Users' Mathboxes Mathbox for Mario Carneiro < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  cvmlift2lem11 Structured version   Unicode version

Theorem cvmlift2lem11 28398
Description: Lemma for cvmlift2 28401. (Contributed by Mario Carneiro, 1-Jun-2015.)
Hypotheses
Ref Expression
cvmlift2.b  |-  B  = 
U. C
cvmlift2.f  |-  ( ph  ->  F  e.  ( C CovMap  J ) )
cvmlift2.g  |-  ( ph  ->  G  e.  ( ( II  tX  II )  Cn  J ) )
cvmlift2.p  |-  ( ph  ->  P  e.  B )
cvmlift2.i  |-  ( ph  ->  ( F `  P
)  =  ( 0 G 0 ) )
cvmlift2.h  |-  H  =  ( iota_ f  e.  ( II  Cn  C ) ( ( F  o.  f )  =  ( z  e.  ( 0 [,] 1 )  |->  ( z G 0 ) )  /\  ( f `
 0 )  =  P ) )
cvmlift2.k  |-  K  =  ( x  e.  ( 0 [,] 1 ) ,  y  e.  ( 0 [,] 1 ) 
|->  ( ( iota_ f  e.  ( II  Cn  C
) ( ( F  o.  f )  =  ( z  e.  ( 0 [,] 1 ) 
|->  ( x G z ) )  /\  (
f `  0 )  =  ( H `  x ) ) ) `
 y ) )
cvmlift2.m  |-  M  =  { z  e.  ( ( 0 [,] 1
)  X.  ( 0 [,] 1 ) )  |  K  e.  ( ( ( II  tX  II )  CnP  C ) `
 z ) }
cvmlift2lem11.1  |-  ( ph  ->  U  e.  II )
cvmlift2lem11.2  |-  ( ph  ->  V  e.  II )
cvmlift2lem11.3  |-  ( ph  ->  Y  e.  V )
cvmlift2lem11.4  |-  ( ph  ->  Z  e.  V )
cvmlift2lem11.5  |-  ( ph  ->  ( E. w  e.  V  ( K  |`  ( U  X.  { w } ) )  e.  ( ( ( II 
tX  II )t  ( U  X.  { w }
) )  Cn  C
)  ->  ( K  |`  ( U  X.  V
) )  e.  ( ( ( II  tX  II )t  ( U  X.  V ) )  Cn  C ) ) )
Assertion
Ref Expression
cvmlift2lem11  |-  ( ph  ->  ( ( U  X.  { Y } )  C_  M  ->  ( U  X.  { Z } )  C_  M ) )
Distinct variable groups:    w, f, x, y, z, F    ph, f, w, x, y, z    x, M, y, z    f, J, w, x, y, z   
w, U, z    f, G, w, x, y, z   
w, V    f, H, w, x, y, z    z, Z    C, f, w, x, y, z    P, f, x, y, z    w, B, x, y, z    f, Y, w, x, y, z   
f, K, w, x, y, z
Allowed substitution hints:    B( f)    P( w)    U( x, y, f)    M( w, f)    V( x, y, z, f)    Z( x, y, w, f)

Proof of Theorem cvmlift2lem11
StepHypRef Expression
1 cvmlift2lem11.1 . . . . . . 7  |-  ( ph  ->  U  e.  II )
21adantr 465 . . . . . 6  |-  ( (
ph  /\  ( U  X.  { Y } ) 
C_  M )  ->  U  e.  II )
3 elssuni 4275 . . . . . . 7  |-  ( U  e.  II  ->  U  C_ 
U. II )
4 iiuni 21120 . . . . . . 7  |-  ( 0 [,] 1 )  = 
U. II
53, 4syl6sseqr 3551 . . . . . 6  |-  ( U  e.  II  ->  U  C_  ( 0 [,] 1
) )
62, 5syl 16 . . . . 5  |-  ( (
ph  /\  ( U  X.  { Y } ) 
C_  M )  ->  U  C_  ( 0 [,] 1 ) )
7 cvmlift2lem11.4 . . . . . . . 8  |-  ( ph  ->  Z  e.  V )
8 cvmlift2lem11.2 . . . . . . . 8  |-  ( ph  ->  V  e.  II )
9 elunii 4250 . . . . . . . . 9  |-  ( ( Z  e.  V  /\  V  e.  II )  ->  Z  e.  U. II )
109, 4syl6eleqr 2566 . . . . . . . 8  |-  ( ( Z  e.  V  /\  V  e.  II )  ->  Z  e.  ( 0 [,] 1 ) )
117, 8, 10syl2anc 661 . . . . . . 7  |-  ( ph  ->  Z  e.  ( 0 [,] 1 ) )
1211adantr 465 . . . . . 6  |-  ( (
ph  /\  ( U  X.  { Y } ) 
C_  M )  ->  Z  e.  ( 0 [,] 1 ) )
1312snssd 4172 . . . . 5  |-  ( (
ph  /\  ( U  X.  { Y } ) 
C_  M )  ->  { Z }  C_  (
0 [,] 1 ) )
14 xpss12 5106 . . . . 5  |-  ( ( U  C_  ( 0 [,] 1 )  /\  { Z }  C_  (
0 [,] 1 ) )  ->  ( U  X.  { Z } ) 
C_  ( ( 0 [,] 1 )  X.  ( 0 [,] 1
) ) )
156, 13, 14syl2anc 661 . . . 4  |-  ( (
ph  /\  ( U  X.  { Y } ) 
C_  M )  -> 
( U  X.  { Z } )  C_  (
( 0 [,] 1
)  X.  ( 0 [,] 1 ) ) )
16 cvmlift2lem11.3 . . . . . . . . . 10  |-  ( ph  ->  Y  e.  V )
1716adantr 465 . . . . . . . . 9  |-  ( (
ph  /\  ( U  X.  { Y } ) 
C_  M )  ->  Y  e.  V )
18 cvmlift2.b . . . . . . . . . . . . 13  |-  B  = 
U. C
19 cvmlift2.f . . . . . . . . . . . . 13  |-  ( ph  ->  F  e.  ( C CovMap  J ) )
20 cvmlift2.g . . . . . . . . . . . . 13  |-  ( ph  ->  G  e.  ( ( II  tX  II )  Cn  J ) )
21 cvmlift2.p . . . . . . . . . . . . 13  |-  ( ph  ->  P  e.  B )
22 cvmlift2.i . . . . . . . . . . . . 13  |-  ( ph  ->  ( F `  P
)  =  ( 0 G 0 ) )
23 cvmlift2.h . . . . . . . . . . . . 13  |-  H  =  ( iota_ f  e.  ( II  Cn  C ) ( ( F  o.  f )  =  ( z  e.  ( 0 [,] 1 )  |->  ( z G 0 ) )  /\  ( f `
 0 )  =  P ) )
24 cvmlift2.k . . . . . . . . . . . . 13  |-  K  =  ( x  e.  ( 0 [,] 1 ) ,  y  e.  ( 0 [,] 1 ) 
|->  ( ( iota_ f  e.  ( II  Cn  C
) ( ( F  o.  f )  =  ( z  e.  ( 0 [,] 1 ) 
|->  ( x G z ) )  /\  (
f `  0 )  =  ( H `  x ) ) ) `
 y ) )
2518, 19, 20, 21, 22, 23, 24cvmlift2lem5 28392 . . . . . . . . . . . 12  |-  ( ph  ->  K : ( ( 0 [,] 1 )  X.  ( 0 [,] 1 ) ) --> B )
2625adantr 465 . . . . . . . . . . 11  |-  ( (
ph  /\  ( U  X.  { Y } ) 
C_  M )  ->  K : ( ( 0 [,] 1 )  X.  ( 0 [,] 1
) ) --> B )
278adantr 465 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  ( U  X.  { Y } ) 
C_  M )  ->  V  e.  II )
28 elssuni 4275 . . . . . . . . . . . . . . . 16  |-  ( V  e.  II  ->  V  C_ 
U. II )
2928, 4syl6sseqr 3551 . . . . . . . . . . . . . . 15  |-  ( V  e.  II  ->  V  C_  ( 0 [,] 1
) )
3027, 29syl 16 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( U  X.  { Y } ) 
C_  M )  ->  V  C_  ( 0 [,] 1 ) )
3130, 17sseldd 3505 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( U  X.  { Y } ) 
C_  M )  ->  Y  e.  ( 0 [,] 1 ) )
3231snssd 4172 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( U  X.  { Y } ) 
C_  M )  ->  { Y }  C_  (
0 [,] 1 ) )
33 xpss12 5106 . . . . . . . . . . . 12  |-  ( ( U  C_  ( 0 [,] 1 )  /\  { Y }  C_  (
0 [,] 1 ) )  ->  ( U  X.  { Y } ) 
C_  ( ( 0 [,] 1 )  X.  ( 0 [,] 1
) ) )
346, 32, 33syl2anc 661 . . . . . . . . . . 11  |-  ( (
ph  /\  ( U  X.  { Y } ) 
C_  M )  -> 
( U  X.  { Y } )  C_  (
( 0 [,] 1
)  X.  ( 0 [,] 1 ) ) )
35 fssres 5749 . . . . . . . . . . 11  |-  ( ( K : ( ( 0 [,] 1 )  X.  ( 0 [,] 1 ) ) --> B  /\  ( U  X.  { Y } )  C_  ( ( 0 [,] 1 )  X.  (
0 [,] 1 ) ) )  ->  ( K  |`  ( U  X.  { Y } ) ) : ( U  X.  { Y } ) --> B )
3626, 34, 35syl2anc 661 . . . . . . . . . 10  |-  ( (
ph  /\  ( U  X.  { Y } ) 
C_  M )  -> 
( K  |`  ( U  X.  { Y }
) ) : ( U  X.  { Y } ) --> B )
3734adantr 465 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  ( U  X.  { Y }
)  C_  M )  /\  z  e.  ( U  X.  { Y }
) )  ->  ( U  X.  { Y }
)  C_  ( (
0 [,] 1 )  X.  ( 0 [,] 1 ) ) )
38 simpr 461 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  ( U  X.  { Y }
)  C_  M )  /\  z  e.  ( U  X.  { Y }
) )  ->  z  e.  ( U  X.  { Y } ) )
39 simpr 461 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  ( U  X.  { Y } ) 
C_  M )  -> 
( U  X.  { Y } )  C_  M
)
40 cvmlift2.m . . . . . . . . . . . . . . 15  |-  M  =  { z  e.  ( ( 0 [,] 1
)  X.  ( 0 [,] 1 ) )  |  K  e.  ( ( ( II  tX  II )  CnP  C ) `
 z ) }
4139, 40syl6sseq 3550 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( U  X.  { Y } ) 
C_  M )  -> 
( U  X.  { Y } )  C_  { z  e.  ( ( 0 [,] 1 )  X.  ( 0 [,] 1
) )  |  K  e.  ( ( ( II 
tX  II )  CnP 
C ) `  z
) } )
42 ssrab 3578 . . . . . . . . . . . . . . 15  |-  ( ( U  X.  { Y } )  C_  { z  e.  ( ( 0 [,] 1 )  X.  ( 0 [,] 1
) )  |  K  e.  ( ( ( II 
tX  II )  CnP 
C ) `  z
) }  <->  ( ( U  X.  { Y }
)  C_  ( (
0 [,] 1 )  X.  ( 0 [,] 1 ) )  /\  A. z  e.  ( U  X.  { Y }
) K  e.  ( ( ( II  tX  II )  CnP  C ) `
 z ) ) )
4342simprbi 464 . . . . . . . . . . . . . 14  |-  ( ( U  X.  { Y } )  C_  { z  e.  ( ( 0 [,] 1 )  X.  ( 0 [,] 1
) )  |  K  e.  ( ( ( II 
tX  II )  CnP 
C ) `  z
) }  ->  A. z  e.  ( U  X.  { Y } ) K  e.  ( ( ( II 
tX  II )  CnP 
C ) `  z
) )
4441, 43syl 16 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( U  X.  { Y } ) 
C_  M )  ->  A. z  e.  ( U  X.  { Y }
) K  e.  ( ( ( II  tX  II )  CnP  C ) `
 z ) )
4544r19.21bi 2833 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  ( U  X.  { Y }
)  C_  M )  /\  z  e.  ( U  X.  { Y }
) )  ->  K  e.  ( ( ( II 
tX  II )  CnP 
C ) `  z
) )
46 iitopon 21118 . . . . . . . . . . . . . . 15  |-  II  e.  (TopOn `  ( 0 [,] 1 ) )
47 txtopon 19827 . . . . . . . . . . . . . . 15  |-  ( ( II  e.  (TopOn `  ( 0 [,] 1
) )  /\  II  e.  (TopOn `  ( 0 [,] 1 ) ) )  ->  ( II  tX  II )  e.  (TopOn `  ( ( 0 [,] 1 )  X.  (
0 [,] 1 ) ) ) )
4846, 46, 47mp2an 672 . . . . . . . . . . . . . 14  |-  ( II 
tX  II )  e.  (TopOn `  ( (
0 [,] 1 )  X.  ( 0 [,] 1 ) ) )
4948toponunii 19200 . . . . . . . . . . . . 13  |-  ( ( 0 [,] 1 )  X.  ( 0 [,] 1 ) )  = 
U. ( II  tX  II )
5049cnpresti 19555 . . . . . . . . . . . 12  |-  ( ( ( U  X.  { Y } )  C_  (
( 0 [,] 1
)  X.  ( 0 [,] 1 ) )  /\  z  e.  ( U  X.  { Y } )  /\  K  e.  ( ( ( II 
tX  II )  CnP 
C ) `  z
) )  ->  ( K  |`  ( U  X.  { Y } ) )  e.  ( ( ( ( II  tX  II )t  ( U  X.  { Y } ) )  CnP 
C ) `  z
) )
5137, 38, 45, 50syl3anc 1228 . . . . . . . . . . 11  |-  ( ( ( ph  /\  ( U  X.  { Y }
)  C_  M )  /\  z  e.  ( U  X.  { Y }
) )  ->  ( K  |`  ( U  X.  { Y } ) )  e.  ( ( ( ( II  tX  II )t  ( U  X.  { Y } ) )  CnP 
C ) `  z
) )
5251ralrimiva 2878 . . . . . . . . . 10  |-  ( (
ph  /\  ( U  X.  { Y } ) 
C_  M )  ->  A. z  e.  ( U  X.  { Y }
) ( K  |`  ( U  X.  { Y } ) )  e.  ( ( ( ( II  tX  II )t  ( U  X.  { Y }
) )  CnP  C
) `  z )
)
53 resttopon 19428 . . . . . . . . . . . 12  |-  ( ( ( II  tX  II )  e.  (TopOn `  (
( 0 [,] 1
)  X.  ( 0 [,] 1 ) ) )  /\  ( U  X.  { Y }
)  C_  ( (
0 [,] 1 )  X.  ( 0 [,] 1 ) ) )  ->  ( ( II 
tX  II )t  ( U  X.  { Y }
) )  e.  (TopOn `  ( U  X.  { Y } ) ) )
5448, 34, 53sylancr 663 . . . . . . . . . . 11  |-  ( (
ph  /\  ( U  X.  { Y } ) 
C_  M )  -> 
( ( II  tX  II )t  ( U  X.  { Y } ) )  e.  (TopOn `  ( U  X.  { Y }
) ) )
55 cvmtop1 28345 . . . . . . . . . . . . . 14  |-  ( F  e.  ( C CovMap  J
)  ->  C  e.  Top )
5619, 55syl 16 . . . . . . . . . . . . 13  |-  ( ph  ->  C  e.  Top )
5756adantr 465 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( U  X.  { Y } ) 
C_  M )  ->  C  e.  Top )
5818toptopon 19201 . . . . . . . . . . . 12  |-  ( C  e.  Top  <->  C  e.  (TopOn `  B ) )
5957, 58sylib 196 . . . . . . . . . . 11  |-  ( (
ph  /\  ( U  X.  { Y } ) 
C_  M )  ->  C  e.  (TopOn `  B
) )
60 cncnp 19547 . . . . . . . . . . 11  |-  ( ( ( ( II  tX  II )t  ( U  X.  { Y } ) )  e.  (TopOn `  ( U  X.  { Y }
) )  /\  C  e.  (TopOn `  B )
)  ->  ( ( K  |`  ( U  X.  { Y } ) )  e.  ( ( ( II  tX  II )t  ( U  X.  { Y }
) )  Cn  C
)  <->  ( ( K  |`  ( U  X.  { Y } ) ) : ( U  X.  { Y } ) --> B  /\  A. z  e.  ( U  X.  { Y }
) ( K  |`  ( U  X.  { Y } ) )  e.  ( ( ( ( II  tX  II )t  ( U  X.  { Y }
) )  CnP  C
) `  z )
) ) )
6154, 59, 60syl2anc 661 . . . . . . . . . 10  |-  ( (
ph  /\  ( U  X.  { Y } ) 
C_  M )  -> 
( ( K  |`  ( U  X.  { Y } ) )  e.  ( ( ( II 
tX  II )t  ( U  X.  { Y }
) )  Cn  C
)  <->  ( ( K  |`  ( U  X.  { Y } ) ) : ( U  X.  { Y } ) --> B  /\  A. z  e.  ( U  X.  { Y }
) ( K  |`  ( U  X.  { Y } ) )  e.  ( ( ( ( II  tX  II )t  ( U  X.  { Y }
) )  CnP  C
) `  z )
) ) )
6236, 52, 61mpbir2and 920 . . . . . . . . 9  |-  ( (
ph  /\  ( U  X.  { Y } ) 
C_  M )  -> 
( K  |`  ( U  X.  { Y }
) )  e.  ( ( ( II  tX  II )t  ( U  X.  { Y } ) )  Cn  C ) )
63 sneq 4037 . . . . . . . . . . . . 13  |-  ( w  =  Y  ->  { w }  =  { Y } )
6463xpeq2d 5023 . . . . . . . . . . . 12  |-  ( w  =  Y  ->  ( U  X.  { w }
)  =  ( U  X.  { Y }
) )
6564reseq2d 5271 . . . . . . . . . . 11  |-  ( w  =  Y  ->  ( K  |`  ( U  X.  { w } ) )  =  ( K  |`  ( U  X.  { Y } ) ) )
6664oveq2d 6298 . . . . . . . . . . . 12  |-  ( w  =  Y  ->  (
( II  tX  II )t  ( U  X.  { w } ) )  =  ( ( II  tX  II )t  ( U  X.  { Y } ) ) )
6766oveq1d 6297 . . . . . . . . . . 11  |-  ( w  =  Y  ->  (
( ( II  tX  II )t  ( U  X.  { w } ) )  Cn  C )  =  ( ( ( II  tX  II )t  ( U  X.  { Y }
) )  Cn  C
) )
6865, 67eleq12d 2549 . . . . . . . . . 10  |-  ( w  =  Y  ->  (
( K  |`  ( U  X.  { w }
) )  e.  ( ( ( II  tX  II )t  ( U  X.  { w } ) )  Cn  C )  <-> 
( K  |`  ( U  X.  { Y }
) )  e.  ( ( ( II  tX  II )t  ( U  X.  { Y } ) )  Cn  C ) ) )
6968rspcev 3214 . . . . . . . . 9  |-  ( ( Y  e.  V  /\  ( K  |`  ( U  X.  { Y }
) )  e.  ( ( ( II  tX  II )t  ( U  X.  { Y } ) )  Cn  C ) )  ->  E. w  e.  V  ( K  |`  ( U  X.  { w }
) )  e.  ( ( ( II  tX  II )t  ( U  X.  { w } ) )  Cn  C ) )
7017, 62, 69syl2anc 661 . . . . . . . 8  |-  ( (
ph  /\  ( U  X.  { Y } ) 
C_  M )  ->  E. w  e.  V  ( K  |`  ( U  X.  { w }
) )  e.  ( ( ( II  tX  II )t  ( U  X.  { w } ) )  Cn  C ) )
71 cvmlift2lem11.5 . . . . . . . . 9  |-  ( ph  ->  ( E. w  e.  V  ( K  |`  ( U  X.  { w } ) )  e.  ( ( ( II 
tX  II )t  ( U  X.  { w }
) )  Cn  C
)  ->  ( K  |`  ( U  X.  V
) )  e.  ( ( ( II  tX  II )t  ( U  X.  V ) )  Cn  C ) ) )
7271imp 429 . . . . . . . 8  |-  ( (
ph  /\  E. w  e.  V  ( K  |`  ( U  X.  {
w } ) )  e.  ( ( ( II  tX  II )t  ( U  X.  { w }
) )  Cn  C
) )  ->  ( K  |`  ( U  X.  V ) )  e.  ( ( ( II 
tX  II )t  ( U  X.  V ) )  Cn  C ) )
7370, 72syldan 470 . . . . . . 7  |-  ( (
ph  /\  ( U  X.  { Y } ) 
C_  M )  -> 
( K  |`  ( U  X.  V ) )  e.  ( ( ( II  tX  II )t  ( U  X.  V ) )  Cn  C ) )
7473adantr 465 . . . . . 6  |-  ( ( ( ph  /\  ( U  X.  { Y }
)  C_  M )  /\  z  e.  ( U  X.  { Z }
) )  ->  ( K  |`  ( U  X.  V ) )  e.  ( ( ( II 
tX  II )t  ( U  X.  V ) )  Cn  C ) )
757adantr 465 . . . . . . . . . 10  |-  ( (
ph  /\  ( U  X.  { Y } ) 
C_  M )  ->  Z  e.  V )
7675snssd 4172 . . . . . . . . 9  |-  ( (
ph  /\  ( U  X.  { Y } ) 
C_  M )  ->  { Z }  C_  V
)
77 xpss2 5110 . . . . . . . . 9  |-  ( { Z }  C_  V  ->  ( U  X.  { Z } )  C_  ( U  X.  V ) )
7876, 77syl 16 . . . . . . . 8  |-  ( (
ph  /\  ( U  X.  { Y } ) 
C_  M )  -> 
( U  X.  { Z } )  C_  ( U  X.  V ) )
79 iitop 21119 . . . . . . . . . 10  |-  II  e.  Top
8079, 79txtopi 19826 . . . . . . . . 9  |-  ( II 
tX  II )  e. 
Top
81 xpss12 5106 . . . . . . . . . 10  |-  ( ( U  C_  ( 0 [,] 1 )  /\  V  C_  ( 0 [,] 1 ) )  -> 
( U  X.  V
)  C_  ( (
0 [,] 1 )  X.  ( 0 [,] 1 ) ) )
826, 30, 81syl2anc 661 . . . . . . . . 9  |-  ( (
ph  /\  ( U  X.  { Y } ) 
C_  M )  -> 
( U  X.  V
)  C_  ( (
0 [,] 1 )  X.  ( 0 [,] 1 ) ) )
8349restuni 19429 . . . . . . . . 9  |-  ( ( ( II  tX  II )  e.  Top  /\  ( U  X.  V )  C_  ( ( 0 [,] 1 )  X.  (
0 [,] 1 ) ) )  ->  ( U  X.  V )  = 
U. ( ( II 
tX  II )t  ( U  X.  V ) ) )
8480, 82, 83sylancr 663 . . . . . . . 8  |-  ( (
ph  /\  ( U  X.  { Y } ) 
C_  M )  -> 
( U  X.  V
)  =  U. (
( II  tX  II )t  ( U  X.  V
) ) )
8578, 84sseqtrd 3540 . . . . . . 7  |-  ( (
ph  /\  ( U  X.  { Y } ) 
C_  M )  -> 
( U  X.  { Z } )  C_  U. (
( II  tX  II )t  ( U  X.  V
) ) )
8685sselda 3504 . . . . . 6  |-  ( ( ( ph  /\  ( U  X.  { Y }
)  C_  M )  /\  z  e.  ( U  X.  { Z }
) )  ->  z  e.  U. ( ( II 
tX  II )t  ( U  X.  V ) ) )
87 eqid 2467 . . . . . . 7  |-  U. (
( II  tX  II )t  ( U  X.  V
) )  =  U. ( ( II  tX  II )t  ( U  X.  V ) )
8887cncnpi 19545 . . . . . 6  |-  ( ( ( K  |`  ( U  X.  V ) )  e.  ( ( ( II  tX  II )t  ( U  X.  V ) )  Cn  C )  /\  z  e.  U. (
( II  tX  II )t  ( U  X.  V
) ) )  -> 
( K  |`  ( U  X.  V ) )  e.  ( ( ( ( II  tX  II )t  ( U  X.  V
) )  CnP  C
) `  z )
)
8974, 86, 88syl2anc 661 . . . . 5  |-  ( ( ( ph  /\  ( U  X.  { Y }
)  C_  M )  /\  z  e.  ( U  X.  { Z }
) )  ->  ( K  |`  ( U  X.  V ) )  e.  ( ( ( ( II  tX  II )t  ( U  X.  V ) )  CnP  C ) `  z ) )
9080a1i 11 . . . . . 6  |-  ( ( ( ph  /\  ( U  X.  { Y }
)  C_  M )  /\  z  e.  ( U  X.  { Z }
) )  ->  (
II  tX  II )  e.  Top )
9182adantr 465 . . . . . 6  |-  ( ( ( ph  /\  ( U  X.  { Y }
)  C_  M )  /\  z  e.  ( U  X.  { Z }
) )  ->  ( U  X.  V )  C_  ( ( 0 [,] 1 )  X.  (
0 [,] 1 ) ) )
9279a1i 11 . . . . . . . . . 10  |-  ( (
ph  /\  ( U  X.  { Y } ) 
C_  M )  ->  II  e.  Top )
93 txopn 19838 . . . . . . . . . 10  |-  ( ( ( II  e.  Top  /\  II  e.  Top )  /\  ( U  e.  II  /\  V  e.  II ) )  ->  ( U  X.  V )  e.  ( II  tX  II ) )
9492, 92, 2, 27, 93syl22anc 1229 . . . . . . . . 9  |-  ( (
ph  /\  ( U  X.  { Y } ) 
C_  M )  -> 
( U  X.  V
)  e.  ( II 
tX  II ) )
95 isopn3i 19349 . . . . . . . . 9  |-  ( ( ( II  tX  II )  e.  Top  /\  ( U  X.  V )  e.  ( II  tX  II ) )  ->  (
( int `  (
II  tX  II )
) `  ( U  X.  V ) )  =  ( U  X.  V
) )
9680, 94, 95sylancr 663 . . . . . . . 8  |-  ( (
ph  /\  ( U  X.  { Y } ) 
C_  M )  -> 
( ( int `  (
II  tX  II )
) `  ( U  X.  V ) )  =  ( U  X.  V
) )
9778, 96sseqtr4d 3541 . . . . . . 7  |-  ( (
ph  /\  ( U  X.  { Y } ) 
C_  M )  -> 
( U  X.  { Z } )  C_  (
( int `  (
II  tX  II )
) `  ( U  X.  V ) ) )
9897sselda 3504 . . . . . 6  |-  ( ( ( ph  /\  ( U  X.  { Y }
)  C_  M )  /\  z  e.  ( U  X.  { Z }
) )  ->  z  e.  ( ( int `  (
II  tX  II )
) `  ( U  X.  V ) ) )
9925ad2antrr 725 . . . . . 6  |-  ( ( ( ph  /\  ( U  X.  { Y }
)  C_  M )  /\  z  e.  ( U  X.  { Z }
) )  ->  K : ( ( 0 [,] 1 )  X.  ( 0 [,] 1
) ) --> B )
10049, 18cnprest 19556 . . . . . 6  |-  ( ( ( ( II  tX  II )  e.  Top  /\  ( U  X.  V
)  C_  ( (
0 [,] 1 )  X.  ( 0 [,] 1 ) ) )  /\  ( z  e.  ( ( int `  (
II  tX  II )
) `  ( U  X.  V ) )  /\  K : ( ( 0 [,] 1 )  X.  ( 0 [,] 1
) ) --> B ) )  ->  ( K  e.  ( ( ( II 
tX  II )  CnP 
C ) `  z
)  <->  ( K  |`  ( U  X.  V
) )  e.  ( ( ( ( II 
tX  II )t  ( U  X.  V ) )  CnP  C ) `  z ) ) )
10190, 91, 98, 99, 100syl22anc 1229 . . . . 5  |-  ( ( ( ph  /\  ( U  X.  { Y }
)  C_  M )  /\  z  e.  ( U  X.  { Z }
) )  ->  ( K  e.  ( (
( II  tX  II )  CnP  C ) `  z )  <->  ( K  |`  ( U  X.  V
) )  e.  ( ( ( ( II 
tX  II )t  ( U  X.  V ) )  CnP  C ) `  z ) ) )
10289, 101mpbird 232 . . . 4  |-  ( ( ( ph  /\  ( U  X.  { Y }
)  C_  M )  /\  z  e.  ( U  X.  { Z }
) )  ->  K  e.  ( ( ( II 
tX  II )  CnP 
C ) `  z
) )
10315, 102ssrabdv 3579 . . 3  |-  ( (
ph  /\  ( U  X.  { Y } ) 
C_  M )  -> 
( U  X.  { Z } )  C_  { z  e.  ( ( 0 [,] 1 )  X.  ( 0 [,] 1
) )  |  K  e.  ( ( ( II 
tX  II )  CnP 
C ) `  z
) } )
104103, 40syl6sseqr 3551 . 2  |-  ( (
ph  /\  ( U  X.  { Y } ) 
C_  M )  -> 
( U  X.  { Z } )  C_  M
)
105104ex 434 1  |-  ( ph  ->  ( ( U  X.  { Y } )  C_  M  ->  ( U  X.  { Z } )  C_  M ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767   A.wral 2814   E.wrex 2815   {crab 2818    C_ wss 3476   {csn 4027   U.cuni 4245    |-> cmpt 4505    X. cxp 4997    |` cres 5001    o. ccom 5003   -->wf 5582   ` cfv 5586   iota_crio 6242  (class class class)co 6282    |-> cmpt2 6284   0cc0 9488   1c1 9489   [,]cicc 11528   ↾t crest 14672   Topctop 19161  TopOnctopon 19162   intcnt 19284    Cn ccn 19491    CnP ccnp 19492    tX ctx 19796   IIcii 21114   CovMap ccvm 28340
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-inf2 8054  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565  ax-pre-sup 9566  ax-addf 9567  ax-mulf 9568
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-iin 4328  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-isom 5595  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-of 6522  df-om 6679  df-1st 6781  df-2nd 6782  df-supp 6899  df-recs 7039  df-rdg 7073  df-1o 7127  df-2o 7128  df-oadd 7131  df-er 7308  df-ec 7310  df-map 7419  df-ixp 7467  df-en 7514  df-dom 7515  df-sdom 7516  df-fin 7517  df-fsupp 7826  df-fi 7867  df-sup 7897  df-oi 7931  df-card 8316  df-cda 8544  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-div 10203  df-nn 10533  df-2 10590  df-3 10591  df-4 10592  df-5 10593  df-6 10594  df-7 10595  df-8 10596  df-9 10597  df-10 10598  df-n0 10792  df-z 10861  df-dec 10973  df-uz 11079  df-q 11179  df-rp 11217  df-xneg 11314  df-xadd 11315  df-xmul 11316  df-ioo 11529  df-ico 11531  df-icc 11532  df-fz 11669  df-fzo 11789  df-fl 11893  df-seq 12072  df-exp 12131  df-hash 12370  df-cj 12891  df-re 12892  df-im 12893  df-sqrt 13027  df-abs 13028  df-clim 13270  df-sum 13468  df-struct 14488  df-ndx 14489  df-slot 14490  df-base 14491  df-sets 14492  df-ress 14493  df-plusg 14564  df-mulr 14565  df-starv 14566  df-sca 14567  df-vsca 14568  df-ip 14569  df-tset 14570  df-ple 14571  df-ds 14573  df-unif 14574  df-hom 14575  df-cco 14576  df-rest 14674  df-topn 14675  df-0g 14693  df-gsum 14694  df-topgen 14695  df-pt 14696  df-prds 14699  df-xrs 14753  df-qtop 14758  df-imas 14759  df-xps 14761  df-mre 14837  df-mrc 14838  df-acs 14840  df-mnd 15728  df-submnd 15778  df-mulg 15861  df-cntz 16150  df-cmn 16596  df-psmet 18182  df-xmet 18183  df-met 18184  df-bl 18185  df-mopn 18186  df-cnfld 18192  df-top 19166  df-bases 19168  df-topon 19169  df-topsp 19170  df-cld 19286  df-ntr 19287  df-cls 19288  df-nei 19365  df-cn 19494  df-cnp 19495  df-cmp 19653  df-con 19679  df-lly 19733  df-nlly 19734  df-tx 19798  df-hmeo 19991  df-xms 20558  df-ms 20559  df-tms 20560  df-ii 21116  df-htpy 21205  df-phtpy 21206  df-phtpc 21227  df-pcon 28306  df-scon 28307  df-cvm 28341
This theorem is referenced by:  cvmlift2lem12  28399
  Copyright terms: Public domain W3C validator