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Theorem cvmliftphtlem 30036
Description: Lemma for cvmliftpht 30037. (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 30020 . . 3  |-  ( ph  ->  ( M  e.  ( II  Cn  C )  /\  ( F  o.  M )  =  G  /\  ( M ` 
0 )  =  P ) )
87simp1d 1017 . 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 22003 . . . . . 6  |-  ( ph  ->  ( ( G ` 
0 )  =  ( H `  0 )  /\  ( G ` 
1 )  =  ( H `  1 ) ) )
1312simpld 460 . . . . 5  |-  ( ph  ->  ( G `  0
)  =  ( H `
 0 ) )
146, 13eqtrd 2463 . . . 4  |-  ( ph  ->  ( F `  P
)  =  ( H `
 0 ) )
151, 9, 3, 10, 5, 14cvmliftiota 30020 . . 3  |-  ( ph  ->  ( N  e.  ( II  Cn  C )  /\  ( F  o.  N )  =  H  /\  ( N ` 
0 )  =  P ) )
1615simp1d 1017 . 2  |-  ( ph  ->  N  e.  ( II 
Cn  C ) )
17 cvmliftphtlem.a . 2  |-  ( ph  ->  A  e.  ( ( II  tX  II )  Cn  C ) )
18 iitop 21899 . . . . . . . . . . . . . . . 16  |-  II  e.  Top
19 iiuni 21900 . . . . . . . . . . . . . . . 16  |-  ( 0 [,] 1 )  = 
U. II
2018, 18, 19, 19txunii 20595 . . . . . . . . . . . . . . 15  |-  ( ( 0 [,] 1 )  X.  ( 0 [,] 1 ) )  = 
U. ( II  tX  II )
2120, 1cnf 20249 . . . . . . . . . . . . . 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 11751 . . . . . . . . . . . . . 14  |-  0  e.  ( 0 [,] 1
)
24 opelxpi 4882 . . . . . . . . . . . . . 14  |-  ( ( s  e.  ( 0 [,] 1 )  /\  0  e.  ( 0 [,] 1 ) )  ->  <. s ,  0
>.  e.  ( ( 0 [,] 1 )  X.  ( 0 [,] 1
) ) )
2523, 24mpan2 675 . . . . . . . . . . . . 13  |-  ( s  e.  ( 0 [,] 1 )  ->  <. s ,  0 >.  e.  ( ( 0 [,] 1
)  X.  ( 0 [,] 1 ) ) )
26 fvco3 5955 . . . . . . . . . . . . 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 479 . . . . . . . . . . . 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 466 . . . . . . . . . . . . 13  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  ( F  o.  A )  =  K )
3029fveq1d 5880 . . . . . . . . . . . 12  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
( F  o.  A
) `  <. s ,  0 >. )  =  ( K `  <. s ,  0 >. )
)
3127, 30eqtr3d 2465 . . . . . . . . . . 11  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  ( F `  ( A `  <. s ,  0
>. ) )  =  ( K `  <. s ,  0 >. )
)
32 df-ov 6305 . . . . . . . . . . . 12  |-  ( s A 0 )  =  ( A `  <. s ,  0 >. )
3332fveq2i 5881 . . . . . . . . . . 11  |-  ( F `
 ( s A 0 ) )  =  ( F `  ( A `  <. s ,  0 >. ) )
34 df-ov 6305 . . . . . . . . . . 11  |-  ( s K 0 )  =  ( K `  <. s ,  0 >. )
3531, 33, 343eqtr4g 2488 . . . . . . . . . 10  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  ( F `  ( s A 0 ) )  =  ( s K 0 ) )
36 iitopon 21898 . . . . . . . . . . . . 13  |-  II  e.  (TopOn `  ( 0 [,] 1 ) )
3736a1i 11 . . . . . . . . . . . 12  |-  ( ph  ->  II  e.  (TopOn `  ( 0 [,] 1
) ) )
384, 10phtpyhtpy 22000 . . . . . . . . . . . . 13  |-  ( ph  ->  ( G ( PHtpy `  J ) H ) 
C_  ( G ( II Htpy  J ) H ) )
3938, 11sseldd 3465 . . . . . . . . . . . 12  |-  ( ph  ->  K  e.  ( G ( II Htpy  J ) H ) )
4037, 4, 10, 39htpyi 21992 . . . . . . . . . . 11  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
( s K 0 )  =  ( G `
 s )  /\  ( s K 1 )  =  ( H `
 s ) ) )
4140simpld 460 . . . . . . . . . 10  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
s K 0 )  =  ( G `  s ) )
4235, 41eqtrd 2463 . . . . . . . . 9  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  ( F `  ( s A 0 ) )  =  ( G `  s ) )
4342mpteq2dva 4507 . . . . . . . 8  |-  ( ph  ->  ( s  e.  ( 0 [,] 1 ) 
|->  ( F `  (
s A 0 ) ) )  =  ( s  e.  ( 0 [,] 1 )  |->  ( G `  s ) ) )
44 fovrn 6450 . . . . . . . . . . 11  |-  ( ( A : ( ( 0 [,] 1 )  X.  ( 0 [,] 1 ) ) --> B  /\  s  e.  ( 0 [,] 1 )  /\  0  e.  ( 0 [,] 1 ) )  ->  ( s A 0 )  e.  B )
4523, 44mp3an3 1349 . . . . . . . . . 10  |-  ( ( A : ( ( 0 [,] 1 )  X.  ( 0 [,] 1 ) ) --> B  /\  s  e.  ( 0 [,] 1 ) )  ->  ( s A 0 )  e.  B )
4622, 45sylan 473 . . . . . . . . 9  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
s A 0 )  e.  B )
47 eqidd 2423 . . . . . . . . 9  |-  ( ph  ->  ( s  e.  ( 0 [,] 1 ) 
|->  ( s A 0 ) )  =  ( s  e.  ( 0 [,] 1 )  |->  ( s A 0 ) ) )
48 cvmcn 29981 . . . . . . . . . . . 12  |-  ( F  e.  ( C CovMap  J
)  ->  F  e.  ( C  Cn  J
) )
493, 48syl 17 . . . . . . . . . . 11  |-  ( ph  ->  F  e.  ( C  Cn  J ) )
50 eqid 2422 . . . . . . . . . . . 12  |-  U. J  =  U. J
511, 50cnf 20249 . . . . . . . . . . 11  |-  ( F  e.  ( C  Cn  J )  ->  F : B --> U. J )
5249, 51syl 17 . . . . . . . . . 10  |-  ( ph  ->  F : B --> U. J
)
5352feqmptd 5931 . . . . . . . . 9  |-  ( ph  ->  F  =  ( x  e.  B  |->  ( F `
 x ) ) )
54 fveq2 5878 . . . . . . . . 9  |-  ( x  =  ( s A 0 )  ->  ( F `  x )  =  ( F `  ( s A 0 ) ) )
5546, 47, 53, 54fmptco 6068 . . . . . . . 8  |-  ( ph  ->  ( F  o.  (
s  e.  ( 0 [,] 1 )  |->  ( s A 0 ) ) )  =  ( s  e.  ( 0 [,] 1 )  |->  ( F `  ( s A 0 ) ) ) )
5619, 50cnf 20249 . . . . . . . . . 10  |-  ( G  e.  ( II  Cn  J )  ->  G : ( 0 [,] 1 ) --> U. J
)
574, 56syl 17 . . . . . . . . 9  |-  ( ph  ->  G : ( 0 [,] 1 ) --> U. J )
5857feqmptd 5931 . . . . . . . 8  |-  ( ph  ->  G  =  ( s  e.  ( 0 [,] 1 )  |->  ( G `
 s ) ) )
5943, 55, 583eqtr4d 2473 . . . . . . 7  |-  ( ph  ->  ( F  o.  (
s  e.  ( 0 [,] 1 )  |->  ( s A 0 ) ) )  =  G )
60 cvmliftphtlem.0 . . . . . . 7  |-  ( ph  ->  ( 0 A 0 )  =  P )
6137cnmptid 20663 . . . . . . . . 9  |-  ( ph  ->  ( s  e.  ( 0 [,] 1 ) 
|->  s )  e.  ( II  Cn  II ) )
6223a1i 11 . . . . . . . . . 10  |-  ( ph  ->  0  e.  ( 0 [,] 1 ) )
6337, 37, 62cnmptc 20664 . . . . . . . . 9  |-  ( ph  ->  ( s  e.  ( 0 [,] 1 ) 
|->  0 )  e.  ( II  Cn  II ) )
6437, 61, 63, 17cnmpt12f 20668 . . . . . . . 8  |-  ( ph  ->  ( s  e.  ( 0 [,] 1 ) 
|->  ( s A 0 ) )  e.  ( II  Cn  C ) )
651cvmlift 30018 . . . . . . . . 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 1265 . . . . . . . 8  |-  ( ph  ->  E! f  e.  ( II  Cn  C ) ( ( F  o.  f )  =  G  /\  ( f ` 
0 )  =  P ) )
67 coeq2 5009 . . . . . . . . . . 11  |-  ( f  =  ( s  e.  ( 0 [,] 1
)  |->  ( s A 0 ) )  -> 
( F  o.  f
)  =  ( F  o.  ( s  e.  ( 0 [,] 1
)  |->  ( s A 0 ) ) ) )
6867eqeq1d 2424 . . . . . . . . . 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 5877 . . . . . . . . . . . 12  |-  ( f  =  ( s  e.  ( 0 [,] 1
)  |->  ( s A 0 ) )  -> 
( f `  0
)  =  ( ( s  e.  ( 0 [,] 1 )  |->  ( s A 0 ) ) `  0 ) )
70 oveq1 6309 . . . . . . . . . . . . . 14  |-  ( s  =  0  ->  (
s A 0 )  =  ( 0 A 0 ) )
71 eqid 2422 . . . . . . . . . . . . . 14  |-  ( s  e.  ( 0 [,] 1 )  |->  ( s A 0 ) )  =  ( s  e.  ( 0 [,] 1
)  |->  ( s A 0 ) )
72 ovex 6330 . . . . . . . . . . . . . 14  |-  ( 0 A 0 )  e. 
_V
7370, 71, 72fvmpt 5961 . . . . . . . . . . . . 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 2479 . . . . . . . . . . 11  |-  ( f  =  ( s  e.  ( 0 [,] 1
)  |->  ( s A 0 ) )  -> 
( f `  0
)  =  ( 0 A 0 ) )
7675eqeq1d 2424 . . . . . . . . . 10  |-  ( f  =  ( s  e.  ( 0 [,] 1
)  |->  ( s A 0 ) )  -> 
( ( f ` 
0 )  =  P  <-> 
( 0 A 0 )  =  P ) )
7768, 76anbi12d 715 . . . . . . . . 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 6286 . . . . . . . 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 665 . . . . . . 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 929 . . . . . 6  |-  ( ph  ->  ( iota_ f  e.  ( II  Cn  C ) ( ( F  o.  f )  =  G  /\  ( f ` 
0 )  =  P ) )  =  ( s  e.  ( 0 [,] 1 )  |->  ( s A 0 ) ) )
812, 80syl5eq 2475 . . . . 5  |-  ( ph  ->  M  =  ( s  e.  ( 0 [,] 1 )  |->  ( s A 0 ) ) )
8219, 1cnf 20249 . . . . . . 7  |-  ( M  e.  ( II  Cn  C )  ->  M : ( 0 [,] 1 ) --> B )
838, 82syl 17 . . . . . 6  |-  ( ph  ->  M : ( 0 [,] 1 ) --> B )
8483feqmptd 5931 . . . . 5  |-  ( ph  ->  M  =  ( s  e.  ( 0 [,] 1 )  |->  ( M `
 s ) ) )
8581, 84eqtr3d 2465 . . . 4  |-  ( ph  ->  ( s  e.  ( 0 [,] 1 ) 
|->  ( s A 0 ) )  =  ( s  e.  ( 0 [,] 1 )  |->  ( M `  s ) ) )
86 mpteqb 5977 . . . . 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 6330 . . . . . 6  |-  ( s A 0 )  e. 
_V
8887a1i 11 . . . . 5  |-  ( s  e.  ( 0 [,] 1 )  ->  (
s A 0 )  e.  _V )
8986, 88mprg 2788 . . . 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 199 . . 3  |-  ( ph  ->  A. s  e.  ( 0 [,] 1 ) ( s A 0 )  =  ( M `
 s ) )
9190r19.21bi 2794 . 2  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
s A 0 )  =  ( M `  s ) )
92 1elunit 11752 . . . . . . . . . . . . . 14  |-  1  e.  ( 0 [,] 1
)
93 opelxpi 4882 . . . . . . . . . . . . . 14  |-  ( ( s  e.  ( 0 [,] 1 )  /\  1  e.  ( 0 [,] 1 ) )  ->  <. s ,  1
>.  e.  ( ( 0 [,] 1 )  X.  ( 0 [,] 1
) ) )
9492, 93mpan2 675 . . . . . . . . . . . . 13  |-  ( s  e.  ( 0 [,] 1 )  ->  <. s ,  1 >.  e.  ( ( 0 [,] 1
)  X.  ( 0 [,] 1 ) ) )
95 fvco3 5955 . . . . . . . . . . . . 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 479 . . . . . . . . . . . 12  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
( F  o.  A
) `  <. s ,  1 >. )  =  ( F `  ( A `
 <. s ,  1
>. ) ) )
9729fveq1d 5880 . . . . . . . . . . . 12  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
( F  o.  A
) `  <. s ,  1 >. )  =  ( K `  <. s ,  1 >. )
)
9896, 97eqtr3d 2465 . . . . . . . . . . 11  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  ( F `  ( A `  <. s ,  1
>. ) )  =  ( K `  <. s ,  1 >. )
)
99 df-ov 6305 . . . . . . . . . . . 12  |-  ( s A 1 )  =  ( A `  <. s ,  1 >. )
10099fveq2i 5881 . . . . . . . . . . 11  |-  ( F `
 ( s A 1 ) )  =  ( F `  ( A `  <. s ,  1 >. ) )
101 df-ov 6305 . . . . . . . . . . 11  |-  ( s K 1 )  =  ( K `  <. s ,  1 >. )
10298, 100, 1013eqtr4g 2488 . . . . . . . . . 10  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  ( F `  ( s A 1 ) )  =  ( s K 1 ) )
10340simprd 464 . . . . . . . . . 10  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
s K 1 )  =  ( H `  s ) )
104102, 103eqtrd 2463 . . . . . . . . 9  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  ( F `  ( s A 1 ) )  =  ( H `  s ) )
105104mpteq2dva 4507 . . . . . . . 8  |-  ( ph  ->  ( s  e.  ( 0 [,] 1 ) 
|->  ( F `  (
s A 1 ) ) )  =  ( s  e.  ( 0 [,] 1 )  |->  ( H `  s ) ) )
106 fovrn 6450 . . . . . . . . . . 11  |-  ( ( A : ( ( 0 [,] 1 )  X.  ( 0 [,] 1 ) ) --> B  /\  s  e.  ( 0 [,] 1 )  /\  1  e.  ( 0 [,] 1 ) )  ->  ( s A 1 )  e.  B )
10792, 106mp3an3 1349 . . . . . . . . . 10  |-  ( ( A : ( ( 0 [,] 1 )  X.  ( 0 [,] 1 ) ) --> B  /\  s  e.  ( 0 [,] 1 ) )  ->  ( s A 1 )  e.  B )
10822, 107sylan 473 . . . . . . . . 9  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
s A 1 )  e.  B )
109 eqidd 2423 . . . . . . . . 9  |-  ( ph  ->  ( s  e.  ( 0 [,] 1 ) 
|->  ( s A 1 ) )  =  ( s  e.  ( 0 [,] 1 )  |->  ( s A 1 ) ) )
110 fveq2 5878 . . . . . . . . 9  |-  ( x  =  ( s A 1 )  ->  ( F `  x )  =  ( F `  ( s A 1 ) ) )
111108, 109, 53, 110fmptco 6068 . . . . . . . 8  |-  ( ph  ->  ( F  o.  (
s  e.  ( 0 [,] 1 )  |->  ( s A 1 ) ) )  =  ( s  e.  ( 0 [,] 1 )  |->  ( F `  ( s A 1 ) ) ) )
11219, 50cnf 20249 . . . . . . . . . 10  |-  ( H  e.  ( II  Cn  J )  ->  H : ( 0 [,] 1 ) --> U. J
)
11310, 112syl 17 . . . . . . . . 9  |-  ( ph  ->  H : ( 0 [,] 1 ) --> U. J )
114113feqmptd 5931 . . . . . . . 8  |-  ( ph  ->  H  =  ( s  e.  ( 0 [,] 1 )  |->  ( H `
 s ) ) )
115105, 111, 1143eqtr4d 2473 . . . . . . 7  |-  ( ph  ->  ( F  o.  (
s  e.  ( 0 [,] 1 )  |->  ( s A 1 ) ) )  =  H )
116 iicon 21906 . . . . . . . . . . . . 13  |-  II  e.  Con
117116a1i 11 . . . . . . . . . . . 12  |-  ( ph  ->  II  e.  Con )
118 iinllycon 29973 . . . . . . . . . . . . 13  |-  II  e. 𝑛Locally  Con
119118a1i 11 . . . . . . . . . . . 12  |-  ( ph  ->  II  e. 𝑛Locally  Con )
12037, 63, 61, 17cnmpt12f 20668 . . . . . . . . . . . 12  |-  ( ph  ->  ( s  e.  ( 0 [,] 1 ) 
|->  ( 0 A s ) )  e.  ( II  Cn  C ) )
121 cvmtop1 29979 . . . . . . . . . . . . . . 15  |-  ( F  e.  ( C CovMap  J
)  ->  C  e.  Top )
1223, 121syl 17 . . . . . . . . . . . . . 14  |-  ( ph  ->  C  e.  Top )
1231toptopon 19935 . . . . . . . . . . . . . 14  |-  ( C  e.  Top  <->  C  e.  (TopOn `  B ) )
124122, 123sylib 199 . . . . . . . . . . . . 13  |-  ( ph  ->  C  e.  (TopOn `  B ) )
125 ffvelrn 6032 . . . . . . . . . . . . . 14  |-  ( ( M : ( 0 [,] 1 ) --> B  /\  0  e.  ( 0 [,] 1 ) )  ->  ( M `  0 )  e.  B )
12683, 23, 125sylancl 666 . . . . . . . . . . . . 13  |-  ( ph  ->  ( M `  0
)  e.  B )
127 cnconst2 20286 . . . . . . . . . . . . 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 1264 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( 0 [,] 1 )  X.  {
( M `  0
) } )  e.  ( II  Cn  C
) )
1294, 10, 11phtpyi 22002 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
( 0 K s )  =  ( G `
 0 )  /\  ( 1 K s )  =  ( G `
 1 ) ) )
130129simpld 460 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
0 K s )  =  ( G ` 
0 ) )
131 opelxpi 4882 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( 0  e.  ( 0 [,] 1 )  /\  s  e.  ( 0 [,] 1 ) )  ->  <. 0 ,  s
>.  e.  ( ( 0 [,] 1 )  X.  ( 0 [,] 1
) ) )
13223, 131mpan 674 . . . . . . . . . . . . . . . . . . 19  |-  ( s  e.  ( 0 [,] 1 )  ->  <. 0 ,  s >.  e.  ( ( 0 [,] 1
)  X.  ( 0 [,] 1 ) ) )
133 fvco3 5955 . . . . . . . . . . . . . . . . . . 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 479 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
( F  o.  A
) `  <. 0 ,  s >. )  =  ( F `  ( A `
 <. 0 ,  s
>. ) ) )
13529fveq1d 5880 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
( F  o.  A
) `  <. 0 ,  s >. )  =  ( K `  <. 0 ,  s >. )
)
136134, 135eqtr3d 2465 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  ( F `  ( A `  <. 0 ,  s
>. ) )  =  ( K `  <. 0 ,  s >. )
)
137 df-ov 6305 . . . . . . . . . . . . . . . . . 18  |-  ( 0 A s )  =  ( A `  <. 0 ,  s >. )
138137fveq2i 5881 . . . . . . . . . . . . . . . . 17  |-  ( F `
 ( 0 A s ) )  =  ( F `  ( A `  <. 0 ,  s >. ) )
139 df-ov 6305 . . . . . . . . . . . . . . . . 17  |-  ( 0 K s )  =  ( K `  <. 0 ,  s >. )
140136, 138, 1393eqtr4g 2488 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  ( F `  ( 0 A s ) )  =  ( 0 K s ) )
1417simp3d 1019 . . . . . . . . . . . . . . . . . . 19  |-  ( ph  ->  ( M `  0
)  =  P )
142141adantr 466 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  ( M `  0 )  =  P )
143142fveq2d 5882 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  ( F `  ( M `  0 ) )  =  ( F `  P ) )
1446adantr 466 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  ( F `  P )  =  ( G ` 
0 ) )
145143, 144eqtrd 2463 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  ( F `  ( M `  0 ) )  =  ( G ` 
0 ) )
146130, 140, 1453eqtr4d 2473 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  ( F `  ( 0 A s ) )  =  ( F `  ( M `  0 ) ) )
147146mpteq2dva 4507 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( s  e.  ( 0 [,] 1 ) 
|->  ( F `  (
0 A s ) ) )  =  ( s  e.  ( 0 [,] 1 )  |->  ( F `  ( M `
 0 ) ) ) )
148 fconstmpt 4894 . . . . . . . . . . . . . 14  |-  ( ( 0 [,] 1 )  X.  { ( F `
 ( M ` 
0 ) ) } )  =  ( s  e.  ( 0 [,] 1 )  |->  ( F `
 ( M ` 
0 ) ) )
149147, 148syl6eqr 2481 . . . . . . . . . . . . 13  |-  ( ph  ->  ( s  e.  ( 0 [,] 1 ) 
|->  ( F `  (
0 A s ) ) )  =  ( ( 0 [,] 1
)  X.  { ( F `  ( M `
 0 ) ) } ) )
150 fovrn 6450 . . . . . . . . . . . . . . . 16  |-  ( ( A : ( ( 0 [,] 1 )  X.  ( 0 [,] 1 ) ) --> B  /\  0  e.  ( 0 [,] 1 )  /\  s  e.  ( 0 [,] 1 ) )  ->  ( 0 A s )  e.  B )
15123, 150mp3an2 1348 . . . . . . . . . . . . . . 15  |-  ( ( A : ( ( 0 [,] 1 )  X.  ( 0 [,] 1 ) ) --> B  /\  s  e.  ( 0 [,] 1 ) )  ->  ( 0 A s )  e.  B )
15222, 151sylan 473 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
0 A s )  e.  B )
153 eqidd 2423 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( s  e.  ( 0 [,] 1 ) 
|->  ( 0 A s ) )  =  ( s  e.  ( 0 [,] 1 )  |->  ( 0 A s ) ) )
154 fveq2 5878 . . . . . . . . . . . . . 14  |-  ( x  =  ( 0 A s )  ->  ( F `  x )  =  ( F `  ( 0 A s ) ) )
155152, 153, 53, 154fmptco 6068 . . . . . . . . . . . . 13  |-  ( ph  ->  ( F  o.  (
s  e.  ( 0 [,] 1 )  |->  ( 0 A s ) ) )  =  ( s  e.  ( 0 [,] 1 )  |->  ( F `  ( 0 A s ) ) ) )
156 ffn 5743 . . . . . . . . . . . . . . 15  |-  ( F : B --> U. J  ->  F  Fn  B )
15752, 156syl 17 . . . . . . . . . . . . . 14  |-  ( ph  ->  F  Fn  B )
158 fcoconst 6072 . . . . . . . . . . . . . 14  |-  ( ( F  Fn  B  /\  ( M `  0 )  e.  B )  -> 
( F  o.  (
( 0 [,] 1
)  X.  { ( M `  0 ) } ) )  =  ( ( 0 [,] 1 )  X.  {
( F `  ( M `  0 )
) } ) )
159157, 126, 158syl2anc 665 . . . . . . . . . . . . 13  |-  ( ph  ->  ( F  o.  (
( 0 [,] 1
)  X.  { ( M `  0 ) } ) )  =  ( ( 0 [,] 1 )  X.  {
( F `  ( M `  0 )
) } ) )
160149, 155, 1593eqtr4d 2473 . . . . . . . . . . . 12  |-  ( ph  ->  ( F  o.  (
s  e.  ( 0 [,] 1 )  |->  ( 0 A s ) ) )  =  ( F  o.  ( ( 0 [,] 1 )  X.  { ( M `
 0 ) } ) ) )
16160, 141eqtr4d 2466 . . . . . . . . . . . . 13  |-  ( ph  ->  ( 0 A 0 )  =  ( M `
 0 ) )
162 oveq2 6310 . . . . . . . . . . . . . . 15  |-  ( s  =  0  ->  (
0 A s )  =  ( 0 A 0 ) )
163 eqid 2422 . . . . . . . . . . . . . . 15  |-  ( s  e.  ( 0 [,] 1 )  |->  ( 0 A s ) )  =  ( s  e.  ( 0 [,] 1
)  |->  ( 0 A s ) )
164162, 163, 72fvmpt 5961 . . . . . . . . . . . . . 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 5888 . . . . . . . . . . . . . . 15  |-  ( M `
 0 )  e. 
_V
167166fvconst2 6132 . . . . . . . . . . . . . 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 2488 . . . . . . . . . . . 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 30002 . . . . . . . . . . 11  |-  ( ph  ->  ( s  e.  ( 0 [,] 1 ) 
|->  ( 0 A s ) )  =  ( ( 0 [,] 1
)  X.  { ( M `  0 ) } ) )
171 fconstmpt 4894 . . . . . . . . . . 11  |-  ( ( 0 [,] 1 )  X.  { ( M `
 0 ) } )  =  ( s  e.  ( 0 [,] 1 )  |->  ( M `
 0 ) )
172170, 171syl6eq 2479 . . . . . . . . . 10  |-  ( ph  ->  ( s  e.  ( 0 [,] 1 ) 
|->  ( 0 A s ) )  =  ( s  e.  ( 0 [,] 1 )  |->  ( M `  0 ) ) )
173 mpteqb 5977 . . . . . . . . . . 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 6330 . . . . . . . . . . . 12  |-  ( 0 A s )  e. 
_V
175174a1i 11 . . . . . . . . . . 11  |-  ( s  e.  ( 0 [,] 1 )  ->  (
0 A s )  e.  _V )
176173, 175mprg 2788 . . . . . . . . . 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 199 . . . . . . . . 9  |-  ( ph  ->  A. s  e.  ( 0 [,] 1 ) ( 0 A s )  =  ( M `
 0 ) )
178 oveq2 6310 . . . . . . . . . . 11  |-  ( s  =  1  ->  (
0 A s )  =  ( 0 A 1 ) )
179178eqeq1d 2424 . . . . . . . . . 10  |-  ( s  =  1  ->  (
( 0 A s )  =  ( M `
 0 )  <->  ( 0 A 1 )  =  ( M `  0
) ) )
180179rspcv 3178 . . . . . . . . 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 2463 . . . . . . 7  |-  ( ph  ->  ( 0 A 1 )  =  P )
18392a1i 11 . . . . . . . . . 10  |-  ( ph  ->  1  e.  ( 0 [,] 1 ) )
18437, 37, 183cnmptc 20664 . . . . . . . . 9  |-  ( ph  ->  ( s  e.  ( 0 [,] 1 ) 
|->  1 )  e.  ( II  Cn  II ) )
18537, 61, 184, 17cnmpt12f 20668 . . . . . . . 8  |-  ( ph  ->  ( s  e.  ( 0 [,] 1 ) 
|->  ( s A 1 ) )  e.  ( II  Cn  C ) )
1861cvmlift 30018 . . . . . . . . 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 1265 . . . . . . . 8  |-  ( ph  ->  E! f  e.  ( II  Cn  C ) ( ( F  o.  f )  =  H  /\  ( f ` 
0 )  =  P ) )
188 coeq2 5009 . . . . . . . . . . 11  |-  ( f  =  ( s  e.  ( 0 [,] 1
)  |->  ( s A 1 ) )  -> 
( F  o.  f
)  =  ( F  o.  ( s  e.  ( 0 [,] 1
)  |->  ( s A 1 ) ) ) )
189188eqeq1d 2424 . . . . . . . . . 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 5877 . . . . . . . . . . . 12  |-  ( f  =  ( s  e.  ( 0 [,] 1
)  |->  ( s A 1 ) )  -> 
( f `  0
)  =  ( ( s  e.  ( 0 [,] 1 )  |->  ( s A 1 ) ) `  0 ) )
191 oveq1 6309 . . . . . . . . . . . . . 14  |-  ( s  =  0  ->  (
s A 1 )  =  ( 0 A 1 ) )
192 eqid 2422 . . . . . . . . . . . . . 14  |-  ( s  e.  ( 0 [,] 1 )  |->  ( s A 1 ) )  =  ( s  e.  ( 0 [,] 1
)  |->  ( s A 1 ) )
193 ovex 6330 . . . . . . . . . . . . . 14  |-  ( 0 A 1 )  e. 
_V
194191, 192, 193fvmpt 5961 . . . . . . . . . . . . 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 2479 . . . . . . . . . . 11  |-  ( f  =  ( s  e.  ( 0 [,] 1
)  |->  ( s A 1 ) )  -> 
( f `  0
)  =  ( 0 A 1 ) )
197196eqeq1d 2424 . . . . . . . . . 10  |-  ( f  =  ( s  e.  ( 0 [,] 1
)  |->  ( s A 1 ) )  -> 
( ( f ` 
0 )  =  P  <-> 
( 0 A 1 )  =  P ) )
198189, 197anbi12d 715 . . . . . . . . 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 6286 . . . . . . . 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 665 . . . . . . 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 929 . . . . . 6  |-  ( ph  ->  ( iota_ f  e.  ( II  Cn  C ) ( ( F  o.  f )  =  H  /\  ( f ` 
0 )  =  P ) )  =  ( s  e.  ( 0 [,] 1 )  |->  ( s A 1 ) ) )
2029, 201syl5eq 2475 . . . . 5  |-  ( ph  ->  N  =  ( s  e.  ( 0 [,] 1 )  |->  ( s A 1 ) ) )
20319, 1cnf 20249 . . . . . . 7  |-  ( N  e.  ( II  Cn  C )  ->  N : ( 0 [,] 1 ) --> B )
20416, 203syl 17 . . . . . 6  |-  ( ph  ->  N : ( 0 [,] 1 ) --> B )
205204feqmptd 5931 . . . . 5  |-  ( ph  ->  N  =  ( s  e.  ( 0 [,] 1 )  |->  ( N `
 s ) ) )
206202, 205eqtr3d 2465 . . . 4  |-  ( ph  ->  ( s  e.  ( 0 [,] 1 ) 
|->  ( s A 1 ) )  =  ( s  e.  ( 0 [,] 1 )  |->  ( N `  s ) ) )
207 mpteqb 5977 . . . . 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 6330 . . . . . 6  |-  ( s A 1 )  e. 
_V
209208a1i 11 . . . . 5  |-  ( s  e.  ( 0 [,] 1 )  ->  (
s A 1 )  e.  _V )
210207, 209mprg 2788 . . . 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 199 . . 3  |-  ( ph  ->  A. s  e.  ( 0 [,] 1 ) ( s A 1 )  =  ( N `
 s ) )
212211r19.21bi 2794 . 2  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
s A 1 )  =  ( N `  s ) )
213177r19.21bi 2794 . 2  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
0 A s )  =  ( M ` 
0 ) )
21437, 184, 61, 17cnmpt12f 20668 . . . . . 6  |-  ( ph  ->  ( s  e.  ( 0 [,] 1 ) 
|->  ( 1 A s ) )  e.  ( II  Cn  C ) )
215 ffvelrn 6032 . . . . . . . 8  |-  ( ( M : ( 0 [,] 1 ) --> B  /\  1  e.  ( 0 [,] 1 ) )  ->  ( M `  1 )  e.  B )
21683, 92, 215sylancl 666 . . . . . . 7  |-  ( ph  ->  ( M `  1
)  e.  B )
217 cnconst2 20286 . . . . . . 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 1264 . . . . . 6  |-  ( ph  ->  ( ( 0 [,] 1 )  X.  {
( M `  1
) } )  e.  ( II  Cn  C
) )
219 opelxpi 4882 . . . . . . . . . . . . . 14  |-  ( ( 1  e.  ( 0 [,] 1 )  /\  s  e.  ( 0 [,] 1 ) )  ->  <. 1 ,  s
>.  e.  ( ( 0 [,] 1 )  X.  ( 0 [,] 1
) ) )
22092, 219mpan 674 . . . . . . . . . . . . 13  |-  ( s  e.  ( 0 [,] 1 )  ->  <. 1 ,  s >.  e.  ( ( 0 [,] 1
)  X.  ( 0 [,] 1 ) ) )
221 fvco3 5955 . . . . . . . . . . . . 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 479 . . . . . . . . . . . 12  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
( F  o.  A
) `  <. 1 ,  s >. )  =  ( F `  ( A `
 <. 1 ,  s
>. ) ) )
22329fveq1d 5880 . . . . . . . . . . . 12  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
( F  o.  A
) `  <. 1 ,  s >. )  =  ( K `  <. 1 ,  s >. )
)
224222, 223eqtr3d 2465 . . . . . . . . . . 11  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  ( F `  ( A `  <. 1 ,  s
>. ) )  =  ( K `  <. 1 ,  s >. )
)
225 df-ov 6305 . . . . . . . . . . . 12  |-  ( 1 A s )  =  ( A `  <. 1 ,  s >. )
226225fveq2i 5881 . . . . . . . . . . 11  |-  ( F `
 ( 1 A s ) )  =  ( F `  ( A `  <. 1 ,  s >. ) )
227 df-ov 6305 . . . . . . . . . . 11  |-  ( 1 K s )  =  ( K `  <. 1 ,  s >. )
228224, 226, 2273eqtr4g 2488 . . . . . . . . . 10  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  ( F `  ( 1 A s ) )  =  ( 1 K s ) )
229129simprd 464 . . . . . . . . . 10  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
1 K s )  =  ( G ` 
1 ) )
2307simp2d 1018 . . . . . . . . . . . . 13  |-  ( ph  ->  ( F  o.  M
)  =  G )
231230adantr 466 . . . . . . . . . . . 12  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  ( F  o.  M )  =  G )
232231fveq1d 5880 . . . . . . . . . . 11  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
( F  o.  M
) `  1 )  =  ( G ` 
1 ) )
23383adantr 466 . . . . . . . . . . . 12  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  M : ( 0 [,] 1 ) --> B )
234 fvco3 5955 . . . . . . . . . . . 12  |-  ( ( M : ( 0 [,] 1 ) --> B  /\  1  e.  ( 0 [,] 1 ) )  ->  ( ( F  o.  M ) `  1 )  =  ( F `  ( M `  1 )
) )
235233, 92, 234sylancl 666 . . . . . . . . . . 11  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
( F  o.  M
) `  1 )  =  ( F `  ( M `  1 ) ) )
236232, 235eqtr3d 2465 . . . . . . . . . 10  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  ( G `  1 )  =  ( F `  ( M `  1 ) ) )
237228, 229, 2363eqtrd 2467 . . . . . . . . 9  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  ( F `  ( 1 A s ) )  =  ( F `  ( M `  1 ) ) )
238237mpteq2dva 4507 . . . . . . . 8  |-  ( ph  ->  ( s  e.  ( 0 [,] 1 ) 
|->  ( F `  (
1 A s ) ) )  =  ( s  e.  ( 0 [,] 1 )  |->  ( F `  ( M `
 1 ) ) ) )
239 fconstmpt 4894 . . . . . . . 8  |-  ( ( 0 [,] 1 )  X.  { ( F `
 ( M ` 
1 ) ) } )  =  ( s  e.  ( 0 [,] 1 )  |->  ( F `
 ( M ` 
1 ) ) )
240238, 239syl6eqr 2481 . . . . . . 7  |-  ( ph  ->  ( s  e.  ( 0 [,] 1 ) 
|->  ( F `  (
1 A s ) ) )  =  ( ( 0 [,] 1
)  X.  { ( F `  ( M `
 1 ) ) } ) )
241 fovrn 6450 . . . . . . . . . 10  |-  ( ( A : ( ( 0 [,] 1 )  X.  ( 0 [,] 1 ) ) --> B  /\  1  e.  ( 0 [,] 1 )  /\  s  e.  ( 0 [,] 1 ) )  ->  ( 1 A s )  e.  B )
24292, 241mp3an2 1348 . . . . . . . . 9  |-  ( ( A : ( ( 0 [,] 1 )  X.  ( 0 [,] 1 ) ) --> B  /\  s  e.  ( 0 [,] 1 ) )  ->  ( 1 A s )  e.  B )
24322, 242sylan 473 . . . . . . . 8  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
1 A s )  e.  B )
244 eqidd 2423 . . . . . . . 8  |-  ( ph  ->  ( s  e.  ( 0 [,] 1 ) 
|->  ( 1 A s ) )  =  ( s  e.  ( 0 [,] 1 )  |->  ( 1 A s ) ) )
245 fveq2 5878 . . . . . . . 8  |-  ( x  =  ( 1 A s )  ->  ( F `  x )  =  ( F `  ( 1 A s ) ) )
246243, 244, 53, 245fmptco 6068 . . . . . . 7  |-  ( ph  ->  ( F  o.  (
s  e.  ( 0 [,] 1 )  |->  ( 1 A s ) ) )  =  ( s  e.  ( 0 [,] 1 )  |->  ( F `  ( 1 A s ) ) ) )
247 fcoconst 6072 . . . . . . . 8  |-  ( ( F  Fn  B  /\  ( M `  1 )  e.  B )  -> 
( F  o.  (
( 0 [,] 1
)  X.  { ( M `  1 ) } ) )  =  ( ( 0 [,] 1 )  X.  {
( F `  ( M `  1 )
) } ) )
248157, 216, 247syl2anc 665 . . . . . . 7  |-  ( ph  ->  ( F  o.  (
( 0 [,] 1
)  X.  { ( M `  1 ) } ) )  =  ( ( 0 [,] 1 )  X.  {
( F `  ( M `  1 )
) } ) )
249240, 246, 2483eqtr4d 2473 . . . . . 6  |-  ( ph  ->  ( F  o.  (
s  e.  ( 0 [,] 1 )  |->  ( 1 A s ) ) )  =  ( F  o.  ( ( 0 [,] 1 )  X.  { ( M `
 1 ) } ) ) )
250 oveq1 6309 . . . . . . . . . 10  |-  ( s  =  1  ->  (
s A 0 )  =  ( 1 A 0 ) )
251 fveq2 5878 . . . . . . . . . 10  |-  ( s  =  1  ->  ( M `  s )  =  ( M ` 
1 ) )
252250, 251eqeq12d 2444 . . . . . . . . 9  |-  ( s  =  1  ->  (
( s A 0 )  =  ( M `
 s )  <->  ( 1 A 0 )  =  ( M `  1
) ) )
253252rspcv 3178 . . . . . . . 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 6310 . . . . . . . . 9  |-  ( s  =  0  ->  (
1 A s )  =  ( 1 A 0 ) )
256 eqid 2422 . . . . . . . . 9  |-  ( s  e.  ( 0 [,] 1 )  |->  ( 1 A s ) )  =  ( s  e.  ( 0 [,] 1
)  |->  ( 1 A s ) )
257 ovex 6330 . . . . . . . . 9  |-  ( 1 A 0 )  e. 
_V
258255, 256, 257fvmpt 5961 . . . . . . . 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 5888 . . . . . . . . 9  |-  ( M `
 1 )  e. 
_V
261260fvconst2 6132 . . . . . . . 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 2488 . . . . . 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 30002 . . . . 5  |-  ( ph  ->  ( s  e.  ( 0 [,] 1 ) 
|->  ( 1 A s ) )  =  ( ( 0 [,] 1
)  X.  { ( M `  1 ) } ) )
265 fconstmpt 4894 . . . . 5  |-  ( ( 0 [,] 1 )  X.  { ( M `
 1 ) } )  =  ( s  e.  ( 0 [,] 1 )  |->  ( M `
 1 ) )
266264, 265syl6eq 2479 . . . 4  |-  ( ph  ->  ( s  e.  ( 0 [,] 1 ) 
|->  ( 1 A s ) )  =  ( s  e.  ( 0 [,] 1 )  |->  ( M `  1 ) ) )
267 mpteqb 5977 . . . . 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 6330 . . . . . 6  |-  ( 1 A s )  e. 
_V
269268a1i 11 . . . . 5  |-  ( s  e.  ( 0 [,] 1 )  ->  (
1 A s )  e.  _V )
270267, 269mprg 2788 . . . 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 199 . . 3  |-  ( ph  ->  A. s  e.  ( 0 [,] 1 ) ( 1 A s )  =  ( M `
 1 ) )
272271r19.21bi 2794 . 2  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
1 A s )  =  ( M ` 
1 ) )
2738, 16, 17, 91, 212, 213, 272isphtpy2d 22005 1  |-  ( ph  ->  A  e.  ( M ( PHtpy `  C ) N ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    = wceq 1437    e. wcel 1868   A.wral 2775   E!wreu 2777   _Vcvv 3081   {csn 3996   <.cop 4002   U.cuni 4216    |-> cmpt 4479    X. cxp 4848    o. ccom 4854    Fn wfn 5593   -->wf 5594   ` cfv 5598   iota_crio 6263  (class class class)co 6302   0cc0 9540   1c1 9541   [,]cicc 11639   Topctop 19904  TopOnctopon 19905    Cn ccn 20227   Conccon 20413  𝑛Locally cnlly 20467    tX ctx 20562   IIcii 21894   Htpy chtpy 21985   PHtpycphtpy 21986   CovMap ccvm 29974
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-8 1870  ax-9 1872  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400  ax-rep 4533  ax-sep 4543  ax-nul 4552  ax-pow 4599  ax-pr 4657  ax-un 6594  ax-inf2 8149  ax-cnex 9596  ax-resscn 9597  ax-1cn 9598  ax-icn 9599  ax-addcl 9600  ax-addrcl 9601  ax-mulcl 9602  ax-mulrcl 9603  ax-mulcom 9604  ax-addass 9605  ax-mulass 9606  ax-distr 9607  ax-i2m1 9608  ax-1ne0 9609  ax-1rid 9610  ax-rnegex 9611  ax-rrecex 9612  ax-cnre 9613  ax-pre-lttri 9614  ax-pre-lttrn 9615  ax-pre-ltadd 9616  ax-pre-mulgt0 9617  ax-pre-sup 9618  ax-addf 9619  ax-mulf 9620
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-fal 1443  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2269  df-mo 2270  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ne 2620  df-nel 2621  df-ral 2780  df-rex 2781  df-reu 2782  df-rmo 2783  df-rab 2784  df-v 3083  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-pss 3452  df-nul 3762  df-if 3910  df-pw 3981  df-sn 3997  df-pr 3999  df-tp 4001  df-op 4003  df-uni 4217  df-int 4253  df-iun 4298  df-iin 4299  df-br 4421  df-opab 4480  df-mpt 4481  df-tr 4516  df-eprel 4761  df-id 4765  df-po 4771  df-so 4772  df-fr 4809  df-se 4810  df-we 4811  df-xp 4856  df-rel 4857  df-cnv 4858  df-co 4859  df-dm 4860  df-rn 4861  df-res 4862  df-ima 4863  df-pred 5396  df-ord 5442  df-on 5443  df-lim 5444  df-suc 5445  df-iota 5562  df-fun 5600  df-fn 5601  df-f 5602  df-f1 5603  df-fo 5604  df-f1o 5605  df-fv 5606  df-isom 5607  df-riota 6264  df-ov 6305  df-oprab 6306  df-mpt2 6307  df-of 6542  df-om 6704  df-1st 6804  df-2nd 6805  df-supp 6923  df-wrecs 7033  df-recs 7095  df-rdg 7133  df-1o 7187  df-2o 7188  df-oadd 7191  df-er 7368  df-ec 7370  df-map 7479  df-ixp 7528  df-en 7575  df-dom 7576  df-sdom 7577  df-fin 7578  df-fsupp 7887  df-fi 7928  df-sup 7959  df-inf 7960  df-oi 8028  df-card 8375  df-cda 8599  df-pnf 9678  df-mnf 9679  df-xr 9680  df-ltxr 9681  df-le 9682  df-sub 9863  df-neg 9864  df-div 10271  df-nn 10611  df-2 10669  df-3 10670  df-4 10671  df-5 10672  df-6 10673  df-7 10674  df-8 10675  df-9 10676  df-10 10677  df-n0 10871  df-z 10939  df-dec 11053  df-uz 11161  df-q 11266  df-rp 11304  df-xneg 11410  df-xadd 11411  df-xmul 11412  df-ioo 11640  df-ico 11642  df-icc 11643  df-fz 11786  df-fzo 11917  df-fl 12028  df-seq 12214  df-exp 12273  df-hash 12516  df-cj 13151  df-re 13152  df-im 13153  df-sqrt 13287  df-abs 13288  df-clim 13540  df-sum 13741  df-struct 15111  df-ndx 15112  df-slot 15113  df-base 15114  df-sets 15115  df-ress 15116  df-plusg 15191  df-mulr 15192  df-starv 15193  df-sca 15194  df-vsca 15195  df-ip 15196  df-tset 15197  df-ple 15198  df-ds 15200  df-unif 15201  df-hom 15202  df-cco 15203  df-rest 15309  df-topn 15310  df-0g 15328  df-gsum 15329  df-topgen 15330  df-pt 15331  df-prds 15334  df-xrs 15388  df-qtop 15394  df-imas 15395  df-xps 15398  df-mre 15480  df-mrc 15481  df-acs 15483  df-mgm 16476  df-sgrp 16515  df-mnd 16525  df-submnd 16571  df-mulg 16664  df-cntz 16959  df-cmn 17420  df-psmet 18950  df-xmet 18951  df-met 18952  df-bl 18953  df-mopn 18954  df-cnfld 18959  df-top 19908  df-bases 19909  df-topon 19910  df-topsp 19911  df-cld 20021  df-ntr 20022  df-cls 20023  df-nei 20101  df-cn 20230  df-cnp 20231  df-cmp 20389  df-con 20414  df-lly 20468  df-nlly 20469  df-tx 20564  df-hmeo 20757  df-xms 21322  df-ms 21323  df-tms 21324  df-ii 21896  df-htpy 21988  df-phtpy 21989  df-phtpc 22010  df-pcon 29940  df-scon 29941  df-cvm 29975
This theorem is referenced by:  cvmliftpht  30037
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