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Theorem pi1xfrcnv 19087
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  pi 1 
( F `  0
) )
pi1xfr.q  |-  Q  =  ( J  pi 1 
( 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  pi 1 
( F `  0
) )
2 pi1xfr.q . . . 4  |-  Q  =  ( J  pi 1 
( 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 19086 . . 3  |-  ( ph  ->  `' G  C_  H )
10 fvex 5745 . . . . . . . 8  |-  (  ~=ph  `  J )  e.  _V
11 ecexg 6912 . . . . . . . 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 6912 . . . . . . . 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 6033 . . . . . 6  |-  ( ph  ->  H  C_  ( _V  X.  _V ) )
16 df-rel 4888 . . . . . 6  |-  ( Rel 
H  <->  H  C_  ( _V 
X.  _V ) )
1715, 16sylibr 205 . . . . 5  |-  ( ph  ->  Rel  H )
18 dfrel2 5324 . . . . 5  |-  ( Rel 
H  <->  `' `' H  =  H
)
1917, 18sylib 190 . . . 4  |-  ( ph  ->  `' `' H  =  H
)
20 0elunit 11020 . . . . . . . . . 10  |-  0  e.  ( 0 [,] 1
)
21 oveq2 6092 . . . . . . . . . . . . 13  |-  ( x  =  0  ->  (
1  -  x )  =  ( 1  -  0 ) )
22 ax-1cn 9053 . . . . . . . . . . . . . 14  |-  1  e.  CC
2322subid1i 9377 . . . . . . . . . . . . 13  |-  ( 1  -  0 )  =  1
2421, 23syl6eq 2486 . . . . . . . . . . . 12  |-  ( x  =  0  ->  (
1  -  x )  =  1 )
2524fveq2d 5735 . . . . . . . . . . 11  |-  ( x  =  0  ->  ( F `  ( 1  -  x ) )  =  ( F `  1
) )
26 fvex 5745 . . . . . . . . . . 11  |-  ( F `
 1 )  e. 
_V
2725, 7, 26fvmpt 5809 . . . . . . . . . 10  |-  ( 0  e.  ( 0 [,] 1 )  ->  (
I `  0 )  =  ( F ` 
1 ) )
2820, 27ax-mp 5 . . . . . . . . 9  |-  ( I `
 0 )  =  ( F `  1
)
2928oveq2i 6095 . . . . . . . 8  |-  ( J  pi 1  ( I `
 0 ) )  =  ( J  pi 1  ( F ` 
1 ) )
302, 29eqtr4i 2461 . . . . . . 7  |-  Q  =  ( J  pi 1 
( I `  0
) )
31 1elunit 11021 . . . . . . . . . 10  |-  1  e.  ( 0 [,] 1
)
32 oveq2 6092 . . . . . . . . . . . . 13  |-  ( x  =  1  ->  (
1  -  x )  =  ( 1  -  1 ) )
3332fveq2d 5735 . . . . . . . . . . . 12  |-  ( x  =  1  ->  ( F `  ( 1  -  x ) )  =  ( F `  (
1  -  1 ) ) )
34 1m1e0 10073 . . . . . . . . . . . . 13  |-  ( 1  -  1 )  =  0
3534fveq2i 5734 . . . . . . . . . . . 12  |-  ( F `
 ( 1  -  1 ) )  =  ( F `  0
)
3633, 35syl6eq 2486 . . . . . . . . . . 11  |-  ( x  =  1  ->  ( F `  ( 1  -  x ) )  =  ( F `  0
) )
37 fvex 5745 . . . . . . . . . . 11  |-  ( F `
 0 )  e. 
_V
3836, 7, 37fvmpt 5809 . . . . . . . . . 10  |-  ( 1  e.  ( 0 [,] 1 )  ->  (
I `  1 )  =  ( F ` 
0 ) )
3931, 38ax-mp 5 . . . . . . . . 9  |-  ( I `
 1 )  =  ( F `  0
)
4039oveq2i 6095 . . . . . . . 8  |-  ( J  pi 1  ( I `
 1 ) )  =  ( J  pi 1  ( F ` 
0 ) )
411, 40eqtr4i 2461 . . . . . . 7  |-  P  =  ( J  pi 1 
( I `  1
) )
42 eqid 2438 . . . . . . 7  |-  ( Base `  Q )  =  (
Base `  Q )
43 eqid 2438 . . . . . . 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 )
>. )
447pcorevcl 19055 . . . . . . . . 9  |-  ( F  e.  ( II  Cn  J )  ->  (
I  e.  ( II 
Cn  J )  /\  ( I `  0
)  =  ( F `
 1 )  /\  ( I `  1
)  =  ( F `
 0 ) ) )
456, 44syl 16 . . . . . . . 8  |-  ( ph  ->  ( I  e.  ( II  Cn  J )  /\  ( I ` 
0 )  =  ( F `  1 )  /\  ( I ` 
1 )  =  ( F `  0 ) ) )
4645simp1d 970 . . . . . . 7  |-  ( ph  ->  I  e.  ( II 
Cn  J ) )
47 oveq2 6092 . . . . . . . . 9  |-  ( z  =  y  ->  (
1  -  z )  =  ( 1  -  y ) )
4847fveq2d 5735 . . . . . . . 8  |-  ( z  =  y  ->  (
I `  ( 1  -  z ) )  =  ( I `  ( 1  -  y
) ) )
4948cbvmptv 4303 . . . . . . 7  |-  ( z  e.  ( 0 [,] 1 )  |->  ( I `
 ( 1  -  z ) ) )  =  ( y  e.  ( 0 [,] 1
)  |->  ( I `  ( 1  -  y
) ) )
50 eqid 2438 . . . . . . 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 )
>. )
5130, 41, 42, 43, 5, 46, 49, 50pi1xfrcnvlem 19086 . . . . . 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 )
>. ) )
52 iitopon 18914 . . . . . . . . . . . . . . . . 17  |-  II  e.  (TopOn `  ( 0 [,] 1 ) )
5352a1i 11 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  II  e.  (TopOn `  ( 0 [,] 1
) ) )
54 cnf2 17318 . . . . . . . . . . . . . . . 16  |-  ( ( II  e.  (TopOn `  ( 0 [,] 1
) )  /\  J  e.  (TopOn `  X )  /\  F  e.  (
II  Cn  J )
)  ->  F :
( 0 [,] 1
) --> X )
5553, 5, 6, 54syl3anc 1185 . . . . . . . . . . . . . . 15  |-  ( ph  ->  F : ( 0 [,] 1 ) --> X )
5655feqmptd 5782 . . . . . . . . . . . . . 14  |-  ( ph  ->  F  =  ( z  e.  ( 0 [,] 1 )  |->  ( F `
 z ) ) )
57 iirev 18959 . . . . . . . . . . . . . . . . 17  |-  ( z  e.  ( 0 [,] 1 )  ->  (
1  -  z )  e.  ( 0 [,] 1 ) )
58 oveq2 6092 . . . . . . . . . . . . . . . . . . 19  |-  ( x  =  ( 1  -  z )  ->  (
1  -  x )  =  ( 1  -  ( 1  -  z
) ) )
5958fveq2d 5735 . . . . . . . . . . . . . . . . . 18  |-  ( x  =  ( 1  -  z )  ->  ( F `  ( 1  -  x ) )  =  ( F `  (
1  -  ( 1  -  z ) ) ) )
60 fvex 5745 . . . . . . . . . . . . . . . . . 18  |-  ( F `
 ( 1  -  ( 1  -  z
) ) )  e. 
_V
6159, 7, 60fvmpt 5809 . . . . . . . . . . . . . . . . 17  |-  ( ( 1  -  z )  e.  ( 0 [,] 1 )  ->  (
I `  ( 1  -  z ) )  =  ( F `  ( 1  -  (
1  -  z ) ) ) )
6257, 61syl 16 . . . . . . . . . . . . . . . 16  |-  ( z  e.  ( 0 [,] 1 )  ->  (
I `  ( 1  -  z ) )  =  ( F `  ( 1  -  (
1  -  z ) ) ) )
63 unitssre 11047 . . . . . . . . . . . . . . . . . . . 20  |-  ( 0 [,] 1 )  C_  RR
6463sseli 3346 . . . . . . . . . . . . . . . . . . 19  |-  ( z  e.  ( 0 [,] 1 )  ->  z  e.  RR )
6564recnd 9119 . . . . . . . . . . . . . . . . . 18  |-  ( z  e.  ( 0 [,] 1 )  ->  z  e.  CC )
66 nncan 9335 . . . . . . . . . . . . . . . . . 18  |-  ( ( 1  e.  CC  /\  z  e.  CC )  ->  ( 1  -  (
1  -  z ) )  =  z )
6722, 65, 66sylancr 646 . . . . . . . . . . . . . . . . 17  |-  ( z  e.  ( 0 [,] 1 )  ->  (
1  -  ( 1  -  z ) )  =  z )
6867fveq2d 5735 . . . . . . . . . . . . . . . 16  |-  ( z  e.  ( 0 [,] 1 )  ->  ( F `  ( 1  -  ( 1  -  z ) ) )  =  ( F `  z ) )
6962, 68eqtrd 2470 . . . . . . . . . . . . . . 15  |-  ( z  e.  ( 0 [,] 1 )  ->  (
I `  ( 1  -  z ) )  =  ( F `  z ) )
7069mpteq2ia 4294 . . . . . . . . . . . . . 14  |-  ( z  e.  ( 0 [,] 1 )  |->  ( I `
 ( 1  -  z ) ) )  =  ( z  e.  ( 0 [,] 1
)  |->  ( F `  z ) )
7156, 70syl6eqr 2488 . . . . . . . . . . . . 13  |-  ( ph  ->  F  =  ( z  e.  ( 0 [,] 1 )  |->  ( I `
 ( 1  -  z ) ) ) )
7271oveq1d 6099 . . . . . . . . . . . 12  |-  ( ph  ->  ( F ( *p
`  J ) ( h ( *p `  J ) I ) )  =  ( ( z  e.  ( 0 [,] 1 )  |->  ( I `  ( 1  -  z ) ) ) ( *p `  J ) ( h ( *p `  J
) I ) ) )
73 eceq1 6944 . . . . . . . . . . . 12  |-  ( ( F ( *p `  J ) ( h ( *p `  J
) I ) )  =  ( ( z  e.  ( 0 [,] 1 )  |->  ( I `
 ( 1  -  z ) ) ) ( *p `  J
) ( h ( *p `  J ) I ) )  ->  [ ( F ( *p `  J ) ( h ( *p
`  J ) I ) ) ] ( 
~=ph  `  J )  =  [ ( ( z  e.  ( 0 [,] 1 )  |->  ( I `
 ( 1  -  z ) ) ) ( *p `  J
) ( h ( *p `  J ) I ) ) ] (  ~=ph  `  J ) )
7472, 73syl 16 . . . . . . . . . . 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 ) )
7574opeq2d 3993 . . . . . . . . . 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 )
>. )
7675mpteq2dv 4299 . . . . . . . . 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 ) >. )
)
7776rneqd 5100 . . . . . . . 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 )
>. ) )
788, 77syl5eq 2482 . . . . . . 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 )
>. ) )
7978cnveqd 5051 . . . . . 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 )
>. ) )
803a1i 11 . . . . . . . . . 10  |-  ( ph  ->  B  =  ( Base `  P ) )
8180unieqd 4028 . . . . . . . . 9  |-  ( ph  ->  U. B  =  U. ( Base `  P )
)
8271oveq2d 6100 . . . . . . . . . . . 12  |-  ( ph  ->  ( g ( *p
`  J ) F )  =  ( g ( *p `  J
) ( z  e.  ( 0 [,] 1
)  |->  ( I `  ( 1  -  z
) ) ) ) )
8382oveq2d 6100 . . . . . . . . . . 11  |-  ( ph  ->  ( I ( *p
`  J ) ( g ( *p `  J ) F ) )  =  ( I ( *p `  J
) ( g ( *p `  J ) ( z  e.  ( 0 [,] 1 ) 
|->  ( I `  (
1  -  z ) ) ) ) ) )
84 eceq1 6944 . . . . . . . . . . 11  |-  ( ( I ( *p `  J ) ( g ( *p `  J
) F ) )  =  ( I ( *p `  J ) ( g ( *p
`  J ) ( z  e.  ( 0 [,] 1 )  |->  ( I `  ( 1  -  z ) ) ) ) )  ->  [ ( I ( *p `  J ) ( g ( *p
`  J ) F ) ) ] ( 
~=ph  `  J )  =  [ ( I ( *p `  J ) ( g ( *p
`  J ) ( z  e.  ( 0 [,] 1 )  |->  ( I `  ( 1  -  z ) ) ) ) ) ] (  ~=ph  `  J ) )
8583, 84syl 16 . . . . . . . . . 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 ) )
8685opeq2d 3993 . . . . . . . . 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
) >. )
8781, 86mpteq12dv 4290 . . . . . . . 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 )
>. ) )
8887rneqd 5100 . . . . . . 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 )
>. ) )
894, 88syl5eq 2482 . . . . . 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 )
>. ) )
9051, 79, 893sstr4d 3393 . . . . 5  |-  ( ph  ->  `' H  C_  G )
91 cnvss 5048 . . . . 5  |-  ( `' H  C_  G  ->  `' `' H  C_  `' G
)
9290, 91syl 16 . . . 4  |-  ( ph  ->  `' `' H  C_  `' G
)
9319, 92eqsstr3d 3385 . . 3  |-  ( ph  ->  H  C_  `' G
)
949, 93eqssd 3367 . 2  |-  ( ph  ->  `' G  =  H
)
9594, 78eqtrd 2470 . . 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 )
>. ) )
9630, 41, 42, 43, 5, 46, 49pi1xfr 19085 . . 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 ) )
9795, 96eqeltrd 2512 . 2  |-  ( ph  ->  `' G  e.  ( Q  GrpHom  P ) )
9894, 97jca 520 1  |-  ( ph  ->  ( `' G  =  H  /\  `' G  e.  ( Q  GrpHom  P ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726   _Vcvv 2958    C_ wss 3322   <.cop 3819   U.cuni 4017    e. cmpt 4269    X. cxp 4879   `'ccnv 4880   ran crn 4882   Rel wrel 4886   -->wf 5453   ` cfv 5457  (class class class)co 6084   [cec 6906   CCcc 8993   RRcr 8994   0cc0 8995   1c1 8996    - cmin 9296   [,]cicc 10924   Basecbs 13474    GrpHom cghm 15008  TopOnctopon 16964    Cn ccn 17293   IIcii 18910    ~=ph cphtpc 18999   *pcpco 19030    pi 1 cpi1 19033
This theorem is referenced by:  pi1xfrgim  19088
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704  ax-inf2 7599  ax-cnex 9051  ax-resscn 9052  ax-1cn 9053  ax-icn 9054  ax-addcl 9055  ax-addrcl 9056  ax-mulcl 9057  ax-mulrcl 9058  ax-mulcom 9059  ax-addass 9060  ax-mulass 9061  ax-distr 9062  ax-i2m1 9063  ax-1ne0 9064  ax-1rid 9065  ax-rnegex 9066  ax-rrecex 9067  ax-cnre 9068  ax-pre-lttri 9069  ax-pre-lttrn 9070  ax-pre-ltadd 9071  ax-pre-mulgt0 9072  ax-pre-sup 9073  ax-addf 9074  ax-mulf 9075
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-iin 4098  df-br 4216  df-opab 4270  df-mpt 4271  df-tr 4306  df-eprel 4497  df-id 4501  df-po 4506  df-so 4507  df-fr 4544  df-se 4545  df-we 4546  df-ord 4587  df-on 4588  df-lim 4589  df-suc 4590  df-om 4849  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-isom 5466  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-of 6308  df-1st 6352  df-2nd 6353  df-riota 6552  df-recs 6636  df-rdg 6671  df-1o 6727  df-2o 6728  df-oadd 6731  df-er 6908  df-ec 6910  df-qs 6914  df-map 7023  df-ixp 7067  df-en 7113  df-dom 7114  df-sdom 7115  df-fin 7116  df-fi 7419  df-sup 7449  df-oi 7482  df-card 7831  df-cda 8053  df-pnf 9127  df-mnf 9128  df-xr 9129  df-ltxr 9130  df-le 9131  df-sub 9298  df-neg 9299  df-div 9683  df-nn 10006  df-2 10063  df-3 10064  df-4 10065  df-5 10066  df-6 10067  df-7 10068  df-8 10069  df-9 10070  df-10 10071  df-n0 10227  df-z 10288  df-dec 10388  df-uz 10494  df-q 10580  df-rp 10618  df-xneg 10715  df-xadd 10716  df-xmul 10717  df-ioo 10925  df-icc 10928  df-fz 11049  df-fzo 11141  df-seq 11329  df-exp 11388  df-hash 11624  df-cj 11909  df-re 11910  df-im 11911  df-sqr 12045  df-abs 12046  df-struct 13476  df-ndx 13477  df-slot 13478  df-base 13479  df-sets 13480  df-ress 13481  df-plusg 13547  df-mulr 13548  df-starv 13549  df-sca 13550  df-vsca 13551  df-tset 13553  df-ple 13554  df-ds 13556  df-unif 13557  df-hom 13558  df-cco 13559  df-rest 13655  df-topn 13656  df-topgen 13672  df-pt 13673  df-prds 13676  df-xrs 13731  df-0g 13732  df-gsum 13733  df-qtop 13738  df-imas 13739  df-divs 13740  df-xps 13741  df-mre 13816  df-mrc 13817  df-acs 13819  df-mnd 14695  df-submnd 14744  df-grp 14817  df-mulg 14820  df-ghm 15009  df-cntz 15121  df-cmn 15419  df-psmet 16699  df-xmet 16700  df-met 16701  df-bl 16702  df-mopn 16703  df-cnfld 16709  df-top 16968  df-bases 16970  df-topon 16971  df-topsp 16972  df-cld 17088  df-cn 17296  df-cnp 17297  df-tx 17599  df-hmeo 17792  df-xms 18355  df-ms 18356  df-tms 18357  df-ii 18912  df-htpy 19000  df-phtpy 19001  df-phtpc 19022  df-pco 19035  df-om1 19036  df-pi1 19038
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