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Theorem cvmliftphtlem 23255
Description: Lemma for cvmliftpht 23256. (Contributed by Mario Carneiro, 6-Jul-2015.)
Hypotheses
Ref Expression
cvmliftpht.b  |-  B  = 
U. C
cvmliftpht.m  |-  M  =  ( iota_ f  e.  ( II  Cn  C ) ( ( F  o.  f )  =  G  /\  ( f ` 
0 )  =  P ) )
cvmliftpht.n  |-  N  =  ( iota_ f  e.  ( II  Cn  C ) ( ( F  o.  f )  =  H  /\  ( f ` 
0 )  =  P ) )
cvmliftpht.f  |-  ( ph  ->  F  e.  ( C CovMap  J ) )
cvmliftpht.p  |-  ( ph  ->  P  e.  B )
cvmliftpht.e  |-  ( ph  ->  ( F `  P
)  =  ( G `
 0 ) )
cvmliftphtlem.g  |-  ( ph  ->  G  e.  ( II 
Cn  J ) )
cvmliftphtlem.h  |-  ( ph  ->  H  e.  ( II 
Cn  J ) )
cvmliftphtlem.k  |-  ( ph  ->  K  e.  ( G ( PHtpy `  J ) H ) )
cvmliftphtlem.a  |-  ( ph  ->  A  e.  ( ( II  tX  II )  Cn  C ) )
cvmliftphtlem.c  |-  ( ph  ->  ( F  o.  A
)  =  K )
cvmliftphtlem.0  |-  ( ph  ->  ( 0 A 0 )  =  P )
Assertion
Ref Expression
cvmliftphtlem  |-  ( ph  ->  A  e.  ( M ( PHtpy `  C ) N ) )
Distinct variable groups:    A, f    B, f    f, F    f, J    C, f    f, G   
f, H    P, f
Allowed substitution hints:    ph( f)    K( f)    M( f)    N( f)

Proof of Theorem cvmliftphtlem
Dummy variables  s  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cvmliftpht.b . . . 4  |-  B  = 
U. C
2 cvmliftpht.m . . . 4  |-  M  =  ( iota_ f  e.  ( II  Cn  C ) ( ( F  o.  f )  =  G  /\  ( f ` 
0 )  =  P ) )
3 cvmliftpht.f . . . 4  |-  ( ph  ->  F  e.  ( C CovMap  J ) )
4 cvmliftphtlem.g . . . 4  |-  ( ph  ->  G  e.  ( II 
Cn  J ) )
5 cvmliftpht.p . . . 4  |-  ( ph  ->  P  e.  B )
6 cvmliftpht.e . . . 4  |-  ( ph  ->  ( F `  P
)  =  ( G `
 0 ) )
71, 2, 3, 4, 5, 6cvmliftiota 23239 . . 3  |-  ( ph  ->  ( M  e.  ( II  Cn  C )  /\  ( F  o.  M )  =  G  /\  ( M ` 
0 )  =  P ) )
87simp1d 967 . 2  |-  ( ph  ->  M  e.  ( II 
Cn  C ) )
9 cvmliftpht.n . . . 4  |-  N  =  ( iota_ f  e.  ( II  Cn  C ) ( ( F  o.  f )  =  H  /\  ( f ` 
0 )  =  P ) )
10 cvmliftphtlem.h . . . 4  |-  ( ph  ->  H  e.  ( II 
Cn  J ) )
11 cvmliftphtlem.k . . . . . . 7  |-  ( ph  ->  K  e.  ( G ( PHtpy `  J ) H ) )
124, 10, 11phtpy01 18479 . . . . . 6  |-  ( ph  ->  ( ( G ` 
0 )  =  ( H `  0 )  /\  ( G ` 
1 )  =  ( H `  1 ) ) )
1312simpld 445 . . . . 5  |-  ( ph  ->  ( G `  0
)  =  ( H `
 0 ) )
146, 13eqtrd 2316 . . . 4  |-  ( ph  ->  ( F `  P
)  =  ( H `
 0 ) )
151, 9, 3, 10, 5, 14cvmliftiota 23239 . . 3  |-  ( ph  ->  ( N  e.  ( II  Cn  C )  /\  ( F  o.  N )  =  H  /\  ( N ` 
0 )  =  P ) )
1615simp1d 967 . 2  |-  ( ph  ->  N  e.  ( II 
Cn  C ) )
17 cvmliftphtlem.a . 2  |-  ( ph  ->  A  e.  ( ( II  tX  II )  Cn  C ) )
18 iitop 18380 . . . . . . . . . . . . . . . 16  |-  II  e.  Top
19 iiuni 18381 . . . . . . . . . . . . . . . 16  |-  ( 0 [,] 1 )  = 
U. II
2018, 18, 19, 19txunii 17284 . . . . . . . . . . . . . . 15  |-  ( ( 0 [,] 1 )  X.  ( 0 [,] 1 ) )  = 
U. ( II  tX  II )
2120, 1cnf 16972 . . . . . . . . . . . . . 14  |-  ( A  e.  ( ( II 
tX  II )  Cn  C )  ->  A : ( ( 0 [,] 1 )  X.  ( 0 [,] 1
) ) --> B )
2217, 21syl 15 . . . . . . . . . . . . 13  |-  ( ph  ->  A : ( ( 0 [,] 1 )  X.  ( 0 [,] 1 ) ) --> B )
23 0elunit 10750 . . . . . . . . . . . . . 14  |-  0  e.  ( 0 [,] 1
)
24 opelxpi 4720 . . . . . . . . . . . . . 14  |-  ( ( s  e.  ( 0 [,] 1 )  /\  0  e.  ( 0 [,] 1 ) )  ->  <. s ,  0
>.  e.  ( ( 0 [,] 1 )  X.  ( 0 [,] 1
) ) )
2523, 24mpan2 652 . . . . . . . . . . . . 13  |-  ( s  e.  ( 0 [,] 1 )  ->  <. s ,  0 >.  e.  ( ( 0 [,] 1
)  X.  ( 0 [,] 1 ) ) )
26 fvco3 5558 . . . . . . . . . . . . 13  |-  ( ( A : ( ( 0 [,] 1 )  X.  ( 0 [,] 1 ) ) --> B  /\  <. s ,  0
>.  e.  ( ( 0 [,] 1 )  X.  ( 0 [,] 1
) ) )  -> 
( ( F  o.  A ) `  <. s ,  0 >. )  =  ( F `  ( A `  <. s ,  0 >. )
) )
2722, 25, 26syl2an 463 . . . . . . . . . . . 12  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
( F  o.  A
) `  <. s ,  0 >. )  =  ( F `  ( A `
 <. s ,  0
>. ) ) )
28 cvmliftphtlem.c . . . . . . . . . . . . . 14  |-  ( ph  ->  ( F  o.  A
)  =  K )
2928adantr 451 . . . . . . . . . . . . 13  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  ( F  o.  A )  =  K )
3029fveq1d 5488 . . . . . . . . . . . 12  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
( F  o.  A
) `  <. s ,  0 >. )  =  ( K `  <. s ,  0 >. )
)
3127, 30eqtr3d 2318 . . . . . . . . . . 11  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  ( F `  ( A `  <. s ,  0
>. ) )  =  ( K `  <. s ,  0 >. )
)
32 df-ov 5823 . . . . . . . . . . . 12  |-  ( s A 0 )  =  ( A `  <. s ,  0 >. )
3332fveq2i 5489 . . . . . . . . . . 11  |-  ( F `
 ( s A 0 ) )  =  ( F `  ( A `  <. s ,  0 >. ) )
34 df-ov 5823 . . . . . . . . . . 11  |-  ( s K 0 )  =  ( K `  <. s ,  0 >. )
3531, 33, 343eqtr4g 2341 . . . . . . . . . 10  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  ( F `  ( s A 0 ) )  =  ( s K 0 ) )
36 iitopon 18379 . . . . . . . . . . . . 13  |-  II  e.  (TopOn `  ( 0 [,] 1 ) )
3736a1i 10 . . . . . . . . . . . 12  |-  ( ph  ->  II  e.  (TopOn `  ( 0 [,] 1
) ) )
384, 10phtpyhtpy 18476 . . . . . . . . . . . . 13  |-  ( ph  ->  ( G ( PHtpy `  J ) H ) 
C_  ( G ( II Htpy  J ) H ) )
3938, 11sseldd 3182 . . . . . . . . . . . 12  |-  ( ph  ->  K  e.  ( G ( II Htpy  J ) H ) )
4037, 4, 10, 39htpyi 18468 . . . . . . . . . . 11  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
( s K 0 )  =  ( G `
 s )  /\  ( s K 1 )  =  ( H `
 s ) ) )
4140simpld 445 . . . . . . . . . 10  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
s K 0 )  =  ( G `  s ) )
4235, 41eqtrd 2316 . . . . . . . . 9  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  ( F `  ( s A 0 ) )  =  ( G `  s ) )
4342mpteq2dva 4107 . . . . . . . 8  |-  ( ph  ->  ( s  e.  ( 0 [,] 1 ) 
|->  ( F `  (
s A 0 ) ) )  =  ( s  e.  ( 0 [,] 1 )  |->  ( G `  s ) ) )
44 fovrn 5952 . . . . . . . . . . 11  |-  ( ( A : ( ( 0 [,] 1 )  X.  ( 0 [,] 1 ) ) --> B  /\  s  e.  ( 0 [,] 1 )  /\  0  e.  ( 0 [,] 1 ) )  ->  ( s A 0 )  e.  B )
4523, 44mp3an3 1266 . . . . . . . . . 10  |-  ( ( A : ( ( 0 [,] 1 )  X.  ( 0 [,] 1 ) ) --> B  /\  s  e.  ( 0 [,] 1 ) )  ->  ( s A 0 )  e.  B )
4622, 45sylan 457 . . . . . . . . 9  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
s A 0 )  e.  B )
47 eqidd 2285 . . . . . . . . 9  |-  ( ph  ->  ( s  e.  ( 0 [,] 1 ) 
|->  ( s A 0 ) )  =  ( s  e.  ( 0 [,] 1 )  |->  ( s A 0 ) ) )
48 cvmcn 23200 . . . . . . . . . . . 12  |-  ( F  e.  ( C CovMap  J
)  ->  F  e.  ( C  Cn  J
) )
493, 48syl 15 . . . . . . . . . . 11  |-  ( ph  ->  F  e.  ( C  Cn  J ) )
50 eqid 2284 . . . . . . . . . . . 12  |-  U. J  =  U. J
511, 50cnf 16972 . . . . . . . . . . 11  |-  ( F  e.  ( C  Cn  J )  ->  F : B --> U. J )
5249, 51syl 15 . . . . . . . . . 10  |-  ( ph  ->  F : B --> U. J
)
5352feqmptd 5537 . . . . . . . . 9  |-  ( ph  ->  F  =  ( x  e.  B  |->  ( F `
 x ) ) )
54 fveq2 5486 . . . . . . . . 9  |-  ( x  =  ( s A 0 )  ->  ( F `  x )  =  ( F `  ( s A 0 ) ) )
5546, 47, 53, 54fmptco 5653 . . . . . . . 8  |-  ( ph  ->  ( F  o.  (
s  e.  ( 0 [,] 1 )  |->  ( s A 0 ) ) )  =  ( s  e.  ( 0 [,] 1 )  |->  ( F `  ( s A 0 ) ) ) )
5619, 50cnf 16972 . . . . . . . . . 10  |-  ( G  e.  ( II  Cn  J )  ->  G : ( 0 [,] 1 ) --> U. J
)
574, 56syl 15 . . . . . . . . 9  |-  ( ph  ->  G : ( 0 [,] 1 ) --> U. J )
5857feqmptd 5537 . . . . . . . 8  |-  ( ph  ->  G  =  ( s  e.  ( 0 [,] 1 )  |->  ( G `
 s ) ) )
5943, 55, 583eqtr4d 2326 . . . . . . 7  |-  ( ph  ->  ( F  o.  (
s  e.  ( 0 [,] 1 )  |->  ( s A 0 ) ) )  =  G )
60 cvmliftphtlem.0 . . . . . . 7  |-  ( ph  ->  ( 0 A 0 )  =  P )
6137cnmptid 17351 . . . . . . . . 9  |-  ( ph  ->  ( s  e.  ( 0 [,] 1 ) 
|->  s )  e.  ( II  Cn  II ) )
6223a1i 10 . . . . . . . . . 10  |-  ( ph  ->  0  e.  ( 0 [,] 1 ) )
6337, 37, 62cnmptc 17352 . . . . . . . . 9  |-  ( ph  ->  ( s  e.  ( 0 [,] 1 ) 
|->  0 )  e.  ( II  Cn  II ) )
6437, 61, 63, 17cnmpt12f 17356 . . . . . . . 8  |-  ( ph  ->  ( s  e.  ( 0 [,] 1 ) 
|->  ( s A 0 ) )  e.  ( II  Cn  C ) )
651cvmlift 23237 . . . . . . . . 9  |-  ( ( ( F  e.  ( C CovMap  J )  /\  G  e.  ( II  Cn  J ) )  /\  ( P  e.  B  /\  ( F `  P
)  =  ( G `
 0 ) ) )  ->  E! f  e.  ( II  Cn  C
) ( ( F  o.  f )  =  G  /\  ( f `
 0 )  =  P ) )
663, 4, 5, 6, 65syl22anc 1183 . . . . . . . 8  |-  ( ph  ->  E! f  e.  ( II  Cn  C ) ( ( F  o.  f )  =  G  /\  ( f ` 
0 )  =  P ) )
67 nfcv 2420 . . . . . . . . 9  |-  F/_ f
( s  e.  ( 0 [,] 1 ) 
|->  ( s A 0 ) )
68 nfv 1605 . . . . . . . . 9  |-  F/ f ( ( F  o.  ( s  e.  ( 0 [,] 1 ) 
|->  ( s A 0 ) ) )  =  G  /\  ( 0 A 0 )  =  P )
69 coeq2 4841 . . . . . . . . . . 11  |-  ( f  =  ( s  e.  ( 0 [,] 1
)  |->  ( s A 0 ) )  -> 
( F  o.  f
)  =  ( F  o.  ( s  e.  ( 0 [,] 1
)  |->  ( s A 0 ) ) ) )
7069eqeq1d 2292 . . . . . . . . . 10  |-  ( f  =  ( s  e.  ( 0 [,] 1
)  |->  ( s A 0 ) )  -> 
( ( F  o.  f )  =  G  <-> 
( F  o.  (
s  e.  ( 0 [,] 1 )  |->  ( s A 0 ) ) )  =  G ) )
71 fveq1 5485 . . . . . . . . . . . 12  |-  ( f  =  ( s  e.  ( 0 [,] 1
)  |->  ( s A 0 ) )  -> 
( f `  0
)  =  ( ( s  e.  ( 0 [,] 1 )  |->  ( s A 0 ) ) `  0 ) )
72 oveq1 5827 . . . . . . . . . . . . . 14  |-  ( s  =  0  ->  (
s A 0 )  =  ( 0 A 0 ) )
73 eqid 2284 . . . . . . . . . . . . . 14  |-  ( s  e.  ( 0 [,] 1 )  |->  ( s A 0 ) )  =  ( s  e.  ( 0 [,] 1
)  |->  ( s A 0 ) )
74 ovex 5845 . . . . . . . . . . . . . 14  |-  ( 0 A 0 )  e. 
_V
7572, 73, 74fvmpt 5564 . . . . . . . . . . . . 13  |-  ( 0  e.  ( 0 [,] 1 )  ->  (
( s  e.  ( 0 [,] 1 ) 
|->  ( s A 0 ) ) `  0
)  =  ( 0 A 0 ) )
7623, 75ax-mp 8 . . . . . . . . . . . 12  |-  ( ( s  e.  ( 0 [,] 1 )  |->  ( s A 0 ) ) `  0 )  =  ( 0 A 0 )
7771, 76syl6eq 2332 . . . . . . . . . . 11  |-  ( f  =  ( s  e.  ( 0 [,] 1
)  |->  ( s A 0 ) )  -> 
( f `  0
)  =  ( 0 A 0 ) )
7877eqeq1d 2292 . . . . . . . . . 10  |-  ( f  =  ( s  e.  ( 0 [,] 1
)  |->  ( s A 0 ) )  -> 
( ( f ` 
0 )  =  P  <-> 
( 0 A 0 )  =  P ) )
7970, 78anbi12d 691 . . . . . . . . 9  |-  ( f  =  ( s  e.  ( 0 [,] 1
)  |->  ( s A 0 ) )  -> 
( ( ( F  o.  f )  =  G  /\  ( f `
 0 )  =  P )  <->  ( ( F  o.  ( s  e.  ( 0 [,] 1
)  |->  ( s A 0 ) ) )  =  G  /\  (
0 A 0 )  =  P ) ) )
8067, 68, 79riota2f 6322 . . . . . . . 8  |-  ( ( ( s  e.  ( 0 [,] 1 ) 
|->  ( s A 0 ) )  e.  ( II  Cn  C )  /\  E! f  e.  ( II  Cn  C
) ( ( F  o.  f )  =  G  /\  ( f `
 0 )  =  P ) )  -> 
( ( ( F  o.  ( s  e.  ( 0 [,] 1
)  |->  ( s A 0 ) ) )  =  G  /\  (
0 A 0 )  =  P )  <->  ( iota_ f  e.  ( II  Cn  C ) ( ( F  o.  f )  =  G  /\  (
f `  0 )  =  P ) )  =  ( s  e.  ( 0 [,] 1 ) 
|->  ( s A 0 ) ) ) )
8164, 66, 80syl2anc 642 . . . . . . 7  |-  ( ph  ->  ( ( ( F  o.  ( s  e.  ( 0 [,] 1
)  |->  ( s A 0 ) ) )  =  G  /\  (
0 A 0 )  =  P )  <->  ( iota_ f  e.  ( II  Cn  C ) ( ( F  o.  f )  =  G  /\  (
f `  0 )  =  P ) )  =  ( s  e.  ( 0 [,] 1 ) 
|->  ( s A 0 ) ) ) )
8259, 60, 81mpbi2and 887 . . . . . 6  |-  ( ph  ->  ( iota_ f  e.  ( II  Cn  C ) ( ( F  o.  f )  =  G  /\  ( f ` 
0 )  =  P ) )  =  ( s  e.  ( 0 [,] 1 )  |->  ( s A 0 ) ) )
832, 82syl5eq 2328 . . . . 5  |-  ( ph  ->  M  =  ( s  e.  ( 0 [,] 1 )  |->  ( s A 0 ) ) )
8419, 1cnf 16972 . . . . . . 7  |-  ( M  e.  ( II  Cn  C )  ->  M : ( 0 [,] 1 ) --> B )
858, 84syl 15 . . . . . 6  |-  ( ph  ->  M : ( 0 [,] 1 ) --> B )
8685feqmptd 5537 . . . . 5  |-  ( ph  ->  M  =  ( s  e.  ( 0 [,] 1 )  |->  ( M `
 s ) ) )
8783, 86eqtr3d 2318 . . . 4  |-  ( ph  ->  ( s  e.  ( 0 [,] 1 ) 
|->  ( s A 0 ) )  =  ( s  e.  ( 0 [,] 1 )  |->  ( M `  s ) ) )
88 mpteqb 5576 . . . . 5  |-  ( A. s  e.  ( 0 [,] 1 ) ( s A 0 )  e.  _V  ->  (
( s  e.  ( 0 [,] 1 ) 
|->  ( s A 0 ) )  =  ( s  e.  ( 0 [,] 1 )  |->  ( M `  s ) )  <->  A. s  e.  ( 0 [,] 1 ) ( s A 0 )  =  ( M `
 s ) ) )
89 ovex 5845 . . . . . 6  |-  ( s A 0 )  e. 
_V
9089a1i 10 . . . . 5  |-  ( s  e.  ( 0 [,] 1 )  ->  (
s A 0 )  e.  _V )
9188, 90mprg 2613 . . . 4  |-  ( ( s  e.  ( 0 [,] 1 )  |->  ( s A 0 ) )  =  ( s  e.  ( 0 [,] 1 )  |->  ( M `
 s ) )  <->  A. s  e.  (
0 [,] 1 ) ( s A 0 )  =  ( M `
 s ) )
9287, 91sylib 188 . . 3  |-  ( ph  ->  A. s  e.  ( 0 [,] 1 ) ( s A 0 )  =  ( M `
 s ) )
9392r19.21bi 2642 . 2  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
s A 0 )  =  ( M `  s ) )
94 1elunit 10751 . . . . . . . . . . . . . 14  |-  1  e.  ( 0 [,] 1
)
95 opelxpi 4720 . . . . . . . . . . . . . 14  |-  ( ( s  e.  ( 0 [,] 1 )  /\  1  e.  ( 0 [,] 1 ) )  ->  <. s ,  1
>.  e.  ( ( 0 [,] 1 )  X.  ( 0 [,] 1
) ) )
9694, 95mpan2 652 . . . . . . . . . . . . 13  |-  ( s  e.  ( 0 [,] 1 )  ->  <. s ,  1 >.  e.  ( ( 0 [,] 1
)  X.  ( 0 [,] 1 ) ) )
97 fvco3 5558 . . . . . . . . . . . . 13  |-  ( ( A : ( ( 0 [,] 1 )  X.  ( 0 [,] 1 ) ) --> B  /\  <. s ,  1
>.  e.  ( ( 0 [,] 1 )  X.  ( 0 [,] 1
) ) )  -> 
( ( F  o.  A ) `  <. s ,  1 >. )  =  ( F `  ( A `  <. s ,  1 >. )
) )
9822, 96, 97syl2an 463 . . . . . . . . . . . 12  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
( F  o.  A
) `  <. s ,  1 >. )  =  ( F `  ( A `
 <. s ,  1
>. ) ) )
9929fveq1d 5488 . . . . . . . . . . . 12  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
( F  o.  A
) `  <. s ,  1 >. )  =  ( K `  <. s ,  1 >. )
)
10098, 99eqtr3d 2318 . . . . . . . . . . 11  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  ( F `  ( A `  <. s ,  1
>. ) )  =  ( K `  <. s ,  1 >. )
)
101 df-ov 5823 . . . . . . . . . . . 12  |-  ( s A 1 )  =  ( A `  <. s ,  1 >. )
102101fveq2i 5489 . . . . . . . . . . 11  |-  ( F `
 ( s A 1 ) )  =  ( F `  ( A `  <. s ,  1 >. ) )
103 df-ov 5823 . . . . . . . . . . 11  |-  ( s K 1 )  =  ( K `  <. s ,  1 >. )
104100, 102, 1033eqtr4g 2341 . . . . . . . . . 10  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  ( F `  ( s A 1 ) )  =  ( s K 1 ) )
10540simprd 449 . . . . . . . . . 10  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
s K 1 )  =  ( H `  s ) )
106104, 105eqtrd 2316 . . . . . . . . 9  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  ( F `  ( s A 1 ) )  =  ( H `  s ) )
107106mpteq2dva 4107 . . . . . . . 8  |-  ( ph  ->  ( s  e.  ( 0 [,] 1 ) 
|->  ( F `  (
s A 1 ) ) )  =  ( s  e.  ( 0 [,] 1 )  |->  ( H `  s ) ) )
108 fovrn 5952 . . . . . . . . . . 11  |-  ( ( A : ( ( 0 [,] 1 )  X.  ( 0 [,] 1 ) ) --> B  /\  s  e.  ( 0 [,] 1 )  /\  1  e.  ( 0 [,] 1 ) )  ->  ( s A 1 )  e.  B )
10994, 108mp3an3 1266 . . . . . . . . . 10  |-  ( ( A : ( ( 0 [,] 1 )  X.  ( 0 [,] 1 ) ) --> B  /\  s  e.  ( 0 [,] 1 ) )  ->  ( s A 1 )  e.  B )
11022, 109sylan 457 . . . . . . . . 9  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
s A 1 )  e.  B )
111 eqidd 2285 . . . . . . . . 9  |-  ( ph  ->  ( s  e.  ( 0 [,] 1 ) 
|->  ( s A 1 ) )  =  ( s  e.  ( 0 [,] 1 )  |->  ( s A 1 ) ) )
112 fveq2 5486 . . . . . . . . 9  |-  ( x  =  ( s A 1 )  ->  ( F `  x )  =  ( F `  ( s A 1 ) ) )
113110, 111, 53, 112fmptco 5653 . . . . . . . 8  |-  ( ph  ->  ( F  o.  (
s  e.  ( 0 [,] 1 )  |->  ( s A 1 ) ) )  =  ( s  e.  ( 0 [,] 1 )  |->  ( F `  ( s A 1 ) ) ) )
11419, 50cnf 16972 . . . . . . . . . 10  |-  ( H  e.  ( II  Cn  J )  ->  H : ( 0 [,] 1 ) --> U. J
)
11510, 114syl 15 . . . . . . . . 9  |-  ( ph  ->  H : ( 0 [,] 1 ) --> U. J )
116115feqmptd 5537 . . . . . . . 8  |-  ( ph  ->  H  =  ( s  e.  ( 0 [,] 1 )  |->  ( H `
 s ) ) )
117107, 113, 1163eqtr4d 2326 . . . . . . 7  |-  ( ph  ->  ( F  o.  (
s  e.  ( 0 [,] 1 )  |->  ( s A 1 ) ) )  =  H )
118 iicon 18387 . . . . . . . . . . . . 13  |-  II  e.  Con
119118a1i 10 . . . . . . . . . . . 12  |-  ( ph  ->  II  e.  Con )
120 iinllycon 23192 . . . . . . . . . . . . 13  |-  II  e. 𝑛Locally  Con
121120a1i 10 . . . . . . . . . . . 12  |-  ( ph  ->  II  e. 𝑛Locally  Con )
12237, 63, 61, 17cnmpt12f 17356 . . . . . . . . . . . 12  |-  ( ph  ->  ( s  e.  ( 0 [,] 1 ) 
|->  ( 0 A s ) )  e.  ( II  Cn  C ) )
123 cvmtop1 23198 . . . . . . . . . . . . . . 15  |-  ( F  e.  ( C CovMap  J
)  ->  C  e.  Top )
1243, 123syl 15 . . . . . . . . . . . . . 14  |-  ( ph  ->  C  e.  Top )
1251toptopon 16667 . . . . . . . . . . . . . 14  |-  ( C  e.  Top  <->  C  e.  (TopOn `  B ) )
126124, 125sylib 188 . . . . . . . . . . . . 13  |-  ( ph  ->  C  e.  (TopOn `  B ) )
127 ffvelrn 5625 . . . . . . . . . . . . . 14  |-  ( ( M : ( 0 [,] 1 ) --> B  /\  0  e.  ( 0 [,] 1 ) )  ->  ( M `  0 )  e.  B )
12885, 23, 127sylancl 643 . . . . . . . . . . . . 13  |-  ( ph  ->  ( M `  0
)  e.  B )
129 cnconst2 17007 . . . . . . . . . . . . 13  |-  ( ( II  e.  (TopOn `  ( 0 [,] 1
) )  /\  C  e.  (TopOn `  B )  /\  ( M `  0
)  e.  B )  ->  ( ( 0 [,] 1 )  X. 
{ ( M ` 
0 ) } )  e.  ( II  Cn  C ) )
13037, 126, 128, 129syl3anc 1182 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( 0 [,] 1 )  X.  {
( M `  0
) } )  e.  ( II  Cn  C
) )
1314, 10, 11phtpyi 18478 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
( 0 K s )  =  ( G `
 0 )  /\  ( 1 K s )  =  ( G `
 1 ) ) )
132131simpld 445 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
0 K s )  =  ( G ` 
0 ) )
133 opelxpi 4720 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( 0  e.  ( 0 [,] 1 )  /\  s  e.  ( 0 [,] 1 ) )  ->  <. 0 ,  s
>.  e.  ( ( 0 [,] 1 )  X.  ( 0 [,] 1
) ) )
13423, 133mpan 651 . . . . . . . . . . . . . . . . . . 19  |-  ( s  e.  ( 0 [,] 1 )  ->  <. 0 ,  s >.  e.  ( ( 0 [,] 1
)  X.  ( 0 [,] 1 ) ) )
135 fvco3 5558 . . . . . . . . . . . . . . . . . . 19  |-  ( ( A : ( ( 0 [,] 1 )  X.  ( 0 [,] 1 ) ) --> B  /\  <. 0 ,  s
>.  e.  ( ( 0 [,] 1 )  X.  ( 0 [,] 1
) ) )  -> 
( ( F  o.  A ) `  <. 0 ,  s >. )  =  ( F `  ( A `  <. 0 ,  s >. )
) )
13622, 134, 135syl2an 463 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
( F  o.  A
) `  <. 0 ,  s >. )  =  ( F `  ( A `
 <. 0 ,  s
>. ) ) )
13729fveq1d 5488 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
( F  o.  A
) `  <. 0 ,  s >. )  =  ( K `  <. 0 ,  s >. )
)
138136, 137eqtr3d 2318 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  ( F `  ( A `  <. 0 ,  s
>. ) )  =  ( K `  <. 0 ,  s >. )
)
139 df-ov 5823 . . . . . . . . . . . . . . . . . 18  |-  ( 0 A s )  =  ( A `  <. 0 ,  s >. )
140139fveq2i 5489 . . . . . . . . . . . . . . . . 17  |-  ( F `
 ( 0 A s ) )  =  ( F `  ( A `  <. 0 ,  s >. ) )
141 df-ov 5823 . . . . . . . . . . . . . . . . 17  |-  ( 0 K s )  =  ( K `  <. 0 ,  s >. )
142138, 140, 1413eqtr4g 2341 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  ( F `  ( 0 A s ) )  =  ( 0 K s ) )
1437simp3d 969 . . . . . . . . . . . . . . . . . . 19  |-  ( ph  ->  ( M `  0
)  =  P )
144143adantr 451 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  ( M `  0 )  =  P )
145144fveq2d 5490 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  ( F `  ( M `  0 ) )  =  ( F `  P ) )
1466adantr 451 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  ( F `  P )  =  ( G ` 
0 ) )
147145, 146eqtrd 2316 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  ( F `  ( M `  0 ) )  =  ( G ` 
0 ) )
148132, 142, 1473eqtr4d 2326 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  ( F `  ( 0 A s ) )  =  ( F `  ( M `  0 ) ) )
149148mpteq2dva 4107 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( s  e.  ( 0 [,] 1 ) 
|->  ( F `  (
0 A s ) ) )  =  ( s  e.  ( 0 [,] 1 )  |->  ( F `  ( M `
 0 ) ) ) )
150 fconstmpt 4731 . . . . . . . . . . . . . 14  |-  ( ( 0 [,] 1 )  X.  { ( F `
 ( M ` 
0 ) ) } )  =  ( s  e.  ( 0 [,] 1 )  |->  ( F `
 ( M ` 
0 ) ) )
151149, 150syl6eqr 2334 . . . . . . . . . . . . 13  |-  ( ph  ->  ( s  e.  ( 0 [,] 1 ) 
|->  ( F `  (
0 A s ) ) )  =  ( ( 0 [,] 1
)  X.  { ( F `  ( M `
 0 ) ) } ) )
152 fovrn 5952 . . . . . . . . . . . . . . . 16  |-  ( ( A : ( ( 0 [,] 1 )  X.  ( 0 [,] 1 ) ) --> B  /\  0  e.  ( 0 [,] 1 )  /\  s  e.  ( 0 [,] 1 ) )  ->  ( 0 A s )  e.  B )
15323, 152mp3an2 1265 . . . . . . . . . . . . . . 15  |-  ( ( A : ( ( 0 [,] 1 )  X.  ( 0 [,] 1 ) ) --> B  /\  s  e.  ( 0 [,] 1 ) )  ->  ( 0 A s )  e.  B )
15422, 153sylan 457 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
0 A s )  e.  B )
155 eqidd 2285 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( s  e.  ( 0 [,] 1 ) 
|->  ( 0 A s ) )  =  ( s  e.  ( 0 [,] 1 )  |->  ( 0 A s ) ) )
156 fveq2 5486 . . . . . . . . . . . . . 14  |-  ( x  =  ( 0 A s )  ->  ( F `  x )  =  ( F `  ( 0 A s ) ) )
157154, 155, 53, 156fmptco 5653 . . . . . . . . . . . . 13  |-  ( ph  ->  ( F  o.  (
s  e.  ( 0 [,] 1 )  |->  ( 0 A s ) ) )  =  ( s  e.  ( 0 [,] 1 )  |->  ( F `  ( 0 A s ) ) ) )
158 ffn 5355 . . . . . . . . . . . . . . 15  |-  ( F : B --> U. J  ->  F  Fn  B )
15952, 158syl 15 . . . . . . . . . . . . . 14  |-  ( ph  ->  F  Fn  B )
160 fcoconst 5657 . . . . . . . . . . . . . 14  |-  ( ( F  Fn  B  /\  ( M `  0 )  e.  B )  -> 
( F  o.  (
( 0 [,] 1
)  X.  { ( M `  0 ) } ) )  =  ( ( 0 [,] 1 )  X.  {
( F `  ( M `  0 )
) } ) )
161159, 128, 160syl2anc 642 . . . . . . . . . . . . 13  |-  ( ph  ->  ( F  o.  (
( 0 [,] 1
)  X.  { ( M `  0 ) } ) )  =  ( ( 0 [,] 1 )  X.  {
( F `  ( M `  0 )
) } ) )
162151, 157, 1613eqtr4d 2326 . . . . . . . . . . . 12  |-  ( ph  ->  ( F  o.  (
s  e.  ( 0 [,] 1 )  |->  ( 0 A s ) ) )  =  ( F  o.  ( ( 0 [,] 1 )  X.  { ( M `
 0 ) } ) ) )
16360, 143eqtr4d 2319 . . . . . . . . . . . . 13  |-  ( ph  ->  ( 0 A 0 )  =  ( M `
 0 ) )
164 oveq2 5828 . . . . . . . . . . . . . . 15  |-  ( s  =  0  ->  (
0 A s )  =  ( 0 A 0 ) )
165 eqid 2284 . . . . . . . . . . . . . . 15  |-  ( s  e.  ( 0 [,] 1 )  |->  ( 0 A s ) )  =  ( s  e.  ( 0 [,] 1
)  |->  ( 0 A s ) )
166164, 165, 74fvmpt 5564 . . . . . . . . . . . . . 14  |-  ( 0  e.  ( 0 [,] 1 )  ->  (
( s  e.  ( 0 [,] 1 ) 
|->  ( 0 A s ) ) `  0
)  =  ( 0 A 0 ) )
16723, 166ax-mp 8 . . . . . . . . . . . . 13  |-  ( ( s  e.  ( 0 [,] 1 )  |->  ( 0 A s ) ) `  0 )  =  ( 0 A 0 )
168 fvex 5500 . . . . . . . . . . . . . . 15  |-  ( M `
 0 )  e. 
_V
169168fvconst2 5691 . . . . . . . . . . . . . 14  |-  ( 0  e.  ( 0 [,] 1 )  ->  (
( ( 0 [,] 1 )  X.  {
( M `  0
) } ) ` 
0 )  =  ( M `  0 ) )
17023, 169ax-mp 8 . . . . . . . . . . . . 13  |-  ( ( ( 0 [,] 1
)  X.  { ( M `  0 ) } ) `  0
)  =  ( M `
 0 )
171163, 167, 1703eqtr4g 2341 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( s  e.  ( 0 [,] 1
)  |->  ( 0 A s ) ) ` 
0 )  =  ( ( ( 0 [,] 1 )  X.  {
( M `  0
) } ) ` 
0 ) )
1721, 19, 3, 119, 121, 62, 122, 130, 162, 171cvmliftmoi 23221 . . . . . . . . . . 11  |-  ( ph  ->  ( s  e.  ( 0 [,] 1 ) 
|->  ( 0 A s ) )  =  ( ( 0 [,] 1
)  X.  { ( M `  0 ) } ) )
173 fconstmpt 4731 . . . . . . . . . . 11  |-  ( ( 0 [,] 1 )  X.  { ( M `
 0 ) } )  =  ( s  e.  ( 0 [,] 1 )  |->  ( M `
 0 ) )
174172, 173syl6eq 2332 . . . . . . . . . 10  |-  ( ph  ->  ( s  e.  ( 0 [,] 1 ) 
|->  ( 0 A s ) )  =  ( s  e.  ( 0 [,] 1 )  |->  ( M `  0 ) ) )
175 mpteqb 5576 . . . . . . . . . . 11  |-  ( A. s  e.  ( 0 [,] 1 ) ( 0 A s )  e.  _V  ->  (
( s  e.  ( 0 [,] 1 ) 
|->  ( 0 A s ) )  =  ( s  e.  ( 0 [,] 1 )  |->  ( M `  0 ) )  <->  A. s  e.  ( 0 [,] 1 ) ( 0 A s )  =  ( M `
 0 ) ) )
176 ovex 5845 . . . . . . . . . . . 12  |-  ( 0 A s )  e. 
_V
177176a1i 10 . . . . . . . . . . 11  |-  ( s  e.  ( 0 [,] 1 )  ->  (
0 A s )  e.  _V )
178175, 177mprg 2613 . . . . . . . . . 10  |-  ( ( s  e.  ( 0 [,] 1 )  |->  ( 0 A s ) )  =  ( s  e.  ( 0 [,] 1 )  |->  ( M `
 0 ) )  <->  A. s  e.  (
0 [,] 1 ) ( 0 A s )  =  ( M `
 0 ) )
179174, 178sylib 188 . . . . . . . . 9  |-  ( ph  ->  A. s  e.  ( 0 [,] 1 ) ( 0 A s )  =  ( M `
 0 ) )
180 oveq2 5828 . . . . . . . . . . 11  |-  ( s  =  1  ->  (
0 A s )  =  ( 0 A 1 ) )
181180eqeq1d 2292 . . . . . . . . . 10  |-  ( s  =  1  ->  (
( 0 A s )  =  ( M `
 0 )  <->  ( 0 A 1 )  =  ( M `  0
) ) )
182181rspcv 2881 . . . . . . . . 9  |-  ( 1  e.  ( 0 [,] 1 )  ->  ( A. s  e.  (
0 [,] 1 ) ( 0 A s )  =  ( M `
 0 )  -> 
( 0 A 1 )  =  ( M `
 0 ) ) )
18394, 179, 182mpsyl 59 . . . . . . . 8  |-  ( ph  ->  ( 0 A 1 )  =  ( M `
 0 ) )
184183, 143eqtrd 2316 . . . . . . 7  |-  ( ph  ->  ( 0 A 1 )  =  P )
18594a1i 10 . . . . . . . . . 10  |-  ( ph  ->  1  e.  ( 0 [,] 1 ) )
18637, 37, 185cnmptc 17352 . . . . . . . . 9  |-  ( ph  ->  ( s  e.  ( 0 [,] 1 ) 
|->  1 )  e.  ( II  Cn  II ) )
18737, 61, 186, 17cnmpt12f 17356 . . . . . . . 8  |-  ( ph  ->  ( s  e.  ( 0 [,] 1 ) 
|->  ( s A 1 ) )  e.  ( II  Cn  C ) )
1881cvmlift 23237 . . . . . . . . 9  |-  ( ( ( F  e.  ( C CovMap  J )  /\  H  e.  ( II  Cn  J ) )  /\  ( P  e.  B  /\  ( F `  P
)  =  ( H `
 0 ) ) )  ->  E! f  e.  ( II  Cn  C
) ( ( F  o.  f )  =  H  /\  ( f `
 0 )  =  P ) )
1893, 10, 5, 14, 188syl22anc 1183 . . . . . . . 8  |-  ( ph  ->  E! f  e.  ( II  Cn  C ) ( ( F  o.  f )  =  H  /\  ( f ` 
0 )  =  P ) )
190 coeq2 4841 . . . . . . . . . . 11  |-  ( f  =  ( s  e.  ( 0 [,] 1
)  |->  ( s A 1 ) )  -> 
( F  o.  f
)  =  ( F  o.  ( s  e.  ( 0 [,] 1
)  |->  ( s A 1 ) ) ) )
191190eqeq1d 2292 . . . . . . . . . 10  |-  ( f  =  ( s  e.  ( 0 [,] 1
)  |->  ( s A 1 ) )  -> 
( ( F  o.  f )  =  H  <-> 
( F  o.  (
s  e.  ( 0 [,] 1 )  |->  ( s A 1 ) ) )  =  H ) )
192 fveq1 5485 . . . . . . . . . . . 12  |-  ( f  =  ( s  e.  ( 0 [,] 1
)  |->  ( s A 1 ) )  -> 
( f `  0
)  =  ( ( s  e.  ( 0 [,] 1 )  |->  ( s A 1 ) ) `  0 ) )
193 oveq1 5827 . . . . . . . . . . . . . 14  |-  ( s  =  0  ->  (
s A 1 )  =  ( 0 A 1 ) )
194 eqid 2284 . . . . . . . . . . . . . 14  |-  ( s  e.  ( 0 [,] 1 )  |->  ( s A 1 ) )  =  ( s  e.  ( 0 [,] 1
)  |->  ( s A 1 ) )
195 ovex 5845 . . . . . . . . . . . . . 14  |-  ( 0 A 1 )  e. 
_V
196193, 194, 195fvmpt 5564 . . . . . . . . . . . . 13  |-  ( 0  e.  ( 0 [,] 1 )  ->  (
( s  e.  ( 0 [,] 1 ) 
|->  ( s A 1 ) ) `  0
)  =  ( 0 A 1 ) )
19723, 196ax-mp 8 . . . . . . . . . . . 12  |-  ( ( s  e.  ( 0 [,] 1 )  |->  ( s A 1 ) ) `  0 )  =  ( 0 A 1 )
198192, 197syl6eq 2332 . . . . . . . . . . 11  |-  ( f  =  ( s  e.  ( 0 [,] 1
)  |->  ( s A 1 ) )  -> 
( f `  0
)  =  ( 0 A 1 ) )
199198eqeq1d 2292 . . . . . . . . . 10  |-  ( f  =  ( s  e.  ( 0 [,] 1
)  |->  ( s A 1 ) )  -> 
( ( f ` 
0 )  =  P  <-> 
( 0 A 1 )  =  P ) )
200191, 199anbi12d 691 . . . . . . . . 9  |-  ( f  =  ( s  e.  ( 0 [,] 1
)  |->  ( s A 1 ) )  -> 
( ( ( F  o.  f )  =  H  /\  ( f `
 0 )  =  P )  <->  ( ( F  o.  ( s  e.  ( 0 [,] 1
)  |->  ( s A 1 ) ) )  =  H  /\  (
0 A 1 )  =  P ) ) )
201200riota2 6323 . . . . . . . 8  |-  ( ( ( s  e.  ( 0 [,] 1 ) 
|->  ( s A 1 ) )  e.  ( II  Cn  C )  /\  E! f  e.  ( II  Cn  C
) ( ( F  o.  f )  =  H  /\  ( f `
 0 )  =  P ) )  -> 
( ( ( F  o.  ( s  e.  ( 0 [,] 1
)  |->  ( s A 1 ) ) )  =  H  /\  (
0 A 1 )  =  P )  <->  ( iota_ f  e.  ( II  Cn  C ) ( ( F  o.  f )  =  H  /\  (
f `  0 )  =  P ) )  =  ( s  e.  ( 0 [,] 1 ) 
|->  ( s A 1 ) ) ) )
202187, 189, 201syl2anc 642 . . . . . . 7  |-  ( ph  ->  ( ( ( F  o.  ( s  e.  ( 0 [,] 1
)  |->  ( s A 1 ) ) )  =  H  /\  (
0 A 1 )  =  P )  <->  ( iota_ f  e.  ( II  Cn  C ) ( ( F  o.  f )  =  H  /\  (
f `  0 )  =  P ) )  =  ( s  e.  ( 0 [,] 1 ) 
|->  ( s A 1 ) ) ) )
203117, 184, 202mpbi2and 887 . . . . . 6  |-  ( ph  ->  ( iota_ f  e.  ( II  Cn  C ) ( ( F  o.  f )  =  H  /\  ( f ` 
0 )  =  P ) )  =  ( s  e.  ( 0 [,] 1 )  |->  ( s A 1 ) ) )
2049, 203syl5eq 2328 . . . . 5  |-  ( ph  ->  N  =  ( s  e.  ( 0 [,] 1 )  |->  ( s A 1 ) ) )
20519, 1cnf 16972 . . . . . . 7  |-  ( N  e.  ( II  Cn  C )  ->  N : ( 0 [,] 1 ) --> B )
20616, 205syl 15 . . . . . 6  |-  ( ph  ->  N : ( 0 [,] 1 ) --> B )
207206feqmptd 5537 . . . . 5  |-  ( ph  ->  N  =  ( s  e.  ( 0 [,] 1 )  |->  ( N `
 s ) ) )
208204, 207eqtr3d 2318 . . . 4  |-  ( ph  ->  ( s  e.  ( 0 [,] 1 ) 
|->  ( s A 1 ) )  =  ( s  e.  ( 0 [,] 1 )  |->  ( N `  s ) ) )
209 mpteqb 5576 . . . . 5  |-  ( A. s  e.  ( 0 [,] 1 ) ( s A 1 )  e.  _V  ->  (
( s  e.  ( 0 [,] 1 ) 
|->  ( s A 1 ) )  =  ( s  e.  ( 0 [,] 1 )  |->  ( N `  s ) )  <->  A. s  e.  ( 0 [,] 1 ) ( s A 1 )  =  ( N `
 s ) ) )
210 ovex 5845 . . . . . 6  |-  ( s A 1 )  e. 
_V
211210a1i 10 . . . . 5  |-  ( s  e.  ( 0 [,] 1 )  ->  (
s A 1 )  e.  _V )
212209, 211mprg 2613 . . . 4  |-  ( ( s  e.  ( 0 [,] 1 )  |->  ( s A 1 ) )  =  ( s  e.  ( 0 [,] 1 )  |->  ( N `
 s ) )  <->  A. s  e.  (
0 [,] 1 ) ( s A 1 )  =  ( N `
 s ) )
213208, 212sylib 188 . . 3  |-  ( ph  ->  A. s  e.  ( 0 [,] 1 ) ( s A 1 )  =  ( N `
 s ) )
214213r19.21bi 2642 . 2  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
s A 1 )  =  ( N `  s ) )
215179r19.21bi 2642 . 2  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
0 A s )  =  ( M ` 
0 ) )
21637, 186, 61, 17cnmpt12f 17356 . . . . . 6  |-  ( ph  ->  ( s  e.  ( 0 [,] 1 ) 
|->  ( 1 A s ) )  e.  ( II  Cn  C ) )
217 ffvelrn 5625 . . . . . . . 8  |-  ( ( M : ( 0 [,] 1 ) --> B  /\  1  e.  ( 0 [,] 1 ) )  ->  ( M `  1 )  e.  B )
21885, 94, 217sylancl 643 . . . . . . 7  |-  ( ph  ->  ( M `  1
)  e.  B )
219 cnconst2 17007 . . . . . . 7  |-  ( ( II  e.  (TopOn `  ( 0 [,] 1
) )  /\  C  e.  (TopOn `  B )  /\  ( M `  1
)  e.  B )  ->  ( ( 0 [,] 1 )  X. 
{ ( M ` 
1 ) } )  e.  ( II  Cn  C ) )
22037, 126, 218, 219syl3anc 1182 . . . . . 6  |-  ( ph  ->  ( ( 0 [,] 1 )  X.  {
( M `  1
) } )  e.  ( II  Cn  C
) )
221 opelxpi 4720 . . . . . . . . . . . . . 14  |-  ( ( 1  e.  ( 0 [,] 1 )  /\  s  e.  ( 0 [,] 1 ) )  ->  <. 1 ,  s
>.  e.  ( ( 0 [,] 1 )  X.  ( 0 [,] 1
) ) )
22294, 221mpan 651 . . . . . . . . . . . . 13  |-  ( s  e.  ( 0 [,] 1 )  ->  <. 1 ,  s >.  e.  ( ( 0 [,] 1
)  X.  ( 0 [,] 1 ) ) )
223 fvco3 5558 . . . . . . . . . . . . 13  |-  ( ( A : ( ( 0 [,] 1 )  X.  ( 0 [,] 1 ) ) --> B  /\  <. 1 ,  s
>.  e.  ( ( 0 [,] 1 )  X.  ( 0 [,] 1
) ) )  -> 
( ( F  o.  A ) `  <. 1 ,  s >. )  =  ( F `  ( A `  <. 1 ,  s >. )
) )
22422, 222, 223syl2an 463 . . . . . . . . . . . 12  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
( F  o.  A
) `  <. 1 ,  s >. )  =  ( F `  ( A `
 <. 1 ,  s
>. ) ) )
22529fveq1d 5488 . . . . . . . . . . . 12  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
( F  o.  A
) `  <. 1 ,  s >. )  =  ( K `  <. 1 ,  s >. )
)
226224, 225eqtr3d 2318 . . . . . . . . . . 11  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  ( F `  ( A `  <. 1 ,  s
>. ) )  =  ( K `  <. 1 ,  s >. )
)
227 df-ov 5823 . . . . . . . . . . . 12  |-  ( 1 A s )  =  ( A `  <. 1 ,  s >. )
228227fveq2i 5489 . . . . . . . . . . 11  |-  ( F `
 ( 1 A s ) )  =  ( F `  ( A `  <. 1 ,  s >. ) )
229 df-ov 5823 . . . . . . . . . . 11  |-  ( 1 K s )  =  ( K `  <. 1 ,  s >. )
230226, 228, 2293eqtr4g 2341 . . . . . . . . . 10  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  ( F `  ( 1 A s ) )  =  ( 1 K s ) )
231131simprd 449 . . . . . . . . . 10  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
1 K s )  =  ( G ` 
1 ) )
2327simp2d 968 . . . . . . . . . . . . 13  |-  ( ph  ->  ( F  o.  M
)  =  G )
233232adantr 451 . . . . . . . . . . . 12  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  ( F  o.  M )  =  G )
234233fveq1d 5488 . . . . . . . . . . 11  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
( F  o.  M
) `  1 )  =  ( G ` 
1 ) )
23585adantr 451 . . . . . . . . . . . 12  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  M : ( 0 [,] 1 ) --> B )
236 fvco3 5558 . . . . . . . . . . . 12  |-  ( ( M : ( 0 [,] 1 ) --> B  /\  1  e.  ( 0 [,] 1 ) )  ->  ( ( F  o.  M ) `  1 )  =  ( F `  ( M `  1 )
) )
237235, 94, 236sylancl 643 . . . . . . . . . . 11  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
( F  o.  M
) `  1 )  =  ( F `  ( M `  1 ) ) )
238234, 237eqtr3d 2318 . . . . . . . . . 10  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  ( G `  1 )  =  ( F `  ( M `  1 ) ) )
239230, 231, 2383eqtrd 2320 . . . . . . . . 9  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  ( F `  ( 1 A s ) )  =  ( F `  ( M `  1 ) ) )
240239mpteq2dva 4107 . . . . . . . 8  |-  ( ph  ->  ( s  e.  ( 0 [,] 1 ) 
|->  ( F `  (
1 A s ) ) )  =  ( s  e.  ( 0 [,] 1 )  |->  ( F `  ( M `
 1 ) ) ) )
241 fconstmpt 4731 . . . . . . . 8  |-  ( ( 0 [,] 1 )  X.  { ( F `
 ( M ` 
1 ) ) } )  =  ( s  e.  ( 0 [,] 1 )  |->  ( F `
 ( M ` 
1 ) ) )
242240, 241syl6eqr 2334 . . . . . . 7  |-  ( ph  ->  ( s  e.  ( 0 [,] 1 ) 
|->  ( F `  (
1 A s ) ) )  =  ( ( 0 [,] 1
)  X.  { ( F `  ( M `
 1 ) ) } ) )
243 fovrn 5952 . . . . . . . . . 10  |-  ( ( A : ( ( 0 [,] 1 )  X.  ( 0 [,] 1 ) ) --> B  /\  1  e.  ( 0 [,] 1 )  /\  s  e.  ( 0 [,] 1 ) )  ->  ( 1 A s )  e.  B )
24494, 243mp3an2 1265 . . . . . . . . 9  |-  ( ( A : ( ( 0 [,] 1 )  X.  ( 0 [,] 1 ) ) --> B  /\  s  e.  ( 0 [,] 1 ) )  ->  ( 1 A s )  e.  B )
24522, 244sylan 457 . . . . . . . 8  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
1 A s )  e.  B )
246 eqidd 2285 . . . . . . . 8  |-  ( ph  ->  ( s  e.  ( 0 [,] 1 ) 
|->  ( 1 A s ) )  =  ( s  e.  ( 0 [,] 1 )  |->  ( 1 A s ) ) )
247 fveq2 5486 . . . . . . . 8  |-  ( x  =  ( 1 A s )  ->  ( F `  x )  =  ( F `  ( 1 A s ) ) )
248245, 246, 53, 247fmptco 5653 . . . . . . 7  |-  ( ph  ->  ( F  o.  (
s  e.  ( 0 [,] 1 )  |->  ( 1 A s ) ) )  =  ( s  e.  ( 0 [,] 1 )  |->  ( F `  ( 1 A s ) ) ) )
249 fcoconst 5657 . . . . . . . 8  |-  ( ( F  Fn  B  /\  ( M `  1 )  e.  B )  -> 
( F  o.  (
( 0 [,] 1
)  X.  { ( M `  1 ) } ) )  =  ( ( 0 [,] 1 )  X.  {
( F `  ( M `  1 )
) } ) )
250159, 218, 249syl2anc 642 . . . . . . 7  |-  ( ph  ->  ( F  o.  (
( 0 [,] 1
)  X.  { ( M `  1 ) } ) )  =  ( ( 0 [,] 1 )  X.  {
( F `  ( M `  1 )
) } ) )
251242, 248, 2503eqtr4d 2326 . . . . . 6  |-  ( ph  ->  ( F  o.  (
s  e.  ( 0 [,] 1 )  |->  ( 1 A s ) ) )  =  ( F  o.  ( ( 0 [,] 1 )  X.  { ( M `
 1 ) } ) ) )
252 oveq1 5827 . . . . . . . . . 10  |-  ( s  =  1  ->  (
s A 0 )  =  ( 1 A 0 ) )
253 fveq2 5486 . . . . . . . . . 10  |-  ( s  =  1  ->  ( M `  s )  =  ( M ` 
1 ) )
254252, 253eqeq12d 2298 . . . . . . . . 9  |-  ( s  =  1  ->  (
( s A 0 )  =  ( M `
 s )  <->  ( 1 A 0 )  =  ( M `  1
) ) )
255254rspcv 2881 . . . . . . . 8  |-  ( 1  e.  ( 0 [,] 1 )  ->  ( A. s  e.  (
0 [,] 1 ) ( s A 0 )  =  ( M `
 s )  -> 
( 1 A 0 )  =  ( M `
 1 ) ) )
25694, 92, 255mpsyl 59 . . . . . . 7  |-  ( ph  ->  ( 1 A 0 )  =  ( M `
 1 ) )
257 oveq2 5828 . . . . . . . . 9  |-  ( s  =  0  ->  (
1 A s )  =  ( 1 A 0 ) )
258 eqid 2284 . . . . . . . . 9  |-  ( s  e.  ( 0 [,] 1 )  |->  ( 1 A s ) )  =  ( s  e.  ( 0 [,] 1
)  |->  ( 1 A s ) )
259 ovex 5845 . . . . . . . . 9  |-  ( 1 A 0 )  e. 
_V
260257, 258, 259fvmpt 5564 . . . . . . . 8  |-  ( 0  e.  ( 0 [,] 1 )  ->  (
( s  e.  ( 0 [,] 1 ) 
|->  ( 1 A s ) ) `  0
)  =  ( 1 A 0 ) )
26123, 260ax-mp 8 . . . . . . 7  |-  ( ( s  e.  ( 0 [,] 1 )  |->  ( 1 A s ) ) `  0 )  =  ( 1 A 0 )
262 fvex 5500 . . . . . . . . 9  |-  ( M `
 1 )  e. 
_V
263262fvconst2 5691 . . . . . . . 8  |-  ( 0  e.  ( 0 [,] 1 )  ->  (
( ( 0 [,] 1 )  X.  {
( M `  1
) } ) ` 
0 )  =  ( M `  1 ) )
26423, 263ax-mp 8 . . . . . . 7  |-  ( ( ( 0 [,] 1
)  X.  { ( M `  1 ) } ) `  0
)  =  ( M `
 1 )
265256, 261, 2643eqtr4g 2341 . . . . . 6  |-  ( ph  ->  ( ( s  e.  ( 0 [,] 1
)  |->  ( 1 A s ) ) ` 
0 )  =  ( ( ( 0 [,] 1 )  X.  {
( M `  1
) } ) ` 
0 ) )
2661, 19, 3, 119, 121, 62, 216, 220, 251, 265cvmliftmoi 23221 . . . . 5  |-  ( ph  ->  ( s  e.  ( 0 [,] 1 ) 
|->  ( 1 A s ) )  =  ( ( 0 [,] 1
)  X.  { ( M `  1 ) } ) )
267 fconstmpt 4731 . . . . 5  |-  ( ( 0 [,] 1 )  X.  { ( M `
 1 ) } )  =  ( s  e.  ( 0 [,] 1 )  |->  ( M `
 1 ) )
268266, 267syl6eq 2332 . . . 4  |-  ( ph  ->  ( s  e.  ( 0 [,] 1 ) 
|->  ( 1 A s ) )  =  ( s  e.  ( 0 [,] 1 )  |->  ( M `  1 ) ) )
269 mpteqb 5576 . . . . 5  |-  ( A. s  e.  ( 0 [,] 1 ) ( 1 A s )  e.  _V  ->  (
( s  e.  ( 0 [,] 1 ) 
|->  ( 1 A s ) )  =  ( s  e.  ( 0 [,] 1 )  |->  ( M `  1 ) )  <->  A. s  e.  ( 0 [,] 1 ) ( 1 A s )  =  ( M `
 1 ) ) )
270 ovex 5845 . . . . . 6  |-  ( 1 A s )  e. 
_V
271270a1i 10 . . . . 5  |-  ( s  e.  ( 0 [,] 1 )  ->  (
1 A s )  e.  _V )
272269, 271mprg 2613 . . . 4  |-  ( ( s  e.  ( 0 [,] 1 )  |->  ( 1 A s ) )  =  ( s  e.  ( 0 [,] 1 )  |->  ( M `
 1 ) )  <->  A. s  e.  (
0 [,] 1 ) ( 1 A s )  =  ( M `
 1 ) )
273268, 272sylib 188 . . 3  |-  ( ph  ->  A. s  e.  ( 0 [,] 1 ) ( 1 A s )  =  ( M `
 1 ) )
274273r19.21bi 2642 . 2  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
1 A s )  =  ( M ` 
1 ) )
2758, 16, 17, 93, 214, 215, 274isphtpy2d 18481 1  |-  ( ph  ->  A  e.  ( M ( PHtpy `  C ) N ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1685   A.wral 2544   E!wreu 2546   _Vcvv 2789   {csn 3641   <.cop 3644   U.cuni 3828    e. cmpt 4078    X. cxp 4686    o. ccom 4692    Fn wfn 5216   -->wf 5217   ` cfv 5221  (class class class)co 5820   iota_crio 6291   0cc0 8733   1c1 8734   [,]cicc 10655   Topctop 16627  TopOnctopon 16628    Cn ccn 16950   Conccon 17133  𝑛Locally cnlly 17187    tX ctx 17251   IIcii 18375   Htpy chtpy 18461   PHtpycphtpy 18462   CovMap ccvm 23193
This theorem is referenced by:  cvmliftpht  23256
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1636  ax-8 1644  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2265  ax-rep 4132  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511  ax-inf2 7338  ax-cnex 8789  ax-resscn 8790  ax-1cn 8791  ax-icn 8792  ax-addcl 8793  ax-addrcl 8794  ax-mulcl 8795  ax-mulrcl 8796  ax-mulcom 8797  ax-addass 8798  ax-mulass 8799  ax-distr 8800  ax-i2m1 8801  ax-1ne0 8802  ax-1rid 8803  ax-rnegex 8804  ax-rrecex 8805  ax-cnre 8806  ax-pre-lttri 8807  ax-pre-lttrn 8808  ax-pre-ltadd 8809  ax-pre-mulgt0 8810  ax-pre-sup 8811  ax-addf 8812  ax-mulf 8813
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1631  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-nel 2450  df-ral 2549  df-rex 2550  df-reu 2551  df-rmo 2552  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-pss 3169  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-tp 3649  df-op 3650  df-uni 3829  df-int 3864  df-iun 3908  df-iin 3909  df-br 4025  df-opab 4079  df-mpt 4080  df-tr 4115  df-eprel 4304  df-id 4308  df-po 4313  df-so 4314  df-fr 4351  df-se 4352  df-we 4353  df-ord 4394  df-on 4395  df-lim 4396  df-suc 4397  df-om 4656  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-isom 5230  df-ov 5823  df-oprab 5824  df-mpt2 5825  df-of 6040  df-1st 6084  df-2nd 6085  df-iota 6253  df-riota 6300  df-recs 6384  df-rdg 6419  df-1o 6475  df-2o 6476  df-oadd 6479  df-er 6656  df-ec 6658  df-map 6770  df-ixp 6814  df-en 6860  df-dom 6861  df-sdom 6862  df-fin 6863  df-fi 7161  df-sup 7190  df-oi 7221  df-card 7568  df-cda 7790  df-pnf 8865  df-mnf 8866  df-xr 8867  df-ltxr 8868  df-le 8869  df-sub 9035  df-neg 9036  df-div 9420  df-nn 9743  df-2 9800  df-3 9801  df-4 9802  df-5 9803  df-6 9804  df-7 9805  df-8 9806  df-9 9807  df-10 9808  df-n0 9962  df-z 10021  df-dec 10121  df-uz 10227  df-q 10313  df-rp 10351  df-xneg 10448  df-xadd 10449  df-xmul 10450  df-ioo 10656  df-ico 10658  df-icc 10659  df-fz 10779  df-fzo 10867  df-fl 10921  df-seq 11043  df-exp 11101  df-hash 11334  df-cj 11580  df-re 11581  df-im 11582  df-sqr 11716  df-abs 11717  df-clim 11958  df-sum 12155  df-struct 13146  df-ndx 13147  df-slot 13148  df-base 13149  df-sets 13150  df-ress 13151  df-plusg 13217  df-mulr 13218  df-starv 13219  df-sca 13220  df-vsca 13221  df-tset 13223  df-ple 13224  df-ds 13226  df-hom 13228  df-cco 13229  df-rest 13323  df-topn 13324  df-topgen 13340  df-pt 13341  df-prds 13344  df-xrs 13399  df-0g 13400  df-gsum 13401  df-qtop 13406  df-imas 13407  df-xps 13409  df-mre 13484  df-mrc 13485  df-acs 13487  df-mnd 14363  df-submnd 14412  df-mulg 14488  df-cntz 14789  df-cmn 15087  df-xmet 16369  df-met 16370  df-bl 16371  df-mopn 16372  df-cnfld 16374  df-top 16632  df-bases 16634  df-topon 16635  df-topsp 16636  df-cld 16752  df-ntr 16753  df-cls 16754  df-nei 16831  df-cn 16953  df-cnp 16954  df-cmp 17110  df-con 17134  df-lly 17188  df-nlly 17189  df-tx 17253  df-hmeo 17442  df-xms 17881  df-ms 17882  df-tms 17883  df-ii 18377  df-htpy 18464  df-phtpy 18465  df-phtpc 18486  df-pcon 23159  df-scon 23160  df-cvm 23194
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