Step |
Hyp |
Ref |
Expression |
1 |
|
cvmliftlem.1 |
|- S = ( k e. J |-> { s e. ( ~P C \ { (/) } ) | ( U. s = ( `' F " k ) /\ A. u e. s ( A. v e. ( s \ { u } ) ( u i^i v ) = (/) /\ ( F |` u ) e. ( ( C |`t u ) Homeo ( J |`t k ) ) ) ) } ) |
2 |
|
cvmliftlem.b |
|- B = U. C |
3 |
|
cvmliftlem.x |
|- X = U. J |
4 |
|
cvmliftlem.f |
|- ( ph -> F e. ( C CovMap J ) ) |
5 |
|
cvmliftlem.g |
|- ( ph -> G e. ( II Cn J ) ) |
6 |
|
cvmliftlem.p |
|- ( ph -> P e. B ) |
7 |
|
cvmliftlem.e |
|- ( ph -> ( F ` P ) = ( G ` 0 ) ) |
8 |
|
cvmliftlem.n |
|- ( ph -> N e. NN ) |
9 |
|
cvmliftlem.t |
|- ( ph -> T : ( 1 ... N ) --> U_ j e. J ( { j } X. ( S ` j ) ) ) |
10 |
|
cvmliftlem.a |
|- ( ph -> A. k e. ( 1 ... N ) ( G " ( ( ( k - 1 ) / N ) [,] ( k / N ) ) ) C_ ( 1st ` ( T ` k ) ) ) |
11 |
|
cvmliftlem.l |
|- L = ( topGen ` ran (,) ) |
12 |
|
cvmliftlem.q |
|- Q = seq 0 ( ( x e. _V , m e. NN |-> ( z e. ( ( ( m - 1 ) / N ) [,] ( m / N ) ) |-> ( `' ( F |` ( iota_ b e. ( 2nd ` ( T ` m ) ) ( x ` ( ( m - 1 ) / N ) ) e. b ) ) ` ( G ` z ) ) ) ) , ( ( _I |` NN ) u. { <. 0 , { <. 0 , P >. } >. } ) ) |
13 |
|
cvmliftlem5.3 |
|- W = ( ( ( M - 1 ) / N ) [,] ( M / N ) ) |
14 |
|
cvmliftlem6.1 |
|- ( ( ph /\ ps ) -> M e. ( 1 ... N ) ) |
15 |
|
cvmliftlem6.2 |
|- ( ( ph /\ ps ) -> ( ( Q ` ( M - 1 ) ) ` ( ( M - 1 ) / N ) ) e. ( `' F " { ( G ` ( ( M - 1 ) / N ) ) } ) ) |
16 |
14
|
adantrr |
|- ( ( ph /\ ( ps /\ z e. W ) ) -> M e. ( 1 ... N ) ) |
17 |
1 2 3 4 5 6 7 8 9 10 11 16
|
cvmliftlem1 |
|- ( ( ph /\ ( ps /\ z e. W ) ) -> ( 2nd ` ( T ` M ) ) e. ( S ` ( 1st ` ( T ` M ) ) ) ) |
18 |
1
|
cvmsss |
|- ( ( 2nd ` ( T ` M ) ) e. ( S ` ( 1st ` ( T ` M ) ) ) -> ( 2nd ` ( T ` M ) ) C_ C ) |
19 |
17 18
|
syl |
|- ( ( ph /\ ( ps /\ z e. W ) ) -> ( 2nd ` ( T ` M ) ) C_ C ) |
20 |
4
|
adantr |
|- ( ( ph /\ ( ps /\ z e. W ) ) -> F e. ( C CovMap J ) ) |
21 |
15
|
adantrr |
|- ( ( ph /\ ( ps /\ z e. W ) ) -> ( ( Q ` ( M - 1 ) ) ` ( ( M - 1 ) / N ) ) e. ( `' F " { ( G ` ( ( M - 1 ) / N ) ) } ) ) |
22 |
|
cvmcn |
|- ( F e. ( C CovMap J ) -> F e. ( C Cn J ) ) |
23 |
2 3
|
cnf |
|- ( F e. ( C Cn J ) -> F : B --> X ) |
24 |
20 22 23
|
3syl |
|- ( ( ph /\ ( ps /\ z e. W ) ) -> F : B --> X ) |
25 |
|
ffn |
|- ( F : B --> X -> F Fn B ) |
26 |
|
fniniseg |
|- ( F Fn B -> ( ( ( Q ` ( M - 1 ) ) ` ( ( M - 1 ) / N ) ) e. ( `' F " { ( G ` ( ( M - 1 ) / N ) ) } ) <-> ( ( ( Q ` ( M - 1 ) ) ` ( ( M - 1 ) / N ) ) e. B /\ ( F ` ( ( Q ` ( M - 1 ) ) ` ( ( M - 1 ) / N ) ) ) = ( G ` ( ( M - 1 ) / N ) ) ) ) ) |
27 |
24 25 26
|
3syl |
|- ( ( ph /\ ( ps /\ z e. W ) ) -> ( ( ( Q ` ( M - 1 ) ) ` ( ( M - 1 ) / N ) ) e. ( `' F " { ( G ` ( ( M - 1 ) / N ) ) } ) <-> ( ( ( Q ` ( M - 1 ) ) ` ( ( M - 1 ) / N ) ) e. B /\ ( F ` ( ( Q ` ( M - 1 ) ) ` ( ( M - 1 ) / N ) ) ) = ( G ` ( ( M - 1 ) / N ) ) ) ) ) |
28 |
21 27
|
mpbid |
|- ( ( ph /\ ( ps /\ z e. W ) ) -> ( ( ( Q ` ( M - 1 ) ) ` ( ( M - 1 ) / N ) ) e. B /\ ( F ` ( ( Q ` ( M - 1 ) ) ` ( ( M - 1 ) / N ) ) ) = ( G ` ( ( M - 1 ) / N ) ) ) ) |
29 |
28
|
simpld |
|- ( ( ph /\ ( ps /\ z e. W ) ) -> ( ( Q ` ( M - 1 ) ) ` ( ( M - 1 ) / N ) ) e. B ) |
30 |
28
|
simprd |
|- ( ( ph /\ ( ps /\ z e. W ) ) -> ( F ` ( ( Q ` ( M - 1 ) ) ` ( ( M - 1 ) / N ) ) ) = ( G ` ( ( M - 1 ) / N ) ) ) |
31 |
|
elfznn |
|- ( M e. ( 1 ... N ) -> M e. NN ) |
32 |
16 31
|
syl |
|- ( ( ph /\ ( ps /\ z e. W ) ) -> M e. NN ) |
33 |
32
|
nnred |
|- ( ( ph /\ ( ps /\ z e. W ) ) -> M e. RR ) |
34 |
|
peano2rem |
|- ( M e. RR -> ( M - 1 ) e. RR ) |
35 |
33 34
|
syl |
|- ( ( ph /\ ( ps /\ z e. W ) ) -> ( M - 1 ) e. RR ) |
36 |
8
|
adantr |
|- ( ( ph /\ ( ps /\ z e. W ) ) -> N e. NN ) |
37 |
35 36
|
nndivred |
|- ( ( ph /\ ( ps /\ z e. W ) ) -> ( ( M - 1 ) / N ) e. RR ) |
38 |
37
|
rexrd |
|- ( ( ph /\ ( ps /\ z e. W ) ) -> ( ( M - 1 ) / N ) e. RR* ) |
39 |
33 36
|
nndivred |
|- ( ( ph /\ ( ps /\ z e. W ) ) -> ( M / N ) e. RR ) |
40 |
39
|
rexrd |
|- ( ( ph /\ ( ps /\ z e. W ) ) -> ( M / N ) e. RR* ) |
41 |
33
|
ltm1d |
|- ( ( ph /\ ( ps /\ z e. W ) ) -> ( M - 1 ) < M ) |
42 |
36
|
nnred |
|- ( ( ph /\ ( ps /\ z e. W ) ) -> N e. RR ) |
43 |
36
|
nngt0d |
|- ( ( ph /\ ( ps /\ z e. W ) ) -> 0 < N ) |
44 |
|
ltdiv1 |
|- ( ( ( M - 1 ) e. RR /\ M e. RR /\ ( N e. RR /\ 0 < N ) ) -> ( ( M - 1 ) < M <-> ( ( M - 1 ) / N ) < ( M / N ) ) ) |
45 |
35 33 42 43 44
|
syl112anc |
|- ( ( ph /\ ( ps /\ z e. W ) ) -> ( ( M - 1 ) < M <-> ( ( M - 1 ) / N ) < ( M / N ) ) ) |
46 |
41 45
|
mpbid |
|- ( ( ph /\ ( ps /\ z e. W ) ) -> ( ( M - 1 ) / N ) < ( M / N ) ) |
47 |
37 39 46
|
ltled |
|- ( ( ph /\ ( ps /\ z e. W ) ) -> ( ( M - 1 ) / N ) <_ ( M / N ) ) |
48 |
|
lbicc2 |
|- ( ( ( ( M - 1 ) / N ) e. RR* /\ ( M / N ) e. RR* /\ ( ( M - 1 ) / N ) <_ ( M / N ) ) -> ( ( M - 1 ) / N ) e. ( ( ( M - 1 ) / N ) [,] ( M / N ) ) ) |
49 |
38 40 47 48
|
syl3anc |
|- ( ( ph /\ ( ps /\ z e. W ) ) -> ( ( M - 1 ) / N ) e. ( ( ( M - 1 ) / N ) [,] ( M / N ) ) ) |
50 |
49 13
|
eleqtrrdi |
|- ( ( ph /\ ( ps /\ z e. W ) ) -> ( ( M - 1 ) / N ) e. W ) |
51 |
1 2 3 4 5 6 7 8 9 10 11 16 13 50
|
cvmliftlem3 |
|- ( ( ph /\ ( ps /\ z e. W ) ) -> ( G ` ( ( M - 1 ) / N ) ) e. ( 1st ` ( T ` M ) ) ) |
52 |
30 51
|
eqeltrd |
|- ( ( ph /\ ( ps /\ z e. W ) ) -> ( F ` ( ( Q ` ( M - 1 ) ) ` ( ( M - 1 ) / N ) ) ) e. ( 1st ` ( T ` M ) ) ) |
53 |
|
eqid |
|- ( iota_ b e. ( 2nd ` ( T ` M ) ) ( ( Q ` ( M - 1 ) ) ` ( ( M - 1 ) / N ) ) e. b ) = ( iota_ b e. ( 2nd ` ( T ` M ) ) ( ( Q ` ( M - 1 ) ) ` ( ( M - 1 ) / N ) ) e. b ) |
54 |
1 2 53
|
cvmsiota |
|- ( ( F e. ( C CovMap J ) /\ ( ( 2nd ` ( T ` M ) ) e. ( S ` ( 1st ` ( T ` M ) ) ) /\ ( ( Q ` ( M - 1 ) ) ` ( ( M - 1 ) / N ) ) e. B /\ ( F ` ( ( Q ` ( M - 1 ) ) ` ( ( M - 1 ) / N ) ) ) e. ( 1st ` ( T ` M ) ) ) ) -> ( ( iota_ b e. ( 2nd ` ( T ` M ) ) ( ( Q ` ( M - 1 ) ) ` ( ( M - 1 ) / N ) ) e. b ) e. ( 2nd ` ( T ` M ) ) /\ ( ( Q ` ( M - 1 ) ) ` ( ( M - 1 ) / N ) ) e. ( iota_ b e. ( 2nd ` ( T ` M ) ) ( ( Q ` ( M - 1 ) ) ` ( ( M - 1 ) / N ) ) e. b ) ) ) |
55 |
20 17 29 52 54
|
syl13anc |
|- ( ( ph /\ ( ps /\ z e. W ) ) -> ( ( iota_ b e. ( 2nd ` ( T ` M ) ) ( ( Q ` ( M - 1 ) ) ` ( ( M - 1 ) / N ) ) e. b ) e. ( 2nd ` ( T ` M ) ) /\ ( ( Q ` ( M - 1 ) ) ` ( ( M - 1 ) / N ) ) e. ( iota_ b e. ( 2nd ` ( T ` M ) ) ( ( Q ` ( M - 1 ) ) ` ( ( M - 1 ) / N ) ) e. b ) ) ) |
56 |
55
|
simpld |
|- ( ( ph /\ ( ps /\ z e. W ) ) -> ( iota_ b e. ( 2nd ` ( T ` M ) ) ( ( Q ` ( M - 1 ) ) ` ( ( M - 1 ) / N ) ) e. b ) e. ( 2nd ` ( T ` M ) ) ) |
57 |
19 56
|
sseldd |
|- ( ( ph /\ ( ps /\ z e. W ) ) -> ( iota_ b e. ( 2nd ` ( T ` M ) ) ( ( Q ` ( M - 1 ) ) ` ( ( M - 1 ) / N ) ) e. b ) e. C ) |
58 |
|
elssuni |
|- ( ( iota_ b e. ( 2nd ` ( T ` M ) ) ( ( Q ` ( M - 1 ) ) ` ( ( M - 1 ) / N ) ) e. b ) e. C -> ( iota_ b e. ( 2nd ` ( T ` M ) ) ( ( Q ` ( M - 1 ) ) ` ( ( M - 1 ) / N ) ) e. b ) C_ U. C ) |
59 |
57 58
|
syl |
|- ( ( ph /\ ( ps /\ z e. W ) ) -> ( iota_ b e. ( 2nd ` ( T ` M ) ) ( ( Q ` ( M - 1 ) ) ` ( ( M - 1 ) / N ) ) e. b ) C_ U. C ) |
60 |
59 2
|
sseqtrrdi |
|- ( ( ph /\ ( ps /\ z e. W ) ) -> ( iota_ b e. ( 2nd ` ( T ` M ) ) ( ( Q ` ( M - 1 ) ) ` ( ( M - 1 ) / N ) ) e. b ) C_ B ) |
61 |
1
|
cvmsf1o |
|- ( ( F e. ( C CovMap J ) /\ ( 2nd ` ( T ` M ) ) e. ( S ` ( 1st ` ( T ` M ) ) ) /\ ( iota_ b e. ( 2nd ` ( T ` M ) ) ( ( Q ` ( M - 1 ) ) ` ( ( M - 1 ) / N ) ) e. b ) e. ( 2nd ` ( T ` M ) ) ) -> ( F |` ( iota_ b e. ( 2nd ` ( T ` M ) ) ( ( Q ` ( M - 1 ) ) ` ( ( M - 1 ) / N ) ) e. b ) ) : ( iota_ b e. ( 2nd ` ( T ` M ) ) ( ( Q ` ( M - 1 ) ) ` ( ( M - 1 ) / N ) ) e. b ) -1-1-onto-> ( 1st ` ( T ` M ) ) ) |
62 |
20 17 56 61
|
syl3anc |
|- ( ( ph /\ ( ps /\ z e. W ) ) -> ( F |` ( iota_ b e. ( 2nd ` ( T ` M ) ) ( ( Q ` ( M - 1 ) ) ` ( ( M - 1 ) / N ) ) e. b ) ) : ( iota_ b e. ( 2nd ` ( T ` M ) ) ( ( Q ` ( M - 1 ) ) ` ( ( M - 1 ) / N ) ) e. b ) -1-1-onto-> ( 1st ` ( T ` M ) ) ) |
63 |
|
f1ocnv |
|- ( ( F |` ( iota_ b e. ( 2nd ` ( T ` M ) ) ( ( Q ` ( M - 1 ) ) ` ( ( M - 1 ) / N ) ) e. b ) ) : ( iota_ b e. ( 2nd ` ( T ` M ) ) ( ( Q ` ( M - 1 ) ) ` ( ( M - 1 ) / N ) ) e. b ) -1-1-onto-> ( 1st ` ( T ` M ) ) -> `' ( F |` ( iota_ b e. ( 2nd ` ( T ` M ) ) ( ( Q ` ( M - 1 ) ) ` ( ( M - 1 ) / N ) ) e. b ) ) : ( 1st ` ( T ` M ) ) -1-1-onto-> ( iota_ b e. ( 2nd ` ( T ` M ) ) ( ( Q ` ( M - 1 ) ) ` ( ( M - 1 ) / N ) ) e. b ) ) |
64 |
|
f1of |
|- ( `' ( F |` ( iota_ b e. ( 2nd ` ( T ` M ) ) ( ( Q ` ( M - 1 ) ) ` ( ( M - 1 ) / N ) ) e. b ) ) : ( 1st ` ( T ` M ) ) -1-1-onto-> ( iota_ b e. ( 2nd ` ( T ` M ) ) ( ( Q ` ( M - 1 ) ) ` ( ( M - 1 ) / N ) ) e. b ) -> `' ( F |` ( iota_ b e. ( 2nd ` ( T ` M ) ) ( ( Q ` ( M - 1 ) ) ` ( ( M - 1 ) / N ) ) e. b ) ) : ( 1st ` ( T ` M ) ) --> ( iota_ b e. ( 2nd ` ( T ` M ) ) ( ( Q ` ( M - 1 ) ) ` ( ( M - 1 ) / N ) ) e. b ) ) |
65 |
62 63 64
|
3syl |
|- ( ( ph /\ ( ps /\ z e. W ) ) -> `' ( F |` ( iota_ b e. ( 2nd ` ( T ` M ) ) ( ( Q ` ( M - 1 ) ) ` ( ( M - 1 ) / N ) ) e. b ) ) : ( 1st ` ( T ` M ) ) --> ( iota_ b e. ( 2nd ` ( T ` M ) ) ( ( Q ` ( M - 1 ) ) ` ( ( M - 1 ) / N ) ) e. b ) ) |
66 |
|
simprr |
|- ( ( ph /\ ( ps /\ z e. W ) ) -> z e. W ) |
67 |
1 2 3 4 5 6 7 8 9 10 11 16 13 66
|
cvmliftlem3 |
|- ( ( ph /\ ( ps /\ z e. W ) ) -> ( G ` z ) e. ( 1st ` ( T ` M ) ) ) |
68 |
65 67
|
ffvelrnd |
|- ( ( ph /\ ( ps /\ z e. W ) ) -> ( `' ( F |` ( iota_ b e. ( 2nd ` ( T ` M ) ) ( ( Q ` ( M - 1 ) ) ` ( ( M - 1 ) / N ) ) e. b ) ) ` ( G ` z ) ) e. ( iota_ b e. ( 2nd ` ( T ` M ) ) ( ( Q ` ( M - 1 ) ) ` ( ( M - 1 ) / N ) ) e. b ) ) |
69 |
60 68
|
sseldd |
|- ( ( ph /\ ( ps /\ z e. W ) ) -> ( `' ( F |` ( iota_ b e. ( 2nd ` ( T ` M ) ) ( ( Q ` ( M - 1 ) ) ` ( ( M - 1 ) / N ) ) e. b ) ) ` ( G ` z ) ) e. B ) |
70 |
69
|
anassrs |
|- ( ( ( ph /\ ps ) /\ z e. W ) -> ( `' ( F |` ( iota_ b e. ( 2nd ` ( T ` M ) ) ( ( Q ` ( M - 1 ) ) ` ( ( M - 1 ) / N ) ) e. b ) ) ` ( G ` z ) ) e. B ) |
71 |
70
|
fmpttd |
|- ( ( ph /\ ps ) -> ( z e. W |-> ( `' ( F |` ( iota_ b e. ( 2nd ` ( T ` M ) ) ( ( Q ` ( M - 1 ) ) ` ( ( M - 1 ) / N ) ) e. b ) ) ` ( G ` z ) ) ) : W --> B ) |
72 |
14 31
|
syl |
|- ( ( ph /\ ps ) -> M e. NN ) |
73 |
1 2 3 4 5 6 7 8 9 10 11 12 13
|
cvmliftlem5 |
|- ( ( ph /\ M e. NN ) -> ( Q ` M ) = ( z e. W |-> ( `' ( F |` ( iota_ b e. ( 2nd ` ( T ` M ) ) ( ( Q ` ( M - 1 ) ) ` ( ( M - 1 ) / N ) ) e. b ) ) ` ( G ` z ) ) ) ) |
74 |
72 73
|
syldan |
|- ( ( ph /\ ps ) -> ( Q ` M ) = ( z e. W |-> ( `' ( F |` ( iota_ b e. ( 2nd ` ( T ` M ) ) ( ( Q ` ( M - 1 ) ) ` ( ( M - 1 ) / N ) ) e. b ) ) ` ( G ` z ) ) ) ) |
75 |
74
|
feq1d |
|- ( ( ph /\ ps ) -> ( ( Q ` M ) : W --> B <-> ( z e. W |-> ( `' ( F |` ( iota_ b e. ( 2nd ` ( T ` M ) ) ( ( Q ` ( M - 1 ) ) ` ( ( M - 1 ) / N ) ) e. b ) ) ` ( G ` z ) ) ) : W --> B ) ) |
76 |
71 75
|
mpbird |
|- ( ( ph /\ ps ) -> ( Q ` M ) : W --> B ) |
77 |
|
fvres |
|- ( z e. W -> ( ( G |` W ) ` z ) = ( G ` z ) ) |
78 |
66 77
|
syl |
|- ( ( ph /\ ( ps /\ z e. W ) ) -> ( ( G |` W ) ` z ) = ( G ` z ) ) |
79 |
|
f1ocnvfv2 |
|- ( ( ( F |` ( iota_ b e. ( 2nd ` ( T ` M ) ) ( ( Q ` ( M - 1 ) ) ` ( ( M - 1 ) / N ) ) e. b ) ) : ( iota_ b e. ( 2nd ` ( T ` M ) ) ( ( Q ` ( M - 1 ) ) ` ( ( M - 1 ) / N ) ) e. b ) -1-1-onto-> ( 1st ` ( T ` M ) ) /\ ( G ` z ) e. ( 1st ` ( T ` M ) ) ) -> ( ( F |` ( iota_ b e. ( 2nd ` ( T ` M ) ) ( ( Q ` ( M - 1 ) ) ` ( ( M - 1 ) / N ) ) e. b ) ) ` ( `' ( F |` ( iota_ b e. ( 2nd ` ( T ` M ) ) ( ( Q ` ( M - 1 ) ) ` ( ( M - 1 ) / N ) ) e. b ) ) ` ( G ` z ) ) ) = ( G ` z ) ) |
80 |
62 67 79
|
syl2anc |
|- ( ( ph /\ ( ps /\ z e. W ) ) -> ( ( F |` ( iota_ b e. ( 2nd ` ( T ` M ) ) ( ( Q ` ( M - 1 ) ) ` ( ( M - 1 ) / N ) ) e. b ) ) ` ( `' ( F |` ( iota_ b e. ( 2nd ` ( T ` M ) ) ( ( Q ` ( M - 1 ) ) ` ( ( M - 1 ) / N ) ) e. b ) ) ` ( G ` z ) ) ) = ( G ` z ) ) |
81 |
|
fvres |
|- ( ( `' ( F |` ( iota_ b e. ( 2nd ` ( T ` M ) ) ( ( Q ` ( M - 1 ) ) ` ( ( M - 1 ) / N ) ) e. b ) ) ` ( G ` z ) ) e. ( iota_ b e. ( 2nd ` ( T ` M ) ) ( ( Q ` ( M - 1 ) ) ` ( ( M - 1 ) / N ) ) e. b ) -> ( ( F |` ( iota_ b e. ( 2nd ` ( T ` M ) ) ( ( Q ` ( M - 1 ) ) ` ( ( M - 1 ) / N ) ) e. b ) ) ` ( `' ( F |` ( iota_ b e. ( 2nd ` ( T ` M ) ) ( ( Q ` ( M - 1 ) ) ` ( ( M - 1 ) / N ) ) e. b ) ) ` ( G ` z ) ) ) = ( F ` ( `' ( F |` ( iota_ b e. ( 2nd ` ( T ` M ) ) ( ( Q ` ( M - 1 ) ) ` ( ( M - 1 ) / N ) ) e. b ) ) ` ( G ` z ) ) ) ) |
82 |
68 81
|
syl |
|- ( ( ph /\ ( ps /\ z e. W ) ) -> ( ( F |` ( iota_ b e. ( 2nd ` ( T ` M ) ) ( ( Q ` ( M - 1 ) ) ` ( ( M - 1 ) / N ) ) e. b ) ) ` ( `' ( F |` ( iota_ b e. ( 2nd ` ( T ` M ) ) ( ( Q ` ( M - 1 ) ) ` ( ( M - 1 ) / N ) ) e. b ) ) ` ( G ` z ) ) ) = ( F ` ( `' ( F |` ( iota_ b e. ( 2nd ` ( T ` M ) ) ( ( Q ` ( M - 1 ) ) ` ( ( M - 1 ) / N ) ) e. b ) ) ` ( G ` z ) ) ) ) |
83 |
78 80 82
|
3eqtr2rd |
|- ( ( ph /\ ( ps /\ z e. W ) ) -> ( F ` ( `' ( F |` ( iota_ b e. ( 2nd ` ( T ` M ) ) ( ( Q ` ( M - 1 ) ) ` ( ( M - 1 ) / N ) ) e. b ) ) ` ( G ` z ) ) ) = ( ( G |` W ) ` z ) ) |
84 |
83
|
anassrs |
|- ( ( ( ph /\ ps ) /\ z e. W ) -> ( F ` ( `' ( F |` ( iota_ b e. ( 2nd ` ( T ` M ) ) ( ( Q ` ( M - 1 ) ) ` ( ( M - 1 ) / N ) ) e. b ) ) ` ( G ` z ) ) ) = ( ( G |` W ) ` z ) ) |
85 |
84
|
mpteq2dva |
|- ( ( ph /\ ps ) -> ( z e. W |-> ( F ` ( `' ( F |` ( iota_ b e. ( 2nd ` ( T ` M ) ) ( ( Q ` ( M - 1 ) ) ` ( ( M - 1 ) / N ) ) e. b ) ) ` ( G ` z ) ) ) ) = ( z e. W |-> ( ( G |` W ) ` z ) ) ) |
86 |
4 22 23
|
3syl |
|- ( ph -> F : B --> X ) |
87 |
86
|
adantr |
|- ( ( ph /\ ps ) -> F : B --> X ) |
88 |
87
|
feqmptd |
|- ( ( ph /\ ps ) -> F = ( y e. B |-> ( F ` y ) ) ) |
89 |
|
fveq2 |
|- ( y = ( `' ( F |` ( iota_ b e. ( 2nd ` ( T ` M ) ) ( ( Q ` ( M - 1 ) ) ` ( ( M - 1 ) / N ) ) e. b ) ) ` ( G ` z ) ) -> ( F ` y ) = ( F ` ( `' ( F |` ( iota_ b e. ( 2nd ` ( T ` M ) ) ( ( Q ` ( M - 1 ) ) ` ( ( M - 1 ) / N ) ) e. b ) ) ` ( G ` z ) ) ) ) |
90 |
70 74 88 89
|
fmptco |
|- ( ( ph /\ ps ) -> ( F o. ( Q ` M ) ) = ( z e. W |-> ( F ` ( `' ( F |` ( iota_ b e. ( 2nd ` ( T ` M ) ) ( ( Q ` ( M - 1 ) ) ` ( ( M - 1 ) / N ) ) e. b ) ) ` ( G ` z ) ) ) ) ) |
91 |
|
iiuni |
|- ( 0 [,] 1 ) = U. II |
92 |
91 3
|
cnf |
|- ( G e. ( II Cn J ) -> G : ( 0 [,] 1 ) --> X ) |
93 |
5 92
|
syl |
|- ( ph -> G : ( 0 [,] 1 ) --> X ) |
94 |
93
|
adantr |
|- ( ( ph /\ ps ) -> G : ( 0 [,] 1 ) --> X ) |
95 |
1 2 3 4 5 6 7 8 9 10 11 14 13
|
cvmliftlem2 |
|- ( ( ph /\ ps ) -> W C_ ( 0 [,] 1 ) ) |
96 |
94 95
|
fssresd |
|- ( ( ph /\ ps ) -> ( G |` W ) : W --> X ) |
97 |
96
|
feqmptd |
|- ( ( ph /\ ps ) -> ( G |` W ) = ( z e. W |-> ( ( G |` W ) ` z ) ) ) |
98 |
85 90 97
|
3eqtr4d |
|- ( ( ph /\ ps ) -> ( F o. ( Q ` M ) ) = ( G |` W ) ) |
99 |
76 98
|
jca |
|- ( ( ph /\ ps ) -> ( ( Q ` M ) : W --> B /\ ( F o. ( Q ` M ) ) = ( G |` W ) ) ) |