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Theorem pi1xfrcnv 19043
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 19042 . . 3  |-  ( ph  ->  `' G  C_  H )
10 fvex 5709 . . . . . . . 8  |-  (  ~=ph  `  J )  e.  _V
11 ecexg 6876 . . . . . . . 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 6876 . . . . . . . 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 5997 . . . . . 6  |-  ( ph  ->  H  C_  ( _V  X.  _V ) )
16 df-rel 4852 . . . . . 6  |-  ( Rel 
H  <->  H  C_  ( _V 
X.  _V ) )
1715, 16sylibr 204 . . . . 5  |-  ( ph  ->  Rel  H )
18 dfrel2 5288 . . . . 5  |-  ( Rel 
H  <->  `' `' H  =  H
)
1917, 18sylib 189 . . . 4  |-  ( ph  ->  `' `' H  =  H
)
20 0elunit 10979 . . . . . . . . . 10  |-  0  e.  ( 0 [,] 1
)
21 oveq2 6056 . . . . . . . . . . . . 13  |-  ( x  =  0  ->  (
1  -  x )  =  ( 1  -  0 ) )
22 ax-1cn 9012 . . . . . . . . . . . . . 14  |-  1  e.  CC
2322subid1i 9336 . . . . . . . . . . . . 13  |-  ( 1  -  0 )  =  1
2421, 23syl6eq 2460 . . . . . . . . . . . 12  |-  ( x  =  0  ->  (
1  -  x )  =  1 )
2524fveq2d 5699 . . . . . . . . . . 11  |-  ( x  =  0  ->  ( F `  ( 1  -  x ) )  =  ( F `  1
) )
26 fvex 5709 . . . . . . . . . . 11  |-  ( F `
 1 )  e. 
_V
2725, 7, 26fvmpt 5773 . . . . . . . . . 10  |-  ( 0  e.  ( 0 [,] 1 )  ->  (
I `  0 )  =  ( F ` 
1 ) )
2820, 27ax-mp 8 . . . . . . . . 9  |-  ( I `
 0 )  =  ( F `  1
)
2928oveq2i 6059 . . . . . . . 8  |-  ( J  pi 1  ( I `
 0 ) )  =  ( J  pi 1  ( F ` 
1 ) )
302, 29eqtr4i 2435 . . . . . . 7  |-  Q  =  ( J  pi 1 
( I `  0
) )
31 1elunit 10980 . . . . . . . . . 10  |-  1  e.  ( 0 [,] 1
)
32 oveq2 6056 . . . . . . . . . . . . 13  |-  ( x  =  1  ->  (
1  -  x )  =  ( 1  -  1 ) )
3332fveq2d 5699 . . . . . . . . . . . 12  |-  ( x  =  1  ->  ( F `  ( 1  -  x ) )  =  ( F `  (
1  -  1 ) ) )
34 1m1e0 10032 . . . . . . . . . . . . 13  |-  ( 1  -  1 )  =  0
3534fveq2i 5698 . . . . . . . . . . . 12  |-  ( F `
 ( 1  -  1 ) )  =  ( F `  0
)
3633, 35syl6eq 2460 . . . . . . . . . . 11  |-  ( x  =  1  ->  ( F `  ( 1  -  x ) )  =  ( F `  0
) )
37 fvex 5709 . . . . . . . . . . 11  |-  ( F `
 0 )  e. 
_V
3836, 7, 37fvmpt 5773 . . . . . . . . . 10  |-  ( 1  e.  ( 0 [,] 1 )  ->  (
I `  1 )  =  ( F ` 
0 ) )
3931, 38ax-mp 8 . . . . . . . . 9  |-  ( I `
 1 )  =  ( F `  0
)
4039oveq2i 6059 . . . . . . . 8  |-  ( J  pi 1  ( I `
 1 ) )  =  ( J  pi 1  ( F ` 
0 ) )
411, 40eqtr4i 2435 . . . . . . 7  |-  P  =  ( J  pi 1 
( I `  1
) )
42 eqid 2412 . . . . . . 7  |-  ( Base `  Q )  =  (
Base `  Q )
43 eqid 2412 . . . . . . 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 19011 . . . . . . . . 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 969 . . . . . . 7  |-  ( ph  ->  I  e.  ( II 
Cn  J ) )
47 oveq2 6056 . . . . . . . . 9  |-  ( z  =  y  ->  (
1  -  z )  =  ( 1  -  y ) )
4847fveq2d 5699 . . . . . . . 8  |-  ( z  =  y  ->  (
I `  ( 1  -  z ) )  =  ( I `  ( 1  -  y
) ) )
4948cbvmptv 4268 . . . . . . 7  |-  ( z  e.  ( 0 [,] 1 )  |->  ( I `
 ( 1  -  z ) ) )  =  ( y  e.  ( 0 [,] 1
)  |->  ( I `  ( 1  -  y
) ) )
50 eqid 2412 . . . . . . 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 19042 . . . . . 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 18870 . . . . . . . . . . . . . . . . 17  |-  II  e.  (TopOn `  ( 0 [,] 1 ) )
5352a1i 11 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  II  e.  (TopOn `  ( 0 [,] 1
) ) )
54 cnf2 17275 . . . . . . . . . . . . . . . 16  |-  ( ( II  e.  (TopOn `  ( 0 [,] 1
) )  /\  J  e.  (TopOn `  X )  /\  F  e.  (
II  Cn  J )
)  ->  F :
( 0 [,] 1
) --> X )
5553, 5, 6, 54syl3anc 1184 . . . . . . . . . . . . . . 15  |-  ( ph  ->  F : ( 0 [,] 1 ) --> X )
5655feqmptd 5746 . . . . . . . . . . . . . 14  |-  ( ph  ->  F  =  ( z  e.  ( 0 [,] 1 )  |->  ( F `
 z ) ) )
57 iirev 18915 . . . . . . . . . . . . . . . . 17  |-  ( z  e.  ( 0 [,] 1 )  ->  (
1  -  z )  e.  ( 0 [,] 1 ) )
58 oveq2 6056 . . . . . . . . . . . . . . . . . . 19  |-  ( x  =  ( 1  -  z )  ->  (
1  -  x )  =  ( 1  -  ( 1  -  z
) ) )
5958fveq2d 5699 . . . . . . . . . . . . . . . . . 18  |-  ( x  =  ( 1  -  z )  ->  ( F `  ( 1  -  x ) )  =  ( F `  (
1  -  ( 1  -  z ) ) ) )
60 fvex 5709 . . . . . . . . . . . . . . . . . 18  |-  ( F `
 ( 1  -  ( 1  -  z
) ) )  e. 
_V
6159, 7, 60fvmpt 5773 . . . . . . . . . . . . . . . . 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 11006 . . . . . . . . . . . . . . . . . . . 20  |-  ( 0 [,] 1 )  C_  RR
6463sseli 3312 . . . . . . . . . . . . . . . . . . 19  |-  ( z  e.  ( 0 [,] 1 )  ->  z  e.  RR )
6564recnd 9078 . . . . . . . . . . . . . . . . . 18  |-  ( z  e.  ( 0 [,] 1 )  ->  z  e.  CC )
66 nncan 9294 . . . . . . . . . . . . . . . . . 18  |-  ( ( 1  e.  CC  /\  z  e.  CC )  ->  ( 1  -  (
1  -  z ) )  =  z )
6722, 65, 66sylancr 645 . . . . . . . . . . . . . . . . 17  |-  ( z  e.  ( 0 [,] 1 )  ->  (
1  -  ( 1  -  z ) )  =  z )
6867fveq2d 5699 . . . . . . . . . . . . . . . 16  |-  ( z  e.  ( 0 [,] 1 )  ->  ( F `  ( 1  -  ( 1  -  z ) ) )  =  ( F `  z ) )
6962, 68eqtrd 2444 . . . . . . . . . . . . . . 15  |-  ( z  e.  ( 0 [,] 1 )  ->  (
I `  ( 1  -  z ) )  =  ( F `  z ) )
7069mpteq2ia 4259 . . . . . . . . . . . . . 14  |-  ( z  e.  ( 0 [,] 1 )  |->  ( I `
 ( 1  -  z ) ) )  =  ( z  e.  ( 0 [,] 1
)  |->  ( F `  z ) )
7156, 70syl6eqr 2462 . . . . . . . . . . . . 13  |-  ( ph  ->  F  =  ( z  e.  ( 0 [,] 1 )  |->  ( I `
 ( 1  -  z ) ) ) )
7271oveq1d 6063 . . . . . . . . . . . 12  |-  ( ph  ->  ( F ( *p
`  J ) ( h ( *p `  J ) I ) )  =  ( ( z  e.  ( 0 [,] 1 )  |->  ( I `  ( 1  -  z ) ) ) ( *p `  J ) ( h ( *p `  J
) I ) ) )
73 eceq1 6908 . . . . . . . . . . . 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 3959 . . . . . . . . . 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 4264 . . . . . . . . 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 5064 . . . . . . . 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 2456 . . . . . . 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 5015 . . . . . 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 3994 . . . . . . . . 9  |-  ( ph  ->  U. B  =  U. ( Base `  P )
)
8271oveq2d 6064 . . . . . . . . . . . 12  |-  ( ph  ->  ( g ( *p
`  J ) F )  =  ( g ( *p `  J
) ( z  e.  ( 0 [,] 1
)  |->  ( I `  ( 1  -  z
) ) ) ) )
8382oveq2d 6064 . . . . . . . . . . 11  |-  ( ph  ->  ( I ( *p
`  J ) ( g ( *p `  J ) F ) )  =  ( I ( *p `  J
) ( g ( *p `  J ) ( z  e.  ( 0 [,] 1 ) 
|->  ( I `  (
1  -  z ) ) ) ) ) )
84 eceq1 6908 . . . . . . . . . . 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 3959 . . . . . . . . 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 4255 . . . . . . . 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 5064 . . . . . . 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 2456 . . . . . 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 3359 . . . . 5  |-  ( ph  ->  `' H  C_  G )
91 cnvss 5012 . . . . 5  |-  ( `' H  C_  G  ->  `' `' H  C_  `' G
)
9290, 91syl 16 . . . 4  |-  ( ph  ->  `' `' H  C_  `' G
)
9319, 92eqsstr3d 3351 . . 3  |-  ( ph  ->  H  C_  `' G
)
949, 93eqssd 3333 . 2  |-  ( ph  ->  `' G  =  H
)
9594, 78eqtrd 2444 . . 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 19041 . . 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 2486 . 2  |-  ( ph  ->  `' G  e.  ( Q  GrpHom  P ) )
9894, 97jca 519 1  |-  ( ph  ->  ( `' G  =  H  /\  `' G  e.  ( Q  GrpHom  P ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721   _Vcvv 2924    C_ wss 3288   <.cop 3785   U.cuni 3983    e. cmpt 4234    X. cxp 4843   `'ccnv 4844   ran crn 4846   Rel wrel 4850   -->wf 5417   ` cfv 5421  (class class class)co 6048   [cec 6870   CCcc 8952   RRcr 8953   0cc0 8954   1c1 8955    - cmin 9255   [,]cicc 10883   Basecbs 13432    GrpHom cghm 14966  TopOnctopon 16922    Cn ccn 17250   IIcii 18866    ~=ph cphtpc 18955   *pcpco 18986    pi 1 cpi1 18989
This theorem is referenced by:  pi1xfrgim  19044
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-rep 4288  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668  ax-inf2 7560  ax-cnex 9010  ax-resscn 9011  ax-1cn 9012  ax-icn 9013  ax-addcl 9014  ax-addrcl 9015  ax-mulcl 9016  ax-mulrcl 9017  ax-mulcom 9018  ax-addass 9019  ax-mulass 9020  ax-distr 9021  ax-i2m1 9022  ax-1ne0 9023  ax-1rid 9024  ax-rnegex 9025  ax-rrecex 9026  ax-cnre 9027  ax-pre-lttri 9028  ax-pre-lttrn 9029  ax-pre-ltadd 9030  ax-pre-mulgt0 9031  ax-pre-sup 9032  ax-addf 9033  ax-mulf 9034
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-nel 2578  df-ral 2679  df-rex 2680  df-reu 2681  df-rmo 2682  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-pss 3304  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-tp 3790  df-op 3791  df-uni 3984  df-int 4019  df-iun 4063  df-iin 4064  df-br 4181  df-opab 4235  df-mpt 4236  df-tr 4271  df-eprel 4462  df-id 4466  df-po 4471  df-so 4472  df-fr 4509  df-se 4510  df-we 4511  df-ord 4552  df-on 4553  df-lim 4554  df-suc 4555  df-om 4813  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-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-isom 5430  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-of 6272  df-1st 6316  df-2nd 6317  df-riota 6516  df-recs 6600  df-rdg 6635  df-1o 6691  df-2o 6692  df-oadd 6695  df-er 6872  df-ec 6874  df-qs 6878  df-map 6987  df-ixp 7031  df-en 7077  df-dom 7078  df-sdom 7079  df-fin 7080  df-fi 7382  df-sup 7412  df-oi 7443  df-card 7790  df-cda 8012  df-pnf 9086  df-mnf 9087  df-xr 9088  df-ltxr 9089  df-le 9090  df-sub 9257  df-neg 9258  df-div 9642  df-nn 9965  df-2 10022  df-3 10023  df-4 10024  df-5 10025  df-6 10026  df-7 10027  df-8 10028  df-9 10029  df-10 10030  df-n0 10186  df-z 10247  df-dec 10347  df-uz 10453  df-q 10539  df-rp 10577  df-xneg 10674  df-xadd 10675  df-xmul 10676  df-ioo 10884  df-icc 10887  df-fz 11008  df-fzo 11099  df-seq 11287  df-exp 11346  df-hash 11582  df-cj 11867  df-re 11868  df-im 11869  df-sqr 12003  df-abs 12004  df-struct 13434  df-ndx 13435  df-slot 13436  df-base 13437  df-sets 13438  df-ress 13439  df-plusg 13505  df-mulr 13506  df-starv 13507  df-sca 13508  df-vsca 13509  df-tset 13511  df-ple 13512  df-ds 13514  df-unif 13515  df-hom 13516  df-cco 13517  df-rest 13613  df-topn 13614  df-topgen 13630  df-pt 13631  df-prds 13634  df-xrs 13689  df-0g 13690  df-gsum 13691  df-qtop 13696  df-imas 13697  df-divs 13698  df-xps 13699  df-mre 13774  df-mrc 13775  df-acs 13777  df-mnd 14653  df-submnd 14702  df-grp 14775  df-mulg 14778  df-ghm 14967  df-cntz 15079  df-cmn 15377  df-psmet 16657  df-xmet 16658  df-met 16659  df-bl 16660  df-mopn 16661  df-cnfld 16667  df-top 16926  df-bases 16928  df-topon 16929  df-topsp 16930  df-cld 17046  df-cn 17253  df-cnp 17254  df-tx 17555  df-hmeo 17748  df-xms 18311  df-ms 18312  df-tms 18313  df-ii 18868  df-htpy 18956  df-phtpy 18957  df-phtpc 18978  df-pco 18991  df-om1 18992  df-pi1 18994
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