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Theorem cvmliftphtlem 24987
Description: Lemma for cvmliftpht 24988. (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 24971 . . 3  |-  ( ph  ->  ( M  e.  ( II  Cn  C )  /\  ( F  o.  M )  =  G  /\  ( M ` 
0 )  =  P ) )
87simp1d 969 . 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 18993 . . . . . 6  |-  ( ph  ->  ( ( G ` 
0 )  =  ( H `  0 )  /\  ( G ` 
1 )  =  ( H `  1 ) ) )
1312simpld 446 . . . . 5  |-  ( ph  ->  ( G `  0
)  =  ( H `
 0 ) )
146, 13eqtrd 2462 . . . 4  |-  ( ph  ->  ( F `  P
)  =  ( H `
 0 ) )
151, 9, 3, 10, 5, 14cvmliftiota 24971 . . 3  |-  ( ph  ->  ( N  e.  ( II  Cn  C )  /\  ( F  o.  N )  =  H  /\  ( N ` 
0 )  =  P ) )
1615simp1d 969 . 2  |-  ( ph  ->  N  e.  ( II 
Cn  C ) )
17 cvmliftphtlem.a . 2  |-  ( ph  ->  A  e.  ( ( II  tX  II )  Cn  C ) )
18 iitop 18893 . . . . . . . . . . . . . . . 16  |-  II  e.  Top
19 iiuni 18894 . . . . . . . . . . . . . . . 16  |-  ( 0 [,] 1 )  = 
U. II
2018, 18, 19, 19txunii 17608 . . . . . . . . . . . . . . 15  |-  ( ( 0 [,] 1 )  X.  ( 0 [,] 1 ) )  = 
U. ( II  tX  II )
2120, 1cnf 17293 . . . . . . . . . . . . . 14  |-  ( A  e.  ( ( II 
tX  II )  Cn  C )  ->  A : ( ( 0 [,] 1 )  X.  ( 0 [,] 1
) ) --> B )
2217, 21syl 16 . . . . . . . . . . . . 13  |-  ( ph  ->  A : ( ( 0 [,] 1 )  X.  ( 0 [,] 1 ) ) --> B )
23 0elunit 10999 . . . . . . . . . . . . . 14  |-  0  e.  ( 0 [,] 1
)
24 opelxpi 4896 . . . . . . . . . . . . . 14  |-  ( ( s  e.  ( 0 [,] 1 )  /\  0  e.  ( 0 [,] 1 ) )  ->  <. s ,  0
>.  e.  ( ( 0 [,] 1 )  X.  ( 0 [,] 1
) ) )
2523, 24mpan2 653 . . . . . . . . . . . . 13  |-  ( s  e.  ( 0 [,] 1 )  ->  <. s ,  0 >.  e.  ( ( 0 [,] 1
)  X.  ( 0 [,] 1 ) ) )
26 fvco3 5786 . . . . . . . . . . . . 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 464 . . . . . . . . . . . 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 452 . . . . . . . . . . . . 13  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  ( F  o.  A )  =  K )
3029fveq1d 5716 . . . . . . . . . . . 12  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
( F  o.  A
) `  <. s ,  0 >. )  =  ( K `  <. s ,  0 >. )
)
3127, 30eqtr3d 2464 . . . . . . . . . . 11  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  ( F `  ( A `  <. s ,  0
>. ) )  =  ( K `  <. s ,  0 >. )
)
32 df-ov 6070 . . . . . . . . . . . 12  |-  ( s A 0 )  =  ( A `  <. s ,  0 >. )
3332fveq2i 5717 . . . . . . . . . . 11  |-  ( F `
 ( s A 0 ) )  =  ( F `  ( A `  <. s ,  0 >. ) )
34 df-ov 6070 . . . . . . . . . . 11  |-  ( s K 0 )  =  ( K `  <. s ,  0 >. )
3531, 33, 343eqtr4g 2487 . . . . . . . . . 10  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  ( F `  ( s A 0 ) )  =  ( s K 0 ) )
36 iitopon 18892 . . . . . . . . . . . . 13  |-  II  e.  (TopOn `  ( 0 [,] 1 ) )
3736a1i 11 . . . . . . . . . . . 12  |-  ( ph  ->  II  e.  (TopOn `  ( 0 [,] 1
) ) )
384, 10phtpyhtpy 18990 . . . . . . . . . . . . 13  |-  ( ph  ->  ( G ( PHtpy `  J ) H ) 
C_  ( G ( II Htpy  J ) H ) )
3938, 11sseldd 3336 . . . . . . . . . . . 12  |-  ( ph  ->  K  e.  ( G ( II Htpy  J ) H ) )
4037, 4, 10, 39htpyi 18982 . . . . . . . . . . 11  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
( s K 0 )  =  ( G `
 s )  /\  ( s K 1 )  =  ( H `
 s ) ) )
4140simpld 446 . . . . . . . . . 10  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
s K 0 )  =  ( G `  s ) )
4235, 41eqtrd 2462 . . . . . . . . 9  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  ( F `  ( s A 0 ) )  =  ( G `  s ) )
4342mpteq2dva 4282 . . . . . . . 8  |-  ( ph  ->  ( s  e.  ( 0 [,] 1 ) 
|->  ( F `  (
s A 0 ) ) )  =  ( s  e.  ( 0 [,] 1 )  |->  ( G `  s ) ) )
44 fovrn 6202 . . . . . . . . . . 11  |-  ( ( A : ( ( 0 [,] 1 )  X.  ( 0 [,] 1 ) ) --> B  /\  s  e.  ( 0 [,] 1 )  /\  0  e.  ( 0 [,] 1 ) )  ->  ( s A 0 )  e.  B )
4523, 44mp3an3 1268 . . . . . . . . . 10  |-  ( ( A : ( ( 0 [,] 1 )  X.  ( 0 [,] 1 ) ) --> B  /\  s  e.  ( 0 [,] 1 ) )  ->  ( s A 0 )  e.  B )
4622, 45sylan 458 . . . . . . . . 9  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
s A 0 )  e.  B )
47 eqidd 2431 . . . . . . . . 9  |-  ( ph  ->  ( s  e.  ( 0 [,] 1 ) 
|->  ( s A 0 ) )  =  ( s  e.  ( 0 [,] 1 )  |->  ( s A 0 ) ) )
48 cvmcn 24932 . . . . . . . . . . . 12  |-  ( F  e.  ( C CovMap  J
)  ->  F  e.  ( C  Cn  J
) )
493, 48syl 16 . . . . . . . . . . 11  |-  ( ph  ->  F  e.  ( C  Cn  J ) )
50 eqid 2430 . . . . . . . . . . . 12  |-  U. J  =  U. J
511, 50cnf 17293 . . . . . . . . . . 11  |-  ( F  e.  ( C  Cn  J )  ->  F : B --> U. J )
5249, 51syl 16 . . . . . . . . . 10  |-  ( ph  ->  F : B --> U. J
)
5352feqmptd 5765 . . . . . . . . 9  |-  ( ph  ->  F  =  ( x  e.  B  |->  ( F `
 x ) ) )
54 fveq2 5714 . . . . . . . . 9  |-  ( x  =  ( s A 0 )  ->  ( F `  x )  =  ( F `  ( s A 0 ) ) )
5546, 47, 53, 54fmptco 5887 . . . . . . . 8  |-  ( ph  ->  ( F  o.  (
s  e.  ( 0 [,] 1 )  |->  ( s A 0 ) ) )  =  ( s  e.  ( 0 [,] 1 )  |->  ( F `  ( s A 0 ) ) ) )
5619, 50cnf 17293 . . . . . . . . . 10  |-  ( G  e.  ( II  Cn  J )  ->  G : ( 0 [,] 1 ) --> U. J
)
574, 56syl 16 . . . . . . . . 9  |-  ( ph  ->  G : ( 0 [,] 1 ) --> U. J )
5857feqmptd 5765 . . . . . . . 8  |-  ( ph  ->  G  =  ( s  e.  ( 0 [,] 1 )  |->  ( G `
 s ) ) )
5943, 55, 583eqtr4d 2472 . . . . . . 7  |-  ( ph  ->  ( F  o.  (
s  e.  ( 0 [,] 1 )  |->  ( s A 0 ) ) )  =  G )
60 cvmliftphtlem.0 . . . . . . 7  |-  ( ph  ->  ( 0 A 0 )  =  P )
6137cnmptid 17676 . . . . . . . . 9  |-  ( ph  ->  ( s  e.  ( 0 [,] 1 ) 
|->  s )  e.  ( II  Cn  II ) )
6223a1i 11 . . . . . . . . . 10  |-  ( ph  ->  0  e.  ( 0 [,] 1 ) )
6337, 37, 62cnmptc 17677 . . . . . . . . 9  |-  ( ph  ->  ( s  e.  ( 0 [,] 1 ) 
|->  0 )  e.  ( II  Cn  II ) )
6437, 61, 63, 17cnmpt12f 17681 . . . . . . . 8  |-  ( ph  ->  ( s  e.  ( 0 [,] 1 ) 
|->  ( s A 0 ) )  e.  ( II  Cn  C ) )
651cvmlift 24969 . . . . . . . . 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 1185 . . . . . . . 8  |-  ( ph  ->  E! f  e.  ( II  Cn  C ) ( ( F  o.  f )  =  G  /\  ( f ` 
0 )  =  P ) )
67 coeq2 5017 . . . . . . . . . . 11  |-  ( f  =  ( s  e.  ( 0 [,] 1
)  |->  ( s A 0 ) )  -> 
( F  o.  f
)  =  ( F  o.  ( s  e.  ( 0 [,] 1
)  |->  ( s A 0 ) ) ) )
6867eqeq1d 2438 . . . . . . . . . 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 5713 . . . . . . . . . . . 12  |-  ( f  =  ( s  e.  ( 0 [,] 1
)  |->  ( s A 0 ) )  -> 
( f `  0
)  =  ( ( s  e.  ( 0 [,] 1 )  |->  ( s A 0 ) ) `  0 ) )
70 oveq1 6074 . . . . . . . . . . . . . 14  |-  ( s  =  0  ->  (
s A 0 )  =  ( 0 A 0 ) )
71 eqid 2430 . . . . . . . . . . . . . 14  |-  ( s  e.  ( 0 [,] 1 )  |->  ( s A 0 ) )  =  ( s  e.  ( 0 [,] 1
)  |->  ( s A 0 ) )
72 ovex 6092 . . . . . . . . . . . . . 14  |-  ( 0 A 0 )  e. 
_V
7370, 71, 72fvmpt 5792 . . . . . . . . . . . . 13  |-  ( 0  e.  ( 0 [,] 1 )  ->  (
( s  e.  ( 0 [,] 1 ) 
|->  ( s A 0 ) ) `  0
)  =  ( 0 A 0 ) )
7423, 73ax-mp 8 . . . . . . . . . . . 12  |-  ( ( s  e.  ( 0 [,] 1 )  |->  ( s A 0 ) ) `  0 )  =  ( 0 A 0 )
7569, 74syl6eq 2478 . . . . . . . . . . 11  |-  ( f  =  ( s  e.  ( 0 [,] 1
)  |->  ( s A 0 ) )  -> 
( f `  0
)  =  ( 0 A 0 ) )
7675eqeq1d 2438 . . . . . . . . . 10  |-  ( f  =  ( s  e.  ( 0 [,] 1
)  |->  ( s A 0 ) )  -> 
( ( f ` 
0 )  =  P  <-> 
( 0 A 0 )  =  P ) )
7768, 76anbi12d 692 . . . . . . . . 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 6558 . . . . . . . 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 643 . . . . . . 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 888 . . . . . 6  |-  ( ph  ->  ( iota_ f  e.  ( II  Cn  C ) ( ( F  o.  f )  =  G  /\  ( f ` 
0 )  =  P ) )  =  ( s  e.  ( 0 [,] 1 )  |->  ( s A 0 ) ) )
812, 80syl5eq 2474 . . . . 5  |-  ( ph  ->  M  =  ( s  e.  ( 0 [,] 1 )  |->  ( s A 0 ) ) )
8219, 1cnf 17293 . . . . . . 7  |-  ( M  e.  ( II  Cn  C )  ->  M : ( 0 [,] 1 ) --> B )
838, 82syl 16 . . . . . 6  |-  ( ph  ->  M : ( 0 [,] 1 ) --> B )
8483feqmptd 5765 . . . . 5  |-  ( ph  ->  M  =  ( s  e.  ( 0 [,] 1 )  |->  ( M `
 s ) ) )
8581, 84eqtr3d 2464 . . . 4  |-  ( ph  ->  ( s  e.  ( 0 [,] 1 ) 
|->  ( s A 0 ) )  =  ( s  e.  ( 0 [,] 1 )  |->  ( M `  s ) ) )
86 mpteqb 5805 . . . . 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 6092 . . . . . 6  |-  ( s A 0 )  e. 
_V
8887a1i 11 . . . . 5  |-  ( s  e.  ( 0 [,] 1 )  ->  (
s A 0 )  e.  _V )
8986, 88mprg 2762 . . . 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 189 . . 3  |-  ( ph  ->  A. s  e.  ( 0 [,] 1 ) ( s A 0 )  =  ( M `
 s ) )
9190r19.21bi 2791 . 2  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
s A 0 )  =  ( M `  s ) )
92 1elunit 11000 . . . . . . . . . . . . . 14  |-  1  e.  ( 0 [,] 1
)
93 opelxpi 4896 . . . . . . . . . . . . . 14  |-  ( ( s  e.  ( 0 [,] 1 )  /\  1  e.  ( 0 [,] 1 ) )  ->  <. s ,  1
>.  e.  ( ( 0 [,] 1 )  X.  ( 0 [,] 1
) ) )
9492, 93mpan2 653 . . . . . . . . . . . . 13  |-  ( s  e.  ( 0 [,] 1 )  ->  <. s ,  1 >.  e.  ( ( 0 [,] 1
)  X.  ( 0 [,] 1 ) ) )
95 fvco3 5786 . . . . . . . . . . . . 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 464 . . . . . . . . . . . 12  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
( F  o.  A
) `  <. s ,  1 >. )  =  ( F `  ( A `
 <. s ,  1
>. ) ) )
9729fveq1d 5716 . . . . . . . . . . . 12  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
( F  o.  A
) `  <. s ,  1 >. )  =  ( K `  <. s ,  1 >. )
)
9896, 97eqtr3d 2464 . . . . . . . . . . 11  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  ( F `  ( A `  <. s ,  1
>. ) )  =  ( K `  <. s ,  1 >. )
)
99 df-ov 6070 . . . . . . . . . . . 12  |-  ( s A 1 )  =  ( A `  <. s ,  1 >. )
10099fveq2i 5717 . . . . . . . . . . 11  |-  ( F `
 ( s A 1 ) )  =  ( F `  ( A `  <. s ,  1 >. ) )
101 df-ov 6070 . . . . . . . . . . 11  |-  ( s K 1 )  =  ( K `  <. s ,  1 >. )
10298, 100, 1013eqtr4g 2487 . . . . . . . . . 10  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  ( F `  ( s A 1 ) )  =  ( s K 1 ) )
10340simprd 450 . . . . . . . . . 10  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
s K 1 )  =  ( H `  s ) )
104102, 103eqtrd 2462 . . . . . . . . 9  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  ( F `  ( s A 1 ) )  =  ( H `  s ) )
105104mpteq2dva 4282 . . . . . . . 8  |-  ( ph  ->  ( s  e.  ( 0 [,] 1 ) 
|->  ( F `  (
s A 1 ) ) )  =  ( s  e.  ( 0 [,] 1 )  |->  ( H `  s ) ) )
106 fovrn 6202 . . . . . . . . . . 11  |-  ( ( A : ( ( 0 [,] 1 )  X.  ( 0 [,] 1 ) ) --> B  /\  s  e.  ( 0 [,] 1 )  /\  1  e.  ( 0 [,] 1 ) )  ->  ( s A 1 )  e.  B )
10792, 106mp3an3 1268 . . . . . . . . . 10  |-  ( ( A : ( ( 0 [,] 1 )  X.  ( 0 [,] 1 ) ) --> B  /\  s  e.  ( 0 [,] 1 ) )  ->  ( s A 1 )  e.  B )
10822, 107sylan 458 . . . . . . . . 9  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
s A 1 )  e.  B )
109 eqidd 2431 . . . . . . . . 9  |-  ( ph  ->  ( s  e.  ( 0 [,] 1 ) 
|->  ( s A 1 ) )  =  ( s  e.  ( 0 [,] 1 )  |->  ( s A 1 ) ) )
110 fveq2 5714 . . . . . . . . 9  |-  ( x  =  ( s A 1 )  ->  ( F `  x )  =  ( F `  ( s A 1 ) ) )
111108, 109, 53, 110fmptco 5887 . . . . . . . 8  |-  ( ph  ->  ( F  o.  (
s  e.  ( 0 [,] 1 )  |->  ( s A 1 ) ) )  =  ( s  e.  ( 0 [,] 1 )  |->  ( F `  ( s A 1 ) ) ) )
11219, 50cnf 17293 . . . . . . . . . 10  |-  ( H  e.  ( II  Cn  J )  ->  H : ( 0 [,] 1 ) --> U. J
)
11310, 112syl 16 . . . . . . . . 9  |-  ( ph  ->  H : ( 0 [,] 1 ) --> U. J )
114113feqmptd 5765 . . . . . . . 8  |-  ( ph  ->  H  =  ( s  e.  ( 0 [,] 1 )  |->  ( H `
 s ) ) )
115105, 111, 1143eqtr4d 2472 . . . . . . 7  |-  ( ph  ->  ( F  o.  (
s  e.  ( 0 [,] 1 )  |->  ( s A 1 ) ) )  =  H )
116 iicon 18900 . . . . . . . . . . . . 13  |-  II  e.  Con
117116a1i 11 . . . . . . . . . . . 12  |-  ( ph  ->  II  e.  Con )
118 iinllycon 24924 . . . . . . . . . . . . 13  |-  II  e. 𝑛Locally  Con
119118a1i 11 . . . . . . . . . . . 12  |-  ( ph  ->  II  e. 𝑛Locally  Con )
12037, 63, 61, 17cnmpt12f 17681 . . . . . . . . . . . 12  |-  ( ph  ->  ( s  e.  ( 0 [,] 1 ) 
|->  ( 0 A s ) )  e.  ( II  Cn  C ) )
121 cvmtop1 24930 . . . . . . . . . . . . . . 15  |-  ( F  e.  ( C CovMap  J
)  ->  C  e.  Top )
1223, 121syl 16 . . . . . . . . . . . . . 14  |-  ( ph  ->  C  e.  Top )
1231toptopon 16981 . . . . . . . . . . . . . 14  |-  ( C  e.  Top  <->  C  e.  (TopOn `  B ) )
124122, 123sylib 189 . . . . . . . . . . . . 13  |-  ( ph  ->  C  e.  (TopOn `  B ) )
125 ffvelrn 5854 . . . . . . . . . . . . . 14  |-  ( ( M : ( 0 [,] 1 ) --> B  /\  0  e.  ( 0 [,] 1 ) )  ->  ( M `  0 )  e.  B )
12683, 23, 125sylancl 644 . . . . . . . . . . . . 13  |-  ( ph  ->  ( M `  0
)  e.  B )
127 cnconst2 17330 . . . . . . . . . . . . 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 1184 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( 0 [,] 1 )  X.  {
( M `  0
) } )  e.  ( II  Cn  C
) )
1294, 10, 11phtpyi 18992 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
( 0 K s )  =  ( G `
 0 )  /\  ( 1 K s )  =  ( G `
 1 ) ) )
130129simpld 446 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
0 K s )  =  ( G ` 
0 ) )
131 opelxpi 4896 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( 0  e.  ( 0 [,] 1 )  /\  s  e.  ( 0 [,] 1 ) )  ->  <. 0 ,  s
>.  e.  ( ( 0 [,] 1 )  X.  ( 0 [,] 1
) ) )
13223, 131mpan 652 . . . . . . . . . . . . . . . . . . 19  |-  ( s  e.  ( 0 [,] 1 )  ->  <. 0 ,  s >.  e.  ( ( 0 [,] 1
)  X.  ( 0 [,] 1 ) ) )
133 fvco3 5786 . . . . . . . . . . . . . . . . . . 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 464 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
( F  o.  A
) `  <. 0 ,  s >. )  =  ( F `  ( A `
 <. 0 ,  s
>. ) ) )
13529fveq1d 5716 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
( F  o.  A
) `  <. 0 ,  s >. )  =  ( K `  <. 0 ,  s >. )
)
136134, 135eqtr3d 2464 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  ( F `  ( A `  <. 0 ,  s
>. ) )  =  ( K `  <. 0 ,  s >. )
)
137 df-ov 6070 . . . . . . . . . . . . . . . . . 18  |-  ( 0 A s )  =  ( A `  <. 0 ,  s >. )
138137fveq2i 5717 . . . . . . . . . . . . . . . . 17  |-  ( F `
 ( 0 A s ) )  =  ( F `  ( A `  <. 0 ,  s >. ) )
139 df-ov 6070 . . . . . . . . . . . . . . . . 17  |-  ( 0 K s )  =  ( K `  <. 0 ,  s >. )
140136, 138, 1393eqtr4g 2487 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  ( F `  ( 0 A s ) )  =  ( 0 K s ) )
1417simp3d 971 . . . . . . . . . . . . . . . . . . 19  |-  ( ph  ->  ( M `  0
)  =  P )
142141adantr 452 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  ( M `  0 )  =  P )
143142fveq2d 5718 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  ( F `  ( M `  0 ) )  =  ( F `  P ) )
1446adantr 452 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  ( F `  P )  =  ( G ` 
0 ) )
145143, 144eqtrd 2462 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  ( F `  ( M `  0 ) )  =  ( G ` 
0 ) )
146130, 140, 1453eqtr4d 2472 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  ( F `  ( 0 A s ) )  =  ( F `  ( M `  0 ) ) )
147146mpteq2dva 4282 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( s  e.  ( 0 [,] 1 ) 
|->  ( F `  (
0 A s ) ) )  =  ( s  e.  ( 0 [,] 1 )  |->  ( F `  ( M `
 0 ) ) ) )
148 fconstmpt 4907 . . . . . . . . . . . . . 14  |-  ( ( 0 [,] 1 )  X.  { ( F `
 ( M ` 
0 ) ) } )  =  ( s  e.  ( 0 [,] 1 )  |->  ( F `
 ( M ` 
0 ) ) )
149147, 148syl6eqr 2480 . . . . . . . . . . . . 13  |-  ( ph  ->  ( s  e.  ( 0 [,] 1 ) 
|->  ( F `  (
0 A s ) ) )  =  ( ( 0 [,] 1
)  X.  { ( F `  ( M `
 0 ) ) } ) )
150 fovrn 6202 . . . . . . . . . . . . . . . 16  |-  ( ( A : ( ( 0 [,] 1 )  X.  ( 0 [,] 1 ) ) --> B  /\  0  e.  ( 0 [,] 1 )  /\  s  e.  ( 0 [,] 1 ) )  ->  ( 0 A s )  e.  B )
15123, 150mp3an2 1267 . . . . . . . . . . . . . . 15  |-  ( ( A : ( ( 0 [,] 1 )  X.  ( 0 [,] 1 ) ) --> B  /\  s  e.  ( 0 [,] 1 ) )  ->  ( 0 A s )  e.  B )
15222, 151sylan 458 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
0 A s )  e.  B )
153 eqidd 2431 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( s  e.  ( 0 [,] 1 ) 
|->  ( 0 A s ) )  =  ( s  e.  ( 0 [,] 1 )  |->  ( 0 A s ) ) )
154 fveq2 5714 . . . . . . . . . . . . . 14  |-  ( x  =  ( 0 A s )  ->  ( F `  x )  =  ( F `  ( 0 A s ) ) )
155152, 153, 53, 154fmptco 5887 . . . . . . . . . . . . 13  |-  ( ph  ->  ( F  o.  (
s  e.  ( 0 [,] 1 )  |->  ( 0 A s ) ) )  =  ( s  e.  ( 0 [,] 1 )  |->  ( F `  ( 0 A s ) ) ) )
156 ffn 5577 . . . . . . . . . . . . . . 15  |-  ( F : B --> U. J  ->  F  Fn  B )
15752, 156syl 16 . . . . . . . . . . . . . 14  |-  ( ph  ->  F  Fn  B )
158 fcoconst 5891 . . . . . . . . . . . . . 14  |-  ( ( F  Fn  B  /\  ( M `  0 )  e.  B )  -> 
( F  o.  (
( 0 [,] 1
)  X.  { ( M `  0 ) } ) )  =  ( ( 0 [,] 1 )  X.  {
( F `  ( M `  0 )
) } ) )
159157, 126, 158syl2anc 643 . . . . . . . . . . . . 13  |-  ( ph  ->  ( F  o.  (
( 0 [,] 1
)  X.  { ( M `  0 ) } ) )  =  ( ( 0 [,] 1 )  X.  {
( F `  ( M `  0 )
) } ) )
160149, 155, 1593eqtr4d 2472 . . . . . . . . . . . 12  |-  ( ph  ->  ( F  o.  (
s  e.  ( 0 [,] 1 )  |->  ( 0 A s ) ) )  =  ( F  o.  ( ( 0 [,] 1 )  X.  { ( M `
 0 ) } ) ) )
16160, 141eqtr4d 2465 . . . . . . . . . . . . 13  |-  ( ph  ->  ( 0 A 0 )  =  ( M `
 0 ) )
162 oveq2 6075 . . . . . . . . . . . . . . 15  |-  ( s  =  0  ->  (
0 A s )  =  ( 0 A 0 ) )
163 eqid 2430 . . . . . . . . . . . . . . 15  |-  ( s  e.  ( 0 [,] 1 )  |->  ( 0 A s ) )  =  ( s  e.  ( 0 [,] 1
)  |->  ( 0 A s ) )
164162, 163, 72fvmpt 5792 . . . . . . . . . . . . . 14  |-  ( 0  e.  ( 0 [,] 1 )  ->  (
( s  e.  ( 0 [,] 1 ) 
|->  ( 0 A s ) ) `  0
)  =  ( 0 A 0 ) )
16523, 164ax-mp 8 . . . . . . . . . . . . 13  |-  ( ( s  e.  ( 0 [,] 1 )  |->  ( 0 A s ) ) `  0 )  =  ( 0 A 0 )
166 fvex 5728 . . . . . . . . . . . . . . 15  |-  ( M `
 0 )  e. 
_V
167166fvconst2 5933 . . . . . . . . . . . . . 14  |-  ( 0  e.  ( 0 [,] 1 )  ->  (
( ( 0 [,] 1 )  X.  {
( M `  0
) } ) ` 
0 )  =  ( M `  0 ) )
16823, 167ax-mp 8 . . . . . . . . . . . . 13  |-  ( ( ( 0 [,] 1
)  X.  { ( M `  0 ) } ) `  0
)  =  ( M `
 0 )
169161, 165, 1683eqtr4g 2487 . . . . . . . . . . . 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 24953 . . . . . . . . . . 11  |-  ( ph  ->  ( s  e.  ( 0 [,] 1 ) 
|->  ( 0 A s ) )  =  ( ( 0 [,] 1
)  X.  { ( M `  0 ) } ) )
171 fconstmpt 4907 . . . . . . . . . . 11  |-  ( ( 0 [,] 1 )  X.  { ( M `
 0 ) } )  =  ( s  e.  ( 0 [,] 1 )  |->  ( M `
 0 ) )
172170, 171syl6eq 2478 . . . . . . . . . 10  |-  ( ph  ->  ( s  e.  ( 0 [,] 1 ) 
|->  ( 0 A s ) )  =  ( s  e.  ( 0 [,] 1 )  |->  ( M `  0 ) ) )
173 mpteqb 5805 . . . . . . . . . . 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 6092 . . . . . . . . . . . 12  |-  ( 0 A s )  e. 
_V
175174a1i 11 . . . . . . . . . . 11  |-  ( s  e.  ( 0 [,] 1 )  ->  (
0 A s )  e.  _V )
176173, 175mprg 2762 . . . . . . . . . 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 189 . . . . . . . . 9  |-  ( ph  ->  A. s  e.  ( 0 [,] 1 ) ( 0 A s )  =  ( M `
 0 ) )
178 oveq2 6075 . . . . . . . . . . 11  |-  ( s  =  1  ->  (
0 A s )  =  ( 0 A 1 ) )
179178eqeq1d 2438 . . . . . . . . . 10  |-  ( s  =  1  ->  (
( 0 A s )  =  ( M `
 0 )  <->  ( 0 A 1 )  =  ( M `  0
) ) )
180179rspcv 3035 . . . . . . . . 9  |-  ( 1  e.  ( 0 [,] 1 )  ->  ( A. s  e.  (
0 [,] 1 ) ( 0 A s )  =  ( M `
 0 )  -> 
( 0 A 1 )  =  ( M `
 0 ) ) )
18192, 177, 180mpsyl 61 . . . . . . . 8  |-  ( ph  ->  ( 0 A 1 )  =  ( M `
 0 ) )
182181, 141eqtrd 2462 . . . . . . 7  |-  ( ph  ->  ( 0 A 1 )  =  P )
18392a1i 11 . . . . . . . . . 10  |-  ( ph  ->  1  e.  ( 0 [,] 1 ) )
18437, 37, 183cnmptc 17677 . . . . . . . . 9  |-  ( ph  ->  ( s  e.  ( 0 [,] 1 ) 
|->  1 )  e.  ( II  Cn  II ) )
18537, 61, 184, 17cnmpt12f 17681 . . . . . . . 8  |-  ( ph  ->  ( s  e.  ( 0 [,] 1 ) 
|->  ( s A 1 ) )  e.  ( II  Cn  C ) )
1861cvmlift 24969 . . . . . . . . 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 1185 . . . . . . . 8  |-  ( ph  ->  E! f  e.  ( II  Cn  C ) ( ( F  o.  f )  =  H  /\  ( f ` 
0 )  =  P ) )
188 coeq2 5017 . . . . . . . . . . 11  |-  ( f  =  ( s  e.  ( 0 [,] 1
)  |->  ( s A 1 ) )  -> 
( F  o.  f
)  =  ( F  o.  ( s  e.  ( 0 [,] 1
)  |->  ( s A 1 ) ) ) )
189188eqeq1d 2438 . . . . . . . . . 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 5713 . . . . . . . . . . . 12  |-  ( f  =  ( s  e.  ( 0 [,] 1
)  |->  ( s A 1 ) )  -> 
( f `  0
)  =  ( ( s  e.  ( 0 [,] 1 )  |->  ( s A 1 ) ) `  0 ) )
191 oveq1 6074 . . . . . . . . . . . . . 14  |-  ( s  =  0  ->  (
s A 1 )  =  ( 0 A 1 ) )
192 eqid 2430 . . . . . . . . . . . . . 14  |-  ( s  e.  ( 0 [,] 1 )  |->  ( s A 1 ) )  =  ( s  e.  ( 0 [,] 1
)  |->  ( s A 1 ) )
193 ovex 6092 . . . . . . . . . . . . . 14  |-  ( 0 A 1 )  e. 
_V
194191, 192, 193fvmpt 5792 . . . . . . . . . . . . 13  |-  ( 0  e.  ( 0 [,] 1 )  ->  (
( s  e.  ( 0 [,] 1 ) 
|->  ( s A 1 ) ) `  0
)  =  ( 0 A 1 ) )
19523, 194ax-mp 8 . . . . . . . . . . . 12  |-  ( ( s  e.  ( 0 [,] 1 )  |->  ( s A 1 ) ) `  0 )  =  ( 0 A 1 )
196190, 195syl6eq 2478 . . . . . . . . . . 11  |-  ( f  =  ( s  e.  ( 0 [,] 1
)  |->  ( s A 1 ) )  -> 
( f `  0
)  =  ( 0 A 1 ) )
197196eqeq1d 2438 . . . . . . . . . 10  |-  ( f  =  ( s  e.  ( 0 [,] 1
)  |->  ( s A 1 ) )  -> 
( ( f ` 
0 )  =  P  <-> 
( 0 A 1 )  =  P ) )
198189, 197anbi12d 692 . . . . . . . . 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 6558 . . . . . . . 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 643 . . . . . . 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 888 . . . . . 6  |-  ( ph  ->  ( iota_ f  e.  ( II  Cn  C ) ( ( F  o.  f )  =  H  /\  ( f ` 
0 )  =  P ) )  =  ( s  e.  ( 0 [,] 1 )  |->  ( s A 1 ) ) )
2029, 201syl5eq 2474 . . . . 5  |-  ( ph  ->  N  =  ( s  e.  ( 0 [,] 1 )  |->  ( s A 1 ) ) )
20319, 1cnf 17293 . . . . . . 7  |-  ( N  e.  ( II  Cn  C )  ->  N : ( 0 [,] 1 ) --> B )
20416, 203syl 16 . . . . . 6  |-  ( ph  ->  N : ( 0 [,] 1 ) --> B )
205204feqmptd 5765 . . . . 5  |-  ( ph  ->  N  =  ( s  e.  ( 0 [,] 1 )  |->  ( N `
 s ) ) )
206202, 205eqtr3d 2464 . . . 4  |-  ( ph  ->  ( s  e.  ( 0 [,] 1 ) 
|->  ( s A 1 ) )  =  ( s  e.  ( 0 [,] 1 )  |->  ( N `  s ) ) )
207 mpteqb 5805 . . . . 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 6092 . . . . . 6  |-  ( s A 1 )  e. 
_V
209208a1i 11 . . . . 5  |-  ( s  e.  ( 0 [,] 1 )  ->  (
s A 1 )  e.  _V )
210207, 209mprg 2762 . . . 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 189 . . 3  |-  ( ph  ->  A. s  e.  ( 0 [,] 1 ) ( s A 1 )  =  ( N `
 s ) )
212211r19.21bi 2791 . 2  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
s A 1 )  =  ( N `  s ) )
213177r19.21bi 2791 . 2  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
0 A s )  =  ( M ` 
0 ) )
21437, 184, 61, 17cnmpt12f 17681 . . . . . 6  |-  ( ph  ->  ( s  e.  ( 0 [,] 1 ) 
|->  ( 1 A s ) )  e.  ( II  Cn  C ) )
215 ffvelrn 5854 . . . . . . . 8  |-  ( ( M : ( 0 [,] 1 ) --> B  /\  1  e.  ( 0 [,] 1 ) )  ->  ( M `  1 )  e.  B )
21683, 92, 215sylancl 644 . . . . . . 7  |-  ( ph  ->  ( M `  1
)  e.  B )
217 cnconst2 17330 . . . . . . 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 1184 . . . . . 6  |-  ( ph  ->  ( ( 0 [,] 1 )  X.  {
( M `  1
) } )  e.  ( II  Cn  C
) )
219 opelxpi 4896 . . . . . . . . . . . . . 14  |-  ( ( 1  e.  ( 0 [,] 1 )  /\  s  e.  ( 0 [,] 1 ) )  ->  <. 1 ,  s
>.  e.  ( ( 0 [,] 1 )  X.  ( 0 [,] 1
) ) )
22092, 219mpan 652 . . . . . . . . . . . . 13  |-  ( s  e.  ( 0 [,] 1 )  ->  <. 1 ,  s >.  e.  ( ( 0 [,] 1
)  X.  ( 0 [,] 1 ) ) )
221 fvco3 5786 . . . . . . . . . . . . 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 464 . . . . . . . . . . . 12  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
( F  o.  A
) `  <. 1 ,  s >. )  =  ( F `  ( A `
 <. 1 ,  s
>. ) ) )
22329fveq1d 5716 . . . . . . . . . . . 12  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
( F  o.  A
) `  <. 1 ,  s >. )  =  ( K `  <. 1 ,  s >. )
)
224222, 223eqtr3d 2464 . . . . . . . . . . 11  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  ( F `  ( A `  <. 1 ,  s
>. ) )  =  ( K `  <. 1 ,  s >. )
)
225 df-ov 6070 . . . . . . . . . . . 12  |-  ( 1 A s )  =  ( A `  <. 1 ,  s >. )
226225fveq2i 5717 . . . . . . . . . . 11  |-  ( F `
 ( 1 A s ) )  =  ( F `  ( A `  <. 1 ,  s >. ) )
227 df-ov 6070 . . . . . . . . . . 11  |-  ( 1 K s )  =  ( K `  <. 1 ,  s >. )
228224, 226, 2273eqtr4g 2487 . . . . . . . . . 10  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  ( F `  ( 1 A s ) )  =  ( 1 K s ) )
229129simprd 450 . . . . . . . . . 10  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
1 K s )  =  ( G ` 
1 ) )
2307simp2d 970 . . . . . . . . . . . . 13  |-  ( ph  ->  ( F  o.  M
)  =  G )
231230adantr 452 . . . . . . . . . . . 12  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  ( F  o.  M )  =  G )
232231fveq1d 5716 . . . . . . . . . . 11  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
( F  o.  M
) `  1 )  =  ( G ` 
1 ) )
23383adantr 452 . . . . . . . . . . . 12  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  M : ( 0 [,] 1 ) --> B )
234 fvco3 5786 . . . . . . . . . . . 12  |-  ( ( M : ( 0 [,] 1 ) --> B  /\  1  e.  ( 0 [,] 1 ) )  ->  ( ( F  o.  M ) `  1 )  =  ( F `  ( M `  1 )
) )
235233, 92, 234sylancl 644 . . . . . . . . . . 11  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
( F  o.  M
) `  1 )  =  ( F `  ( M `  1 ) ) )
236232, 235eqtr3d 2464 . . . . . . . . . 10  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  ( G `  1 )  =  ( F `  ( M `  1 ) ) )
237228, 229, 2363eqtrd 2466 . . . . . . . . 9  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  ( F `  ( 1 A s ) )  =  ( F `  ( M `  1 ) ) )
238237mpteq2dva 4282 . . . . . . . 8  |-  ( ph  ->  ( s  e.  ( 0 [,] 1 ) 
|->  ( F `  (
1 A s ) ) )  =  ( s  e.  ( 0 [,] 1 )  |->  ( F `  ( M `
 1 ) ) ) )
239 fconstmpt 4907 . . . . . . . 8  |-  ( ( 0 [,] 1 )  X.  { ( F `
 ( M ` 
1 ) ) } )  =  ( s  e.  ( 0 [,] 1 )  |->  ( F `
 ( M ` 
1 ) ) )
240238, 239syl6eqr 2480 . . . . . . 7  |-  ( ph  ->  ( s  e.  ( 0 [,] 1 ) 
|->  ( F `  (
1 A s ) ) )  =  ( ( 0 [,] 1
)  X.  { ( F `  ( M `
 1 ) ) } ) )
241 fovrn 6202 . . . . . . . . . 10  |-  ( ( A : ( ( 0 [,] 1 )  X.  ( 0 [,] 1 ) ) --> B  /\  1  e.  ( 0 [,] 1 )  /\  s  e.  ( 0 [,] 1 ) )  ->  ( 1 A s )  e.  B )
24292, 241mp3an2 1267 . . . . . . . . 9  |-  ( ( A : ( ( 0 [,] 1 )  X.  ( 0 [,] 1 ) ) --> B  /\  s  e.  ( 0 [,] 1 ) )  ->  ( 1 A s )  e.  B )
24322, 242sylan 458 . . . . . . . 8  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
1 A s )  e.  B )
244 eqidd 2431 . . . . . . . 8  |-  ( ph  ->  ( s  e.  ( 0 [,] 1 ) 
|->  ( 1 A s ) )  =  ( s  e.  ( 0 [,] 1 )  |->  ( 1 A s ) ) )
245 fveq2 5714 . . . . . . . 8  |-  ( x  =  ( 1 A s )  ->  ( F `  x )  =  ( F `  ( 1 A s ) ) )
246243, 244, 53, 245fmptco 5887 . . . . . . 7  |-  ( ph  ->  ( F  o.  (
s  e.  ( 0 [,] 1 )  |->  ( 1 A s ) ) )  =  ( s  e.  ( 0 [,] 1 )  |->  ( F `  ( 1 A s ) ) ) )
247 fcoconst 5891 . . . . . . . 8  |-  ( ( F  Fn  B  /\  ( M `  1 )  e.  B )  -> 
( F  o.  (
( 0 [,] 1
)  X.  { ( M `  1 ) } ) )  =  ( ( 0 [,] 1 )  X.  {
( F `  ( M `  1 )
) } ) )
248157, 216, 247syl2anc 643 . . . . . . 7  |-  ( ph  ->  ( F  o.  (
( 0 [,] 1
)  X.  { ( M `  1 ) } ) )  =  ( ( 0 [,] 1 )  X.  {
( F `  ( M `  1 )
) } ) )
249240, 246, 2483eqtr4d 2472 . . . . . 6  |-  ( ph  ->  ( F  o.  (
s  e.  ( 0 [,] 1 )  |->  ( 1 A s ) ) )  =  ( F  o.  ( ( 0 [,] 1 )  X.  { ( M `
 1 ) } ) ) )
250 oveq1 6074 . . . . . . . . . 10  |-  ( s  =  1  ->  (
s A 0 )  =  ( 1 A 0 ) )
251 fveq2 5714 . . . . . . . . . 10  |-  ( s  =  1  ->  ( M `  s )  =  ( M ` 
1 ) )
252250, 251eqeq12d 2444 . . . . . . . . 9  |-  ( s  =  1  ->  (
( s A 0 )  =  ( M `
 s )  <->  ( 1 A 0 )  =  ( M `  1
) ) )
253252rspcv 3035 . . . . . . . 8  |-  ( 1  e.  ( 0 [,] 1 )  ->  ( A. s  e.  (
0 [,] 1 ) ( s A 0 )  =  ( M `
 s )  -> 
( 1 A 0 )  =  ( M `
 1 ) ) )
25492, 90, 253mpsyl 61 . . . . . . 7  |-  ( ph  ->  ( 1 A 0 )  =  ( M `
 1 ) )
255 oveq2 6075 . . . . . . . . 9  |-  ( s  =  0  ->  (
1 A s )  =  ( 1 A 0 ) )
256 eqid 2430 . . . . . . . . 9  |-  ( s  e.  ( 0 [,] 1 )  |->  ( 1 A s ) )  =  ( s  e.  ( 0 [,] 1
)  |->  ( 1 A s ) )
257 ovex 6092 . . . . . . . . 9  |-  ( 1 A 0 )  e. 
_V
258255, 256, 257fvmpt 5792 . . . . . . . 8  |-  ( 0  e.  ( 0 [,] 1 )  ->  (
( s  e.  ( 0 [,] 1 ) 
|->  ( 1 A s ) ) `  0
)  =  ( 1 A 0 ) )
25923, 258ax-mp 8 . . . . . . 7  |-  ( ( s  e.  ( 0 [,] 1 )  |->  ( 1 A s ) ) `  0 )  =  ( 1 A 0 )
260 fvex 5728 . . . . . . . . 9  |-  ( M `
 1 )  e. 
_V
261260fvconst2 5933 . . . . . . . 8  |-  ( 0  e.  ( 0 [,] 1 )  ->  (
( ( 0 [,] 1 )  X.  {
( M `  1
) } ) ` 
0 )  =  ( M `  1 ) )
26223, 261ax-mp 8 . . . . . . 7  |-  ( ( ( 0 [,] 1
)  X.  { ( M `  1 ) } ) `  0
)  =  ( M `
 1 )
263254, 259, 2623eqtr4g 2487 . . . . . 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 24953 . . . . 5  |-  ( ph  ->  ( s  e.  ( 0 [,] 1 ) 
|->  ( 1 A s ) )  =  ( ( 0 [,] 1
)  X.  { ( M `  1 ) } ) )
265 fconstmpt 4907 . . . . 5  |-  ( ( 0 [,] 1 )  X.  { ( M `
 1 ) } )  =  ( s  e.  ( 0 [,] 1 )  |->  ( M `
 1 ) )
266264, 265syl6eq 2478 . . . 4  |-  ( ph  ->  ( s  e.  ( 0 [,] 1 ) 
|->  ( 1 A s ) )  =  ( s  e.  ( 0 [,] 1 )  |->  ( M `  1 ) ) )
267 mpteqb 5805 . . . . 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 6092 . . . . . 6  |-  ( 1 A s )  e. 
_V
269268a1i 11 . . . . 5  |-  ( s  e.  ( 0 [,] 1 )  ->  (
1 A s )  e.  _V )
270267, 269mprg 2762 . . . 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 189 . . 3  |-  ( ph  ->  A. s  e.  ( 0 [,] 1 ) ( 1 A s )  =  ( M `
 1 ) )
272271r19.21bi 2791 . 2  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
1 A s )  =  ( M ` 
1 ) )
2738, 16, 17, 91, 212, 213, 272isphtpy2d 18995 1  |-  ( ph  ->  A  e.  ( M ( PHtpy `  C ) N ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725   A.wral 2692   E!wreu 2694   _Vcvv 2943   {csn 3801   <.cop 3804   U.cuni 4002    e. cmpt 4253    X. cxp 4862    o. ccom 4868    Fn wfn 5435   -->wf 5436   ` cfv 5440  (class class class)co 6067   iota_crio 6528   0cc0 8974   1c1 8975   [,]cicc 10903   Topctop 16941  TopOnctopon 16942    Cn ccn 17271   Conccon 17457  𝑛Locally cnlly 17511    tX ctx 17575   IIcii 18888   Htpy chtpy 18975   PHtpycphtpy 18976   CovMap ccvm 24925
This theorem is referenced by:  cvmliftpht  24988
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2411  ax-rep 4307  ax-sep 4317  ax-nul 4325  ax-pow 4364  ax-pr 4390  ax-un 4687  ax-inf2 7580  ax-cnex 9030  ax-resscn 9031  ax-1cn 9032  ax-icn 9033  ax-addcl 9034  ax-addrcl 9035  ax-mulcl 9036  ax-mulrcl 9037  ax-mulcom 9038  ax-addass 9039  ax-mulass 9040  ax-distr 9041  ax-i2m1 9042  ax-1ne0 9043  ax-1rid 9044  ax-rnegex 9045  ax-rrecex 9046  ax-cnre 9047  ax-pre-lttri 9048  ax-pre-lttrn 9049  ax-pre-ltadd 9050  ax-pre-mulgt0 9051  ax-pre-sup 9052  ax-addf 9053  ax-mulf 9054
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2417  df-cleq 2423  df-clel 2426  df-nfc 2555  df-ne 2595  df-nel 2596  df-ral 2697  df-rex 2698  df-reu 2699  df-rmo 2700  df-rab 2701  df-v 2945  df-sbc 3149  df-csb 3239  df-dif 3310  df-un 3312  df-in 3314  df-ss 3321  df-pss 3323  df-nul 3616  df-if 3727  df-pw 3788  df-sn 3807  df-pr 3808  df-tp 3809  df-op 3810  df-uni 4003  df-int 4038  df-iun 4082  df-iin 4083  df-br 4200  df-opab 4254  df-mpt 4255  df-tr 4290  df-eprel 4481  df-id 4485  df-po 4490  df-so 4491  df-fr 4528  df-se 4529  df-we 4530  df-ord 4571  df-on 4572  df-lim 4573  df-suc 4574  df-om 4832  df-xp 4870  df-rel 4871  df-cnv 4872  df-co 4873  df-dm 4874  df-rn 4875  df-res 4876  df-ima 4877  df-iota 5404  df-fun 5442  df-fn 5443  df-f 5444  df-f1 5445  df-fo 5446  df-f1o 5447  df-fv 5448  df-isom 5449  df-ov 6070  df-oprab 6071  df-mpt2 6072  df-of 6291  df-1st 6335  df-2nd 6336  df-riota 6535  df-recs 6619  df-rdg 6654  df-1o 6710  df-2o 6711  df-oadd 6714  df-er 6891  df-ec 6893  df-map 7006  df-ixp 7050  df-en 7096  df-dom 7097  df-sdom 7098  df-fin 7099  df-fi 7402  df-sup 7432  df-oi 7463  df-card 7810  df-cda 8032  df-pnf 9106  df-mnf 9107  df-xr 9108  df-ltxr 9109  df-le 9110  df-sub 9277  df-neg 9278  df-div 9662  df-nn 9985  df-2 10042  df-3 10043  df-4 10044  df-5 10045  df-6 10046  df-7 10047  df-8 10048  df-9 10049  df-10 10050  df-n0 10206  df-z 10267  df-dec 10367  df-uz 10473  df-q 10559  df-rp 10597  df-xneg 10694  df-xadd 10695  df-xmul 10696  df-ioo 10904  df-ico 10906  df-icc 10907  df-fz 11028  df-fzo 11119  df-fl 11185  df-seq 11307  df-exp 11366  df-hash 11602  df-cj 11887  df-re 11888  df-im 11889  df-sqr 12023  df-abs 12024  df-clim 12265  df-sum 12463  df-struct 13454  df-ndx 13455  df-slot 13456  df-base 13457  df-sets 13458  df-ress 13459  df-plusg 13525  df-mulr 13526  df-starv 13527  df-sca 13528  df-vsca 13529  df-tset 13531  df-ple 13532  df-ds 13534  df-unif 13535  df-hom 13536  df-cco 13537  df-rest 13633  df-topn 13634  df-topgen 13650  df-pt 13651  df-prds 13654  df-xrs 13709  df-0g 13710  df-gsum 13711  df-qtop 13716  df-imas 13717  df-xps 13719  df-mre 13794  df-mrc 13795  df-acs 13797  df-mnd 14673  df-submnd 14722  df-mulg 14798  df-cntz 15099  df-cmn 15397  df-psmet 16677  df-xmet 16678  df-met 16679  df-bl 16680  df-mopn 16681  df-cnfld 16687  df-top 16946  df-bases 16948  df-topon 16949  df-topsp 16950  df-cld 17066  df-ntr 17067  df-cls 17068  df-nei 17145  df-cn 17274  df-cnp 17275  df-cmp 17433  df-con 17458  df-lly 17512  df-nlly 17513  df-tx 17577  df-hmeo 17770  df-xms 18333  df-ms 18334  df-tms 18335  df-ii 18890  df-htpy 18978  df-phtpy 18979  df-phtpc 19000  df-pcon 24891  df-scon 24892  df-cvm 24926
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