider M = addLoopStr(# I,(add| (I,R)),(In ((0. R),I)) #) as non empty addLoopStr ;
now__::_thesis:_for_u_being_set_st_u_in_[:I,I:]_holds_
u_in_dom_the_addF_of_R
let u be set ; ::_thesis: ( u in [:I,I:] implies u in dom the addF of R )
A1: dom the addF of R = [: the carrier of R, the carrier of R:] by FUNCT_2:def_1;
assume u in [:I,I:] ; ::_thesis: u in dom the addF of R
hence u in dom the addF of R by A1; ::_thesis: verum
end;
then [:I,I:] c= dom the addF of R by TARSKI:def_3;
then A2: dom ( the addF of R || I) = [:I,I:] by RELAT_1:62;
A3: for a, b being Element of M
for a9, b9 being Element of I st a9 = a & b9 = b holds
a + b = a9 + b9
proof
let a, b be Element of M; ::_thesis: for a9, b9 being Element of I st a9 = a & b9 = b holds
a + b = a9 + b9
let a9, b9 be Element of I; ::_thesis: ( a9 = a & b9 = b implies a + b = a9 + b9 )
assume A4: ( a9 = a & b9 = b ) ; ::_thesis: a + b = a9 + b9
[a9,b9] in dom ( the addF of R || I) by A2;
hence a + b = a9 + b9 by A4, FUNCT_1:47; ::_thesis: verum
end;
reconsider I = I as RightIdeal of R ;
M is right_complementable
proof
let a be Element of M; :: according to ALGSTR_0:def_16 ::_thesis: a is right_complementable
reconsider a9 = a as Element of I ;
reconsider b = - a9 as Element of M by Th14;
a + b = a9 + (- a9) by A3
.= 0. R by RLVECT_1:5
.= 0. M by Th3, FUNCT_7:def_1 ;
hence ex b being Element of M st a + b = 0. M ; :: according to ALGSTR_0:def_11 ::_thesis: verum
end;
hence Gr (I,R) is right_complementable ; ::_thesis: verum
end;
end;
Lm1: for R being comRing
for a being Element of R holds { (a * r) where r is Element of R : verum } is Ideal of R
proof
let R be comRing; ::_thesis: for a being Element of R holds { (a * r) where r is Element of R : verum } is Ideal of R
let a be Element of R; ::_thesis: { (a * r) where r is Element of R : verum } is Ideal of R
set M = { (a * r) where r is Element of R : verum } ;
A1: now__::_thesis:_for_u_being_set_st_u_in__{__(a_*_r)_where_r_is_Element_of_R_:_verum__}__holds_
u_in_the_carrier_of_R
let u be set ; ::_thesis: ( u in { (a * r) where r is Element of R : verum } implies u in the carrier of R )
assume u in { (a * r) where r is Element of R : verum } ; ::_thesis: u in the carrier of R
then ex r being Element of R st u = a * r ;
hence u in the carrier of R ; ::_thesis: verum
end;
a * (1. R) in { (a * r) where r is Element of R : verum } ;
then reconsider M = { (a * r) where r is Element of R : verum } as non empty Subset of R by A1, TARSKI:def_3;
reconsider M = M as non empty Subset of R ;
A2: now__::_thesis:_for_b,_c_being_Element_of_R_st_b_in_M_&_c_in_M_holds_
b_+_c_in_M
let b, c be Element of R; ::_thesis: ( b in M & c in M implies b + c in M )
assume that
A3: b in M and
A4: c in M ; ::_thesis: b + c in M
consider r being Element of R such that
A5: a * r = b by A3;
consider s being Element of R such that
A6: a * s = c by A4;
b + c = a * (r + s) by A5, A6, VECTSP_1:def_7;
hence b + c in M ; ::_thesis: verum
end;
A7: now__::_thesis:_for_s,_b_being_Element_of_R_st_b_in_M_holds_
s_*_b_in_M
let s, b be Element of R; ::_thesis: ( b in M implies s * b in M )
assume b in M ; ::_thesis: s * b in M
then consider r being Element of R such that
A8: a * r = b ;
s * b = (s * r) * a by A8, GROUP_1:def_3;
hence s * b in M ; ::_thesis: verum
end;
now__::_thesis:_for_s,_b_being_Element_of_R_st_b_in_M_holds_
b_*_s_in_M
let s, b be Element of R; ::_thesis: ( b in M implies b * s in M )
assume b in M ; ::_thesis: b * s in M
then consider r being Element of R such that
A9: a * r = b ;
b * s = a * (r * s) by A9, GROUP_1:def_3;
hence b * s in M ; ::_thesis: verum
end;
hence { (a * r) where r is Element of R : verum } is Ideal of R by A2, A7, Def1, Def2, Def3; ::_thesis: verum
end;
theorem Th19: :: IDEAL_1:19
for R being non empty right_unital doubleLoopStr
for I being non empty left-ideal Subset of R holds
( I is proper iff not 1. R in I )
proof
let R be non empty right_unital doubleLoopStr ; ::_thesis: for I being non empty left-ideal Subset of R holds
( I is proper iff not 1. R in I )
let I be non empty left-ideal Subset of R; ::_thesis: ( I is proper iff not 1. R in I )
A1: now__::_thesis:_(_I_is_proper_implies_not_1._R_in_I_)
assume A2: I is proper ; ::_thesis: not 1. R in I
thus not 1. R in I ::_thesis: verum
proof
assume A3: 1. R in I ; ::_thesis: contradiction
A4: now__::_thesis:_for_u_being_set_st_u_in_the_carrier_of_R_holds_
u_in_I
let u be set ; ::_thesis: ( u in the carrier of R implies u in I )
assume u in the carrier of R ; ::_thesis: u in I
then reconsider u9 = u as Element of R ;
u9 * (1. R) = u9 by VECTSP_1:def_4;
hence u in I by A3, Def2; ::_thesis: verum
end;
for u being set st u in I holds
u in the carrier of R ;
then I = the carrier of R by A4, TARSKI:1;
hence contradiction by A2, SUBSET_1:def_6; ::_thesis: verum
end;
end;
now__::_thesis:_(_not_1._R_in_I_implies_I_is_proper_)
assume not 1. R in I ; ::_thesis: I is proper
then I <> the carrier of R ;
hence I is proper by SUBSET_1:def_6; ::_thesis: verum
end;
hence ( I is proper iff not 1. R in I ) by A1; ::_thesis: verum
end;
theorem :: IDEAL_1:20
for R being non empty right_unital left_unital doubleLoopStr
for I being non empty right-ideal Subset of R holds
( I is proper iff for u being Element of R st u is unital holds
not u in I )
proof
let R be non empty right_unital left_unital doubleLoopStr ; ::_thesis: for I being non empty right-ideal Subset of R holds
( I is proper iff for u being Element of R st u is unital holds
not u in I )
let I be non empty right-ideal Subset of R; ::_thesis: ( I is proper iff for u being Element of R st u is unital holds
not u in I )
A1: now__::_thesis:_(_I_is_proper_implies_for_u_being_Element_of_R_st_u_is_unital_holds_
not_u_in_I_)
assume A2: I is proper ; ::_thesis: for u being Element of R st u is unital holds
not u in I
A3: not 1. R in I
proof
assume A4: 1. R in I ; ::_thesis: contradiction
A5: now__::_thesis:_for_u_being_set_st_u_in_the_carrier_of_R_holds_
u_in_I
let u be set ; ::_thesis: ( u in the carrier of R implies u in I )
assume u in the carrier of R ; ::_thesis: u in I
then reconsider u9 = u as Element of R ;
(1. R) * u9 = u9 by VECTSP_1:def_8;
hence u in I by A4, Def3; ::_thesis: verum
end;
for u being set st u in I holds
u in the carrier of R ;
then I = the carrier of R by A5, TARSKI:1;
hence contradiction by A2, SUBSET_1:def_6; ::_thesis: verum
end;
thus for u being Element of R st u is unital holds
not u in I ::_thesis: verum
proof
let u be Element of R; ::_thesis: ( u is unital implies not u in I )
assume u is unital ; ::_thesis: not u in I
then u divides 1. R by GCD_1:def_2;
then ex b being Element of R st 1. R = u * b by GCD_1:def_1;
hence not u in I by A3, Def3; ::_thesis: verum
end;
end;
now__::_thesis:_(_(_for_u_being_Element_of_R_st_u_is_unital_holds_
not_u_in_I_)_implies_I_is_proper_)
1. R divides 1. R ;
then A6: 1. R is unital by GCD_1:def_2;
assume for u being Element of R st u is unital holds
not u in I ; ::_thesis: I is proper
then I <> the carrier of R by A6;
hence I is proper by SUBSET_1:def_6; ::_thesis: verum
end;
hence ( I is proper iff for u being Element of R st u is unital holds
not u in I ) by A1; ::_thesis: verum
end;
theorem :: IDEAL_1:21
for R being non empty right_unital doubleLoopStr
for I being non empty left-ideal right-ideal Subset of R holds
( I is proper iff for u being Element of R st u is unital holds
not u in I )
proof
let R be non empty right_unital doubleLoopStr ; ::_thesis: for I being non empty left-ideal right-ideal Subset of R holds
( I is proper iff for u being Element of R st u is unital holds
not u in I )
let I be non empty left-ideal right-ideal Subset of R; ::_thesis: ( I is proper iff for u being Element of R st u is unital holds
not u in I )
A1: now__::_thesis:_(_I_is_proper_implies_for_u_being_Element_of_R_st_u_is_unital_holds_
not_u_in_I_)
assume A2: I is proper ; ::_thesis: for u being Element of R st u is unital holds
not u in I
A3: not 1. R in I
proof
assume A4: 1. R in I ; ::_thesis: contradiction
A5: now__::_thesis:_for_u_being_set_st_u_in_the_carrier_of_R_holds_
u_in_I
let u be set ; ::_thesis: ( u in the carrier of R implies u in I )
assume u in the carrier of R ; ::_thesis: u in I
then reconsider u9 = u as Element of R ;
u9 * (1. R) = u9 by VECTSP_1:def_4;
hence u in I by A4, Def2; ::_thesis: verum
end;
for u being set st u in I holds
u in the carrier of R ;
then I = the carrier of R by A5, TARSKI:1;
hence contradiction by A2, SUBSET_1:def_6; ::_thesis: verum
end;
thus for u being Element of R st u is unital holds
not u in I ::_thesis: verum
proof
let u be Element of R; ::_thesis: ( u is unital implies not u in I )
assume u is unital ; ::_thesis: not u in I
then u divides 1. R by GCD_1:def_2;
then ex b being Element of R st 1. R = u * b by GCD_1:def_1;
hence not u in I by A3, Def3; ::_thesis: verum
end;
end;
now__::_thesis:_(_(_for_u_being_Element_of_R_st_u_is_unital_holds_
not_u_in_I_)_implies_I_is_proper_)
1. R divides 1. R ;
then A6: 1. R is unital by GCD_1:def_2;
assume for u being Element of R st u is unital holds
not u in I ; ::_thesis: I is proper
then I <> the carrier of R by A6;
hence I is proper by SUBSET_1:def_6; ::_thesis: verum
end;
hence ( I is proper iff for u being Element of R st u is unital holds
not u in I ) by A1; ::_thesis: verum
end;
theorem :: IDEAL_1:22
for R being non degenerated comRing holds
( R is Field iff for I being Ideal of R holds
( I = {(0. R)} or I = the carrier of R ) )
proof
let R be non degenerated comRing; ::_thesis: ( R is Field iff for I being Ideal of R holds
( I = {(0. R)} or I = the carrier of R ) )
A1: now__::_thesis:_(_R_is_Field_implies_for_I_being_Ideal_of_R_holds_
(_I_=_{(0._R)}_or_I_=_the_carrier_of_R_)_)
assume A2: R is Field ; ::_thesis: for I being Ideal of R holds
( I = {(0. R)} or I = the carrier of R )
thus for I being Ideal of R holds
( I = {(0. R)} or I = the carrier of R ) ::_thesis: verum
proof
let I be Ideal of R; ::_thesis: ( I = {(0. R)} or I = the carrier of R )
assume A3: I <> {(0. R)} ; ::_thesis: I = the carrier of R
reconsider R = R as Field by A2;
ex a being Element of R st
( a in I & a <> 0. R )
proof
assume A4: for a being Element of R holds
( not a in I or not a <> 0. R ) ; ::_thesis: contradiction
A5: now__::_thesis:_for_u_being_set_st_u_in_I_holds_
u_in_{(0._R)}
let u be set ; ::_thesis: ( u in I implies u in {(0. R)} )
assume u in I ; ::_thesis: u in {(0. R)}
then reconsider u9 = u as Element of I ;
u9 = 0. R by A4;
hence u in {(0. R)} by TARSKI:def_1; ::_thesis: verum
end;
now__::_thesis:_for_u_being_set_st_u_in_{(0._R)}_holds_
u_in_I
let u be set ; ::_thesis: ( u in {(0. R)} implies u in I )
assume A6: u in {(0. R)} ; ::_thesis: u in I
then reconsider u9 = u as Element of R ;
u9 = 0. R by A6, TARSKI:def_1;
hence u in I by Th3; ::_thesis: verum
end;
hence contradiction by A3, A5, TARSKI:1; ::_thesis: verum
end;
then consider a being Element of R such that
A7: a in I and
A8: a <> 0. R ;
ex b being Element of R st b * a = 1. R by A8, VECTSP_1:def_9;
then 1. R in I by A7, Def3;
then not I is proper by Th19;
hence I = the carrier of R by SUBSET_1:def_6; ::_thesis: verum
end;
end;
now__::_thesis:_(_(_for_I_being_Ideal_of_R_holds_
(_I_=_{(0._R)}_or_I_=_the_carrier_of_R_)_)_implies_R_is_Field_)
assume A9: for I being Ideal of R holds
( I = {(0. R)} or I = the carrier of R ) ; ::_thesis: R is Field
now__::_thesis:_for_a_being_Element_of_R_st_a_<>_0._R_holds_
ex_b_being_Element_of_R_st_b_*_a_=_1._R
let a be Element of R; ::_thesis: ( a <> 0. R implies ex b being Element of R st b * a = 1. R )
reconsider a9 = a as Element of R ;
reconsider M = { (a9 * r) where r is Element of R : verum } as Ideal of R by Lm1;
a * (1. R) = a by VECTSP_1:def_8;
then A10: a in M ;
assume a <> 0. R ; ::_thesis: ex b being Element of R st b * a = 1. R
then M <> {(0. R)} by A10, TARSKI:def_1;
then M = the carrier of R by A9;
then 1. R in M ;
then ex b being Element of R st a * b = 1. R ;
hence ex b being Element of R st b * a = 1. R ; ::_thesis: verum
end;
hence R is Field by VECTSP_1:def_9; ::_thesis: verum
end;
hence ( R is Field iff for I being Ideal of R holds
( I = {(0. R)} or I = the carrier of R ) ) by A1; ::_thesis: verum
end;
begin
definition
let R be non empty multLoopStr ;
let A be non empty Subset of R;
mode LinearCombination of A -> FinSequence of the carrier of R means :Def8: :: IDEAL_1:def 8
for i being set st i in dom it holds
ex u, v being Element of R ex a being Element of A st it /. i = (u * a) * v;
existence
ex b1 being FinSequence of the carrier of R st
for i being set st i in dom b1 holds
ex u, v being Element of R ex a being Element of A st b1 /. i = (u * a) * v
proof
set p = <*> the carrier of R;
take <*> the carrier of R ; ::_thesis: for i being set st i in dom (<*> the carrier of R) holds
ex u, v being Element of R ex a being Element of A st (<*> the carrier of R) /. i = (u * a) * v
let i be set ; ::_thesis: ( i in dom (<*> the carrier of R) implies ex u, v being Element of R ex a being Element of A st (<*> the carrier of R) /. i = (u * a) * v )
assume i in dom (<*> the carrier of R) ; ::_thesis: ex u, v being Element of R ex a being Element of A st (<*> the carrier of R) /. i = (u * a) * v
hence ex u, v being Element of R ex a being Element of A st (<*> the carrier of R) /. i = (u * a) * v ; ::_thesis: verum
end;
mode LeftLinearCombination of A -> FinSequence of the carrier of R means :Def9: :: IDEAL_1:def 9
for i being set st i in dom it holds
ex u being Element of R ex a being Element of A st it /. i = u * a;
existence
ex b1 being FinSequence of the carrier of R st
for i being set st i in dom b1 holds
ex u being Element of R ex a being Element of A st b1 /. i = u * a
proof
set a = the Element of A;
reconsider aR = the Element of A as Element of R ;
reconsider a9 = the Element of A * the Element of A as Element of R ;
set p = <*a9*>;
take <*a9*> ; ::_thesis: for i being set st i in dom <*a9*> holds
ex u being Element of R ex a being Element of A st <*a9*> /. i = u * a
let i be set ; ::_thesis: ( i in dom <*a9*> implies ex u being Element of R ex a being Element of A st <*a9*> /. i = u * a )
assume A1: i in dom <*a9*> ; ::_thesis: ex u being Element of R ex a being Element of A st <*a9*> /. i = u * a
take aR ; ::_thesis: ex a being Element of A st <*a9*> /. i = aR * a
take the Element of A ; ::_thesis: <*a9*> /. i = aR * the Element of A
dom <*a9*> = {1} by FINSEQ_1:2, FINSEQ_1:38;
then A2: i = 1 by A1, TARSKI:def_1;
thus <*a9*> /. i = <*a9*> . i by A1, PARTFUN1:def_6
.= aR * the Element of A by A2, FINSEQ_1:40 ; ::_thesis: verum
end;
mode RightLinearCombination of A -> FinSequence of the carrier of R means :Def10: :: IDEAL_1:def 10
for i being set st i in dom it holds
ex u being Element of R ex a being Element of A st it /. i = a * u;
existence
ex b1 being FinSequence of the carrier of R st
for i being set st i in dom b1 holds
ex u being Element of R ex a being Element of A st b1 /. i = a * u
proof
set a = the Element of A;
reconsider aR = the Element of A as Element of R ;
reconsider a9 = the Element of A * the Element of A as Element of R ;
set p = <*a9*>;
take <*a9*> ; ::_thesis: for i being set st i in dom <*a9*> holds
ex u being Element of R ex a being Element of A st <*a9*> /. i = a * u
let i be set ; ::_thesis: ( i in dom <*a9*> implies ex u being Element of R ex a being Element of A st <*a9*> /. i = a * u )
assume A3: i in dom <*a9*> ; ::_thesis: ex u being Element of R ex a being Element of A st <*a9*> /. i = a * u
take aR ; ::_thesis: ex a being Element of A st <*a9*> /. i = a * aR
take the Element of A ; ::_thesis: <*a9*> /. i = the Element of A * aR
dom <*a9*> = {1} by FINSEQ_1:2, FINSEQ_1:38;
then A4: i = 1 by A3, TARSKI:def_1;
thus <*a9*> /. i = <*a9*> . i by A3, PARTFUN1:def_6
.= the Element of A * aR by A4, FINSEQ_1:40 ; ::_thesis: verum
end;
end;
:: deftheorem Def8 defines LinearCombination IDEAL_1:def_8_:_
for R being non empty multLoopStr
for A being non empty Subset of R
for b3 being FinSequence of the carrier of R holds
( b3 is LinearCombination of A iff for i being set st i in dom b3 holds
ex u, v being Element of R ex a being Element of A st b3 /. i = (u * a) * v );
:: deftheorem Def9 defines LeftLinearCombination IDEAL_1:def_9_:_
for R being non empty multLoopStr
for A being non empty Subset of R
for b3 being FinSequence of the carrier of R holds
( b3 is LeftLinearCombination of A iff for i being set st i in dom b3 holds
ex u being Element of R ex a being Element of A st b3 /. i = u * a );
:: deftheorem Def10 defines RightLinearCombination IDEAL_1:def_10_:_
for R being non empty multLoopStr
for A being non empty Subset of R
for b3 being FinSequence of the carrier of R holds
( b3 is RightLinearCombination of A iff for i being set st i in dom b3 holds
ex u being Element of R ex a being Element of A st b3 /. i = a * u );
registration
let R be non empty multLoopStr ;
let A be non empty Subset of R;
cluster non empty Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like for LinearCombination of A;
existence
not for b1 being LinearCombination of A holds b1 is empty
proof
set a = the Element of A;
set u = the Element of R;
set v = the Element of R;
reconsider p = <*(( the Element of R * the Element of A) * the Element of R)*> as FinSequence of the carrier of R ;
take p ; ::_thesis: ( p is LinearCombination of A & not p is empty )
now__::_thesis:_for_i_being_set_st_i_in_dom_p_holds_
ex_u,_v_being_Element_of_R_ex_a_being_Element_of_A_st_p_/._i_=_(u_*_a)_*_v
let i be set ; ::_thesis: ( i in dom p implies ex u, v being Element of R ex a being Element of A st p /. i = (u * a) * v )
assume A1: i in dom p ; ::_thesis: ex u, v being Element of R ex a being Element of A st p /. i = (u * a) * v
take u = the Element of R; ::_thesis: ex v being Element of R ex a being Element of A st p /. i = (u * a) * v
take v = the Element of R; ::_thesis: ex a being Element of A st p /. i = (u * a) * v
take a = the Element of A; ::_thesis: p /. i = (u * a) * v
i in {1} by A1, FINSEQ_1:2, FINSEQ_1:38;
then i = 1 by TARSKI:def_1;
hence p /. i = (u * a) * v by FINSEQ_4:16; ::_thesis: verum
end;
hence ( p is LinearCombination of A & not p is empty ) by Def8; ::_thesis: verum
end;
cluster non empty Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like for LeftLinearCombination of A;
existence
not for b1 being LeftLinearCombination of A holds b1 is empty
proof
set a = the Element of A;
set u = the Element of R;
reconsider p = <*( the Element of R * the Element of A)*> as FinSequence of the carrier of R ;
take p ; ::_thesis: ( p is LeftLinearCombination of A & not p is empty )
now__::_thesis:_for_i_being_set_st_i_in_dom_p_holds_
ex_u_being_Element_of_R_ex_a_being_Element_of_A_st_p_/._i_=_u_*_a
let i be set ; ::_thesis: ( i in dom p implies ex u being Element of R ex a being Element of A st p /. i = u * a )
assume A2: i in dom p ; ::_thesis: ex u being Element of R ex a being Element of A st p /. i = u * a
take u = the Element of R; ::_thesis: ex a being Element of A st p /. i = u * a
take a = the Element of A; ::_thesis: p /. i = u * a
i in {1} by A2, FINSEQ_1:2, FINSEQ_1:38;
then i = 1 by TARSKI:def_1;
hence p /. i = u * a by FINSEQ_4:16; ::_thesis: verum
end;
hence ( p is LeftLinearCombination of A & not p is empty ) by Def9; ::_thesis: verum
end;
cluster non empty Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like for RightLinearCombination of A;
existence
not for b1 being RightLinearCombination of A holds b1 is empty
proof
set a = the Element of A;
set v = the Element of R;
reconsider p = <*( the Element of A * the Element of R)*> as FinSequence of the carrier of R ;
take p ; ::_thesis: ( p is RightLinearCombination of A & not p is empty )
now__::_thesis:_for_i_being_set_st_i_in_dom_p_holds_
ex_v_being_Element_of_R_ex_a_being_Element_of_A_st_p_/._i_=_a_*_v
let i be set ; ::_thesis: ( i in dom p implies ex v being Element of R ex a being Element of A st p /. i = a * v )
assume A3: i in dom p ; ::_thesis: ex v being Element of R ex a being Element of A st p /. i = a * v
take v = the Element of R; ::_thesis: ex a being Element of A st p /. i = a * v
take a = the Element of A; ::_thesis: p /. i = a * v
i in {1} by A3, FINSEQ_1:2, FINSEQ_1:38;
then i = 1 by TARSKI:def_1;
hence p /. i = a * v by FINSEQ_4:16; ::_thesis: verum
end;
hence ( p is RightLinearCombination of A & not p is empty ) by Def10; ::_thesis: verum
end;
end;
definition
let R be non empty multLoopStr ;
let A, B be non empty Subset of R;
let F be LinearCombination of A;
let G be LinearCombination of B;
:: original: ^
redefine funcF ^ G -> LinearCombination of A \/ B;
coherence
F ^ G is LinearCombination of A \/ B
proof
set H = F ^ G;
thus F ^ G is LinearCombination of A \/ B ::_thesis: verum
proof
let i be set ; :: according to IDEAL_1:def_8 ::_thesis: ( i in dom (F ^ G) implies ex u, v being Element of R ex a being Element of A \/ B st (F ^ G) /. i = (u * a) * v )
assume A1: i in dom (F ^ G) ; ::_thesis: ex u, v being Element of R ex a being Element of A \/ B st (F ^ G) /. i = (u * a) * v
then reconsider i = i as Element of NAT ;
percases ( i in dom F or not i in dom F ) ;
supposeA2: i in dom F ; ::_thesis: ex u, v being Element of R ex a being Element of A \/ B st (F ^ G) /. i = (u * a) * v
then A3: ( F /. i = F . i & F . i = (F ^ G) . i ) by FINSEQ_1:def_7, PARTFUN1:def_6;
consider u, v being Element of R, a being Element of A such that
A4: F /. i = (u * a) * v by A2, Def8;
a in A \/ B by XBOOLE_0:def_3;
hence ex u, v being Element of R ex a being Element of A \/ B st (F ^ G) /. i = (u * a) * v by A1, A4, A3, PARTFUN1:def_6; ::_thesis: verum
end;
suppose not i in dom F ; ::_thesis: ex u, v being Element of R ex a being Element of A \/ B st (F ^ G) /. i = (u * a) * v
then consider n being Nat such that
A5: n in dom G and
A6: i = (len F) + n by A1, FINSEQ_1:25;
A7: ( G /. n = G . n & G . n = (F ^ G) . i ) by A5, A6, FINSEQ_1:def_7, PARTFUN1:def_6;
consider u, v being Element of R, b being Element of B such that
A8: G /. n = (u * b) * v by A5, Def8;
b in A \/ B by XBOOLE_0:def_3;
hence ex u, v being Element of R ex a being Element of A \/ B st (F ^ G) /. i = (u * a) * v by A1, A8, A7, PARTFUN1:def_6; ::_thesis: verum
end;
end;
end;
end;
end;
theorem Th23: :: IDEAL_1:23
for R being non empty associative multLoopStr
for A being non empty Subset of R
for a being Element of R
for F being LinearCombination of A holds a * F is LinearCombination of A
proof
let R be non empty associative multLoopStr ; ::_thesis: for A being non empty Subset of R
for a being Element of R
for F being LinearCombination of A holds a * F is LinearCombination of A
let A be non empty Subset of R; ::_thesis: for a being Element of R
for F being LinearCombination of A holds a * F is LinearCombination of A
let a be Element of R; ::_thesis: for F being LinearCombination of A holds a * F is LinearCombination of A
let F be LinearCombination of A; ::_thesis: a * F is LinearCombination of A
let i be set ; :: according to IDEAL_1:def_8 ::_thesis: ( i in dom (a * F) implies ex u, v being Element of R ex a being Element of A st (a * F) /. i = (u * a) * v )
assume i in dom (a * F) ; ::_thesis: ex u, v being Element of R ex a being Element of A st (a * F) /. i = (u * a) * v
then A1: i in dom F by POLYNOM1:def_1;
then consider u, v being Element of R, b being Element of A such that
A2: F /. i = (u * b) * v by Def8;
take x = a * u; ::_thesis: ex v being Element of R ex a being Element of A st (a * F) /. i = (x * a) * v
take v ; ::_thesis: ex a being Element of A st (a * F) /. i = (x * a) * v
take b ; ::_thesis: (a * F) /. i = (x * b) * v
thus (a * F) /. i = a * (F /. i) by A1, POLYNOM1:def_1
.= (a * (u * b)) * v by A2, GROUP_1:def_3
.= (x * b) * v by GROUP_1:def_3 ; ::_thesis: verum
end;
theorem Th24: :: IDEAL_1:24
for R being non empty associative multLoopStr
for A being non empty Subset of R
for a being Element of R
for F being LinearCombination of A holds F * a is LinearCombination of A
proof
let R be non empty associative multLoopStr ; ::_thesis: for A being non empty Subset of R
for a being Element of R
for F being LinearCombination of A holds F * a is LinearCombination of A
let A be non empty Subset of R; ::_thesis: for a being Element of R
for F being LinearCombination of A holds F * a is LinearCombination of A
let a be Element of R; ::_thesis: for F being LinearCombination of A holds F * a is LinearCombination of A
let F be LinearCombination of A; ::_thesis: F * a is LinearCombination of A
let i be set ; :: according to IDEAL_1:def_8 ::_thesis: ( i in dom (F * a) implies ex u, v being Element of R ex a being Element of A st (F * a) /. i = (u * a) * v )
assume i in dom (F * a) ; ::_thesis: ex u, v being Element of R ex a being Element of A st (F * a) /. i = (u * a) * v
then A1: i in dom F by POLYNOM1:def_2;
then consider u, v being Element of R, b being Element of A such that
A2: F /. i = (u * b) * v by Def8;
take u ; ::_thesis: ex v being Element of R ex a being Element of A st (F * a) /. i = (u * a) * v
take x = v * a; ::_thesis: ex a being Element of A st (F * a) /. i = (u * a) * x
take b ; ::_thesis: (F * a) /. i = (u * b) * x
thus (F * a) /. i = (F /. i) * a by A1, POLYNOM1:def_2
.= (u * (b * v)) * a by A2, GROUP_1:def_3
.= u * ((b * v) * a) by GROUP_1:def_3
.= u * (b * (v * a)) by GROUP_1:def_3
.= (u * b) * x by GROUP_1:def_3 ; ::_thesis: verum
end;
theorem Th25: :: IDEAL_1:25
for R being non empty right_unital multLoopStr
for A being non empty Subset of R
for f being LeftLinearCombination of A holds f is LinearCombination of A
proof
let R be non empty right_unital multLoopStr ; ::_thesis: for A being non empty Subset of R
for f being LeftLinearCombination of A holds f is LinearCombination of A
let A be non empty Subset of R; ::_thesis: for f being LeftLinearCombination of A holds f is LinearCombination of A
let f be LeftLinearCombination of A; ::_thesis: f is LinearCombination of A
let i be set ; :: according to IDEAL_1:def_8 ::_thesis: ( i in dom f implies ex u, v being Element of R ex a being Element of A st f /. i = (u * a) * v )
assume i in dom f ; ::_thesis: ex u, v being Element of R ex a being Element of A st f /. i = (u * a) * v
then consider r being Element of R, a being Element of A such that
A1: f /. i = r * a by Def9;
f /. i = (r * a) * (1. R) by A1, VECTSP_1:def_4;
hence ex u, v being Element of R ex a being Element of A st f /. i = (u * a) * v ; ::_thesis: verum
end;
definition
let R be non empty multLoopStr ;
let A, B be non empty Subset of R;
let F be LeftLinearCombination of A;
let G be LeftLinearCombination of B;
:: original: ^
redefine funcF ^ G -> LeftLinearCombination of A \/ B;
coherence
F ^ G is LeftLinearCombination of A \/ B
proof
set H = F ^ G;
thus F ^ G is LeftLinearCombination of A \/ B ::_thesis: verum
proof
let i be set ; :: according to IDEAL_1:def_9 ::_thesis: ( i in dom (F ^ G) implies ex u being Element of R ex a being Element of A \/ B st (F ^ G) /. i = u * a )
assume A1: i in dom (F ^ G) ; ::_thesis: ex u being Element of R ex a being Element of A \/ B st (F ^ G) /. i = u * a
then reconsider i = i as Element of NAT ;
percases ( i in dom F or not i in dom F ) ;
supposeA2: i in dom F ; ::_thesis: ex u being Element of R ex a being Element of A \/ B st (F ^ G) /. i = u * a
then A3: ( F /. i = F . i & F . i = (F ^ G) . i ) by FINSEQ_1:def_7, PARTFUN1:def_6;
consider u being Element of R, a being Element of A such that
A4: F /. i = u * a by A2, Def9;
a in A \/ B by XBOOLE_0:def_3;
hence ex u being Element of R ex a being Element of A \/ B st (F ^ G) /. i = u * a by A1, A4, A3, PARTFUN1:def_6; ::_thesis: verum
end;
suppose not i in dom F ; ::_thesis: ex u being Element of R ex a being Element of A \/ B st (F ^ G) /. i = u * a
then consider n being Nat such that
A5: n in dom G and
A6: i = (len F) + n by A1, FINSEQ_1:25;
A7: ( G /. n = G . n & G . n = (F ^ G) . i ) by A5, A6, FINSEQ_1:def_7, PARTFUN1:def_6;
consider u being Element of R, b being Element of B such that
A8: G /. n = u * b by A5, Def9;
b in A \/ B by XBOOLE_0:def_3;
hence ex u being Element of R ex a being Element of A \/ B st (F ^ G) /. i = u * a by A1, A8, A7, PARTFUN1:def_6; ::_thesis: verum
end;
end;
end;
end;
end;
theorem Th26: :: IDEAL_1:26
for R being non empty associative multLoopStr
for A being non empty Subset of R
for a being Element of R
for F being LeftLinearCombination of A holds a * F is LeftLinearCombination of A
proof
let R be non empty associative multLoopStr ; ::_thesis: for A being non empty Subset of R
for a being Element of R
for F being LeftLinearCombination of A holds a * F is LeftLinearCombination of A
let A be non empty Subset of R; ::_thesis: for a being Element of R
for F being LeftLinearCombination of A holds a * F is LeftLinearCombination of A
let a be Element of R; ::_thesis: for F being LeftLinearCombination of A holds a * F is LeftLinearCombination of A
let F be LeftLinearCombination of A; ::_thesis: a * F is LeftLinearCombination of A
let i be set ; :: according to IDEAL_1:def_9 ::_thesis: ( i in dom (a * F) implies ex u being Element of R ex a being Element of A st (a * F) /. i = u * a )
assume i in dom (a * F) ; ::_thesis: ex u being Element of R ex a being Element of A st (a * F) /. i = u * a
then A1: i in dom F by POLYNOM1:def_1;
then consider u being Element of R, b being Element of A such that
A2: F /. i = u * b by Def9;
take x = a * u; ::_thesis: ex a being Element of A st (a * F) /. i = x * a
take b ; ::_thesis: (a * F) /. i = x * b
thus (a * F) /. i = a * (F /. i) by A1, POLYNOM1:def_1
.= x * b by A2, GROUP_1:def_3 ; ::_thesis: verum
end;
theorem :: IDEAL_1:27
for R being non empty multLoopStr
for A being non empty Subset of R
for a being Element of R
for F being LeftLinearCombination of A holds F * a is LinearCombination of A
proof
let R be non empty multLoopStr ; ::_thesis: for A being non empty Subset of R
for a being Element of R
for F being LeftLinearCombination of A holds F * a is LinearCombination of A
let A be non empty Subset of R; ::_thesis: for a being Element of R
for F being LeftLinearCombination of A holds F * a is LinearCombination of A
let a be Element of R; ::_thesis: for F being LeftLinearCombination of A holds F * a is LinearCombination of A
let F be LeftLinearCombination of A; ::_thesis: F * a is LinearCombination of A
let i be set ; :: according to IDEAL_1:def_8 ::_thesis: ( i in dom (F * a) implies ex u, v being Element of R ex a being Element of A st (F * a) /. i = (u * a) * v )
reconsider c = a as Element of R ;
assume i in dom (F * a) ; ::_thesis: ex u, v being Element of R ex a being Element of A st (F * a) /. i = (u * a) * v
then A1: i in dom F by POLYNOM1:def_2;
then consider u being Element of R, b being Element of A such that
A2: F /. i = u * b by Def9;
take u ; ::_thesis: ex v being Element of R ex a being Element of A st (F * a) /. i = (u * a) * v
take c ; ::_thesis: ex a being Element of A st (F * a) /. i = (u * a) * c
take b ; ::_thesis: (F * a) /. i = (u * b) * c
thus (F * a) /. i = (u * b) * c by A1, A2, POLYNOM1:def_2; ::_thesis: verum
end;
theorem Th28: :: IDEAL_1:28
for R being non empty left_unital multLoopStr
for A being non empty Subset of R
for f being RightLinearCombination of A holds f is LinearCombination of A
proof
let R be non empty left_unital multLoopStr ; ::_thesis: for A being non empty Subset of R
for f being RightLinearCombination of A holds f is LinearCombination of A
let A be non empty Subset of R; ::_thesis: for f being RightLinearCombination of A holds f is LinearCombination of A
let f be RightLinearCombination of A; ::_thesis: f is LinearCombination of A
let i be set ; :: according to IDEAL_1:def_8 ::_thesis: ( i in dom f implies ex u, v being Element of R ex a being Element of A st f /. i = (u * a) * v )
assume i in dom f ; ::_thesis: ex u, v being Element of R ex a being Element of A st f /. i = (u * a) * v
then consider r being Element of R, a being Element of A such that
A1: f /. i = a * r by Def10;
f /. i = ((1. R) * a) * r by A1, VECTSP_1:def_8;
hence ex u, v being Element of R ex a being Element of A st f /. i = (u * a) * v ; ::_thesis: verum
end;
definition
let R be non empty multLoopStr ;
let A, B be non empty Subset of R;
let F be RightLinearCombination of A;
let G be RightLinearCombination of B;
:: original: ^
redefine funcF ^ G -> RightLinearCombination of A \/ B;
coherence
F ^ G is RightLinearCombination of A \/ B
proof
set H = F ^ G;
thus F ^ G is RightLinearCombination of A \/ B ::_thesis: verum
proof
let i be set ; :: according to IDEAL_1:def_10 ::_thesis: ( i in dom (F ^ G) implies ex u being Element of R ex a being Element of A \/ B st (F ^ G) /. i = a * u )
assume A1: i in dom (F ^ G) ; ::_thesis: ex u being Element of R ex a being Element of A \/ B st (F ^ G) /. i = a * u
then reconsider i = i as Element of NAT ;
percases ( i in dom F or not i in dom F ) ;
supposeA2: i in dom F ; ::_thesis: ex u being Element of R ex a being Element of A \/ B st (F ^ G) /. i = a * u
then A3: ( F /. i = F . i & F . i = (F ^ G) . i ) by FINSEQ_1:def_7, PARTFUN1:def_6;
consider u being Element of R, a being Element of A such that
A4: F /. i = a * u by A2, Def10;
a in A \/ B by XBOOLE_0:def_3;
hence ex u being Element of R ex a being Element of A \/ B st (F ^ G) /. i = a * u by A1, A4, A3, PARTFUN1:def_6; ::_thesis: verum
end;
suppose not i in dom F ; ::_thesis: ex u being Element of R ex a being Element of A \/ B st (F ^ G) /. i = a * u
then consider n being Nat such that
A5: n in dom G and
A6: i = (len F) + n by A1, FINSEQ_1:25;
A7: ( G /. n = G . n & G . n = (F ^ G) . i ) by A5, A6, FINSEQ_1:def_7, PARTFUN1:def_6;
consider u being Element of R, b being Element of B such that
A8: G /. n = b * u by A5, Def10;
b in A \/ B by XBOOLE_0:def_3;
hence ex u being Element of R ex a being Element of A \/ B st (F ^ G) /. i = a * u by A1, A8, A7, PARTFUN1:def_6; ::_thesis: verum
end;
end;
end;
end;
end;
theorem Th29: :: IDEAL_1:29
for R being non empty associative multLoopStr
for A being non empty Subset of R
for a being Element of R
for F being RightLinearCombination of A holds F * a is RightLinearCombination of A
proof
let R be non empty associative multLoopStr ; ::_thesis: for A being non empty Subset of R
for a being Element of R
for F being RightLinearCombination of A holds F * a is RightLinearCombination of A
let A be non empty Subset of R; ::_thesis: for a being Element of R
for F being RightLinearCombination of A holds F * a is RightLinearCombination of A
let a be Element of R; ::_thesis: for F being RightLinearCombination of A holds F * a is RightLinearCombination of A
let F be RightLinearCombination of A; ::_thesis: F * a is RightLinearCombination of A
let i be set ; :: according to IDEAL_1:def_10 ::_thesis: ( i in dom (F * a) implies ex u being Element of R ex a being Element of A st (F * a) /. i = a * u )
assume i in dom (F * a) ; ::_thesis: ex u being Element of R ex a being Element of A st (F * a) /. i = a * u
then A1: i in dom F by POLYNOM1:def_2;
then consider u being Element of R, b being Element of A such that
A2: F /. i = b * u by Def10;
take x = u * a; ::_thesis: ex a being Element of A st (F * a) /. i = a * x
take b ; ::_thesis: (F * a) /. i = b * x
thus (F * a) /. i = (F /. i) * a by A1, POLYNOM1:def_2
.= b * x by A2, GROUP_1:def_3 ; ::_thesis: verum
end;
theorem :: IDEAL_1:30
for R being non empty associative multLoopStr
for A being non empty Subset of R
for a being Element of R
for F being RightLinearCombination of A holds a * F is LinearCombination of A
proof
let R be non empty associative multLoopStr ; ::_thesis: for A being non empty Subset of R
for a being Element of R
for F being RightLinearCombination of A holds a * F is LinearCombination of A
let A be non empty Subset of R; ::_thesis: for a being Element of R
for F being RightLinearCombination of A holds a * F is LinearCombination of A
let a be Element of R; ::_thesis: for F being RightLinearCombination of A holds a * F is LinearCombination of A
let F be RightLinearCombination of A; ::_thesis: a * F is LinearCombination of A
let i be set ; :: according to IDEAL_1:def_8 ::_thesis: ( i in dom (a * F) implies ex u, v being Element of R ex a being Element of A st (a * F) /. i = (u * a) * v )
reconsider c = a as Element of R ;
assume i in dom (a * F) ; ::_thesis: ex u, v being Element of R ex a being Element of A st (a * F) /. i = (u * a) * v
then A1: i in dom F by POLYNOM1:def_1;
then consider u being Element of R, b being Element of A such that
A2: F /. i = b * u by Def10;
take c ; ::_thesis: ex v being Element of R ex a being Element of A st (a * F) /. i = (c * a) * v
take u ; ::_thesis: ex a being Element of A st (a * F) /. i = (c * a) * u
take b ; ::_thesis: (a * F) /. i = (c * b) * u
thus (a * F) /. i = a * (F /. i) by A1, POLYNOM1:def_1
.= (c * b) * u by A2, GROUP_1:def_3 ; ::_thesis: verum
end;
theorem Th31: :: IDEAL_1:31
for R being non empty associative commutative multLoopStr
for A being non empty Subset of R
for f being LinearCombination of A holds
( f is LeftLinearCombination of A & f is RightLinearCombination of A )
proof
let R be non empty associative commutative multLoopStr ; ::_thesis: for A being non empty Subset of R
for f being LinearCombination of A holds
( f is LeftLinearCombination of A & f is RightLinearCombination of A )
let A be non empty Subset of R; ::_thesis: for f being LinearCombination of A holds
( f is LeftLinearCombination of A & f is RightLinearCombination of A )
let f be LinearCombination of A; ::_thesis: ( f is LeftLinearCombination of A & f is RightLinearCombination of A )
hereby :: according to IDEAL_1:def_9 ::_thesis: f is RightLinearCombination of A
let i be set ; ::_thesis: ( i in dom f implies ex r being Element of R ex a being Element of A st f /. i = r * a )
assume i in dom f ; ::_thesis: ex r being Element of R ex a being Element of A st f /. i = r * a
then consider r, s being Element of R, a being Element of A such that
A1: f /. i = (r * a) * s by Def8;
f /. i = (r * s) * a by A1, GROUP_1:def_3;
hence ex r being Element of R ex a being Element of A st f /. i = r * a ; ::_thesis: verum
end;
let i be set ; :: according to IDEAL_1:def_10 ::_thesis: ( i in dom f implies ex u being Element of R ex a being Element of A st f /. i = a * u )
assume i in dom f ; ::_thesis: ex u being Element of R ex a being Element of A st f /. i = a * u
then consider r, s being Element of R, a being Element of A such that
A2: f /. i = (r * a) * s by Def8;
f /. i = a * (r * s) by A2, GROUP_1:def_3;
hence ex r being Element of R ex a being Element of A st f /. i = a * r ; ::_thesis: verum
end;
theorem Th32: :: IDEAL_1:32
for S being non empty doubleLoopStr
for F being non empty Subset of S
for lc being non empty LinearCombination of F ex p being LinearCombination of F ex e being Element of S st
( lc = p ^ <*e*> & <*e*> is LinearCombination of F )
proof
let S be non empty doubleLoopStr ; ::_thesis: for F being non empty Subset of S
for lc being non empty LinearCombination of F ex p being LinearCombination of F ex e being Element of S st
( lc = p ^ <*e*> & <*e*> is LinearCombination of F )
let F be non empty Subset of S; ::_thesis: for lc being non empty LinearCombination of F ex p being LinearCombination of F ex e being Element of S st
( lc = p ^ <*e*> & <*e*> is LinearCombination of F )
let lc be non empty LinearCombination of F; ::_thesis: ex p being LinearCombination of F ex e being Element of S st
( lc = p ^ <*e*> & <*e*> is LinearCombination of F )
len lc <> 0 ;
then consider p being FinSequence of the carrier of S, e being Element of S such that
A1: lc = p ^ <*e*> by FINSEQ_2:19;
now__::_thesis:_for_i_being_set_st_i_in_dom_p_holds_
ex_u,_v_being_Element_of_S_ex_a_being_Element_of_F_st_p_/._i_=_(u_*_a)_*_v
let i be set ; ::_thesis: ( i in dom p implies ex u, v being Element of S ex a being Element of F st p /. i = (u * a) * v )
assume A2: i in dom p ; ::_thesis: ex u, v being Element of S ex a being Element of F st p /. i = (u * a) * v
then reconsider i1 = i as Element of NAT ;
A3: dom p c= dom lc by A1, FINSEQ_1:26;
then consider u, v being Element of S, a being Element of F such that
A4: lc /. i = (u * a) * v by A2, Def8;
take u = u; ::_thesis: ex v being Element of S ex a being Element of F st p /. i = (u * a) * v
take v = v; ::_thesis: ex a being Element of F st p /. i = (u * a) * v
take a = a; ::_thesis: p /. i = (u * a) * v
thus p /. i = p . i by A2, PARTFUN1:def_6
.= lc . i1 by A1, A2, FINSEQ_1:def_7
.= (u * a) * v by A2, A3, A4, PARTFUN1:def_6 ; ::_thesis: verum
end;
then reconsider p = p as LinearCombination of F by Def8;
A5: len lc = (len p) + 1 by A1, FINSEQ_2:16;
take p ; ::_thesis: ex e being Element of S st
( lc = p ^ <*e*> & <*e*> is LinearCombination of F )
take e ; ::_thesis: ( lc = p ^ <*e*> & <*e*> is LinearCombination of F )
thus lc = p ^ <*e*> by A1; ::_thesis: <*e*> is LinearCombination of F
let i be set ; :: according to IDEAL_1:def_8 ::_thesis: ( i in dom <*e*> implies ex u, v being Element of S ex a being Element of F st <*e*> /. i = (u * a) * v )
assume A6: i in dom <*e*> ; ::_thesis: ex u, v being Element of S ex a being Element of F st <*e*> /. i = (u * a) * v
A7: len lc in dom lc by FINSEQ_5:6;
then A8: lc /. (len lc) = lc . (len lc) by PARTFUN1:def_6;
dom <*e*> = {1} by FINSEQ_1:2, FINSEQ_1:38;
then A9: i = 1 by A6, TARSKI:def_1;
consider u, v being Element of S, a being Element of F such that
A10: lc /. (len lc) = (u * a) * v by A7, Def8;
take u ; ::_thesis: ex v being Element of S ex a being Element of F st <*e*> /. i = (u * a) * v
take v ; ::_thesis: ex a being Element of F st <*e*> /. i = (u * a) * v
take a ; ::_thesis: <*e*> /. i = (u * a) * v
thus <*e*> /. i = <*e*> . i by A6, PARTFUN1:def_6
.= e by A9, FINSEQ_1:40
.= (u * a) * v by A1, A5, A10, A8, FINSEQ_1:42 ; ::_thesis: verum
end;
theorem Th33: :: IDEAL_1:33
for S being non empty doubleLoopStr
for F being non empty Subset of S
for lc being non empty LeftLinearCombination of F ex p being LeftLinearCombination of F ex e being Element of S st
( lc = p ^ <*e*> & <*e*> is LeftLinearCombination of F )
proof
let S be non empty doubleLoopStr ; ::_thesis: for F being non empty Subset of S
for lc being non empty LeftLinearCombination of F ex p being LeftLinearCombination of F ex e being Element of S st
( lc = p ^ <*e*> & <*e*> is LeftLinearCombination of F )
let F be non empty Subset of S; ::_thesis: for lc being non empty LeftLinearCombination of F ex p being LeftLinearCombination of F ex e being Element of S st
( lc = p ^ <*e*> & <*e*> is LeftLinearCombination of F )
let lc be non empty LeftLinearCombination of F; ::_thesis: ex p being LeftLinearCombination of F ex e being Element of S st
( lc = p ^ <*e*> & <*e*> is LeftLinearCombination of F )
len lc <> 0 ;
then consider p being FinSequence of the carrier of S, e being Element of S such that
A1: lc = p ^ <*e*> by FINSEQ_2:19;
now__::_thesis:_for_i_being_set_st_i_in_dom_p_holds_
ex_u_being_Element_of_S_ex_a_being_Element_of_F_st_p_/._i_=_u_*_a
let i be set ; ::_thesis: ( i in dom p implies ex u being Element of S ex a being Element of F st p /. i = u * a )
assume A2: i in dom p ; ::_thesis: ex u being Element of S ex a being Element of F st p /. i = u * a
then reconsider i1 = i as Element of NAT ;
A3: dom p c= dom lc by A1, FINSEQ_1:26;
then consider u being Element of S, a being Element of F such that
A4: lc /. i = u * a by A2, Def9;
take u = u; ::_thesis: ex a being Element of F st p /. i = u * a
take a = a; ::_thesis: p /. i = u * a
thus p /. i = p . i by A2, PARTFUN1:def_6
.= lc . i1 by A1, A2, FINSEQ_1:def_7
.= u * a by A2, A3, A4, PARTFUN1:def_6 ; ::_thesis: verum
end;
then reconsider p = p as LeftLinearCombination of F by Def9;
A5: len lc = (len p) + 1 by A1, FINSEQ_2:16;
take p ; ::_thesis: ex e being Element of S st
( lc = p ^ <*e*> & <*e*> is LeftLinearCombination of F )
take e ; ::_thesis: ( lc = p ^ <*e*> & <*e*> is LeftLinearCombination of F )
thus lc = p ^ <*e*> by A1; ::_thesis: <*e*> is LeftLinearCombination of F
let i be set ; :: according to IDEAL_1:def_9 ::_thesis: ( i in dom <*e*> implies ex u being Element of S ex a being Element of F st <*e*> /. i = u * a )
assume A6: i in dom <*e*> ; ::_thesis: ex u being Element of S ex a being Element of F st <*e*> /. i = u * a
A7: len lc in dom lc by FINSEQ_5:6;
then A8: lc /. (len lc) = lc . (len lc) by PARTFUN1:def_6;
dom <*e*> = {1} by FINSEQ_1:2, FINSEQ_1:38;
then A9: i = 1 by A6, TARSKI:def_1;
consider u being Element of S, a being Element of F such that
A10: lc /. (len lc) = u * a by A7, Def9;
take u ; ::_thesis: ex a being Element of F st <*e*> /. i = u * a
take a ; ::_thesis: <*e*> /. i = u * a
thus <*e*> /. i = <*e*> . i by A6, PARTFUN1:def_6
.= e by A9, FINSEQ_1:40
.= u * a by A1, A5, A10, A8, FINSEQ_1:42 ; ::_thesis: verum
end;
theorem Th34: :: IDEAL_1:34
for S being non empty doubleLoopStr
for F being non empty Subset of S
for lc being non empty RightLinearCombination of F ex p being RightLinearCombination of F ex e being Element of S st
( lc = p ^ <*e*> & <*e*> is RightLinearCombination of F )
proof
let S be non empty doubleLoopStr ; ::_thesis: for F being non empty Subset of S
for lc being non empty RightLinearCombination of F ex p being RightLinearCombination of F ex e being Element of S st
( lc = p ^ <*e*> & <*e*> is RightLinearCombination of F )
let F be non empty Subset of S; ::_thesis: for lc being non empty RightLinearCombination of F ex p being RightLinearCombination of F ex e being Element of S st
( lc = p ^ <*e*> & <*e*> is RightLinearCombination of F )
let lc be non empty RightLinearCombination of F; ::_thesis: ex p being RightLinearCombination of F ex e being Element of S st
( lc = p ^ <*e*> & <*e*> is RightLinearCombination of F )
len lc <> 0 ;
then consider p being FinSequence of the carrier of S, e being Element of S such that
A1: lc = p ^ <*e*> by FINSEQ_2:19;
now__::_thesis:_for_i_being_set_st_i_in_dom_p_holds_
ex_u_being_Element_of_S_ex_a_being_Element_of_F_st_p_/._i_=_a_*_u
let i be set ; ::_thesis: ( i in dom p implies ex u being Element of S ex a being Element of F st p /. i = a * u )
assume A2: i in dom p ; ::_thesis: ex u being Element of S ex a being Element of F st p /. i = a * u
then reconsider i1 = i as Element of NAT ;
A3: dom p c= dom lc by A1, FINSEQ_1:26;
then consider u being Element of S, a being Element of F such that
A4: lc /. i = a * u by A2, Def10;
take u = u; ::_thesis: ex a being Element of F st p /. i = a * u
take a = a; ::_thesis: p /. i = a * u
thus p /. i = p . i by A2, PARTFUN1:def_6
.= lc . i1 by A1, A2, FINSEQ_1:def_7
.= a * u by A2, A3, A4, PARTFUN1:def_6 ; ::_thesis: verum
end;
then reconsider p = p as RightLinearCombination of F by Def10;
A5: len lc = (len p) + 1 by A1, FINSEQ_2:16;
take p ; ::_thesis: ex e being Element of S st
( lc = p ^ <*e*> & <*e*> is RightLinearCombination of F )
take e ; ::_thesis: ( lc = p ^ <*e*> & <*e*> is RightLinearCombination of F )
thus lc = p ^ <*e*> by A1; ::_thesis: <*e*> is RightLinearCombination of F
let i be set ; :: according to IDEAL_1:def_10 ::_thesis: ( i in dom <*e*> implies ex u being Element of S ex a being Element of F st <*e*> /. i = a * u )
assume A6: i in dom <*e*> ; ::_thesis: ex u being Element of S ex a being Element of F st <*e*> /. i = a * u
A7: len lc in dom lc by FINSEQ_5:6;
then A8: lc /. (len lc) = lc . (len lc) by PARTFUN1:def_6;
dom <*e*> = {1} by FINSEQ_1:2, FINSEQ_1:38;
then A9: i = 1 by A6, TARSKI:def_1;
consider u being Element of S, a being Element of F such that
A10: lc /. (len lc) = a * u by A7, Def10;
take u ; ::_thesis: ex a being Element of F st <*e*> /. i = a * u
take a ; ::_thesis: <*e*> /. i = a * u
thus <*e*> /. i = <*e*> . i by A6, PARTFUN1:def_6
.= e by A9, FINSEQ_1:40
.= a * u by A1, A5, A10, A8, FINSEQ_1:42 ; ::_thesis: verum
end;
definition
let R be non empty multLoopStr ;
let A be non empty Subset of R;
let L be LinearCombination of A;
let E be FinSequence of [: the carrier of R, the carrier of R, the carrier of R:];
predE represents L means :Def11: :: IDEAL_1:def 11
( len E = len L & ( for i being set st i in dom L holds
( L . i = (((E /. i) `1_3) * ((E /. i) `2_3)) * ((E /. i) `3_3) & (E /. i) `2_3 in A ) ) );
end;
:: deftheorem Def11 defines represents IDEAL_1:def_11_:_
for R being non empty multLoopStr
for A being non empty Subset of R
for L being LinearCombination of A
for E being FinSequence of [: the carrier of R, the carrier of R, the carrier of R:] holds
( E represents L iff ( len E = len L & ( for i being set st i in dom L holds
( L . i = (((E /. i) `1_3) * ((E /. i) `2_3)) * ((E /. i) `3_3) & (E /. i) `2_3 in A ) ) ) );
theorem :: IDEAL_1:35
for R being non empty multLoopStr
for A being non empty Subset of R
for L being LinearCombination of A ex E being FinSequence of [: the carrier of R, the carrier of R, the carrier of R:] st E represents L
proof
let R be non empty multLoopStr ; ::_thesis: for A being non empty Subset of R
for L being LinearCombination of A ex E being FinSequence of [: the carrier of R, the carrier of R, the carrier of R:] st E represents L
let A be non empty Subset of R; ::_thesis: for L being LinearCombination of A ex E being FinSequence of [: the carrier of R, the carrier of R, the carrier of R:] st E represents L
let L be LinearCombination of A; ::_thesis: ex E being FinSequence of [: the carrier of R, the carrier of R, the carrier of R:] st E represents L
set D = [: the carrier of R, the carrier of R, the carrier of R:];
defpred S1[ set , set ] means ex x, y, z being Element of R st
( $2 = [x,y,z] & y in A & L /. $1 = (x * y) * z );
A1: now__::_thesis:_for_k_being_Element_of_NAT_st_k_in_Seg_(len_L)_holds_
ex_d_being_Element_of_[:_the_carrier_of_R,_the_carrier_of_R,_the_carrier_of_R:]_st_S1[k,d]
let k be Element of NAT ; ::_thesis: ( k in Seg (len L) implies ex d being Element of [: the carrier of R, the carrier of R, the carrier of R:] st S1[k,d] )
assume k in Seg (len L) ; ::_thesis: ex d being Element of [: the carrier of R, the carrier of R, the carrier of R:] st S1[k,d]
then k in dom L by FINSEQ_1:def_3;
then consider u, v being Element of R, a being Element of A such that
A2: L /. k = (u * a) * v by Def8;
reconsider b = a as Element of R ;
reconsider d = [u,b,v] as Element of [: the carrier of R, the carrier of R, the carrier of R:] ;
take d = d; ::_thesis: S1[k,d]
thus S1[k,d] by A2; ::_thesis: verum
end;
consider E being FinSequence of [: the carrier of R, the carrier of R, the carrier of R:] such that
A3: dom E = Seg (len L) and
A4: for k being Element of NAT st k in Seg (len L) holds
S1[k,E /. k] from RECDEF_1:sch_17(A1);
take E ; ::_thesis: E represents L
thus len E = len L by A3, FINSEQ_1:def_3; :: according to IDEAL_1:def_11 ::_thesis: for i being set st i in dom L holds
( L . i = (((E /. i) `1_3) * ((E /. i) `2_3)) * ((E /. i) `3_3) & (E /. i) `2_3 in A )
let i be set ; ::_thesis: ( i in dom L implies ( L . i = (((E /. i) `1_3) * ((E /. i) `2_3)) * ((E /. i) `3_3) & (E /. i) `2_3 in A ) )
assume A5: i in dom L ; ::_thesis: ( L . i = (((E /. i) `1_3) * ((E /. i) `2_3)) * ((E /. i) `3_3) & (E /. i) `2_3 in A )
reconsider k = i as Element of NAT by A5;
dom L = Seg (len L) by FINSEQ_1:def_3;
then consider x, y, z being Element of R such that
A6: E /. k = [x,y,z] and
A7: y in A and
A8: L /. k = (x * y) * z by A4, A5;
A9: [x,y,z] `3_3 = z ;
( [x,y,z] `1_3 = x & [x,y,z] `2_3 = y ) ;
hence L . i = (((E /. i) `1_3) * ((E /. i) `2_3)) * ((E /. i) `3_3) by A5, A6, A8, A9, PARTFUN1:def_6; ::_thesis: (E /. i) `2_3 in A
thus (E /. i) `2_3 in A by A6, A7, MCART_1:def_6; ::_thesis: verum
end;
theorem :: IDEAL_1:36
for R, S being non empty multLoopStr
for F being non empty Subset of R
for lc being LinearCombination of F
for G being non empty Subset of S
for P being Function of the carrier of R, the carrier of S
for E being FinSequence of [: the carrier of R, the carrier of R, the carrier of R:] st P .: F c= G & E represents lc holds
ex LC being LinearCombination of G st
( len lc = len LC & ( for i being set st i in dom LC holds
LC . i = ((P . ((E /. i) `1_3)) * (P . ((E /. i) `2_3))) * (P . ((E /. i) `3_3)) ) )
proof
let R, S be non empty multLoopStr ; ::_thesis: for F being non empty Subset of R
for lc being LinearCombination of F
for G being non empty Subset of S
for P being Function of the carrier of R, the carrier of S
for E being FinSequence of [: the carrier of R, the carrier of R, the carrier of R:] st P .: F c= G & E represents lc holds
ex LC being LinearCombination of G st
( len lc = len LC & ( for i being set st i in dom LC holds
LC . i = ((P . ((E /. i) `1_3)) * (P . ((E /. i) `2_3))) * (P . ((E /. i) `3_3)) ) )
let F be non empty Subset of R; ::_thesis: for lc being LinearCombination of F
for G being non empty Subset of S
for P being Function of the carrier of R, the carrier of S
for E being FinSequence of [: the carrier of R, the carrier of R, the carrier of R:] st P .: F c= G & E represents lc holds
ex LC being LinearCombination of G st
( len lc = len LC & ( for i being set st i in dom LC holds
LC . i = ((P . ((E /. i) `1_3)) * (P . ((E /. i) `2_3))) * (P . ((E /. i) `3_3)) ) )
let lc be LinearCombination of F; ::_thesis: for G being non empty Subset of S
for P being Function of the carrier of R, the carrier of S
for E being FinSequence of [: the carrier of R, the carrier of R, the carrier of R:] st P .: F c= G & E represents lc holds
ex LC being LinearCombination of G st
( len lc = len LC & ( for i being set st i in dom LC holds
LC . i = ((P . ((E /. i) `1_3)) * (P . ((E /. i) `2_3))) * (P . ((E /. i) `3_3)) ) )
let G be non empty Subset of S; ::_thesis: for P being Function of the carrier of R, the carrier of S
for E being FinSequence of [: the carrier of R, the carrier of R, the carrier of R:] st P .: F c= G & E represents lc holds
ex LC being LinearCombination of G st
( len lc = len LC & ( for i being set st i in dom LC holds
LC . i = ((P . ((E /. i) `1_3)) * (P . ((E /. i) `2_3))) * (P . ((E /. i) `3_3)) ) )
let P be Function of the carrier of R, the carrier of S; ::_thesis: for E being FinSequence of [: the carrier of R, the carrier of R, the carrier of R:] st P .: F c= G & E represents lc holds
ex LC being LinearCombination of G st
( len lc = len LC & ( for i being set st i in dom LC holds
LC . i = ((P . ((E /. i) `1_3)) * (P . ((E /. i) `2_3))) * (P . ((E /. i) `3_3)) ) )
let E be FinSequence of [: the carrier of R, the carrier of R, the carrier of R:]; ::_thesis: ( P .: F c= G & E represents lc implies ex LC being LinearCombination of G st
( len lc = len LC & ( for i being set st i in dom LC holds
LC . i = ((P . ((E /. i) `1_3)) * (P . ((E /. i) `2_3))) * (P . ((E /. i) `3_3)) ) ) )
assume A1: P .: F c= G ; ::_thesis: ( not E represents lc or ex LC being LinearCombination of G st
( len lc = len LC & ( for i being set st i in dom LC holds
LC . i = ((P . ((E /. i) `1_3)) * (P . ((E /. i) `2_3))) * (P . ((E /. i) `3_3)) ) ) )
deffunc H1( Nat) -> Element of the carrier of S = ((P . ((E /. $1) `1_3)) * (P . ((E /. $1) `2_3))) * (P . ((E /. $1) `3_3));
consider LC being FinSequence of the carrier of S such that
A2: len LC = len lc and
A3: for k being Nat st k in dom LC holds
LC . k = H1(k) from FINSEQ_2:sch_1();
assume A4: E represents lc ; ::_thesis: ex LC being LinearCombination of G st
( len lc = len LC & ( for i being set st i in dom LC holds
LC . i = ((P . ((E /. i) `1_3)) * (P . ((E /. i) `2_3))) * (P . ((E /. i) `3_3)) ) )
now__::_thesis:_for_i_being_set_st_i_in_dom_LC_holds_
ex_u,_v_being_Element_of_S_ex_a_being_Element_of_G_st_LC_/._i_=_(u_*_a)_*_v
let i be set ; ::_thesis: ( i in dom LC implies ex u, v being Element of S ex a being Element of G st LC /. i = (u * a) * v )
assume A5: i in dom LC ; ::_thesis: ex u, v being Element of S ex a being Element of G st LC /. i = (u * a) * v
dom lc = dom LC by A2, FINSEQ_3:29;
then ( dom P = the carrier of R & (E /. i) `2_3 in F ) by A4, A5, Def11, FUNCT_2:def_1;
then P . ((E /. i) `2_3) in P .: F by FUNCT_1:def_6;
then reconsider a = P . ((E /. i) `2_3) as Element of G by A1;
reconsider u = P . ((E /. i) `1_3), v = P . ((E /. i) `3_3) as Element of S ;
take u = u; ::_thesis: ex v being Element of S ex a being Element of G st LC /. i = (u * a) * v
take v = v; ::_thesis: ex a being Element of G st LC /. i = (u * a) * v
take a = a; ::_thesis: LC /. i = (u * a) * v
LC . i = LC /. i by A5, PARTFUN1:def_6;
hence LC /. i = (u * a) * v by A3, A5; ::_thesis: verum
end;
then reconsider LC = LC as LinearCombination of G by Def8;
take LC ; ::_thesis: ( len lc = len LC & ( for i being set st i in dom LC holds
LC . i = ((P . ((E /. i) `1_3)) * (P . ((E /. i) `2_3))) * (P . ((E /. i) `3_3)) ) )
thus len lc = len LC by A2; ::_thesis: for i being set st i in dom LC holds
LC . i = ((P . ((E /. i) `1_3)) * (P . ((E /. i) `2_3))) * (P . ((E /. i) `3_3))
let i be set ; ::_thesis: ( i in dom LC implies LC . i = ((P . ((E /. i) `1_3)) * (P . ((E /. i) `2_3))) * (P . ((E /. i) `3_3)) )
assume i in dom LC ; ::_thesis: LC . i = ((P . ((E /. i) `1_3)) * (P . ((E /. i) `2_3))) * (P . ((E /. i) `3_3))
hence LC . i = ((P . ((E /. i) `1_3)) * (P . ((E /. i) `2_3))) * (P . ((E /. i) `3_3)) by A3; ::_thesis: verum
end;
definition
let R be non empty multLoopStr ;
let A be non empty Subset of R;
let L be LeftLinearCombination of A;
let E be FinSequence of [: the carrier of R, the carrier of R:];
predE represents L means :Def12: :: IDEAL_1:def 12
( len E = len L & ( for i being set st i in dom L holds
( L . i = ((E /. i) `1) * ((E /. i) `2) & (E /. i) `2 in A ) ) );
end;
:: deftheorem Def12 defines represents IDEAL_1:def_12_:_
for R being non empty multLoopStr
for A being non empty Subset of R
for L being LeftLinearCombination of A
for E being FinSequence of [: the carrier of R, the carrier of R:] holds
( E represents L iff ( len E = len L & ( for i being set st i in dom L holds
( L . i = ((E /. i) `1) * ((E /. i) `2) & (E /. i) `2 in A ) ) ) );
theorem :: IDEAL_1:37
for R being non empty multLoopStr
for A being non empty Subset of R
for L being LeftLinearCombination of A ex E being FinSequence of [: the carrier of R, the carrier of R:] st E represents L
proof
let R be non empty multLoopStr ; ::_thesis: for A being non empty Subset of R
for L being LeftLinearCombination of A ex E being FinSequence of [: the carrier of R, the carrier of R:] st E represents L
let A be non empty Subset of R; ::_thesis: for L being LeftLinearCombination of A ex E being FinSequence of [: the carrier of R, the carrier of R:] st E represents L
let L be LeftLinearCombination of A; ::_thesis: ex E being FinSequence of [: the carrier of R, the carrier of R:] st E represents L
set D = [: the carrier of R, the carrier of R:];
defpred S1[ set , set ] means ex x, y being Element of R st
( $2 = [x,y] & y in A & L /. $1 = x * y );
A1: now__::_thesis:_for_k_being_Element_of_NAT_st_k_in_Seg_(len_L)_holds_
ex_d_being_Element_of_[:_the_carrier_of_R,_the_carrier_of_R:]_st_S1[k,d]
let k be Element of NAT ; ::_thesis: ( k in Seg (len L) implies ex d being Element of [: the carrier of R, the carrier of R:] st S1[k,d] )
assume k in Seg (len L) ; ::_thesis: ex d being Element of [: the carrier of R, the carrier of R:] st S1[k,d]
then k in dom L by FINSEQ_1:def_3;
then consider u being Element of R, a being Element of A such that
A2: L /. k = u * a by Def9;
reconsider b = a as Element of R ;
reconsider d = [u,b] as Element of [: the carrier of R, the carrier of R:] ;
take d = d; ::_thesis: S1[k,d]
thus S1[k,d] by A2; ::_thesis: verum
end;
consider E being FinSequence of [: the carrier of R, the carrier of R:] such that
A3: dom E = Seg (len L) and
A4: for k being Element of NAT st k in Seg (len L) holds
S1[k,E /. k] from RECDEF_1:sch_17(A1);
take E ; ::_thesis: E represents L
thus len E = len L by A3, FINSEQ_1:def_3; :: according to IDEAL_1:def_12 ::_thesis: for i being set st i in dom L holds
( L . i = ((E /. i) `1) * ((E /. i) `2) & (E /. i) `2 in A )
let i be set ; ::_thesis: ( i in dom L implies ( L . i = ((E /. i) `1) * ((E /. i) `2) & (E /. i) `2 in A ) )
assume A5: i in dom L ; ::_thesis: ( L . i = ((E /. i) `1) * ((E /. i) `2) & (E /. i) `2 in A )
reconsider k = i as Element of NAT by A5;
dom L = Seg (len L) by FINSEQ_1:def_3;
then consider x, y being Element of R such that
A6: E /. k = [x,y] and
A7: y in A and
A8: L /. k = x * y by A4, A5;
A9: ( [x,y] `1 = x & [x,y] `2 = y ) ;
hence L . i = ((E /. i) `1) * ((E /. i) `2) by A5, A6, A8, PARTFUN1:def_6; ::_thesis: (E /. i) `2 in A
thus (E /. i) `2 in A by A6, A7, A9; ::_thesis: verum
end;
theorem :: IDEAL_1:38
for R, S being non empty multLoopStr
for F being non empty Subset of R
for lc being LeftLinearCombination of F
for G being non empty Subset of S
for P being Function of the carrier of R, the carrier of S
for E being FinSequence of [: the carrier of R, the carrier of R:] st P .: F c= G & E represents lc holds
ex LC being LeftLinearCombination of G st
( len lc = len LC & ( for i being set st i in dom LC holds
LC . i = (P . ((E /. i) `1)) * (P . ((E /. i) `2)) ) )
proof
let R, S be non empty multLoopStr ; ::_thesis: for F being non empty Subset of R
for lc being LeftLinearCombination of F
for G being non empty Subset of S
for P being Function of the carrier of R, the carrier of S
for E being FinSequence of [: the carrier of R, the carrier of R:] st P .: F c= G & E represents lc holds
ex LC being LeftLinearCombination of G st
( len lc = len LC & ( for i being set st i in dom LC holds
LC . i = (P . ((E /. i) `1)) * (P . ((E /. i) `2)) ) )
let F be non empty Subset of R; ::_thesis: for lc being LeftLinearCombination of F
for G being non empty Subset of S
for P being Function of the carrier of R, the carrier of S
for E being FinSequence of [: the carrier of R, the carrier of R:] st P .: F c= G & E represents lc holds
ex LC being LeftLinearCombination of G st
( len lc = len LC & ( for i being set st i in dom LC holds
LC . i = (P . ((E /. i) `1)) * (P . ((E /. i) `2)) ) )
let lc be LeftLinearCombination of F; ::_thesis: for G being non empty Subset of S
for P being Function of the carrier of R, the carrier of S
for E being FinSequence of [: the carrier of R, the carrier of R:] st P .: F c= G & E represents lc holds
ex LC being LeftLinearCombination of G st
( len lc = len LC & ( for i being set st i in dom LC holds
LC . i = (P . ((E /. i) `1)) * (P . ((E /. i) `2)) ) )
let G be non empty Subset of S; ::_thesis: for P being Function of the carrier of R, the carrier of S
for E being FinSequence of [: the carrier of R, the carrier of R:] st P .: F c= G & E represents lc holds
ex LC being LeftLinearCombination of G st
( len lc = len LC & ( for i being set st i in dom LC holds
LC . i = (P . ((E /. i) `1)) * (P . ((E /. i) `2)) ) )
let P be Function of the carrier of R, the carrier of S; ::_thesis: for E being FinSequence of [: the carrier of R, the carrier of R:] st P .: F c= G & E represents lc holds
ex LC being LeftLinearCombination of G st
( len lc = len LC & ( for i being set st i in dom LC holds
LC . i = (P . ((E /. i) `1)) * (P . ((E /. i) `2)) ) )
let E be FinSequence of [: the carrier of R, the carrier of R:]; ::_thesis: ( P .: F c= G & E represents lc implies ex LC being LeftLinearCombination of G st
( len lc = len LC & ( for i being set st i in dom LC holds
LC . i = (P . ((E /. i) `1)) * (P . ((E /. i) `2)) ) ) )
assume A1: P .: F c= G ; ::_thesis: ( not E represents lc or ex LC being LeftLinearCombination of G st
( len lc = len LC & ( for i being set st i in dom LC holds
LC . i = (P . ((E /. i) `1)) * (P . ((E /. i) `2)) ) ) )
deffunc H1( Nat) -> Element of the carrier of S = (P . ((E /. $1) `1)) * (P . ((E /. $1) `2));
consider LC being FinSequence of the carrier of S such that
A2: len LC = len lc and
A3: for k being Nat st k in dom LC holds
LC . k = H1(k) from FINSEQ_2:sch_1();
assume A4: E represents lc ; ::_thesis: ex LC being LeftLinearCombination of G st
( len lc = len LC & ( for i being set st i in dom LC holds
LC . i = (P . ((E /. i) `1)) * (P . ((E /. i) `2)) ) )
now__::_thesis:_for_i_being_set_st_i_in_dom_LC_holds_
ex_u_being_Element_of_S_ex_a_being_Element_of_G_st_LC_/._i_=_u_*_a
let i be set ; ::_thesis: ( i in dom LC implies ex u being Element of S ex a being Element of G st LC /. i = u * a )
assume A5: i in dom LC ; ::_thesis: ex u being Element of S ex a being Element of G st LC /. i = u * a
dom lc = dom LC by A2, FINSEQ_3:29;
then ( dom P = the carrier of R & (E /. i) `2 in F ) by A4, A5, Def12, FUNCT_2:def_1;
then P . ((E /. i) `2) in P .: F by FUNCT_1:def_6;
then reconsider a = P . ((E /. i) `2) as Element of G by A1;
reconsider u = P . ((E /. i) `1) as Element of S ;
take u = u; ::_thesis: ex a being Element of G st LC /. i = u * a
take a = a; ::_thesis: LC /. i = u * a
LC . i = LC /. i by A5, PARTFUN1:def_6;
hence LC /. i = u * a by A3, A5; ::_thesis: verum
end;
then reconsider LC = LC as LeftLinearCombination of G by Def9;
take LC ; ::_thesis: ( len lc = len LC & ( for i being set st i in dom LC holds
LC . i = (P . ((E /. i) `1)) * (P . ((E /. i) `2)) ) )
thus len lc = len LC by A2; ::_thesis: for i being set st i in dom LC holds
LC . i = (P . ((E /. i) `1)) * (P . ((E /. i) `2))
let i be set ; ::_thesis: ( i in dom LC implies LC . i = (P . ((E /. i) `1)) * (P . ((E /. i) `2)) )
assume i in dom LC ; ::_thesis: LC . i = (P . ((E /. i) `1)) * (P . ((E /. i) `2))
hence LC . i = (P . ((E /. i) `1)) * (P . ((E /. i) `2)) by A3; ::_thesis: verum
end;
definition
let R be non empty multLoopStr ;
let A be non empty Subset of R;
let L be RightLinearCombination of A;
let E be FinSequence of [: the carrier of R, the carrier of R:];
predE represents L means :Def13: :: IDEAL_1:def 13
( len E = len L & ( for i being set st i in dom L holds
( L . i = ((E /. i) `1) * ((E /. i) `2) & (E /. i) `1 in A ) ) );
end;
:: deftheorem Def13 defines represents IDEAL_1:def_13_:_
for R being non empty multLoopStr
for A being non empty Subset of R
for L being RightLinearCombination of A
for E being FinSequence of [: the carrier of R, the carrier of R:] holds
( E represents L iff ( len E = len L & ( for i being set st i in dom L holds
( L . i = ((E /. i) `1) * ((E /. i) `2) & (E /. i) `1 in A ) ) ) );
theorem :: IDEAL_1:39
for R being non empty multLoopStr
for A being non empty Subset of R
for L being RightLinearCombination of A ex E being FinSequence of [: the carrier of R, the carrier of R:] st E represents L
proof
let R be non empty multLoopStr ; ::_thesis: for A being non empty Subset of R
for L being RightLinearCombination of A ex E being FinSequence of [: the carrier of R, the carrier of R:] st E represents L
let A be non empty Subset of R; ::_thesis: for L being RightLinearCombination of A ex E being FinSequence of [: the carrier of R, the carrier of R:] st E represents L
let L be RightLinearCombination of A; ::_thesis: ex E being FinSequence of [: the carrier of R, the carrier of R:] st E represents L
set D = [: the carrier of R, the carrier of R:];
defpred S1[ set , set ] means ex x, y being Element of R st
( $2 = [x,y] & x in A & L /. $1 = x * y );
A1: now__::_thesis:_for_k_being_Element_of_NAT_st_k_in_Seg_(len_L)_holds_
ex_d_being_Element_of_[:_the_carrier_of_R,_the_carrier_of_R:]_st_S1[k,d]
let k be Element of NAT ; ::_thesis: ( k in Seg (len L) implies ex d being Element of [: the carrier of R, the carrier of R:] st S1[k,d] )
assume k in Seg (len L) ; ::_thesis: ex d being Element of [: the carrier of R, the carrier of R:] st S1[k,d]
then k in dom L by FINSEQ_1:def_3;
then consider v being Element of R, a being Element of A such that
A2: L /. k = a * v by Def10;
reconsider b = a as Element of R ;
reconsider d = [b,v] as Element of [: the carrier of R, the carrier of R:] ;
take d = d; ::_thesis: S1[k,d]
thus S1[k,d] by A2; ::_thesis: verum
end;
consider E being FinSequence of [: the carrier of R, the carrier of R:] such that
A3: dom E = Seg (len L) and
A4: for k being Element of NAT st k in Seg (len L) holds
S1[k,E /. k] from RECDEF_1:sch_17(A1);
take E ; ::_thesis: E represents L
thus len E = len L by A3, FINSEQ_1:def_3; :: according to IDEAL_1:def_13 ::_thesis: for i being set st i in dom L holds
( L . i = ((E /. i) `1) * ((E /. i) `2) & (E /. i) `1 in A )
let i be set ; ::_thesis: ( i in dom L implies ( L . i = ((E /. i) `1) * ((E /. i) `2) & (E /. i) `1 in A ) )
assume A5: i in dom L ; ::_thesis: ( L . i = ((E /. i) `1) * ((E /. i) `2) & (E /. i) `1 in A )
reconsider k = i as Element of NAT by A5;
dom L = Seg (len L) by FINSEQ_1:def_3;
then consider x, y being Element of R such that
A6: E /. k = [x,y] and
A7: x in A and
A8: L /. k = x * y by A4, A5;
A9: ( [x,y] `1 = x & [x,y] `2 = y ) ;
hence L . i = ((E /. i) `1) * ((E /. i) `2) by A5, A6, A8, PARTFUN1:def_6; ::_thesis: (E /. i) `1 in A
thus (E /. i) `1 in A by A6, A7, A9; ::_thesis: verum
end;
theorem :: IDEAL_1:40
for R, S being non empty multLoopStr
for F being non empty Subset of R
for lc being RightLinearCombination of F
for G being non empty Subset of S
for P being Function of the carrier of R, the carrier of S
for E being FinSequence of [: the carrier of R, the carrier of R:] st P .: F c= G & E represents lc holds
ex LC being RightLinearCombination of G st
( len lc = len LC & ( for i being set st i in dom LC holds
LC . i = (P . ((E /. i) `1)) * (P . ((E /. i) `2)) ) )
proof
let R, S be non empty multLoopStr ; ::_thesis: for F being non empty Subset of R
for lc being RightLinearCombination of F
for G being non empty Subset of S
for P being Function of the carrier of R, the carrier of S
for E being FinSequence of [: the carrier of R, the carrier of R:] st P .: F c= G & E represents lc holds
ex LC being RightLinearCombination of G st
( len lc = len LC & ( for i being set st i in dom LC holds
LC . i = (P . ((E /. i) `1)) * (P . ((E /. i) `2)) ) )
let F be non empty Subset of R; ::_thesis: for lc being RightLinearCombination of F
for G being non empty Subset of S
for P being Function of the carrier of R, the carrier of S
for E being FinSequence of [: the carrier of R, the carrier of R:] st P .: F c= G & E represents lc holds
ex LC being RightLinearCombination of G st
( len lc = len LC & ( for i being set st i in dom LC holds
LC . i = (P . ((E /. i) `1)) * (P . ((E /. i) `2)) ) )
let lc be RightLinearCombination of F; ::_thesis: for G being non empty Subset of S
for P being Function of the carrier of R, the carrier of S
for E being FinSequence of [: the carrier of R, the carrier of R:] st P .: F c= G & E represents lc holds
ex LC being RightLinearCombination of G st
( len lc = len LC & ( for i being set st i in dom LC holds
LC . i = (P . ((E /. i) `1)) * (P . ((E /. i) `2)) ) )
let G be non empty Subset of S; ::_thesis: for P being Function of the carrier of R, the carrier of S
for E being FinSequence of [: the carrier of R, the carrier of R:] st P .: F c= G & E represents lc holds
ex LC being RightLinearCombination of G st
( len lc = len LC & ( for i being set st i in dom LC holds
LC . i = (P . ((E /. i) `1)) * (P . ((E /. i) `2)) ) )
let P be Function of the carrier of R, the carrier of S; ::_thesis: for E being FinSequence of [: the carrier of R, the carrier of R:] st P .: F c= G & E represents lc holds
ex LC being RightLinearCombination of G st
( len lc = len LC & ( for i being set st i in dom LC holds
LC . i = (P . ((E /. i) `1)) * (P . ((E /. i) `2)) ) )
let E be FinSequence of [: the carrier of R, the carrier of R:]; ::_thesis: ( P .: F c= G & E represents lc implies ex LC being RightLinearCombination of G st
( len lc = len LC & ( for i being set st i in dom LC holds
LC . i = (P . ((E /. i) `1)) * (P . ((E /. i) `2)) ) ) )
assume A1: P .: F c= G ; ::_thesis: ( not E represents lc or ex LC being RightLinearCombination of G st
( len lc = len LC & ( for i being set st i in dom LC holds
LC . i = (P . ((E /. i) `1)) * (P . ((E /. i) `2)) ) ) )
deffunc H1( Nat) -> Element of the carrier of S = (P . ((E /. $1) `1)) * (P . ((E /. $1) `2));
consider LC being FinSequence of the carrier of S such that
A2: len LC = len lc and
A3: for k being Nat st k in dom LC holds
LC . k = H1(k) from FINSEQ_2:sch_1();
assume A4: E represents lc ; ::_thesis: ex LC being RightLinearCombination of G st
( len lc = len LC & ( for i being set st i in dom LC holds
LC . i = (P . ((E /. i) `1)) * (P . ((E /. i) `2)) ) )
now__::_thesis:_for_i_being_set_st_i_in_dom_LC_holds_
ex_v_being_Element_of_S_ex_a_being_Element_of_G_st_LC_/._i_=_a_*_v
let i be set ; ::_thesis: ( i in dom LC implies ex v being Element of S ex a being Element of G st LC /. i = a * v )
assume A5: i in dom LC ; ::_thesis: ex v being Element of S ex a being Element of G st LC /. i = a * v
dom lc = dom LC by A2, FINSEQ_3:29;
then ( dom P = the carrier of R & (E /. i) `1 in F ) by A4, A5, Def13, FUNCT_2:def_1;
then P . ((E /. i) `1) in P .: F by FUNCT_1:def_6;
then reconsider a = P . ((E /. i) `1) as Element of G by A1;
reconsider v = P . ((E /. i) `2) as Element of S ;
take v = v; ::_thesis: ex a being Element of G st LC /. i = a * v
take a = a; ::_thesis: LC /. i = a * v
LC . i = LC /. i by A5, PARTFUN1:def_6;
hence LC /. i = a * v by A3, A5; ::_thesis: verum
end;
then reconsider LC = LC as RightLinearCombination of G by Def10;
take LC ; ::_thesis: ( len lc = len LC & ( for i being set st i in dom LC holds
LC . i = (P . ((E /. i) `1)) * (P . ((E /. i) `2)) ) )
thus len lc = len LC by A2; ::_thesis: for i being set st i in dom LC holds
LC . i = (P . ((E /. i) `1)) * (P . ((E /. i) `2))
let i be set ; ::_thesis: ( i in dom LC implies LC . i = (P . ((E /. i) `1)) * (P . ((E /. i) `2)) )
assume i in dom LC ; ::_thesis: LC . i = (P . ((E /. i) `1)) * (P . ((E /. i) `2))
hence LC . i = (P . ((E /. i) `1)) * (P . ((E /. i) `2)) by A3; ::_thesis: verum
end;
theorem :: IDEAL_1:41
for R being non empty multLoopStr
for A being non empty Subset of R
for l being LinearCombination of A
for n being Element of NAT holds l | (Seg n) is LinearCombination of A
proof
let R be non empty multLoopStr ; ::_thesis: for A being non empty Subset of R
for l being LinearCombination of A
for n being Element of NAT holds l | (Seg n) is LinearCombination of A
let A be non empty Subset of R; ::_thesis: for l being LinearCombination of A
for n being Element of NAT holds l | (Seg n) is LinearCombination of A
let l be LinearCombination of A; ::_thesis: for n being Element of NAT holds l | (Seg n) is LinearCombination of A
let n be Element of NAT ; ::_thesis: l | (Seg n) is LinearCombination of A
reconsider ln = l | (Seg n) as FinSequence of the carrier of R by FINSEQ_1:18;
now__::_thesis:_for_i_being_set_st_i_in_dom_ln_holds_
ex_u,_v_being_Element_of_R_ex_a_being_Element_of_A_st_ln_/._i_=_(u_*_a)_*_v
let i be set ; ::_thesis: ( i in dom ln implies ex u, v being Element of R ex a being Element of A st ln /. i = (u * a) * v )
assume A1: i in dom ln ; ::_thesis: ex u, v being Element of R ex a being Element of A st ln /. i = (u * a) * v
A2: dom ln c= dom l by RELAT_1:60;
then consider u, v being Element of R, a being Element of A such that
A3: l /. i = (u * a) * v by A1, Def8;
take u = u; ::_thesis: ex v being Element of R ex a being Element of A st ln /. i = (u * a) * v
take v = v; ::_thesis: ex a being Element of A st ln /. i = (u * a) * v
take a = a; ::_thesis: ln /. i = (u * a) * v
thus ln /. i = ln . i by A1, PARTFUN1:def_6
.= l . i by A1, FUNCT_1:47
.= (u * a) * v by A1, A2, A3, PARTFUN1:def_6 ; ::_thesis: verum
end;
hence l | (Seg n) is LinearCombination of A by Def8; ::_thesis: verum
end;
theorem :: IDEAL_1:42
for R being non empty multLoopStr
for A being non empty Subset of R
for l being LeftLinearCombination of A
for n being Element of NAT holds l | (Seg n) is LeftLinearCombination of A
proof
let R be non empty multLoopStr ; ::_thesis: for A being non empty Subset of R
for l being LeftLinearCombination of A
for n being Element of NAT holds l | (Seg n) is LeftLinearCombination of A
let A be non empty Subset of R; ::_thesis: for l being LeftLinearCombination of A
for n being Element of NAT holds l | (Seg n) is LeftLinearCombination of A
let l be LeftLinearCombination of A; ::_thesis: for n being Element of NAT holds l | (Seg n) is LeftLinearCombination of A
let n be Element of NAT ; ::_thesis: l | (Seg n) is LeftLinearCombination of A
reconsider ln = l | (Seg n) as FinSequence of the carrier of R by FINSEQ_1:18;
now__::_thesis:_for_i_being_set_st_i_in_dom_ln_holds_
ex_u_being_Element_of_R_ex_a_being_Element_of_A_st_ln_/._i_=_u_*_a
let i be set ; ::_thesis: ( i in dom ln implies ex u being Element of R ex a being Element of A st ln /. i = u * a )
assume A1: i in dom ln ; ::_thesis: ex u being Element of R ex a being Element of A st ln /. i = u * a
A2: dom ln c= dom l by RELAT_1:60;
then consider u being Element of R, a being Element of A such that
A3: l /. i = u * a by A1, Def9;
take u = u; ::_thesis: ex a being Element of A st ln /. i = u * a
take a = a; ::_thesis: ln /. i = u * a
thus ln /. i = ln . i by A1, PARTFUN1:def_6
.= l . i by A1, FUNCT_1:47
.= u * a by A1, A2, A3, PARTFUN1:def_6 ; ::_thesis: verum
end;
hence l | (Seg n) is LeftLinearCombination of A by Def9; ::_thesis: verum
end;
theorem :: IDEAL_1:43
for R being non empty multLoopStr
for A being non empty Subset of R
for l being RightLinearCombination of A
for n being Element of NAT holds l | (Seg n) is RightLinearCombination of A
proof
let R be non empty multLoopStr ; ::_thesis: for A being non empty Subset of R
for l being RightLinearCombination of A
for n being Element of NAT holds l | (Seg n) is RightLinearCombination of A
let A be non empty Subset of R; ::_thesis: for l being RightLinearCombination of A
for n being Element of NAT holds l | (Seg n) is RightLinearCombination of A
let l be RightLinearCombination of A; ::_thesis: for n being Element of NAT holds l | (Seg n) is RightLinearCombination of A
let n be Element of NAT ; ::_thesis: l | (Seg n) is RightLinearCombination of A
reconsider ln = l | (Seg n) as FinSequence of the carrier of R by FINSEQ_1:18;
now__::_thesis:_for_i_being_set_st_i_in_dom_ln_holds_
ex_v_being_Element_of_R_ex_a_being_Element_of_A_st_ln_/._i_=_a_*_v
let i be set ; ::_thesis: ( i in dom ln implies ex v being Element of R ex a being Element of A st ln /. i = a * v )
assume A1: i in dom ln ; ::_thesis: ex v being Element of R ex a being Element of A st ln /. i = a * v
A2: dom ln c= dom l by RELAT_1:60;
then consider v being Element of R, a being Element of A such that
A3: l /. i = a * v by A1, Def10;
take v = v; ::_thesis: ex a being Element of A st ln /. i = a * v
take a = a; ::_thesis: ln /. i = a * v
thus ln /. i = ln . i by A1, PARTFUN1:def_6
.= l . i by A1, FUNCT_1:47
.= a * v by A1, A2, A3, PARTFUN1:def_6 ; ::_thesis: verum
end;
hence l | (Seg n) is RightLinearCombination of A by Def10; ::_thesis: verum
end;
begin
definition
let L be non empty doubleLoopStr ;
let F be Subset of L;
assume A1: not F is empty ;
funcF -Ideal -> Ideal of L means :Def14: :: IDEAL_1:def 14
( F c= it & ( for I being Ideal of L st F c= I holds
it c= I ) );
existence
ex b1 being Ideal of L st
( F c= b1 & ( for I being Ideal of L st F c= I holds
b1 c= I ) )
proof
set Id = { I where I is Subset of L : ( F c= I & I is Ideal of L ) } ;
set I = meet { I where I is Subset of L : ( F c= I & I is Ideal of L ) } ;
the carrier of L is Ideal of L by Th10;
then A2: the carrier of L in { I where I is Subset of L : ( F c= I & I is Ideal of L ) } ;
A3: now__::_thesis:_for_X_being_set_st_X_in__{__I_where_I_is_Subset_of_L_:_(_F_c=_I_&_I_is_Ideal_of_L_)__}__holds_
F_c=_X
let X be set ; ::_thesis: ( X in { I where I is Subset of L : ( F c= I & I is Ideal of L ) } implies F c= X )
assume X in { I where I is Subset of L : ( F c= I & I is Ideal of L ) } ; ::_thesis: F c= X
then ex X9 being Subset of L st
( X9 = X & F c= X9 & X9 is Ideal of L ) ;
hence F c= X ; ::_thesis: verum
end;
then F c= meet { I where I is Subset of L : ( F c= I & I is Ideal of L ) } by A2, SETFAM_1:5;
then reconsider I = meet { I where I is Subset of L : ( F c= I & I is Ideal of L ) } as non empty Subset of L by A1, A2, SETFAM_1:3;
A4: I is add-closed
proof
let x, y be Element of L; :: according to IDEAL_1:def_1 ::_thesis: ( x in I & y in I implies x + y in I )
assume A5: ( x in I & y in I ) ; ::_thesis: x + y in I
now__::_thesis:_for_X_being_set_st_X_in__{__I_where_I_is_Subset_of_L_:_(_F_c=_I_&_I_is_Ideal_of_L_)__}__holds_
{(x_+_y)}_c=_X
let X be set ; ::_thesis: ( X in { I where I is Subset of L : ( F c= I & I is Ideal of L ) } implies {(x + y)} c= X )
assume A6: X in { I where I is Subset of L : ( F c= I & I is Ideal of L ) } ; ::_thesis: {(x + y)} c= X
then consider X9 being Subset of L such that
A7: X9 = X and
F c= X9 and
A8: X9 is Ideal of L ;
( x in X & y in X ) by A5, A6, SETFAM_1:def_1;
then x + y in X9 by A7, A8, Def1;
hence {(x + y)} c= X by A7, ZFMISC_1:31; ::_thesis: verum
end;
then {(x + y)} c= I by A2, SETFAM_1:5;
hence x + y in I by ZFMISC_1:31; ::_thesis: verum
end;
A9: I is left-ideal
proof
let p, x be Element of L; :: according to IDEAL_1:def_2 ::_thesis: ( x in I implies p * x in I )
assume A10: x in I ; ::_thesis: p * x in I
now__::_thesis:_for_X_being_set_st_X_in__{__I_where_I_is_Subset_of_L_:_(_F_c=_I_&_I_is_Ideal_of_L_)__}__holds_
{(p_*_x)}_c=_X
let X be set ; ::_thesis: ( X in { I where I is Subset of L : ( F c= I & I is Ideal of L ) } implies {(p * x)} c= X )
assume A11: X in { I where I is Subset of L : ( F c= I & I is Ideal of L ) } ; ::_thesis: {(p * x)} c= X
then consider X9 being Subset of L such that
A12: X9 = X and
F c= X9 and
A13: X9 is Ideal of L ;
x in X by A10, A11, SETFAM_1:def_1;
then p * x in X9 by A12, A13, Def2;
hence {(p * x)} c= X by A12, ZFMISC_1:31; ::_thesis: verum
end;
then {(p * x)} c= I by A2, SETFAM_1:5;
hence p * x in I by ZFMISC_1:31; ::_thesis: verum
end;
I is right-ideal
proof
let p, x be Element of L; :: according to IDEAL_1:def_3 ::_thesis: ( x in I implies x * p in I )
assume A14: x in I ; ::_thesis: x * p in I
now__::_thesis:_for_X_being_set_st_X_in__{__I_where_I_is_Subset_of_L_:_(_F_c=_I_&_I_is_Ideal_of_L_)__}__holds_
{(x_*_p)}_c=_X
let X be set ; ::_thesis: ( X in { I where I is Subset of L : ( F c= I & I is Ideal of L ) } implies {(x * p)} c= X )
assume A15: X in { I where I is Subset of L : ( F c= I & I is Ideal of L ) } ; ::_thesis: {(x * p)} c= X
then consider X9 being Subset of L such that
A16: X9 = X and
F c= X9 and
A17: X9 is Ideal of L ;
x in X by A14, A15, SETFAM_1:def_1;
then x * p in X9 by A16, A17, Def3;
hence {(x * p)} c= X by A16, ZFMISC_1:31; ::_thesis: verum
end;
then {(x * p)} c= I by A2, SETFAM_1:5;
hence x * p in I by ZFMISC_1:31; ::_thesis: verum
end;
then reconsider I = I as Ideal of L by A4, A9;
take I ; ::_thesis: ( F c= I & ( for I being Ideal of L st F c= I holds
I c= I ) )
now__::_thesis:_for_X_being_Ideal_of_L_st_F_c=_X_holds_
I_c=_X
let X be Ideal of L; ::_thesis: ( F c= X implies I c= X )
assume F c= X ; ::_thesis: I c= X
then X in { I where I is Subset of L : ( F c= I & I is Ideal of L ) } ;
hence I c= X by SETFAM_1:3; ::_thesis: verum
end;
hence ( F c= I & ( for I being Ideal of L st F c= I holds
I c= I ) ) by A2, A3, SETFAM_1:5; ::_thesis: verum
end;
uniqueness
for b1, b2 being Ideal of L st F c= b1 & ( for I being Ideal of L st F c= I holds
b1 c= I ) & F c= b2 & ( for I being Ideal of L st F c= I holds
b2 c= I ) holds
b1 = b2
proof
let X, Y be Ideal of L; ::_thesis: ( F c= X & ( for I being Ideal of L st F c= I holds
X c= I ) & F c= Y & ( for I being Ideal of L st F c= I holds
Y c= I ) implies X = Y )
assume ( F c= X & ( for I being Ideal of L st F c= I holds
X c= I ) & F c= Y & ( for I being Ideal of L st F c= I holds
Y c= I ) ) ; ::_thesis: X = Y
then ( X c= Y & Y c= X ) ;
hence X = Y by XBOOLE_0:def_10; ::_thesis: verum
end;
funcF -LeftIdeal -> LeftIdeal of L means :Def15: :: IDEAL_1:def 15
( F c= it & ( for I being LeftIdeal of L st F c= I holds
it c= I ) );
existence
ex b1 being LeftIdeal of L st
( F c= b1 & ( for I being LeftIdeal of L st F c= I holds
b1 c= I ) )
proof
set Id = { I where I is Subset of L : ( F c= I & I is LeftIdeal of L ) } ;
set I = meet { I where I is Subset of L : ( F c= I & I is LeftIdeal of L ) } ;
the carrier of L is LeftIdeal of L by Th11;
then A18: the carrier of L in { I where I is Subset of L : ( F c= I & I is LeftIdeal of L ) } ;
A19: now__::_thesis:_for_X_being_set_st_X_in__{__I_where_I_is_Subset_of_L_:_(_F_c=_I_&_I_is_LeftIdeal_of_L_)__}__holds_
F_c=_X
let X be set ; ::_thesis: ( X in { I where I is Subset of L : ( F c= I & I is LeftIdeal of L ) } implies F c= X )
assume X in { I where I is Subset of L : ( F c= I & I is LeftIdeal of L ) } ; ::_thesis: F c= X
then ex X9 being Subset of L st
( X9 = X & F c= X9 & X9 is LeftIdeal of L ) ;
hence F c= X ; ::_thesis: verum
end;
then F c= meet { I where I is Subset of L : ( F c= I & I is LeftIdeal of L ) } by A18, SETFAM_1:5;
then reconsider I = meet { I where I is Subset of L : ( F c= I & I is LeftIdeal of L ) } as non empty Subset of L by A1, A18, SETFAM_1:3;
A20: I is add-closed
proof
let x, y be Element of L; :: according to IDEAL_1:def_1 ::_thesis: ( x in I & y in I implies x + y in I )
assume A21: ( x in I & y in I ) ; ::_thesis: x + y in I
now__::_thesis:_for_X_being_set_st_X_in__{__I_where_I_is_Subset_of_L_:_(_F_c=_I_&_I_is_LeftIdeal_of_L_)__}__holds_
{(x_+_y)}_c=_X
let X be set ; ::_thesis: ( X in { I where I is Subset of L : ( F c= I & I is LeftIdeal of L ) } implies {(x + y)} c= X )
assume A22: X in { I where I is Subset of L : ( F c= I & I is LeftIdeal of L ) } ; ::_thesis: {(x + y)} c= X
then consider X9 being Subset of L such that
A23: X9 = X and
F c= X9 and
A24: X9 is LeftIdeal of L ;
( x in X & y in X ) by A21, A22, SETFAM_1:def_1;
then x + y in X9 by A23, A24, Def1;
hence {(x + y)} c= X by A23, ZFMISC_1:31; ::_thesis: verum
end;
then {(x + y)} c= I by A18, SETFAM_1:5;
hence x + y in I by ZFMISC_1:31; ::_thesis: verum
end;
I is left-ideal
proof
let p, x be Element of L; :: according to IDEAL_1:def_2 ::_thesis: ( x in I implies p * x in I )
assume A25: x in I ; ::_thesis: p * x in I
now__::_thesis:_for_X_being_set_st_X_in__{__I_where_I_is_Subset_of_L_:_(_F_c=_I_&_I_is_LeftIdeal_of_L_)__}__holds_
{(p_*_x)}_c=_X
let X be set ; ::_thesis: ( X in { I where I is Subset of L : ( F c= I & I is LeftIdeal of L ) } implies {(p * x)} c= X )
assume A26: X in { I where I is Subset of L : ( F c= I & I is LeftIdeal of L ) } ; ::_thesis: {(p * x)} c= X
then consider X9 being Subset of L such that
A27: X9 = X and
F c= X9 and
A28: X9 is LeftIdeal of L ;
x in X by A25, A26, SETFAM_1:def_1;
then p * x in X9 by A27, A28, Def2;
hence {(p * x)} c= X by A27, ZFMISC_1:31; ::_thesis: verum
end;
then {(p * x)} c= I by A18, SETFAM_1:5;
hence p * x in I by ZFMISC_1:31; ::_thesis: verum
end;
then reconsider I = I as LeftIdeal of L by A20;
take I ; ::_thesis: ( F c= I & ( for I being LeftIdeal of L st F c= I holds
I c= I ) )
now__::_thesis:_for_X_being_LeftIdeal_of_L_st_F_c=_X_holds_
I_c=_X
let X be LeftIdeal of L; ::_thesis: ( F c= X implies I c= X )
assume F c= X ; ::_thesis: I c= X
then X in { I where I is Subset of L : ( F c= I & I is LeftIdeal of L ) } ;
hence I c= X by SETFAM_1:3; ::_thesis: verum
end;
hence ( F c= I & ( for I being LeftIdeal of L st F c= I holds
I c= I ) ) by A18, A19, SETFAM_1:5; ::_thesis: verum
end;
uniqueness
for b1, b2 being LeftIdeal of L st F c= b1 & ( for I being LeftIdeal of L st F c= I holds
b1 c= I ) & F c= b2 & ( for I being LeftIdeal of L st F c= I holds
b2 c= I ) holds
b1 = b2
proof
let X, Y be LeftIdeal of L; ::_thesis: ( F c= X & ( for I being LeftIdeal of L st F c= I holds
X c= I ) & F c= Y & ( for I being LeftIdeal of L st F c= I holds
Y c= I ) implies X = Y )
assume ( F c= X & ( for I being LeftIdeal of L st F c= I holds
X c= I ) & F c= Y & ( for I being LeftIdeal of L st F c= I holds
Y c= I ) ) ; ::_thesis: X = Y
then ( X c= Y & Y c= X ) ;
hence X = Y by XBOOLE_0:def_10; ::_thesis: verum
end;
funcF -RightIdeal -> RightIdeal of L means :Def16: :: IDEAL_1:def 16
( F c= it & ( for I being RightIdeal of L st F c= I holds
it c= I ) );
existence
ex b1 being RightIdeal of L st
( F c= b1 & ( for I being RightIdeal of L st F c= I holds
b1 c= I ) )
proof
set Id = { I where I is Subset of L : ( F c= I & I is RightIdeal of L ) } ;
set I = meet { I where I is Subset of L : ( F c= I & I is RightIdeal of L ) } ;
the carrier of L is RightIdeal of L by Th12;
then A29: the carrier of L in { I where I is Subset of L : ( F c= I & I is RightIdeal of L ) } ;
A30: now__::_thesis:_for_X_being_set_st_X_in__{__I_where_I_is_Subset_of_L_:_(_F_c=_I_&_I_is_RightIdeal_of_L_)__}__holds_
F_c=_X
let X be set ; ::_thesis: ( X in { I where I is Subset of L : ( F c= I & I is RightIdeal of L ) } implies F c= X )
assume X in { I where I is Subset of L : ( F c= I & I is RightIdeal of L ) } ; ::_thesis: F c= X
then ex X9 being Subset of L st
( X9 = X & F c= X9 & X9 is RightIdeal of L ) ;
hence F c= X ; ::_thesis: verum
end;
then F c= meet { I where I is Subset of L : ( F c= I & I is RightIdeal of L ) } by A29, SETFAM_1:5;
then reconsider I = meet { I where I is Subset of L : ( F c= I & I is RightIdeal of L ) } as non empty Subset of L by A1, A29, SETFAM_1:3;
A31: I is add-closed
proof
let x, y be Element of L; :: according to IDEAL_1:def_1 ::_thesis: ( x in I & y in I implies x + y in I )
assume A32: ( x in I & y in I ) ; ::_thesis: x + y in I
now__::_thesis:_for_X_being_set_st_X_in__{__I_where_I_is_Subset_of_L_:_(_F_c=_I_&_I_is_RightIdeal_of_L_)__}__holds_
{(x_+_y)}_c=_X
let X be set ; ::_thesis: ( X in { I where I is Subset of L : ( F c= I & I is RightIdeal of L ) } implies {(x + y)} c= X )
assume A33: X in { I where I is Subset of L : ( F c= I & I is RightIdeal of L ) } ; ::_thesis: {(x + y)} c= X
then consider X9 being Subset of L such that
A34: X9 = X and
F c= X9 and
A35: X9 is RightIdeal of L ;
( x in X & y in X ) by A32, A33, SETFAM_1:def_1;
then x + y in X9 by A34, A35, Def1;
hence {(x + y)} c= X by A34, ZFMISC_1:31; ::_thesis: verum
end;
then {(x + y)} c= I by A29, SETFAM_1:5;
hence x + y in I by ZFMISC_1:31; ::_thesis: verum
end;
I is right-ideal
proof
let p, x be Element of L; :: according to IDEAL_1:def_3 ::_thesis: ( x in I implies x * p in I )
assume A36: x in I ; ::_thesis: x * p in I
now__::_thesis:_for_X_being_set_st_X_in__{__I_where_I_is_Subset_of_L_:_(_F_c=_I_&_I_is_RightIdeal_of_L_)__}__holds_
{(x_*_p)}_c=_X
let X be set ; ::_thesis: ( X in { I where I is Subset of L : ( F c= I & I is RightIdeal of L ) } implies {(x * p)} c= X )
assume A37: X in { I where I is Subset of L : ( F c= I & I is RightIdeal of L ) } ; ::_thesis: {(x * p)} c= X
then consider X9 being Subset of L such that
A38: X9 = X and
F c= X9 and
A39: X9 is RightIdeal of L ;
x in X by A36, A37, SETFAM_1:def_1;
then x * p in X9 by A38, A39, Def3;
hence {(x * p)} c= X by A38, ZFMISC_1:31; ::_thesis: verum
end;
then {(x * p)} c= I by A29, SETFAM_1:5;
hence x * p in I by ZFMISC_1:31; ::_thesis: verum
end;
then reconsider I = I as RightIdeal of L by A31;
take I ; ::_thesis: ( F c= I & ( for I being RightIdeal of L st F c= I holds
I c= I ) )
now__::_thesis:_for_X_being_RightIdeal_of_L_st_F_c=_X_holds_
I_c=_X
let X be RightIdeal of L; ::_thesis: ( F c= X implies I c= X )
assume F c= X ; ::_thesis: I c= X
then X in { I where I is Subset of L : ( F c= I & I is RightIdeal of L ) } ;
hence I c= X by SETFAM_1:3; ::_thesis: verum
end;
hence ( F c= I & ( for I being RightIdeal of L st F c= I holds
I c= I ) ) by A29, A30, SETFAM_1:5; ::_thesis: verum
end;
uniqueness
for b1, b2 being RightIdeal of L st F c= b1 & ( for I being RightIdeal of L st F c= I holds
b1 c= I ) & F c= b2 & ( for I being RightIdeal of L st F c= I holds
b2 c= I ) holds
b1 = b2
proof
let X, Y be RightIdeal of L; ::_thesis: ( F c= X & ( for I being RightIdeal of L st F c= I holds
X c= I ) & F c= Y & ( for I being RightIdeal of L st F c= I holds
Y c= I ) implies X = Y )
assume ( F c= X & ( for I being RightIdeal of L st F c= I holds
X c= I ) & F c= Y & ( for I being RightIdeal of L st F c= I holds
Y c= I ) ) ; ::_thesis: X = Y
then ( X c= Y & Y c= X ) ;
hence X = Y by XBOOLE_0:def_10; ::_thesis: verum
end;
end;
:: deftheorem Def14 defines -Ideal IDEAL_1:def_14_:_
for L being non empty doubleLoopStr
for F being Subset of L st not F is empty holds
for b3 being Ideal of L holds
( b3 = F -Ideal iff ( F c= b3 & ( for I being Ideal of L st F c= I holds
b3 c= I ) ) );
:: deftheorem Def15 defines -LeftIdeal IDEAL_1:def_15_:_
for L being non empty doubleLoopStr
for F being Subset of L st not F is empty holds
for b3 being LeftIdeal of L holds
( b3 = F -LeftIdeal iff ( F c= b3 & ( for I being LeftIdeal of L st F c= I holds
b3 c= I ) ) );
:: deftheorem Def16 defines -RightIdeal IDEAL_1:def_16_:_
for L being non empty doubleLoopStr
for F being Subset of L st not F is empty holds
for b3 being RightIdeal of L holds
( b3 = F -RightIdeal iff ( F c= b3 & ( for I being RightIdeal of L st F c= I holds
b3 c= I ) ) );
theorem Th44: :: IDEAL_1:44
for L being non empty doubleLoopStr
for I being Ideal of L holds I -Ideal = I
proof
let L be non empty doubleLoopStr ; ::_thesis: for I being Ideal of L holds I -Ideal = I
let I be Ideal of L; ::_thesis: I -Ideal = I
( I c= I -Ideal & I -Ideal c= I ) by Def14;
hence I -Ideal = I by XBOOLE_0:def_10; ::_thesis: verum
end;
theorem Th45: :: IDEAL_1:45
for L being non empty doubleLoopStr
for I being LeftIdeal of L holds I -LeftIdeal = I
proof
let L be non empty doubleLoopStr ; ::_thesis: for I being LeftIdeal of L holds I -LeftIdeal = I
let I be LeftIdeal of L; ::_thesis: I -LeftIdeal = I
( I c= I -LeftIdeal & I -LeftIdeal c= I ) by Def15;
hence I -LeftIdeal = I by XBOOLE_0:def_10; ::_thesis: verum
end;
theorem Th46: :: IDEAL_1:46
for L being non empty doubleLoopStr
for I being RightIdeal of L holds I -RightIdeal = I
proof
let L be non empty doubleLoopStr ; ::_thesis: for I being RightIdeal of L holds I -RightIdeal = I
let I be RightIdeal of L; ::_thesis: I -RightIdeal = I
( I c= I -RightIdeal & I -RightIdeal c= I ) by Def16;
hence I -RightIdeal = I by XBOOLE_0:def_10; ::_thesis: verum
end;
definition
let L be non empty doubleLoopStr ;
let I be Ideal of L;
mode Basis of I -> non empty Subset of L means :: IDEAL_1:def 17
it -Ideal = I;
existence
ex b1 being non empty Subset of L st b1 -Ideal = I
proof
take I ; ::_thesis: I -Ideal = I
thus I -Ideal = I by Th44; ::_thesis: verum
end;
end;
:: deftheorem defines Basis IDEAL_1:def_17_:_
for L being non empty doubleLoopStr
for I being Ideal of L
for b3 being non empty Subset of L holds
( b3 is Basis of I iff b3 -Ideal = I );
theorem :: IDEAL_1:47
for L being non empty right_complementable add-associative right_zeroed distributive doubleLoopStr holds {(0. L)} -Ideal = {(0. L)} by Th44;
theorem :: IDEAL_1:48
for L being non empty add-cancelable right_zeroed distributive left_zeroed doubleLoopStr holds {(0. L)} -Ideal = {(0. L)} by Th44;
theorem :: IDEAL_1:49
for L being non empty right_add-cancelable right_zeroed right-distributive left_zeroed doubleLoopStr holds {(0. L)} -LeftIdeal = {(0. L)} by Th45;
theorem :: IDEAL_1:50
for L being non empty left_add-cancelable right_zeroed left-distributive doubleLoopStr holds {(0. L)} -RightIdeal = {(0. L)} by Th46;
theorem :: IDEAL_1:51
for L being non empty well-unital doubleLoopStr holds {(1. L)} -Ideal = the carrier of L
proof
let L be non empty well-unital doubleLoopStr ; ::_thesis: {(1. L)} -Ideal = the carrier of L
the carrier of L c= {(1. L)} -Ideal
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in the carrier of L or x in {(1. L)} -Ideal )
assume x in the carrier of L ; ::_thesis: x in {(1. L)} -Ideal
then reconsider x9 = x as Element of L ;
( 1. L in {(1. L)} & {(1. L)} c= {(1. L)} -Ideal ) by Def14, TARSKI:def_1;
then x9 * (1. L) in {(1. L)} -Ideal by Def2;
hence x in {(1. L)} -Ideal by VECTSP_1:def_6; ::_thesis: verum
end;
hence {(1. L)} -Ideal = the carrier of L by XBOOLE_0:def_10; ::_thesis: verum
end;
theorem :: IDEAL_1:52
for L being non empty right_unital doubleLoopStr holds {(1. L)} -LeftIdeal = the carrier of L
proof
let L be non empty right_unital doubleLoopStr ; ::_thesis: {(1. L)} -LeftIdeal = the carrier of L
the carrier of L c= {(1. L)} -LeftIdeal
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in the carrier of L or x in {(1. L)} -LeftIdeal )
assume x in the carrier of L ; ::_thesis: x in {(1. L)} -LeftIdeal
then reconsider x9 = x as Element of L ;
( 1. L in {(1. L)} & {(1. L)} c= {(1. L)} -LeftIdeal ) by Def15, TARSKI:def_1;
then x9 * (1. L) in {(1. L)} -LeftIdeal by Def2;
hence x in {(1. L)} -LeftIdeal by VECTSP_1:def_4; ::_thesis: verum
end;
hence {(1. L)} -LeftIdeal = the carrier of L by XBOOLE_0:def_10; ::_thesis: verum
end;
theorem :: IDEAL_1:53
for L being non empty left_unital doubleLoopStr holds {(1. L)} -RightIdeal = the carrier of L
proof
let L be non empty left_unital doubleLoopStr ; ::_thesis: {(1. L)} -RightIdeal = the carrier of L
the carrier of L c= {(1. L)} -RightIdeal
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in the carrier of L or x in {(1. L)} -RightIdeal )
assume x in the carrier of L ; ::_thesis: x in {(1. L)} -RightIdeal
then reconsider x9 = x as Element of L ;
( 1. L in {(1. L)} & {(1. L)} c= {(1. L)} -RightIdeal ) by Def16, TARSKI:def_1;
then (1. L) * x9 in {(1. L)} -RightIdeal by Def3;
hence x in {(1. L)} -RightIdeal by VECTSP_1:def_8; ::_thesis: verum
end;
hence {(1. L)} -RightIdeal = the carrier of L by XBOOLE_0:def_10; ::_thesis: verum
end;
theorem :: IDEAL_1:54
for L being non empty doubleLoopStr holds ([#] L) -Ideal = the carrier of L
proof
let L be non empty doubleLoopStr ; ::_thesis: ([#] L) -Ideal = the carrier of L
[#] L c= ([#] L) -Ideal by Def14;
hence ([#] L) -Ideal = the carrier of L by XBOOLE_0:def_10; ::_thesis: verum
end;
theorem :: IDEAL_1:55
for L being non empty doubleLoopStr holds ([#] L) -LeftIdeal = the carrier of L
proof
let L be non empty doubleLoopStr ; ::_thesis: ([#] L) -LeftIdeal = the carrier of L
[#] L c= ([#] L) -LeftIdeal by Def15;
hence ([#] L) -LeftIdeal = the carrier of L by XBOOLE_0:def_10; ::_thesis: verum
end;
theorem :: IDEAL_1:56
for L being non empty doubleLoopStr holds ([#] L) -RightIdeal = the carrier of L
proof
let L be non empty doubleLoopStr ; ::_thesis: ([#] L) -RightIdeal = the carrier of L
[#] L c= ([#] L) -RightIdeal by Def16;
hence ([#] L) -RightIdeal = the carrier of L by XBOOLE_0:def_10; ::_thesis: verum
end;
theorem Th57: :: IDEAL_1:57
for L being non empty doubleLoopStr
for A, B being non empty Subset of L st A c= B holds
A -Ideal c= B -Ideal
proof
let L be non empty doubleLoopStr ; ::_thesis: for A, B being non empty Subset of L st A c= B holds
A -Ideal c= B -Ideal
let A, B be non empty Subset of L; ::_thesis: ( A c= B implies A -Ideal c= B -Ideal )
assume A1: A c= B ; ::_thesis: A -Ideal c= B -Ideal
B c= B -Ideal by Def14;
then A c= B -Ideal by A1, XBOOLE_1:1;
hence A -Ideal c= B -Ideal by Def14; ::_thesis: verum
end;
theorem :: IDEAL_1:58
for L being non empty doubleLoopStr
for A, B being non empty Subset of L st A c= B holds
A -LeftIdeal c= B -LeftIdeal
proof
let L be non empty doubleLoopStr ; ::_thesis: for A, B being non empty Subset of L st A c= B holds
A -LeftIdeal c= B -LeftIdeal
let A, B be non empty Subset of L; ::_thesis: ( A c= B implies A -LeftIdeal c= B -LeftIdeal )
assume A1: A c= B ; ::_thesis: A -LeftIdeal c= B -LeftIdeal
B c= B -LeftIdeal by Def15;
then A c= B -LeftIdeal by A1, XBOOLE_1:1;
hence A -LeftIdeal c= B -LeftIdeal by Def15; ::_thesis: verum
end;
theorem :: IDEAL_1:59
for L being non empty doubleLoopStr
for A, B being non empty Subset of L st A c= B holds
A -RightIdeal c= B -RightIdeal
proof
let L be non empty doubleLoopStr ; ::_thesis: for A, B being non empty Subset of L st A c= B holds
A -RightIdeal c= B -RightIdeal
let A, B be non empty Subset of L; ::_thesis: ( A c= B implies A -RightIdeal c= B -RightIdeal )
assume A1: A c= B ; ::_thesis: A -RightIdeal c= B -RightIdeal
B c= B -RightIdeal by Def16;
then A c= B -RightIdeal by A1, XBOOLE_1:1;
hence A -RightIdeal c= B -RightIdeal by Def16; ::_thesis: verum
end;
theorem Th60: :: IDEAL_1:60
for L being non empty add-cancelable add-associative right_zeroed associative well-unital distributive left_zeroed doubleLoopStr
for F being non empty Subset of L
for x being set holds
( x in F -Ideal iff ex f being LinearCombination of F st x = Sum f )
proof
let L be non empty add-cancelable add-associative right_zeroed associative well-unital distributive left_zeroed doubleLoopStr ; ::_thesis: for F being non empty Subset of L
for x being set holds
( x in F -Ideal iff ex f being LinearCombination of F st x = Sum f )
let F be non empty Subset of L; ::_thesis: for x being set holds
( x in F -Ideal iff ex f being LinearCombination of F st x = Sum f )
set I = { x where x is Element of L : ex lc being LinearCombination of F st Sum lc = x } ;
A1: { x where x is Element of L : ex lc being LinearCombination of F st Sum lc = x } c= the carrier of L
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in { x where x is Element of L : ex lc being LinearCombination of F st Sum lc = x } or x in the carrier of L )
assume x in { x where x is Element of L : ex lc being LinearCombination of F st Sum lc = x } ; ::_thesis: x in the carrier of L
then ex x9 being Element of L st
( x9 = x & ex lc being LinearCombination of F st Sum lc = x9 ) ;
hence x in the carrier of L ; ::_thesis: verum
end;
let x be set ; ::_thesis: ( x in F -Ideal iff ex f being LinearCombination of F st x = Sum f )
A2: F c= { x where x is Element of L : ex lc being LinearCombination of F st Sum lc = x }
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in F or x in { x where x is Element of L : ex lc being LinearCombination of F st Sum lc = x } )
assume A3: x in F ; ::_thesis: x in { x where x is Element of L : ex lc being LinearCombination of F st Sum lc = x }
then reconsider x = x as Element of L ;
set lc = <*x*>;
now__::_thesis:_for_i_being_set_st_i_in_dom_<*x*>_holds_
ex_u,_v_being_Element_of_L_ex_a_being_Element_of_F_st_<*x*>_/._i_=_(u_*_a)_*_v
let i be set ; ::_thesis: ( i in dom <*x*> implies ex u, v being Element of L ex a being Element of F st <*x*> /. i = (u * a) * v )
assume A4: i in dom <*x*> ; ::_thesis: ex u, v being Element of L ex a being Element of F st <*x*> /. i = (u * a) * v
dom <*x*> = {1} by FINSEQ_1:2, FINSEQ_1:38;
then i = 1 by A4, TARSKI:def_1;
then <*x*> . i = x by FINSEQ_1:40
.= (1. L) * x by VECTSP_1:def_8
.= ((1. L) * x) * (1. L) by VECTSP_1:def_4 ;
hence ex u, v being Element of L ex a being Element of F st <*x*> /. i = (u * a) * v by A3, A4, PARTFUN1:def_6; ::_thesis: verum
end;
then reconsider lc = <*x*> as LinearCombination of F by Def8;
Sum lc = x by BINOM:3;
hence x in { x where x is Element of L : ex lc being LinearCombination of F st Sum lc = x } ; ::_thesis: verum
end;
A5: { x where x is Element of L : ex lc being LinearCombination of F st Sum lc = x } c= F -Ideal
proof
defpred S1[ Nat] means for lc being LinearCombination of F st len lc <= $1 holds
Sum lc in F -Ideal ;
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in { x where x is Element of L : ex lc being LinearCombination of F st Sum lc = x } or x in F -Ideal )
assume x in { x where x is Element of L : ex lc being LinearCombination of F st Sum lc = x } ; ::_thesis: x in F -Ideal
then consider x9 being Element of L such that
A6: x9 = x and
A7: ex lc being LinearCombination of F st Sum lc = x9 ;
consider lc being LinearCombination of F such that
A8: Sum lc = x9 by A7;
A9: for k being Element of NAT st S1[k] holds
S1[k + 1]
proof
let k be Element of NAT ; ::_thesis: ( S1[k] implies S1[k + 1] )
assume A10: S1[k] ; ::_thesis: S1[k + 1]
thus S1[k + 1] ::_thesis: verum
proof
let lc be LinearCombination of F; ::_thesis: ( len lc <= k + 1 implies Sum lc in F -Ideal )
assume A11: len lc <= k + 1 ; ::_thesis: Sum lc in F -Ideal
percases ( len lc <= k or len lc = k + 1 ) by A11, NAT_1:8;
suppose len lc <= k ; ::_thesis: Sum lc in F -Ideal
hence Sum lc in F -Ideal by A10; ::_thesis: verum
end;
supposeA12: len lc = k + 1 ; ::_thesis: Sum lc in F -Ideal
then not lc is empty ;
then consider q being LinearCombination of F, r being Element of L such that
A13: lc = q ^ <*r*> and
A14: <*r*> is LinearCombination of F by Th32;
k + 1 = (len q) + (len <*r*>) by A12, A13, FINSEQ_1:22
.= (len q) + 1 by FINSEQ_1:39 ;
then A15: Sum q in F -Ideal by A10;
dom <*r*> = {1} by FINSEQ_1:2, FINSEQ_1:38;
then A16: 1 in dom <*r*> by TARSKI:def_1;
then consider u, v being Element of L, a being Element of F such that
A17: <*r*> /. 1 = (u * a) * v by A14, Def8;
F c= F -Ideal by Def14;
then a in F -Ideal by TARSKI:def_3;
then A18: u * a in F -Ideal by Def2;
A19: <*r*> /. 1 = <*r*> . 1 by A16, PARTFUN1:def_6;
Sum <*r*> = r by BINOM:3
.= (u * a) * v by A17, A19, FINSEQ_1:40 ;
then A20: Sum <*r*> in F -Ideal by A18, Def3;
Sum lc = (Sum q) + (Sum <*r*>) by A13, RLVECT_1:41;
hence Sum lc in F -Ideal by A15, A20, Def1; ::_thesis: verum
end;
end;
end;
end;
A21: S1[ 0 ]
proof
set y = the Element of F;
let lc be LinearCombination of F; ::_thesis: ( len lc <= 0 implies Sum lc in F -Ideal )
assume len lc <= 0 ; ::_thesis: Sum lc in F -Ideal
then lc = <*> the carrier of L ;
then A22: Sum lc = 0. L by RLVECT_1:43;
F c= F -Ideal by Def14;
then A23: the Element of F in F -Ideal by TARSKI:def_3;
(0. L) * the Element of F = 0. L by BINOM:1;
hence Sum lc in F -Ideal by A22, A23, Def2; ::_thesis: verum
end;
for k being Element of NAT holds S1[k] from NAT_1:sch_1(A21, A9);
then S1[ len lc] ;
hence x in F -Ideal by A6, A8; ::_thesis: verum
end;
reconsider I = { x where x is Element of L : ex lc being LinearCombination of F st Sum lc = x } as non empty Subset of L by A2, A1;
reconsider I9 = I as non empty Subset of L ;
A24: I9 is add-closed
proof
let x, y be Element of L; :: according to IDEAL_1:def_1 ::_thesis: ( x in I9 & y in I9 implies x + y in I9 )
assume that
A25: x in I9 and
A26: y in I9 ; ::_thesis: x + y in I9
consider x9 being Element of L such that
A27: x9 = x and
A28: ex lc being LinearCombination of F st Sum lc = x9 by A25;
consider lcx being LinearCombination of F such that
A29: Sum lcx = x9 by A28;
consider y9 being Element of L such that
A30: y9 = y and
A31: ex lc being LinearCombination of F st Sum lc = y9 by A26;
consider lcy being LinearCombination of F such that
A32: Sum lcy = y9 by A31;
Sum (lcx ^ lcy) = x9 + y9 by A29, A32, RLVECT_1:41;
hence x + y in I9 by A27, A30; ::_thesis: verum
end;
A33: I9 is right-ideal
proof
let p, x be Element of L; :: according to IDEAL_1:def_3 ::_thesis: ( x in I9 implies x * p in I9 )
assume x in I9 ; ::_thesis: x * p in I9
then consider x9 being Element of L such that
A34: x9 = x and
A35: ex lc being LinearCombination of F st Sum lc = x9 ;
consider lcx being LinearCombination of F such that
A36: Sum lcx = x9 by A35;
reconsider lcxp = lcx * p as LinearCombination of F by Th24;
x * p = Sum lcxp by A34, A36, BINOM:5;
hence x * p in I9 ; ::_thesis: verum
end;
I9 is left-ideal
proof
let p, x be Element of L; :: according to IDEAL_1:def_2 ::_thesis: ( x in I9 implies p * x in I9 )
assume x in I9 ; ::_thesis: p * x in I9
then consider x9 being Element of L such that
A37: x9 = x and
A38: ex lc being LinearCombination of F st Sum lc = x9 ;
consider lcx being LinearCombination of F such that
A39: Sum lcx = x9 by A38;
reconsider plcx = p * lcx as LinearCombination of F by Th23;
p * x = Sum plcx by A37, A39, BINOM:4;
hence p * x in I9 ; ::_thesis: verum
end;
then F -Ideal c= I by A2, A24, A33, Def14;
then A40: I = F -Ideal by A5, XBOOLE_0:def_10;
hereby ::_thesis: ( ex f being LinearCombination of F st x = Sum f implies x in F -Ideal )
assume x in F -Ideal ; ::_thesis: ex f being LinearCombination of F st x = Sum f
then ex x9 being Element of L st
( x9 = x & ex lc being LinearCombination of F st Sum lc = x9 ) by A40;
hence ex f being LinearCombination of F st x = Sum f ; ::_thesis: verum
end;
assume ex f being LinearCombination of F st x = Sum f ; ::_thesis: x in F -Ideal
hence x in F -Ideal by A40; ::_thesis: verum
end;
theorem Th61: :: IDEAL_1:61
for L being non empty add-cancelable add-associative right_zeroed associative well-unital distributive left_zeroed doubleLoopStr
for F being non empty Subset of L
for x being set holds
( x in F -LeftIdeal iff ex f being LeftLinearCombination of F st x = Sum f )
proof
let L be non empty add-cancelable add-associative right_zeroed associative well-unital distributive left_zeroed doubleLoopStr ; ::_thesis: for F being non empty Subset of L
for x being set holds
( x in F -LeftIdeal iff ex f being LeftLinearCombination of F st x = Sum f )
let F be non empty Subset of L; ::_thesis: for x being set holds
( x in F -LeftIdeal iff ex f being LeftLinearCombination of F st x = Sum f )
set I = { x where x is Element of L : ex lc being LeftLinearCombination of F st Sum lc = x } ;
A1: { x where x is Element of L : ex lc being LeftLinearCombination of F st Sum lc = x } c= the carrier of L
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in { x where x is Element of L : ex lc being LeftLinearCombination of F st Sum lc = x } or x in the carrier of L )
assume x in { x where x is Element of L : ex lc being LeftLinearCombination of F st Sum lc = x } ; ::_thesis: x in the carrier of L
then ex x9 being Element of L st
( x9 = x & ex lc being LeftLinearCombination of F st Sum lc = x9 ) ;
hence x in the carrier of L ; ::_thesis: verum
end;
let x be set ; ::_thesis: ( x in F -LeftIdeal iff ex f being LeftLinearCombination of F st x = Sum f )
A2: F c= { x where x is Element of L : ex lc being LeftLinearCombination of F st Sum lc = x }
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in F or x in { x where x is Element of L : ex lc being LeftLinearCombination of F st Sum lc = x } )
assume A3: x in F ; ::_thesis: x in { x where x is Element of L : ex lc being LeftLinearCombination of F st Sum lc = x }
then reconsider x = x as Element of L ;
set lc = <*x*>;
now__::_thesis:_for_i_being_set_st_i_in_dom_<*x*>_holds_
ex_u_being_Element_of_L_ex_a_being_Element_of_F_st_<*x*>_/._i_=_u_*_a
let i be set ; ::_thesis: ( i in dom <*x*> implies ex u being Element of L ex a being Element of F st <*x*> /. i = u * a )
assume A4: i in dom <*x*> ; ::_thesis: ex u being Element of L ex a being Element of F st <*x*> /. i = u * a
dom <*x*> = {1} by FINSEQ_1:2, FINSEQ_1:38;
then i = 1 by A4, TARSKI:def_1;
then <*x*> . i = x by FINSEQ_1:40
.= (1. L) * x by VECTSP_1:def_8 ;
hence ex u being Element of L ex a being Element of F st <*x*> /. i = u * a by A3, A4, PARTFUN1:def_6; ::_thesis: verum
end;
then reconsider lc = <*x*> as LeftLinearCombination of F by Def9;
Sum lc = x by BINOM:3;
hence x in { x where x is Element of L : ex lc being LeftLinearCombination of F st Sum lc = x } ; ::_thesis: verum
end;
A5: { x where x is Element of L : ex lc being LeftLinearCombination of F st Sum lc = x } c= F -LeftIdeal
proof
defpred S1[ Nat] means for lc being LeftLinearCombination of F st len lc <= $1 holds
Sum lc in F -LeftIdeal ;
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in { x where x is Element of L : ex lc being LeftLinearCombination of F st Sum lc = x } or x in F -LeftIdeal )
assume x in { x where x is Element of L : ex lc being LeftLinearCombination of F st Sum lc = x } ; ::_thesis: x in F -LeftIdeal
then consider x9 being Element of L such that
A6: x9 = x and
A7: ex lc being LeftLinearCombination of F st Sum lc = x9 ;
consider lc being LeftLinearCombination of F such that
A8: Sum lc = x9 by A7;
A9: for k being Element of NAT st S1[k] holds
S1[k + 1]
proof
let k be Element of NAT ; ::_thesis: ( S1[k] implies S1[k + 1] )
assume A10: S1[k] ; ::_thesis: S1[k + 1]
thus S1[k + 1] ::_thesis: verum
proof
let lc be LeftLinearCombination of F; ::_thesis: ( len lc <= k + 1 implies Sum lc in F -LeftIdeal )
assume A11: len lc <= k + 1 ; ::_thesis: Sum lc in F -LeftIdeal
percases ( len lc <= k or len lc = k + 1 ) by A11, NAT_1:8;
suppose len lc <= k ; ::_thesis: Sum lc in F -LeftIdeal
hence Sum lc in F -LeftIdeal by A10; ::_thesis: verum
end;
supposeA12: len lc = k + 1 ; ::_thesis: Sum lc in F -LeftIdeal
then not lc is empty ;
then consider q being LeftLinearCombination of F, r being Element of L such that
A13: lc = q ^ <*r*> and
A14: <*r*> is LeftLinearCombination of F by Th33;
k + 1 = (len q) + (len <*r*>) by A12, A13, FINSEQ_1:22
.= (len q) + 1 by FINSEQ_1:39 ;
then A15: Sum q in F -LeftIdeal by A10;
dom <*r*> = {1} by FINSEQ_1:2, FINSEQ_1:38;
then A16: 1 in dom <*r*> by TARSKI:def_1;
then consider u being Element of L, a being Element of F such that
A17: <*r*> /. 1 = u * a by A14, Def9;
F c= F -LeftIdeal by Def15;
then A18: a in F -LeftIdeal by TARSKI:def_3;
A19: <*r*> /. 1 = <*r*> . 1 by A16, PARTFUN1:def_6;
Sum <*r*> = r by BINOM:3
.= u * a by A17, A19, FINSEQ_1:40 ;
then A20: Sum <*r*> in F -LeftIdeal by A18, Def2;
Sum lc = (Sum q) + (Sum <*r*>) by A13, RLVECT_1:41;
hence Sum lc in F -LeftIdeal by A15, A20, Def1; ::_thesis: verum
end;
end;
end;
end;
A21: S1[ 0 ]
proof
set y = the Element of F;
let lc be LeftLinearCombination of F; ::_thesis: ( len lc <= 0 implies Sum lc in F -LeftIdeal )
assume len lc <= 0 ; ::_thesis: Sum lc in F -LeftIdeal
then lc = <*> the carrier of L ;
then A22: Sum lc = 0. L by RLVECT_1:43;
F c= F -LeftIdeal by Def15;
then A23: the Element of F in F -LeftIdeal by TARSKI:def_3;
(0. L) * the Element of F = 0. L by BINOM:1;
hence Sum lc in F -LeftIdeal by A22, A23, Def2; ::_thesis: verum
end;
for k being Element of NAT holds S1[k] from NAT_1:sch_1(A21, A9);
then S1[ len lc] ;
hence x in F -LeftIdeal by A6, A8; ::_thesis: verum
end;
reconsider I = { x where x is Element of L : ex lc being LeftLinearCombination of F st Sum lc = x } as non empty Subset of L by A2, A1;
reconsider I9 = I as non empty Subset of L ;
A24: I9 is add-closed
proof
let x, y be Element of L; :: according to IDEAL_1:def_1 ::_thesis: ( x in I9 & y in I9 implies x + y in I9 )
assume that
A25: x in I9 and
A26: y in I9 ; ::_thesis: x + y in I9
consider x9 being Element of L such that
A27: x9 = x and
A28: ex lc being LeftLinearCombination of F st Sum lc = x9 by A25;
consider lcx being LeftLinearCombination of F such that
A29: Sum lcx = x9 by A28;
consider y9 being Element of L such that
A30: y9 = y and
A31: ex lc being LeftLinearCombination of F st Sum lc = y9 by A26;
consider lcy being LeftLinearCombination of F such that
A32: Sum lcy = y9 by A31;
Sum (lcx ^ lcy) = x9 + y9 by A29, A32, RLVECT_1:41;
hence x + y in I9 by A27, A30; ::_thesis: verum
end;
I9 is left-ideal
proof
let p, x be Element of L; :: according to IDEAL_1:def_2 ::_thesis: ( x in I9 implies p * x in I9 )
assume x in I9 ; ::_thesis: p * x in I9
then consider x9 being Element of L such that
A33: x9 = x and
A34: ex lc being LeftLinearCombination of F st Sum lc = x9 ;
consider lcx being LeftLinearCombination of F such that
A35: Sum lcx = x9 by A34;
reconsider plcx = p * lcx as LeftLinearCombination of F by Th26;
p * x = Sum plcx by A33, A35, BINOM:4;
hence p * x in I9 ; ::_thesis: verum
end;
then F -LeftIdeal c= I by A2, A24, Def15;
then A36: I = F -LeftIdeal by A5, XBOOLE_0:def_10;
hereby ::_thesis: ( ex f being LeftLinearCombination of F st x = Sum f implies x in F -LeftIdeal )
assume x in F -LeftIdeal ; ::_thesis: ex f being LeftLinearCombination of F st x = Sum f
then ex x9 being Element of L st
( x9 = x & ex lc being LeftLinearCombination of F st Sum lc = x9 ) by A36;
hence ex f being LeftLinearCombination of F st x = Sum f ; ::_thesis: verum
end;
assume ex f being LeftLinearCombination of F st x = Sum f ; ::_thesis: x in F -LeftIdeal
hence x in F -LeftIdeal by A36; ::_thesis: verum
end;
theorem Th62: :: IDEAL_1:62
for L being non empty add-cancelable add-associative right_zeroed associative well-unital distributive left_zeroed doubleLoopStr
for F being non empty Subset of L
for x being set holds
( x in F -RightIdeal iff ex f being RightLinearCombination of F st x = Sum f )
proof
let L be non empty add-cancelable add-associative right_zeroed associative well-unital distributive left_zeroed doubleLoopStr ; ::_thesis: for F being non empty Subset of L
for x being set holds
( x in F -RightIdeal iff ex f being RightLinearCombination of F st x = Sum f )
let F be non empty Subset of L; ::_thesis: for x being set holds
( x in F -RightIdeal iff ex f being RightLinearCombination of F st x = Sum f )
set I = { x where x is Element of L : ex lc being RightLinearCombination of F st Sum lc = x } ;
A1: { x where x is Element of L : ex lc being RightLinearCombination of F st Sum lc = x } c= the carrier of L
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in { x where x is Element of L : ex lc being RightLinearCombination of F st Sum lc = x } or x in the carrier of L )
assume x in { x where x is Element of L : ex lc being RightLinearCombination of F st Sum lc = x } ; ::_thesis: x in the carrier of L
then ex x9 being Element of L st
( x9 = x & ex lc being RightLinearCombination of F st Sum lc = x9 ) ;
hence x in the carrier of L ; ::_thesis: verum
end;
let x be set ; ::_thesis: ( x in F -RightIdeal iff ex f being RightLinearCombination of F st x = Sum f )
A2: F c= { x where x is Element of L : ex lc being RightLinearCombination of F st Sum lc = x }
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in F or x in { x where x is Element of L : ex lc being RightLinearCombination of F st Sum lc = x } )
assume A3: x in F ; ::_thesis: x in { x where x is Element of L : ex lc being RightLinearCombination of F st Sum lc = x }
then reconsider x = x as Element of L ;
set lc = <*x*>;
now__::_thesis:_for_i_being_set_st_i_in_dom_<*x*>_holds_
ex_v_being_Element_of_L_ex_a_being_Element_of_F_st_<*x*>_/._i_=_a_*_v
let i be set ; ::_thesis: ( i in dom <*x*> implies ex v being Element of L ex a being Element of F st <*x*> /. i = a * v )
assume A4: i in dom <*x*> ; ::_thesis: ex v being Element of L ex a being Element of F st <*x*> /. i = a * v
dom <*x*> = {1} by FINSEQ_1:2, FINSEQ_1:38;
then i = 1 by A4, TARSKI:def_1;
then <*x*> . i = x by FINSEQ_1:40
.= x * (1. L) by VECTSP_1:def_4 ;
hence ex v being Element of L ex a being Element of F st <*x*> /. i = a * v by A3, A4, PARTFUN1:def_6; ::_thesis: verum
end;
then reconsider lc = <*x*> as RightLinearCombination of F by Def10;
Sum lc = x by BINOM:3;
hence x in { x where x is Element of L : ex lc being RightLinearCombination of F st Sum lc = x } ; ::_thesis: verum
end;
A5: { x where x is Element of L : ex lc being RightLinearCombination of F st Sum lc = x } c= F -RightIdeal
proof
defpred S1[ Nat] means for lc being RightLinearCombination of F st len lc <= $1 holds
Sum lc in F -RightIdeal ;
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in { x where x is Element of L : ex lc being RightLinearCombination of F st Sum lc = x } or x in F -RightIdeal )
assume x in { x where x is Element of L : ex lc being RightLinearCombination of F st Sum lc = x } ; ::_thesis: x in F -RightIdeal
then consider x9 being Element of L such that
A6: x9 = x and
A7: ex lc being RightLinearCombination of F st Sum lc = x9 ;
consider lc being RightLinearCombination of F such that
A8: Sum lc = x9 by A7;
A9: for k being Element of NAT st S1[k] holds
S1[k + 1]
proof
let k be Element of NAT ; ::_thesis: ( S1[k] implies S1[k + 1] )
assume A10: S1[k] ; ::_thesis: S1[k + 1]
thus S1[k + 1] ::_thesis: verum
proof
let lc be RightLinearCombination of F; ::_thesis: ( len lc <= k + 1 implies Sum lc in F -RightIdeal )
assume A11: len lc <= k + 1 ; ::_thesis: Sum lc in F -RightIdeal
percases ( len lc <= k or len lc = k + 1 ) by A11, NAT_1:8;
suppose len lc <= k ; ::_thesis: Sum lc in F -RightIdeal
hence Sum lc in F -RightIdeal by A10; ::_thesis: verum
end;
supposeA12: len lc = k + 1 ; ::_thesis: Sum lc in F -RightIdeal
then not lc is empty ;
then consider q being RightLinearCombination of F, r being Element of L such that
A13: lc = q ^ <*r*> and
A14: <*r*> is RightLinearCombination of F by Th34;
k + 1 = (len q) + (len <*r*>) by A12, A13, FINSEQ_1:22
.= (len q) + 1 by FINSEQ_1:39 ;
then A15: Sum q in F -RightIdeal by A10;
dom <*r*> = {1} by FINSEQ_1:2, FINSEQ_1:38;
then A16: 1 in dom <*r*> by TARSKI:def_1;
then consider v being Element of L, a being Element of F such that
A17: <*r*> /. 1 = a * v by A14, Def10;
F c= F -RightIdeal by Def16;
then A18: a in F -RightIdeal by TARSKI:def_3;
A19: <*r*> /. 1 = <*r*> . 1 by A16, PARTFUN1:def_6;
Sum <*r*> = r by BINOM:3
.= a * v by A17, A19, FINSEQ_1:40 ;
then A20: Sum <*r*> in F -RightIdeal by A18, Def3;
Sum lc = (Sum q) + (Sum <*r*>) by A13, RLVECT_1:41;
hence Sum lc in F -RightIdeal by A15, A20, Def1; ::_thesis: verum
end;
end;
end;
end;
A21: S1[ 0 ]
proof
set y = the Element of F;
let lc be RightLinearCombination of F; ::_thesis: ( len lc <= 0 implies Sum lc in F -RightIdeal )
assume len lc <= 0 ; ::_thesis: Sum lc in F -RightIdeal
then lc = <*> the carrier of L ;
then A22: Sum lc = 0. L by RLVECT_1:43;
F c= F -RightIdeal by Def16;
then A23: the Element of F in F -RightIdeal by TARSKI:def_3;
the Element of F * (0. L) = 0. L by BINOM:2;
hence Sum lc in F -RightIdeal by A22, A23, Def3; ::_thesis: verum
end;
for k being Element of NAT holds S1[k] from NAT_1:sch_1(A21, A9);
then S1[ len lc] ;
hence x in F -RightIdeal by A6, A8; ::_thesis: verum
end;
reconsider I = { x where x is Element of L : ex lc being RightLinearCombination of F st Sum lc = x } as non empty Subset of L by A2, A1;
reconsider I9 = I as non empty Subset of L ;
A24: I9 is add-closed
proof
let x, y be Element of L; :: according to IDEAL_1:def_1 ::_thesis: ( x in I9 & y in I9 implies x + y in I9 )
assume that
A25: x in I9 and
A26: y in I9 ; ::_thesis: x + y in I9
consider x9 being Element of L such that
A27: x9 = x and
A28: ex lc being RightLinearCombination of F st Sum lc = x9 by A25;
consider lcx being RightLinearCombination of F such that
A29: Sum lcx = x9 by A28;
consider y9 being Element of L such that
A30: y9 = y and
A31: ex lc being RightLinearCombination of F st Sum lc = y9 by A26;
consider lcy being RightLinearCombination of F such that
A32: Sum lcy = y9 by A31;
Sum (lcx ^ lcy) = x9 + y9 by A29, A32, RLVECT_1:41;
hence x + y in I9 by A27, A30; ::_thesis: verum
end;
I9 is right-ideal
proof
let p, x be Element of L; :: according to IDEAL_1:def_3 ::_thesis: ( x in I9 implies x * p in I9 )
assume x in I9 ; ::_thesis: x * p in I9
then consider x9 being Element of L such that
A33: x9 = x and
A34: ex lc being RightLinearCombination of F st Sum lc = x9 ;
consider lcx being RightLinearCombination of F such that
A35: Sum lcx = x9 by A34;
reconsider lcxp = lcx * p as RightLinearCombination of F by Th29;
x * p = Sum lcxp by A33, A35, BINOM:5;
hence x * p in I9 ; ::_thesis: verum
end;
then F -RightIdeal c= I by A2, A24, Def16;
then A36: I = F -RightIdeal by A5, XBOOLE_0:def_10;
hereby ::_thesis: ( ex f being RightLinearCombination of F st x = Sum f implies x in F -RightIdeal )
assume x in F -RightIdeal ; ::_thesis: ex f being RightLinearCombination of F st x = Sum f
then ex x9 being Element of L st
( x9 = x & ex lc being RightLinearCombination of F st Sum lc = x9 ) by A36;
hence ex f being RightLinearCombination of F st x = Sum f ; ::_thesis: verum
end;
assume ex f being RightLinearCombination of F st x = Sum f ; ::_thesis: x in F -RightIdeal
hence x in F -RightIdeal by A36; ::_thesis: verum
end;
theorem Th63: :: IDEAL_1:63
for R being non empty add-cancelable add-associative right_zeroed associative commutative well-unital distributive left_zeroed doubleLoopStr
for F being non empty Subset of R holds
( F -Ideal = F -LeftIdeal & F -Ideal = F -RightIdeal )
proof
let R be non empty add-cancelable add-associative right_zeroed associative commutative well-unital distributive left_zeroed doubleLoopStr ; ::_thesis: for F being non empty Subset of R holds
( F -Ideal = F -LeftIdeal & F -Ideal = F -RightIdeal )
let F be non empty Subset of R; ::_thesis: ( F -Ideal = F -LeftIdeal & F -Ideal = F -RightIdeal )
now__::_thesis:_for_x_being_set_holds_
(_(_x_in_F_-Ideal_implies_x_in_F_-LeftIdeal_)_&_(_x_in_F_-LeftIdeal_implies_x_in_F_-Ideal_)_)
let x be set ; ::_thesis: ( ( x in F -Ideal implies x in F -LeftIdeal ) & ( x in F -LeftIdeal implies x in F -Ideal ) )
hereby ::_thesis: ( x in F -LeftIdeal implies x in F -Ideal )
assume x in F -Ideal ; ::_thesis: x in F -LeftIdeal
then consider lc being LinearCombination of F such that
A1: x = Sum lc by Th60;
lc is LeftLinearCombination of F by Th31;
hence x in F -LeftIdeal by A1, Th61; ::_thesis: verum
end;
assume x in F -LeftIdeal ; ::_thesis: x in F -Ideal
then consider lc being LeftLinearCombination of F such that
A2: x = Sum lc by Th61;
lc is LinearCombination of F by Th25;
hence x in F -Ideal by A2, Th60; ::_thesis: verum
end;
hence F -Ideal = F -LeftIdeal by TARSKI:1; ::_thesis: F -Ideal = F -RightIdeal
now__::_thesis:_for_x_being_set_holds_
(_(_x_in_F_-Ideal_implies_x_in_F_-RightIdeal_)_&_(_x_in_F_-RightIdeal_implies_x_in_F_-Ideal_)_)
let x be set ; ::_thesis: ( ( x in F -Ideal implies x in F -RightIdeal ) & ( x in F -RightIdeal implies x in F -Ideal ) )
hereby ::_thesis: ( x in F -RightIdeal implies x in F -Ideal )
assume x in F -Ideal ; ::_thesis: x in F -RightIdeal
then consider lc being LinearCombination of F such that
A3: x = Sum lc by Th60;
lc is RightLinearCombination of F by Th31;
hence x in F -RightIdeal by A3, Th62; ::_thesis: verum
end;
assume x in F -RightIdeal ; ::_thesis: x in F -Ideal
then consider lc being RightLinearCombination of F such that
A4: x = Sum lc by Th62;
lc is LinearCombination of F by Th28;
hence x in F -Ideal by A4, Th60; ::_thesis: verum
end;
hence F -Ideal = F -RightIdeal by TARSKI:1; ::_thesis: verum
end;
theorem Th64: :: IDEAL_1:64
for R being non empty add-cancelable add-associative right_zeroed associative commutative well-unital distributive left_zeroed doubleLoopStr
for a being Element of R holds {a} -Ideal = { (a * r) where r is Element of R : verum }
proof
let R be non empty add-cancelable add-associative right_zeroed associative commutative well-unital distributive left_zeroed doubleLoopStr ; ::_thesis: for a being Element of R holds {a} -Ideal = { (a * r) where r is Element of R : verum }
let a be Element of R; ::_thesis: {a} -Ideal = { (a * r) where r is Element of R : verum }
set A = {a};
reconsider a9 = a as Element of {a} by TARSKI:def_1;
set M = { (Sum s) where s is LinearCombination of {a} : verum } ;
set N = { (a * r) where r is Element of R : verum } ;
A1: for u being set st u in { (Sum s) where s is LinearCombination of {a} : verum } holds
u in { (a * r) where r is Element of R : verum }
proof
let u be set ; ::_thesis: ( u in { (Sum s) where s is LinearCombination of {a} : verum } implies u in { (a * r) where r is Element of R : verum } )
assume u in { (Sum s) where s is LinearCombination of {a} : verum } ; ::_thesis: u in { (a * r) where r is Element of R : verum }
then consider s being LinearCombination of {a} such that
A2: u = Sum s ;
consider f being Function of NAT, the carrier of R such that
A3: Sum s = f . (len s) and
A4: f . 0 = 0. R and
A5: for j being Element of NAT
for v being Element of R st j < len s & v = s . (j + 1) holds
f . (j + 1) = (f . j) + v by RLVECT_1:def_12;
defpred S1[ Element of NAT ] means ex r being Element of R st f . $1 = a * r;
A6: now__::_thesis:_for_j_being_Element_of_NAT_st_0_<=_j_&_j_<_len_s_&_S1[j]_holds_
S1[j_+_1]
let j be Element of NAT ; ::_thesis: ( 0 <= j & j < len s & S1[j] implies S1[j + 1] )
assume that
0 <= j and
A7: j < len s ; ::_thesis: ( S1[j] implies S1[j + 1] )
thus ( S1[j] implies S1[j + 1] ) ::_thesis: verum
proof
assume ex r being Element of R st f . j = a * r ; ::_thesis: S1[j + 1]
then consider r1 being Element of R such that
A8: f . j = a * r1 ;
( 0 + 1 <= j + 1 & j + 1 <= len s ) by A7, NAT_1:13;
then j + 1 in Seg (len s) by FINSEQ_1:1;
then A9: j + 1 in dom s by FINSEQ_1:def_3;
then consider r2, r3 being Element of R, a9 being Element of {a} such that
A10: s /. (j + 1) = (r2 * a9) * r3 by Def8;
s . (j + 1) = s /. (j + 1) by A9, PARTFUN1:def_6;
then f . (j + 1) = (f . j) + (s /. (j + 1)) by A5, A7;
then f . (j + 1) = (a * r1) + ((r2 * a) * r3) by A8, A10, TARSKI:def_1
.= (a * r1) + (a * (r2 * r3)) by GROUP_1:def_3
.= a * (r1 + (r2 * r3)) by VECTSP_1:def_7 ;
hence S1[j + 1] ; ::_thesis: verum
end;
end;
f . 0 = a * (0. R) by A4, BINOM:2;
then A11: S1[ 0 ] ;
for k being Element of NAT st 0 <= k & k <= len s holds
S1[k] from INT_1:sch_7(A11, A6);
then ex r being Element of R st Sum s = a * r by A3;
hence u in { (a * r) where r is Element of R : verum } by A2; ::_thesis: verum
end;
A12: now__::_thesis:_for_x_being_set_holds_
(_(_x_in_{a}_-Ideal_implies_x_in__{__(Sum_s)_where_s_is_LinearCombination_of_{a}_:_verum__}__)_&_(_x_in__{__(Sum_s)_where_s_is_LinearCombination_of_{a}_:_verum__}__implies_x_in_{a}_-Ideal_)_)
let x be set ; ::_thesis: ( ( x in {a} -Ideal implies x in { (Sum s) where s is LinearCombination of {a} : verum } ) & ( x in { (Sum s) where s is LinearCombination of {a} : verum } implies x in {a} -Ideal ) )
hereby ::_thesis: ( x in { (Sum s) where s is LinearCombination of {a} : verum } implies x in {a} -Ideal )
assume x in {a} -Ideal ; ::_thesis: x in { (Sum s) where s is LinearCombination of {a} : verum }
then x in {a} -RightIdeal by Th63;
then consider f being RightLinearCombination of {a} such that
A13: x = Sum f by Th62;
f is LinearCombination of {a} by Th28;
hence x in { (Sum s) where s is LinearCombination of {a} : verum } by A13; ::_thesis: verum
end;
assume x in { (Sum s) where s is LinearCombination of {a} : verum } ; ::_thesis: x in {a} -Ideal
then ex s being LinearCombination of {a} st x = Sum s ;
hence x in {a} -Ideal by Th60; ::_thesis: verum
end;
for u being set st u in { (a * r) where r is Element of R : verum } holds
u in { (Sum s) where s is LinearCombination of {a} : verum }
proof
let u be set ; ::_thesis: ( u in { (a * r) where r is Element of R : verum } implies u in { (Sum s) where s is LinearCombination of {a} : verum } )
assume u in { (a * r) where r is Element of R : verum } ; ::_thesis: u in { (Sum s) where s is LinearCombination of {a} : verum }
then consider r being Element of R such that
A14: u = a * r ;
set s = <*(a * r)*>;
for i being set st i in dom <*(a * r)*> holds
ex r, t being Element of R ex a being Element of {a} st <*(a * r)*> /. i = (r * a) * t
proof
let i be set ; ::_thesis: ( i in dom <*(a * r)*> implies ex r, t being Element of R ex a being Element of {a} st <*(a * r)*> /. i = (r * a) * t )
A15: len <*(a * r)*> = 1 by FINSEQ_1:40;
assume i in dom <*(a * r)*> ; ::_thesis: ex r, t being Element of R ex a being Element of {a} st <*(a * r)*> /. i = (r * a) * t
then i in {1} by A15, FINSEQ_1:2, FINSEQ_1:def_3;
then i = 1 by TARSKI:def_1;
then <*(a * r)*> /. i = a * r by FINSEQ_4:16
.= (r * a9) * (1. R) by VECTSP_1:def_4 ;
hence ex r, t being Element of R ex a being Element of {a} st <*(a * r)*> /. i = (r * a) * t ; ::_thesis: verum
end;
then reconsider s = <*(a * r)*> as LinearCombination of {a} by Def8;
Sum s = a * r by BINOM:3;
hence u in { (Sum s) where s is LinearCombination of {a} : verum } by A14; ::_thesis: verum
end;
then { (Sum s) where s is LinearCombination of {a} : verum } = { (a * r) where r is Element of R : verum } by A1, TARSKI:1;
hence {a} -Ideal = { (a * r) where r is Element of R : verum } by A12, TARSKI:1; ::_thesis: verum
end;
theorem Th65: :: IDEAL_1:65
for R being non empty add-cancelable Abelian add-associative right_zeroed associative commutative well-unital distributive left_zeroed doubleLoopStr
for a, b being Element of R holds {a,b} -Ideal = { ((a * r) + (b * s)) where r, s is Element of R : verum }
proof
let R be non empty add-cancelable Abelian add-associative right_zeroed associative commutative well-unital distributive left_zeroed doubleLoopStr ; ::_thesis: for a, b being Element of R holds {a,b} -Ideal = { ((a * r) + (b * s)) where r, s is Element of R : verum }
let a, b be Element of R; ::_thesis: {a,b} -Ideal = { ((a * r) + (b * s)) where r, s is Element of R : verum }
set A = {a,b};
reconsider a9 = a, b9 = b as Element of {a,b} by TARSKI:def_2;
set M = { (Sum s) where s is LinearCombination of {a,b} : verum } ;
set N = { ((a * r) + (b * s)) where r, s is Element of R : verum } ;
A1: for u being set st u in { (Sum s) where s is LinearCombination of {a,b} : verum } holds
u in { ((a * r) + (b * s)) where r, s is Element of R : verum }
proof
let u be set ; ::_thesis: ( u in { (Sum s) where s is LinearCombination of {a,b} : verum } implies u in { ((a * r) + (b * s)) where r, s is Element of R : verum } )
assume u in { (Sum s) where s is LinearCombination of {a,b} : verum } ; ::_thesis: u in { ((a * r) + (b * s)) where r, s is Element of R : verum }
then consider s being LinearCombination of {a,b} such that
A2: u = Sum s ;
consider f being Function of NAT, the carrier of R such that
A3: Sum s = f . (len s) and
A4: f . 0 = 0. R and
A5: for j being Element of NAT
for v being Element of R st j < len s & v = s . (j + 1) holds
f . (j + 1) = (f . j) + v by RLVECT_1:def_12;
defpred S1[ Element of NAT ] means ex r, s being Element of R st f . $1 = (a * r) + (b * s);
A6: now__::_thesis:_for_j_being_Element_of_NAT_st_0_<=_j_&_j_<_len_s_&_S1[j]_holds_
S1[j_+_1]
let j be Element of NAT ; ::_thesis: ( 0 <= j & j < len s & S1[j] implies S1[j + 1] )
assume that
0 <= j and
A7: j < len s ; ::_thesis: ( S1[j] implies S1[j + 1] )
thus ( S1[j] implies S1[j + 1] ) ::_thesis: verum
proof
( 0 + 1 <= j + 1 & j + 1 <= len s ) by A7, NAT_1:13;
then j + 1 in Seg (len s) by FINSEQ_1:1;
then A8: j + 1 in dom s by FINSEQ_1:def_3;
then A9: s /. (j + 1) = s . (j + 1) by PARTFUN1:def_6;
assume ex r, s being Element of R st f . j = (a * r) + (b * s) ; ::_thesis: S1[j + 1]
then consider r1, s1 being Element of R such that
A10: f . j = (a * r1) + (b * s1) ;
consider r2, r3 being Element of R, a9 being Element of {a,b} such that
A11: s /. (j + 1) = (r2 * a9) * r3 by A8, Def8;
percases ( a9 = a or a9 = b ) by TARSKI:def_2;
suppose a9 = a ; ::_thesis: S1[j + 1]
then f . (j + 1) = ((a * r1) + (b * s1)) + ((r2 * a) * r3) by A5, A7, A10, A11, A9
.= ((a * r1) + ((a * r2) * r3)) + (b * s1) by RLVECT_1:def_3
.= ((a * r1) + (a * (r2 * r3))) + (b * s1) by GROUP_1:def_3
.= (a * (r1 + (r2 * r3))) + (b * s1) by VECTSP_1:def_7 ;
hence S1[j + 1] ; ::_thesis: verum
end;
suppose a9 = b ; ::_thesis: S1[j + 1]
then f . (j + 1) = ((a * r1) + (b * s1)) + ((r2 * b) * r3) by A5, A7, A10, A11, A9
.= (a * r1) + ((b * s1) + ((b * r2) * r3)) by RLVECT_1:def_3
.= (a * r1) + ((b * s1) + (b * (r2 * r3))) by GROUP_1:def_3
.= (a * r1) + (b * (s1 + (r2 * r3))) by VECTSP_1:def_7 ;
hence S1[j + 1] ; ::_thesis: verum
end;
end;
end;
end;
f . 0 = a * (0. R) by A4, BINOM:2
.= (a * (0. R)) + (0. R) by RLVECT_1:def_4
.= (a * (0. R)) + (b * (0. R)) by BINOM:2 ;
then A12: S1[ 0 ] ;
for k being Element of NAT st 0 <= k & k <= len s holds
S1[k] from INT_1:sch_7(A12, A6);
then ex r, t being Element of R st Sum s = (a * r) + (b * t) by A3;
hence u in { ((a * r) + (b * s)) where r, s is Element of R : verum } by A2; ::_thesis: verum
end;
A13: now__::_thesis:_for_x_being_set_holds_
(_(_x_in_{a,b}_-Ideal_implies_x_in__{__(Sum_s)_where_s_is_LinearCombination_of_{a,b}_:_verum__}__)_&_(_x_in__{__(Sum_s)_where_s_is_LinearCombination_of_{a,b}_:_verum__}__implies_x_in_{a,b}_-Ideal_)_)
let x be set ; ::_thesis: ( ( x in {a,b} -Ideal implies x in { (Sum s) where s is LinearCombination of {a,b} : verum } ) & ( x in { (Sum s) where s is LinearCombination of {a,b} : verum } implies x in {a,b} -Ideal ) )
hereby ::_thesis: ( x in { (Sum s) where s is LinearCombination of {a,b} : verum } implies x in {a,b} -Ideal )
assume x in {a,b} -Ideal ; ::_thesis: x in { (Sum s) where s is LinearCombination of {a,b} : verum }
then x in {a,b} -RightIdeal by Th63;
then consider f being RightLinearCombination of {a,b} such that
A14: x = Sum f by Th62;
f is LinearCombination of {a,b} by Th28;
hence x in { (Sum s) where s is LinearCombination of {a,b} : verum } by A14; ::_thesis: verum
end;
assume x in { (Sum s) where s is LinearCombination of {a,b} : verum } ; ::_thesis: x in {a,b} -Ideal
then ex s being LinearCombination of {a,b} st x = Sum s ;
hence x in {a,b} -Ideal by Th60; ::_thesis: verum
end;
for u being set st u in { ((a * r) + (b * s)) where r, s is Element of R : verum } holds
u in { (Sum s) where s is LinearCombination of {a,b} : verum }
proof
let u be set ; ::_thesis: ( u in { ((a * r) + (b * s)) where r, s is Element of R : verum } implies u in { (Sum s) where s is LinearCombination of {a,b} : verum } )
assume u in { ((a * r) + (b * s)) where r, s is Element of R : verum } ; ::_thesis: u in { (Sum s) where s is LinearCombination of {a,b} : verum }
then consider r, t being Element of R such that
A15: u = (a * r) + (b * t) ;
set s = <*(a * r),(b * t)*>;
for i being set st i in dom <*(a * r),(b * t)*> holds
ex r, t being Element of R ex a being Element of {a,b} st <*(a * r),(b * t)*> /. i = (r * a) * t
proof
let i be set ; ::_thesis: ( i in dom <*(a * r),(b * t)*> implies ex r, t being Element of R ex a being Element of {a,b} st <*(a * r),(b * t)*> /. i = (r * a) * t )
assume A16: i in dom <*(a * r),(b * t)*> ; ::_thesis: ex r, t being Element of R ex a being Element of {a,b} st <*(a * r),(b * t)*> /. i = (r * a) * t
then i in Seg (len <*(a * r),(b * t)*>) by FINSEQ_1:def_3;
then A17: i in {1,2} by FINSEQ_1:2, FINSEQ_1:44;
percases ( i = 1 or i = 2 ) by A17, TARSKI:def_2;
suppose i = 1 ; ::_thesis: ex r, t being Element of R ex a being Element of {a,b} st <*(a * r),(b * t)*> /. i = (r * a) * t
then <*(a * r),(b * t)*> /. i = <*(a * r),(b * t)*> . 1 by A16, PARTFUN1:def_6
.= a * r by FINSEQ_1:44
.= ((1. R) * a9) * r by VECTSP_1:def_8 ;
hence ex r, t being Element of R ex a being Element of {a,b} st <*(a * r),(b * t)*> /. i = (r * a) * t ; ::_thesis: verum
end;
suppose i = 2 ; ::_thesis: ex r, t being Element of R ex a being Element of {a,b} st <*(a * r),(b * t)*> /. i = (r * a) * t
then <*(a * r),(b * t)*> /. i = <*(a * r),(b * t)*> . 2 by A16, PARTFUN1:def_6
.= b * t by FINSEQ_1:44
.= ((1. R) * b9) * t by VECTSP_1:def_8 ;
hence ex r, t being Element of R ex a being Element of {a,b} st <*(a * r),(b * t)*> /. i = (r * a) * t ; ::_thesis: verum
end;
end;
end;
then reconsider s = <*(a * r),(b * t)*> as LinearCombination of {a,b} by Def8;
Sum s = (a * r) + (b * t) by Th1;
hence u in { (Sum s) where s is LinearCombination of {a,b} : verum } by A15; ::_thesis: verum
end;
then { (Sum s) where s is LinearCombination of {a,b} : verum } = { ((a * r) + (b * s)) where r, s is Element of R : verum } by A1, TARSKI:1;
hence {a,b} -Ideal = { ((a * r) + (b * s)) where r, s is Element of R : verum } by A13, TARSKI:1; ::_thesis: verum
end;
theorem Th66: :: IDEAL_1:66
for R being non empty doubleLoopStr
for a being Element of R holds a in {a} -Ideal
proof
let R be non empty doubleLoopStr ; ::_thesis: for a being Element of R holds a in {a} -Ideal
let a be Element of R; ::_thesis: a in {a} -Ideal
( a in {a} & {a} c= {a} -Ideal ) by Def14, TARSKI:def_1;
hence a in {a} -Ideal ; ::_thesis: verum
end;
theorem :: IDEAL_1:67
for R being non empty right_complementable Abelian add-associative right_zeroed associative commutative well-unital distributive left_zeroed doubleLoopStr
for A being non empty Subset of R
for a being Element of R st a in A -Ideal holds
{a} -Ideal c= A -Ideal
proof
let R be non empty right_complementable Abelian add-associative right_zeroed associative commutative well-unital distributive left_zeroed doubleLoopStr ; ::_thesis: for A being non empty Subset of R
for a being Element of R st a in A -Ideal holds
{a} -Ideal c= A -Ideal
let A be non empty Subset of R; ::_thesis: for a being Element of R st a in A -Ideal holds
{a} -Ideal c= A -Ideal
let a be Element of R; ::_thesis: ( a in A -Ideal implies {a} -Ideal c= A -Ideal )
assume a in A -Ideal ; ::_thesis: {a} -Ideal c= A -Ideal
then consider s being LinearCombination of A such that
A1: a = Sum s by Th60;
now__::_thesis:_for_u_being_set_st_u_in_{a}_-Ideal_holds_
u_in_A_-Ideal
let u be set ; ::_thesis: ( u in {a} -Ideal implies u in A -Ideal )
assume u in {a} -Ideal ; ::_thesis: u in A -Ideal
then u in { (a * r) where r is Element of R : verum } by Th64;
then consider r being Element of R such that
A2: u = a * r ;
set t = s * r;
A3: dom s = dom (s * r) by POLYNOM1:def_2;
for i being set st i in dom (s * r) holds
ex u, v being Element of R ex a being Element of A st (s * r) /. i = (u * a) * v
proof
let i be set ; ::_thesis: ( i in dom (s * r) implies ex u, v being Element of R ex a being Element of A st (s * r) /. i = (u * a) * v )
assume A4: i in dom (s * r) ; ::_thesis: ex u, v being Element of R ex a being Element of A st (s * r) /. i = (u * a) * v
then consider u, v being Element of R, b being Element of A such that
A5: s /. i = (u * b) * v by A3, Def8;
(s * r) /. i = ((u * b) * v) * r by A3, A4, A5, POLYNOM1:def_2
.= (u * b) * (v * r) by GROUP_1:def_3 ;
hence ex u, v being Element of R ex a being Element of A st (s * r) /. i = (u * a) * v ; ::_thesis: verum
end;
then A6: s * r is LinearCombination of A by Def8;
Sum (s * r) = u by A1, A2, BINOM:5;
hence u in A -Ideal by A6, Th60; ::_thesis: verum
end;
hence {a} -Ideal c= A -Ideal by TARSKI:def_3; ::_thesis: verum
end;
Lm2: for a, b being set holds {a} c= {a,b}
proof
let a, b be set ; ::_thesis: {a} c= {a,b}
now__::_thesis:_for_u_being_set_st_u_in_{a}_holds_
u_in_{a,b}
let u be set ; ::_thesis: ( u in {a} implies u in {a,b} )
assume u in {a} ; ::_thesis: u in {a,b}
then u = a by TARSKI:def_1;
hence u in {a,b} by TARSKI:def_2; ::_thesis: verum
end;
hence {a} c= {a,b} by TARSKI:def_3; ::_thesis: verum
end;
theorem :: IDEAL_1:68
for R being non empty doubleLoopStr
for a, b being Element of R holds
( a in {a,b} -Ideal & b in {a,b} -Ideal )
proof
let R be non empty doubleLoopStr ; ::_thesis: for a, b being Element of R holds
( a in {a,b} -Ideal & b in {a,b} -Ideal )
let a, b be Element of R; ::_thesis: ( a in {a,b} -Ideal & b in {a,b} -Ideal )
( {a} -Ideal c= {a,b} -Ideal & a in {a} -Ideal ) by Lm2, Th57, Th66;
hence a in {a,b} -Ideal ; ::_thesis: b in {a,b} -Ideal
( {b} -Ideal c= {a,b} -Ideal & b in {b} -Ideal ) by Lm2, Th57, Th66;
hence b in {a,b} -Ideal ; ::_thesis: verum
end;
theorem :: IDEAL_1:69
for R being non empty doubleLoopStr
for a, b being Element of R holds
( {a} -Ideal c= {a,b} -Ideal & {b} -Ideal c= {a,b} -Ideal ) by Lm2, Th57;
begin
definition
let R be non empty multMagma ;
let I be Subset of R;
let a be Element of R;
funca * I -> Subset of R equals :: IDEAL_1:def 18
{ (a * i) where i is Element of R : i in I } ;
coherence
{ (a * i) where i is Element of R : i in I } is Subset of R
proof
set M = { (a * i) where i is Element of R : i in I } ;
{ (a * i) where i is Element of R : i in I } is Subset of R
proof
percases ( I is empty or not I is empty ) ;
supposeA1: I is empty ; ::_thesis: { (a * i) where i is Element of R : i in I } is Subset of R
{ (a * i) where i is Element of R : i in I } is empty
proof
assume not { (a * i) where i is Element of R : i in I } is empty ; ::_thesis: contradiction
then reconsider M = { (a * i) where i is Element of R : i in I } as non empty set ;
set b = the Element of M;
the Element of M in { (a * i) where i is Element of R : i in I } ;
then ex i being Element of R st
( the Element of M = a * i & i in I ) ;
hence contradiction by A1; ::_thesis: verum
end;
then for u being set st u in { (a * i) where i is Element of R : i in I } holds
u in the carrier of R ;
hence { (a * i) where i is Element of R : i in I } is Subset of R by TARSKI:def_3; ::_thesis: verum
end;
suppose not I is empty ; ::_thesis: { (a * i) where i is Element of R : i in I } is Subset of R
then reconsider I = I as non empty set ;
set j9 = the Element of I;
the Element of I in I ;
then reconsider j = the Element of I as Element of R ;
a * j in { (a * i) where i is Element of R : i in I } ;
then reconsider M = { (a * i) where i is Element of R : i in I } as non empty set ;
for x being set st x in M holds
x in the carrier of R
proof
let x be set ; ::_thesis: ( x in M implies x in the carrier of R )
assume x in M ; ::_thesis: x in the carrier of R
then ex i being Element of R st
( x = a * i & i in I ) ;
hence x in the carrier of R ; ::_thesis: verum
end;
hence { (a * i) where i is Element of R : i in I } is Subset of R by TARSKI:def_3; ::_thesis: verum
end;
end;
end;
hence { (a * i) where i is Element of R : i in I } is Subset of R ; ::_thesis: verum
end;
end;
:: deftheorem defines * IDEAL_1:def_18_:_
for R being non empty multMagma
for I being Subset of R
for a being Element of R holds a * I = { (a * i) where i is Element of R : i in I } ;
registration
let R be non empty multLoopStr ;
let I be non empty Subset of R;
let a be Element of R;
clustera * I -> non empty ;
coherence
not a * I is empty
proof
set j = the Element of I;
a * the Element of I in { (a * i) where i is Element of R : i in I } ;
hence not a * I is empty ; ::_thesis: verum
end;
end;
registration
let R be non empty distributive doubleLoopStr ;
let I be add-closed Subset of R;
let a be Element of R;
clustera * I -> add-closed ;
coherence
a * I is add-closed
proof
set M = { (a * i) where i is Element of R : i in I } ;
for x, y being Element of R st x in { (a * i) where i is Element of R : i in I } & y in { (a * i) where i is Element of R : i in I } holds
x + y in { (a * i) where i is Element of R : i in I }
proof
let x, y be Element of R; ::_thesis: ( x in { (a * i) where i is Element of R : i in I } & y in { (a * i) where i is Element of R : i in I } implies x + y in { (a * i) where i is Element of R : i in I } )
assume that
A1: x in { (a * i) where i is Element of R : i in I } and
A2: y in { (a * i) where i is Element of R : i in I } ; ::_thesis: x + y in { (a * i) where i is Element of R : i in I }
consider i being Element of R such that
A3: ( x = a * i & i in I ) by A1;
consider j being Element of R such that
A4: ( y = a * j & j in I ) by A2;
reconsider k = i + j as Element of R ;
( k in I & x + y = a * k ) by A3, A4, Def1, VECTSP_1:def_7;
hence x + y in { (a * i) where i is Element of R : i in I } ; ::_thesis: verum
end;
hence a * I is add-closed by Def1; ::_thesis: verum
end;
end;
registration
let R be non empty associative doubleLoopStr ;
let I be right-ideal Subset of R;
let a be Element of R;
clustera * I -> right-ideal ;
coherence
a * I is right-ideal
proof
set M = { (a * i) where i is Element of R : i in I } ;
for y, x being Element of R st x in { (a * i) where i is Element of R : i in I } holds
x * y in { (a * i) where i is Element of R : i in I }
proof
let y, x be Element of R; ::_thesis: ( x in { (a * i) where i is Element of R : i in I } implies x * y in { (a * i) where i is Element of R : i in I } )
assume x in { (a * i) where i is Element of R : i in I } ; ::_thesis: x * y in { (a * i) where i is Element of R : i in I }
then consider i being Element of R such that
A1: ( x = a * i & i in I ) ;
( x * y = a * (i * y) & i * y in I ) by A1, Def3, GROUP_1:def_3;
hence x * y in { (a * i) where i is Element of R : i in I } ; ::_thesis: verum
end;
hence a * I is right-ideal by Def3; ::_thesis: verum
end;
end;
theorem Th70: :: IDEAL_1:70
for R being non empty left_add-cancelable right_zeroed left-distributive doubleLoopStr
for I being non empty Subset of R holds (0. R) * I = {(0. R)}
proof
let R be non empty left_add-cancelable right_zeroed left-distributive doubleLoopStr ; ::_thesis: for I being non empty Subset of R holds (0. R) * I = {(0. R)}
let I be non empty Subset of R; ::_thesis: (0. R) * I = {(0. R)}
A1: now__::_thesis:_for_u_being_set_st_u_in_{(0._R)}_holds_
u_in_(0._R)_*_I
set j = the Element of I;
let u be set ; ::_thesis: ( u in {(0. R)} implies u in (0. R) * I )
assume u in {(0. R)} ; ::_thesis: u in (0. R) * I
then A2: u = 0. R by TARSKI:def_1;
(0. R) * the Element of I = 0. R by BINOM:1;
hence u in (0. R) * I by A2; ::_thesis: verum
end;
now__::_thesis:_for_u_being_set_st_u_in_(0._R)_*_I_holds_
u_in_{(0._R)}
let u be set ; ::_thesis: ( u in (0. R) * I implies u in {(0. R)} )
assume u in (0. R) * I ; ::_thesis: u in {(0. R)}
then ex i being Element of R st
( u = (0. R) * i & i in I ) ;
then u = 0. R by BINOM:1;
hence u in {(0. R)} by TARSKI:def_1; ::_thesis: verum
end;
hence (0. R) * I = {(0. R)} by A1, TARSKI:1; ::_thesis: verum
end;
theorem :: IDEAL_1:71
for R being non empty left_unital doubleLoopStr
for I being Subset of R holds (1. R) * I = I
proof
let R be non empty left_unital doubleLoopStr ; ::_thesis: for I being Subset of R holds (1. R) * I = I
let I be Subset of R; ::_thesis: (1. R) * I = I
A1: now__::_thesis:_for_u_being_set_st_u_in_I_holds_
u_in_(1._R)_*_I
let u be set ; ::_thesis: ( u in I implies u in (1. R) * I )
assume A2: u in I ; ::_thesis: u in (1. R) * I
then reconsider u9 = u as Element of R ;
(1. R) * u9 = u9 by VECTSP_1:def_8;
hence u in (1. R) * I by A2; ::_thesis: verum
end;
now__::_thesis:_for_u_being_set_st_u_in_(1._R)_*_I_holds_
u_in_I
let u be set ; ::_thesis: ( u in (1. R) * I implies u in I )
assume u in (1. R) * I ; ::_thesis: u in I
then ex i being Element of R st
( u = (1. R) * i & i in I ) ;
hence u in I by VECTSP_1:def_8; ::_thesis: verum
end;
hence (1. R) * I = I by A1, TARSKI:1; ::_thesis: verum
end;
definition
let R be non empty addLoopStr ;
let I, J be Subset of R;
funcI + J -> Subset of R equals :: IDEAL_1:def 19
{ (a + b) where a, b is Element of R : ( a in I & b in J ) } ;
coherence
{ (a + b) where a, b is Element of R : ( a in I & b in J ) } is Subset of R
proof
set M = { (a + b) where a, b is Element of R : ( a in I & b in J ) } ;
{ (a + b) where a, b is Element of R : ( a in I & b in J ) } is Subset of R
proof
percases ( I is empty or J is empty or ( not I is empty & not J is empty ) ) ;
supposeA1: ( I is empty or J is empty ) ; ::_thesis: { (a + b) where a, b is Element of R : ( a in I & b in J ) } is Subset of R
now__::_thesis:_(_(_I_is_empty_&__{__(a_+_b)_where_a,_b_is_Element_of_R_:_(_a_in_I_&_b_in_J_)__}__is_Subset_of_R_)_or_(_J_is_empty_&__{__(a_+_b)_where_a,_b_is_Element_of_R_:_(_a_in_I_&_b_in_J_)__}__is_Subset_of_R_)_)
percases ( I is empty or J is empty ) by A1;
caseA2: I is empty ; ::_thesis: { (a + b) where a, b is Element of R : ( a in I & b in J ) } is Subset of R
{ (a + b) where a, b is Element of R : ( a in I & b in J ) } is empty
proof
assume not { (a + b) where a, b is Element of R : ( a in I & b in J ) } is empty ; ::_thesis: contradiction
then reconsider M = { (a + b) where a, b is Element of R : ( a in I & b in J ) } as non empty set ;
set x = the Element of M;
the Element of M in { (a + b) where a, b is Element of R : ( a in I & b in J ) } ;
then ex a, b being Element of R st
( the Element of M = a + b & a in I & b in J ) ;
hence contradiction by A2; ::_thesis: verum
end;
then for u being set st u in { (a + b) where a, b is Element of R : ( a in I & b in J ) } holds
u in the carrier of R ;
hence { (a + b) where a, b is Element of R : ( a in I & b in J ) } is Subset of R by TARSKI:def_3; ::_thesis: verum
end;
caseA3: J is empty ; ::_thesis: { (a + b) where a, b is Element of R : ( a in I & b in J ) } is Subset of R
{ (a + b) where a, b is Element of R : ( a in I & b in J ) } is empty
proof
assume not { (a + b) where a, b is Element of R : ( a in I & b in J ) } is empty ; ::_thesis: contradiction
then reconsider M = { (a + b) where a, b is Element of R : ( a in I & b in J ) } as non empty set ;
set x = the Element of M;
the Element of M in { (a + b) where a, b is Element of R : ( a in I & b in J ) } ;
then ex a, b being Element of R st
( the Element of M = a + b & a in I & b in J ) ;
hence contradiction by A3; ::_thesis: verum
end;
then for u being set st u in { (a + b) where a, b is Element of R : ( a in I & b in J ) } holds
u in the carrier of R ;
hence { (a + b) where a, b is Element of R : ( a in I & b in J ) } is Subset of R by TARSKI:def_3; ::_thesis: verum
end;
end;
end;
hence { (a + b) where a, b is Element of R : ( a in I & b in J ) } is Subset of R ; ::_thesis: verum
end;
supposeA4: ( not I is empty & not J is empty ) ; ::_thesis: { (a + b) where a, b is Element of R : ( a in I & b in J ) } is Subset of R
then reconsider J = J as non empty set ;
reconsider I = I as non empty set by A4;
set x9 = the Element of I;
set y9 = the Element of J;
( the Element of I in I & the Element of J in J ) ;
then reconsider x = the Element of I, y = the Element of J as Element of R ;
x + y in { (a + b) where a, b is Element of R : ( a in I & b in J ) } ;
then reconsider M = { (a + b) where a, b is Element of R : ( a in I & b in J ) } as non empty set ;
for x being set st x in M holds
x in the carrier of R
proof
let x be set ; ::_thesis: ( x in M implies x in the carrier of R )
assume x in M ; ::_thesis: x in the carrier of R
then ex a, b being Element of R st
( x = a + b & a in I & b in J ) ;
hence x in the carrier of R ; ::_thesis: verum
end;
hence { (a + b) where a, b is Element of R : ( a in I & b in J ) } is Subset of R by TARSKI:def_3; ::_thesis: verum
end;
end;
end;
hence { (a + b) where a, b is Element of R : ( a in I & b in J ) } is Subset of R ; ::_thesis: verum
end;
end;
:: deftheorem defines + IDEAL_1:def_19_:_
for R being non empty addLoopStr
for I, J being Subset of R holds I + J = { (a + b) where a, b is Element of R : ( a in I & b in J ) } ;
registration
let R be non empty addLoopStr ;
let I, J be non empty Subset of R;
clusterI + J -> non empty ;
coherence
not I + J is empty
proof
not { (x + y) where x, y is Element of R : ( x in I & y in J ) } is empty
proof
set y = the Element of J;
set x = the Element of I;
the Element of I + the Element of J in { (a + b) where a, b is Element of R : ( a in I & b in J ) } ;
hence not { (x + y) where x, y is Element of R : ( x in I & y in J ) } is empty ; ::_thesis: verum
end;
hence not I + J is empty ; ::_thesis: verum
end;
end;
definition
let R be non empty Abelian addLoopStr ;
let I, J be Subset of R;
:: original: +
redefine funcI + J -> Subset of R;
commutativity
for I, J being Subset of R holds I + J = J + I
proof
now__::_thesis:_for_I,_J_being_Subset_of_R_holds_I_+_J_=_J_+_I
let I, J be Subset of R; ::_thesis: I + J = J + I
A1: now__::_thesis:_for_u_being_set_st_u_in_J_+_I_holds_
u_in_I_+_J
let u be set ; ::_thesis: ( u in J + I implies u in I + J )
assume u in J + I ; ::_thesis: u in I + J
then ex a, b being Element of R st
( u = a + b & a in J & b in I ) ;
hence u in I + J ; ::_thesis: verum
end;
now__::_thesis:_for_u_being_set_st_u_in_I_+_J_holds_
u_in_J_+_I
let u be set ; ::_thesis: ( u in I + J implies u in J + I )
assume u in I + J ; ::_thesis: u in J + I
then ex a, b being Element of R st
( u = a + b & a in I & b in J ) ;
hence u in J + I ; ::_thesis: verum
end;
hence I + J = J + I by A1, TARSKI:1; ::_thesis: verum
end;
hence for I, J being Subset of R holds I + J = J + I ; ::_thesis: verum
end;
end;
registration
let R be non empty Abelian add-associative addLoopStr ;
let I, J be add-closed Subset of R;
clusterI + J -> add-closed ;
coherence
I + J is add-closed
proof
set M = { (a + b) where a, b is Element of R : ( a in I & b in J ) } ;
for x, y being Element of R st x in { (a + b) where a, b is Element of R : ( a in I & b in J ) } & y in { (a + b) where a, b is Element of R : ( a in I & b in J ) } holds
x + y in { (a + b) where a, b is Element of R : ( a in I & b in J ) }
proof
let x, y be Element of R; ::_thesis: ( x in { (a + b) where a, b is Element of R : ( a in I & b in J ) } & y in { (a + b) where a, b is Element of R : ( a in I & b in J ) } implies x + y in { (a + b) where a, b is Element of R : ( a in I & b in J ) } )
assume that
A1: x in { (a + b) where a, b is Element of R : ( a in I & b in J ) } and
A2: y in { (a + b) where a, b is Element of R : ( a in I & b in J ) } ; ::_thesis: x + y in { (a + b) where a, b is Element of R : ( a in I & b in J ) }
consider a9, b9 being Element of R such that
A3: x = a9 + b9 and
A4: ( a9 in I & b9 in J ) by A1;
consider c, d being Element of R such that
A5: y = c + d and
A6: ( c in I & d in J ) by A2;
A7: (a9 + c) + (b9 + d) = ((a9 + c) + b9) + d by RLVECT_1:def_3
.= (c + x) + d by A3, RLVECT_1:def_3
.= x + y by A5, RLVECT_1:def_3 ;
( a9 + c in I & b9 + d in J ) by A4, A6, Def1;
hence x + y in { (a + b) where a, b is Element of R : ( a in I & b in J ) } by A7; ::_thesis: verum
end;
hence I + J is add-closed by Def1; ::_thesis: verum
end;
end;
registration
let R be non empty left-distributive doubleLoopStr ;
let I, J be right-ideal Subset of R;
clusterI + J -> right-ideal ;
coherence
I + J is right-ideal
proof
set M = { (a + b) where a, b is Element of R : ( a in I & b in J ) } ;
for y, x being Element of R st x in { (a + b) where a, b is Element of R : ( a in I & b in J ) } holds
x * y in { (a + b) where a, b is Element of R : ( a in I & b in J ) }
proof
let y, x be Element of R; ::_thesis: ( x in { (a + b) where a, b is Element of R : ( a in I & b in J ) } implies x * y in { (a + b) where a, b is Element of R : ( a in I & b in J ) } )
assume x in { (a + b) where a, b is Element of R : ( a in I & b in J ) } ; ::_thesis: x * y in { (a + b) where a, b is Element of R : ( a in I & b in J ) }
then consider a9, b9 being Element of R such that
A1: x = a9 + b9 and
A2: ( a9 in I & b9 in J ) ;
A3: (a9 * y) + (b9 * y) = x * y by A1, VECTSP_1:def_3;
( a9 * y in I & b9 * y in J ) by A2, Def3;
hence x * y in { (a + b) where a, b is Element of R : ( a in I & b in J ) } by A3; ::_thesis: verum
end;
hence I + J is right-ideal by Def3; ::_thesis: verum
end;
end;
registration
let R be non empty right-distributive doubleLoopStr ;
let I, J be left-ideal Subset of R;
clusterI + J -> left-ideal ;
coherence
I + J is left-ideal
proof
set M = { (a + b) where a, b is Element of R : ( a in I & b in J ) } ;
for y, x being Element of R st x in { (a + b) where a, b is Element of R : ( a in I & b in J ) } holds
y * x in { (a + b) where a, b is Element of R : ( a in I & b in J ) }
proof
let y, x be Element of R; ::_thesis: ( x in { (a + b) where a, b is Element of R : ( a in I & b in J ) } implies y * x in { (a + b) where a, b is Element of R : ( a in I & b in J ) } )
assume x in { (a + b) where a, b is Element of R : ( a in I & b in J ) } ; ::_thesis: y * x in { (a + b) where a, b is Element of R : ( a in I & b in J ) }
then consider a9, b9 being Element of R such that
A1: x = a9 + b9 and
A2: ( a9 in I & b9 in J ) ;
A3: (y * a9) + (y * b9) = y * x by A1, VECTSP_1:def_2;
( y * a9 in I & y * b9 in J ) by A2, Def2;
hence y * x in { (a + b) where a, b is Element of R : ( a in I & b in J ) } by A3; ::_thesis: verum
end;
hence I + J is left-ideal by Def2; ::_thesis: verum
end;
end;
theorem :: IDEAL_1:72
for R being non empty add-associative addLoopStr
for I, J, K being Subset of R holds I + (J + K) = (I + J) + K
proof
let R be non empty add-associative addLoopStr ; ::_thesis: for I, J, K being Subset of R holds I + (J + K) = (I + J) + K
let I, J, K be Subset of R; ::_thesis: I + (J + K) = (I + J) + K
A1: now__::_thesis:_for_u_being_set_st_u_in_(I_+_J)_+_K_holds_
u_in_I_+_(J_+_K)
let u be set ; ::_thesis: ( u in (I + J) + K implies u in I + (J + K) )
assume u in (I + J) + K ; ::_thesis: u in I + (J + K)
then consider a, b being Element of R such that
A2: u = a + b and
A3: a in I + J and
A4: b in K ;
consider c, d being Element of R such that
A5: a = c + d and
A6: c in I and
A7: d in J by A3;
d + b in { (a9 + b9) where a9, b9 is Element of R : ( a9 in J & b9 in K ) } by A4, A7;
then c + (d + b) in { (a9 + b9) where a9, b9 is Element of R : ( a9 in I & b9 in J + K ) } by A6;
hence u in I + (J + K) by A2, A5, RLVECT_1:def_3; ::_thesis: verum
end;
now__::_thesis:_for_u_being_set_st_u_in_I_+_(J_+_K)_holds_
u_in_(I_+_J)_+_K
let u be set ; ::_thesis: ( u in I + (J + K) implies u in (I + J) + K )
assume u in I + (J + K) ; ::_thesis: u in (I + J) + K
then consider a, b being Element of R such that
A8: u = a + b and
A9: a in I and
A10: b in J + K ;
consider c, d being Element of R such that
A11: b = c + d and
A12: c in J and
A13: d in K by A10;
a + c in { (a9 + b9) where a9, b9 is Element of R : ( a9 in I & b9 in J ) } by A9, A12;
then (a + c) + d in { (a9 + b9) where a9, b9 is Element of R : ( a9 in I + J & b9 in K ) } by A13;
hence u in (I + J) + K by A8, A11, RLVECT_1:def_3; ::_thesis: verum
end;
hence I + (J + K) = (I + J) + K by A1, TARSKI:1; ::_thesis: verum
end;
theorem Th73: :: IDEAL_1:73
for R being non empty right_add-cancelable right_zeroed right-distributive left_zeroed doubleLoopStr
for I, J being non empty right-ideal Subset of R holds I c= I + J
proof
let R be non empty right_add-cancelable right_zeroed right-distributive left_zeroed doubleLoopStr ; ::_thesis: for I, J being non empty right-ideal Subset of R holds I c= I + J
let I, J be non empty right-ideal Subset of R; ::_thesis: I c= I + J
now__::_thesis:_for_u_being_set_st_u_in_I_holds_
u_in_I_+_J
let u be set ; ::_thesis: ( u in I implies u in I + J )
assume u in I ; ::_thesis: u in I + J
then reconsider u9 = u as Element of I ;
0. R is Element of J by Th3;
then u9 + (0. R) in { (a + b) where a, b is Element of R : ( a in I & b in J ) } ;
hence u in I + J by RLVECT_1:def_4; ::_thesis: verum
end;
hence I c= I + J by TARSKI:def_3; ::_thesis: verum
end;
theorem Th74: :: IDEAL_1:74
for R being non empty right_add-cancelable right-distributive left_zeroed doubleLoopStr
for I, J being non empty right-ideal Subset of R holds J c= I + J
proof
let R be non empty right_add-cancelable right-distributive left_zeroed doubleLoopStr ; ::_thesis: for I, J being non empty right-ideal Subset of R holds J c= I + J
let I, J be non empty right-ideal Subset of R; ::_thesis: J c= I + J
now__::_thesis:_for_u_being_set_st_u_in_J_holds_
u_in_I_+_J
let u be set ; ::_thesis: ( u in J implies u in I + J )
assume u in J ; ::_thesis: u in I + J
then reconsider u9 = u as Element of J ;
0. R is Element of I by Th3;
then (0. R) + u9 in { (a + b) where a, b is Element of R : ( a in I & b in J ) } ;
hence u in I + J by ALGSTR_1:def_2; ::_thesis: verum
end;
hence J c= I + J by TARSKI:def_3; ::_thesis: verum
end;
theorem :: IDEAL_1:75
for R being non empty addLoopStr
for I, J being Subset of R
for K being add-closed Subset of R st I c= K & J c= K holds
I + J c= K
proof
let R be non empty addLoopStr ; ::_thesis: for I, J being Subset of R
for K being add-closed Subset of R st I c= K & J c= K holds
I + J c= K
let I, J be Subset of R; ::_thesis: for K being add-closed Subset of R st I c= K & J c= K holds
I + J c= K
let K be add-closed Subset of R; ::_thesis: ( I c= K & J c= K implies I + J c= K )
assume A1: ( I c= K & J c= K ) ; ::_thesis: I + J c= K
now__::_thesis:_for_u_being_set_st_u_in_I_+_J_holds_
u_in_K
let u be set ; ::_thesis: ( u in I + J implies u in K )
assume u in I + J ; ::_thesis: u in K
then ex i, j being Element of R st
( u = i + j & i in I & j in J ) ;
hence u in K by A1, Def1; ::_thesis: verum
end;
hence I + J c= K by TARSKI:def_3; ::_thesis: verum
end;
theorem :: IDEAL_1:76
for R being non empty add-cancelable Abelian add-associative right_zeroed associative commutative well-unital distributive left_zeroed doubleLoopStr
for a, b being Element of R holds {a,b} -Ideal = ({a} -Ideal) + ({b} -Ideal)
proof
let R be non empty add-cancelable Abelian add-associative right_zeroed associative commutative well-unital distributive left_zeroed doubleLoopStr ; ::_thesis: for a, b being Element of R holds {a,b} -Ideal = ({a} -Ideal) + ({b} -Ideal)
let a, b be Element of R; ::_thesis: {a,b} -Ideal = ({a} -Ideal) + ({b} -Ideal)
A1: now__::_thesis:_for_u_being_set_st_u_in_{a,b}_-Ideal_holds_
u_in_({a}_-Ideal)_+_({b}_-Ideal)
let u be set ; ::_thesis: ( u in {a,b} -Ideal implies u in ({a} -Ideal) + ({b} -Ideal) )
assume u in {a,b} -Ideal ; ::_thesis: u in ({a} -Ideal) + ({b} -Ideal)
then u in { ((a * r) + (b * s)) where r, s is Element of R : verum } by Th65;
then consider r, s being Element of R such that
A2: u = (a * r) + (b * s) ;
b * s in { (b * v) where v is Element of R : verum } ;
then reconsider b9 = b * s as Element of {b} -Ideal by Th64;
a * r in { (a * v) where v is Element of R : verum } ;
then reconsider a9 = a * r as Element of {a} -Ideal by Th64;
a9 + b9 in { (x + y) where x, y is Element of R : ( x in {a} -Ideal & y in {b} -Ideal ) } ;
hence u in ({a} -Ideal) + ({b} -Ideal) by A2; ::_thesis: verum
end;
now__::_thesis:_for_u_being_set_st_u_in_({a}_-Ideal)_+_({b}_-Ideal)_holds_
u_in_{a,b}_-Ideal
let u be set ; ::_thesis: ( u in ({a} -Ideal) + ({b} -Ideal) implies u in {a,b} -Ideal )
assume u in ({a} -Ideal) + ({b} -Ideal) ; ::_thesis: u in {a,b} -Ideal
then consider x, y being Element of R such that
A3: u = x + y and
A4: x in {a} -Ideal and
A5: y in {b} -Ideal ;
y in { (b * v) where v is Element of R : verum } by A5, Th64;
then A6: ex s being Element of R st y = b * s ;
x in { (a * v) where v is Element of R : verum } by A4, Th64;
then ex r being Element of R st x = a * r ;
then u in { ((a * v) + (b * d)) where v, d is Element of R : verum } by A3, A6;
hence u in {a,b} -Ideal by Th65; ::_thesis: verum
end;
hence {a,b} -Ideal = ({a} -Ideal) + ({b} -Ideal) by A1, TARSKI:1; ::_thesis: verum
end;
definition
let R be non empty 1-sorted ;
let I, J be Subset of R;
:: original: /\
redefine funcI /\ J -> Subset of R equals :: IDEAL_1:def 20
{ x where x is Element of R : ( x in I & x in J ) } ;
coherence
I /\ J is Subset of R
proof
I /\ J is Subset of R ;
hence I /\ J is Subset of R ; ::_thesis: verum
end;
compatibility
for b1 being Subset of R holds
( b1 = I /\ J iff b1 = { x where x is Element of R : ( x in I & x in J ) } )
proof
defpred S1[ set ] means $1 in J;
defpred S2[ set ] means $1 in I;
set X = { x where x is Element of R : ( S2[x] & S1[x] ) } ;
set Y = { x where x is Element of R : S2[x] } ;
set Z = { x where x is Element of R : S1[x] } ;
A1: { x where x is Element of R : S2[x] } = I by DOMAIN_1:22;
{ x where x is Element of R : ( S2[x] & S1[x] ) } = { x where x is Element of R : S2[x] } /\ { x where x is Element of R : S1[x] } from DOMAIN_1:sch_10();
hence for b1 being Subset of R holds
( b1 = I /\ J iff b1 = { x where x is Element of R : ( x in I & x in J ) } ) by A1, DOMAIN_1:22; ::_thesis: verum
end;
end;
:: deftheorem defines /\ IDEAL_1:def_20_:_
for R being non empty 1-sorted
for I, J being Subset of R holds I /\ J = { x where x is Element of R : ( x in I & x in J ) } ;
registration
let R be non empty left_add-cancelable right_zeroed left-distributive doubleLoopStr ;
let I, J be non empty left-ideal Subset of R;
clusterI /\ J -> non empty ;
coherence
not I /\ J is empty
proof
( 0. R in I & 0. R in J ) by Th2;
then 0. R in { x where x is Element of R : ( x in I & x in J ) } ;
hence not I /\ J is empty ; ::_thesis: verum
end;
end;
registration
let R be non empty addLoopStr ;
let I, J be add-closed Subset of R;
clusterI /\ J -> add-closed for Subset of R;
coherence
for b1 being Subset of R st b1 = I /\ J holds
b1 is add-closed
proof
set M = { x where x is Element of R : ( x in I & x in J ) } ;
{ x where x is Element of R : ( x in I & x in J ) } = I /\ J ;
then reconsider M = { x where x is Element of R : ( x in I & x in J ) } as Subset of R ;
for x, y being Element of R st x in M & y in M holds
x + y in M
proof
let x, y be Element of R; ::_thesis: ( x in M & y in M implies x + y in M )
assume that
A1: x in M and
A2: y in M ; ::_thesis: x + y in M
consider c being Element of R such that
A3: c = y and
A4: ( c in I & c in J ) by A2;
consider a being Element of R such that
A5: x = a and
A6: ( a in I & a in J ) by A1;
( a + c in I & a + c in J ) by A6, A4, Def1;
hence x + y in M by A5, A3; ::_thesis: verum
end;
hence for b1 being Subset of R st b1 = I /\ J holds
b1 is add-closed by Def1; ::_thesis: verum
end;
end;
registration
let R be non empty multLoopStr ;
let I, J be left-ideal Subset of R;
clusterI /\ J -> left-ideal for Subset of R;
coherence
for b1 being Subset of R st b1 = I /\ J holds
b1 is left-ideal
proof
set M = { x where x is Element of R : ( x in I & x in J ) } ;
{ x where x is Element of R : ( x in I & x in J ) } = I /\ J ;
then reconsider M = { x where x is Element of R : ( x in I & x in J ) } as Subset of R ;
for y, x being Element of R st x in M holds
y * x in M
proof
let y, x be Element of R; ::_thesis: ( x in M implies y * x in M )
assume x in M ; ::_thesis: y * x in M
then consider a being Element of R such that
A1: x = a and
A2: ( a in I & a in J ) ;
( y * a in I & y * a in J ) by A2, Def2;
hence y * x in M by A1; ::_thesis: verum
end;
hence for b1 being Subset of R st b1 = I /\ J holds
b1 is left-ideal by Def2; ::_thesis: verum
end;
end;
registration
let R be non empty multLoopStr ;
let I, J be right-ideal Subset of R;
clusterI /\ J -> right-ideal for Subset of R;
coherence
for b1 being Subset of R st b1 = I /\ J holds
b1 is right-ideal
proof
set M = { x where x is Element of R : ( x in I & x in J ) } ;
{ x where x is Element of R : ( x in I & x in J ) } = I /\ J ;
then reconsider M = { x where x is Element of R : ( x in I & x in J ) } as Subset of R ;
for y, x being Element of R st x in M holds
x * y in M
proof
let y, x be Element of R; ::_thesis: ( x in M implies x * y in M )
assume x in M ; ::_thesis: x * y in M
then consider a being Element of R such that
A1: x = a and
A2: ( a in I & a in J ) ;
( a * y in I & a * y in J ) by A2, Def3;
hence x * y in M by A1; ::_thesis: verum
end;
hence for b1 being Subset of R st b1 = I /\ J holds
b1 is right-ideal by Def3; ::_thesis: verum
end;
end;
theorem :: IDEAL_1:77
for R being non empty right_complementable Abelian add-associative right_zeroed left-distributive left_unital left_zeroed doubleLoopStr
for I being non empty add-closed left-ideal Subset of R
for J being Subset of R
for K being non empty Subset of R st J c= I holds
I /\ (J + K) = J + (I /\ K)
proof
let R be non empty right_complementable Abelian add-associative right_zeroed left-distributive left_unital left_zeroed doubleLoopStr ; ::_thesis: for I being non empty add-closed left-ideal Subset of R
for J being Subset of R
for K being non empty Subset of R st J c= I holds
I /\ (J + K) = J + (I /\ K)
let I be non empty add-closed left-ideal Subset of R; ::_thesis: for J being Subset of R
for K being non empty Subset of R st J c= I holds
I /\ (J + K) = J + (I /\ K)
let J be Subset of R; ::_thesis: for K being non empty Subset of R st J c= I holds
I /\ (J + K) = J + (I /\ K)
let K be non empty Subset of R; ::_thesis: ( J c= I implies I /\ (J + K) = J + (I /\ K) )
assume A1: J c= I ; ::_thesis: I /\ (J + K) = J + (I /\ K)
A2: now__::_thesis:_for_u_being_set_st_u_in_J_+_(I_/\_K)_holds_
u_in_I_/\_(J_+_K)
let u be set ; ::_thesis: ( u in J + (I /\ K) implies u in I /\ (J + K) )
assume u in J + (I /\ K) ; ::_thesis: u in I /\ (J + K)
then consider j, ik being Element of R such that
A3: u = j + ik and
A4: j in J and
A5: ik in I /\ K ;
A6: ex z being Element of R st
( z = ik & z in I & z in K ) by A5;
then reconsider k9 = ik as Element of K ;
u = j + k9 by A3;
then A7: u in J + K by A4;
reconsider j9 = j as Element of I by A1, A4;
reconsider i9 = ik as Element of I by A6;
u = j9 + i9 by A3;
then u in I by Def1;
hence u in I /\ (J + K) by A7; ::_thesis: verum
end;
now__::_thesis:_for_u_being_set_st_u_in_I_/\_(J_+_K)_holds_
u_in_J_+_(I_/\_K)
let u be set ; ::_thesis: ( u in I /\ (J + K) implies u in J + (I /\ K) )
assume u in I /\ (J + K) ; ::_thesis: u in J + (I /\ K)
then consider v being Element of R such that
A8: u = v and
A9: v in I and
A10: v in J + K ;
consider j, k being Element of R such that
A11: v = j + k and
A12: j in J and
A13: k in K by A10;
reconsider j9 = j as Element of I by A1, A12;
- j9 in I by Th13;
then (j9 + k) + (- j9) in I by A9, A11, Def1;
then (j9 + (- j9)) + k in I by RLVECT_1:def_3;
then (0. R) + k in I by RLVECT_1:5;
then k in I by ALGSTR_1:def_2;
then k in I /\ K by A13;
hence u in J + (I /\ K) by A8, A11, A12; ::_thesis: verum
end;
hence I /\ (J + K) = J + (I /\ K) by A2, TARSKI:1; ::_thesis: verum
end;
definition
let R be non empty doubleLoopStr ;
let I, J be Subset of R;
funcI *' J -> Subset of R equals :: IDEAL_1:def 21
{ (Sum s) where s is FinSequence of the carrier of R : for i being Element of NAT st 1 <= i & i <= len s holds
ex a, b being Element of R st
( s . i = a * b & a in I & b in J ) } ;
coherence
{ (Sum s) where s is FinSequence of the carrier of R : for i being Element of NAT st 1 <= i & i <= len s holds
ex a, b being Element of R st
( s . i = a * b & a in I & b in J ) } is Subset of R
proof
set M = { (Sum s) where s is FinSequence of the carrier of R : for i being Element of NAT st 1 <= i & i <= len s holds
ex a, b being Element of R st
( s . i = a * b & a in I & b in J ) } ;
now__::_thesis:_for_u_being_set_st_u_in__{__(Sum_s)_where_s_is_FinSequence_of_the_carrier_of_R_:_for_i_being_Element_of_NAT_st_1_<=_i_&_i_<=_len_s_holds_
ex_a,_b_being_Element_of_R_st_
(_s_._i_=_a_*_b_&_a_in_I_&_b_in_J_)__}__holds_
u_in_the_carrier_of_R
let u be set ; ::_thesis: ( u in { (Sum s) where s is FinSequence of the carrier of R : for i being Element of NAT st 1 <= i & i <= len s holds
ex a, b being Element of R st
( s . i = a * b & a in I & b in J ) } implies u in the carrier of R )
assume u in { (Sum s) where s is FinSequence of the carrier of R : for i being Element of NAT st 1 <= i & i <= len s holds
ex a, b being Element of R st
( s . i = a * b & a in I & b in J ) } ; ::_thesis: u in the carrier of R
then ex s being FinSequence of the carrier of R st
( u = Sum s & ( for i being Element of NAT st 1 <= i & i <= len s holds
ex a, b being Element of R st
( s . i = a * b & a in I & b in J ) ) ) ;
hence u in the carrier of R ; ::_thesis: verum
end;
then reconsider M = { (Sum s) where s is FinSequence of the carrier of R : for i being Element of NAT st 1 <= i & i <= len s holds
ex a, b being Element of R st
( s . i = a * b & a in I & b in J ) } as Subset of R by TARSKI:def_3;
M is Subset of R ;
hence { (Sum s) where s is FinSequence of the carrier of R : for i being Element of NAT st 1 <= i & i <= len s holds
ex a, b being Element of R st
( s . i = a * b & a in I & b in J ) } is Subset of R ; ::_thesis: verum
end;
end;
:: deftheorem defines *' IDEAL_1:def_21_:_
for R being non empty doubleLoopStr
for I, J being Subset of R holds I *' J = { (Sum s) where s is FinSequence of the carrier of R : for i being Element of NAT st 1 <= i & i <= len s holds
ex a, b being Element of R st
( s . i = a * b & a in I & b in J ) } ;
registration
let R be non empty doubleLoopStr ;
let I, J be Subset of R;
clusterI *' J -> non empty ;
coherence
not I *' J is empty
proof
set M = { (Sum s) where s is FinSequence of the carrier of R : for i being Element of NAT st 1 <= i & i <= len s holds
ex a, b being Element of R st
( s . i = a * b & a in I & b in J ) } ;
not { (Sum s) where s is FinSequence of the carrier of R : for i being Element of NAT st 1 <= i & i <= len s holds
ex a, b being Element of R st
( s . i = a * b & a in I & b in J ) } is empty
proof
set p = <*> the carrier of R;
for i being Element of NAT st 1 <= i & i <= len (<*> the carrier of R) holds
ex a, b being Element of R st
( (<*> the carrier of R) . i = a * b & a in I & b in J ) ;
then Sum (<*> the carrier of R) in { (Sum s) where s is FinSequence of the carrier of R : for i being Element of NAT st 1 <= i & i <= len s holds
ex a, b being Element of R st
( s . i = a * b & a in I & b in J ) } ;
hence not { (Sum s) where s is FinSequence of the carrier of R : for i being Element of NAT st 1 <= i & i <= len s holds
ex a, b being Element of R st
( s . i = a * b & a in I & b in J ) } is empty ; ::_thesis: verum
end;
hence not I *' J is empty ; ::_thesis: verum
end;
end;
definition
let R be non empty commutative doubleLoopStr ;
let I, J be Subset of R;
:: original: *'
redefine funcI *' J -> Subset of R;
commutativity
for I, J being Subset of R holds I *' J = J *' I
proof
now__::_thesis:_for_I,_J_being_Subset_of_R_holds_I_*'_J_=_J_*'_I
let I, J be Subset of R; ::_thesis: I *' J = J *' I
A1: now__::_thesis:_for_u_being_set_st_u_in_J_*'_I_holds_
u_in_I_*'_J
let u be set ; ::_thesis: ( u in J *' I implies u in I *' J )
assume u in J *' I ; ::_thesis: u in I *' J
then consider s being FinSequence of the carrier of R such that
A2: u = Sum s and
A3: for i being Element of NAT st 1 <= i & i <= len s holds
ex a, b being Element of R st
( s . i = a * b & a in J & b in I ) ;
for i being Element of NAT st 1 <= i & i <= len s holds
ex a, b being Element of R st
( s . i = a * b & a in I & b in J )
proof
let i be Element of NAT ; ::_thesis: ( 1 <= i & i <= len s implies ex a, b being Element of R st
( s . i = a * b & a in I & b in J ) )
assume ( 1 <= i & i <= len s ) ; ::_thesis: ex a, b being Element of R st
( s . i = a * b & a in I & b in J )
then ex a, b being Element of R st
( s . i = a * b & a in J & b in I ) by A3;
hence ex a, b being Element of R st
( s . i = a * b & a in I & b in J ) ; ::_thesis: verum
end;
hence u in I *' J by A2; ::_thesis: verum
end;
now__::_thesis:_for_u_being_set_st_u_in_I_*'_J_holds_
u_in_J_*'_I
let u be set ; ::_thesis: ( u in I *' J implies u in J *' I )
assume u in I *' J ; ::_thesis: u in J *' I
then consider s being FinSequence of the carrier of R such that
A4: u = Sum s and
A5: for i being Element of NAT st 1 <= i & i <= len s holds
ex a, b being Element of R st
( s . i = a * b & a in I & b in J ) ;
for i being Element of NAT st 1 <= i & i <= len s holds
ex a, b being Element of R st
( s . i = a * b & a in J & b in I )
proof
let i be Element of NAT ; ::_thesis: ( 1 <= i & i <= len s implies ex a, b being Element of R st
( s . i = a * b & a in J & b in I ) )
assume ( 1 <= i & i <= len s ) ; ::_thesis: ex a, b being Element of R st
( s . i = a * b & a in J & b in I )
then ex a, b being Element of R st
( s . i = a * b & a in I & b in J ) by A5;
hence ex a, b being Element of R st
( s . i = a * b & a in J & b in I ) ; ::_thesis: verum
end;
hence u in J *' I by A4; ::_thesis: verum
end;
hence I *' J = J *' I by A1, TARSKI:1; ::_thesis: verum
end;
hence for I, J being Subset of R holds I *' J = J *' I ; ::_thesis: verum
end;
end;
registration
let R be non empty add-associative right_zeroed doubleLoopStr ;
let I, J be Subset of R;
clusterI *' J -> add-closed ;
coherence
I *' J is add-closed
proof
set M = { (Sum s) where s is FinSequence of the carrier of R : for i being Element of NAT st 1 <= i & i <= len s holds
ex a, b being Element of R st
( s . i = a * b & a in I & b in J ) } ;
{ (Sum s) where s is FinSequence of the carrier of R : for i being Element of NAT st 1 <= i & i <= len s holds
ex a, b being Element of R st
( s . i = a * b & a in I & b in J ) } = I *' J ;
then reconsider M = { (Sum s) where s is FinSequence of the carrier of R : for i being Element of NAT st 1 <= i & i <= len s holds
ex a, b being Element of R st
( s . i = a * b & a in I & b in J ) } as non empty Subset of R ;
for x, y being Element of R st x in M & y in M holds
x + y in M
proof
let x, y be Element of R; ::_thesis: ( x in M & y in M implies x + y in M )
assume that
A1: x in M and
A2: y in M ; ::_thesis: x + y in M
consider s being FinSequence of the carrier of R such that
A3: x = Sum s and
A4: for i being Element of NAT st 1 <= i & i <= len s holds
ex a, b being Element of R st
( s . i = a * b & a in I & b in J ) by A1;
consider t being FinSequence of the carrier of R such that
A5: y = Sum t and
A6: for i being Element of NAT st 1 <= i & i <= len t holds
ex a, b being Element of R st
( t . i = a * b & a in I & b in J ) by A2;
set q = s ^ t;
A7: now__::_thesis:_for_i_being_Element_of_NAT_st_1_<=_i_&_i_<=_len_(s_^_t)_holds_
ex_a,_r_being_Element_of_R_st_
(_(s_^_t)_._i_=_a_*_r_&_a_in_I_&_r_in_J_)
let i be Element of NAT ; ::_thesis: ( 1 <= i & i <= len (s ^ t) implies ex a, r being Element of R st
( (s ^ t) . i = a * r & a in I & r in J ) )
assume that
A8: 1 <= i and
A9: i <= len (s ^ t) ; ::_thesis: ex a, r being Element of R st
( (s ^ t) . i = a * r & a in I & r in J )
thus ex a, r being Element of R st
( (s ^ t) . i = a * r & a in I & r in J ) ::_thesis: verum
proof
percases ( i <= len s or len s < i ) ;
supposeA10: i <= len s ; ::_thesis: ex a, r being Element of R st
( (s ^ t) . i = a * r & a in I & r in J )
then i in Seg (len s) by A8, FINSEQ_1:1;
then i in dom s by FINSEQ_1:def_3;
then (s ^ t) . i = s . i by FINSEQ_1:def_7;
hence ex a, r being Element of R st
( (s ^ t) . i = a * r & a in I & r in J ) by A4, A8, A10; ::_thesis: verum
end;
supposeA11: len s < i ; ::_thesis: ex a, r being Element of R st
( (s ^ t) . i = a * r & a in I & r in J )
then reconsider j = i - (len s) as Element of NAT by INT_1:5;
(len s) - (len s) < j by A11, XREAL_1:9;
then A12: 1 <= j by NAT_1:14;
i <= (len s) + (len t) by A9, FINSEQ_1:22;
then A13: j <= ((len s) + (len t)) - (len s) by XREAL_1:9;
t . j = (s ^ t) . i by A9, A11, FINSEQ_1:24;
hence ex a, r being Element of R st
( (s ^ t) . i = a * r & a in I & r in J ) by A6, A12, A13; ::_thesis: verum
end;
end;
end;
end;
Sum (s ^ t) = x + y by A3, A5, RLVECT_1:41;
hence x + y in M by A7; ::_thesis: verum
end;
hence I *' J is add-closed by Def1; ::_thesis: verum
end;
end;
registration
let R be non empty left_add-cancelable right_zeroed associative left-distributive doubleLoopStr ;
let I, J be right-ideal Subset of R;
clusterI *' J -> right-ideal ;
coherence
I *' J is right-ideal
proof
set M = { (Sum s) where s is FinSequence of the carrier of R : for i being Element of NAT st 1 <= i & i <= len s holds
ex a, b being Element of R st
( s . i = a * b & a in I & b in J ) } ;
{ (Sum s) where s is FinSequence of the carrier of R : for i being Element of NAT st 1 <= i & i <= len s holds
ex a, b being Element of R st
( s . i = a * b & a in I & b in J ) } = I *' J ;
then reconsider M = { (Sum s) where s is FinSequence of the carrier of R : for i being Element of NAT st 1 <= i & i <= len s holds
ex a, b being Element of R st
( s . i = a * b & a in I & b in J ) } as non empty Subset of R ;
for y, x being Element of R st x in M holds
x * y in M
proof
let y, x be Element of R; ::_thesis: ( x in M implies x * y in M )
assume x in M ; ::_thesis: x * y in M
then consider s being FinSequence of the carrier of R such that
A1: x = Sum s and
A2: for i being Element of NAT st 1 <= i & i <= len s holds
ex a, b being Element of R st
( s . i = a * b & a in I & b in J ) ;
set q = s * y;
A3: Seg (len (s * y)) = dom (s * y) by FINSEQ_1:def_3
.= dom s by POLYNOM1:def_2
.= Seg (len s) by FINSEQ_1:def_3 ;
then A4: len (s * y) = len s by FINSEQ_1:6;
A5: now__::_thesis:_for_i_being_Element_of_NAT_st_1_<=_i_&_i_<=_len_(s_*_y)_holds_
ex_b,_r_being_Element_of_R_st_
(_(s_*_y)_._i_=_b_*_r_&_b_in_I_&_r_in_J_)
let i be Element of NAT ; ::_thesis: ( 1 <= i & i <= len (s * y) implies ex b, r being Element of R st
( (s * y) . i = b * r & b in I & r in J ) )
assume A6: ( 1 <= i & i <= len (s * y) ) ; ::_thesis: ex b, r being Element of R st
( (s * y) . i = b * r & b in I & r in J )
then consider c, r9 being Element of R such that
A7: s . i = c * r9 and
A8: ( c in I & r9 in J ) by A2, A4;
i in Seg (len s) by A3, A6, FINSEQ_1:1;
then A9: i in dom s by FINSEQ_1:def_3;
then A10: s /. i = c * r9 by A7, PARTFUN1:def_6;
i in Seg (len (s * y)) by A6, FINSEQ_1:1;
then i in dom (s * y) by FINSEQ_1:def_3;
then A11: (s * y) . i = (s * y) /. i by PARTFUN1:def_6
.= (c * r9) * y by A9, A10, POLYNOM1:def_2
.= c * (r9 * y) by GROUP_1:def_3 ;
thus ex b, r being Element of R st
( (s * y) . i = b * r & b in I & r in J ) ::_thesis: verum
proof
take c ; ::_thesis: ex r being Element of R st
( (s * y) . i = c * r & c in I & r in J )
take r9 * y ; ::_thesis: ( (s * y) . i = c * (r9 * y) & c in I & r9 * y in J )
thus ( (s * y) . i = c * (r9 * y) & c in I & r9 * y in J ) by A8, A11, Def3; ::_thesis: verum
end;
end;
Sum (s * y) = (Sum s) * y by BINOM:5;
hence x * y in M by A1, A5; ::_thesis: verum
end;
hence I *' J is right-ideal by Def3; ::_thesis: verum
end;
end;
registration
let R be non empty right_add-cancelable associative right-distributive left_zeroed doubleLoopStr ;
let I, J be left-ideal Subset of R;
clusterI *' J -> left-ideal ;
coherence
I *' J is left-ideal
proof
set M = { (Sum s) where s is FinSequence of the carrier of R : for i being Element of NAT st 1 <= i & i <= len s holds
ex a, b being Element of R st
( s . i = a * b & a in I & b in J ) } ;
{ (Sum s) where s is FinSequence of the carrier of R : for i being Element of NAT st 1 <= i & i <= len s holds
ex a, b being Element of R st
( s . i = a * b & a in I & b in J ) } = I *' J ;
then reconsider M = { (Sum s) where s is FinSequence of the carrier of R : for i being Element of NAT st 1 <= i & i <= len s holds
ex a, b being Element of R st
( s . i = a * b & a in I & b in J ) } as non empty Subset of R ;
for y, x being Element of R st x in M holds
y * x in M
proof
let y, x be Element of R; ::_thesis: ( x in M implies y * x in M )
assume x in M ; ::_thesis: y * x in M
then consider s being FinSequence of the carrier of R such that
A1: x = Sum s and
A2: for i being Element of NAT st 1 <= i & i <= len s holds
ex a, b being Element of R st
( s . i = a * b & a in I & b in J ) ;
set q = y * s;
A3: Seg (len (y * s)) = dom (y * s) by FINSEQ_1:def_3
.= dom s by POLYNOM1:def_1
.= Seg (len s) by FINSEQ_1:def_3 ;
then A4: len (y * s) = len s by FINSEQ_1:6;
A5: now__::_thesis:_for_i_being_Element_of_NAT_st_1_<=_i_&_i_<=_len_(y_*_s)_holds_
ex_b,_r_being_Element_of_R_st_
(_(y_*_s)_._i_=_b_*_r_&_b_in_I_&_r_in_J_)
let i be Element of NAT ; ::_thesis: ( 1 <= i & i <= len (y * s) implies ex b, r being Element of R st
( (y * s) . i = b * r & b in I & r in J ) )
assume A6: ( 1 <= i & i <= len (y * s) ) ; ::_thesis: ex b, r being Element of R st
( (y * s) . i = b * r & b in I & r in J )
then consider c, r9 being Element of R such that
A7: s . i = c * r9 and
A8: ( c in I & r9 in J ) by A2, A4;
i in Seg (len s) by A3, A6, FINSEQ_1:1;
then A9: i in dom s by FINSEQ_1:def_3;
then A10: s /. i = c * r9 by A7, PARTFUN1:def_6;
i in Seg (len (y * s)) by A6, FINSEQ_1:1;
then i in dom (y * s) by FINSEQ_1:def_3;
then A11: (y * s) . i = (y * s) /. i by PARTFUN1:def_6
.= y * (c * r9) by A9, A10, POLYNOM1:def_1
.= (y * c) * r9 by GROUP_1:def_3 ;
thus ex b, r being Element of R st
( (y * s) . i = b * r & b in I & r in J ) ::_thesis: verum
proof
take y * c ; ::_thesis: ex r being Element of R st
( (y * s) . i = (y * c) * r & y * c in I & r in J )
take r9 ; ::_thesis: ( (y * s) . i = (y * c) * r9 & y * c in I & r9 in J )
thus ( (y * s) . i = (y * c) * r9 & y * c in I & r9 in J ) by A8, A11, Def2; ::_thesis: verum
end;
end;
Sum (y * s) = y * (Sum s) by BINOM:4;
hence y * x in M by A1, A5; ::_thesis: verum
end;
hence I *' J is left-ideal by Def2; ::_thesis: verum
end;
end;
theorem :: IDEAL_1:78
for R being non empty left_add-cancelable right_zeroed left-distributive left_zeroed doubleLoopStr
for I being non empty Subset of R holds {(0. R)} *' I = {(0. R)}
proof
let R be non empty left_add-cancelable right_zeroed left-distributive left_zeroed doubleLoopStr ; ::_thesis: for I being non empty Subset of R holds {(0. R)} *' I = {(0. R)}
let I be non empty Subset of R; ::_thesis: {(0. R)} *' I = {(0. R)}
A1: now__::_thesis:_for_u_being_set_st_u_in_{(0._R)}_*'_I_holds_
u_in_{(0._R)}
let u be set ; ::_thesis: ( u in {(0. R)} *' I implies u in {(0. R)} )
assume u in {(0. R)} *' I ; ::_thesis: u in {(0. R)}
then consider s being FinSequence of the carrier of R such that
A2: Sum s = u and
A3: for i being Element of NAT st 1 <= i & i <= len s holds
ex a, b being Element of R st
( s . i = a * b & a in {(0. R)} & b in I ) ;
now__::_thesis:_(_(_len_s_=_0_&_Sum_s_=_0._R_)_or_(_len_s_<>_0_&_Sum_s_=_0._R_)_)
percases ( len s = 0 or len s <> 0 ) ;
case len s = 0 ; ::_thesis: Sum s = 0. R
then s = <*> the carrier of R ;
hence Sum s = 0. R by RLVECT_1:43; ::_thesis: verum
end;
case len s <> 0 ; ::_thesis: Sum s = 0. R
then 1 <= len s by NAT_1:14;
then 1 in Seg (len s) by FINSEQ_1:1;
then A4: 1 in dom s by FINSEQ_1:def_3;
A5: for i being Element of NAT st i in dom s holds
s /. i = 0. R
proof
let i be Element of NAT ; ::_thesis: ( i in dom s implies s /. i = 0. R )
assume A6: i in dom s ; ::_thesis: s /. i = 0. R
then i in Seg (len s) by FINSEQ_1:def_3;
then ( 1 <= i & i <= len s ) by FINSEQ_1:1;
then consider a, b being Element of R such that
A7: s . i = a * b and
A8: a in {(0. R)} and
b in I by A3;
A9: a = 0. R by A8, TARSKI:def_1;
s /. i = a * b by A6, A7, PARTFUN1:def_6;
hence s /. i = 0. R by A9, BINOM:1; ::_thesis: verum
end;
then for i being Element of NAT st i in dom s & i <> 1 holds
s /. i = 0. R ;
hence Sum s = s /. 1 by A4, POLYNOM2:3
.= 0. R by A4, A5 ;
::_thesis: verum
end;
end;
end;
hence u in {(0. R)} by A2, TARSKI:def_1; ::_thesis: verum
end;
now__::_thesis:_for_u_being_set_st_u_in_{(0._R)}_holds_
u_in_{(0._R)}_*'_I
reconsider o = 0. R as Element of {(0. R)} by TARSKI:def_1;
set a = the Element of I;
let u be set ; ::_thesis: ( u in {(0. R)} implies u in {(0. R)} *' I )
assume A10: u in {(0. R)} ; ::_thesis: u in {(0. R)} *' I
set q = <*((0. R) * the Element of I)*>;
A11: ( len <*((0. R) * the Element of I)*> = 1 & <*((0. R) * the Element of I)*> . 1 = (0. R) * the Element of I ) by FINSEQ_1:40;
A12: for i being Element of NAT st 1 <= i & i <= len <*((0. R) * the Element of I)*> holds
ex b, r being Element of R st
( <*((0. R) * the Element of I)*> . i = b * r & b in {(0. R)} & r in I )
proof
let i be Element of NAT ; ::_thesis: ( 1 <= i & i <= len <*((0. R) * the Element of I)*> implies ex b, r being Element of R st
( <*((0. R) * the Element of I)*> . i = b * r & b in {(0. R)} & r in I ) )
assume ( 1 <= i & i <= len <*((0. R) * the Element of I)*> ) ; ::_thesis: ex b, r being Element of R st
( <*((0. R) * the Element of I)*> . i = b * r & b in {(0. R)} & r in I )
then <*((0. R) * the Element of I)*> . i = o * the Element of I by A11, XXREAL_0:1;
hence ex b, r being Element of R st
( <*((0. R) * the Element of I)*> . i = b * r & b in {(0. R)} & r in I ) ; ::_thesis: verum
end;
Sum <*((0. R) * the Element of I)*> = (0. R) * the Element of I by BINOM:3
.= 0. R by BINOM:1
.= u by A10, TARSKI:def_1 ;
hence u in {(0. R)} *' I by A12; ::_thesis: verum
end;
hence {(0. R)} *' I = {(0. R)} by A1, TARSKI:1; ::_thesis: verum
end;
theorem Th79: :: IDEAL_1:79
for R being non empty add-cancelable right_zeroed distributive left_zeroed doubleLoopStr
for I being non empty add-closed right-ideal Subset of R
for J being non empty add-closed left-ideal Subset of R holds I *' J c= I /\ J
proof
let R be non empty add-cancelable right_zeroed distributive left_zeroed doubleLoopStr ; ::_thesis: for I being non empty add-closed right-ideal Subset of R
for J being non empty add-closed left-ideal Subset of R holds I *' J c= I /\ J
let I be non empty add-closed right-ideal Subset of R; ::_thesis: for J being non empty add-closed left-ideal Subset of R holds I *' J c= I /\ J
let J be non empty add-closed left-ideal Subset of R; ::_thesis: I *' J c= I /\ J
now__::_thesis:_for_u_being_set_st_u_in_I_*'_J_holds_
u_in_I_/\_J
let u be set ; ::_thesis: ( u in I *' J implies u in I /\ J )
assume u in I *' J ; ::_thesis: u in I /\ J
then consider s being FinSequence of the carrier of R such that
A1: Sum s = u and
A2: for i being Element of NAT st 1 <= i & i <= len s holds
ex a, b being Element of R st
( s . i = a * b & a in I & b in J ) ;
consider f being Function of NAT, the carrier of R such that
A3: Sum s = f . (len s) and
A4: f . 0 = 0. R and
A5: for j being Element of NAT
for v being Element of R st j < len s & v = s . (j + 1) holds
f . (j + 1) = (f . j) + v by RLVECT_1:def_12;
defpred S1[ Element of NAT ] means ( f . $1 in I & f . $1 in J );
A6: now__::_thesis:_for_j_being_Element_of_NAT_st_0_<=_j_&_j_<_len_s_&_S1[j]_holds_
S1[j_+_1]
let j be Element of NAT ; ::_thesis: ( 0 <= j & j < len s & S1[j] implies S1[j + 1] )
assume that
0 <= j and
A7: j < len s ; ::_thesis: ( S1[j] implies S1[j + 1] )
thus ( S1[j] implies S1[j + 1] ) ::_thesis: verum
proof
A8: ( j + 1 <= len s & 0 + 1 <= j + 1 ) by A7, NAT_1:13;
then j + 1 in Seg (len s) by FINSEQ_1:1;
then j + 1 in dom s by FINSEQ_1:def_3;
then A9: s . (j + 1) = s /. (j + 1) by PARTFUN1:def_6;
ex a, b being Element of R st
( s . (j + 1) = a * b & a in I & b in J ) by A2, A8;
then A10: ( s /. (j + 1) in I & s /. (j + 1) in J ) by A9, Def2, Def3;
assume A11: ( f . j in I & f . j in J ) ; ::_thesis: S1[j + 1]
f . (j + 1) = (f . j) + (s /. (j + 1)) by A5, A7, A9;
hence S1[j + 1] by A11, A10, Def1; ::_thesis: verum
end;
end;
A12: S1[ 0 ] by A4, Th2, Th3;
for j being Element of NAT st 0 <= j & j <= len s holds
S1[j] from INT_1:sch_7(A12, A6);
then ( Sum s in I & Sum s in J ) by A3;
hence u in I /\ J by A1; ::_thesis: verum
end;
hence I *' J c= I /\ J by TARSKI:def_3; ::_thesis: verum
end;
theorem Th80: :: IDEAL_1:80
for R being non empty add-cancelable Abelian add-associative right_zeroed associative distributive left_zeroed doubleLoopStr
for I, J, K being non empty right-ideal Subset of R holds I *' (J + K) = (I *' J) + (I *' K)
proof
let R be non empty add-cancelable Abelian add-associative right_zeroed associative distributive left_zeroed doubleLoopStr ; ::_thesis: for I, J, K being non empty right-ideal Subset of R holds I *' (J + K) = (I *' J) + (I *' K)
let I, J, K be non empty right-ideal Subset of R; ::_thesis: I *' (J + K) = (I *' J) + (I *' K)
A1: now__::_thesis:_for_u_being_set_st_u_in_I_*'_(J_+_K)_holds_
u_in_(I_*'_J)_+_(I_*'_K)
let u be set ; ::_thesis: ( u in I *' (J + K) implies u in (I *' J) + (I *' K) )
assume u in I *' (J + K) ; ::_thesis: u in (I *' J) + (I *' K)
then consider s being FinSequence of the carrier of R such that
A2: Sum s = u and
A3: for i being Element of NAT st 1 <= i & i <= len s holds
ex a, b being Element of R st
( s . i = a * b & a in I & b in J + K ) ;
consider f being Function of NAT, the carrier of R such that
A4: Sum s = f . (len s) and
A5: f . 0 = 0. R and
A6: for j being Element of NAT
for v being Element of R st j < len s & v = s . (j + 1) holds
f . (j + 1) = (f . j) + v by RLVECT_1:def_12;
defpred S1[ Element of NAT ] means ex x, y being Element of R st
( f . $1 = x + y & x in I *' J & y in I *' K );
A7: now__::_thesis:_for_n_being_Element_of_NAT_st_0_<=_n_&_n_<_len_s_&_S1[n]_holds_
S1[n_+_1]
let n be Element of NAT ; ::_thesis: ( 0 <= n & n < len s & S1[n] implies S1[n + 1] )
assume that
0 <= n and
A8: n < len s ; ::_thesis: ( S1[n] implies S1[n + 1] )
thus ( S1[n] implies S1[n + 1] ) ::_thesis: verum
proof
assume ex x, y being Element of R st
( f . n = x + y & x in I *' J & y in I *' K ) ; ::_thesis: S1[n + 1]
then consider x, y being Element of R such that
A9: f . n = x + y and
A10: x in I *' J and
A11: y in I *' K ;
consider p being FinSequence of the carrier of R such that
A12: Sum p = y and
A13: for i being Element of NAT st 1 <= i & i <= len p holds
ex a, b being Element of R st
( p . i = a * b & a in I & b in K ) by A11;
consider q being FinSequence of the carrier of R such that
A14: Sum q = x and
A15: for i being Element of NAT st 1 <= i & i <= len q holds
ex a, b being Element of R st
( q . i = a * b & a in I & b in J ) by A10;
A16: ( 0 + 1 <= n + 1 & n + 1 <= len s ) by A8, NAT_1:13;
then consider a, b being Element of R such that
A17: s . (n + 1) = a * b and
A18: a in I and
A19: b in J + K by A3;
consider c, d being Element of R such that
A20: b = c + d and
A21: c in J and
A22: d in K by A19;
set q1 = q ^ <*(a * c)*>;
set p1 = p ^ <*(a * d)*>;
n + 1 in Seg (len s) by A16, FINSEQ_1:1;
then A23: n + 1 in dom s by FINSEQ_1:def_3;
then A24: s . (n + 1) = s /. (n + 1) by PARTFUN1:def_6;
A25: len (p ^ <*(a * d)*>) = (len p) + (len <*(a * d)*>) by FINSEQ_1:22
.= (len p) + 1 by FINSEQ_1:40 ;
for i being Element of NAT st 1 <= i & i <= len (p ^ <*(a * d)*>) holds
ex a, b being Element of R st
( (p ^ <*(a * d)*>) . i = a * b & a in I & b in K )
proof
let i be Element of NAT ; ::_thesis: ( 1 <= i & i <= len (p ^ <*(a * d)*>) implies ex a, b being Element of R st
( (p ^ <*(a * d)*>) . i = a * b & a in I & b in K ) )
assume that
A26: 1 <= i and
A27: i <= len (p ^ <*(a * d)*>) ; ::_thesis: ex a, b being Element of R st
( (p ^ <*(a * d)*>) . i = a * b & a in I & b in K )
percases ( i = len (p ^ <*(a * d)*>) or i <> len (p ^ <*(a * d)*>) ) ;
suppose i = len (p ^ <*(a * d)*>) ; ::_thesis: ex a, b being Element of R st
( (p ^ <*(a * d)*>) . i = a * b & a in I & b in K )
hence ex a, b being Element of R st
( (p ^ <*(a * d)*>) . i = a * b & a in I & b in K ) by A18, A22, A25, FINSEQ_1:42; ::_thesis: verum
end;
suppose i <> len (p ^ <*(a * d)*>) ; ::_thesis: ex a, b being Element of R st
( (p ^ <*(a * d)*>) . i = a * b & a in I & b in K )
then i < len (p ^ <*(a * d)*>) by A27, XXREAL_0:1;
then A28: i <= len p by A25, NAT_1:13;
then consider a, b being Element of R such that
A29: p . i = a * b and
A30: ( a in I & b in K ) by A13, A26;
i in Seg (len p) by A26, A28, FINSEQ_1:1;
then i in dom p by FINSEQ_1:def_3;
then (p ^ <*(a * d)*>) . i = a * b by A29, FINSEQ_1:def_7;
hence ex a, b being Element of R st
( (p ^ <*(a * d)*>) . i = a * b & a in I & b in K ) by A30; ::_thesis: verum
end;
end;
end;
then A31: Sum (p ^ <*(a * d)*>) in I *' K ;
A32: len (q ^ <*(a * c)*>) = (len q) + (len <*(a * c)*>) by FINSEQ_1:22
.= (len q) + 1 by FINSEQ_1:40 ;
for i being Element of NAT st 1 <= i & i <= len (q ^ <*(a * c)*>) holds
ex a, b being Element of R st
( (q ^ <*(a * c)*>) . i = a * b & a in I & b in J )
proof
let i be Element of NAT ; ::_thesis: ( 1 <= i & i <= len (q ^ <*(a * c)*>) implies ex a, b being Element of R st
( (q ^ <*(a * c)*>) . i = a * b & a in I & b in J ) )
assume that
A33: 1 <= i and
A34: i <= len (q ^ <*(a * c)*>) ; ::_thesis: ex a, b being Element of R st
( (q ^ <*(a * c)*>) . i = a * b & a in I & b in J )
percases ( i = len (q ^ <*(a * c)*>) or i <> len (q ^ <*(a * c)*>) ) ;
suppose i = len (q ^ <*(a * c)*>) ; ::_thesis: ex a, b being Element of R st
( (q ^ <*(a * c)*>) . i = a * b & a in I & b in J )
hence ex a, b being Element of R st
( (q ^ <*(a * c)*>) . i = a * b & a in I & b in J ) by A18, A21, A32, FINSEQ_1:42; ::_thesis: verum
end;
suppose i <> len (q ^ <*(a * c)*>) ; ::_thesis: ex a, b being Element of R st
( (q ^ <*(a * c)*>) . i = a * b & a in I & b in J )
then i < len (q ^ <*(a * c)*>) by A34, XXREAL_0:1;
then A35: i <= len q by A32, NAT_1:13;
then consider a, b being Element of R such that
A36: q . i = a * b and
A37: ( a in I & b in J ) by A15, A33;
i in Seg (len q) by A33, A35, FINSEQ_1:1;
then i in dom q by FINSEQ_1:def_3;
then (q ^ <*(a * c)*>) . i = a * b by A36, FINSEQ_1:def_7;
hence ex a, b being Element of R st
( (q ^ <*(a * c)*>) . i = a * b & a in I & b in J ) by A37; ::_thesis: verum
end;
end;
end;
then A38: Sum (q ^ <*(a * c)*>) in { (Sum t) where t is FinSequence of the carrier of R : for i being Element of NAT st 1 <= i & i <= len t holds
ex a, b being Element of R st
( t . i = a * b & a in I & b in J ) } ;
A39: s /. (n + 1) = a * (c + d) by A23, A17, A20, PARTFUN1:def_6
.= (a * c) + (a * d) by VECTSP_1:def_7 ;
(Sum (q ^ <*(a * c)*>)) + (Sum (p ^ <*(a * d)*>)) = ((Sum q) + (Sum <*(a * c)*>)) + (Sum (p ^ <*(a * d)*>)) by RLVECT_1:41
.= ((Sum q) + (a * c)) + (Sum (p ^ <*(a * d)*>)) by BINOM:3
.= ((Sum q) + (a * c)) + ((Sum p) + (Sum <*(a * d)*>)) by RLVECT_1:41
.= ((Sum q) + (a * c)) + ((Sum p) + (a * d)) by BINOM:3
.= (((Sum q) + (a * c)) + (Sum p)) + (a * d) by RLVECT_1:def_3
.= ((a * c) + ((Sum q) + (Sum p))) + (a * d) by RLVECT_1:def_3
.= (f . n) + ((a * c) + (a * d)) by A9, A14, A12, RLVECT_1:def_3
.= f . (n + 1) by A6, A8, A24, A39 ;
hence S1[n + 1] by A38, A31; ::_thesis: verum
end;
end;
A40: S1[ 0 ]
proof
take 0. R ; ::_thesis: ex y being Element of R st
( f . 0 = (0. R) + y & 0. R in I *' J & y in I *' K )
take 0. R ; ::_thesis: ( f . 0 = (0. R) + (0. R) & 0. R in I *' J & 0. R in I *' K )
thus ( f . 0 = (0. R) + (0. R) & 0. R in I *' J & 0. R in I *' K ) by A5, Th3, RLVECT_1:def_4; ::_thesis: verum
end;
for n being Element of NAT st 0 <= n & n <= len s holds
S1[n] from INT_1:sch_7(A40, A7);
then ex x, y being Element of R st
( Sum s = x + y & x in I *' J & y in I *' K ) by A4;
hence u in (I *' J) + (I *' K) by A2; ::_thesis: verum
end;
now__::_thesis:_for_u_being_set_st_u_in_(I_*'_J)_+_(I_*'_K)_holds_
u_in_I_*'_(J_+_K)
let u be set ; ::_thesis: ( u in (I *' J) + (I *' K) implies u in I *' (J + K) )
assume u in (I *' J) + (I *' K) ; ::_thesis: u in I *' (J + K)
then consider a, b being Element of R such that
A41: u = a + b and
A42: a in I *' J and
A43: b in I *' K ;
consider p being FinSequence of the carrier of R such that
A44: b = Sum p and
A45: for i being Element of NAT st 1 <= i & i <= len p holds
ex a, b being Element of R st
( p . i = a * b & a in I & b in K ) by A43;
consider q being FinSequence of the carrier of R such that
A46: a = Sum q and
A47: for i being Element of NAT st 1 <= i & i <= len q holds
ex a, b being Element of R st
( q . i = a * b & a in I & b in J ) by A42;
set s = p ^ q;
A48: for i being Element of NAT st 1 <= i & i <= len (p ^ q) holds
ex a, b being Element of R st
( (p ^ q) . i = a * b & a in I & b in J + K )
proof
let i be Element of NAT ; ::_thesis: ( 1 <= i & i <= len (p ^ q) implies ex a, b being Element of R st
( (p ^ q) . i = a * b & a in I & b in J + K ) )
assume that
A49: 1 <= i and
A50: i <= len (p ^ q) ; ::_thesis: ex a, b being Element of R st
( (p ^ q) . i = a * b & a in I & b in J + K )
i in Seg (len (p ^ q)) by A49, A50, FINSEQ_1:1;
then A51: i in dom (p ^ q) by FINSEQ_1:def_3;
now__::_thesis:_(_(_i_<=_len_p_&_ex_a,_b_being_Element_of_R_st_
(_(p_^_q)_._i_=_a_*_b_&_a_in_I_&_b_in_J_+_K_)_)_or_(_i_>_len_p_&_ex_a,_b_being_Element_of_R_st_
(_(p_^_q)_._i_=_a_*_b_&_a_in_I_&_b_in_J_+_K_)_)_)
percases ( i <= len p or i > len p ) ;
caseA52: i <= len p ; ::_thesis: ex a, b being Element of R st
( (p ^ q) . i = a * b & a in I & b in J + K )
then consider a, b being Element of R such that
A53: p . i = a * b and
A54: a in I and
A55: b in K by A45, A49;
i in Seg (len p) by A49, A52, FINSEQ_1:1;
then i in dom p by FINSEQ_1:def_3;
then A56: (p ^ q) . i = a * b by A53, FINSEQ_1:def_7
.= a * ((0. R) + b) by ALGSTR_1:def_2 ;
0. R in J by Th3;
then (0. R) + b in { (a9 + b9) where a9, b9 is Element of R : ( a9 in J & b9 in K ) } by A55;
hence ex a, b being Element of R st
( (p ^ q) . i = a * b & a in I & b in J + K ) by A54, A56; ::_thesis: verum
end;
case i > len p ; ::_thesis: ex a, b being Element of R st
( (p ^ q) . i = a * b & a in I & b in J + K )
then not i in Seg (len p) by FINSEQ_1:1;
then not i in dom p by FINSEQ_1:def_3;
then consider n being Nat such that
A57: n in dom q and
A58: i = (len p) + n by A51, FINSEQ_1:25;
n in Seg (len q) by A57, FINSEQ_1:def_3;
then ( 1 <= n & n <= len q ) by FINSEQ_1:1;
then consider a, b being Element of R such that
A59: q . n = a * b and
A60: a in I and
A61: b in J by A47, A57;
0. R in K by Th3;
then A62: b + (0. R) in { (a9 + b9) where a9, b9 is Element of R : ( a9 in J & b9 in K ) } by A61;
(p ^ q) . i = q . n by A57, A58, FINSEQ_1:def_7
.= a * (b + (0. R)) by A59, RLVECT_1:def_4 ;
hence ex a, b being Element of R st
( (p ^ q) . i = a * b & a in I & b in J + K ) by A60, A62; ::_thesis: verum
end;
end;
end;
hence ex a, b being Element of R st
( (p ^ q) . i = a * b & a in I & b in J + K ) ; ::_thesis: verum
end;
Sum (p ^ q) = u by A41, A46, A44, RLVECT_1:41;
hence u in I *' (J + K) by A48; ::_thesis: verum
end;
hence I *' (J + K) = (I *' J) + (I *' K) by A1, TARSKI:1; ::_thesis: verum
end;
theorem Th81: :: IDEAL_1:81
for R being non empty add-cancelable Abelian add-associative right_zeroed associative commutative distributive left_zeroed doubleLoopStr
for I, J being non empty right-ideal Subset of R holds (I + J) *' (I /\ J) c= I *' J
proof
let R be non empty add-cancelable Abelian add-associative right_zeroed associative commutative distributive left_zeroed doubleLoopStr ; ::_thesis: for I, J being non empty right-ideal Subset of R holds (I + J) *' (I /\ J) c= I *' J
let I, J be non empty right-ideal Subset of R; ::_thesis: (I + J) *' (I /\ J) c= I *' J
A1: now__::_thesis:_for_u_being_set_st_u_in_(I_*'_(I_/\_J))_+_(J_*'_(I_/\_J))_holds_
u_in_I_*'_J
let u be set ; ::_thesis: ( u in (I *' (I /\ J)) + (J *' (I /\ J)) implies u in I *' J )
assume u in (I *' (I /\ J)) + (J *' (I /\ J)) ; ::_thesis: u in I *' J
then consider a, b being Element of R such that
A2: u = a + b and
A3: a in I *' (I /\ J) and
A4: b in J *' (I /\ J) ;
consider s being FinSequence of the carrier of R such that
A5: b = Sum s and
A6: for i being Element of NAT st 1 <= i & i <= len s holds
ex a, b being Element of R st
( s . i = a * b & a in J & b in I /\ J ) by A4;
for i being Element of NAT st 1 <= i & i <= len s holds
ex x, y being Element of R st
( s . i = x * y & x in I & y in J )
proof
let i be Element of NAT ; ::_thesis: ( 1 <= i & i <= len s implies ex x, y being Element of R st
( s . i = x * y & x in I & y in J ) )
assume ( 1 <= i & i <= len s ) ; ::_thesis: ex x, y being Element of R st
( s . i = x * y & x in I & y in J )
then A7: ex x, y being Element of R st
( s . i = x * y & x in J & y in I /\ J ) by A6;
I /\ J c= I by XBOOLE_1:17;
hence ex x, y being Element of R st
( s . i = x * y & x in I & y in J ) by A7; ::_thesis: verum
end;
then A8: Sum s in { (Sum t) where t is FinSequence of the carrier of R : for i being Element of NAT st 1 <= i & i <= len t holds
ex a, b being Element of R st
( t . i = a * b & a in I & b in J ) } ;
consider q being FinSequence of the carrier of R such that
A9: a = Sum q and
A10: for i being Element of NAT st 1 <= i & i <= len q holds
ex a, b being Element of R st
( q . i = a * b & a in I & b in I /\ J ) by A3;
for i being Element of NAT st 1 <= i & i <= len q holds
ex x, y being Element of R st
( q . i = x * y & x in I & y in J )
proof
let i be Element of NAT ; ::_thesis: ( 1 <= i & i <= len q implies ex x, y being Element of R st
( q . i = x * y & x in I & y in J ) )
assume ( 1 <= i & i <= len q ) ; ::_thesis: ex x, y being Element of R st
( q . i = x * y & x in I & y in J )
then A11: ex x, y being Element of R st
( q . i = x * y & x in I & y in I /\ J ) by A10;
I /\ J c= J by XBOOLE_1:17;
hence ex x, y being Element of R st
( q . i = x * y & x in I & y in J ) by A11; ::_thesis: verum
end;
then Sum q in { (Sum t) where t is FinSequence of the carrier of R : for i being Element of NAT st 1 <= i & i <= len t holds
ex a, b being Element of R st
( t . i = a * b & a in I & b in J ) } ;
hence u in I *' J by A2, A9, A5, A8, Def1; ::_thesis: verum
end;
(I + J) *' (I /\ J) = (I *' (I /\ J)) + (J *' (I /\ J)) by Th80;
hence (I + J) *' (I /\ J) c= I *' J by A1, TARSKI:def_3; ::_thesis: verum
end;
theorem :: IDEAL_1:82
for R being non empty left_add-cancelable right_zeroed left-distributive doubleLoopStr
for I, J being non empty add-closed left-ideal Subset of R holds (I + J) *' (I /\ J) c= I /\ J
proof
let R be non empty left_add-cancelable right_zeroed left-distributive doubleLoopStr ; ::_thesis: for I, J being non empty add-closed left-ideal Subset of R holds (I + J) *' (I /\ J) c= I /\ J
let I, J be non empty add-closed left-ideal Subset of R; ::_thesis: (I + J) *' (I /\ J) c= I /\ J
now__::_thesis:_for_u_being_set_st_u_in_(I_+_J)_*'_(I_/\_J)_holds_
u_in_I_/\_J
let u be set ; ::_thesis: ( u in (I + J) *' (I /\ J) implies u in I /\ J )
assume u in (I + J) *' (I /\ J) ; ::_thesis: u in I /\ J
then consider s being FinSequence of the carrier of R such that
A1: u = Sum s and
A2: for i being Element of NAT st 1 <= i & i <= len s holds
ex a, b being Element of R st
( s . i = a * b & a in I + J & b in I /\ J ) ;
consider f being Function of NAT, the carrier of R such that
A3: Sum s = f . (len s) and
A4: f . 0 = 0. R and
A5: for j being Element of NAT
for v being Element of R st j < len s & v = s . (j + 1) holds
f . (j + 1) = (f . j) + v by RLVECT_1:def_12;
defpred S1[ Element of NAT ] means f . $1 in I /\ J;
A6: now__::_thesis:_for_n_being_Element_of_NAT_st_0_<=_n_&_n_<_len_s_&_S1[n]_holds_
S1[n_+_1]
let n be Element of NAT ; ::_thesis: ( 0 <= n & n < len s & S1[n] implies S1[n + 1] )
assume that
0 <= n and
A7: n < len s ; ::_thesis: ( S1[n] implies S1[n + 1] )
thus ( S1[n] implies S1[n + 1] ) ::_thesis: verum
proof
A8: ( 0 + 1 <= n + 1 & n + 1 <= len s ) by A7, NAT_1:13;
then n + 1 in Seg (len s) by FINSEQ_1:1;
then n + 1 in dom s by FINSEQ_1:def_3;
then A9: s . (n + 1) = s /. (n + 1) by PARTFUN1:def_6;
assume A10: f . n in I /\ J ; ::_thesis: S1[n + 1]
ex x, y being Element of R st
( s . (n + 1) = x * y & x in I + J & y in I /\ J ) by A2, A8;
then s /. (n + 1) in I /\ J by A9, Def2;
then (f . n) + (s /. (n + 1)) in I /\ J by A10, Def1;
hence S1[n + 1] by A5, A7, A9; ::_thesis: verum
end;
end;
A11: S1[ 0 ] by A4, Th2;
for n being Element of NAT st 0 <= n & n <= len s holds
S1[n] from INT_1:sch_7(A11, A6);
hence u in I /\ J by A1, A3; ::_thesis: verum
end;
hence (I + J) *' (I /\ J) c= I /\ J by TARSKI:def_3; ::_thesis: verum
end;
definition
let R be non empty addLoopStr ;
let I, J be Subset of R;
predI,J are_co-prime means :Def22: :: IDEAL_1:def 22
I + J = the carrier of R;
end;
:: deftheorem Def22 defines are_co-prime IDEAL_1:def_22_:_
for R being non empty addLoopStr
for I, J being Subset of R holds
( I,J are_co-prime iff I + J = the carrier of R );
theorem Th83: :: IDEAL_1:83
for R being non empty left_unital left_zeroed doubleLoopStr
for I, J being non empty Subset of R st I,J are_co-prime holds
I /\ J c= (I + J) *' (I /\ J)
proof
let R be non empty left_unital left_zeroed doubleLoopStr ; ::_thesis: for I, J being non empty Subset of R st I,J are_co-prime holds
I /\ J c= (I + J) *' (I /\ J)
let I, J be non empty Subset of R; ::_thesis: ( I,J are_co-prime implies I /\ J c= (I + J) *' (I /\ J) )
assume I,J are_co-prime ; ::_thesis: I /\ J c= (I + J) *' (I /\ J)
then A1: I + J = the carrier of R by Def22;
now__::_thesis:_for_u_being_set_st_u_in_I_/\_J_holds_
u_in_(I_+_J)_*'_(I_/\_J)
let u be set ; ::_thesis: ( u in I /\ J implies u in (I + J) *' (I /\ J) )
assume A2: u in I /\ J ; ::_thesis: u in (I + J) *' (I /\ J)
then reconsider u9 = u as Element of R ;
set q = <*((1. R) * u9)*>;
A3: len <*((1. R) * u9)*> = 1 by FINSEQ_1:39;
A4: for i being Element of NAT st 1 <= i & i <= len <*((1. R) * u9)*> holds
ex x, y being Element of R st
( <*((1. R) * u9)*> . i = x * y & x in I + J & y in I /\ J )
proof
let i be Element of NAT ; ::_thesis: ( 1 <= i & i <= len <*((1. R) * u9)*> implies ex x, y being Element of R st
( <*((1. R) * u9)*> . i = x * y & x in I + J & y in I /\ J ) )
assume A5: ( 1 <= i & i <= len <*((1. R) * u9)*> ) ; ::_thesis: ex x, y being Element of R st
( <*((1. R) * u9)*> . i = x * y & x in I + J & y in I /\ J )
take 1. R ; ::_thesis: ex y being Element of R st
( <*((1. R) * u9)*> . i = (1. R) * y & 1. R in I + J & y in I /\ J )
take u9 ; ::_thesis: ( <*((1. R) * u9)*> . i = (1. R) * u9 & 1. R in I + J & u9 in I /\ J )
i = 1 by A3, A5, XXREAL_0:1;
hence ( <*((1. R) * u9)*> . i = (1. R) * u9 & 1. R in I + J & u9 in I /\ J ) by A1, A2, FINSEQ_1:40; ::_thesis: verum
end;
Sum <*((1. R) * u9)*> = (1. R) * u9 by BINOM:3
.= u9 by VECTSP_1:def_8 ;
hence u in (I + J) *' (I /\ J) by A4; ::_thesis: verum
end;
hence I /\ J c= (I + J) *' (I /\ J) by TARSKI:def_3; ::_thesis: verum
end;
theorem :: IDEAL_1:84
for R being non empty add-cancelable Abelian add-associative right_zeroed associative commutative distributive left_unital left_zeroed doubleLoopStr
for I being non empty add-closed left-ideal right-ideal Subset of R
for J being non empty add-closed left-ideal Subset of R st I,J are_co-prime holds
I *' J = I /\ J
proof
let R be non empty add-cancelable Abelian add-associative right_zeroed associative commutative distributive left_unital left_zeroed doubleLoopStr ; ::_thesis: for I being non empty add-closed left-ideal right-ideal Subset of R
for J being non empty add-closed left-ideal Subset of R st I,J are_co-prime holds
I *' J = I /\ J
let I be non empty add-closed left-ideal right-ideal Subset of R; ::_thesis: for J being non empty add-closed left-ideal Subset of R st I,J are_co-prime holds
I *' J = I /\ J
let J be non empty add-closed left-ideal Subset of R; ::_thesis: ( I,J are_co-prime implies I *' J = I /\ J )
A1: I *' J c= I /\ J by Th79;
assume I,J are_co-prime ; ::_thesis: I *' J = I /\ J
then A2: I /\ J c= (I + J) *' (I /\ J) by Th83;
(I + J) *' (I /\ J) c= I *' J by Th81;
then I /\ J c= I *' J by A2, XBOOLE_1:1;
hence I *' J = I /\ J by A1, XBOOLE_0:def_10; ::_thesis: verum
end;
definition
let R be non empty multMagma ;
let I, J be Subset of R;
funcI % J -> Subset of R equals :: IDEAL_1:def 23
{ a where a is Element of R : a * J c= I } ;
coherence
{ a where a is Element of R : a * J c= I } is Subset of R
proof
set M = { a where a is Element of R : a * J c= I } ;
for x being set st x in { a where a is Element of R : a * J c= I } holds
x in the carrier of R
proof
let x be set ; ::_thesis: ( x in { a where a is Element of R : a * J c= I } implies x in the carrier of R )
assume x in { a where a is Element of R : a * J c= I } ; ::_thesis: x in the carrier of R
then ex a being Element of R st
( x = a & a * J c= I ) ;
hence x in the carrier of R ; ::_thesis: verum
end;
hence { a where a is Element of R : a * J c= I } is Subset of R by TARSKI:def_3; ::_thesis: verum
end;
end;
:: deftheorem defines % IDEAL_1:def_23_:_
for R being non empty multMagma
for I, J being Subset of R holds I % J = { a where a is Element of R : a * J c= I } ;
registration
let R be non empty left_add-cancelable right_zeroed left-distributive doubleLoopStr ;
let I, J be non empty left-ideal Subset of R;
clusterI % J -> non empty ;
coherence
not I % J is empty
proof
set M = { a where a is Element of R : a * J c= I } ;
0. R in I by Th2;
then for u being set st u in {(0. R)} holds
u in I by TARSKI:def_1;
then A1: {(0. R)} c= I by TARSKI:def_3;
(0. R) * J = {(0. R)} by Th70;
then 0. R in { a where a is Element of R : a * J c= I } by A1;
hence not I % J is empty ; ::_thesis: verum
end;
end;
registration
let R be non empty left_add-cancelable right_zeroed left-distributive doubleLoopStr ;
let I, J be non empty add-closed left-ideal Subset of R;
clusterI % J -> add-closed ;
coherence
I % J is add-closed
proof
set M = { a where a is Element of R : a * J c= I } ;
{ a where a is Element of R : a * J c= I } = I % J ;
then reconsider M = { a where a is Element of R : a * J c= I } as non empty Subset of R ;
for x, y being Element of R st x in M & y in M holds
x + y in M
proof
let x, y be Element of R; ::_thesis: ( x in M & y in M implies x + y in M )
assume that
A1: x in M and
A2: y in M ; ::_thesis: x + y in M
consider b being Element of R such that
A3: y = b and
A4: b * J c= I by A2;
consider a being Element of R such that
A5: x = a and
A6: a * J c= I by A1;
now__::_thesis:_for_u_being_set_st_u_in_(a_+_b)_*_J_holds_
u_in_I
let u be set ; ::_thesis: ( u in (a + b) * J implies u in I )
assume u in (a + b) * J ; ::_thesis: u in I
then consider c being Element of R such that
A7: u = (a + b) * c and
A8: c in J ;
A9: b * c in { (b * i) where i is Element of R : i in J } by A8;
( u = (a * c) + (b * c) & a * c in a * J ) by A7, A8, VECTSP_1:def_3;
hence u in I by A6, A4, A9, Def1; ::_thesis: verum
end;
then (a + b) * J c= I by TARSKI:def_3;
hence x + y in M by A5, A3; ::_thesis: verum
end;
hence I % J is add-closed by Def1; ::_thesis: verum
end;
end;
registration
let R be non empty left_add-cancelable right_zeroed associative commutative left-distributive doubleLoopStr ;
let I, J be non empty left-ideal Subset of R;
clusterI % J -> left-ideal ;
coherence
I % J is left-ideal
proof
set M = { a where a is Element of R : a * J c= I } ;
{ a where a is Element of R : a * J c= I } = I % J ;
then reconsider M = { a where a is Element of R : a * J c= I } as non empty Subset of R ;
for y, x being Element of R st x in M holds
y * x in M
proof
let y, x be Element of R; ::_thesis: ( x in M implies y * x in M )
assume x in M ; ::_thesis: y * x in M
then consider a being Element of R such that
A1: x = a and
A2: a * J c= I ;
now__::_thesis:_for_u_being_set_st_u_in_(y_*_a)_*_J_holds_
u_in_I
let u be set ; ::_thesis: ( u in (y * a) * J implies u in I )
assume u in (y * a) * J ; ::_thesis: u in I
then consider c being Element of R such that
A3: u = (y * a) * c and
A4: c in J ;
y * c in J by A4, Def2;
then A5: a * (y * c) in { (a * i) where i is Element of R : i in J } ;
u = a * (y * c) by A3, GROUP_1:def_3;
hence u in I by A2, A5; ::_thesis: verum
end;
then (y * a) * J c= I by TARSKI:def_3;
hence y * x in M by A1; ::_thesis: verum
end;
hence I % J is left-ideal by Def2; ::_thesis: verum
end;
clusterI % J -> right-ideal ;
coherence
I % J is right-ideal ;
end;
theorem :: IDEAL_1:85
for R being non empty multLoopStr
for I being non empty right-ideal Subset of R
for J being Subset of R holds I c= I % J
proof
let R be non empty multLoopStr ; ::_thesis: for I being non empty right-ideal Subset of R
for J being Subset of R holds I c= I % J
let I be non empty right-ideal Subset of R; ::_thesis: for J being Subset of R holds I c= I % J
let J be Subset of R; ::_thesis: I c= I % J
now__::_thesis:_for_u_being_set_st_u_in_I_holds_
u_in_I_%_J
let u be set ; ::_thesis: ( u in I implies u in I % J )
assume A1: u in I ; ::_thesis: u in I % J
then reconsider u9 = u as Element of R ;
now__::_thesis:_for_v_being_set_st_v_in_u9_*_J_holds_
v_in_I
let v be set ; ::_thesis: ( v in u9 * J implies v in I )
assume v in u9 * J ; ::_thesis: v in I
then ex j being Element of R st
( v = u9 * j & j in J ) ;
hence v in I by A1, Def3; ::_thesis: verum
end;
then u9 * J c= I by TARSKI:def_3;
hence u in I % J ; ::_thesis: verum
end;
hence I c= I % J by TARSKI:def_3; ::_thesis: verum
end;
theorem :: IDEAL_1:86
for R being non empty left_add-cancelable right_zeroed left-distributive doubleLoopStr
for I being non empty add-closed left-ideal Subset of R
for J being Subset of R holds (I % J) *' J c= I
proof
let R be non empty left_add-cancelable right_zeroed left-distributive doubleLoopStr ; ::_thesis: for I being non empty add-closed left-ideal Subset of R
for J being Subset of R holds (I % J) *' J c= I
let I be non empty add-closed left-ideal Subset of R; ::_thesis: for J being Subset of R holds (I % J) *' J c= I
let J be Subset of R; ::_thesis: (I % J) *' J c= I
now__::_thesis:_for_u_being_set_st_u_in_(I_%_J)_*'_J_holds_
u_in_I
let u be set ; ::_thesis: ( u in (I % J) *' J implies u in I )
assume u in (I % J) *' J ; ::_thesis: u in I
then consider s being FinSequence of the carrier of R such that
A1: Sum s = u and
A2: for i being Element of NAT st 1 <= i & i <= len s holds
ex a, b being Element of R st
( s . i = a * b & a in I % J & b in J ) ;
consider f being Function of NAT, the carrier of R such that
A3: Sum s = f . (len s) and
A4: f . 0 = 0. R and
A5: for j being Element of NAT
for v being Element of R st j < len s & v = s . (j + 1) holds
f . (j + 1) = (f . j) + v by RLVECT_1:def_12;
defpred S1[ Element of NAT ] means f . $1 in I;
A6: now__::_thesis:_for_j_being_Element_of_NAT_st_0_<=_j_&_j_<_len_s_&_S1[j]_holds_
S1[j_+_1]
let j be Element of NAT ; ::_thesis: ( 0 <= j & j < len s & S1[j] implies S1[j + 1] )
assume that
0 <= j and
A7: j < len s ; ::_thesis: ( S1[j] implies S1[j + 1] )
thus ( S1[j] implies S1[j + 1] ) ::_thesis: verum
proof
A8: ( j + 1 <= len s & 0 + 1 <= j + 1 ) by A7, NAT_1:13;
then consider a, b being Element of R such that
A9: s . (j + 1) = a * b and
A10: a in I % J and
A11: b in J by A2;
j + 1 in Seg (len s) by A8, FINSEQ_1:1;
then j + 1 in dom s by FINSEQ_1:def_3;
then A12: s . (j + 1) = s /. (j + 1) by PARTFUN1:def_6;
then A13: f . (j + 1) = (f . j) + (s /. (j + 1)) by A5, A7;
assume A14: f . j in I ; ::_thesis: S1[j + 1]
consider d being Element of R such that
A15: a = d and
A16: d * J c= I by A10;
a * b in { (d * i) where i is Element of R : i in J } by A11, A15;
hence S1[j + 1] by A14, A12, A13, A9, A16, Def1; ::_thesis: verum
end;
end;
A17: S1[ 0 ] by A4, Th2;
for j being Element of NAT st 0 <= j & j <= len s holds
S1[j] from INT_1:sch_7(A17, A6);
hence u in I by A1, A3; ::_thesis: verum
end;
hence (I % J) *' J c= I by TARSKI:def_3; ::_thesis: verum
end;
theorem Th87: :: IDEAL_1:87
for R being non empty right_add-cancelable right-distributive left_zeroed doubleLoopStr
for I being non empty add-closed right-ideal Subset of R
for J being Subset of R holds (I % J) *' J c= I
proof
let R be non empty right_add-cancelable right-distributive left_zeroed doubleLoopStr ; ::_thesis: for I being non empty add-closed right-ideal Subset of R
for J being Subset of R holds (I % J) *' J c= I
let I be non empty add-closed right-ideal Subset of R; ::_thesis: for J being Subset of R holds (I % J) *' J c= I
let J be Subset of R; ::_thesis: (I % J) *' J c= I
now__::_thesis:_for_u_being_set_st_u_in_(I_%_J)_*'_J_holds_
u_in_I
let u be set ; ::_thesis: ( u in (I % J) *' J implies u in I )
assume u in (I % J) *' J ; ::_thesis: u in I
then consider s being FinSequence of the carrier of R such that
A1: Sum s = u and
A2: for i being Element of NAT st 1 <= i & i <= len s holds
ex a, b being Element of R st
( s . i = a * b & a in I % J & b in J ) ;
consider f being Function of NAT, the carrier of R such that
A3: Sum s = f . (len s) and
A4: f . 0 = 0. R and
A5: for j being Element of NAT
for v being Element of R st j < len s & v = s . (j + 1) holds
f . (j + 1) = (f . j) + v by RLVECT_1:def_12;
defpred S1[ Element of NAT ] means f . $1 in I;
A6: now__::_thesis:_for_j_being_Element_of_NAT_st_0_<=_j_&_j_<_len_s_&_S1[j]_holds_
S1[j_+_1]
let j be Element of NAT ; ::_thesis: ( 0 <= j & j < len s & S1[j] implies S1[j + 1] )
assume that
0 <= j and
A7: j < len s ; ::_thesis: ( S1[j] implies S1[j + 1] )
thus ( S1[j] implies S1[j + 1] ) ::_thesis: verum
proof
A8: ( j + 1 <= len s & 0 + 1 <= j + 1 ) by A7, NAT_1:13;
then consider a, b being Element of R such that
A9: s . (j + 1) = a * b and
A10: a in I % J and
A11: b in J by A2;
j + 1 in Seg (len s) by A8, FINSEQ_1:1;
then j + 1 in dom s by FINSEQ_1:def_3;
then A12: s . (j + 1) = s /. (j + 1) by PARTFUN1:def_6;
then A13: f . (j + 1) = (f . j) + (s /. (j + 1)) by A5, A7;
assume A14: f . j in I ; ::_thesis: S1[j + 1]
consider d being Element of R such that
A15: a = d and
A16: d * J c= I by A10;
a * b in { (d * i) where i is Element of R : i in J } by A11, A15;
hence S1[j + 1] by A14, A12, A13, A9, A16, Def1; ::_thesis: verum
end;
end;
A17: S1[ 0 ] by A4, Th3;
for j being Element of NAT st 0 <= j & j <= len s holds
S1[j] from INT_1:sch_7(A17, A6);
hence u in I by A1, A3; ::_thesis: verum
end;
hence (I % J) *' J c= I by TARSKI:def_3; ::_thesis: verum
end;
theorem :: IDEAL_1:88
for R being non empty right_add-cancelable associative commutative right-distributive left_zeroed doubleLoopStr
for I being non empty add-closed right-ideal Subset of R
for J, K being Subset of R holds (I % J) % K = I % (J *' K)
proof
let R be non empty right_add-cancelable associative commutative right-distributive left_zeroed doubleLoopStr ; ::_thesis: for I being non empty add-closed right-ideal Subset of R
for J, K being Subset of R holds (I % J) % K = I % (J *' K)
let I be non empty add-closed right-ideal Subset of R; ::_thesis: for J, K being Subset of R holds (I % J) % K = I % (J *' K)
let J, K be Subset of R; ::_thesis: (I % J) % K = I % (J *' K)
A1: now__::_thesis:_for_u_being_set_st_u_in_(I_%_J)_%_K_holds_
u_in_I_%_(J_*'_K)
let u be set ; ::_thesis: ( u in (I % J) % K implies u in I % (J *' K) )
assume u in (I % J) % K ; ::_thesis: u in I % (J *' K)
then consider a being Element of R such that
A2: u = a and
A3: a * K c= I % J ;
now__::_thesis:_for_v_being_set_st_v_in_a_*_(J_*'_K)_holds_
v_in_I
let v be set ; ::_thesis: ( v in a * (J *' K) implies v in I )
assume v in a * (J *' K) ; ::_thesis: v in I
then consider b being Element of R such that
A4: v = a * b and
A5: b in J *' K ;
consider s being FinSequence of the carrier of R such that
A6: Sum s = b and
A7: for i being Element of NAT st 1 <= i & i <= len s holds
ex a, b being Element of R st
( s . i = a * b & a in J & b in K ) by A5;
set q = a * s;
A8: dom (a * s) = dom s by POLYNOM1:def_1;
A9: Seg (len (a * s)) = dom (a * s) by FINSEQ_1:def_3
.= dom s by POLYNOM1:def_1
.= Seg (len s) by FINSEQ_1:def_3 ;
then A10: len (a * s) = len s by FINSEQ_1:6;
for j being Element of NAT st 1 <= j & j <= len (a * s) holds
ex c, d being Element of R st
( (a * s) . j = c * d & c in I % J & d in J )
proof
let j be Element of NAT ; ::_thesis: ( 1 <= j & j <= len (a * s) implies ex c, d being Element of R st
( (a * s) . j = c * d & c in I % J & d in J ) )
assume A11: ( 1 <= j & j <= len (a * s) ) ; ::_thesis: ex c, d being Element of R st
( (a * s) . j = c * d & c in I % J & d in J )
then consider c, d being Element of R such that
A12: s . j = c * d and
A13: c in J and
A14: d in K by A7, A10;
A15: a * d in { (a * b9) where b9 is Element of R : b9 in K } by A14;
j in Seg (len s) by A9, A11, FINSEQ_1:1;
then A16: j in dom s by FINSEQ_1:def_3;
then A17: s /. j = c * d by A12, PARTFUN1:def_6;
(a * s) . j = (a * s) /. j by A8, A16, PARTFUN1:def_6
.= a * (c * d) by A16, A17, POLYNOM1:def_1
.= (a * d) * c by GROUP_1:def_3 ;
hence ex c, d being Element of R st
( (a * s) . j = c * d & c in I % J & d in J ) by A3, A13, A15; ::_thesis: verum
end;
then A18: Sum (a * s) in { (Sum t) where t is FinSequence of the carrier of R : for i being Element of NAT st 1 <= i & i <= len t holds
ex a, b being Element of R st
( t . i = a * b & a in I % J & b in J ) } ;
A19: (I % J) *' J c= I by Th87;
Sum (a * s) = v by A4, A6, BINOM:4;
hence v in I by A18, A19; ::_thesis: verum
end;
then a * (J *' K) c= I by TARSKI:def_3;
hence u in I % (J *' K) by A2; ::_thesis: verum
end;
now__::_thesis:_for_u_being_set_st_u_in_I_%_(J_*'_K)_holds_
u_in_(I_%_J)_%_K
let u be set ; ::_thesis: ( u in I % (J *' K) implies u in (I % J) % K )
assume u in I % (J *' K) ; ::_thesis: u in (I % J) % K
then consider a being Element of R such that
A20: u = a and
A21: a * (J *' K) c= I ;
now__::_thesis:_for_v_being_set_st_v_in_a_*_K_holds_
v_in_I_%_J
let v be set ; ::_thesis: ( v in a * K implies v in I % J )
assume v in a * K ; ::_thesis: v in I % J
then consider b being Element of R such that
A22: v = a * b and
A23: b in K ;
now__::_thesis:_for_z_being_set_st_z_in_(a_*_b)_*_J_holds_
z_in_I
let z be set ; ::_thesis: ( z in (a * b) * J implies z in I )
assume z in (a * b) * J ; ::_thesis: z in I
then consider c being Element of R such that
A24: z = (a * b) * c and
A25: c in J ;
A26: z = a * (c * b) by A24, GROUP_1:def_3;
set q = <*(c * b)*>;
A27: len <*(c * b)*> = 1 by FINSEQ_1:40;
A28: for i being Element of NAT st 1 <= i & i <= len <*(c * b)*> holds
ex x, y being Element of R st
( <*(c * b)*> . i = x * y & x in J & y in K )
proof
let i be Element of NAT ; ::_thesis: ( 1 <= i & i <= len <*(c * b)*> implies ex x, y being Element of R st
( <*(c * b)*> . i = x * y & x in J & y in K ) )
assume ( 1 <= i & i <= len <*(c * b)*> ) ; ::_thesis: ex x, y being Element of R st
( <*(c * b)*> . i = x * y & x in J & y in K )
then <*(c * b)*> . i = <*(c * b)*> . 1 by A27, XXREAL_0:1
.= c * b by FINSEQ_1:40 ;
hence ex x, y being Element of R st
( <*(c * b)*> . i = x * y & x in J & y in K ) by A23, A25; ::_thesis: verum
end;
Sum <*(c * b)*> = c * b by BINOM:3;
then c * b in { (Sum t) where t is FinSequence of the carrier of R : for i being Element of NAT st 1 <= i & i <= len t holds
ex a, b being Element of R st
( t . i = a * b & a in J & b in K ) } by A28;
then z in { (a * f) where f is Element of R : f in J *' K } by A26;
hence z in I by A21; ::_thesis: verum
end;
then (a * b) * J c= I by TARSKI:def_3;
hence v in I % J by A22; ::_thesis: verum
end;
then a * K c= I % J by TARSKI:def_3;
hence u in (I % J) % K by A20; ::_thesis: verum
end;
hence (I % J) % K = I % (J *' K) by A1, TARSKI:1; ::_thesis: verum
end;
theorem :: IDEAL_1:89
for R being non empty multLoopStr
for I, J, K being Subset of R holds (J /\ K) % I = (J % I) /\ (K % I)
proof
let R be non empty multLoopStr ; ::_thesis: for I, J, K being Subset of R holds (J /\ K) % I = (J % I) /\ (K % I)
let I, J, K be Subset of R; ::_thesis: (J /\ K) % I = (J % I) /\ (K % I)
A1: now__::_thesis:_for_u_being_set_st_u_in_(J_/\_K)_%_I_holds_
u_in_(J_%_I)_/\_(K_%_I)
let u be set ; ::_thesis: ( u in (J /\ K) % I implies u in (J % I) /\ (K % I) )
assume u in (J /\ K) % I ; ::_thesis: u in (J % I) /\ (K % I)
then consider a being Element of R such that
A2: u = a and
A3: a * I c= J /\ K ;
now__::_thesis:_for_v_being_set_st_v_in_a_*_I_holds_
v_in_K
let v be set ; ::_thesis: ( v in a * I implies v in K )
assume v in a * I ; ::_thesis: v in K
then v in J /\ K by A3;
then ex x being Element of R st
( v = x & x in J & x in K ) ;
hence v in K ; ::_thesis: verum
end;
then a * I c= K by TARSKI:def_3;
then A4: u in K % I by A2;
now__::_thesis:_for_v_being_set_st_v_in_a_*_I_holds_
v_in_J
let v be set ; ::_thesis: ( v in a * I implies v in J )
assume v in a * I ; ::_thesis: v in J
then v in J /\ K by A3;
then ex x being Element of R st
( v = x & x in J & x in K ) ;
hence v in J ; ::_thesis: verum
end;
then a * I c= J by TARSKI:def_3;
then u in J % I by A2;
hence u in (J % I) /\ (K % I) by A4; ::_thesis: verum
end;
now__::_thesis:_for_u_being_set_st_u_in_(J_%_I)_/\_(K_%_I)_holds_
u_in_(J_/\_K)_%_I
let u be set ; ::_thesis: ( u in (J % I) /\ (K % I) implies u in (J /\ K) % I )
assume u in (J % I) /\ (K % I) ; ::_thesis: u in (J /\ K) % I
then A5: ex x being Element of R st
( x = u & x in J % I & x in K % I ) ;
then consider a being Element of R such that
A6: u = a and
A7: a * I c= J ;
ex b being Element of R st
( u = b & b * I c= K ) by A5;
then for v being set st v in a * I holds
v in J /\ K by A6, A7;
then a * I c= J /\ K by TARSKI:def_3;
hence u in (J /\ K) % I by A6; ::_thesis: verum
end;
hence (J /\ K) % I = (J % I) /\ (K % I) by A1, TARSKI:1; ::_thesis: verum
end;
theorem :: IDEAL_1:90
for R being non empty right_add-cancelable right_zeroed right-distributive left_zeroed doubleLoopStr
for I being add-closed Subset of R
for J, K being non empty right-ideal Subset of R holds I % (J + K) = (I % J) /\ (I % K)
proof
let R be non empty right_add-cancelable right_zeroed right-distributive left_zeroed doubleLoopStr ; ::_thesis: for I being add-closed Subset of R
for J, K being non empty right-ideal Subset of R holds I % (J + K) = (I % J) /\ (I % K)
let I be add-closed Subset of R; ::_thesis: for J, K being non empty right-ideal Subset of R holds I % (J + K) = (I % J) /\ (I % K)
let J, K be non empty right-ideal Subset of R; ::_thesis: I % (J + K) = (I % J) /\ (I % K)
A1: now__::_thesis:_for_u_being_set_st_u_in_I_%_(J_+_K)_holds_
u_in_(I_%_J)_/\_(I_%_K)
let u be set ; ::_thesis: ( u in I % (J + K) implies u in (I % J) /\ (I % K) )
assume u in I % (J + K) ; ::_thesis: u in (I % J) /\ (I % K)
then consider a being Element of R such that
A2: u = a and
A3: a * (J + K) c= I ;
now__::_thesis:_for_u_being_set_st_u_in_a_*_J_holds_
u_in_I
let u be set ; ::_thesis: ( u in a * J implies u in I )
assume u in a * J ; ::_thesis: u in I
then A4: ex j being Element of R st
( u = a * j & j in J ) ;
J c= J + K by Th73;
then u in { (a * j9) where j9 is Element of R : j9 in J + K } by A4;
hence u in I by A3; ::_thesis: verum
end;
then a * J c= I by TARSKI:def_3;
then A5: u in I % J by A2;
now__::_thesis:_for_u_being_set_st_u_in_a_*_K_holds_
u_in_I
let u be set ; ::_thesis: ( u in a * K implies u in I )
assume u in a * K ; ::_thesis: u in I
then A6: ex j being Element of R st
( u = a * j & j in K ) ;
K c= J + K by Th74;
then u in { (a * j9) where j9 is Element of R : j9 in J + K } by A6;
hence u in I by A3; ::_thesis: verum
end;
then a * K c= I by TARSKI:def_3;
then u in I % K by A2;
hence u in (I % J) /\ (I % K) by A5; ::_thesis: verum
end;
now__::_thesis:_for_u_being_set_st_u_in_(I_%_J)_/\_(I_%_K)_holds_
u_in_I_%_(J_+_K)
let u be set ; ::_thesis: ( u in (I % J) /\ (I % K) implies u in I % (J + K) )
assume u in (I % J) /\ (I % K) ; ::_thesis: u in I % (J + K)
then A7: ex x being Element of R st
( u = x & x in I % J & x in I % K ) ;
then consider a being Element of R such that
A8: u = a and
A9: a * J c= I ;
consider b being Element of R such that
A10: u = b and
A11: b * K c= I by A7;
now__::_thesis:_for_v_being_set_st_v_in_a_*_(J_+_K)_holds_
v_in_I
let v be set ; ::_thesis: ( v in a * (J + K) implies v in I )
assume v in a * (J + K) ; ::_thesis: v in I
then consider j being Element of R such that
A12: v = a * j and
A13: j in J + K ;
consider x9, y being Element of R such that
A14: j = x9 + y and
A15: ( x9 in J & y in K ) by A13;
A16: ( a * x9 in a * J & b * y in { (b * j9) where j9 is Element of R : j9 in K } ) by A15;
v = (a * x9) + (b * y) by A8, A10, A12, A14, VECTSP_1:def_2;
hence v in I by A9, A11, A16, Def1; ::_thesis: verum
end;
then a * (J + K) c= I by TARSKI:def_3;
hence u in I % (J + K) by A8; ::_thesis: verum
end;
hence I % (J + K) = (I % J) /\ (I % K) by A1, TARSKI:1; ::_thesis: verum
end;
definition
let R be non empty well-unital doubleLoopStr ;
let I be Subset of R;
func sqrt I -> Subset of R equals :: IDEAL_1:def 24
{ a where a is Element of R : ex n being Element of NAT st a |^ n in I } ;
coherence
{ a where a is Element of R : ex n being Element of NAT st a |^ n in I } is Subset of R
proof
set M = { a where a is Element of R : ex n being Element of NAT st a |^ n in I } ;
for x being set st x in { a where a is Element of R : ex n being Element of NAT st a |^ n in I } holds
x in the carrier of R
proof
let x be set ; ::_thesis: ( x in { a where a is Element of R : ex n being Element of NAT st a |^ n in I } implies x in the carrier of R )
assume x in { a where a is Element of R : ex n being Element of NAT st a |^ n in I } ; ::_thesis: x in the carrier of R
then ex a being Element of R st
( a = x & ex n being Element of NAT st a |^ n in I ) ;
hence x in the carrier of R ; ::_thesis: verum
end;
hence { a where a is Element of R : ex n being Element of NAT st a |^ n in I } is Subset of R by TARSKI:def_3; ::_thesis: verum
end;
end;
:: deftheorem defines sqrt IDEAL_1:def_24_:_
for R being non empty well-unital doubleLoopStr
for I being Subset of R holds sqrt I = { a where a is Element of R : ex n being Element of NAT st a |^ n in I } ;
registration
let R be non empty well-unital doubleLoopStr ;
let I be non empty Subset of R;
cluster sqrt I -> non empty ;
coherence
not sqrt I is empty
proof
set M = { a where a is Element of R : ex n being Element of NAT st a |^ n in I } ;
not { a where a is Element of R : ex n being Element of NAT st a |^ n in I } is empty
proof
set a = the Element of I;
the Element of I |^ 1 = the Element of I by BINOM:8;
then the Element of I in { a where a is Element of R : ex n being Element of NAT st a |^ n in I } ;
hence not { a where a is Element of R : ex n being Element of NAT st a |^ n in I } is empty ; ::_thesis: verum
end;
hence not sqrt I is empty ; ::_thesis: verum
end;
end;
registration
let R be non empty add-cancelable Abelian add-associative right_zeroed associative commutative well-unital distributive left_zeroed doubleLoopStr ;
let I be non empty add-closed right-ideal Subset of R;
cluster sqrt I -> add-closed ;
coherence
sqrt I is add-closed
proof
set M = { a where a is Element of R : ex n being Element of NAT st a |^ n in I } ;
{ a where a is Element of R : ex n being Element of NAT st a |^ n in I } = sqrt I ;
then reconsider M = { a where a is Element of R : ex n being Element of NAT st a |^ n in I } as non empty Subset of R ;
for x, y being Element of R st x in M & y in M holds
x + y in M
proof
let x, y be Element of R; ::_thesis: ( x in M & y in M implies x + y in M )
assume that
A1: x in M and
A2: y in M ; ::_thesis: x + y in M
consider a being Element of R such that
A3: x = a and
A4: ex n being Element of NAT st a |^ n in I by A1;
consider n being Element of NAT such that
A5: a |^ n in I by A4;
consider b being Element of R such that
A6: y = b and
A7: ex m being Element of NAT st b |^ m in I by A2;
consider m being Element of NAT such that
A8: b |^ m in I by A7;
set p = (a,b) In_Power (n + m);
consider f being Function of NAT, the carrier of R such that
A9: Sum ((a,b) In_Power (n + m)) = f . (len ((a,b) In_Power (n + m))) and
A10: f . 0 = 0. R and
A11: for j being Element of NAT
for v being Element of R st j < len ((a,b) In_Power (n + m)) & v = ((a,b) In_Power (n + m)) . (j + 1) holds
f . (j + 1) = (f . j) + v by RLVECT_1:def_12;
defpred S1[ Element of NAT ] means f . R in I;
A12: for i being Element of NAT st 1 <= i & i <= len ((a,b) In_Power (n + m)) holds
((a,b) In_Power (n + m)) . i in I
proof
let i be Element of NAT ; ::_thesis: ( 1 <= i & i <= len ((a,b) In_Power (n + m)) implies ((a,b) In_Power (n + m)) . i in I )
assume that
A13: 1 <= i and
A14: i <= len ((a,b) In_Power (n + m)) ; ::_thesis: ((a,b) In_Power (n + m)) . i in I
set r = i - 1;
set l = (n + m) - (i - 1);
1 - 1 <= i - 1 by A13, XREAL_1:9;
then reconsider r = i - 1 as Element of NAT by INT_1:3;
i <= (n + m) + 1 by A14, BINOM:def_7;
then r <= ((n + m) + 1) - 1 by XREAL_1:9;
then r - r <= (n + m) - r by XREAL_1:9;
then reconsider l = (n + m) - (i - 1) as Element of NAT by INT_1:3;
i in Seg (len ((a,b) In_Power (n + m))) by A13, A14, FINSEQ_1:1;
then A15: i in dom ((a,b) In_Power (n + m)) by FINSEQ_1:def_3;
then A16: ((a,b) In_Power (n + m)) . i = ((a,b) In_Power (n + m)) /. i by PARTFUN1:def_6
.= (((n + m) choose r) * (a |^ l)) * (b |^ r) by A15, BINOM:def_7 ;
percases ( n <= l or l < n ) ;
suppose n <= l ; ::_thesis: ((a,b) In_Power (n + m)) . i in I
then consider k being Nat such that
A17: l = n + k by NAT_1:10;
reconsider k = k as Element of NAT by ORDINAL1:def_12;
a |^ l = (a |^ n) * (a |^ k) by A17, BINOM:10;
then a |^ l in I by A5, Def3;
then ((n + m) choose r) * (a |^ l) in I by Th17;
hence ((a,b) In_Power (n + m)) . i in I by A16, Def3; ::_thesis: verum
end;
suppose l < n ; ::_thesis: ((a,b) In_Power (n + m)) . i in I
then ((n + m) + (- r)) + r < n + r by XREAL_1:6;
then (- n) + (n + m) < (- n) + (n + r) by XREAL_1:6;
then consider k being Nat such that
A18: r = m + k by NAT_1:10;
reconsider k = k as Element of NAT by ORDINAL1:def_12;
b |^ r = (b |^ m) * (b |^ k) by A18, BINOM:10;
then b |^ r in I by A8, Def3;
hence ((a,b) In_Power (n + m)) . i in I by A16, Def3; ::_thesis: verum
end;
end;
end;
A19: now__::_thesis:_for_j_being_Element_of_NAT_st_0_<=_j_&_j_<_len_((a,b)_In_Power_(n_+_m))_&_S1[j]_holds_
S1[j_+_1]
let j be Element of NAT ; ::_thesis: ( 0 <= j & j < len ((a,b) In_Power (n + m)) & S1[j] implies S1[j + 1] )
assume that
0 <= j and
A20: j < len ((a,b) In_Power (n + m)) ; ::_thesis: ( S1[j] implies S1[j + 1] )
thus ( S1[j] implies S1[j + 1] ) ::_thesis: verum
proof
assume A21: f . j in I ; ::_thesis: S1[j + 1]
A22: j + 1 <= len ((a,b) In_Power (n + m)) by A20, NAT_1:13;
1 <= j + 1 by NAT_1:11;
then j + 1 in Seg (len ((a,b) In_Power (n + m))) by A22, FINSEQ_1:1;
then j + 1 in dom ((a,b) In_Power (n + m)) by FINSEQ_1:def_3;
then A23: ((a,b) In_Power (n + m)) /. (j + 1) = ((a,b) In_Power (n + m)) . (j + 1) by PARTFUN1:def_6;
then A24: ((a,b) In_Power (n + m)) /. (j + 1) in I by A12, A22, NAT_1:11;
f . (j + 1) = (f . j) + (((a,b) In_Power (n + m)) /. (j + 1)) by A11, A20, A23;
hence S1[j + 1] by A21, A24, Def1; ::_thesis: verum
end;
end;
A25: (a + b) |^ (n + m) = Sum ((a,b) In_Power (n + m)) by BINOM:25;
A26: S1[ 0 ] by A10, Th2;
for i being Element of NAT st 0 <= i & i <= len ((a,b) In_Power (n + m)) holds
S1[i] from INT_1:sch_7(A26, A19);
then (a + b) |^ (n + m) in I by A25, A9;
hence x + y in M by A3, A6; ::_thesis: verum
end;
hence sqrt I is add-closed by Def1; ::_thesis: verum
end;
end;
registration
let R be non empty associative commutative well-unital doubleLoopStr ;
let I be non empty left-ideal Subset of R;
cluster sqrt I -> left-ideal ;
coherence
sqrt I is left-ideal
proof
set M = { a where a is Element of R : ex n being Element of NAT st a |^ n in I } ;
{ a where a is Element of R : ex n being Element of NAT st a |^ n in I } = sqrt I ;
then reconsider M = { a where a is Element of R : ex n being Element of NAT st a |^ n in I } as non empty Subset of R ;
for y, x being Element of R st x in M holds
y * x in M
proof
let y9, x9 be Element of R; ::_thesis: ( x9 in M implies y9 * x9 in M )
reconsider x = x9, y = y9 as Element of R ;
assume x9 in M ; ::_thesis: y9 * x9 in M
then consider a being Element of R such that
A1: x = a and
A2: ex n being Element of NAT st a |^ n in I ;
consider n being Element of NAT such that
A3: a |^ n in I by A2;
A4: (y * a) |^ n = (y |^ n) * (a |^ n) by BINOM:9;
(y |^ n) * (a |^ n) in I by A3, Def2;
hence y9 * x9 in M by A1, A4; ::_thesis: verum
end;
hence sqrt I is left-ideal by Def2; ::_thesis: verum
end;
cluster sqrt I -> right-ideal ;
coherence
sqrt I is right-ideal ;
end;
theorem :: IDEAL_1:91
for R being non empty associative well-unital doubleLoopStr
for I being non empty Subset of R
for a being Element of R holds
( a in sqrt I iff ex n being Element of NAT st a |^ n in sqrt I )
proof
let R be non empty associative well-unital doubleLoopStr ; ::_thesis: for I being non empty Subset of R
for a being Element of R holds
( a in sqrt I iff ex n being Element of NAT st a |^ n in sqrt I )
let I be non empty Subset of R; ::_thesis: for a being Element of R holds
( a in sqrt I iff ex n being Element of NAT st a |^ n in sqrt I )
let a be Element of R; ::_thesis: ( a in sqrt I iff ex n being Element of NAT st a |^ n in sqrt I )
A1: now__::_thesis:_(_ex_n_being_Element_of_NAT_st_a_|^_n_in_sqrt_I_implies_a_in_sqrt_I_)
assume ex n being Element of NAT st a |^ n in sqrt I ; ::_thesis: a in sqrt I
then consider n being Element of NAT such that
A2: a |^ n in sqrt I ;
consider d being Element of R such that
A3: a |^ n = d and
A4: ex m being Element of NAT st d |^ m in I by A2;
consider m being Element of NAT such that
A5: d |^ m in I by A4;
a |^ (n * m) = d |^ m by A3, BINOM:11;
hence a in sqrt I by A5; ::_thesis: verum
end;
now__::_thesis:_(_a_in_sqrt_I_implies_ex_n_being_Element_of_NAT_st_a_|^_n_in_sqrt_I_)
A6: a |^ 1 = a by BINOM:8;
assume a in sqrt I ; ::_thesis: ex n being Element of NAT st a |^ n in sqrt I
hence ex n being Element of NAT st a |^ n in sqrt I by A6; ::_thesis: verum
end;
hence ( a in sqrt I iff ex n being Element of NAT st a |^ n in sqrt I ) by A1; ::_thesis: verum
end;
theorem :: IDEAL_1:92
for R being non empty add-cancelable right_zeroed associative well-unital distributive left_zeroed doubleLoopStr
for I being non empty add-closed right-ideal Subset of R
for J being non empty add-closed left-ideal Subset of R holds sqrt (I *' J) = sqrt (I /\ J)
proof
let R be non empty add-cancelable right_zeroed associative well-unital distributive left_zeroed doubleLoopStr ; ::_thesis: for I being non empty add-closed right-ideal Subset of R
for J being non empty add-closed left-ideal Subset of R holds sqrt (I *' J) = sqrt (I /\ J)
let I be non empty add-closed right-ideal Subset of R; ::_thesis: for J being non empty add-closed left-ideal Subset of R holds sqrt (I *' J) = sqrt (I /\ J)
let J be non empty add-closed left-ideal Subset of R; ::_thesis: sqrt (I *' J) = sqrt (I /\ J)
A1: now__::_thesis:_for_u_being_set_st_u_in_sqrt_(I_*'_J)_holds_
u_in_sqrt_(I_/\_J)
let u be set ; ::_thesis: ( u in sqrt (I *' J) implies u in sqrt (I /\ J) )
assume u in sqrt (I *' J) ; ::_thesis: u in sqrt (I /\ J)
then consider d being Element of R such that
A2: u = d and
A3: ex m being Element of NAT st d |^ m in I *' J ;
consider m being Element of NAT such that
A4: d |^ m in I *' J by A3;
consider s being FinSequence of the carrier of R such that
A5: d |^ m = Sum s and
A6: for i being Element of NAT st 1 <= i & i <= len s holds
ex a, b being Element of R st
( s . i = a * b & a in I & b in J ) by A4;
consider f being Function of NAT, the carrier of R such that
A7: Sum s = f . (len s) and
A8: f . 0 = 0. R and
A9: for j being Element of NAT
for v being Element of R st j < len s & v = s . (j + 1) holds
f . (j + 1) = (f . j) + v by RLVECT_1:def_12;
defpred S1[ Element of NAT ] means f . $1 in I /\ J;
A10: now__::_thesis:_for_j_being_Element_of_NAT_st_0_<=_j_&_j_<_len_s_&_S1[j]_holds_
S1[j_+_1]
let j be Element of NAT ; ::_thesis: ( 0 <= j & j < len s & S1[j] implies S1[j + 1] )
assume that
0 <= j and
A11: j < len s ; ::_thesis: ( S1[j] implies S1[j + 1] )
thus ( S1[j] implies S1[j + 1] ) ::_thesis: verum
proof
assume f . j in I /\ J ; ::_thesis: S1[j + 1]
then A12: ex g being Element of R st
( g = f . j & g in I & g in J ) ;
A13: ( j + 1 <= len s & 0 + 1 <= j + 1 ) by A11, NAT_1:13;
then A14: ex a, b being Element of R st
( s . (j + 1) = a * b & a in I & b in J ) by A6;
j + 1 in Seg (len s) by A13, FINSEQ_1:1;
then j + 1 in dom s by FINSEQ_1:def_3;
then A15: s . (j + 1) = s /. (j + 1) by PARTFUN1:def_6;
then A16: f . (j + 1) = (f . j) + (s /. (j + 1)) by A9, A11;
s /. (j + 1) in J by A15, A14, Def2;
then A17: f . (j + 1) in J by A12, A16, Def1;
s /. (j + 1) in I by A15, A14, Def3;
then f . (j + 1) in I by A12, A16, Def1;
hence S1[j + 1] by A17; ::_thesis: verum
end;
end;
( f . 0 in I & f . 0 in J ) by A8, Th2, Th3;
then A18: S1[ 0 ] ;
for j being Element of NAT st 0 <= j & j <= len s holds
S1[j] from INT_1:sch_7(A18, A10);
then Sum s in I /\ J by A7;
hence u in sqrt (I /\ J) by A2, A5; ::_thesis: verum
end;
now__::_thesis:_for_u_being_set_st_u_in_sqrt_(I_/\_J)_holds_
u_in_sqrt_(I_*'_J)
let u be set ; ::_thesis: ( u in sqrt (I /\ J) implies u in sqrt (I *' J) )
assume u in sqrt (I /\ J) ; ::_thesis: u in sqrt (I *' J)
then consider d being Element of R such that
A19: u = d and
A20: ex m being Element of NAT st d |^ m in I /\ J ;
consider m being Element of NAT such that
A21: d |^ m in I /\ J by A20;
set q = <*((d |^ m) * (d |^ m))*>;
A22: len <*((d |^ m) * (d |^ m))*> = 1 by FINSEQ_1:40;
A23: ex g being Element of R st
( d |^ m = g & g in I & g in J ) by A21;
A24: for i being Element of NAT st 1 <= i & i <= len <*((d |^ m) * (d |^ m))*> holds
ex x, y being Element of R st
( <*((d |^ m) * (d |^ m))*> . i = x * y & x in I & y in J )
proof
let i be Element of NAT ; ::_thesis: ( 1 <= i & i <= len <*((d |^ m) * (d |^ m))*> implies ex x, y being Element of R st
( <*((d |^ m) * (d |^ m))*> . i = x * y & x in I & y in J ) )
assume A25: ( 1 <= i & i <= len <*((d |^ m) * (d |^ m))*> ) ; ::_thesis: ex x, y being Element of R st
( <*((d |^ m) * (d |^ m))*> . i = x * y & x in I & y in J )
then i in Seg (len <*((d |^ m) * (d |^ m))*>) by FINSEQ_1:1;
then i in dom <*((d |^ m) * (d |^ m))*> by FINSEQ_1:def_3;
then A26: <*((d |^ m) * (d |^ m))*> . i = <*((d |^ m) * (d |^ m))*> /. i by PARTFUN1:def_6;
then <*((d |^ m) * (d |^ m))*> /. i = <*((d |^ m) * (d |^ m))*> . 1 by A22, A25, XXREAL_0:1
.= (d |^ m) * (d |^ m) by FINSEQ_1:40 ;
hence ex x, y being Element of R st
( <*((d |^ m) * (d |^ m))*> . i = x * y & x in I & y in J ) by A23, A26; ::_thesis: verum
end;
d |^ (m + m) = (d |^ m) * (d |^ m) by BINOM:10
.= Sum <*((d |^ m) * (d |^ m))*> by BINOM:3 ;
then d |^ (m + m) in I *' J by A24;
hence u in sqrt (I *' J) by A19; ::_thesis: verum
end;
hence sqrt (I *' J) = sqrt (I /\ J) by A1, TARSKI:1; ::_thesis: verum
end;
begin
definition
let L be non empty doubleLoopStr ;
let I be Ideal of L;
attrI is finitely_generated means :Def25: :: IDEAL_1:def 25
ex F being non empty finite Subset of L st I = F -Ideal ;
end;
:: deftheorem Def25 defines finitely_generated IDEAL_1:def_25_:_
for L being non empty doubleLoopStr
for I being Ideal of L holds
( I is finitely_generated iff ex F being non empty finite Subset of L st I = F -Ideal );
registration
let L be non empty doubleLoopStr ;
cluster non empty add-closed left-ideal right-ideal finitely_generated for Element of bool the carrier of L;
existence
ex b1 being Ideal of L st b1 is finitely_generated
proof
consider x being set such that
A1: x in the carrier of L by XBOOLE_0:def_1;
reconsider x = x as Element of L by A1;
take {x} -Ideal ; ::_thesis: {x} -Ideal is finitely_generated
thus {x} -Ideal is finitely_generated by Def25; ::_thesis: verum
end;
end;
registration
let L be non empty doubleLoopStr ;
let F be non empty finite Subset of L;
clusterF -Ideal -> finitely_generated ;
coherence
F -Ideal is finitely_generated by Def25;
end;
definition
let L be non empty doubleLoopStr ;
attrL is Noetherian means :Def26: :: IDEAL_1:def 26
for I being Ideal of L holds I is finitely_generated ;
end;
:: deftheorem Def26 defines Noetherian IDEAL_1:def_26_:_
for L being non empty doubleLoopStr holds
( L is Noetherian iff for I being Ideal of L holds I is finitely_generated );
registration
cluster non empty non degenerated right_complementable Abelian add-associative right_zeroed associative commutative well-unital distributive Euclidian for doubleLoopStr ;
existence
ex b1 being non empty doubleLoopStr st
( b1 is Euclidian & b1 is Abelian & b1 is add-associative & b1 is right_zeroed & b1 is right_complementable & b1 is well-unital & b1 is distributive & b1 is commutative & b1 is associative & not b1 is degenerated )
proof
take INT.Ring ; ::_thesis: ( INT.Ring is Euclidian & INT.Ring is Abelian & INT.Ring is add-associative & INT.Ring is right_zeroed & INT.Ring is right_complementable & INT.Ring is well-unital & INT.Ring is distributive & INT.Ring is commutative & INT.Ring is associative & not INT.Ring is degenerated )
thus ( INT.Ring is Euclidian & INT.Ring is Abelian & INT.Ring is add-associative & INT.Ring is right_zeroed & INT.Ring is right_complementable & INT.Ring is well-unital & INT.Ring is distributive & INT.Ring is commutative & INT.Ring is associative & not INT.Ring is degenerated ) ; ::_thesis: verum
end;
end;
definition
let L be non empty doubleLoopStr ;
let I be Ideal of L;
attrI is principal means :Def27: :: IDEAL_1:def 27
ex e being Element of L st I = {e} -Ideal ;
end;
:: deftheorem Def27 defines principal IDEAL_1:def_27_:_
for L being non empty doubleLoopStr
for I being Ideal of L holds
( I is principal iff ex e being Element of L st I = {e} -Ideal );
definition
let L be non empty doubleLoopStr ;
attrL is PID means :Def28: :: IDEAL_1:def 28
for I being Ideal of L holds I is principal ;
end;
:: deftheorem Def28 defines PID IDEAL_1:def_28_:_
for L being non empty doubleLoopStr holds
( L is PID iff for I being Ideal of L holds I is principal );
theorem Th93: :: IDEAL_1:93
for L being non empty doubleLoopStr
for F being non empty Subset of L st F <> {(0. L)} holds
ex x being Element of L st
( x <> 0. L & x in F )
proof
let L be non empty doubleLoopStr ; ::_thesis: for F being non empty Subset of L st F <> {(0. L)} holds
ex x being Element of L st
( x <> 0. L & x in F )
let F be non empty Subset of L; ::_thesis: ( F <> {(0. L)} implies ex x being Element of L st
( x <> 0. L & x in F ) )
assume A1: F <> {(0. L)} ; ::_thesis: ex x being Element of L st
( x <> 0. L & x in F )
now__::_thesis:_ex_x_being_set_st_
(_x_in_F_&_not_x_=_0._L_)
assume A2: for x being set st x in F holds
x = 0. L ; ::_thesis: contradiction
for x being set holds
( x in F iff x = 0. L )
proof
let e be set ; ::_thesis: ( e in F iff e = 0. L )
A3: ex a being set st a in F by XBOOLE_0:def_1;
thus ( e in F implies e = 0. L ) by A2; ::_thesis: ( e = 0. L implies e in F )
assume e = 0. L ; ::_thesis: e in F
hence e in F by A2, A3; ::_thesis: verum
end;
hence contradiction by A1, TARSKI:def_1; ::_thesis: verum
end;
hence ex x being Element of L st
( x <> 0. L & x in F ) ; ::_thesis: verum
end;
theorem Th94: :: IDEAL_1:94
for R being non empty right_complementable add-associative right_zeroed well-unital distributive left_zeroed Euclidian doubleLoopStr holds R is PID
proof
let R be non empty right_complementable add-associative right_zeroed well-unital distributive left_zeroed Euclidian doubleLoopStr ; ::_thesis: R is PID
let I be Ideal of R; :: according to IDEAL_1:def_28 ::_thesis: I is principal
percases ( I = {(0. R)} or I <> {(0. R)} ) ;
supposeA1: I = {(0. R)} ; ::_thesis: I is principal
set e = 0. R;
take 0. R ; :: according to IDEAL_1:def_27 ::_thesis: I = {(0. R)} -Ideal
thus I = {(0. R)} -Ideal by A1, Th44; ::_thesis: verum
end;
suppose I <> {(0. R)} ; ::_thesis: I is principal
then consider x being Element of R such that
A2: ( x <> 0. R & x in I ) by Th93;
set I9 = { y where y is Element of I : y <> 0. R } ;
A3: { y where y is Element of I : y <> 0. R } c= the carrier of R
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in { y where y is Element of I : y <> 0. R } or x in the carrier of R )
assume x in { y where y is Element of I : y <> 0. R } ; ::_thesis: x in the carrier of R
then ex y being Element of I st
( x = y & y <> 0. R ) ;
hence x in the carrier of R ; ::_thesis: verum
end;
x in { y where y is Element of I : y <> 0. R } by A2;
then reconsider I9 = { y where y is Element of I : y <> 0. R } as non empty Subset of R by A3;
consider f being Function of the carrier of R,NAT such that
A4: for a, b being Element of R st b <> 0. R holds
ex q, r being Element of R st
( a = (q * b) + r & ( r = 0. R or f . r < f . b ) ) by INT_3:def_8;
set K = { (f . i) where i is Element of I9 : verum } ;
set i = the Element of I9;
A5: f . the Element of I9 in { (f . i) where i is Element of I9 : verum } ;
{ (f . i) where i is Element of I9 : verum } c= NAT
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in { (f . i) where i is Element of I9 : verum } or x in NAT )
assume x in { (f . i) where i is Element of I9 : verum } ; ::_thesis: x in NAT
then ex e being Element of I9 st f . e = x ;
hence x in NAT ; ::_thesis: verum
end;
then reconsider K = { (f . i) where i is Element of I9 : verum } as non empty Subset of NAT by A5;
set k = min K;
min K in K by XXREAL_2:def_7;
then consider e being Element of I9 such that
A6: f . e = min K ;
e in I9 ;
then A7: ex e9 being Element of I st
( e9 = e & e9 <> 0. R ) ;
reconsider e = e as Element of R ;
take e ; :: according to IDEAL_1:def_27 ::_thesis: I = {e} -Ideal
now__::_thesis:_for_x_being_set_holds_
(_(_x_in_I_implies_x_in_{e}_-Ideal_)_&_(_x_in_{e}_-Ideal_implies_x_in_I_)_)
let x be set ; ::_thesis: ( ( x in I implies x in {e} -Ideal ) & ( x in {e} -Ideal implies x in I ) )
{e} c= I by A7, ZFMISC_1:31;
then A8: {e} -Ideal c= I by Def14;
hereby ::_thesis: ( x in {e} -Ideal implies x in I )
assume A9: x in I ; ::_thesis: x in {e} -Ideal
then reconsider x9 = x as Element of R ;
consider q, r being Element of R such that
A10: x9 = (q * e) + r and
A11: ( r = 0. R or f . r < min K ) by A4, A6, A7;
now__::_thesis:_not_r_<>_0._R
q * e in I by A7, Def2;
then A12: - (q * e) in I by Th13;
assume A13: r <> 0. R ; ::_thesis: contradiction
(- (q * e)) + x9 = ((- (q * e)) + (q * e)) + r by A10, RLVECT_1:def_3
.= (0. R) + r by RLVECT_1:5
.= r by ALGSTR_1:def_2 ;
then r in I by A9, A12, Def1;
then r in I9 by A13;
then f . r in K ;
hence contradiction by A11, A13, XXREAL_2:def_7; ::_thesis: verum
end;
then A14: x9 = q * e by A10, RLVECT_1:def_4;
( e in {e} & {e} c= {e} -Ideal ) by Def14, TARSKI:def_1;
hence x in {e} -Ideal by A14, Def2; ::_thesis: verum
end;
assume x in {e} -Ideal ; ::_thesis: x in I
hence x in I by A8; ::_thesis: verum
end;
hence I = {e} -Ideal by TARSKI:1; ::_thesis: verum
end;
end;
end;
theorem Th95: :: IDEAL_1:95
for L being non empty doubleLoopStr st L is PID holds
L is Noetherian
proof
let L be non empty doubleLoopStr ; ::_thesis: ( L is PID implies L is Noetherian )
assume A1: L is PID ; ::_thesis: L is Noetherian
let I be Ideal of L; :: according to IDEAL_1:def_26 ::_thesis: I is finitely_generated
I is principal by A1, Def28;
then consider e being Element of L such that
A2: I = {e} -Ideal by Def27;
take {e} ; :: according to IDEAL_1:def_25 ::_thesis: I = {e} -Ideal
thus I = {e} -Ideal by A2; ::_thesis: verum
end;
registration
cluster INT.Ring -> Noetherian ;
coherence
INT.Ring is Noetherian
proof
INT.Ring is PID by Th94;
hence INT.Ring is Noetherian by Th95; ::_thesis: verum
end;
end;
registration
cluster non empty non degenerated right_complementable Abelian add-associative right_zeroed associative commutative well-unital distributive Noetherian for doubleLoopStr ;
existence
ex b1 being non empty doubleLoopStr st
( b1 is Noetherian & b1 is Abelian & b1 is add-associative & b1 is right_zeroed & b1 is right_complementable & b1 is well-unital & b1 is distributive & b1 is commutative & b1 is associative & not b1 is degenerated )
proof
take INT.Ring ; ::_thesis: ( INT.Ring is Noetherian & INT.Ring is Abelian & INT.Ring is add-associative & INT.Ring is right_zeroed & INT.Ring is right_complementable & INT.Ring is well-unital & INT.Ring is distributive & INT.Ring is commutative & INT.Ring is associative & not INT.Ring is degenerated )
thus ( INT.Ring is Noetherian & INT.Ring is Abelian & INT.Ring is add-associative & INT.Ring is right_zeroed & INT.Ring is right_complementable & INT.Ring is well-unital & INT.Ring is distributive & INT.Ring is commutative & INT.Ring is associative & not INT.Ring is degenerated ) ; ::_thesis: verum
end;
end;
theorem :: IDEAL_1:96
for R being non empty add-cancelable add-associative right_zeroed associative well-unital distributive left_zeroed Noetherian doubleLoopStr
for B being non empty Subset of R ex C being non empty finite Subset of R st
( C c= B & C -Ideal = B -Ideal )
proof
let R be non empty add-cancelable add-associative right_zeroed associative well-unital distributive left_zeroed Noetherian doubleLoopStr ; ::_thesis: for B being non empty Subset of R ex C being non empty finite Subset of R st
( C c= B & C -Ideal = B -Ideal )
let B be non empty Subset of R; ::_thesis: ex C being non empty finite Subset of R st
( C c= B & C -Ideal = B -Ideal )
defpred S1[ set , set ] means ex cL being non empty LinearCombination of B st
( $1 = Sum cL & ex fsB being FinSequence of B st
( dom fsB = dom cL & $2 = rng fsB & ( for i being Element of NAT st i in dom cL holds
ex u, v being Element of R st cL /. i = (u * (fsB /. i)) * v ) ) );
B -Ideal is finitely_generated by Def26;
then consider D being non empty finite Subset of R such that
A1: D -Ideal = B -Ideal by Def25;
A2: D c= B -Ideal by A1, Def14;
A3: for e being set st e in D holds
ex u being set st
( u in bool B & S1[e,u] )
proof
let e be set ; ::_thesis: ( e in D implies ex u being set st
( u in bool B & S1[e,u] ) )
assume e in D ; ::_thesis: ex u being set st
( u in bool B & S1[e,u] )
then consider cL being LinearCombination of B such that
A4: e = Sum cL by A2, Th60;
percases ( not cL is empty or cL is empty ) ;
supposeA5: not cL is empty ; ::_thesis: ex u being set st
( u in bool B & S1[e,u] )
defpred S2[ set , Element of B] means ex u, v being Element of R st cL /. $1 = (u * $2) * v;
A6: now__::_thesis:_for_k_being_Element_of_NAT_st_k_in_Seg_(len_cL)_holds_
ex_d_being_Element_of_B_st_S2[k,d]
let k be Element of NAT ; ::_thesis: ( k in Seg (len cL) implies ex d being Element of B st S2[k,d] )
assume k in Seg (len cL) ; ::_thesis: ex d being Element of B st S2[k,d]
then k in dom cL by FINSEQ_1:def_3;
then consider u, v being Element of R, a being Element of B such that
A7: cL /. k = (u * a) * v by Def8;
take d = a; ::_thesis: S2[k,d]
thus S2[k,d] by A7; ::_thesis: verum
end;
consider fsB being FinSequence of B such that
A8: ( dom fsB = Seg (len cL) & ( for k being Element of NAT st k in Seg (len cL) holds
S2[k,fsB /. k] ) ) from RECDEF_1:sch_17(A6);
take u = rng fsB; ::_thesis: ( u in bool B & S1[e,u] )
thus u in bool B ; ::_thesis: S1[e,u]
dom cL = Seg (len cL) by FINSEQ_1:def_3;
hence S1[e,u] by A4, A5, A8; ::_thesis: verum
end;
supposeA9: cL is empty ; ::_thesis: ex u being set st
( u in bool B & S1[e,u] )
set b = the Element of B;
set kL = <*(((0. R) * the Element of B) * (0. R))*>;
now__::_thesis:_for_i_being_set_st_i_in_dom_<*(((0._R)_*_the_Element_of_B)_*_(0._R))*>_holds_
ex_u,_v_being_Element_of_the_carrier_of_R_ex_b_being_Element_of_B_st_<*(((0._R)_*_the_Element_of_B)_*_(0._R))*>_/._i_=_(u_*_b)_*_v
let i be set ; ::_thesis: ( i in dom <*(((0. R) * the Element of B) * (0. R))*> implies ex u, v being Element of the carrier of R ex b being Element of B st <*(((0. R) * the Element of B) * (0. R))*> /. i = (u * b) * v )
assume A10: i in dom <*(((0. R) * the Element of B) * (0. R))*> ; ::_thesis: ex u, v being Element of the carrier of R ex b being Element of B st <*(((0. R) * the Element of B) * (0. R))*> /. i = (u * b) * v
take u = 0. R; ::_thesis: ex v being Element of the carrier of R ex b being Element of B st <*(((0. R) * the Element of B) * (0. R))*> /. i = (u * b) * v
take v = 0. R; ::_thesis: ex b being Element of B st <*(((0. R) * the Element of B) * (0. R))*> /. i = (u * b) * v
take b = the Element of B; ::_thesis: <*(((0. R) * the Element of B) * (0. R))*> /. i = (u * b) * v
i in Seg (len <*(((0. R) * the Element of B) * (0. R))*>) by A10, FINSEQ_1:def_3;
then i in {1} by FINSEQ_1:2, FINSEQ_1:40;
then i = 1 by TARSKI:def_1;
hence <*(((0. R) * the Element of B) * (0. R))*> /. i = (u * b) * v by FINSEQ_4:16; ::_thesis: verum
end;
then reconsider kL = <*(((0. R) * the Element of B) * (0. R))*> as non empty LinearCombination of B by Def8;
cL = <*> the carrier of R by A9;
then A11: e = 0. R by A4, RLVECT_1:43
.= ((0. R) * the Element of B) * (0. R) by BINOM:2
.= Sum kL by BINOM:3 ;
defpred S2[ Element of NAT , Element of B] means ex u, v being Element of R st kL /. $1 = (u * $2) * v;
A12: now__::_thesis:_for_k_being_Element_of_NAT_st_k_in_Seg_(len_kL)_holds_
ex_b_being_Element_of_B_st_S2[k,b]
let k be Element of NAT ; ::_thesis: ( k in Seg (len kL) implies ex b being Element of B st S2[k,b] )
assume A13: k in Seg (len kL) ; ::_thesis: ex b being Element of B st S2[k,b]
take b = the Element of B; ::_thesis: S2[k,b]
k in {1} by A13, FINSEQ_1:2, FINSEQ_1:40;
then k = 1 by TARSKI:def_1;
hence S2[k,b] by FINSEQ_4:16; ::_thesis: verum
end;
consider fsB being FinSequence of B such that
A14: ( dom fsB = Seg (len kL) & ( for k being Element of NAT st k in Seg (len kL) holds
S2[k,fsB /. k] ) ) from RECDEF_1:sch_17(A12);
take u = rng fsB; ::_thesis: ( u in bool B & S1[e,u] )
thus u in bool B ; ::_thesis: S1[e,u]
dom kL = Seg (len kL) by FINSEQ_1:def_3;
hence S1[e,u] by A11, A14; ::_thesis: verum
end;
end;
end;
consider f being Function of D,(bool B) such that
A15: for e being set st e in D holds
S1[e,f . e] from FUNCT_2:sch_1(A3);
A16: now__::_thesis:_for_r_being_set_st_r_in_rng_f_holds_
r_is_finite
let r be set ; ::_thesis: ( r in rng f implies r is finite )
assume r in rng f ; ::_thesis: r is finite
then consider x being set such that
A17: ( x in dom f & r = f . x ) by FUNCT_1:def_3;
ex cL being non empty LinearCombination of B st
( x = Sum cL & ex fsB being FinSequence of B st
( dom fsB = dom cL & r = rng fsB & ( for i being Element of NAT st i in dom cL holds
ex u, v being Element of R st cL /. i = (u * (fsB /. i)) * v ) ) ) by A15, A17;
hence r is finite ; ::_thesis: verum
end;
reconsider rf = rng f as Subset-Family of B ;
reconsider C = union rf as Subset of B ;
consider r being set such that
A18: r in rng f by XBOOLE_0:def_1;
consider x being set such that
A19: ( x in dom f & r = f . x ) by A18, FUNCT_1:def_3;
ex cL being non empty LinearCombination of B st
( x = Sum cL & ex fsB being FinSequence of B st
( dom fsB = dom cL & r = rng fsB & ( for i being Element of NAT st i in dom cL holds
ex u, v being Element of R st cL /. i = (u * (fsB /. i)) * v ) ) ) by A15, A19;
then not r is empty by RELAT_1:42;
then ex x being set st x in r by XBOOLE_0:def_1;
then reconsider C = C as non empty finite Subset of R by A18, A16, FINSET_1:7, TARSKI:def_4, XBOOLE_1:1;
now__::_thesis:_for_d_being_set_st_d_in_D_holds_
d_in_C_-Ideal
let d be set ; ::_thesis: ( d in D implies d in C -Ideal )
assume A20: d in D ; ::_thesis: d in C -Ideal
then consider cL being non empty LinearCombination of B such that
A21: d = Sum cL and
A22: ex fsB being FinSequence of B st
( dom fsB = dom cL & f . d = rng fsB & ( for i being Element of NAT st i in dom cL holds
ex u, v being Element of R st cL /. i = (u * (fsB /. i)) * v ) ) by A15;
d in dom f by A20, FUNCT_2:def_1;
then f . d in rng f by FUNCT_1:def_3;
then A23: f . d c= C by ZFMISC_1:74;
now__::_thesis:_for_i_being_set_st_i_in_dom_cL_holds_
ex_u,_v_being_Element_of_R_ex_a_being_Element_of_C_st_cL_/._i_=_(u_*_a)_*_v
let i be set ; ::_thesis: ( i in dom cL implies ex u, v being Element of R ex a being Element of C st cL /. i = (u * a) * v )
consider fsB being FinSequence of B such that
A24: dom fsB = dom cL and
A25: f . d = rng fsB and
A26: for i being Element of NAT st i in dom cL holds
ex u, v being Element of R st cL /. i = (u * (fsB /. i)) * v by A22;
assume A27: i in dom cL ; ::_thesis: ex u, v being Element of R ex a being Element of C st cL /. i = (u * a) * v
then fsB /. i = fsB . i by A24, PARTFUN1:def_6;
then A28: fsB /. i in f . d by A27, A24, A25, FUNCT_1:def_3;
ex u, v being Element of R st cL /. i = (u * (fsB /. i)) * v by A27, A26;
hence ex u, v being Element of R ex a being Element of C st cL /. i = (u * a) * v by A23, A28; ::_thesis: verum
end;
then reconsider cL9 = cL as LinearCombination of C by Def8;
d = Sum cL9 by A21;
hence d in C -Ideal by Th60; ::_thesis: verum
end;
then D c= C -Ideal by TARSKI:def_3;
then D -Ideal c= (C -Ideal) -Ideal by Th57;
then A29: B -Ideal c= C -Ideal by A1, Th44;
take C ; ::_thesis: ( C c= B & C -Ideal = B -Ideal )
C -Ideal c= B -Ideal by Th57;
hence ( C c= B & C -Ideal = B -Ideal ) by A29, XBOOLE_0:def_10; ::_thesis: verum
end;
theorem :: IDEAL_1:97
for R being non empty doubleLoopStr st ( for B being non empty Subset of R ex C being non empty finite Subset of R st
( C c= B & C -Ideal = B -Ideal ) ) holds
for a being sequence of R ex m being Element of NAT st a . (m + 1) in (rng (a | (m + 1))) -Ideal
proof
let R be non empty doubleLoopStr ; ::_thesis: ( ( for B being non empty Subset of R ex C being non empty finite Subset of R st
( C c= B & C -Ideal = B -Ideal ) ) implies for a being sequence of R ex m being Element of NAT st a . (m + 1) in (rng (a | (m + 1))) -Ideal )
assume A1: for B being non empty Subset of R ex C being non empty finite Subset of R st
( C c= B & C -Ideal = B -Ideal ) ; ::_thesis: for a being sequence of R ex m being Element of NAT st a . (m + 1) in (rng (a | (m + 1))) -Ideal
let a be sequence of R; ::_thesis: ex m being Element of NAT st a . (m + 1) in (rng (a | (m + 1))) -Ideal
reconsider B = rng a as non empty Subset of R ;
consider C being non empty finite Subset of R such that
A2: C c= B and
A3: C -Ideal = B -Ideal by A1;
defpred S1[ set , set ] means $1 = a . $2;
A4: dom a = NAT by FUNCT_2:def_1;
A5: for e being set st e in C holds
ex u being set st
( u in NAT & S1[e,u] )
proof
let e be set ; ::_thesis: ( e in C implies ex u being set st
( u in NAT & S1[e,u] ) )
assume e in C ; ::_thesis: ex u being set st
( u in NAT & S1[e,u] )
then consider u being set such that
A6: u in dom a and
A7: e = a . u by A2, FUNCT_1:def_3;
take u ; ::_thesis: ( u in NAT & S1[e,u] )
thus u in NAT by A6; ::_thesis: S1[e,u]
thus S1[e,u] by A7; ::_thesis: verum
end;
consider f being Function of C,NAT such that
A8: for e being set st e in C holds
S1[e,f . e] from FUNCT_2:sch_1(A5);
set Rf = rng f;
reconsider Rf = rng f as non empty finite Subset of NAT ;
reconsider m = max Rf as Element of NAT by ORDINAL1:def_12;
set D = rng (a | (Segm (m + 1)));
A9: dom f = C by FUNCT_2:def_1;
A10: C c= rng (a | (Segm (m + 1)))
proof
let X be set ; :: according to TARSKI:def_3 ::_thesis: ( not X in C or X in rng (a | (Segm (m + 1))) )
set fx = f . X;
assume A11: X in C ; ::_thesis: X in rng (a | (Segm (m + 1)))
then f . X in Rf by A9, FUNCT_1:def_3;
then f . X <= m by XXREAL_2:def_8;
then f . X < m + 1 by NAT_1:13;
then f . X in Segm (m + 1) by NAT_1:44;
then a . (f . X) in rng (a | (Segm (m + 1))) by A4, FUNCT_1:50;
hence X in rng (a | (Segm (m + 1))) by A8, A11; ::_thesis: verum
end;
then reconsider D = rng (a | (Segm (m + 1))) as non empty Subset of R ;
A12: D -Ideal c= B -Ideal by Th57, RELAT_1:70;
B -Ideal c= D -Ideal by A3, A10, Th57;
then A13: D -Ideal = B -Ideal by A12, XBOOLE_0:def_10;
take m ; ::_thesis: a . (m + 1) in (rng (a | (m + 1))) -Ideal
( B c= B -Ideal & a . (m + 1) in B ) by Def14, FUNCT_2:4;
hence a . (m + 1) in (rng (a | (m + 1))) -Ideal by A13; ::_thesis: verum
end;
registration
let X, Y be non empty set ;
let f be Function of X,Y;
let A be non empty Subset of X;
clusterf | A -> non empty ;
coherence
not f | A is empty
proof
dom f = X by FUNCT_2:def_1;
then not (dom f) /\ A is empty by XBOOLE_1:28;
then dom f meets A by XBOOLE_0:def_7;
hence not f | A is empty by RELAT_1:66; ::_thesis: verum
end;
end;
theorem :: IDEAL_1:98
for R being non empty doubleLoopStr st ( for a being sequence of R ex m being Element of NAT st a . (m + 1) in (rng (a | (m + 1))) -Ideal ) holds
for F being Function of NAT,(bool the carrier of R) holds
( ex i being Element of NAT st F . i is not Ideal of R or ex j, k being Element of NAT st
( j < k & not F . j c< F . k ) )
proof
let R be non empty doubleLoopStr ; ::_thesis: ( ( for a being sequence of R ex m being Element of NAT st a . (m + 1) in (rng (a | (m + 1))) -Ideal ) implies for F being Function of NAT,(bool the carrier of R) holds
( ex i being Element of NAT st F . i is not Ideal of R or ex j, k being Element of NAT st
( j < k & not F . j c< F . k ) ) )
assume A1: for a being sequence of R ex m being Element of NAT st a . (m + 1) in (rng (a | (m + 1))) -Ideal ; ::_thesis: for F being Function of NAT,(bool the carrier of R) holds
( ex i being Element of NAT st F . i is not Ideal of R or ex j, k being Element of NAT st
( j < k & not F . j c< F . k ) )
given F being Function of NAT,(bool the carrier of R) such that A2: for i being Element of NAT holds F . i is Ideal of R and
A3: for j, k being Element of NAT st j < k holds
F . j c< F . k ; ::_thesis: contradiction
defpred S1[ set , set ] means ex n being Element of NAT st
( n = $1 & ( n = 0 implies $2 in F . 0 ) & ( n > 0 implies ex k being Element of NAT st
( n = k + 1 & $2 in (F . n) \ (F . k) ) ) );
A4: for e being set st e in NAT holds
ex u being set st
( u in the carrier of R & S1[e,u] )
proof
let e be set ; ::_thesis: ( e in NAT implies ex u being set st
( u in the carrier of R & S1[e,u] ) )
assume e in NAT ; ::_thesis: ex u being set st
( u in the carrier of R & S1[e,u] )
then reconsider n = e as Element of NAT ;
percases ( n = 0 or n > 0 ) ;
supposeA5: n = 0 ; ::_thesis: ex u being set st
( u in the carrier of R & S1[e,u] )
F . 0 is Ideal of R by A2;
then consider u being set such that
A6: u in F . 0 by XBOOLE_0:def_1;
take u ; ::_thesis: ( u in the carrier of R & S1[e,u] )
thus u in the carrier of R by A6; ::_thesis: S1[e,u]
take n ; ::_thesis: ( n = e & ( n = 0 implies u in F . 0 ) & ( n > 0 implies ex k being Element of NAT st
( n = k + 1 & u in (F . n) \ (F . k) ) ) )
thus n = e ; ::_thesis: ( ( n = 0 implies u in F . 0 ) & ( n > 0 implies ex k being Element of NAT st
( n = k + 1 & u in (F . n) \ (F . k) ) ) )
thus ( ( n = 0 implies u in F . 0 ) & ( n > 0 implies ex k being Element of NAT st
( n = k + 1 & u in (F . n) \ (F . k) ) ) ) by A5, A6; ::_thesis: verum
end;
suppose n > 0 ; ::_thesis: ex u being set st
( u in the carrier of R & S1[e,u] )
then consider k being Nat such that
A7: n = k + 1 by NAT_1:6;
reconsider k = k as Element of NAT by ORDINAL1:def_12;
n > k by A7, NAT_1:13;
then not F . n c= F . k by A3, XBOOLE_1:60;
then not (F . n) \ (F . k) is empty by XBOOLE_1:37;
then consider u being set such that
A8: u in (F . n) \ (F . k) by XBOOLE_0:def_1;
take u ; ::_thesis: ( u in the carrier of R & S1[e,u] )
thus u in the carrier of R by A8; ::_thesis: S1[e,u]
take n ; ::_thesis: ( n = e & ( n = 0 implies u in F . 0 ) & ( n > 0 implies ex k being Element of NAT st
( n = k + 1 & u in (F . n) \ (F . k) ) ) )
thus n = e ; ::_thesis: ( ( n = 0 implies u in F . 0 ) & ( n > 0 implies ex k being Element of NAT st
( n = k + 1 & u in (F . n) \ (F . k) ) ) )
thus ( ( n = 0 implies u in F . 0 ) & ( n > 0 implies ex k being Element of NAT st
( n = k + 1 & u in (F . n) \ (F . k) ) ) ) by A7, A8; ::_thesis: verum
end;
end;
end;
consider f being Function of NAT, the carrier of R such that
A9: for e being set st e in NAT holds
S1[e,f . e] from FUNCT_2:sch_1(A4);
consider m being Element of NAT such that
A10: f . (m + 1) in (rng (f | (m + 1))) -Ideal by A1;
reconsider m1 = m + 1 as non zero Nat ;
A11: ex n being Element of NAT st
( n = m + 1 & ( n = 0 implies f . (m + 1) in F . 0 ) & ( n > 0 implies ex k being Element of NAT st
( n = k + 1 & f . (m + 1) in (F . n) \ (F . k) ) ) ) by A9;
defpred S2[ Element of NAT ] means rng (f | (Segm ($1 + 1))) c= F . $1;
A12: for k being Element of NAT st S2[k] holds
S2[k + 1]
proof
let k be Element of NAT ; ::_thesis: ( S2[k] implies S2[k + 1] )
assume A13: rng (f | (Segm (k + 1))) c= F . k ; ::_thesis: S2[k + 1]
let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in rng (f | (Segm ((k + 1) + 1))) or y in F . (k + 1) )
assume y in rng (f | (Segm ((k + 1) + 1))) ; ::_thesis: y in F . (k + 1)
then consider x being set such that
A14: x in dom (f | (Segm ((k + 1) + 1))) and
A15: y = (f | (Segm ((k + 1) + 1))) . x by FUNCT_1:def_3;
A16: x in dom f by A14, RELAT_1:57;
reconsider nx = x as Element of NAT by A14;
x in Segm ((k + 1) + 1) by A14, RELAT_1:57;
then nx < (k + 1) + 1 by NAT_1:44;
then A17: nx <= k + 1 by NAT_1:13;
percases ( nx < k + 1 or nx = k + 1 ) by A17, XXREAL_0:1;
suppose nx < k + 1 ; ::_thesis: y in F . (k + 1)
then A18: nx in Segm (k + 1) by NAT_1:44;
k < k + 1 by NAT_1:13;
then F . k c< F . (k + 1) by A3;
then A19: F . k c= F . (k + 1) by XBOOLE_0:def_8;
y = f . nx by A14, A15, FUNCT_1:47;
then y in rng (f | (Segm (k + 1))) by A16, A18, FUNCT_1:50;
then y in F . k by A13;
hence y in F . (k + 1) by A19; ::_thesis: verum
end;
supposeA20: nx = k + 1 ; ::_thesis: y in F . (k + 1)
( y = f . nx & ex n being Element of NAT st
( n = nx & ( n = 0 implies f . nx in F . 0 ) & ( n > 0 implies ex k being Element of NAT st
( n = k + 1 & f . nx in (F . n) \ (F . k) ) ) ) ) by A9, A14, A15, FUNCT_1:47;
hence y in F . (k + 1) by A20, XBOOLE_0:def_5; ::_thesis: verum
end;
end;
end;
F . m is Ideal of R by A2;
then A21: F . m = (F . m) -Ideal by Th44;
A22: S2[ 0 ]
proof
let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in rng (f | (Segm (0 + 1))) or y in F . 0 )
assume y in rng (f | (Segm (0 + 1))) ; ::_thesis: y in F . 0
then consider x being set such that
A23: x in dom (f | (Segm 1)) and
A24: y = (f | (Segm 1)) . x by FUNCT_1:def_3;
( x in Segm 1 & ex n being Element of NAT st
( n = x & ( n = 0 implies f . x in F . 0 ) & ( n > 0 implies ex k being Element of NAT st
( n = k + 1 & f . x in (F . n) \ (F . k) ) ) ) ) by A9, A23, RELAT_1:57;
hence y in F . 0 by A24, CARD_1:49, FUNCT_1:49, TARSKI:def_1; ::_thesis: verum
end;
for m being Element of NAT holds S2[m] from NAT_1:sch_1(A22, A12);
then (rng (f | (Segm m1))) -Ideal c= F . m by A21, Th57;
hence contradiction by A10, A11, XBOOLE_0:def_5; ::_thesis: verum
end;
theorem :: IDEAL_1:99
for R being non empty doubleLoopStr st ( for F being Function of NAT,(bool the carrier of R) holds
( ex i being Element of NAT st F . i is not Ideal of R or ex j, k being Element of NAT st
( j < k & not F . j c< F . k ) ) ) holds
R is Noetherian
proof
let R be non empty doubleLoopStr ; ::_thesis: ( ( for F being Function of NAT,(bool the carrier of R) holds
( ex i being Element of NAT st F . i is not Ideal of R or ex j, k being Element of NAT st
( j < k & not F . j c< F . k ) ) ) implies R is Noetherian )
assume that
A1: for F being Function of NAT,(bool the carrier of R) holds
( ex i being Element of NAT st F . i is not Ideal of R or ex j, k being Element of NAT st
( j < k & not F . j c< F . k ) ) and
A2: not R is Noetherian ; ::_thesis: contradiction
consider I being Ideal of R such that
A3: not I is finitely_generated by A2, Def26;
set D = { S where S is Subset of R : S is non empty finite Subset of I } ;
consider e being set such that
A4: e in I by XBOOLE_0:def_1;
reconsider e = e as Element of R by A4;
{e} c= I by A4, ZFMISC_1:31;
then A5: {e} in { S where S is Subset of R : S is non empty finite Subset of I } ;
{ S where S is Subset of R : S is non empty finite Subset of I } c= bool the carrier of R
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in { S where S is Subset of R : S is non empty finite Subset of I } or x in bool the carrier of R )
assume x in { S where S is Subset of R : S is non empty finite Subset of I } ; ::_thesis: x in bool the carrier of R
then ex s being Subset of R st
( x = s & s is non empty finite Subset of I ) ;
hence x in bool the carrier of R ; ::_thesis: verum
end;
then reconsider D = { S where S is Subset of R : S is non empty finite Subset of I } as non empty Subset-Family of R by A5;
reconsider e9 = {e} as Element of D by A5;
defpred S1[ set , Element of D, set ] means ex r being Element of R st
( r in I \ ($2 -Ideal) & $3 = $2 \/ {r} );
A6: for n being Element of NAT
for x being Element of D ex y being Element of D st S1[n,x,y]
proof
let n be Element of NAT ; ::_thesis: for x being Element of D ex y being Element of D st S1[n,x,y]
let x be Element of D; ::_thesis: ex y being Element of D st S1[n,x,y]
x in D ;
then consider x9 being Subset of R such that
A7: x9 = x and
A8: x9 is non empty finite Subset of I ;
reconsider x19 = x9 as non empty finite Subset of I by A8;
x9 -Ideal c= I -Ideal by A8, Th57;
then x9 -Ideal c= I by Th44;
then not I c= x9 -Ideal by A3, A8, XBOOLE_0:def_10;
then consider r being set such that
A9: r in I and
A10: not r in x9 -Ideal by TARSKI:def_3;
set y = x19 \/ {r};
A11: x19 \/ {r} c= I
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in x19 \/ {r} or x in I )
assume x in x19 \/ {r} ; ::_thesis: x in I
then ( x in x19 or x in {r} ) by XBOOLE_0:def_3;
hence x in I by A9, TARSKI:def_1; ::_thesis: verum
end;
then x19 \/ {r} is Subset of R by XBOOLE_1:1;
then A12: x19 \/ {r} in D by A11;
reconsider r = r as Element of R by A9;
reconsider y = x19 \/ {r} as Element of D by A12;
take y ; ::_thesis: S1[n,x,y]
take r ; ::_thesis: ( r in I \ (x -Ideal) & y = x \/ {r} )
thus ( r in I \ (x -Ideal) & y = x \/ {r} ) by A7, A9, A10, XBOOLE_0:def_5; ::_thesis: verum
end;
consider f being Function of NAT,D such that
A13: ( f . 0 = e9 & ( for n being Element of NAT holds S1[n,f . n,f . (n + 1)] ) ) from RECDEF_1:sch_2(A6);
defpred S2[ Element of NAT , Subset of R] means ex c being Subset of R st
( c = f . $1 & $2 = c -Ideal );
A14: for x being Element of NAT ex y being Subset of R st S2[x,y]
proof
let x be Element of NAT ; ::_thesis: ex y being Subset of R st S2[x,y]
f . x in D ;
then consider c being Subset of R such that
A15: c = f . x and
c is non empty finite Subset of I ;
reconsider y = c -Ideal as Subset of R ;
take y ; ::_thesis: S2[x,y]
take c ; ::_thesis: ( c = f . x & y = c -Ideal )
thus ( c = f . x & y = c -Ideal ) by A15; ::_thesis: verum
end;
consider F being Function of NAT,(bool the carrier of R) such that
A16: for x being Element of NAT holds S2[x,F . x] from FUNCT_2:sch_3(A14);
A17: for j, k being Element of NAT st j < k holds
F . j c< F . k
proof
let j, k be Element of NAT ; ::_thesis: ( j < k implies F . j c< F . k )
defpred S3[ Element of NAT ] means F . j c< F . ((j + 1) + $1);
assume j < k ; ::_thesis: F . j c< F . k
then j + 1 <= k by NAT_1:13;
then consider i being Nat such that
A18: k = (j + 1) + i by NAT_1:10;
A19: for i being Element of NAT holds F . i c< F . (i + 1)
proof
let i be Element of NAT ; ::_thesis: F . i c< F . (i + 1)
consider c being Subset of R such that
A20: c = f . i and
A21: F . i = c -Ideal by A16;
consider c1 being Subset of R such that
A22: c1 = f . (i + 1) and
A23: F . (i + 1) = c1 -Ideal by A16;
c1 in D by A22;
then ex c19 being Subset of R st
( c19 = c1 & c19 is non empty finite Subset of I ) ;
then A24: c1 c= c1 -Ideal by Def14;
consider r being Element of R such that
A25: r in I \ (c -Ideal) and
A26: c1 = c \/ {r} by A13, A20, A22;
c in D by A20;
then ex c9 being Subset of R st
( c9 = c & c9 is non empty finite Subset of I ) ;
hence F . i c= F . (i + 1) by A21, A23, A26, Th57, XBOOLE_1:7; :: according to XBOOLE_0:def_8 ::_thesis: not F . i = F . (i + 1)
r in {r} by TARSKI:def_1;
then r in c1 by A26, XBOOLE_0:def_3;
hence not F . i = F . (i + 1) by A21, A23, A25, A24, XBOOLE_0:def_5; ::_thesis: verum
end;
A27: for i being Element of NAT st S3[i] holds
S3[i + 1]
proof
let i be Element of NAT ; ::_thesis: ( S3[i] implies S3[i + 1] )
assume that
A28: F . j c= F . ((j + 1) + i) and
F . j <> F . ((j + 1) + i) ; :: according to XBOOLE_0:def_8 ::_thesis: S3[i + 1]
A29: F . ((j + 1) + i) c< F . (((j + 1) + i) + 1) by A19;
then F . ((j + 1) + i) c= F . (((j + 1) + i) + 1) by XBOOLE_0:def_8;
hence F . j c= F . ((j + 1) + (i + 1)) by A28, XBOOLE_1:1; :: according to XBOOLE_0:def_8 ::_thesis: not F . j = F . ((j + 1) + (i + 1))
thus not F . j = F . ((j + 1) + (i + 1)) by A28, A29, XBOOLE_0:def_8; ::_thesis: verum
end;
A30: S3[ 0 ] by A19;
A31: for i being Element of NAT holds S3[i] from NAT_1:sch_1(A30, A27);
i in NAT by ORDINAL1:def_12;
hence F . j c< F . k by A31, A18; ::_thesis: verum
end;
for i being Element of NAT holds F . i is Ideal of R
proof
let i be Element of NAT ; ::_thesis: F . i is Ideal of R
ex c being Subset of R st
( c = f . i & F . i = c -Ideal ) by A16;
hence F . i is Ideal of R ; ::_thesis: verum
end;
hence contradiction by A1, A17; ::_thesis: verum
end;