Users' Mathboxes Mathbox for Mario Carneiro < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  cvmlift3lem9 Structured version   Unicode version

Theorem cvmlift3lem9 28398
Description: Lemma for cvmlift2 28387. (Contributed by Mario Carneiro, 7-May-2015.)
Hypotheses
Ref Expression
cvmlift3.b  |-  B  = 
U. C
cvmlift3.y  |-  Y  = 
U. K
cvmlift3.f  |-  ( ph  ->  F  e.  ( C CovMap  J ) )
cvmlift3.k  |-  ( ph  ->  K  e. SCon )
cvmlift3.l  |-  ( ph  ->  K  e. 𝑛Locally PCon )
cvmlift3.o  |-  ( ph  ->  O  e.  Y )
cvmlift3.g  |-  ( ph  ->  G  e.  ( K  Cn  J ) )
cvmlift3.p  |-  ( ph  ->  P  e.  B )
cvmlift3.e  |-  ( ph  ->  ( F `  P
)  =  ( G `
 O ) )
cvmlift3.h  |-  H  =  ( x  e.  Y  |->  ( iota_ z  e.  B  E. f  e.  (
II  Cn  K )
( ( f ` 
0 )  =  O  /\  ( f ` 
1 )  =  x  /\  ( ( iota_ g  e.  ( II  Cn  C ) ( ( F  o.  g )  =  ( G  o.  f )  /\  (
g `  0 )  =  P ) ) ` 
1 )  =  z ) ) )
cvmlift3lem7.s  |-  S  =  ( k  e.  J  |->  { s  e.  ( ~P C  \  { (/)
} )  |  ( U. s  =  ( `' F " k )  /\  A. c  e.  s  ( A. d  e.  ( s  \  {
c } ) ( c  i^i  d )  =  (/)  /\  ( F  |`  c )  e.  ( ( Ct  c )
Homeo ( Jt  k ) ) ) ) } )
Assertion
Ref Expression
cvmlift3lem9  |-  ( ph  ->  E. f  e.  ( K  Cn  C ) ( ( F  o.  f )  =  G  /\  ( f `  O )  =  P ) )
Distinct variable groups:    c, d,
f, k, s, z, g, x    J, c   
g, d, x, J, f, k, s    F, c, d, f, g, k, s    x, z, F    H, c, d, f, g, x, z    S, f, x    B, d, f, g, x, z    G, c, d, f, g, k, x, z    C, c, d, f, g, k, s, x, z    ph, f, x    K, c, f, g, x, z    P, c, d, f, g, x, z    O, c, f, g, x, z    f, Y, g, x, z
Allowed substitution hints:    ph( z, g, k, s, c, d)    B( k, s, c)    P( k, s)    S( z, g, k, s, c, d)    G( s)    H( k, s)    J( z)    K( k, s, d)    O( k, s, d)    Y( k, s, c, d)

Proof of Theorem cvmlift3lem9
StepHypRef Expression
1 cvmlift3.b . . 3  |-  B  = 
U. C
2 cvmlift3.y . . 3  |-  Y  = 
U. K
3 cvmlift3.f . . 3  |-  ( ph  ->  F  e.  ( C CovMap  J ) )
4 cvmlift3.k . . 3  |-  ( ph  ->  K  e. SCon )
5 cvmlift3.l . . 3  |-  ( ph  ->  K  e. 𝑛Locally PCon )
6 cvmlift3.o . . 3  |-  ( ph  ->  O  e.  Y )
7 cvmlift3.g . . 3  |-  ( ph  ->  G  e.  ( K  Cn  J ) )
8 cvmlift3.p . . 3  |-  ( ph  ->  P  e.  B )
9 cvmlift3.e . . 3  |-  ( ph  ->  ( F `  P
)  =  ( G `
 O ) )
10 cvmlift3.h . . 3  |-  H  =  ( x  e.  Y  |->  ( iota_ z  e.  B  E. f  e.  (
II  Cn  K )
( ( f ` 
0 )  =  O  /\  ( f ` 
1 )  =  x  /\  ( ( iota_ g  e.  ( II  Cn  C ) ( ( F  o.  g )  =  ( G  o.  f )  /\  (
g `  0 )  =  P ) ) ` 
1 )  =  z ) ) )
11 cvmlift3lem7.s . . 3  |-  S  =  ( k  e.  J  |->  { s  e.  ( ~P C  \  { (/)
} )  |  ( U. s  =  ( `' F " k )  /\  A. c  e.  s  ( A. d  e.  ( s  \  {
c } ) ( c  i^i  d )  =  (/)  /\  ( F  |`  c )  e.  ( ( Ct  c )
Homeo ( Jt  k ) ) ) ) } )
121, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11cvmlift3lem8 28397 . 2  |-  ( ph  ->  H  e.  ( K  Cn  C ) )
131, 2, 3, 4, 5, 6, 7, 8, 9, 10cvmlift3lem5 28394 . 2  |-  ( ph  ->  ( F  o.  H
)  =  G )
14 iitopon 21111 . . . . . 6  |-  II  e.  (TopOn `  ( 0 [,] 1 ) )
1514a1i 11 . . . . 5  |-  ( ph  ->  II  e.  (TopOn `  ( 0 [,] 1
) ) )
16 scontop 28299 . . . . . . 7  |-  ( K  e. SCon  ->  K  e.  Top )
174, 16syl 16 . . . . . 6  |-  ( ph  ->  K  e.  Top )
182toptopon 19194 . . . . . 6  |-  ( K  e.  Top  <->  K  e.  (TopOn `  Y ) )
1917, 18sylib 196 . . . . 5  |-  ( ph  ->  K  e.  (TopOn `  Y ) )
20 cnconst2 19543 . . . . 5  |-  ( ( II  e.  (TopOn `  ( 0 [,] 1
) )  /\  K  e.  (TopOn `  Y )  /\  O  e.  Y
)  ->  ( (
0 [,] 1 )  X.  { O }
)  e.  ( II 
Cn  K ) )
2115, 19, 6, 20syl3anc 1223 . . . 4  |-  ( ph  ->  ( ( 0 [,] 1 )  X.  { O } )  e.  ( II  Cn  K ) )
22 0elunit 11627 . . . . 5  |-  0  e.  ( 0 [,] 1
)
23 fvconst2g 6105 . . . . 5  |-  ( ( O  e.  Y  /\  0  e.  ( 0 [,] 1 ) )  ->  ( ( ( 0 [,] 1 )  X.  { O }
) `  0 )  =  O )
246, 22, 23sylancl 662 . . . 4  |-  ( ph  ->  ( ( ( 0 [,] 1 )  X. 
{ O } ) `
 0 )  =  O )
25 1elunit 11628 . . . . 5  |-  1  e.  ( 0 [,] 1
)
26 fvconst2g 6105 . . . . 5  |-  ( ( O  e.  Y  /\  1  e.  ( 0 [,] 1 ) )  ->  ( ( ( 0 [,] 1 )  X.  { O }
) `  1 )  =  O )
276, 25, 26sylancl 662 . . . 4  |-  ( ph  ->  ( ( ( 0 [,] 1 )  X. 
{ O } ) `
 1 )  =  O )
289sneqd 4032 . . . . . . . . 9  |-  ( ph  ->  { ( F `  P ) }  =  { ( G `  O ) } )
2928xpeq2d 5016 . . . . . . . 8  |-  ( ph  ->  ( ( 0 [,] 1 )  X.  {
( F `  P
) } )  =  ( ( 0 [,] 1 )  X.  {
( G `  O
) } ) )
30 cvmcn 28333 . . . . . . . . . 10  |-  ( F  e.  ( C CovMap  J
)  ->  F  e.  ( C  Cn  J
) )
31 eqid 2460 . . . . . . . . . . 11  |-  U. J  =  U. J
321, 31cnf 19506 . . . . . . . . . 10  |-  ( F  e.  ( C  Cn  J )  ->  F : B --> U. J )
33 ffn 5722 . . . . . . . . . 10  |-  ( F : B --> U. J  ->  F  Fn  B )
343, 30, 32, 334syl 21 . . . . . . . . 9  |-  ( ph  ->  F  Fn  B )
35 fcoconst 6049 . . . . . . . . 9  |-  ( ( F  Fn  B  /\  P  e.  B )  ->  ( F  o.  (
( 0 [,] 1
)  X.  { P } ) )  =  ( ( 0 [,] 1 )  X.  {
( F `  P
) } ) )
3634, 8, 35syl2anc 661 . . . . . . . 8  |-  ( ph  ->  ( F  o.  (
( 0 [,] 1
)  X.  { P } ) )  =  ( ( 0 [,] 1 )  X.  {
( F `  P
) } ) )
372, 31cnf 19506 . . . . . . . . . . 11  |-  ( G  e.  ( K  Cn  J )  ->  G : Y --> U. J )
387, 37syl 16 . . . . . . . . . 10  |-  ( ph  ->  G : Y --> U. J
)
39 ffn 5722 . . . . . . . . . 10  |-  ( G : Y --> U. J  ->  G  Fn  Y )
4038, 39syl 16 . . . . . . . . 9  |-  ( ph  ->  G  Fn  Y )
41 fcoconst 6049 . . . . . . . . 9  |-  ( ( G  Fn  Y  /\  O  e.  Y )  ->  ( G  o.  (
( 0 [,] 1
)  X.  { O } ) )  =  ( ( 0 [,] 1 )  X.  {
( G `  O
) } ) )
4240, 6, 41syl2anc 661 . . . . . . . 8  |-  ( ph  ->  ( G  o.  (
( 0 [,] 1
)  X.  { O } ) )  =  ( ( 0 [,] 1 )  X.  {
( G `  O
) } ) )
4329, 36, 423eqtr4d 2511 . . . . . . 7  |-  ( ph  ->  ( F  o.  (
( 0 [,] 1
)  X.  { P } ) )  =  ( G  o.  (
( 0 [,] 1
)  X.  { O } ) ) )
44 fvconst2g 6105 . . . . . . . 8  |-  ( ( P  e.  B  /\  0  e.  ( 0 [,] 1 ) )  ->  ( ( ( 0 [,] 1 )  X.  { P }
) `  0 )  =  P )
458, 22, 44sylancl 662 . . . . . . 7  |-  ( ph  ->  ( ( ( 0 [,] 1 )  X. 
{ P } ) `
 0 )  =  P )
46 cvmtop1 28331 . . . . . . . . . . 11  |-  ( F  e.  ( C CovMap  J
)  ->  C  e.  Top )
473, 46syl 16 . . . . . . . . . 10  |-  ( ph  ->  C  e.  Top )
481toptopon 19194 . . . . . . . . . 10  |-  ( C  e.  Top  <->  C  e.  (TopOn `  B ) )
4947, 48sylib 196 . . . . . . . . 9  |-  ( ph  ->  C  e.  (TopOn `  B ) )
50 cnconst2 19543 . . . . . . . . 9  |-  ( ( II  e.  (TopOn `  ( 0 [,] 1
) )  /\  C  e.  (TopOn `  B )  /\  P  e.  B
)  ->  ( (
0 [,] 1 )  X.  { P }
)  e.  ( II 
Cn  C ) )
5115, 49, 8, 50syl3anc 1223 . . . . . . . 8  |-  ( ph  ->  ( ( 0 [,] 1 )  X.  { P } )  e.  ( II  Cn  C ) )
52 cvmtop2 28332 . . . . . . . . . . . . 13  |-  ( F  e.  ( C CovMap  J
)  ->  J  e.  Top )
533, 52syl 16 . . . . . . . . . . . 12  |-  ( ph  ->  J  e.  Top )
5431toptopon 19194 . . . . . . . . . . . 12  |-  ( J  e.  Top  <->  J  e.  (TopOn `  U. J ) )
5553, 54sylib 196 . . . . . . . . . . 11  |-  ( ph  ->  J  e.  (TopOn `  U. J ) )
5638, 6ffvelrnd 6013 . . . . . . . . . . 11  |-  ( ph  ->  ( G `  O
)  e.  U. J
)
57 cnconst2 19543 . . . . . . . . . . 11  |-  ( ( II  e.  (TopOn `  ( 0 [,] 1
) )  /\  J  e.  (TopOn `  U. J )  /\  ( G `  O )  e.  U. J )  ->  (
( 0 [,] 1
)  X.  { ( G `  O ) } )  e.  ( II  Cn  J ) )
5815, 55, 56, 57syl3anc 1223 . . . . . . . . . 10  |-  ( ph  ->  ( ( 0 [,] 1 )  X.  {
( G `  O
) } )  e.  ( II  Cn  J
) )
5942, 58eqeltrd 2548 . . . . . . . . 9  |-  ( ph  ->  ( G  o.  (
( 0 [,] 1
)  X.  { O } ) )  e.  ( II  Cn  J
) )
60 fvconst2g 6105 . . . . . . . . . . 11  |-  ( ( ( G `  O
)  e.  U. J  /\  0  e.  (
0 [,] 1 ) )  ->  ( (
( 0 [,] 1
)  X.  { ( G `  O ) } ) `  0
)  =  ( G `
 O ) )
6156, 22, 60sylancl 662 . . . . . . . . . 10  |-  ( ph  ->  ( ( ( 0 [,] 1 )  X. 
{ ( G `  O ) } ) `
 0 )  =  ( G `  O
) )
6242fveq1d 5859 . . . . . . . . . 10  |-  ( ph  ->  ( ( G  o.  ( ( 0 [,] 1 )  X.  { O } ) ) ` 
0 )  =  ( ( ( 0 [,] 1 )  X.  {
( G `  O
) } ) ` 
0 ) )
6361, 62, 93eqtr4rd 2512 . . . . . . . . 9  |-  ( ph  ->  ( F `  P
)  =  ( ( G  o.  ( ( 0 [,] 1 )  X.  { O }
) ) `  0
) )
641cvmlift 28370 . . . . . . . . 9  |-  ( ( ( F  e.  ( C CovMap  J )  /\  ( G  o.  (
( 0 [,] 1
)  X.  { O } ) )  e.  ( II  Cn  J
) )  /\  ( P  e.  B  /\  ( F `  P )  =  ( ( G  o.  ( ( 0 [,] 1 )  X. 
{ O } ) ) `  0 ) ) )  ->  E! g  e.  ( II  Cn  C ) ( ( F  o.  g )  =  ( G  o.  ( ( 0 [,] 1 )  X.  { O } ) )  /\  ( g `  0
)  =  P ) )
653, 59, 8, 63, 64syl22anc 1224 . . . . . . . 8  |-  ( ph  ->  E! g  e.  ( II  Cn  C ) ( ( F  o.  g )  =  ( G  o.  ( ( 0 [,] 1 )  X.  { O }
) )  /\  (
g `  0 )  =  P ) )
66 coeq2 5152 . . . . . . . . . . 11  |-  ( g  =  ( ( 0 [,] 1 )  X. 
{ P } )  ->  ( F  o.  g )  =  ( F  o.  ( ( 0 [,] 1 )  X.  { P }
) ) )
6766eqeq1d 2462 . . . . . . . . . 10  |-  ( g  =  ( ( 0 [,] 1 )  X. 
{ P } )  ->  ( ( F  o.  g )  =  ( G  o.  (
( 0 [,] 1
)  X.  { O } ) )  <->  ( F  o.  ( ( 0 [,] 1 )  X.  { P } ) )  =  ( G  o.  (
( 0 [,] 1
)  X.  { O } ) ) ) )
68 fveq1 5856 . . . . . . . . . . 11  |-  ( g  =  ( ( 0 [,] 1 )  X. 
{ P } )  ->  ( g ` 
0 )  =  ( ( ( 0 [,] 1 )  X.  { P } ) `  0
) )
6968eqeq1d 2462 . . . . . . . . . 10  |-  ( g  =  ( ( 0 [,] 1 )  X. 
{ P } )  ->  ( ( g `
 0 )  =  P  <->  ( ( ( 0 [,] 1 )  X.  { P }
) `  0 )  =  P ) )
7067, 69anbi12d 710 . . . . . . . . 9  |-  ( g  =  ( ( 0 [,] 1 )  X. 
{ P } )  ->  ( ( ( F  o.  g )  =  ( G  o.  ( ( 0 [,] 1 )  X.  { O } ) )  /\  ( g `  0
)  =  P )  <-> 
( ( F  o.  ( ( 0 [,] 1 )  X.  { P } ) )  =  ( G  o.  (
( 0 [,] 1
)  X.  { O } ) )  /\  ( ( ( 0 [,] 1 )  X. 
{ P } ) `
 0 )  =  P ) ) )
7170riota2 6259 . . . . . . . 8  |-  ( ( ( ( 0 [,] 1 )  X.  { P } )  e.  ( II  Cn  C )  /\  E! g  e.  ( II  Cn  C
) ( ( F  o.  g )  =  ( G  o.  (
( 0 [,] 1
)  X.  { O } ) )  /\  ( g `  0
)  =  P ) )  ->  ( (
( F  o.  (
( 0 [,] 1
)  X.  { P } ) )  =  ( G  o.  (
( 0 [,] 1
)  X.  { O } ) )  /\  ( ( ( 0 [,] 1 )  X. 
{ P } ) `
 0 )  =  P )  <->  ( iota_ g  e.  ( II  Cn  C ) ( ( F  o.  g )  =  ( G  o.  ( ( 0 [,] 1 )  X.  { O } ) )  /\  ( g `  0
)  =  P ) )  =  ( ( 0 [,] 1 )  X.  { P }
) ) )
7251, 65, 71syl2anc 661 . . . . . . 7  |-  ( ph  ->  ( ( ( F  o.  ( ( 0 [,] 1 )  X. 
{ P } ) )  =  ( G  o.  ( ( 0 [,] 1 )  X. 
{ O } ) )  /\  ( ( ( 0 [,] 1
)  X.  { P } ) `  0
)  =  P )  <-> 
( iota_ g  e.  ( II  Cn  C ) ( ( F  o.  g )  =  ( G  o.  ( ( 0 [,] 1 )  X.  { O }
) )  /\  (
g `  0 )  =  P ) )  =  ( ( 0 [,] 1 )  X.  { P } ) ) )
7343, 45, 72mpbi2and 914 . . . . . 6  |-  ( ph  ->  ( iota_ g  e.  ( II  Cn  C ) ( ( F  o.  g )  =  ( G  o.  ( ( 0 [,] 1 )  X.  { O }
) )  /\  (
g `  0 )  =  P ) )  =  ( ( 0 [,] 1 )  X.  { P } ) )
7473fveq1d 5859 . . . . 5  |-  ( ph  ->  ( ( iota_ g  e.  ( II  Cn  C
) ( ( F  o.  g )  =  ( G  o.  (
( 0 [,] 1
)  X.  { O } ) )  /\  ( g `  0
)  =  P ) ) `  1 )  =  ( ( ( 0 [,] 1 )  X.  { P }
) `  1 )
)
75 fvconst2g 6105 . . . . . 6  |-  ( ( P  e.  B  /\  1  e.  ( 0 [,] 1 ) )  ->  ( ( ( 0 [,] 1 )  X.  { P }
) `  1 )  =  P )
768, 25, 75sylancl 662 . . . . 5  |-  ( ph  ->  ( ( ( 0 [,] 1 )  X. 
{ P } ) `
 1 )  =  P )
7774, 76eqtrd 2501 . . . 4  |-  ( ph  ->  ( ( iota_ g  e.  ( II  Cn  C
) ( ( F  o.  g )  =  ( G  o.  (
( 0 [,] 1
)  X.  { O } ) )  /\  ( g `  0
)  =  P ) ) `  1 )  =  P )
78 fveq1 5856 . . . . . . 7  |-  ( f  =  ( ( 0 [,] 1 )  X. 
{ O } )  ->  ( f ` 
0 )  =  ( ( ( 0 [,] 1 )  X.  { O } ) `  0
) )
7978eqeq1d 2462 . . . . . 6  |-  ( f  =  ( ( 0 [,] 1 )  X. 
{ O } )  ->  ( ( f `
 0 )  =  O  <->  ( ( ( 0 [,] 1 )  X.  { O }
) `  0 )  =  O ) )
80 fveq1 5856 . . . . . . 7  |-  ( f  =  ( ( 0 [,] 1 )  X. 
{ O } )  ->  ( f ` 
1 )  =  ( ( ( 0 [,] 1 )  X.  { O } ) `  1
) )
8180eqeq1d 2462 . . . . . 6  |-  ( f  =  ( ( 0 [,] 1 )  X. 
{ O } )  ->  ( ( f `
 1 )  =  O  <->  ( ( ( 0 [,] 1 )  X.  { O }
) `  1 )  =  O ) )
82 coeq2 5152 . . . . . . . . . . 11  |-  ( f  =  ( ( 0 [,] 1 )  X. 
{ O } )  ->  ( G  o.  f )  =  ( G  o.  ( ( 0 [,] 1 )  X.  { O }
) ) )
8382eqeq2d 2474 . . . . . . . . . 10  |-  ( f  =  ( ( 0 [,] 1 )  X. 
{ O } )  ->  ( ( F  o.  g )  =  ( G  o.  f
)  <->  ( F  o.  g )  =  ( G  o.  ( ( 0 [,] 1 )  X.  { O }
) ) ) )
8483anbi1d 704 . . . . . . . . 9  |-  ( f  =  ( ( 0 [,] 1 )  X. 
{ O } )  ->  ( ( ( F  o.  g )  =  ( G  o.  f )  /\  (
g `  0 )  =  P )  <->  ( ( F  o.  g )  =  ( G  o.  ( ( 0 [,] 1 )  X.  { O } ) )  /\  ( g `  0
)  =  P ) ) )
8584riotabidv 6238 . . . . . . . 8  |-  ( f  =  ( ( 0 [,] 1 )  X. 
{ O } )  ->  ( iota_ g  e.  ( II  Cn  C
) ( ( F  o.  g )  =  ( G  o.  f
)  /\  ( g `  0 )  =  P ) )  =  ( iota_ g  e.  ( II  Cn  C ) ( ( F  o.  g )  =  ( G  o.  ( ( 0 [,] 1 )  X.  { O }
) )  /\  (
g `  0 )  =  P ) ) )
8685fveq1d 5859 . . . . . . 7  |-  ( f  =  ( ( 0 [,] 1 )  X. 
{ O } )  ->  ( ( iota_ g  e.  ( II  Cn  C ) ( ( F  o.  g )  =  ( G  o.  f )  /\  (
g `  0 )  =  P ) ) ` 
1 )  =  ( ( iota_ g  e.  ( II  Cn  C ) ( ( F  o.  g )  =  ( G  o.  ( ( 0 [,] 1 )  X.  { O }
) )  /\  (
g `  0 )  =  P ) ) ` 
1 ) )
8786eqeq1d 2462 . . . . . 6  |-  ( f  =  ( ( 0 [,] 1 )  X. 
{ O } )  ->  ( ( (
iota_ g  e.  (
II  Cn  C )
( ( F  o.  g )  =  ( G  o.  f )  /\  ( g ` 
0 )  =  P ) ) `  1
)  =  P  <->  ( ( iota_ g  e.  ( II 
Cn  C ) ( ( F  o.  g
)  =  ( G  o.  ( ( 0 [,] 1 )  X. 
{ O } ) )  /\  ( g `
 0 )  =  P ) ) ` 
1 )  =  P ) )
8879, 81, 873anbi123d 1294 . . . . 5  |-  ( f  =  ( ( 0 [,] 1 )  X. 
{ O } )  ->  ( ( ( f `  0 )  =  O  /\  (
f `  1 )  =  O  /\  (
( iota_ g  e.  ( II  Cn  C ) ( ( F  o.  g )  =  ( G  o.  f )  /\  ( g ` 
0 )  =  P ) ) `  1
)  =  P )  <-> 
( ( ( ( 0 [,] 1 )  X.  { O }
) `  0 )  =  O  /\  (
( ( 0 [,] 1 )  X.  { O } ) `  1
)  =  O  /\  ( ( iota_ g  e.  ( II  Cn  C
) ( ( F  o.  g )  =  ( G  o.  (
( 0 [,] 1
)  X.  { O } ) )  /\  ( g `  0
)  =  P ) ) `  1 )  =  P ) ) )
8988rspcev 3207 . . . 4  |-  ( ( ( ( 0 [,] 1 )  X.  { O } )  e.  ( II  Cn  K )  /\  ( ( ( ( 0 [,] 1
)  X.  { O } ) `  0
)  =  O  /\  ( ( ( 0 [,] 1 )  X. 
{ O } ) `
 1 )  =  O  /\  ( (
iota_ g  e.  (
II  Cn  C )
( ( F  o.  g )  =  ( G  o.  ( ( 0 [,] 1 )  X.  { O }
) )  /\  (
g `  0 )  =  P ) ) ` 
1 )  =  P ) )  ->  E. f  e.  ( II  Cn  K
) ( ( f `
 0 )  =  O  /\  ( f `
 1 )  =  O  /\  ( (
iota_ g  e.  (
II  Cn  C )
( ( F  o.  g )  =  ( G  o.  f )  /\  ( g ` 
0 )  =  P ) ) `  1
)  =  P ) )
9021, 24, 27, 77, 89syl13anc 1225 . . 3  |-  ( ph  ->  E. f  e.  ( II  Cn  K ) ( ( f ` 
0 )  =  O  /\  ( f ` 
1 )  =  O  /\  ( ( iota_ g  e.  ( II  Cn  C ) ( ( F  o.  g )  =  ( G  o.  f )  /\  (
g `  0 )  =  P ) ) ` 
1 )  =  P ) )
911, 2, 3, 4, 5, 6, 7, 8, 9, 10cvmlift3lem4 28393 . . . 4  |-  ( (
ph  /\  O  e.  Y )  ->  (
( H `  O
)  =  P  <->  E. f  e.  ( II  Cn  K
) ( ( f `
 0 )  =  O  /\  ( f `
 1 )  =  O  /\  ( (
iota_ g  e.  (
II  Cn  C )
( ( F  o.  g )  =  ( G  o.  f )  /\  ( g ` 
0 )  =  P ) ) `  1
)  =  P ) ) )
926, 91mpdan 668 . . 3  |-  ( ph  ->  ( ( H `  O )  =  P  <->  E. f  e.  (
II  Cn  K )
( ( f ` 
0 )  =  O  /\  ( f ` 
1 )  =  O  /\  ( ( iota_ g  e.  ( II  Cn  C ) ( ( F  o.  g )  =  ( G  o.  f )  /\  (
g `  0 )  =  P ) ) ` 
1 )  =  P ) ) )
9390, 92mpbird 232 . 2  |-  ( ph  ->  ( H `  O
)  =  P )
94 coeq2 5152 . . . . 5  |-  ( f  =  H  ->  ( F  o.  f )  =  ( F  o.  H ) )
9594eqeq1d 2462 . . . 4  |-  ( f  =  H  ->  (
( F  o.  f
)  =  G  <->  ( F  o.  H )  =  G ) )
96 fveq1 5856 . . . . 5  |-  ( f  =  H  ->  (
f `  O )  =  ( H `  O ) )
9796eqeq1d 2462 . . . 4  |-  ( f  =  H  ->  (
( f `  O
)  =  P  <->  ( H `  O )  =  P ) )
9895, 97anbi12d 710 . . 3  |-  ( f  =  H  ->  (
( ( F  o.  f )  =  G  /\  ( f `  O )  =  P )  <->  ( ( F  o.  H )  =  G  /\  ( H `
 O )  =  P ) ) )
9998rspcev 3207 . 2  |-  ( ( H  e.  ( K  Cn  C )  /\  ( ( F  o.  H )  =  G  /\  ( H `  O )  =  P ) )  ->  E. f  e.  ( K  Cn  C
) ( ( F  o.  f )  =  G  /\  ( f `
 O )  =  P ) )
10012, 13, 93, 99syl12anc 1221 1  |-  ( ph  ->  E. f  e.  ( K  Cn  C ) ( ( F  o.  f )  =  G  /\  ( f `  O )  =  P ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 968    = wceq 1374    e. wcel 1762   A.wral 2807   E.wrex 2808   E!wreu 2809   {crab 2811    \ cdif 3466    i^i cin 3468   (/)c0 3778   ~Pcpw 4003   {csn 4020   U.cuni 4238    |-> cmpt 4498    X. cxp 4990   `'ccnv 4991    |` cres 4994   "cima 4995    o. ccom 4996    Fn wfn 5574   -->wf 5575   ` cfv 5579   iota_crio 6235  (class class class)co 6275   0cc0 9481   1c1 9482   [,]cicc 11521   ↾t crest 14665   Topctop 19154  TopOnctopon 19155    Cn ccn 19484  𝑛Locally cnlly 19725   Homeochmeo 19982   IIcii 21107  PConcpcon 28290  SConcscon 28291   CovMap ccvm 28326
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-inf2 8047  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558  ax-pre-sup 9559  ax-addf 9560  ax-mulf 9561
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-fal 1380  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-nel 2658  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-int 4276  df-iun 4320  df-iin 4321  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-se 4832  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-isom 5588  df-riota 6236  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-of 6515  df-om 6672  df-1st 6774  df-2nd 6775  df-supp 6892  df-recs 7032  df-rdg 7066  df-1o 7120  df-2o 7121  df-oadd 7124  df-er 7301  df-ec 7303  df-map 7412  df-ixp 7460  df-en 7507  df-dom 7508  df-sdom 7509  df-fin 7510  df-fsupp 7819  df-fi 7860  df-sup 7890  df-oi 7924  df-card 8309  df-cda 8537  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9796  df-neg 9797  df-div 10196  df-nn 10526  df-2 10583  df-3 10584  df-4 10585  df-5 10586  df-6 10587  df-7 10588  df-8 10589  df-9 10590  df-10 10591  df-n0 10785  df-z 10854  df-dec 10966  df-uz 11072  df-q 11172  df-rp 11210  df-xneg 11307  df-xadd 11308  df-xmul 11309  df-ioo 11522  df-ico 11524  df-icc 11525  df-fz 11662  df-fzo 11782  df-fl 11886  df-seq 12064  df-exp 12123  df-hash 12361  df-cj 12882  df-re 12883  df-im 12884  df-sqr 13018  df-abs 13019  df-clim 13260  df-sum 13458  df-struct 14481  df-ndx 14482  df-slot 14483  df-base 14484  df-sets 14485  df-ress 14486  df-plusg 14557  df-mulr 14558  df-starv 14559  df-sca 14560  df-vsca 14561  df-ip 14562  df-tset 14563  df-ple 14564  df-ds 14566  df-unif 14567  df-hom 14568  df-cco 14569  df-rest 14667  df-topn 14668  df-0g 14686  df-gsum 14687  df-topgen 14688  df-pt 14689  df-prds 14692  df-xrs 14746  df-qtop 14751  df-imas 14752  df-xps 14754  df-mre 14830  df-mrc 14831  df-acs 14833  df-mnd 15721  df-submnd 15771  df-mulg 15854  df-cntz 16143  df-cmn 16589  df-psmet 18175  df-xmet 18176  df-met 18177  df-bl 18178  df-mopn 18179  df-cnfld 18185  df-top 19159  df-bases 19161  df-topon 19162  df-topsp 19163  df-cld 19279  df-ntr 19280  df-cls 19281  df-nei 19358  df-cn 19487  df-cnp 19488  df-cmp 19646  df-con 19672  df-lly 19726  df-nlly 19727  df-tx 19791  df-hmeo 19984  df-xms 20551  df-ms 20552  df-tms 20553  df-ii 21109  df-htpy 21198  df-phtpy 21199  df-phtpc 21220  df-pco 21233  df-pcon 28292  df-scon 28293  df-cvm 28327
This theorem is referenced by:  cvmlift3  28399
  Copyright terms: Public domain W3C validator