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Theorem pi1xfrcnv 21320
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 21319 . . 3  |-  ( ph  ->  `' G  C_  H )
10 fvex 5876 . . . . . . . 8  |-  (  ~=ph  `  J )  e.  _V
11 ecexg 7315 . . . . . . . 8  |-  ( ( 
~=ph  `  J )  e. 
_V  ->  [ h ]
(  ~=ph  `  J )  e.  _V )
1210, 11mp1i 12 . . . . . . 7  |-  ( (
ph  /\  h  e.  U. ( Base `  Q
) )  ->  [ h ] (  ~=ph  `  J
)  e.  _V )
13 ecexg 7315 . . . . . . . 8  |-  ( ( 
~=ph  `  J )  e. 
_V  ->  [ ( F ( *p `  J
) ( h ( *p `  J ) I ) ) ] (  ~=ph  `  J )  e.  _V )
1410, 13mp1i 12 . . . . . . 7  |-  ( (
ph  /\  h  e.  U. ( Base `  Q
) )  ->  [ ( F ( *p `  J ) ( h ( *p `  J
) I ) ) ] (  ~=ph  `  J
)  e.  _V )
158, 12, 14fliftrel 6194 . . . . . 6  |-  ( ph  ->  H  C_  ( _V  X.  _V ) )
16 df-rel 5006 . . . . . 6  |-  ( Rel 
H  <->  H  C_  ( _V 
X.  _V ) )
1715, 16sylibr 212 . . . . 5  |-  ( ph  ->  Rel  H )
18 dfrel2 5457 . . . . 5  |-  ( Rel 
H  <->  `' `' H  =  H
)
1917, 18sylib 196 . . . 4  |-  ( ph  ->  `' `' H  =  H
)
20 0elunit 11638 . . . . . . . . . 10  |-  0  e.  ( 0 [,] 1
)
21 oveq2 6292 . . . . . . . . . . . . 13  |-  ( x  =  0  ->  (
1  -  x )  =  ( 1  -  0 ) )
22 1m0e1 10646 . . . . . . . . . . . . 13  |-  ( 1  -  0 )  =  1
2321, 22syl6eq 2524 . . . . . . . . . . . 12  |-  ( x  =  0  ->  (
1  -  x )  =  1 )
2423fveq2d 5870 . . . . . . . . . . 11  |-  ( x  =  0  ->  ( F `  ( 1  -  x ) )  =  ( F `  1
) )
25 fvex 5876 . . . . . . . . . . 11  |-  ( F `
 1 )  e. 
_V
2624, 7, 25fvmpt 5950 . . . . . . . . . 10  |-  ( 0  e.  ( 0 [,] 1 )  ->  (
I `  0 )  =  ( F ` 
1 ) )
2720, 26ax-mp 5 . . . . . . . . 9  |-  ( I `
 0 )  =  ( F `  1
)
2827oveq2i 6295 . . . . . . . 8  |-  ( J  pi1  ( I `
 0 ) )  =  ( J  pi1  ( F ` 
1 ) )
292, 28eqtr4i 2499 . . . . . . 7  |-  Q  =  ( J  pi1 
( I `  0
) )
30 1elunit 11639 . . . . . . . . . 10  |-  1  e.  ( 0 [,] 1
)
31 oveq2 6292 . . . . . . . . . . . . 13  |-  ( x  =  1  ->  (
1  -  x )  =  ( 1  -  1 ) )
3231fveq2d 5870 . . . . . . . . . . . 12  |-  ( x  =  1  ->  ( F `  ( 1  -  x ) )  =  ( F `  (
1  -  1 ) ) )
33 1m1e0 10604 . . . . . . . . . . . . 13  |-  ( 1  -  1 )  =  0
3433fveq2i 5869 . . . . . . . . . . . 12  |-  ( F `
 ( 1  -  1 ) )  =  ( F `  0
)
3532, 34syl6eq 2524 . . . . . . . . . . 11  |-  ( x  =  1  ->  ( F `  ( 1  -  x ) )  =  ( F `  0
) )
36 fvex 5876 . . . . . . . . . . 11  |-  ( F `
 0 )  e. 
_V
3735, 7, 36fvmpt 5950 . . . . . . . . . 10  |-  ( 1  e.  ( 0 [,] 1 )  ->  (
I `  1 )  =  ( F ` 
0 ) )
3830, 37ax-mp 5 . . . . . . . . 9  |-  ( I `
 1 )  =  ( F `  0
)
3938oveq2i 6295 . . . . . . . 8  |-  ( J  pi1  ( I `
 1 ) )  =  ( J  pi1  ( F ` 
0 ) )
401, 39eqtr4i 2499 . . . . . . 7  |-  P  =  ( J  pi1 
( I `  1
) )
41 eqid 2467 . . . . . . 7  |-  ( Base `  Q )  =  (
Base `  Q )
42 eqid 2467 . . . . . . 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 21288 . . . . . . . . 9  |-  ( F  e.  ( II  Cn  J )  ->  (
I  e.  ( II 
Cn  J )  /\  ( I `  0
)  =  ( F `
 1 )  /\  ( I `  1
)  =  ( F `
 0 ) ) )
446, 43syl 16 . . . . . . . 8  |-  ( ph  ->  ( I  e.  ( II  Cn  J )  /\  ( I ` 
0 )  =  ( F `  1 )  /\  ( I ` 
1 )  =  ( F `  0 ) ) )
4544simp1d 1008 . . . . . . 7  |-  ( ph  ->  I  e.  ( II 
Cn  J ) )
46 oveq2 6292 . . . . . . . . 9  |-  ( z  =  y  ->  (
1  -  z )  =  ( 1  -  y ) )
4746fveq2d 5870 . . . . . . . 8  |-  ( z  =  y  ->  (
I `  ( 1  -  z ) )  =  ( I `  ( 1  -  y
) ) )
4847cbvmptv 4538 . . . . . . 7  |-  ( z  e.  ( 0 [,] 1 )  |->  ( I `
 ( 1  -  z ) ) )  =  ( y  e.  ( 0 [,] 1
)  |->  ( I `  ( 1  -  y
) ) )
49 eqid 2467 . . . . . . 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 21319 . . . . . 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 21146 . . . . . . . . . . . . . . . . 17  |-  II  e.  (TopOn `  ( 0 [,] 1 ) )
5251a1i 11 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  II  e.  (TopOn `  ( 0 [,] 1
) ) )
53 cnf2 19544 . . . . . . . . . . . . . . . 16  |-  ( ( II  e.  (TopOn `  ( 0 [,] 1
) )  /\  J  e.  (TopOn `  X )  /\  F  e.  (
II  Cn  J )
)  ->  F :
( 0 [,] 1
) --> X )
5452, 5, 6, 53syl3anc 1228 . . . . . . . . . . . . . . 15  |-  ( ph  ->  F : ( 0 [,] 1 ) --> X )
5554feqmptd 5920 . . . . . . . . . . . . . 14  |-  ( ph  ->  F  =  ( z  e.  ( 0 [,] 1 )  |->  ( F `
 z ) ) )
56 iirev 21192 . . . . . . . . . . . . . . . . 17  |-  ( z  e.  ( 0 [,] 1 )  ->  (
1  -  z )  e.  ( 0 [,] 1 ) )
57 oveq2 6292 . . . . . . . . . . . . . . . . . . 19  |-  ( x  =  ( 1  -  z )  ->  (
1  -  x )  =  ( 1  -  ( 1  -  z
) ) )
5857fveq2d 5870 . . . . . . . . . . . . . . . . . 18  |-  ( x  =  ( 1  -  z )  ->  ( F `  ( 1  -  x ) )  =  ( F `  (
1  -  ( 1  -  z ) ) ) )
59 fvex 5876 . . . . . . . . . . . . . . . . . 18  |-  ( F `
 ( 1  -  ( 1  -  z
) ) )  e. 
_V
6058, 7, 59fvmpt 5950 . . . . . . . . . . . . . . . . 17  |-  ( ( 1  -  z )  e.  ( 0 [,] 1 )  ->  (
I `  ( 1  -  z ) )  =  ( F `  ( 1  -  (
1  -  z ) ) ) )
6156, 60syl 16 . . . . . . . . . . . . . . . 16  |-  ( z  e.  ( 0 [,] 1 )  ->  (
I `  ( 1  -  z ) )  =  ( F `  ( 1  -  (
1  -  z ) ) ) )
62 ax-1cn 9550 . . . . . . . . . . . . . . . . . 18  |-  1  e.  CC
63 unitssre 11667 . . . . . . . . . . . . . . . . . . . 20  |-  ( 0 [,] 1 )  C_  RR
6463sseli 3500 . . . . . . . . . . . . . . . . . . 19  |-  ( z  e.  ( 0 [,] 1 )  ->  z  e.  RR )
6564recnd 9622 . . . . . . . . . . . . . . . . . 18  |-  ( z  e.  ( 0 [,] 1 )  ->  z  e.  CC )
66 nncan 9848 . . . . . . . . . . . . . . . . . 18  |-  ( ( 1  e.  CC  /\  z  e.  CC )  ->  ( 1  -  (
1  -  z ) )  =  z )
6762, 65, 66sylancr 663 . . . . . . . . . . . . . . . . 17  |-  ( z  e.  ( 0 [,] 1 )  ->  (
1  -  ( 1  -  z ) )  =  z )
6867fveq2d 5870 . . . . . . . . . . . . . . . 16  |-  ( z  e.  ( 0 [,] 1 )  ->  ( F `  ( 1  -  ( 1  -  z ) ) )  =  ( F `  z ) )
6961, 68eqtrd 2508 . . . . . . . . . . . . . . 15  |-  ( z  e.  ( 0 [,] 1 )  ->  (
I `  ( 1  -  z ) )  =  ( F `  z ) )
7069mpteq2ia 4529 . . . . . . . . . . . . . 14  |-  ( z  e.  ( 0 [,] 1 )  |->  ( I `
 ( 1  -  z ) ) )  =  ( z  e.  ( 0 [,] 1
)  |->  ( F `  z ) )
7155, 70syl6eqr 2526 . . . . . . . . . . . . 13  |-  ( ph  ->  F  =  ( z  e.  ( 0 [,] 1 )  |->  ( I `
 ( 1  -  z ) ) ) )
7271oveq1d 6299 . . . . . . . . . . . 12  |-  ( ph  ->  ( F ( *p
`  J ) ( h ( *p `  J ) I ) )  =  ( ( z  e.  ( 0 [,] 1 )  |->  ( I `  ( 1  -  z ) ) ) ( *p `  J ) ( h ( *p `  J
) I ) ) )
7372eceq1d 7348 . . . . . . . . . . 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 4220 . . . . . . . . . 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 4534 . . . . . . . . 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 5230 . . . . . . . 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 2520 . . . . . . 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 5178 . . . . . 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 4255 . . . . . . . . 9  |-  ( ph  ->  U. B  =  U. ( Base `  P )
)
8171oveq2d 6300 . . . . . . . . . . . 12  |-  ( ph  ->  ( g ( *p
`  J ) F )  =  ( g ( *p `  J
) ( z  e.  ( 0 [,] 1
)  |->  ( I `  ( 1  -  z
) ) ) ) )
8281oveq2d 6300 . . . . . . . . . . 11  |-  ( ph  ->  ( I ( *p
`  J ) ( g ( *p `  J ) F ) )  =  ( I ( *p `  J
) ( g ( *p `  J ) ( z  e.  ( 0 [,] 1 ) 
|->  ( I `  (
1  -  z ) ) ) ) ) )
8382eceq1d 7348 . . . . . . . . . 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 4220 . . . . . . . . 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 4525 . . . . . . . 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 5230 . . . . . . 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 2520 . . . . . 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 3547 . . . . 5  |-  ( ph  ->  `' H  C_  G )
89 cnvss 5175 . . . . 5  |-  ( `' H  C_  G  ->  `' `' H  C_  `' G
)
9088, 89syl 16 . . . 4  |-  ( ph  ->  `' `' H  C_  `' G
)
9119, 90eqsstr3d 3539 . . 3  |-  ( ph  ->  H  C_  `' G
)
929, 91eqssd 3521 . 2  |-  ( ph  ->  `' G  =  H
)
9392, 77eqtrd 2508 . . 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 21318 . . 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 2555 . 2  |-  ( ph  ->  `' G  e.  ( Q  GrpHom  P ) )
9692, 95jca 532 1  |-  ( ph  ->  ( `' G  =  H  /\  `' G  e.  ( Q  GrpHom  P ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   _Vcvv 3113    C_ wss 3476   <.cop 4033   U.cuni 4245    |-> cmpt 4505    X. cxp 4997   `'ccnv 4998   ran crn 5000   Rel wrel 5004   -->wf 5584   ` cfv 5588  (class class class)co 6284   [cec 7309   CCcc 9490   RRcr 9491   0cc0 9492   1c1 9493    - cmin 9805   [,]cicc 11532   Basecbs 14490    GrpHom cghm 16069  TopOnctopon 19190    Cn ccn 19519   IIcii 21142    ~=ph cphtpc 21232   *pcpco 21263    pi1 cpi1 21266
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576  ax-inf2 8058  ax-cnex 9548  ax-resscn 9549  ax-1cn 9550  ax-icn 9551  ax-addcl 9552  ax-addrcl 9553  ax-mulcl 9554  ax-mulrcl 9555  ax-mulcom 9556  ax-addass 9557  ax-mulass 9558  ax-distr 9559  ax-i2m1 9560  ax-1ne0 9561  ax-1rid 9562  ax-rnegex 9563  ax-rrecex 9564  ax-cnre 9565  ax-pre-lttri 9566  ax-pre-lttrn 9567  ax-pre-ltadd 9568  ax-pre-mulgt0 9569  ax-pre-sup 9570  ax-addf 9571  ax-mulf 9572
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-iin 4328  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-isom 5597  df-riota 6245  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-of 6524  df-om 6685  df-1st 6784  df-2nd 6785  df-supp 6902  df-recs 7042  df-rdg 7076  df-1o 7130  df-2o 7131  df-oadd 7134  df-er 7311  df-ec 7313  df-qs 7317  df-map 7422  df-ixp 7470  df-en 7517  df-dom 7518  df-sdom 7519  df-fin 7520  df-fsupp 7830  df-fi 7871  df-sup 7901  df-oi 7935  df-card 8320  df-cda 8548  df-pnf 9630  df-mnf 9631  df-xr 9632  df-ltxr 9633  df-le 9634  df-sub 9807  df-neg 9808  df-div 10207  df-nn 10537  df-2 10594  df-3 10595  df-4 10596  df-5 10597  df-6 10598  df-7 10599  df-8 10600  df-9 10601  df-10 10602  df-n0 10796  df-z 10865  df-dec 10977  df-uz 11083  df-q 11183  df-rp 11221  df-xneg 11318  df-xadd 11319  df-xmul 11320  df-ioo 11533  df-icc 11536  df-fz 11673  df-fzo 11793  df-seq 12076  df-exp 12135  df-hash 12374  df-cj 12895  df-re 12896  df-im 12897  df-sqrt 13031  df-abs 13032  df-struct 14492  df-ndx 14493  df-slot 14494  df-base 14495  df-sets 14496  df-ress 14497  df-plusg 14568  df-mulr 14569  df-starv 14570  df-sca 14571  df-vsca 14572  df-ip 14573  df-tset 14574  df-ple 14575  df-ds 14577  df-unif 14578  df-hom 14579  df-cco 14580  df-rest 14678  df-topn 14679  df-0g 14697  df-gsum 14698  df-topgen 14699  df-pt 14700  df-prds 14703  df-xrs 14757  df-qtop 14762  df-imas 14763  df-divs 14764  df-xps 14765  df-mre 14841  df-mrc 14842  df-acs 14844  df-mnd 15732  df-submnd 15787  df-grp 15867  df-mulg 15870  df-ghm 16070  df-cntz 16160  df-cmn 16606  df-psmet 18210  df-xmet 18211  df-met 18212  df-bl 18213  df-mopn 18214  df-cnfld 18220  df-top 19194  df-bases 19196  df-topon 19197  df-topsp 19198  df-cld 19314  df-cn 19522  df-cnp 19523  df-tx 19826  df-hmeo 20019  df-xms 20586  df-ms 20587  df-tms 20588  df-ii 21144  df-htpy 21233  df-phtpy 21234  df-phtpc 21255  df-pco 21268  df-om1 21269  df-pi1 21271
This theorem is referenced by:  pi1xfrgim  21321
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