Step |
Hyp |
Ref |
Expression |
1 |
|
pi1xfr.p |
|- P = ( J pi1 ( F ` 0 ) ) |
2 |
|
pi1xfr.q |
|- Q = ( J pi1 ( F ` 1 ) ) |
3 |
|
pi1xfr.b |
|- B = ( Base ` P ) |
4 |
|
pi1xfr.g |
|- G = ran ( g e. U. B |-> <. [ g ] ( ~=ph ` J ) , [ ( I ( *p ` J ) ( g ( *p ` J ) F ) ) ] ( ~=ph ` J ) >. ) |
5 |
|
pi1xfr.j |
|- ( ph -> J e. ( TopOn ` X ) ) |
6 |
|
pi1xfr.f |
|- ( ph -> F e. ( II Cn J ) ) |
7 |
|
pi1xfr.i |
|- I = ( x e. ( 0 [,] 1 ) |-> ( F ` ( 1 - x ) ) ) |
8 |
|
pi1xfrcnv.h |
|- H = ran ( h e. U. ( Base ` Q ) |-> <. [ h ] ( ~=ph ` J ) , [ ( F ( *p ` J ) ( h ( *p ` J ) I ) ) ] ( ~=ph ` J ) >. ) |
9 |
1 2 3 4 5 6 7 8
|
pi1xfrcnvlem |
|- ( ph -> `' G C_ H ) |
10 |
|
fvex |
|- ( ~=ph ` J ) e. _V |
11 |
|
ecexg |
|- ( ( ~=ph ` J ) e. _V -> [ h ] ( ~=ph ` J ) e. _V ) |
12 |
10 11
|
mp1i |
|- ( ( ph /\ h e. U. ( Base ` Q ) ) -> [ h ] ( ~=ph ` J ) e. _V ) |
13 |
|
ecexg |
|- ( ( ~=ph ` J ) e. _V -> [ ( F ( *p ` J ) ( h ( *p ` J ) I ) ) ] ( ~=ph ` J ) e. _V ) |
14 |
10 13
|
mp1i |
|- ( ( ph /\ h e. U. ( Base ` Q ) ) -> [ ( F ( *p ` J ) ( h ( *p ` J ) I ) ) ] ( ~=ph ` J ) e. _V ) |
15 |
8 12 14
|
fliftrel |
|- ( ph -> H C_ ( _V X. _V ) ) |
16 |
|
df-rel |
|- ( Rel H <-> H C_ ( _V X. _V ) ) |
17 |
15 16
|
sylibr |
|- ( ph -> Rel H ) |
18 |
|
dfrel2 |
|- ( Rel H <-> `' `' H = H ) |
19 |
17 18
|
sylib |
|- ( ph -> `' `' H = H ) |
20 |
|
0elunit |
|- 0 e. ( 0 [,] 1 ) |
21 |
|
oveq2 |
|- ( x = 0 -> ( 1 - x ) = ( 1 - 0 ) ) |
22 |
|
1m0e1 |
|- ( 1 - 0 ) = 1 |
23 |
21 22
|
eqtrdi |
|- ( x = 0 -> ( 1 - x ) = 1 ) |
24 |
23
|
fveq2d |
|- ( x = 0 -> ( F ` ( 1 - x ) ) = ( F ` 1 ) ) |
25 |
|
fvex |
|- ( F ` 1 ) e. _V |
26 |
24 7 25
|
fvmpt |
|- ( 0 e. ( 0 [,] 1 ) -> ( I ` 0 ) = ( F ` 1 ) ) |
27 |
20 26
|
ax-mp |
|- ( I ` 0 ) = ( F ` 1 ) |
28 |
27
|
oveq2i |
|- ( J pi1 ( I ` 0 ) ) = ( J pi1 ( F ` 1 ) ) |
29 |
2 28
|
eqtr4i |
|- Q = ( J pi1 ( I ` 0 ) ) |
30 |
|
1elunit |
|- 1 e. ( 0 [,] 1 ) |
31 |
|
oveq2 |
|- ( x = 1 -> ( 1 - x ) = ( 1 - 1 ) ) |
32 |
31
|
fveq2d |
|- ( x = 1 -> ( F ` ( 1 - x ) ) = ( F ` ( 1 - 1 ) ) ) |
33 |
|
1m1e0 |
|- ( 1 - 1 ) = 0 |
34 |
33
|
fveq2i |
|- ( F ` ( 1 - 1 ) ) = ( F ` 0 ) |
35 |
32 34
|
eqtrdi |
|- ( x = 1 -> ( F ` ( 1 - x ) ) = ( F ` 0 ) ) |
36 |
|
fvex |
|- ( F ` 0 ) e. _V |
37 |
35 7 36
|
fvmpt |
|- ( 1 e. ( 0 [,] 1 ) -> ( I ` 1 ) = ( F ` 0 ) ) |
38 |
30 37
|
ax-mp |
|- ( I ` 1 ) = ( F ` 0 ) |
39 |
38
|
oveq2i |
|- ( J pi1 ( I ` 1 ) ) = ( J pi1 ( F ` 0 ) ) |
40 |
1 39
|
eqtr4i |
|- P = ( J pi1 ( I ` 1 ) ) |
41 |
|
eqid |
|- ( Base ` Q ) = ( Base ` Q ) |
42 |
|
eqid |
|- 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 ) >. ) |
43 |
7
|
pcorevcl |
|- ( F e. ( II Cn J ) -> ( I e. ( II Cn J ) /\ ( I ` 0 ) = ( F ` 1 ) /\ ( I ` 1 ) = ( F ` 0 ) ) ) |
44 |
6 43
|
syl |
|- ( ph -> ( I e. ( II Cn J ) /\ ( I ` 0 ) = ( F ` 1 ) /\ ( I ` 1 ) = ( F ` 0 ) ) ) |
45 |
44
|
simp1d |
|- ( ph -> I e. ( II Cn J ) ) |
46 |
|
oveq2 |
|- ( z = y -> ( 1 - z ) = ( 1 - y ) ) |
47 |
46
|
fveq2d |
|- ( z = y -> ( I ` ( 1 - z ) ) = ( I ` ( 1 - y ) ) ) |
48 |
47
|
cbvmptv |
|- ( z e. ( 0 [,] 1 ) |-> ( I ` ( 1 - z ) ) ) = ( y e. ( 0 [,] 1 ) |-> ( I ` ( 1 - y ) ) ) |
49 |
|
eqid |
|- 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 ) >. ) |
50 |
29 40 41 42 5 45 48 49
|
pi1xfrcnvlem |
|- ( 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 |
|- II e. ( TopOn ` ( 0 [,] 1 ) ) |
52 |
|
cnf2 |
|- ( ( II e. ( TopOn ` ( 0 [,] 1 ) ) /\ J e. ( TopOn ` X ) /\ F e. ( II Cn J ) ) -> F : ( 0 [,] 1 ) --> X ) |
53 |
51 5 6 52
|
mp3an2i |
|- ( ph -> F : ( 0 [,] 1 ) --> X ) |
54 |
53
|
feqmptd |
|- ( ph -> F = ( z e. ( 0 [,] 1 ) |-> ( F ` z ) ) ) |
55 |
|
iirev |
|- ( z e. ( 0 [,] 1 ) -> ( 1 - z ) e. ( 0 [,] 1 ) ) |
56 |
|
oveq2 |
|- ( x = ( 1 - z ) -> ( 1 - x ) = ( 1 - ( 1 - z ) ) ) |
57 |
56
|
fveq2d |
|- ( x = ( 1 - z ) -> ( F ` ( 1 - x ) ) = ( F ` ( 1 - ( 1 - z ) ) ) ) |
58 |
|
fvex |
|- ( F ` ( 1 - ( 1 - z ) ) ) e. _V |
59 |
57 7 58
|
fvmpt |
|- ( ( 1 - z ) e. ( 0 [,] 1 ) -> ( I ` ( 1 - z ) ) = ( F ` ( 1 - ( 1 - z ) ) ) ) |
60 |
55 59
|
syl |
|- ( z e. ( 0 [,] 1 ) -> ( I ` ( 1 - z ) ) = ( F ` ( 1 - ( 1 - z ) ) ) ) |
61 |
|
ax-1cn |
|- 1 e. CC |
62 |
|
unitssre |
|- ( 0 [,] 1 ) C_ RR |
63 |
62
|
sseli |
|- ( z e. ( 0 [,] 1 ) -> z e. RR ) |
64 |
63
|
recnd |
|- ( z e. ( 0 [,] 1 ) -> z e. CC ) |
65 |
|
nncan |
|- ( ( 1 e. CC /\ z e. CC ) -> ( 1 - ( 1 - z ) ) = z ) |
66 |
61 64 65
|
sylancr |
|- ( z e. ( 0 [,] 1 ) -> ( 1 - ( 1 - z ) ) = z ) |
67 |
66
|
fveq2d |
|- ( z e. ( 0 [,] 1 ) -> ( F ` ( 1 - ( 1 - z ) ) ) = ( F ` z ) ) |
68 |
60 67
|
eqtrd |
|- ( z e. ( 0 [,] 1 ) -> ( I ` ( 1 - z ) ) = ( F ` z ) ) |
69 |
68
|
mpteq2ia |
|- ( z e. ( 0 [,] 1 ) |-> ( I ` ( 1 - z ) ) ) = ( z e. ( 0 [,] 1 ) |-> ( F ` z ) ) |
70 |
54 69
|
eqtr4di |
|- ( ph -> F = ( z e. ( 0 [,] 1 ) |-> ( I ` ( 1 - z ) ) ) ) |
71 |
70
|
oveq1d |
|- ( ph -> ( F ( *p ` J ) ( h ( *p ` J ) I ) ) = ( ( z e. ( 0 [,] 1 ) |-> ( I ` ( 1 - z ) ) ) ( *p ` J ) ( h ( *p ` J ) I ) ) ) |
72 |
71
|
eceq1d |
|- ( 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 ) ) |
73 |
72
|
opeq2d |
|- ( 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 ) >. ) |
74 |
73
|
mpteq2dv |
|- ( 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 ) >. ) ) |
75 |
74
|
rneqd |
|- ( 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 ) >. ) ) |
76 |
8 75
|
eqtrid |
|- ( 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 ) >. ) ) |
77 |
76
|
cnveqd |
|- ( 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 ) >. ) ) |
78 |
3
|
a1i |
|- ( ph -> B = ( Base ` P ) ) |
79 |
78
|
unieqd |
|- ( ph -> U. B = U. ( Base ` P ) ) |
80 |
70
|
oveq2d |
|- ( ph -> ( g ( *p ` J ) F ) = ( g ( *p ` J ) ( z e. ( 0 [,] 1 ) |-> ( I ` ( 1 - z ) ) ) ) ) |
81 |
80
|
oveq2d |
|- ( ph -> ( I ( *p ` J ) ( g ( *p ` J ) F ) ) = ( I ( *p ` J ) ( g ( *p ` J ) ( z e. ( 0 [,] 1 ) |-> ( I ` ( 1 - z ) ) ) ) ) ) |
82 |
81
|
eceq1d |
|- ( 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 ) ) |
83 |
82
|
opeq2d |
|- ( 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 ) >. ) |
84 |
79 83
|
mpteq12dv |
|- ( 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 ) >. ) ) |
85 |
84
|
rneqd |
|- ( 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 ) >. ) ) |
86 |
4 85
|
eqtrid |
|- ( 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 ) >. ) ) |
87 |
50 77 86
|
3sstr4d |
|- ( ph -> `' H C_ G ) |
88 |
|
cnvss |
|- ( `' H C_ G -> `' `' H C_ `' G ) |
89 |
87 88
|
syl |
|- ( ph -> `' `' H C_ `' G ) |
90 |
19 89
|
eqsstrrd |
|- ( ph -> H C_ `' G ) |
91 |
9 90
|
eqssd |
|- ( ph -> `' G = H ) |
92 |
91 76
|
eqtrd |
|- ( 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 ) >. ) ) |
93 |
29 40 41 42 5 45 48
|
pi1xfr |
|- ( 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 ) ) |
94 |
92 93
|
eqeltrd |
|- ( ph -> `' G e. ( Q GrpHom P ) ) |
95 |
91 94
|
jca |
|- ( ph -> ( `' G = H /\ `' G e. ( Q GrpHom P ) ) ) |