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Theorem cvmliftphtlem 30033
Description: Lemma for cvmliftpht 30034. (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 30017 . . 3  |-  ( ph  ->  ( M  e.  ( II  Cn  C )  /\  ( F  o.  M )  =  G  /\  ( M ` 
0 )  =  P ) )
87simp1d 1019 . 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 22009 . . . . . 6  |-  ( ph  ->  ( ( G ` 
0 )  =  ( H `  0 )  /\  ( G ` 
1 )  =  ( H `  1 ) ) )
1312simpld 461 . . . . 5  |-  ( ph  ->  ( G `  0
)  =  ( H `
 0 ) )
146, 13eqtrd 2484 . . . 4  |-  ( ph  ->  ( F `  P
)  =  ( H `
 0 ) )
151, 9, 3, 10, 5, 14cvmliftiota 30017 . . 3  |-  ( ph  ->  ( N  e.  ( II  Cn  C )  /\  ( F  o.  N )  =  H  /\  ( N ` 
0 )  =  P ) )
1615simp1d 1019 . 2  |-  ( ph  ->  N  e.  ( II 
Cn  C ) )
17 cvmliftphtlem.a . 2  |-  ( ph  ->  A  e.  ( ( II  tX  II )  Cn  C ) )
18 iitop 21905 . . . . . . . . . . . . . . . 16  |-  II  e.  Top
19 iiuni 21906 . . . . . . . . . . . . . . . 16  |-  ( 0 [,] 1 )  = 
U. II
2018, 18, 19, 19txunii 20601 . . . . . . . . . . . . . . 15  |-  ( ( 0 [,] 1 )  X.  ( 0 [,] 1 ) )  = 
U. ( II  tX  II )
2120, 1cnf 20255 . . . . . . . . . . . . . 14  |-  ( A  e.  ( ( II 
tX  II )  Cn  C )  ->  A : ( ( 0 [,] 1 )  X.  ( 0 [,] 1
) ) --> B )
2217, 21syl 17 . . . . . . . . . . . . 13  |-  ( ph  ->  A : ( ( 0 [,] 1 )  X.  ( 0 [,] 1 ) ) --> B )
23 0elunit 11747 . . . . . . . . . . . . . 14  |-  0  e.  ( 0 [,] 1
)
24 opelxpi 4865 . . . . . . . . . . . . . 14  |-  ( ( s  e.  ( 0 [,] 1 )  /\  0  e.  ( 0 [,] 1 ) )  ->  <. s ,  0
>.  e.  ( ( 0 [,] 1 )  X.  ( 0 [,] 1
) ) )
2523, 24mpan2 676 . . . . . . . . . . . . 13  |-  ( s  e.  ( 0 [,] 1 )  ->  <. s ,  0 >.  e.  ( ( 0 [,] 1
)  X.  ( 0 [,] 1 ) ) )
26 fvco3 5940 . . . . . . . . . . . . 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 480 . . . . . . . . . . . 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 467 . . . . . . . . . . . . 13  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  ( F  o.  A )  =  K )
3029fveq1d 5865 . . . . . . . . . . . 12  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
( F  o.  A
) `  <. s ,  0 >. )  =  ( K `  <. s ,  0 >. )
)
3127, 30eqtr3d 2486 . . . . . . . . . . 11  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  ( F `  ( A `  <. s ,  0
>. ) )  =  ( K `  <. s ,  0 >. )
)
32 df-ov 6291 . . . . . . . . . . . 12  |-  ( s A 0 )  =  ( A `  <. s ,  0 >. )
3332fveq2i 5866 . . . . . . . . . . 11  |-  ( F `
 ( s A 0 ) )  =  ( F `  ( A `  <. s ,  0 >. ) )
34 df-ov 6291 . . . . . . . . . . 11  |-  ( s K 0 )  =  ( K `  <. s ,  0 >. )
3531, 33, 343eqtr4g 2509 . . . . . . . . . 10  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  ( F `  ( s A 0 ) )  =  ( s K 0 ) )
36 iitopon 21904 . . . . . . . . . . . . 13  |-  II  e.  (TopOn `  ( 0 [,] 1 ) )
3736a1i 11 . . . . . . . . . . . 12  |-  ( ph  ->  II  e.  (TopOn `  ( 0 [,] 1
) ) )
384, 10phtpyhtpy 22006 . . . . . . . . . . . . 13  |-  ( ph  ->  ( G ( PHtpy `  J ) H ) 
C_  ( G ( II Htpy  J ) H ) )
3938, 11sseldd 3432 . . . . . . . . . . . 12  |-  ( ph  ->  K  e.  ( G ( II Htpy  J ) H ) )
4037, 4, 10, 39htpyi 21998 . . . . . . . . . . 11  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
( s K 0 )  =  ( G `
 s )  /\  ( s K 1 )  =  ( H `
 s ) ) )
4140simpld 461 . . . . . . . . . 10  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
s K 0 )  =  ( G `  s ) )
4235, 41eqtrd 2484 . . . . . . . . 9  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  ( F `  ( s A 0 ) )  =  ( G `  s ) )
4342mpteq2dva 4488 . . . . . . . 8  |-  ( ph  ->  ( s  e.  ( 0 [,] 1 ) 
|->  ( F `  (
s A 0 ) ) )  =  ( s  e.  ( 0 [,] 1 )  |->  ( G `  s ) ) )
44 fovrn 6436 . . . . . . . . . . 11  |-  ( ( A : ( ( 0 [,] 1 )  X.  ( 0 [,] 1 ) ) --> B  /\  s  e.  ( 0 [,] 1 )  /\  0  e.  ( 0 [,] 1 ) )  ->  ( s A 0 )  e.  B )
4523, 44mp3an3 1352 . . . . . . . . . 10  |-  ( ( A : ( ( 0 [,] 1 )  X.  ( 0 [,] 1 ) ) --> B  /\  s  e.  ( 0 [,] 1 ) )  ->  ( s A 0 )  e.  B )
4622, 45sylan 474 . . . . . . . . 9  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
s A 0 )  e.  B )
47 eqidd 2451 . . . . . . . . 9  |-  ( ph  ->  ( s  e.  ( 0 [,] 1 ) 
|->  ( s A 0 ) )  =  ( s  e.  ( 0 [,] 1 )  |->  ( s A 0 ) ) )
48 cvmcn 29978 . . . . . . . . . . . 12  |-  ( F  e.  ( C CovMap  J
)  ->  F  e.  ( C  Cn  J
) )
493, 48syl 17 . . . . . . . . . . 11  |-  ( ph  ->  F  e.  ( C  Cn  J ) )
50 eqid 2450 . . . . . . . . . . . 12  |-  U. J  =  U. J
511, 50cnf 20255 . . . . . . . . . . 11  |-  ( F  e.  ( C  Cn  J )  ->  F : B --> U. J )
5249, 51syl 17 . . . . . . . . . 10  |-  ( ph  ->  F : B --> U. J
)
5352feqmptd 5916 . . . . . . . . 9  |-  ( ph  ->  F  =  ( x  e.  B  |->  ( F `
 x ) ) )
54 fveq2 5863 . . . . . . . . 9  |-  ( x  =  ( s A 0 )  ->  ( F `  x )  =  ( F `  ( s A 0 ) ) )
5546, 47, 53, 54fmptco 6054 . . . . . . . 8  |-  ( ph  ->  ( F  o.  (
s  e.  ( 0 [,] 1 )  |->  ( s A 0 ) ) )  =  ( s  e.  ( 0 [,] 1 )  |->  ( F `  ( s A 0 ) ) ) )
5619, 50cnf 20255 . . . . . . . . . 10  |-  ( G  e.  ( II  Cn  J )  ->  G : ( 0 [,] 1 ) --> U. J
)
574, 56syl 17 . . . . . . . . 9  |-  ( ph  ->  G : ( 0 [,] 1 ) --> U. J )
5857feqmptd 5916 . . . . . . . 8  |-  ( ph  ->  G  =  ( s  e.  ( 0 [,] 1 )  |->  ( G `
 s ) ) )
5943, 55, 583eqtr4d 2494 . . . . . . 7  |-  ( ph  ->  ( F  o.  (
s  e.  ( 0 [,] 1 )  |->  ( s A 0 ) ) )  =  G )
60 cvmliftphtlem.0 . . . . . . 7  |-  ( ph  ->  ( 0 A 0 )  =  P )
6137cnmptid 20669 . . . . . . . . 9  |-  ( ph  ->  ( s  e.  ( 0 [,] 1 ) 
|->  s )  e.  ( II  Cn  II ) )
6223a1i 11 . . . . . . . . . 10  |-  ( ph  ->  0  e.  ( 0 [,] 1 ) )
6337, 37, 62cnmptc 20670 . . . . . . . . 9  |-  ( ph  ->  ( s  e.  ( 0 [,] 1 ) 
|->  0 )  e.  ( II  Cn  II ) )
6437, 61, 63, 17cnmpt12f 20674 . . . . . . . 8  |-  ( ph  ->  ( s  e.  ( 0 [,] 1 ) 
|->  ( s A 0 ) )  e.  ( II  Cn  C ) )
651cvmlift 30015 . . . . . . . . 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 1268 . . . . . . . 8  |-  ( ph  ->  E! f  e.  ( II  Cn  C ) ( ( F  o.  f )  =  G  /\  ( f ` 
0 )  =  P ) )
67 coeq2 4992 . . . . . . . . . . 11  |-  ( f  =  ( s  e.  ( 0 [,] 1
)  |->  ( s A 0 ) )  -> 
( F  o.  f
)  =  ( F  o.  ( s  e.  ( 0 [,] 1
)  |->  ( s A 0 ) ) ) )
6867eqeq1d 2452 . . . . . . . . . 10  |-  ( f  =  ( s  e.  ( 0 [,] 1
)  |->  ( s A 0 ) )  -> 
( ( F  o.  f )  =  G  <-> 
( F  o.  (
s  e.  ( 0 [,] 1 )  |->  ( s A 0 ) ) )  =  G ) )
69 fveq1 5862 . . . . . . . . . . . 12  |-  ( f  =  ( s  e.  ( 0 [,] 1
)  |->  ( s A 0 ) )  -> 
( f `  0
)  =  ( ( s  e.  ( 0 [,] 1 )  |->  ( s A 0 ) ) `  0 ) )
70 oveq1 6295 . . . . . . . . . . . . . 14  |-  ( s  =  0  ->  (
s A 0 )  =  ( 0 A 0 ) )
71 eqid 2450 . . . . . . . . . . . . . 14  |-  ( s  e.  ( 0 [,] 1 )  |->  ( s A 0 ) )  =  ( s  e.  ( 0 [,] 1
)  |->  ( s A 0 ) )
72 ovex 6316 . . . . . . . . . . . . . 14  |-  ( 0 A 0 )  e. 
_V
7370, 71, 72fvmpt 5946 . . . . . . . . . . . . 13  |-  ( 0  e.  ( 0 [,] 1 )  ->  (
( s  e.  ( 0 [,] 1 ) 
|->  ( s A 0 ) ) `  0
)  =  ( 0 A 0 ) )
7423, 73ax-mp 5 . . . . . . . . . . . 12  |-  ( ( s  e.  ( 0 [,] 1 )  |->  ( s A 0 ) ) `  0 )  =  ( 0 A 0 )
7569, 74syl6eq 2500 . . . . . . . . . . 11  |-  ( f  =  ( s  e.  ( 0 [,] 1
)  |->  ( s A 0 ) )  -> 
( f `  0
)  =  ( 0 A 0 ) )
7675eqeq1d 2452 . . . . . . . . . 10  |-  ( f  =  ( s  e.  ( 0 [,] 1
)  |->  ( s A 0 ) )  -> 
( ( f ` 
0 )  =  P  <-> 
( 0 A 0 )  =  P ) )
7768, 76anbi12d 716 . . . . . . . . 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 ) ) )
7877riota2 6272 . . . . . . . 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 ) ) ) )
7964, 66, 78syl2anc 666 . . . . . . 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 ) ) ) )
8059, 60, 79mpbi2and 931 . . . . . 6  |-  ( ph  ->  ( iota_ f  e.  ( II  Cn  C ) ( ( F  o.  f )  =  G  /\  ( f ` 
0 )  =  P ) )  =  ( s  e.  ( 0 [,] 1 )  |->  ( s A 0 ) ) )
812, 80syl5eq 2496 . . . . 5  |-  ( ph  ->  M  =  ( s  e.  ( 0 [,] 1 )  |->  ( s A 0 ) ) )
8219, 1cnf 20255 . . . . . . 7  |-  ( M  e.  ( II  Cn  C )  ->  M : ( 0 [,] 1 ) --> B )
838, 82syl 17 . . . . . 6  |-  ( ph  ->  M : ( 0 [,] 1 ) --> B )
8483feqmptd 5916 . . . . 5  |-  ( ph  ->  M  =  ( s  e.  ( 0 [,] 1 )  |->  ( M `
 s ) ) )
8581, 84eqtr3d 2486 . . . 4  |-  ( ph  ->  ( s  e.  ( 0 [,] 1 ) 
|->  ( s A 0 ) )  =  ( s  e.  ( 0 [,] 1 )  |->  ( M `  s ) ) )
86 mpteqb 5962 . . . . 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 ) ) )
87 ovex 6316 . . . . . 6  |-  ( s A 0 )  e. 
_V
8887a1i 11 . . . . 5  |-  ( s  e.  ( 0 [,] 1 )  ->  (
s A 0 )  e.  _V )
8986, 88mprg 2750 . . . 4  |-  ( ( s  e.  ( 0 [,] 1 )  |->  ( s A 0 ) )  =  ( s  e.  ( 0 [,] 1 )  |->  ( M `
 s ) )  <->  A. s  e.  (
0 [,] 1 ) ( s A 0 )  =  ( M `
 s ) )
9085, 89sylib 200 . . 3  |-  ( ph  ->  A. s  e.  ( 0 [,] 1 ) ( s A 0 )  =  ( M `
 s ) )
9190r19.21bi 2756 . 2  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
s A 0 )  =  ( M `  s ) )
92 1elunit 11748 . . . . . . . . . . . . . 14  |-  1  e.  ( 0 [,] 1
)
93 opelxpi 4865 . . . . . . . . . . . . . 14  |-  ( ( s  e.  ( 0 [,] 1 )  /\  1  e.  ( 0 [,] 1 ) )  ->  <. s ,  1
>.  e.  ( ( 0 [,] 1 )  X.  ( 0 [,] 1
) ) )
9492, 93mpan2 676 . . . . . . . . . . . . 13  |-  ( s  e.  ( 0 [,] 1 )  ->  <. s ,  1 >.  e.  ( ( 0 [,] 1
)  X.  ( 0 [,] 1 ) ) )
95 fvco3 5940 . . . . . . . . . . . . 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 >. )
) )
9622, 94, 95syl2an 480 . . . . . . . . . . . 12  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
( F  o.  A
) `  <. s ,  1 >. )  =  ( F `  ( A `
 <. s ,  1
>. ) ) )
9729fveq1d 5865 . . . . . . . . . . . 12  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
( F  o.  A
) `  <. s ,  1 >. )  =  ( K `  <. s ,  1 >. )
)
9896, 97eqtr3d 2486 . . . . . . . . . . 11  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  ( F `  ( A `  <. s ,  1
>. ) )  =  ( K `  <. s ,  1 >. )
)
99 df-ov 6291 . . . . . . . . . . . 12  |-  ( s A 1 )  =  ( A `  <. s ,  1 >. )
10099fveq2i 5866 . . . . . . . . . . 11  |-  ( F `
 ( s A 1 ) )  =  ( F `  ( A `  <. s ,  1 >. ) )
101 df-ov 6291 . . . . . . . . . . 11  |-  ( s K 1 )  =  ( K `  <. s ,  1 >. )
10298, 100, 1013eqtr4g 2509 . . . . . . . . . 10  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  ( F `  ( s A 1 ) )  =  ( s K 1 ) )
10340simprd 465 . . . . . . . . . 10  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
s K 1 )  =  ( H `  s ) )
104102, 103eqtrd 2484 . . . . . . . . 9  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  ( F `  ( s A 1 ) )  =  ( H `  s ) )
105104mpteq2dva 4488 . . . . . . . 8  |-  ( ph  ->  ( s  e.  ( 0 [,] 1 ) 
|->  ( F `  (
s A 1 ) ) )  =  ( s  e.  ( 0 [,] 1 )  |->  ( H `  s ) ) )
106 fovrn 6436 . . . . . . . . . . 11  |-  ( ( A : ( ( 0 [,] 1 )  X.  ( 0 [,] 1 ) ) --> B  /\  s  e.  ( 0 [,] 1 )  /\  1  e.  ( 0 [,] 1 ) )  ->  ( s A 1 )  e.  B )
10792, 106mp3an3 1352 . . . . . . . . . 10  |-  ( ( A : ( ( 0 [,] 1 )  X.  ( 0 [,] 1 ) ) --> B  /\  s  e.  ( 0 [,] 1 ) )  ->  ( s A 1 )  e.  B )
10822, 107sylan 474 . . . . . . . . 9  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
s A 1 )  e.  B )
109 eqidd 2451 . . . . . . . . 9  |-  ( ph  ->  ( s  e.  ( 0 [,] 1 ) 
|->  ( s A 1 ) )  =  ( s  e.  ( 0 [,] 1 )  |->  ( s A 1 ) ) )
110 fveq2 5863 . . . . . . . . 9  |-  ( x  =  ( s A 1 )  ->  ( F `  x )  =  ( F `  ( s A 1 ) ) )
111108, 109, 53, 110fmptco 6054 . . . . . . . 8  |-  ( ph  ->  ( F  o.  (
s  e.  ( 0 [,] 1 )  |->  ( s A 1 ) ) )  =  ( s  e.  ( 0 [,] 1 )  |->  ( F `  ( s A 1 ) ) ) )
11219, 50cnf 20255 . . . . . . . . . 10  |-  ( H  e.  ( II  Cn  J )  ->  H : ( 0 [,] 1 ) --> U. J
)
11310, 112syl 17 . . . . . . . . 9  |-  ( ph  ->  H : ( 0 [,] 1 ) --> U. J )
114113feqmptd 5916 . . . . . . . 8  |-  ( ph  ->  H  =  ( s  e.  ( 0 [,] 1 )  |->  ( H `
 s ) ) )
115105, 111, 1143eqtr4d 2494 . . . . . . 7  |-  ( ph  ->  ( F  o.  (
s  e.  ( 0 [,] 1 )  |->  ( s A 1 ) ) )  =  H )
116 iicon 21912 . . . . . . . . . . . . 13  |-  II  e.  Con
117116a1i 11 . . . . . . . . . . . 12  |-  ( ph  ->  II  e.  Con )
118 iinllycon 29970 . . . . . . . . . . . . 13  |-  II  e. 𝑛Locally  Con
119118a1i 11 . . . . . . . . . . . 12  |-  ( ph  ->  II  e. 𝑛Locally  Con )
12037, 63, 61, 17cnmpt12f 20674 . . . . . . . . . . . 12  |-  ( ph  ->  ( s  e.  ( 0 [,] 1 ) 
|->  ( 0 A s ) )  e.  ( II  Cn  C ) )
121 cvmtop1 29976 . . . . . . . . . . . . . . 15  |-  ( F  e.  ( C CovMap  J
)  ->  C  e.  Top )
1223, 121syl 17 . . . . . . . . . . . . . 14  |-  ( ph  ->  C  e.  Top )
1231toptopon 19941 . . . . . . . . . . . . . 14  |-  ( C  e.  Top  <->  C  e.  (TopOn `  B ) )
124122, 123sylib 200 . . . . . . . . . . . . 13  |-  ( ph  ->  C  e.  (TopOn `  B ) )
125 ffvelrn 6018 . . . . . . . . . . . . . 14  |-  ( ( M : ( 0 [,] 1 ) --> B  /\  0  e.  ( 0 [,] 1 ) )  ->  ( M `  0 )  e.  B )
12683, 23, 125sylancl 667 . . . . . . . . . . . . 13  |-  ( ph  ->  ( M `  0
)  e.  B )
127 cnconst2 20292 . . . . . . . . . . . . 13  |-  ( ( II  e.  (TopOn `  ( 0 [,] 1
) )  /\  C  e.  (TopOn `  B )  /\  ( M `  0
)  e.  B )  ->  ( ( 0 [,] 1 )  X. 
{ ( M ` 
0 ) } )  e.  ( II  Cn  C ) )
12837, 124, 126, 127syl3anc 1267 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( 0 [,] 1 )  X.  {
( M `  0
) } )  e.  ( II  Cn  C
) )
1294, 10, 11phtpyi 22008 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
( 0 K s )  =  ( G `
 0 )  /\  ( 1 K s )  =  ( G `
 1 ) ) )
130129simpld 461 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
0 K s )  =  ( G ` 
0 ) )
131 opelxpi 4865 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( 0  e.  ( 0 [,] 1 )  /\  s  e.  ( 0 [,] 1 ) )  ->  <. 0 ,  s
>.  e.  ( ( 0 [,] 1 )  X.  ( 0 [,] 1
) ) )
13223, 131mpan 675 . . . . . . . . . . . . . . . . . . 19  |-  ( s  e.  ( 0 [,] 1 )  ->  <. 0 ,  s >.  e.  ( ( 0 [,] 1
)  X.  ( 0 [,] 1 ) ) )
133 fvco3 5940 . . . . . . . . . . . . . . . . . . 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 >. )
) )
13422, 132, 133syl2an 480 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
( F  o.  A
) `  <. 0 ,  s >. )  =  ( F `  ( A `
 <. 0 ,  s
>. ) ) )
13529fveq1d 5865 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
( F  o.  A
) `  <. 0 ,  s >. )  =  ( K `  <. 0 ,  s >. )
)
136134, 135eqtr3d 2486 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  ( F `  ( A `  <. 0 ,  s
>. ) )  =  ( K `  <. 0 ,  s >. )
)
137 df-ov 6291 . . . . . . . . . . . . . . . . . 18  |-  ( 0 A s )  =  ( A `  <. 0 ,  s >. )
138137fveq2i 5866 . . . . . . . . . . . . . . . . 17  |-  ( F `
 ( 0 A s ) )  =  ( F `  ( A `  <. 0 ,  s >. ) )
139 df-ov 6291 . . . . . . . . . . . . . . . . 17  |-  ( 0 K s )  =  ( K `  <. 0 ,  s >. )
140136, 138, 1393eqtr4g 2509 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  ( F `  ( 0 A s ) )  =  ( 0 K s ) )
1417simp3d 1021 . . . . . . . . . . . . . . . . . . 19  |-  ( ph  ->  ( M `  0
)  =  P )
142141adantr 467 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  ( M `  0 )  =  P )
143142fveq2d 5867 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  ( F `  ( M `  0 ) )  =  ( F `  P ) )
1446adantr 467 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  ( F `  P )  =  ( G ` 
0 ) )
145143, 144eqtrd 2484 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  ( F `  ( M `  0 ) )  =  ( G ` 
0 ) )
146130, 140, 1453eqtr4d 2494 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  ( F `  ( 0 A s ) )  =  ( F `  ( M `  0 ) ) )
147146mpteq2dva 4488 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( s  e.  ( 0 [,] 1 ) 
|->  ( F `  (
0 A s ) ) )  =  ( s  e.  ( 0 [,] 1 )  |->  ( F `  ( M `
 0 ) ) ) )
148 fconstmpt 4877 . . . . . . . . . . . . . 14  |-  ( ( 0 [,] 1 )  X.  { ( F `
 ( M ` 
0 ) ) } )  =  ( s  e.  ( 0 [,] 1 )  |->  ( F `
 ( M ` 
0 ) ) )
149147, 148syl6eqr 2502 . . . . . . . . . . . . 13  |-  ( ph  ->  ( s  e.  ( 0 [,] 1 ) 
|->  ( F `  (
0 A s ) ) )  =  ( ( 0 [,] 1
)  X.  { ( F `  ( M `
 0 ) ) } ) )
150 fovrn 6436 . . . . . . . . . . . . . . . 16  |-  ( ( A : ( ( 0 [,] 1 )  X.  ( 0 [,] 1 ) ) --> B  /\  0  e.  ( 0 [,] 1 )  /\  s  e.  ( 0 [,] 1 ) )  ->  ( 0 A s )  e.  B )
15123, 150mp3an2 1351 . . . . . . . . . . . . . . 15  |-  ( ( A : ( ( 0 [,] 1 )  X.  ( 0 [,] 1 ) ) --> B  /\  s  e.  ( 0 [,] 1 ) )  ->  ( 0 A s )  e.  B )
15222, 151sylan 474 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
0 A s )  e.  B )
153 eqidd 2451 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( s  e.  ( 0 [,] 1 ) 
|->  ( 0 A s ) )  =  ( s  e.  ( 0 [,] 1 )  |->  ( 0 A s ) ) )
154 fveq2 5863 . . . . . . . . . . . . . 14  |-  ( x  =  ( 0 A s )  ->  ( F `  x )  =  ( F `  ( 0 A s ) ) )
155152, 153, 53, 154fmptco 6054 . . . . . . . . . . . . 13  |-  ( ph  ->  ( F  o.  (
s  e.  ( 0 [,] 1 )  |->  ( 0 A s ) ) )  =  ( s  e.  ( 0 [,] 1 )  |->  ( F `  ( 0 A s ) ) ) )
156 ffn 5726 . . . . . . . . . . . . . . 15  |-  ( F : B --> U. J  ->  F  Fn  B )
15752, 156syl 17 . . . . . . . . . . . . . 14  |-  ( ph  ->  F  Fn  B )
158 fcoconst 6058 . . . . . . . . . . . . . 14  |-  ( ( F  Fn  B  /\  ( M `  0 )  e.  B )  -> 
( F  o.  (
( 0 [,] 1
)  X.  { ( M `  0 ) } ) )  =  ( ( 0 [,] 1 )  X.  {
( F `  ( M `  0 )
) } ) )
159157, 126, 158syl2anc 666 . . . . . . . . . . . . 13  |-  ( ph  ->  ( F  o.  (
( 0 [,] 1
)  X.  { ( M `  0 ) } ) )  =  ( ( 0 [,] 1 )  X.  {
( F `  ( M `  0 )
) } ) )
160149, 155, 1593eqtr4d 2494 . . . . . . . . . . . 12  |-  ( ph  ->  ( F  o.  (
s  e.  ( 0 [,] 1 )  |->  ( 0 A s ) ) )  =  ( F  o.  ( ( 0 [,] 1 )  X.  { ( M `
 0 ) } ) ) )
16160, 141eqtr4d 2487 . . . . . . . . . . . . 13  |-  ( ph  ->  ( 0 A 0 )  =  ( M `
 0 ) )
162 oveq2 6296 . . . . . . . . . . . . . . 15  |-  ( s  =  0  ->  (
0 A s )  =  ( 0 A 0 ) )
163 eqid 2450 . . . . . . . . . . . . . . 15  |-  ( s  e.  ( 0 [,] 1 )  |->  ( 0 A s ) )  =  ( s  e.  ( 0 [,] 1
)  |->  ( 0 A s ) )
164162, 163, 72fvmpt 5946 . . . . . . . . . . . . . 14  |-  ( 0  e.  ( 0 [,] 1 )  ->  (
( s  e.  ( 0 [,] 1 ) 
|->  ( 0 A s ) ) `  0
)  =  ( 0 A 0 ) )
16523, 164ax-mp 5 . . . . . . . . . . . . 13  |-  ( ( s  e.  ( 0 [,] 1 )  |->  ( 0 A s ) ) `  0 )  =  ( 0 A 0 )
166 fvex 5873 . . . . . . . . . . . . . . 15  |-  ( M `
 0 )  e. 
_V
167166fvconst2 6118 . . . . . . . . . . . . . 14  |-  ( 0  e.  ( 0 [,] 1 )  ->  (
( ( 0 [,] 1 )  X.  {
( M `  0
) } ) ` 
0 )  =  ( M `  0 ) )
16823, 167ax-mp 5 . . . . . . . . . . . . 13  |-  ( ( ( 0 [,] 1
)  X.  { ( M `  0 ) } ) `  0
)  =  ( M `
 0 )
169161, 165, 1683eqtr4g 2509 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( s  e.  ( 0 [,] 1
)  |->  ( 0 A s ) ) ` 
0 )  =  ( ( ( 0 [,] 1 )  X.  {
( M `  0
) } ) ` 
0 ) )
1701, 19, 3, 117, 119, 62, 120, 128, 160, 169cvmliftmoi 29999 . . . . . . . . . . 11  |-  ( ph  ->  ( s  e.  ( 0 [,] 1 ) 
|->  ( 0 A s ) )  =  ( ( 0 [,] 1
)  X.  { ( M `  0 ) } ) )
171 fconstmpt 4877 . . . . . . . . . . 11  |-  ( ( 0 [,] 1 )  X.  { ( M `
 0 ) } )  =  ( s  e.  ( 0 [,] 1 )  |->  ( M `
 0 ) )
172170, 171syl6eq 2500 . . . . . . . . . 10  |-  ( ph  ->  ( s  e.  ( 0 [,] 1 ) 
|->  ( 0 A s ) )  =  ( s  e.  ( 0 [,] 1 )  |->  ( M `  0 ) ) )
173 mpteqb 5962 . . . . . . . . . . 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 ) ) )
174 ovex 6316 . . . . . . . . . . . 12  |-  ( 0 A s )  e. 
_V
175174a1i 11 . . . . . . . . . . 11  |-  ( s  e.  ( 0 [,] 1 )  ->  (
0 A s )  e.  _V )
176173, 175mprg 2750 . . . . . . . . . 10  |-  ( ( s  e.  ( 0 [,] 1 )  |->  ( 0 A s ) )  =  ( s  e.  ( 0 [,] 1 )  |->  ( M `
 0 ) )  <->  A. s  e.  (
0 [,] 1 ) ( 0 A s )  =  ( M `
 0 ) )
177172, 176sylib 200 . . . . . . . . 9  |-  ( ph  ->  A. s  e.  ( 0 [,] 1 ) ( 0 A s )  =  ( M `
 0 ) )
178 oveq2 6296 . . . . . . . . . . 11  |-  ( s  =  1  ->  (
0 A s )  =  ( 0 A 1 ) )
179178eqeq1d 2452 . . . . . . . . . 10  |-  ( s  =  1  ->  (
( 0 A s )  =  ( M `
 0 )  <->  ( 0 A 1 )  =  ( M `  0
) ) )
180179rspcv 3145 . . . . . . . . 9  |-  ( 1  e.  ( 0 [,] 1 )  ->  ( A. s  e.  (
0 [,] 1 ) ( 0 A s )  =  ( M `
 0 )  -> 
( 0 A 1 )  =  ( M `
 0 ) ) )
18192, 177, 180mpsyl 65 . . . . . . . 8  |-  ( ph  ->  ( 0 A 1 )  =  ( M `
 0 ) )
182181, 141eqtrd 2484 . . . . . . 7  |-  ( ph  ->  ( 0 A 1 )  =  P )
18392a1i 11 . . . . . . . . . 10  |-  ( ph  ->  1  e.  ( 0 [,] 1 ) )
18437, 37, 183cnmptc 20670 . . . . . . . . 9  |-  ( ph  ->  ( s  e.  ( 0 [,] 1 ) 
|->  1 )  e.  ( II  Cn  II ) )
18537, 61, 184, 17cnmpt12f 20674 . . . . . . . 8  |-  ( ph  ->  ( s  e.  ( 0 [,] 1 ) 
|->  ( s A 1 ) )  e.  ( II  Cn  C ) )
1861cvmlift 30015 . . . . . . . . 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 ) )
1873, 10, 5, 14, 186syl22anc 1268 . . . . . . . 8  |-  ( ph  ->  E! f  e.  ( II  Cn  C ) ( ( F  o.  f )  =  H  /\  ( f ` 
0 )  =  P ) )
188 coeq2 4992 . . . . . . . . . . 11  |-  ( f  =  ( s  e.  ( 0 [,] 1
)  |->  ( s A 1 ) )  -> 
( F  o.  f
)  =  ( F  o.  ( s  e.  ( 0 [,] 1
)  |->  ( s A 1 ) ) ) )
189188eqeq1d 2452 . . . . . . . . . 10  |-  ( f  =  ( s  e.  ( 0 [,] 1
)  |->  ( s A 1 ) )  -> 
( ( F  o.  f )  =  H  <-> 
( F  o.  (
s  e.  ( 0 [,] 1 )  |->  ( s A 1 ) ) )  =  H ) )
190 fveq1 5862 . . . . . . . . . . . 12  |-  ( f  =  ( s  e.  ( 0 [,] 1
)  |->  ( s A 1 ) )  -> 
( f `  0
)  =  ( ( s  e.  ( 0 [,] 1 )  |->  ( s A 1 ) ) `  0 ) )
191 oveq1 6295 . . . . . . . . . . . . . 14  |-  ( s  =  0  ->  (
s A 1 )  =  ( 0 A 1 ) )
192 eqid 2450 . . . . . . . . . . . . . 14  |-  ( s  e.  ( 0 [,] 1 )  |->  ( s A 1 ) )  =  ( s  e.  ( 0 [,] 1
)  |->  ( s A 1 ) )
193 ovex 6316 . . . . . . . . . . . . . 14  |-  ( 0 A 1 )  e. 
_V
194191, 192, 193fvmpt 5946 . . . . . . . . . . . . 13  |-  ( 0  e.  ( 0 [,] 1 )  ->  (
( s  e.  ( 0 [,] 1 ) 
|->  ( s A 1 ) ) `  0
)  =  ( 0 A 1 ) )
19523, 194ax-mp 5 . . . . . . . . . . . 12  |-  ( ( s  e.  ( 0 [,] 1 )  |->  ( s A 1 ) ) `  0 )  =  ( 0 A 1 )
196190, 195syl6eq 2500 . . . . . . . . . . 11  |-  ( f  =  ( s  e.  ( 0 [,] 1
)  |->  ( s A 1 ) )  -> 
( f `  0
)  =  ( 0 A 1 ) )
197196eqeq1d 2452 . . . . . . . . . 10  |-  ( f  =  ( s  e.  ( 0 [,] 1
)  |->  ( s A 1 ) )  -> 
( ( f ` 
0 )  =  P  <-> 
( 0 A 1 )  =  P ) )
198189, 197anbi12d 716 . . . . . . . . 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 ) ) )
199198riota2 6272 . . . . . . . 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 ) ) ) )
200185, 187, 199syl2anc 666 . . . . . . 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 ) ) ) )
201115, 182, 200mpbi2and 931 . . . . . 6  |-  ( ph  ->  ( iota_ f  e.  ( II  Cn  C ) ( ( F  o.  f )  =  H  /\  ( f ` 
0 )  =  P ) )  =  ( s  e.  ( 0 [,] 1 )  |->  ( s A 1 ) ) )
2029, 201syl5eq 2496 . . . . 5  |-  ( ph  ->  N  =  ( s  e.  ( 0 [,] 1 )  |->  ( s A 1 ) ) )
20319, 1cnf 20255 . . . . . . 7  |-  ( N  e.  ( II  Cn  C )  ->  N : ( 0 [,] 1 ) --> B )
20416, 203syl 17 . . . . . 6  |-  ( ph  ->  N : ( 0 [,] 1 ) --> B )
205204feqmptd 5916 . . . . 5  |-  ( ph  ->  N  =  ( s  e.  ( 0 [,] 1 )  |->  ( N `
 s ) ) )
206202, 205eqtr3d 2486 . . . 4  |-  ( ph  ->  ( s  e.  ( 0 [,] 1 ) 
|->  ( s A 1 ) )  =  ( s  e.  ( 0 [,] 1 )  |->  ( N `  s ) ) )
207 mpteqb 5962 . . . . 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 ) ) )
208 ovex 6316 . . . . . 6  |-  ( s A 1 )  e. 
_V
209208a1i 11 . . . . 5  |-  ( s  e.  ( 0 [,] 1 )  ->  (
s A 1 )  e.  _V )
210207, 209mprg 2750 . . . 4  |-  ( ( s  e.  ( 0 [,] 1 )  |->  ( s A 1 ) )  =  ( s  e.  ( 0 [,] 1 )  |->  ( N `
 s ) )  <->  A. s  e.  (
0 [,] 1 ) ( s A 1 )  =  ( N `
 s ) )
211206, 210sylib 200 . . 3  |-  ( ph  ->  A. s  e.  ( 0 [,] 1 ) ( s A 1 )  =  ( N `
 s ) )
212211r19.21bi 2756 . 2  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
s A 1 )  =  ( N `  s ) )
213177r19.21bi 2756 . 2  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
0 A s )  =  ( M ` 
0 ) )
21437, 184, 61, 17cnmpt12f 20674 . . . . . 6  |-  ( ph  ->  ( s  e.  ( 0 [,] 1 ) 
|->  ( 1 A s ) )  e.  ( II  Cn  C ) )
215 ffvelrn 6018 . . . . . . . 8  |-  ( ( M : ( 0 [,] 1 ) --> B  /\  1  e.  ( 0 [,] 1 ) )  ->  ( M `  1 )  e.  B )
21683, 92, 215sylancl 667 . . . . . . 7  |-  ( ph  ->  ( M `  1
)  e.  B )
217 cnconst2 20292 . . . . . . 7  |-  ( ( II  e.  (TopOn `  ( 0 [,] 1
) )  /\  C  e.  (TopOn `  B )  /\  ( M `  1
)  e.  B )  ->  ( ( 0 [,] 1 )  X. 
{ ( M ` 
1 ) } )  e.  ( II  Cn  C ) )
21837, 124, 216, 217syl3anc 1267 . . . . . 6  |-  ( ph  ->  ( ( 0 [,] 1 )  X.  {
( M `  1
) } )  e.  ( II  Cn  C
) )
219 opelxpi 4865 . . . . . . . . . . . . . 14  |-  ( ( 1  e.  ( 0 [,] 1 )  /\  s  e.  ( 0 [,] 1 ) )  ->  <. 1 ,  s
>.  e.  ( ( 0 [,] 1 )  X.  ( 0 [,] 1
) ) )
22092, 219mpan 675 . . . . . . . . . . . . 13  |-  ( s  e.  ( 0 [,] 1 )  ->  <. 1 ,  s >.  e.  ( ( 0 [,] 1
)  X.  ( 0 [,] 1 ) ) )
221 fvco3 5940 . . . . . . . . . . . . 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 >. )
) )
22222, 220, 221syl2an 480 . . . . . . . . . . . 12  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
( F  o.  A
) `  <. 1 ,  s >. )  =  ( F `  ( A `
 <. 1 ,  s
>. ) ) )
22329fveq1d 5865 . . . . . . . . . . . 12  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
( F  o.  A
) `  <. 1 ,  s >. )  =  ( K `  <. 1 ,  s >. )
)
224222, 223eqtr3d 2486 . . . . . . . . . . 11  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  ( F `  ( A `  <. 1 ,  s
>. ) )  =  ( K `  <. 1 ,  s >. )
)
225 df-ov 6291 . . . . . . . . . . . 12  |-  ( 1 A s )  =  ( A `  <. 1 ,  s >. )
226225fveq2i 5866 . . . . . . . . . . 11  |-  ( F `
 ( 1 A s ) )  =  ( F `  ( A `  <. 1 ,  s >. ) )
227 df-ov 6291 . . . . . . . . . . 11  |-  ( 1 K s )  =  ( K `  <. 1 ,  s >. )
228224, 226, 2273eqtr4g 2509 . . . . . . . . . 10  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  ( F `  ( 1 A s ) )  =  ( 1 K s ) )
229129simprd 465 . . . . . . . . . 10  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
1 K s )  =  ( G ` 
1 ) )
2307simp2d 1020 . . . . . . . . . . . . 13  |-  ( ph  ->  ( F  o.  M
)  =  G )
231230adantr 467 . . . . . . . . . . . 12  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  ( F  o.  M )  =  G )
232231fveq1d 5865 . . . . . . . . . . 11  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
( F  o.  M
) `  1 )  =  ( G ` 
1 ) )
23383adantr 467 . . . . . . . . . . . 12  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  M : ( 0 [,] 1 ) --> B )
234 fvco3 5940 . . . . . . . . . . . 12  |-  ( ( M : ( 0 [,] 1 ) --> B  /\  1  e.  ( 0 [,] 1 ) )  ->  ( ( F  o.  M ) `  1 )  =  ( F `  ( M `  1 )
) )
235233, 92, 234sylancl 667 . . . . . . . . . . 11  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
( F  o.  M
) `  1 )  =  ( F `  ( M `  1 ) ) )
236232, 235eqtr3d 2486 . . . . . . . . . 10  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  ( G `  1 )  =  ( F `  ( M `  1 ) ) )
237228, 229, 2363eqtrd 2488 . . . . . . . . 9  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  ( F `  ( 1 A s ) )  =  ( F `  ( M `  1 ) ) )
238237mpteq2dva 4488 . . . . . . . 8  |-  ( ph  ->  ( s  e.  ( 0 [,] 1 ) 
|->  ( F `  (
1 A s ) ) )  =  ( s  e.  ( 0 [,] 1 )  |->  ( F `  ( M `
 1 ) ) ) )
239 fconstmpt 4877 . . . . . . . 8  |-  ( ( 0 [,] 1 )  X.  { ( F `
 ( M ` 
1 ) ) } )  =  ( s  e.  ( 0 [,] 1 )  |->  ( F `
 ( M ` 
1 ) ) )
240238, 239syl6eqr 2502 . . . . . . 7  |-  ( ph  ->  ( s  e.  ( 0 [,] 1 ) 
|->  ( F `  (
1 A s ) ) )  =  ( ( 0 [,] 1
)  X.  { ( F `  ( M `
 1 ) ) } ) )
241 fovrn 6436 . . . . . . . . . 10  |-  ( ( A : ( ( 0 [,] 1 )  X.  ( 0 [,] 1 ) ) --> B  /\  1  e.  ( 0 [,] 1 )  /\  s  e.  ( 0 [,] 1 ) )  ->  ( 1 A s )  e.  B )
24292, 241mp3an2 1351 . . . . . . . . 9  |-  ( ( A : ( ( 0 [,] 1 )  X.  ( 0 [,] 1 ) ) --> B  /\  s  e.  ( 0 [,] 1 ) )  ->  ( 1 A s )  e.  B )
24322, 242sylan 474 . . . . . . . 8  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
1 A s )  e.  B )
244 eqidd 2451 . . . . . . . 8  |-  ( ph  ->  ( s  e.  ( 0 [,] 1 ) 
|->  ( 1 A s ) )  =  ( s  e.  ( 0 [,] 1 )  |->  ( 1 A s ) ) )
245 fveq2 5863 . . . . . . . 8  |-  ( x  =  ( 1 A s )  ->  ( F `  x )  =  ( F `  ( 1 A s ) ) )
246243, 244, 53, 245fmptco 6054 . . . . . . 7  |-  ( ph  ->  ( F  o.  (
s  e.  ( 0 [,] 1 )  |->  ( 1 A s ) ) )  =  ( s  e.  ( 0 [,] 1 )  |->  ( F `  ( 1 A s ) ) ) )
247 fcoconst 6058 . . . . . . . 8  |-  ( ( F  Fn  B  /\  ( M `  1 )  e.  B )  -> 
( F  o.  (
( 0 [,] 1
)  X.  { ( M `  1 ) } ) )  =  ( ( 0 [,] 1 )  X.  {
( F `  ( M `  1 )
) } ) )
248157, 216, 247syl2anc 666 . . . . . . 7  |-  ( ph  ->  ( F  o.  (
( 0 [,] 1
)  X.  { ( M `  1 ) } ) )  =  ( ( 0 [,] 1 )  X.  {
( F `  ( M `  1 )
) } ) )
249240, 246, 2483eqtr4d 2494 . . . . . 6  |-  ( ph  ->  ( F  o.  (
s  e.  ( 0 [,] 1 )  |->  ( 1 A s ) ) )  =  ( F  o.  ( ( 0 [,] 1 )  X.  { ( M `
 1 ) } ) ) )
250 oveq1 6295 . . . . . . . . . 10  |-  ( s  =  1  ->  (
s A 0 )  =  ( 1 A 0 ) )
251 fveq2 5863 . . . . . . . . . 10  |-  ( s  =  1  ->  ( M `  s )  =  ( M ` 
1 ) )
252250, 251eqeq12d 2465 . . . . . . . . 9  |-  ( s  =  1  ->  (
( s A 0 )  =  ( M `
 s )  <->  ( 1 A 0 )  =  ( M `  1
) ) )
253252rspcv 3145 . . . . . . . 8  |-  ( 1  e.  ( 0 [,] 1 )  ->  ( A. s  e.  (
0 [,] 1 ) ( s A 0 )  =  ( M `
 s )  -> 
( 1 A 0 )  =  ( M `
 1 ) ) )
25492, 90, 253mpsyl 65 . . . . . . 7  |-  ( ph  ->  ( 1 A 0 )  =  ( M `
 1 ) )
255 oveq2 6296 . . . . . . . . 9  |-  ( s  =  0  ->  (
1 A s )  =  ( 1 A 0 ) )
256 eqid 2450 . . . . . . . . 9  |-  ( s  e.  ( 0 [,] 1 )  |->  ( 1 A s ) )  =  ( s  e.  ( 0 [,] 1
)  |->  ( 1 A s ) )
257 ovex 6316 . . . . . . . . 9  |-  ( 1 A 0 )  e. 
_V
258255, 256, 257fvmpt 5946 . . . . . . . 8  |-  ( 0  e.  ( 0 [,] 1 )  ->  (
( s  e.  ( 0 [,] 1 ) 
|->  ( 1 A s ) ) `  0
)  =  ( 1 A 0 ) )
25923, 258ax-mp 5 . . . . . . 7  |-  ( ( s  e.  ( 0 [,] 1 )  |->  ( 1 A s ) ) `  0 )  =  ( 1 A 0 )
260 fvex 5873 . . . . . . . . 9  |-  ( M `
 1 )  e. 
_V
261260fvconst2 6118 . . . . . . . 8  |-  ( 0  e.  ( 0 [,] 1 )  ->  (
( ( 0 [,] 1 )  X.  {
( M `  1
) } ) ` 
0 )  =  ( M `  1 ) )
26223, 261ax-mp 5 . . . . . . 7  |-  ( ( ( 0 [,] 1
)  X.  { ( M `  1 ) } ) `  0
)  =  ( M `
 1 )
263254, 259, 2623eqtr4g 2509 . . . . . 6  |-  ( ph  ->  ( ( s  e.  ( 0 [,] 1
)  |->  ( 1 A s ) ) ` 
0 )  =  ( ( ( 0 [,] 1 )  X.  {
( M `  1
) } ) ` 
0 ) )
2641, 19, 3, 117, 119, 62, 214, 218, 249, 263cvmliftmoi 29999 . . . . 5  |-  ( ph  ->  ( s  e.  ( 0 [,] 1 ) 
|->  ( 1 A s ) )  =  ( ( 0 [,] 1
)  X.  { ( M `  1 ) } ) )
265 fconstmpt 4877 . . . . 5  |-  ( ( 0 [,] 1 )  X.  { ( M `
 1 ) } )  =  ( s  e.  ( 0 [,] 1 )  |->  ( M `
 1 ) )
266264, 265syl6eq 2500 . . . 4  |-  ( ph  ->  ( s  e.  ( 0 [,] 1 ) 
|->  ( 1 A s ) )  =  ( s  e.  ( 0 [,] 1 )  |->  ( M `  1 ) ) )
267 mpteqb 5962 . . . . 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 ) ) )
268 ovex 6316 . . . . . 6  |-  ( 1 A s )  e. 
_V
269268a1i 11 . . . . 5  |-  ( s  e.  ( 0 [,] 1 )  ->  (
1 A s )  e.  _V )
270267, 269mprg 2750 . . . 4  |-  ( ( s  e.  ( 0 [,] 1 )  |->  ( 1 A s ) )  =  ( s  e.  ( 0 [,] 1 )  |->  ( M `
 1 ) )  <->  A. s  e.  (
0 [,] 1 ) ( 1 A s )  =  ( M `
 1 ) )
271266, 270sylib 200 . . 3  |-  ( ph  ->  A. s  e.  ( 0 [,] 1 ) ( 1 A s )  =  ( M `
 1 ) )
272271r19.21bi 2756 . 2  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
1 A s )  =  ( M ` 
1 ) )
2738, 16, 17, 91, 212, 213, 272isphtpy2d 22011 1  |-  ( ph  ->  A  e.  ( M ( PHtpy `  C ) N ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 188    /\ wa 371    = wceq 1443    e. wcel 1886   A.wral 2736   E!wreu 2738   _Vcvv 3044   {csn 3967   <.cop 3973   U.cuni 4197    |-> cmpt 4460    X. cxp 4831    o. ccom 4837    Fn wfn 5576   -->wf 5577   ` cfv 5581   iota_crio 6249  (class class class)co 6288   0cc0 9536   1c1 9537   [,]cicc 11635   Topctop 19910  TopOnctopon 19911    Cn ccn 20233   Conccon 20419  𝑛Locally cnlly 20473    tX ctx 20568   IIcii 21900   Htpy chtpy 21991   PHtpycphtpy 21992   CovMap ccvm 29971
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-8 1888  ax-9 1895  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430  ax-rep 4514  ax-sep 4524  ax-nul 4533  ax-pow 4580  ax-pr 4638  ax-un 6580  ax-inf2 8143  ax-cnex 9592  ax-resscn 9593  ax-1cn 9594  ax-icn 9595  ax-addcl 9596  ax-addrcl 9597  ax-mulcl 9598  ax-mulrcl 9599  ax-mulcom 9600  ax-addass 9601  ax-mulass 9602  ax-distr 9603  ax-i2m1 9604  ax-1ne0 9605  ax-1rid 9606  ax-rnegex 9607  ax-rrecex 9608  ax-cnre 9609  ax-pre-lttri 9610  ax-pre-lttrn 9611  ax-pre-ltadd 9612  ax-pre-mulgt0 9613  ax-pre-sup 9614  ax-addf 9615  ax-mulf 9616
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 985  df-3an 986  df-tru 1446  df-fal 1449  df-ex 1663  df-nf 1667  df-sb 1797  df-eu 2302  df-mo 2303  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-ne 2623  df-nel 2624  df-ral 2741  df-rex 2742  df-reu 2743  df-rmo 2744  df-rab 2745  df-v 3046  df-sbc 3267  df-csb 3363  df-dif 3406  df-un 3408  df-in 3410  df-ss 3417  df-pss 3419  df-nul 3731  df-if 3881  df-pw 3952  df-sn 3968  df-pr 3970  df-tp 3972  df-op 3974  df-uni 4198  df-int 4234  df-iun 4279  df-iin 4280  df-br 4402  df-opab 4461  df-mpt 4462  df-tr 4497  df-eprel 4744  df-id 4748  df-po 4754  df-so 4755  df-fr 4792  df-se 4793  df-we 4794  df-xp 4839  df-rel 4840  df-cnv 4841  df-co 4842  df-dm 4843  df-rn 4844  df-res 4845  df-ima 4846  df-pred 5379  df-ord 5425  df-on 5426  df-lim 5427  df-suc 5428  df-iota 5545  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-isom 5590  df-riota 6250  df-ov 6291  df-oprab 6292  df-mpt2 6293  df-of 6528  df-om 6690  df-1st 6790  df-2nd 6791  df-supp 6912  df-wrecs 7025  df-recs 7087  df-rdg 7125  df-1o 7179  df-2o 7180  df-oadd 7183  df-er 7360  df-ec 7362  df-map 7471  df-ixp 7520  df-en 7567  df-dom 7568  df-sdom 7569  df-fin 7570  df-fsupp 7881  df-fi 7922  df-sup 7953  df-inf 7954  df-oi 8022  df-card 8370  df-cda 8595  df-pnf 9674  df-mnf 9675  df-xr 9676  df-ltxr 9677  df-le 9678  df-sub 9859  df-neg 9860  df-div 10267  df-nn 10607  df-2 10665  df-3 10666  df-4 10667  df-5 10668  df-6 10669  df-7 10670  df-8 10671  df-9 10672  df-10 10673  df-n0 10867  df-z 10935  df-dec 11049  df-uz 11157  df-q 11262  df-rp 11300  df-xneg 11406  df-xadd 11407  df-xmul 11408  df-ioo 11636  df-ico 11638  df-icc 11639  df-fz 11782  df-fzo 11913  df-fl 12025  df-seq 12211  df-exp 12270  df-hash 12513  df-cj 13155  df-re 13156  df-im 13157  df-sqrt 13291  df-abs 13292  df-clim 13545  df-sum 13746  df-struct 15116  df-ndx 15117  df-slot 15118  df-base 15119  df-sets 15120  df-ress 15121  df-plusg 15196  df-mulr 15197  df-starv 15198  df-sca 15199  df-vsca 15200  df-ip 15201  df-tset 15202  df-ple 15203  df-ds 15205  df-unif 15206  df-hom 15207  df-cco 15208  df-rest 15314  df-topn 15315  df-0g 15333  df-gsum 15334  df-topgen 15335  df-pt 15336  df-prds 15339  df-xrs 15393  df-qtop 15399  df-imas 15400  df-xps 15403  df-mre 15485  df-mrc 15486  df-acs 15488  df-mgm 16481  df-sgrp 16520  df-mnd 16530  df-submnd 16576  df-mulg 16669  df-cntz 16964  df-cmn 17425  df-psmet 18955  df-xmet 18956  df-met 18957  df-bl 18958  df-mopn 18959  df-cnfld 18964  df-top 19914  df-bases 19915  df-topon 19916  df-topsp 19917  df-cld 20027  df-ntr 20028  df-cls 20029  df-nei 20107  df-cn 20236  df-cnp 20237  df-cmp 20395  df-con 20420  df-lly 20474  df-nlly 20475  df-tx 20570  df-hmeo 20763  df-xms 21328  df-ms 21329  df-tms 21330  df-ii 21902  df-htpy 21994  df-phtpy 21995  df-phtpc 22016  df-pcon 29937  df-scon 29938  df-cvm 29972
This theorem is referenced by:  cvmliftpht  30034
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