MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  pi1xfrcnv Structured version   Unicode version

Theorem pi1xfrcnv 20627
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 20626 . . 3  |-  ( ph  ->  `' G  C_  H )
10 fvex 5699 . . . . . . . 8  |-  (  ~=ph  `  J )  e.  _V
11 ecexg 7103 . . . . . . . 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 7103 . . . . . . . 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 5999 . . . . . 6  |-  ( ph  ->  H  C_  ( _V  X.  _V ) )
16 df-rel 4845 . . . . . 6  |-  ( Rel 
H  <->  H  C_  ( _V 
X.  _V ) )
1715, 16sylibr 212 . . . . 5  |-  ( ph  ->  Rel  H )
18 dfrel2 5286 . . . . 5  |-  ( Rel 
H  <->  `' `' H  =  H
)
1917, 18sylib 196 . . . 4  |-  ( ph  ->  `' `' H  =  H
)
20 0elunit 11401 . . . . . . . . . 10  |-  0  e.  ( 0 [,] 1
)
21 oveq2 6097 . . . . . . . . . . . . 13  |-  ( x  =  0  ->  (
1  -  x )  =  ( 1  -  0 ) )
22 1m0e1 10430 . . . . . . . . . . . . 13  |-  ( 1  -  0 )  =  1
2321, 22syl6eq 2489 . . . . . . . . . . . 12  |-  ( x  =  0  ->  (
1  -  x )  =  1 )
2423fveq2d 5693 . . . . . . . . . . 11  |-  ( x  =  0  ->  ( F `  ( 1  -  x ) )  =  ( F `  1
) )
25 fvex 5699 . . . . . . . . . . 11  |-  ( F `
 1 )  e. 
_V
2624, 7, 25fvmpt 5772 . . . . . . . . . 10  |-  ( 0  e.  ( 0 [,] 1 )  ->  (
I `  0 )  =  ( F ` 
1 ) )
2720, 26ax-mp 5 . . . . . . . . 9  |-  ( I `
 0 )  =  ( F `  1
)
2827oveq2i 6100 . . . . . . . 8  |-  ( J  pi1  ( I `
 0 ) )  =  ( J  pi1  ( F ` 
1 ) )
292, 28eqtr4i 2464 . . . . . . 7  |-  Q  =  ( J  pi1 
( I `  0
) )
30 1elunit 11402 . . . . . . . . . 10  |-  1  e.  ( 0 [,] 1
)
31 oveq2 6097 . . . . . . . . . . . . 13  |-  ( x  =  1  ->  (
1  -  x )  =  ( 1  -  1 ) )
3231fveq2d 5693 . . . . . . . . . . . 12  |-  ( x  =  1  ->  ( F `  ( 1  -  x ) )  =  ( F `  (
1  -  1 ) ) )
33 1m1e0 10388 . . . . . . . . . . . . 13  |-  ( 1  -  1 )  =  0
3433fveq2i 5692 . . . . . . . . . . . 12  |-  ( F `
 ( 1  -  1 ) )  =  ( F `  0
)
3532, 34syl6eq 2489 . . . . . . . . . . 11  |-  ( x  =  1  ->  ( F `  ( 1  -  x ) )  =  ( F `  0
) )
36 fvex 5699 . . . . . . . . . . 11  |-  ( F `
 0 )  e. 
_V
3735, 7, 36fvmpt 5772 . . . . . . . . . 10  |-  ( 1  e.  ( 0 [,] 1 )  ->  (
I `  1 )  =  ( F ` 
0 ) )
3830, 37ax-mp 5 . . . . . . . . 9  |-  ( I `
 1 )  =  ( F `  0
)
3938oveq2i 6100 . . . . . . . 8  |-  ( J  pi1  ( I `
 1 ) )  =  ( J  pi1  ( F ` 
0 ) )
401, 39eqtr4i 2464 . . . . . . 7  |-  P  =  ( J  pi1 
( I `  1
) )
41 eqid 2441 . . . . . . 7  |-  ( Base `  Q )  =  (
Base `  Q )
42 eqid 2441 . . . . . . 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 20595 . . . . . . . . 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 1000 . . . . . . 7  |-  ( ph  ->  I  e.  ( II 
Cn  J ) )
46 oveq2 6097 . . . . . . . . 9  |-  ( z  =  y  ->  (
1  -  z )  =  ( 1  -  y ) )
4746fveq2d 5693 . . . . . . . 8  |-  ( z  =  y  ->  (
I `  ( 1  -  z ) )  =  ( I `  ( 1  -  y
) ) )
4847cbvmptv 4381 . . . . . . 7  |-  ( z  e.  ( 0 [,] 1 )  |->  ( I `
 ( 1  -  z ) ) )  =  ( y  e.  ( 0 [,] 1
)  |->  ( I `  ( 1  -  y
) ) )
49 eqid 2441 . . . . . . 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 20626 . . . . . 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 20453 . . . . . . . . . . . . . . . . 17  |-  II  e.  (TopOn `  ( 0 [,] 1 ) )
5251a1i 11 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  II  e.  (TopOn `  ( 0 [,] 1
) ) )
53 cnf2 18851 . . . . . . . . . . . . . . . 16  |-  ( ( II  e.  (TopOn `  ( 0 [,] 1
) )  /\  J  e.  (TopOn `  X )  /\  F  e.  (
II  Cn  J )
)  ->  F :
( 0 [,] 1
) --> X )
5452, 5, 6, 53syl3anc 1218 . . . . . . . . . . . . . . 15  |-  ( ph  ->  F : ( 0 [,] 1 ) --> X )
5554feqmptd 5742 . . . . . . . . . . . . . 14  |-  ( ph  ->  F  =  ( z  e.  ( 0 [,] 1 )  |->  ( F `
 z ) ) )
56 iirev 20499 . . . . . . . . . . . . . . . . 17  |-  ( z  e.  ( 0 [,] 1 )  ->  (
1  -  z )  e.  ( 0 [,] 1 ) )
57 oveq2 6097 . . . . . . . . . . . . . . . . . . 19  |-  ( x  =  ( 1  -  z )  ->  (
1  -  x )  =  ( 1  -  ( 1  -  z
) ) )
5857fveq2d 5693 . . . . . . . . . . . . . . . . . 18  |-  ( x  =  ( 1  -  z )  ->  ( F `  ( 1  -  x ) )  =  ( F `  (
1  -  ( 1  -  z ) ) ) )
59 fvex 5699 . . . . . . . . . . . . . . . . . 18  |-  ( F `
 ( 1  -  ( 1  -  z
) ) )  e. 
_V
6058, 7, 59fvmpt 5772 . . . . . . . . . . . . . . . . 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 9338 . . . . . . . . . . . . . . . . . 18  |-  1  e.  CC
63 unitssre 11430 . . . . . . . . . . . . . . . . . . . 20  |-  ( 0 [,] 1 )  C_  RR
6463sseli 3350 . . . . . . . . . . . . . . . . . . 19  |-  ( z  e.  ( 0 [,] 1 )  ->  z  e.  RR )
6564recnd 9410 . . . . . . . . . . . . . . . . . 18  |-  ( z  e.  ( 0 [,] 1 )  ->  z  e.  CC )
66 nncan 9636 . . . . . . . . . . . . . . . . . 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 5693 . . . . . . . . . . . . . . . 16  |-  ( z  e.  ( 0 [,] 1 )  ->  ( F `  ( 1  -  ( 1  -  z ) ) )  =  ( F `  z ) )
6961, 68eqtrd 2473 . . . . . . . . . . . . . . 15  |-  ( z  e.  ( 0 [,] 1 )  ->  (
I `  ( 1  -  z ) )  =  ( F `  z ) )
7069mpteq2ia 4372 . . . . . . . . . . . . . 14  |-  ( z  e.  ( 0 [,] 1 )  |->  ( I `
 ( 1  -  z ) ) )  =  ( z  e.  ( 0 [,] 1
)  |->  ( F `  z ) )
7155, 70syl6eqr 2491 . . . . . . . . . . . . 13  |-  ( ph  ->  F  =  ( z  e.  ( 0 [,] 1 )  |->  ( I `
 ( 1  -  z ) ) ) )
7271oveq1d 6104 . . . . . . . . . . . 12  |-  ( ph  ->  ( F ( *p
`  J ) ( h ( *p `  J ) I ) )  =  ( ( z  e.  ( 0 [,] 1 )  |->  ( I `  ( 1  -  z ) ) ) ( *p `  J ) ( h ( *p `  J
) I ) ) )
73 eceq1 7135 . . . . . . . . . . . 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 4064 . . . . . . . . . 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 4377 . . . . . . . . 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 5065 . . . . . . . 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 2485 . . . . . . 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 5013 . . . . . 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 4099 . . . . . . . . 9  |-  ( ph  ->  U. B  =  U. ( Base `  P )
)
8271oveq2d 6105 . . . . . . . . . . . 12  |-  ( ph  ->  ( g ( *p
`  J ) F )  =  ( g ( *p `  J
) ( z  e.  ( 0 [,] 1
)  |->  ( I `  ( 1  -  z
) ) ) ) )
8382oveq2d 6105 . . . . . . . . . . 11  |-  ( ph  ->  ( I ( *p
`  J ) ( g ( *p `  J ) F ) )  =  ( I ( *p `  J
) ( g ( *p `  J ) ( z  e.  ( 0 [,] 1 ) 
|->  ( I `  (
1  -  z ) ) ) ) ) )
84 eceq1 7135 . . . . . . . . . . 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 4064 . . . . . . . . 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 4368 . . . . . . . 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 5065 . . . . . . 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 2485 . . . . . 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 )
>. ) )
9050, 79, 893sstr4d 3397 . . . . 5  |-  ( ph  ->  `' H  C_  G )
91 cnvss 5010 . . . . 5  |-  ( `' H  C_  G  ->  `' `' H  C_  `' G
)
9290, 91syl 16 . . . 4  |-  ( ph  ->  `' `' H  C_  `' G
)
9319, 92eqsstr3d 3389 . . 3  |-  ( ph  ->  H  C_  `' G
)
949, 93eqssd 3371 . 2  |-  ( ph  ->  `' G  =  H
)
9594, 78eqtrd 2473 . . 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 )
>. ) )
9629, 40, 41, 42, 5, 45, 48pi1xfr 20625 . . 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 2515 . 2  |-  ( ph  ->  `' G  e.  ( Q  GrpHom  P ) )
9894, 97jca 532 1  |-  ( ph  ->  ( `' G  =  H  /\  `' G  e.  ( Q  GrpHom  P ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   _Vcvv 2970    C_ wss 3326   <.cop 3881   U.cuni 4089    e. cmpt 4348    X. cxp 4836   `'ccnv 4837   ran crn 4839   Rel wrel 4843   -->wf 5412   ` cfv 5416  (class class class)co 6089   [cec 7097   CCcc 9278   RRcr 9279   0cc0 9280   1c1 9281    - cmin 9593   [,]cicc 11301   Basecbs 14172    GrpHom cghm 15742  TopOnctopon 18497    Cn ccn 18826   IIcii 20449    ~=ph cphtpc 20539   *pcpco 20570    pi1 cpi1 20573
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2422  ax-rep 4401  ax-sep 4411  ax-nul 4419  ax-pow 4468  ax-pr 4529  ax-un 6370  ax-inf2 7845  ax-cnex 9336  ax-resscn 9337  ax-1cn 9338  ax-icn 9339  ax-addcl 9340  ax-addrcl 9341  ax-mulcl 9342  ax-mulrcl 9343  ax-mulcom 9344  ax-addass 9345  ax-mulass 9346  ax-distr 9347  ax-i2m1 9348  ax-1ne0 9349  ax-1rid 9350  ax-rnegex 9351  ax-rrecex 9352  ax-cnre 9353  ax-pre-lttri 9354  ax-pre-lttrn 9355  ax-pre-ltadd 9356  ax-pre-mulgt0 9357  ax-pre-sup 9358  ax-addf 9359  ax-mulf 9360
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3185  df-csb 3287  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-pss 3342  df-nul 3636  df-if 3790  df-pw 3860  df-sn 3876  df-pr 3878  df-tp 3880  df-op 3882  df-uni 4090  df-int 4127  df-iun 4171  df-iin 4172  df-br 4291  df-opab 4349  df-mpt 4350  df-tr 4384  df-eprel 4630  df-id 4634  df-po 4639  df-so 4640  df-fr 4677  df-se 4678  df-we 4679  df-ord 4720  df-on 4721  df-lim 4722  df-suc 4723  df-xp 4844  df-rel 4845  df-cnv 4846  df-co 4847  df-dm 4848  df-rn 4849  df-res 4850  df-ima 4851  df-iota 5379  df-fun 5418  df-fn 5419  df-f 5420  df-f1 5421  df-fo 5422  df-f1o 5423  df-fv 5424  df-isom 5425  df-riota 6050  df-ov 6092  df-oprab 6093  df-mpt2 6094  df-of 6318  df-om 6475  df-1st 6575  df-2nd 6576  df-supp 6689  df-recs 6830  df-rdg 6864  df-1o 6918  df-2o 6919  df-oadd 6922  df-er 7099  df-ec 7101  df-qs 7105  df-map 7214  df-ixp 7262  df-en 7309  df-dom 7310  df-sdom 7311  df-fin 7312  df-fsupp 7619  df-fi 7659  df-sup 7689  df-oi 7722  df-card 8107  df-cda 8335  df-pnf 9418  df-mnf 9419  df-xr 9420  df-ltxr 9421  df-le 9422  df-sub 9595  df-neg 9596  df-div 9992  df-nn 10321  df-2 10378  df-3 10379  df-4 10380  df-5 10381  df-6 10382  df-7 10383  df-8 10384  df-9 10385  df-10 10386  df-n0 10578  df-z 10645  df-dec 10754  df-uz 10860  df-q 10952  df-rp 10990  df-xneg 11087  df-xadd 11088  df-xmul 11089  df-ioo 11302  df-icc 11305  df-fz 11436  df-fzo 11547  df-seq 11805  df-exp 11864  df-hash 12102  df-cj 12586  df-re 12587  df-im 12588  df-sqr 12722  df-abs 12723  df-struct 14174  df-ndx 14175  df-slot 14176  df-base 14177  df-sets 14178  df-ress 14179  df-plusg 14249  df-mulr 14250  df-starv 14251  df-sca 14252  df-vsca 14253  df-ip 14254  df-tset 14255  df-ple 14256  df-ds 14258  df-unif 14259  df-hom 14260  df-cco 14261  df-rest 14359  df-topn 14360  df-0g 14378  df-gsum 14379  df-topgen 14380  df-pt 14381  df-prds 14384  df-xrs 14438  df-qtop 14443  df-imas 14444  df-divs 14445  df-xps 14446  df-mre 14522  df-mrc 14523  df-acs 14525  df-mnd 15413  df-submnd 15463  df-grp 15543  df-mulg 15546  df-ghm 15743  df-cntz 15833  df-cmn 16277  df-psmet 17807  df-xmet 17808  df-met 17809  df-bl 17810  df-mopn 17811  df-cnfld 17817  df-top 18501  df-bases 18503  df-topon 18504  df-topsp 18505  df-cld 18621  df-cn 18829  df-cnp 18830  df-tx 19133  df-hmeo 19326  df-xms 19893  df-ms 19894  df-tms 19895  df-ii 20451  df-htpy 20540  df-phtpy 20541  df-phtpc 20562  df-pco 20575  df-om1 20576  df-pi1 20578
This theorem is referenced by:  pi1xfrgim  20628
  Copyright terms: Public domain W3C validator