:: 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 c1,
c2,
c3,
c4 be non
empty HGrStr ;
let c5 be
Function of
c1,
c3;
let c6 be
Function of
c2,
c4;
func Gr2Iso c5,
c6 -> Function of
(product <*a1,a2*>),
(product <*a3,a4*>) means :
Def1:
:: TOPALG_4:def 1
for
b1 being
Element of
(product <*a1,a2*>)ex
b2 being
Element of
a1ex
b3 being
Element of
a2 st
(
b1 = <*b2,b3*> &
a7 . b1 = <*(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*>)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*>) st ( for b3 being Element of (product <*c1,c2*>)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*>)ex b4 being Element of c1ex b5 being Element of c2 st
( b3 = <*b4,b5*> & b2 . b3 = <*(c5 . b4),(c6 . b5)*> ) ) holds
b1 = b2
end;
:: deftheorem Def1 defines Gr2Iso TOPALG_4:def 1 :
for
b1,
b2,
b3,
b4 being non
empty HGrStr for
b5 being
Function of
b1,
b3for
b6 being
Function of
b2,
b4for
b7 being
Function of
(product <*b1,b2*>),
(product <*b3,b4*>) holds
(
b7 = Gr2Iso b5,
b6 iff for
b8 being
Element of
(product <*b1,b2*>)ex
b9 being
Element of
b1ex
b10 being
Element of
b2 st
(
b8 = <*b9,b10*> &
b7 . b8 = <*(b5 . b9),(b6 . b10)*> ) );
theorem Th2: :: TOPALG_4:2
definition
let c1,
c2,
c3,
c4 be
Group;
let c5 be
Homomorphism of
c1,
c3;
let c6 be
Homomorphism of
c2,
c4;
redefine func Gr2Iso as
Gr2Iso c5,
c6 -> 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 c4,
c5,
c6 be non
empty TopSpace;
let c7 be
Function of
c6,
c4;
let c8 be
Function of
c6,
c5;
redefine func <: as
<:c4,c5:> -> Function of
a3,
[: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
b1,
b2 being non
empty TopSpacefor
b3,
b4 being
Point of
b1for
b5,
b6 being
Point of
b2 st
[b3,b5],
[b4,b6] are_connected holds
for
b7 being
Path of
[b3,b5],
[b4,b6] holds
pr1 b7 is
Path of
b3,
b4
theorem Th14: :: TOPALG_4:14
for
b1,
b2 being non
empty TopSpacefor
b3,
b4 being
Point of
b1for
b5,
b6 being
Point of
b2 st
[b3,b5],
[b4,b6] are_connected holds
for
b7 being
Path of
[b3,b5],
[b4,b6] holds
pr2 b7 is
Path of
b5,
b6
theorem Th15: :: TOPALG_4:15
theorem Th16: :: TOPALG_4:16
for
b1,
b2 being non
empty TopSpacefor
b3,
b4 being
Point of
b1for
b5,
b6 being
Point of
b2 st
b3,
b4 are_connected &
b5,
b6 are_connected holds
for
b7 being
Path of
b3,
b4for
b8 being
Path of
b5,
b6 holds
<:b7,b8:> is
Path of
[b3,b5],
[b4,b6]
definition
let c4,
c5 be non
empty arcwise_connected TopSpace;
let c6,
c7 be
Point of
c4;
let c8,
c9 be
Point of
c5;
let c10 be
Path of
c6,
c7;
let c11 be
Path of
c8,
c9;
redefine func <: as
<:c7,c8:> -> Path of
[a3,a5],
[a4,a6];
coherence
<:c10,c11:> is Path of [c6,c8],[c7,c9]
end;
definition
let c4,
c5 be non
empty arcwise_connected TopSpace;
let c6,
c7 be
Point of
c4;
let c8,
c9 be
Point of
c5;
let c10 be
Path of
[c6,c8],
[c7,c9];
redefine func pr1 as
pr1 c7 -> Path of
a3,
a4;
coherence
pr1 c10 is Path of c6,c7
redefine func pr2 as
pr2 c7 -> Path of
a5,
a6;
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 st b7,b8 are_homotopic holds
( 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 st b7,b8 are_homotopic holds
( 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
b1,
b2 being non
empty TopSpacefor
b3,
b4 being
Point of
b1for
b5,
b6 being
Point of
b2for
b7,
b8 being
Path of
[b3,b5],
[b4,b6]for
b9 being
Homotopy of
b7,
b8for
b10,
b11 being
Path of
b3,
b4 st
b10 = pr1 b7 &
b11 = pr1 b8 &
b7,
b8 are_homotopic holds
pr1 b9 is
Homotopy of
b10,
b11
theorem Th18: :: TOPALG_4:18
for
b1,
b2 being non
empty TopSpacefor
b3,
b4 being
Point of
b1for
b5,
b6 being
Point of
b2for
b7,
b8 being
Path of
[b3,b5],
[b4,b6]for
b9 being
Homotopy of
b7,
b8for
b10,
b11 being
Path of
b5,
b6 st
b10 = pr2 b7 &
b11 = pr2 b8 &
b7,
b8 are_homotopic holds
pr2 b9 is
Homotopy of
b10,
b11
theorem Th19: :: TOPALG_4:19
for
b1,
b2 being non
empty TopSpacefor
b3,
b4 being
Point of
b1for
b5,
b6 being
Point of
b2for
b7,
b8 being
Path of
[b3,b5],
[b4,b6]for
b9,
b10 being
Path of
b3,
b4 st
b9 = pr1 b7 &
b10 = pr1 b8 &
b7,
b8 are_homotopic holds
b9,
b10 are_homotopic
theorem Th20: :: TOPALG_4:20
for
b1,
b2 being non
empty TopSpacefor
b3,
b4 being
Point of
b1for
b5,
b6 being
Point of
b2for
b7,
b8 being
Path of
[b3,b5],
[b4,b6]for
b9,
b10 being
Path of
b5,
b6 st
b9 = pr2 b7 &
b10 = pr2 b8 &
b7,
b8 are_homotopic holds
b9,
b10 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 st b9 = pr1 b7 & b10 = pr1 b8 & b11 = pr2 b7 & b12 = pr2 b8 & b9,b10 are_homotopic & b11,b12 are_homotopic holds
( <: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
b1,
b2 being non
empty TopSpacefor
b3,
b4 being
Point of
b1for
b5,
b6 being
Point of
b2for
b7,
b8 being
Path of
[b3,b5],
[b4,b6]for
b9,
b10 being
Path of
b3,
b4for
b11,
b12 being
Path of
b5,
b6for
b13 being
Homotopy of
b9,
b10for
b14 being
Homotopy of
b11,
b12 st
b9 = pr1 b7 &
b10 = pr1 b8 &
b11 = pr2 b7 &
b12 = pr2 b8 &
b9,
b10 are_homotopic &
b11,
b12 are_homotopic holds
<:b13,b14:> is
Homotopy of
b7,
b8
theorem Th22: :: TOPALG_4:22
for
b1,
b2 being non
empty TopSpacefor
b3,
b4 being
Point of
b1for
b5,
b6 being
Point of
b2for
b7,
b8 being
Path of
[b3,b5],
[b4,b6]for
b9,
b10 being
Path of
b3,
b4for
b11,
b12 being
Path of
b5,
b6 st
b9 = pr1 b7 &
b10 = pr1 b8 &
b11 = pr2 b7 &
b12 = pr2 b8 &
b9,
b10 are_homotopic &
b11,
b12 are_homotopic holds
b7,
b8 are_homotopic
theorem Th23: :: TOPALG_4:23
for
b1,
b2 being non
empty TopSpacefor
b3,
b4,
b5 being
Point of
b1for
b6,
b7,
b8 being
Point of
b2for
b9 being
Path of
[b3,b6],
[b4,b7]for
b10 being
Path of
[b4,b7],
[b5,b8]for
b11 being
Path of
b3,
b4for
b12 being
Path of
b4,
b5 st
[b3,b6],
[b4,b7] are_connected &
[b4,b7],
[b5,b8] are_connected &
b11 = pr1 b9 &
b12 = pr1 b10 holds
pr1 (b9 + b10) = b11 + b12
theorem Th24: :: TOPALG_4:24
for
b1,
b2 being non
empty arcwise_connected TopSpacefor
b3,
b4,
b5 being
Point of
b1for
b6,
b7,
b8 being
Point of
b2for
b9 being
Path of
[b3,b6],
[b4,b7]for
b10 being
Path of
[b4,b7],
[b5,b8] holds
pr1 (b9 + b10) = (pr1 b9) + (pr1 b10)
theorem Th25: :: TOPALG_4:25
for
b1,
b2 being non
empty TopSpacefor
b3,
b4,
b5 being
Point of
b1for
b6,
b7,
b8 being
Point of
b2for
b9 being
Path of
[b3,b6],
[b4,b7]for
b10 being
Path of
[b4,b7],
[b5,b8]for
b11 being
Path of
b6,
b7for
b12 being
Path of
b7,
b8 st
[b3,b6],
[b4,b7] are_connected &
[b4,b7],
[b5,b8] are_connected &
b11 = pr2 b9 &
b12 = pr2 b10 holds
pr2 (b9 + b10) = b11 + b12
theorem Th26: :: TOPALG_4:26
for
b1,
b2 being non
empty arcwise_connected TopSpacefor
b3,
b4,
b5 being
Point of
b1for
b6,
b7,
b8 being
Point of
b2for
b9 being
Path of
[b3,b6],
[b4,b7]for
b10 being
Path of
[b4,b7],
[b5,b8] holds
pr2 (b9 + b10) = (pr2 b9) + (pr2 b10)
definition
let c4,
c5 be non
empty TopSpace;
let c6 be
Point of
c4;
let c7 be
Point of
c5;
set c8 =
pi_1 [:c4,c5:],
[c6,c7];
set c9 =
<*(pi_1 c4,c6),(pi_1 c5,c7)*>;
set c10 =
product <*(pi_1 c4,c6),(pi_1 c5,c7)*>;
func FGPrIso c3,
c4 -> Function of
(pi_1 [:a1,a2:],[a3,a4]),
(product <*(pi_1 a1,a3),(pi_1 a2,a4)*>) means :
Def2:
:: TOPALG_4:def 2
for
b1 being
Point of
(pi_1 [:a1,a2:],[a3,a4])ex
b2 being
Loop of
[a3,a4] st
(
b1 = Class (EqRel [:a1,a2:],[a3,a4]),
b2 &
a5 . b1 = <*(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])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)*>) st ( for b3 being Point of (pi_1 [:c4,c5:],[c6,c7])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])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))*> ) ) holds
b1 = b2
end;
:: deftheorem Def2 defines FGPrIso TOPALG_4:def 2 :
for
b1,
b2 being non
empty TopSpacefor
b3 being
Point of
b1for
b4 being
Point of
b2for
b5 being
Function of
(pi_1 [:b1,b2:],[b3,b4]),
(product <*(pi_1 b1,b3),(pi_1 b2,b4)*>) holds
(
b5 = FGPrIso b3,
b4 iff for
b6 being
Point of
(pi_1 [:b1,b2:],[b3,b4])ex
b7 being
Loop of
[b3,b4] st
(
b6 = Class (EqRel [:b1,b2:],[b3,b4]),
b7 &
b5 . b6 = <*(Class (EqRel b1,b3),(pr1 b7)),(Class (EqRel b2,b4),(pr2 b7))*> ) );
theorem Th27: :: TOPALG_4:27
for
b1,
b2 being non
empty TopSpacefor
b3 being
Point of
b1for
b4 being
Point of
b2for
b5 being
Point of
(pi_1 [:b1,b2:],[b3,b4])for
b6 being
Loop of
[b3,b4] st
b5 = Class (EqRel [:b1,b2:],[b3,b4]),
b6 holds
(FGPrIso b3,b4) . b5 = <*(Class (EqRel b1,b3),(pr1 b6)),(Class (EqRel b2,b4),(pr2 b6))*>
theorem Th28: :: TOPALG_4:28
for
b1,
b2 being non
empty TopSpacefor
b3 being
Point of
b1for
b4 being
Point of
b2for
b5 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 c4,
c5 be non
empty TopSpace;
let c6 be
Point of
c4;
let c7 be
Point of
c5;
redefine func FGPrIso as
FGPrIso c3,
c4 -> 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
b1,
b2 being non
empty TopSpacefor
b3,
b4 being
Point of
b1for
b5,
b6 being
Point of
b2for
b7 being
Homomorphism of
(pi_1 b1,b3),
(pi_1 b1,b4)for
b8 being
Homomorphism of
(pi_1 b2,b5),
(pi_1 b2,b6) st
b7 is_isomorphism &
b8 is_isomorphism holds
(Gr2Iso b7,b8) * (FGPrIso b3,b5) is_isomorphism
theorem Th32: :: TOPALG_4:32
for
b1,
b2 being non
empty arcwise_connected TopSpacefor
b3,
b4 being
Point of
b1for
b5,
b6 being
Point of
b2 holds
pi_1 [:b1,b2:],
[b3,b5],
product <*(pi_1 b1,b4),(pi_1 b2,b6)*> are_isomorphic