:: TOPALG_4 semantic presentation
Lemma1:
1 in {1,2}
by TARSKI:def 2;
Lemma2:
2 in {1,2}
by TARSKI:def 2;
theorem Th1: :: TOPALG_4:1
definition
let c
1, c
2, c
3, c
4 be non
empty HGrStr ;
let c
5 be
Function of c
1,c
3;
let c
6 be
Function of c
2,c
4;
func Gr2Iso c
5,c
6 -> Function of
(product <*a1,a2*>),
(product <*a3,a4*>) means :
Def1:
:: TOPALG_4:def 1
for b
1 being
Element of
(product <*a1,a2*>) holds
ex b
2 being
Element of a
1ex b
3 being
Element of a
2 st
( b
1 = <*b2,b3*> & a
7 . b
1 = <*(a5 . b2),(a6 . b3)*> );
existence
ex b1 being Function of (product <*c1,c2*>),(product <*c3,c4*>) st
for b2 being Element of (product <*c1,c2*>) holds
ex b3 being Element of c1ex b4 being Element of c2 st
( b2 = <*b3,b4*> & b1 . b2 = <*(c5 . b3),(c6 . b4)*> )
uniqueness
for b1, b2 being Function of (product <*c1,c2*>),(product <*c3,c4*>) holds
( ( for b3 being Element of (product <*c1,c2*>) holds
ex b4 being Element of c1ex b5 being Element of c2 st
( b3 = <*b4,b5*> & b1 . b3 = <*(c5 . b4),(c6 . b5)*> ) ) & ( for b3 being Element of (product <*c1,c2*>) holds
ex b4 being Element of c1ex b5 being Element of c2 st
( b3 = <*b4,b5*> & b2 . b3 = <*(c5 . b4),(c6 . b5)*> ) ) implies b1 = b2 )
end;
:: deftheorem Def1 defines Gr2Iso TOPALG_4:def 1 :
for b
1, b
2, b
3, b
4 being non
empty HGrStr for b
5 being
Function of b
1,b
3for b
6 being
Function of b
2,b
4for b
7 being
Function of
(product <*b1,b2*>),
(product <*b3,b4*>) holds
( b
7 = Gr2Iso b
5,b
6 iff for b
8 being
Element of
(product <*b1,b2*>) holds
ex b
9 being
Element of b
1ex b
10 being
Element of b
2 st
( b
8 = <*b9,b10*> & b
7 . b
8 = <*(b5 . b9),(b6 . b10)*> ) );
theorem Th2: :: TOPALG_4:2
definition
let c
1, c
2, c
3, c
4 be
Group;
let c
5 be
Homomorphism of c
1,c
3;
let c
6 be
Homomorphism of c
2,c
4;
redefine func Gr2Iso as
Gr2Iso c
5,c
6 -> Homomorphism of
(product <*a1,a2*>),
(product <*a3,a4*>);
coherence
Gr2Iso c5,c6 is Homomorphism of (product <*c1,c2*>),(product <*c3,c4*>)
end;
theorem Th3: :: TOPALG_4:3
theorem Th4: :: TOPALG_4:4
theorem Th5: :: TOPALG_4:5
theorem Th6: :: TOPALG_4:6
set c1 = the carrier of I[01] ;
reconsider c2 = 0, c3 = 1 as Point of I[01] by BORSUK_1:def 17, BORSUK_1:def 18;
theorem Th7: :: TOPALG_4:7
theorem Th8: :: TOPALG_4:8
definition
let c
4, c
5, c
6 be non
empty TopSpace;
let c
7 be
Function of c
6,c
4;
let c
8 be
Function of c
6,c
5;
redefine func <: as
<:c4,c5:> -> Function of a
3,
[:a1,a2:];
coherence
<:c7,c8:> is Function of c6,[:c4,c5:]
end;
theorem Th9: :: TOPALG_4:9
theorem Th10: :: TOPALG_4:10
theorem Th11: :: TOPALG_4:11
theorem Th12: :: TOPALG_4:12
theorem Th13: :: TOPALG_4:13
for b
1, b
2 being non
empty TopSpacefor b
3, b
4 being
Point of b
1for b
5, b
6 being
Point of b
2 holds
(
[b3,b5],
[b4,b6] are_connected implies for b
7 being
Path of
[b3,b5],
[b4,b6] holds
pr1 b
7 is
Path of b
3,b
4 )
theorem Th14: :: TOPALG_4:14
for b
1, b
2 being non
empty TopSpacefor b
3, b
4 being
Point of b
1for b
5, b
6 being
Point of b
2 holds
(
[b3,b5],
[b4,b6] are_connected implies for b
7 being
Path of
[b3,b5],
[b4,b6] holds
pr2 b
7 is
Path of b
5,b
6 )
theorem Th15: :: TOPALG_4:15
theorem Th16: :: TOPALG_4:16
for b
1, b
2 being non
empty TopSpacefor b
3, b
4 being
Point of b
1for b
5, b
6 being
Point of b
2 holds
( b
3,b
4 are_connected & b
5,b
6 are_connected implies for b
7 being
Path of b
3,b
4for b
8 being
Path of b
5,b
6 holds
<:b7,b8:> is
Path of
[b3,b5],
[b4,b6] )
definition
let c
4, c
5 be non
empty arcwise_connected TopSpace;
let c
6, c
7 be
Point of c
4;
let c
8, c
9 be
Point of c
5;
let c
10 be
Path of c
6,c
7;
let c
11 be
Path of c
8,c
9;
redefine func <: as
<:c7,c8:> -> Path of
[a3,a5],
[a4,a6];
coherence
<:c10,c11:> is Path of [c6,c8],[c7,c9]
end;
definition
let c
4, c
5 be non
empty arcwise_connected TopSpace;
let c
6, c
7 be
Point of c
4;
let c
8, c
9 be
Point of c
5;
let c
10 be
Path of
[c6,c8],
[c7,c9];
redefine func pr1 as
pr1 c
7 -> Path of a
3,a
4;
coherence
pr1 c10 is Path of c6,c7
redefine func pr2 as
pr2 c
7 -> Path of a
5,a
6;
coherence
pr2 c10 is Path of c8,c9
end;
Lemma19:
for b1, b2 being non empty TopSpace
for b3, b4 being Point of b1
for b5, b6 being Point of b2
for b7, b8 being Path of [b3,b5],[b4,b6]
for b9 being Homotopy of b7,b8 holds
( b7,b8 are_homotopic implies ( pr1 b9 is continuous & ( for b10 being Point of I[01] holds
( (pr1 b9) . b10,0 = (pr1 b7) . b10 & (pr1 b9) . b10,1 = (pr1 b8) . b10 & ( for b11 being Point of I[01] holds
( (pr1 b9) . 0,b11 = b3 & (pr1 b9) . 1,b11 = b4 ) ) ) ) ) )
Lemma20:
for b1, b2 being non empty TopSpace
for b3, b4 being Point of b1
for b5, b6 being Point of b2
for b7, b8 being Path of [b3,b5],[b4,b6]
for b9 being Homotopy of b7,b8 holds
( b7,b8 are_homotopic implies ( pr2 b9 is continuous & ( for b10 being Point of I[01] holds
( (pr2 b9) . b10,0 = (pr2 b7) . b10 & (pr2 b9) . b10,1 = (pr2 b8) . b10 & ( for b11 being Point of I[01] holds
( (pr2 b9) . 0,b11 = b5 & (pr2 b9) . 1,b11 = b6 ) ) ) ) ) )
theorem Th17: :: TOPALG_4:17
for b
1, b
2 being non
empty TopSpacefor b
3, b
4 being
Point of b
1for b
5, b
6 being
Point of b
2for b
7, b
8 being
Path of
[b3,b5],
[b4,b6]for b
9 being
Homotopy of b
7,b
8for b
10, b
11 being
Path of b
3,b
4 holds
( b
10 = pr1 b
7 & b
11 = pr1 b
8 & b
7,b
8 are_homotopic implies
pr1 b
9 is
Homotopy of b
10,b
11 )
theorem Th18: :: TOPALG_4:18
for b
1, b
2 being non
empty TopSpacefor b
3, b
4 being
Point of b
1for b
5, b
6 being
Point of b
2for b
7, b
8 being
Path of
[b3,b5],
[b4,b6]for b
9 being
Homotopy of b
7,b
8for b
10, b
11 being
Path of b
5,b
6 holds
( b
10 = pr2 b
7 & b
11 = pr2 b
8 & b
7,b
8 are_homotopic implies
pr2 b
9 is
Homotopy of b
10,b
11 )
theorem Th19: :: TOPALG_4:19
for b
1, b
2 being non
empty TopSpacefor b
3, b
4 being
Point of b
1for b
5, b
6 being
Point of b
2for b
7, b
8 being
Path of
[b3,b5],
[b4,b6]for b
9, b
10 being
Path of b
3,b
4 holds
( b
9 = pr1 b
7 & b
10 = pr1 b
8 & b
7,b
8 are_homotopic implies b
9,b
10 are_homotopic )
theorem Th20: :: TOPALG_4:20
for b
1, b
2 being non
empty TopSpacefor b
3, b
4 being
Point of b
1for b
5, b
6 being
Point of b
2for b
7, b
8 being
Path of
[b3,b5],
[b4,b6]for b
9, b
10 being
Path of b
5,b
6 holds
( b
9 = pr2 b
7 & b
10 = pr2 b
8 & b
7,b
8 are_homotopic implies b
9,b
10 are_homotopic )
Lemma23:
for b1, b2 being non empty TopSpace
for b3, b4 being Point of b1
for b5, b6 being Point of b2
for b7, b8 being Path of [b3,b5],[b4,b6]
for b9, b10 being Path of b3,b4
for b11, b12 being Path of b5,b6
for b13 being Homotopy of b9,b10
for b14 being Homotopy of b11,b12 holds
( b9 = pr1 b7 & b10 = pr1 b8 & b11 = pr2 b7 & b12 = pr2 b8 & b9,b10 are_homotopic & b11,b12 are_homotopic implies ( <:b13,b14:> is continuous & ( for b15 being Point of I[01] holds
( <:b13,b14:> . b15,0 = b7 . b15 & <:b13,b14:> . b15,1 = b8 . b15 & ( for b16 being Point of I[01] holds
( <:b13,b14:> . 0,b16 = [b3,b5] & <:b13,b14:> . 1,b16 = [b4,b6] ) ) ) ) ) )
theorem Th21: :: TOPALG_4:21
for b
1, b
2 being non
empty TopSpacefor b
3, b
4 being
Point of b
1for b
5, b
6 being
Point of b
2for b
7, b
8 being
Path of
[b3,b5],
[b4,b6]for b
9, b
10 being
Path of b
3,b
4for b
11, b
12 being
Path of b
5,b
6for b
13 being
Homotopy of b
9,b
10for b
14 being
Homotopy of b
11,b
12 holds
( b
9 = pr1 b
7 & b
10 = pr1 b
8 & b
11 = pr2 b
7 & b
12 = pr2 b
8 & b
9,b
10 are_homotopic & b
11,b
12 are_homotopic implies
<:b13,b14:> is
Homotopy of b
7,b
8 )
theorem Th22: :: TOPALG_4:22
for b
1, b
2 being non
empty TopSpacefor b
3, b
4 being
Point of b
1for b
5, b
6 being
Point of b
2for b
7, b
8 being
Path of
[b3,b5],
[b4,b6]for b
9, b
10 being
Path of b
3,b
4for b
11, b
12 being
Path of b
5,b
6 holds
( b
9 = pr1 b
7 & b
10 = pr1 b
8 & b
11 = pr2 b
7 & b
12 = pr2 b
8 & b
9,b
10 are_homotopic & b
11,b
12 are_homotopic implies b
7,b
8 are_homotopic )
theorem Th23: :: TOPALG_4:23
for b
1, b
2 being non
empty TopSpacefor b
3, b
4, b
5 being
Point of b
1for b
6, b
7, b
8 being
Point of b
2for b
9 being
Path of
[b3,b6],
[b4,b7]for b
10 being
Path of
[b4,b7],
[b5,b8]for b
11 being
Path of b
3,b
4for b
12 being
Path of b
4,b
5 holds
(
[b3,b6],
[b4,b7] are_connected &
[b4,b7],
[b5,b8] are_connected & b
11 = pr1 b
9 & b
12 = pr1 b
10 implies
pr1 (b9 + b10) = b
11 + b
12 )
theorem Th24: :: TOPALG_4:24
for b
1, b
2 being non
empty arcwise_connected TopSpacefor b
3, b
4, b
5 being
Point of b
1for b
6, b
7, b
8 being
Point of b
2for b
9 being
Path of
[b3,b6],
[b4,b7]for b
10 being
Path of
[b4,b7],
[b5,b8] holds
pr1 (b9 + b10) = (pr1 b9) + (pr1 b10)
theorem Th25: :: TOPALG_4:25
for b
1, b
2 being non
empty TopSpacefor b
3, b
4, b
5 being
Point of b
1for b
6, b
7, b
8 being
Point of b
2for b
9 being
Path of
[b3,b6],
[b4,b7]for b
10 being
Path of
[b4,b7],
[b5,b8]for b
11 being
Path of b
6,b
7for b
12 being
Path of b
7,b
8 holds
(
[b3,b6],
[b4,b7] are_connected &
[b4,b7],
[b5,b8] are_connected & b
11 = pr2 b
9 & b
12 = pr2 b
10 implies
pr2 (b9 + b10) = b
11 + b
12 )
theorem Th26: :: TOPALG_4:26
for b
1, b
2 being non
empty arcwise_connected TopSpacefor b
3, b
4, b
5 being
Point of b
1for b
6, b
7, b
8 being
Point of b
2for b
9 being
Path of
[b3,b6],
[b4,b7]for b
10 being
Path of
[b4,b7],
[b5,b8] holds
pr2 (b9 + b10) = (pr2 b9) + (pr2 b10)
definition
let c
4, c
5 be non
empty TopSpace;
let c
6 be
Point of c
4;
let c
7 be
Point of c
5;
set c
8 =
pi_1 [:c4,c5:],
[c6,c7];
set c
9 =
<*(pi_1 c4,c6),(pi_1 c5,c7)*>;
set c
10 =
product <*(pi_1 c4,c6),(pi_1 c5,c7)*>;
func FGPrIso c
3,c
4 -> Function of
(pi_1 [:a1,a2:],[a3,a4]),
(product <*(pi_1 a1,a3),(pi_1 a2,a4)*>) means :
Def2:
:: TOPALG_4:def 2
for b
1 being
Point of
(pi_1 [:a1,a2:],[a3,a4]) holds
ex b
2 being
Loop of
[a3,a4] st
( b
1 = Class (EqRel [:a1,a2:],[a3,a4]),b
2 & a
5 . b
1 = <*(Class (EqRel a1,a3),(pr1 b2)),(Class (EqRel a2,a4),(pr2 b2))*> );
existence
ex b1 being Function of (pi_1 [:c4,c5:],[c6,c7]),(product <*(pi_1 c4,c6),(pi_1 c5,c7)*>) st
for b2 being Point of (pi_1 [:c4,c5:],[c6,c7]) holds
ex b3 being Loop of [c6,c7] st
( b2 = Class (EqRel [:c4,c5:],[c6,c7]),b3 & b1 . b2 = <*(Class (EqRel c4,c6),(pr1 b3)),(Class (EqRel c5,c7),(pr2 b3))*> )
uniqueness
for b1, b2 being Function of (pi_1 [:c4,c5:],[c6,c7]),(product <*(pi_1 c4,c6),(pi_1 c5,c7)*>) holds
( ( for b3 being Point of (pi_1 [:c4,c5:],[c6,c7]) holds
ex b4 being Loop of [c6,c7] st
( b3 = Class (EqRel [:c4,c5:],[c6,c7]),b4 & b1 . b3 = <*(Class (EqRel c4,c6),(pr1 b4)),(Class (EqRel c5,c7),(pr2 b4))*> ) ) & ( for b3 being Point of (pi_1 [:c4,c5:],[c6,c7]) holds
ex b4 being Loop of [c6,c7] st
( b3 = Class (EqRel [:c4,c5:],[c6,c7]),b4 & b2 . b3 = <*(Class (EqRel c4,c6),(pr1 b4)),(Class (EqRel c5,c7),(pr2 b4))*> ) ) implies b1 = b2 )
end;
:: deftheorem Def2 defines FGPrIso TOPALG_4:def 2 :
for b
1, b
2 being non
empty TopSpacefor b
3 being
Point of b
1for b
4 being
Point of b
2for b
5 being
Function of
(pi_1 [:b1,b2:],[b3,b4]),
(product <*(pi_1 b1,b3),(pi_1 b2,b4)*>) holds
( b
5 = FGPrIso b
3,b
4 iff for b
6 being
Point of
(pi_1 [:b1,b2:],[b3,b4]) holds
ex b
7 being
Loop of
[b3,b4] st
( b
6 = Class (EqRel [:b1,b2:],[b3,b4]),b
7 & b
5 . b
6 = <*(Class (EqRel b1,b3),(pr1 b7)),(Class (EqRel b2,b4),(pr2 b7))*> ) );
theorem Th27: :: TOPALG_4:27
for b
1, b
2 being non
empty TopSpacefor b
3 being
Point of b
1for b
4 being
Point of b
2for b
5 being
Point of
(pi_1 [:b1,b2:],[b3,b4])for b
6 being
Loop of
[b3,b4] holds
( b
5 = Class (EqRel [:b1,b2:],[b3,b4]),b
6 implies
(FGPrIso b3,b4) . b
5 = <*(Class (EqRel b1,b3),(pr1 b6)),(Class (EqRel b2,b4),(pr2 b6))*> )
theorem Th28: :: TOPALG_4:28
for b
1, b
2 being non
empty TopSpacefor b
3 being
Point of b
1for b
4 being
Point of b
2for b
5 being
Loop of
[b3,b4] holds
(FGPrIso b3,b4) . (Class (EqRel [:b1,b2:],[b3,b4]),b5) = <*(Class (EqRel b1,b3),(pr1 b5)),(Class (EqRel b2,b4),(pr2 b5))*>
definition
let c
4, c
5 be non
empty TopSpace;
let c
6 be
Point of c
4;
let c
7 be
Point of c
5;
redefine func FGPrIso as
FGPrIso c
3,c
4 -> Homomorphism of
(pi_1 [:a1,a2:],[a3,a4]),
(product <*(pi_1 a1,a3),(pi_1 a2,a4)*>);
coherence
FGPrIso c6,c7 is Homomorphism of (pi_1 [:c4,c5:],[c6,c7]),(product <*(pi_1 c4,c6),(pi_1 c5,c7)*>)
end;
theorem Th29: :: TOPALG_4:29
theorem Th30: :: TOPALG_4:30
theorem Th31: :: TOPALG_4:31
for b
1, b
2 being non
empty TopSpacefor b
3, b
4 being
Point of b
1for b
5, b
6 being
Point of b
2for b
7 being
Homomorphism of
(pi_1 b1,b3),
(pi_1 b1,b4)for b
8 being
Homomorphism of
(pi_1 b2,b5),
(pi_1 b2,b6) holds
( b
7 is_isomorphism & b
8 is_isomorphism implies
(Gr2Iso b7,b8) * (FGPrIso b3,b5) is_isomorphism )
theorem Th32: :: TOPALG_4:32
for b
1, b
2 being non
empty arcwise_connected TopSpacefor b
3, b
4 being
Point of b
1for b
5, b
6 being
Point of b
2 holds
pi_1 [:b1,b2:],
[b3,b5],
product <*(pi_1 b1,b4),(pi_1 b2,b6)*> are_isomorphic