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Theorem pi1xfrcnv 21974
Description: Given a path  F between two basepoints, there is an induced group homomorphism on the fundamental groups. (Contributed by Mario Carneiro, 12-Feb-2015.) (Proof shortened by Mario Carneiro, 23-Dec-2016.)
Hypotheses
Ref Expression
pi1xfr.p  |-  P  =  ( J  pi1 
( F `  0
) )
pi1xfr.q  |-  Q  =  ( J  pi1 
( F `  1
) )
pi1xfr.b  |-  B  =  ( Base `  P
)
pi1xfr.g  |-  G  =  ran  ( g  e. 
U. B  |->  <. [ g ] (  ~=ph  `  J
) ,  [ ( I ( *p `  J ) ( g ( *p `  J
) F ) ) ] (  ~=ph  `  J
) >. )
pi1xfr.j  |-  ( ph  ->  J  e.  (TopOn `  X ) )
pi1xfr.f  |-  ( ph  ->  F  e.  ( II 
Cn  J ) )
pi1xfr.i  |-  I  =  ( x  e.  ( 0 [,] 1 ) 
|->  ( F `  (
1  -  x ) ) )
pi1xfrcnv.h  |-  H  =  ran  ( h  e. 
U. ( Base `  Q
)  |->  <. [ h ]
(  ~=ph  `  J ) ,  [ ( F ( *p `  J ) ( h ( *p
`  J ) I ) ) ] ( 
~=ph  `  J ) >.
)
Assertion
Ref Expression
pi1xfrcnv  |-  ( ph  ->  ( `' G  =  H  /\  `' G  e.  ( Q  GrpHom  P ) ) )
Distinct variable groups:    g, h, x, B    g, F, h, x    g, I, h, x    h, G    ph, g, h, x    g, J, h, x    P, g, h, x    Q, g, h, x
Allowed substitution hints:    G( x, g)    H( x, g, h)    X( x, g, h)

Proof of Theorem pi1xfrcnv
Dummy variables  y 
z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 pi1xfr.p . . . 4  |-  P  =  ( J  pi1 
( F `  0
) )
2 pi1xfr.q . . . 4  |-  Q  =  ( J  pi1 
( F `  1
) )
3 pi1xfr.b . . . 4  |-  B  =  ( Base `  P
)
4 pi1xfr.g . . . 4  |-  G  =  ran  ( g  e. 
U. B  |->  <. [ g ] (  ~=ph  `  J
) ,  [ ( I ( *p `  J ) ( g ( *p `  J
) F ) ) ] (  ~=ph  `  J
) >. )
5 pi1xfr.j . . . 4  |-  ( ph  ->  J  e.  (TopOn `  X ) )
6 pi1xfr.f . . . 4  |-  ( ph  ->  F  e.  ( II 
Cn  J ) )
7 pi1xfr.i . . . 4  |-  I  =  ( x  e.  ( 0 [,] 1 ) 
|->  ( F `  (
1  -  x ) ) )
8 pi1xfrcnv.h . . . 4  |-  H  =  ran  ( h  e. 
U. ( Base `  Q
)  |->  <. [ h ]
(  ~=ph  `  J ) ,  [ ( F ( *p `  J ) ( h ( *p
`  J ) I ) ) ] ( 
~=ph  `  J ) >.
)
91, 2, 3, 4, 5, 6, 7, 8pi1xfrcnvlem 21973 . . 3  |-  ( ph  ->  `' G  C_  H )
10 fvex 5882 . . . . . . . 8  |-  (  ~=ph  `  J )  e.  _V
11 ecexg 7366 . . . . . . . 8  |-  ( ( 
~=ph  `  J )  e. 
_V  ->  [ h ]
(  ~=ph  `  J )  e.  _V )
1210, 11mp1i 13 . . . . . . 7  |-  ( (
ph  /\  h  e.  U. ( Base `  Q
) )  ->  [ h ] (  ~=ph  `  J
)  e.  _V )
13 ecexg 7366 . . . . . . . 8  |-  ( ( 
~=ph  `  J )  e. 
_V  ->  [ ( F ( *p `  J
) ( h ( *p `  J ) I ) ) ] (  ~=ph  `  J )  e.  _V )
1410, 13mp1i 13 . . . . . . 7  |-  ( (
ph  /\  h  e.  U. ( Base `  Q
) )  ->  [ ( F ( *p `  J ) ( h ( *p `  J
) I ) ) ] (  ~=ph  `  J
)  e.  _V )
158, 12, 14fliftrel 6207 . . . . . 6  |-  ( ph  ->  H  C_  ( _V  X.  _V ) )
16 df-rel 4852 . . . . . 6  |-  ( Rel 
H  <->  H  C_  ( _V 
X.  _V ) )
1715, 16sylibr 215 . . . . 5  |-  ( ph  ->  Rel  H )
18 dfrel2 5297 . . . . 5  |-  ( Rel 
H  <->  `' `' H  =  H
)
1917, 18sylib 199 . . . 4  |-  ( ph  ->  `' `' H  =  H
)
20 0elunit 11737 . . . . . . . . . 10  |-  0  e.  ( 0 [,] 1
)
21 oveq2 6304 . . . . . . . . . . . . 13  |-  ( x  =  0  ->  (
1  -  x )  =  ( 1  -  0 ) )
22 1m0e1 10709 . . . . . . . . . . . . 13  |-  ( 1  -  0 )  =  1
2321, 22syl6eq 2477 . . . . . . . . . . . 12  |-  ( x  =  0  ->  (
1  -  x )  =  1 )
2423fveq2d 5876 . . . . . . . . . . 11  |-  ( x  =  0  ->  ( F `  ( 1  -  x ) )  =  ( F `  1
) )
25 fvex 5882 . . . . . . . . . . 11  |-  ( F `
 1 )  e. 
_V
2624, 7, 25fvmpt 5955 . . . . . . . . . 10  |-  ( 0  e.  ( 0 [,] 1 )  ->  (
I `  0 )  =  ( F ` 
1 ) )
2720, 26ax-mp 5 . . . . . . . . 9  |-  ( I `
 0 )  =  ( F `  1
)
2827oveq2i 6307 . . . . . . . 8  |-  ( J  pi1  ( I `
 0 ) )  =  ( J  pi1  ( F ` 
1 ) )
292, 28eqtr4i 2452 . . . . . . 7  |-  Q  =  ( J  pi1 
( I `  0
) )
30 1elunit 11738 . . . . . . . . . 10  |-  1  e.  ( 0 [,] 1
)
31 oveq2 6304 . . . . . . . . . . . . 13  |-  ( x  =  1  ->  (
1  -  x )  =  ( 1  -  1 ) )
3231fveq2d 5876 . . . . . . . . . . . 12  |-  ( x  =  1  ->  ( F `  ( 1  -  x ) )  =  ( F `  (
1  -  1 ) ) )
33 1m1e0 10667 . . . . . . . . . . . . 13  |-  ( 1  -  1 )  =  0
3433fveq2i 5875 . . . . . . . . . . . 12  |-  ( F `
 ( 1  -  1 ) )  =  ( F `  0
)
3532, 34syl6eq 2477 . . . . . . . . . . 11  |-  ( x  =  1  ->  ( F `  ( 1  -  x ) )  =  ( F `  0
) )
36 fvex 5882 . . . . . . . . . . 11  |-  ( F `
 0 )  e. 
_V
3735, 7, 36fvmpt 5955 . . . . . . . . . 10  |-  ( 1  e.  ( 0 [,] 1 )  ->  (
I `  1 )  =  ( F ` 
0 ) )
3830, 37ax-mp 5 . . . . . . . . 9  |-  ( I `
 1 )  =  ( F `  0
)
3938oveq2i 6307 . . . . . . . 8  |-  ( J  pi1  ( I `
 1 ) )  =  ( J  pi1  ( F ` 
0 ) )
401, 39eqtr4i 2452 . . . . . . 7  |-  P  =  ( J  pi1 
( I `  1
) )
41 eqid 2420 . . . . . . 7  |-  ( Base `  Q )  =  (
Base `  Q )
42 eqid 2420 . . . . . . 7  |-  ran  (
h  e.  U. ( Base `  Q )  |->  <. [ h ] ( 
~=ph  `  J ) ,  [ ( ( z  e.  ( 0 [,] 1 )  |->  ( I `
 ( 1  -  z ) ) ) ( *p `  J
) ( h ( *p `  J ) I ) ) ] (  ~=ph  `  J )
>. )  =  ran  ( h  e.  U. ( Base `  Q )  |->  <. [ h ] ( 
~=ph  `  J ) ,  [ ( ( z  e.  ( 0 [,] 1 )  |->  ( I `
 ( 1  -  z ) ) ) ( *p `  J
) ( h ( *p `  J ) I ) ) ] (  ~=ph  `  J )
>. )
437pcorevcl 21942 . . . . . . . . 9  |-  ( F  e.  ( II  Cn  J )  ->  (
I  e.  ( II 
Cn  J )  /\  ( I `  0
)  =  ( F `
 1 )  /\  ( I `  1
)  =  ( F `
 0 ) ) )
446, 43syl 17 . . . . . . . 8  |-  ( ph  ->  ( I  e.  ( II  Cn  J )  /\  ( I ` 
0 )  =  ( F `  1 )  /\  ( I ` 
1 )  =  ( F `  0 ) ) )
4544simp1d 1017 . . . . . . 7  |-  ( ph  ->  I  e.  ( II 
Cn  J ) )
46 oveq2 6304 . . . . . . . . 9  |-  ( z  =  y  ->  (
1  -  z )  =  ( 1  -  y ) )
4746fveq2d 5876 . . . . . . . 8  |-  ( z  =  y  ->  (
I `  ( 1  -  z ) )  =  ( I `  ( 1  -  y
) ) )
4847cbvmptv 4509 . . . . . . 7  |-  ( z  e.  ( 0 [,] 1 )  |->  ( I `
 ( 1  -  z ) ) )  =  ( y  e.  ( 0 [,] 1
)  |->  ( I `  ( 1  -  y
) ) )
49 eqid 2420 . . . . . . 7  |-  ran  (
g  e.  U. ( Base `  P )  |->  <. [ g ] ( 
~=ph  `  J ) ,  [ ( I ( *p `  J ) ( g ( *p
`  J ) ( z  e.  ( 0 [,] 1 )  |->  ( I `  ( 1  -  z ) ) ) ) ) ] (  ~=ph  `  J )
>. )  =  ran  ( g  e.  U. ( Base `  P )  |-> 
<. [ g ] ( 
~=ph  `  J ) ,  [ ( I ( *p `  J ) ( g ( *p
`  J ) ( z  e.  ( 0 [,] 1 )  |->  ( I `  ( 1  -  z ) ) ) ) ) ] (  ~=ph  `  J )
>. )
5029, 40, 41, 42, 5, 45, 48, 49pi1xfrcnvlem 21973 . . . . . 6  |-  ( ph  ->  `' ran  ( h  e. 
U. ( Base `  Q
)  |->  <. [ h ]
(  ~=ph  `  J ) ,  [ ( ( z  e.  ( 0 [,] 1 )  |->  ( I `
 ( 1  -  z ) ) ) ( *p `  J
) ( h ( *p `  J ) I ) ) ] (  ~=ph  `  J )
>. )  C_  ran  (
g  e.  U. ( Base `  P )  |->  <. [ g ] ( 
~=ph  `  J ) ,  [ ( I ( *p `  J ) ( g ( *p
`  J ) ( z  e.  ( 0 [,] 1 )  |->  ( I `  ( 1  -  z ) ) ) ) ) ] (  ~=ph  `  J )
>. ) )
51 iitopon 21800 . . . . . . . . . . . . . . . . 17  |-  II  e.  (TopOn `  ( 0 [,] 1 ) )
5251a1i 11 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  II  e.  (TopOn `  ( 0 [,] 1
) ) )
53 cnf2 20189 . . . . . . . . . . . . . . . 16  |-  ( ( II  e.  (TopOn `  ( 0 [,] 1
) )  /\  J  e.  (TopOn `  X )  /\  F  e.  (
II  Cn  J )
)  ->  F :
( 0 [,] 1
) --> X )
5452, 5, 6, 53syl3anc 1264 . . . . . . . . . . . . . . 15  |-  ( ph  ->  F : ( 0 [,] 1 ) --> X )
5554feqmptd 5925 . . . . . . . . . . . . . 14  |-  ( ph  ->  F  =  ( z  e.  ( 0 [,] 1 )  |->  ( F `
 z ) ) )
56 iirev 21846 . . . . . . . . . . . . . . . . 17  |-  ( z  e.  ( 0 [,] 1 )  ->  (
1  -  z )  e.  ( 0 [,] 1 ) )
57 oveq2 6304 . . . . . . . . . . . . . . . . . . 19  |-  ( x  =  ( 1  -  z )  ->  (
1  -  x )  =  ( 1  -  ( 1  -  z
) ) )
5857fveq2d 5876 . . . . . . . . . . . . . . . . . 18  |-  ( x  =  ( 1  -  z )  ->  ( F `  ( 1  -  x ) )  =  ( F `  (
1  -  ( 1  -  z ) ) ) )
59 fvex 5882 . . . . . . . . . . . . . . . . . 18  |-  ( F `
 ( 1  -  ( 1  -  z
) ) )  e. 
_V
6058, 7, 59fvmpt 5955 . . . . . . . . . . . . . . . . 17  |-  ( ( 1  -  z )  e.  ( 0 [,] 1 )  ->  (
I `  ( 1  -  z ) )  =  ( F `  ( 1  -  (
1  -  z ) ) ) )
6156, 60syl 17 . . . . . . . . . . . . . . . 16  |-  ( z  e.  ( 0 [,] 1 )  ->  (
I `  ( 1  -  z ) )  =  ( F `  ( 1  -  (
1  -  z ) ) ) )
62 ax-1cn 9586 . . . . . . . . . . . . . . . . . 18  |-  1  e.  CC
63 unitssre 11766 . . . . . . . . . . . . . . . . . . . 20  |-  ( 0 [,] 1 )  C_  RR
6463sseli 3457 . . . . . . . . . . . . . . . . . . 19  |-  ( z  e.  ( 0 [,] 1 )  ->  z  e.  RR )
6564recnd 9658 . . . . . . . . . . . . . . . . . 18  |-  ( z  e.  ( 0 [,] 1 )  ->  z  e.  CC )
66 nncan 9892 . . . . . . . . . . . . . . . . . 18  |-  ( ( 1  e.  CC  /\  z  e.  CC )  ->  ( 1  -  (
1  -  z ) )  =  z )
6762, 65, 66sylancr 667 . . . . . . . . . . . . . . . . 17  |-  ( z  e.  ( 0 [,] 1 )  ->  (
1  -  ( 1  -  z ) )  =  z )
6867fveq2d 5876 . . . . . . . . . . . . . . . 16  |-  ( z  e.  ( 0 [,] 1 )  ->  ( F `  ( 1  -  ( 1  -  z ) ) )  =  ( F `  z ) )
6961, 68eqtrd 2461 . . . . . . . . . . . . . . 15  |-  ( z  e.  ( 0 [,] 1 )  ->  (
I `  ( 1  -  z ) )  =  ( F `  z ) )
7069mpteq2ia 4499 . . . . . . . . . . . . . 14  |-  ( z  e.  ( 0 [,] 1 )  |->  ( I `
 ( 1  -  z ) ) )  =  ( z  e.  ( 0 [,] 1
)  |->  ( F `  z ) )
7155, 70syl6eqr 2479 . . . . . . . . . . . . 13  |-  ( ph  ->  F  =  ( z  e.  ( 0 [,] 1 )  |->  ( I `
 ( 1  -  z ) ) ) )
7271oveq1d 6311 . . . . . . . . . . . 12  |-  ( ph  ->  ( F ( *p
`  J ) ( h ( *p `  J ) I ) )  =  ( ( z  e.  ( 0 [,] 1 )  |->  ( I `  ( 1  -  z ) ) ) ( *p `  J ) ( h ( *p `  J
) I ) ) )
7372eceq1d 7399 . . . . . . . . . . 11  |-  ( ph  ->  [ ( F ( *p `  J ) ( h ( *p
`  J ) I ) ) ] ( 
~=ph  `  J )  =  [ ( ( z  e.  ( 0 [,] 1 )  |->  ( I `
 ( 1  -  z ) ) ) ( *p `  J
) ( h ( *p `  J ) I ) ) ] (  ~=ph  `  J ) )
7473opeq2d 4188 . . . . . . . . . 10  |-  ( ph  -> 
<. [ h ] ( 
~=ph  `  J ) ,  [ ( F ( *p `  J ) ( h ( *p
`  J ) I ) ) ] ( 
~=ph  `  J ) >.  =  <. [ h ]
(  ~=ph  `  J ) ,  [ ( ( z  e.  ( 0 [,] 1 )  |->  ( I `
 ( 1  -  z ) ) ) ( *p `  J
) ( h ( *p `  J ) I ) ) ] (  ~=ph  `  J )
>. )
7574mpteq2dv 4504 . . . . . . . . 9  |-  ( ph  ->  ( h  e.  U. ( Base `  Q )  |-> 
<. [ h ] ( 
~=ph  `  J ) ,  [ ( F ( *p `  J ) ( h ( *p
`  J ) I ) ) ] ( 
~=ph  `  J ) >.
)  =  ( h  e.  U. ( Base `  Q )  |->  <. [ h ] (  ~=ph  `  J
) ,  [ ( ( z  e.  ( 0 [,] 1 ) 
|->  ( I `  (
1  -  z ) ) ) ( *p
`  J ) ( h ( *p `  J ) I ) ) ] (  ~=ph  `  J ) >. )
)
7675rneqd 5073 . . . . . . . 8  |-  ( ph  ->  ran  ( h  e. 
U. ( Base `  Q
)  |->  <. [ h ]
(  ~=ph  `  J ) ,  [ ( F ( *p `  J ) ( h ( *p
`  J ) I ) ) ] ( 
~=ph  `  J ) >.
)  =  ran  (
h  e.  U. ( Base `  Q )  |->  <. [ h ] ( 
~=ph  `  J ) ,  [ ( ( z  e.  ( 0 [,] 1 )  |->  ( I `
 ( 1  -  z ) ) ) ( *p `  J
) ( h ( *p `  J ) I ) ) ] (  ~=ph  `  J )
>. ) )
778, 76syl5eq 2473 . . . . . . 7  |-  ( ph  ->  H  =  ran  (
h  e.  U. ( Base `  Q )  |->  <. [ h ] ( 
~=ph  `  J ) ,  [ ( ( z  e.  ( 0 [,] 1 )  |->  ( I `
 ( 1  -  z ) ) ) ( *p `  J
) ( h ( *p `  J ) I ) ) ] (  ~=ph  `  J )
>. ) )
7877cnveqd 5021 . . . . . 6  |-  ( ph  ->  `' H  =  `' ran  ( h  e.  U. ( Base `  Q )  |-> 
<. [ h ] ( 
~=ph  `  J ) ,  [ ( ( z  e.  ( 0 [,] 1 )  |->  ( I `
 ( 1  -  z ) ) ) ( *p `  J
) ( h ( *p `  J ) I ) ) ] (  ~=ph  `  J )
>. ) )
793a1i 11 . . . . . . . . . 10  |-  ( ph  ->  B  =  ( Base `  P ) )
8079unieqd 4223 . . . . . . . . 9  |-  ( ph  ->  U. B  =  U. ( Base `  P )
)
8171oveq2d 6312 . . . . . . . . . . . 12  |-  ( ph  ->  ( g ( *p
`  J ) F )  =  ( g ( *p `  J
) ( z  e.  ( 0 [,] 1
)  |->  ( I `  ( 1  -  z
) ) ) ) )
8281oveq2d 6312 . . . . . . . . . . 11  |-  ( ph  ->  ( I ( *p
`  J ) ( g ( *p `  J ) F ) )  =  ( I ( *p `  J
) ( g ( *p `  J ) ( z  e.  ( 0 [,] 1 ) 
|->  ( I `  (
1  -  z ) ) ) ) ) )
8382eceq1d 7399 . . . . . . . . . 10  |-  ( ph  ->  [ ( I ( *p `  J ) ( g ( *p
`  J ) F ) ) ] ( 
~=ph  `  J )  =  [ ( I ( *p `  J ) ( g ( *p
`  J ) ( z  e.  ( 0 [,] 1 )  |->  ( I `  ( 1  -  z ) ) ) ) ) ] (  ~=ph  `  J ) )
8483opeq2d 4188 . . . . . . . . 9  |-  ( ph  -> 
<. [ g ] ( 
~=ph  `  J ) ,  [ ( I ( *p `  J ) ( g ( *p
`  J ) F ) ) ] ( 
~=ph  `  J ) >.  =  <. [ g ] (  ~=ph  `  J ) ,  [ ( I ( *p `  J
) ( g ( *p `  J ) ( z  e.  ( 0 [,] 1 ) 
|->  ( I `  (
1  -  z ) ) ) ) ) ] (  ~=ph  `  J
) >. )
8580, 84mpteq12dv 4495 . . . . . . . 8  |-  ( ph  ->  ( g  e.  U. B  |->  <. [ g ] (  ~=ph  `  J ) ,  [ ( I ( *p `  J
) ( g ( *p `  J ) F ) ) ] (  ~=ph  `  J )
>. )  =  (
g  e.  U. ( Base `  P )  |->  <. [ g ] ( 
~=ph  `  J ) ,  [ ( I ( *p `  J ) ( g ( *p
`  J ) ( z  e.  ( 0 [,] 1 )  |->  ( I `  ( 1  -  z ) ) ) ) ) ] (  ~=ph  `  J )
>. ) )
8685rneqd 5073 . . . . . . 7  |-  ( ph  ->  ran  ( g  e. 
U. B  |->  <. [ g ] (  ~=ph  `  J
) ,  [ ( I ( *p `  J ) ( g ( *p `  J
) F ) ) ] (  ~=ph  `  J
) >. )  =  ran  ( g  e.  U. ( Base `  P )  |-> 
<. [ g ] ( 
~=ph  `  J ) ,  [ ( I ( *p `  J ) ( g ( *p
`  J ) ( z  e.  ( 0 [,] 1 )  |->  ( I `  ( 1  -  z ) ) ) ) ) ] (  ~=ph  `  J )
>. ) )
874, 86syl5eq 2473 . . . . . 6  |-  ( ph  ->  G  =  ran  (
g  e.  U. ( Base `  P )  |->  <. [ g ] ( 
~=ph  `  J ) ,  [ ( I ( *p `  J ) ( g ( *p
`  J ) ( z  e.  ( 0 [,] 1 )  |->  ( I `  ( 1  -  z ) ) ) ) ) ] (  ~=ph  `  J )
>. ) )
8850, 78, 873sstr4d 3504 . . . . 5  |-  ( ph  ->  `' H  C_  G )
89 cnvss 5018 . . . . 5  |-  ( `' H  C_  G  ->  `' `' H  C_  `' G
)
9088, 89syl 17 . . . 4  |-  ( ph  ->  `' `' H  C_  `' G
)
9119, 90eqsstr3d 3496 . . 3  |-  ( ph  ->  H  C_  `' G
)
929, 91eqssd 3478 . 2  |-  ( ph  ->  `' G  =  H
)
9392, 77eqtrd 2461 . . 3  |-  ( ph  ->  `' G  =  ran  ( h  e.  U. ( Base `  Q )  |->  <. [ h ] ( 
~=ph  `  J ) ,  [ ( ( z  e.  ( 0 [,] 1 )  |->  ( I `
 ( 1  -  z ) ) ) ( *p `  J
) ( h ( *p `  J ) I ) ) ] (  ~=ph  `  J )
>. ) )
9429, 40, 41, 42, 5, 45, 48pi1xfr 21972 . . 3  |-  ( ph  ->  ran  ( h  e. 
U. ( Base `  Q
)  |->  <. [ h ]
(  ~=ph  `  J ) ,  [ ( ( z  e.  ( 0 [,] 1 )  |->  ( I `
 ( 1  -  z ) ) ) ( *p `  J
) ( h ( *p `  J ) I ) ) ] (  ~=ph  `  J )
>. )  e.  ( Q  GrpHom  P ) )
9593, 94eqeltrd 2508 . 2  |-  ( ph  ->  `' G  e.  ( Q  GrpHom  P ) )
9692, 95jca 534 1  |-  ( ph  ->  ( `' G  =  H  /\  `' G  e.  ( Q  GrpHom  P ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1867   _Vcvv 3078    C_ wss 3433   <.cop 3999   U.cuni 4213    |-> cmpt 4475    X. cxp 4843   `'ccnv 4844   ran crn 4846   Rel wrel 4850   -->wf 5588   ` cfv 5592  (class class class)co 6296   [cec 7360   CCcc 9526   RRcr 9527   0cc0 9528   1c1 9529    - cmin 9849   [,]cicc 11627   Basecbs 15073    GrpHom cghm 16824  TopOnctopon 19842    Cn ccn 20164   IIcii 21796    ~=ph cphtpc 21886   *pcpco 21917    pi1 cpi1 21920
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-8 1869  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-rep 4529  ax-sep 4539  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6588  ax-inf2 8137  ax-cnex 9584  ax-resscn 9585  ax-1cn 9586  ax-icn 9587  ax-addcl 9588  ax-addrcl 9589  ax-mulcl 9590  ax-mulrcl 9591  ax-mulcom 9592  ax-addass 9593  ax-mulass 9594  ax-distr 9595  ax-i2m1 9596  ax-1ne0 9597  ax-1rid 9598  ax-rnegex 9599  ax-rrecex 9600  ax-cnre 9601  ax-pre-lttri 9602  ax-pre-lttrn 9603  ax-pre-ltadd 9604  ax-pre-mulgt0 9605  ax-pre-sup 9606  ax-addf 9607  ax-mulf 9608
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-nel 2619  df-ral 2778  df-rex 2779  df-reu 2780  df-rmo 2781  df-rab 2782  df-v 3080  df-sbc 3297  df-csb 3393  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-pss 3449  df-nul 3759  df-if 3907  df-pw 3978  df-sn 3994  df-pr 3996  df-tp 3998  df-op 4000  df-uni 4214  df-int 4250  df-iun 4295  df-iin 4296  df-br 4418  df-opab 4476  df-mpt 4477  df-tr 4512  df-eprel 4756  df-id 4760  df-po 4766  df-so 4767  df-fr 4804  df-se 4805  df-we 4806  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-pred 5390  df-ord 5436  df-on 5437  df-lim 5438  df-suc 5439  df-iota 5556  df-fun 5594  df-fn 5595  df-f 5596  df-f1 5597  df-fo 5598  df-f1o 5599  df-fv 5600  df-isom 5601  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-of 6536  df-om 6698  df-1st 6798  df-2nd 6799  df-supp 6917  df-wrecs 7027  df-recs 7089  df-rdg 7127  df-1o 7181  df-2o 7182  df-oadd 7185  df-er 7362  df-ec 7364  df-qs 7368  df-map 7473  df-ixp 7522  df-en 7569  df-dom 7570  df-sdom 7571  df-fin 7572  df-fsupp 7881  df-fi 7922  df-sup 7953  df-oi 8016  df-card 8363  df-cda 8587  df-pnf 9666  df-mnf 9667  df-xr 9668  df-ltxr 9669  df-le 9670  df-sub 9851  df-neg 9852  df-div 10259  df-nn 10599  df-2 10657  df-3 10658  df-4 10659  df-5 10660  df-6 10661  df-7 10662  df-8 10663  df-9 10664  df-10 10665  df-n0 10859  df-z 10927  df-dec 11041  df-uz 11149  df-q 11254  df-rp 11292  df-xneg 11398  df-xadd 11399  df-xmul 11400  df-ioo 11628  df-icc 11631  df-fz 11772  df-fzo 11903  df-seq 12200  df-exp 12259  df-hash 12502  df-cj 13130  df-re 13131  df-im 13132  df-sqrt 13266  df-abs 13267  df-struct 15075  df-ndx 15076  df-slot 15077  df-base 15078  df-sets 15079  df-ress 15080  df-plusg 15155  df-mulr 15156  df-starv 15157  df-sca 15158  df-vsca 15159  df-ip 15160  df-tset 15161  df-ple 15162  df-ds 15164  df-unif 15165  df-hom 15166  df-cco 15167  df-rest 15273  df-topn 15274  df-0g 15292  df-gsum 15293  df-topgen 15294  df-pt 15295  df-prds 15298  df-xrs 15352  df-qtop 15357  df-imas 15358  df-qus 15359  df-xps 15360  df-mre 15436  df-mrc 15437  df-acs 15439  df-mgm 16432  df-sgrp 16471  df-mnd 16481  df-submnd 16527  df-grp 16617  df-mulg 16620  df-ghm 16825  df-cntz 16915  df-cmn 17360  df-psmet 18890  df-xmet 18891  df-met 18892  df-bl 18893  df-mopn 18894  df-cnfld 18899  df-top 19845  df-bases 19846  df-topon 19847  df-topsp 19848  df-cld 19958  df-cn 20167  df-cnp 20168  df-tx 20501  df-hmeo 20694  df-xms 21259  df-ms 21260  df-tms 21261  df-ii 21798  df-htpy 21887  df-phtpy 21888  df-phtpc 21909  df-pco 21922  df-om1 21923  df-pi1 21925
This theorem is referenced by:  pi1xfrgim  21975
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