Users' Mathboxes Mathbox for Mario Carneiro < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  cvmliftlem8 Structured version   Unicode version

Theorem cvmliftlem8 28227
Description: Lemma for cvmlift 28234. The functions  Q are continuous functions because they are defined as  `' ( F  |`  I )  o.  G where  G is continuous and  ( F  |`  I ) is a homeomorphism. (Contributed by Mario Carneiro, 16-Feb-2015.)
Hypotheses
Ref Expression
cvmliftlem.1  |-  S  =  ( k  e.  J  |->  { s  e.  ( ~P C  \  { (/)
} )  |  ( U. s  =  ( `' F " k )  /\  A. u  e.  s  ( A. v  e.  ( s  \  {
u } ) ( u  i^i  v )  =  (/)  /\  ( F  |`  u )  e.  ( ( Ct  u )
Homeo ( Jt  k ) ) ) ) } )
cvmliftlem.b  |-  B  = 
U. C
cvmliftlem.x  |-  X  = 
U. J
cvmliftlem.f  |-  ( ph  ->  F  e.  ( C CovMap  J ) )
cvmliftlem.g  |-  ( ph  ->  G  e.  ( II 
Cn  J ) )
cvmliftlem.p  |-  ( ph  ->  P  e.  B )
cvmliftlem.e  |-  ( ph  ->  ( F `  P
)  =  ( G `
 0 ) )
cvmliftlem.n  |-  ( ph  ->  N  e.  NN )
cvmliftlem.t  |-  ( ph  ->  T : ( 1 ... N ) --> U_ j  e.  J  ( { j }  X.  ( S `  j ) ) )
cvmliftlem.a  |-  ( ph  ->  A. k  e.  ( 1 ... N ) ( G " (
( ( k  - 
1 )  /  N
) [,] ( k  /  N ) ) )  C_  ( 1st `  ( T `  k
) ) )
cvmliftlem.l  |-  L  =  ( topGen `  ran  (,) )
cvmliftlem.q  |-  Q  =  seq 0 ( ( x  e.  _V ,  m  e.  NN  |->  ( z  e.  ( ( ( m  -  1 )  /  N ) [,] ( m  /  N
) )  |->  ( `' ( F  |`  ( iota_ b  e.  ( 2nd `  ( T `  m
) ) ( x `
 ( ( m  -  1 )  /  N ) )  e.  b ) ) `  ( G `  z ) ) ) ) ,  ( (  _I  |`  NN )  u.  { <. 0 ,  { <. 0 ,  P >. } >. } ) )
cvmliftlem5.3  |-  W  =  ( ( ( M  -  1 )  /  N ) [,] ( M  /  N ) )
Assertion
Ref Expression
cvmliftlem8  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  ( Q `  M )  e.  ( ( Lt  W )  Cn  C ) )
Distinct variable groups:    v, b,
z, B    j, b,
k, m, s, u, x, F, v, z   
z, L    M, b,
j, k, m, s, u, v, x, z    P, b, k, m, u, v, x, z    C, b, j, k, s, u, v, z    ph, j,
s, x, z    N, b, k, m, u, v, x, z    S, b, j, k, s, u, v, x, z    j, X    G, b, j, k, m, s, u, v, x, z    T, b, j, k, m, s, u, v, x, z    J, b, j, k, s, u, v, x, z    Q, b, k, m, u, v, x, z    k, W, m, x, z
Allowed substitution hints:    ph( v, u, k, m, b)    B( x, u, j, k, m, s)    C( x, m)    P( j, s)    Q( j, s)    S( m)    J( m)    L( x, v, u, j, k, m, s, b)    N( j, s)    W( v, u, j, s, b)    X( x, z, v, u, k, m, s, b)

Proof of Theorem cvmliftlem8
StepHypRef Expression
1 elfznn 11703 . . 3  |-  ( M  e.  ( 1 ... N )  ->  M  e.  NN )
2 cvmliftlem.1 . . . 4  |-  S  =  ( k  e.  J  |->  { s  e.  ( ~P C  \  { (/)
} )  |  ( U. s  =  ( `' F " k )  /\  A. u  e.  s  ( A. v  e.  ( s  \  {
u } ) ( u  i^i  v )  =  (/)  /\  ( F  |`  u )  e.  ( ( Ct  u )
Homeo ( Jt  k ) ) ) ) } )
3 cvmliftlem.b . . . 4  |-  B  = 
U. C
4 cvmliftlem.x . . . 4  |-  X  = 
U. J
5 cvmliftlem.f . . . 4  |-  ( ph  ->  F  e.  ( C CovMap  J ) )
6 cvmliftlem.g . . . 4  |-  ( ph  ->  G  e.  ( II 
Cn  J ) )
7 cvmliftlem.p . . . 4  |-  ( ph  ->  P  e.  B )
8 cvmliftlem.e . . . 4  |-  ( ph  ->  ( F `  P
)  =  ( G `
 0 ) )
9 cvmliftlem.n . . . 4  |-  ( ph  ->  N  e.  NN )
10 cvmliftlem.t . . . 4  |-  ( ph  ->  T : ( 1 ... N ) --> U_ j  e.  J  ( { j }  X.  ( S `  j ) ) )
11 cvmliftlem.a . . . 4  |-  ( ph  ->  A. k  e.  ( 1 ... N ) ( G " (
( ( k  - 
1 )  /  N
) [,] ( k  /  N ) ) )  C_  ( 1st `  ( T `  k
) ) )
12 cvmliftlem.l . . . 4  |-  L  =  ( topGen `  ran  (,) )
13 cvmliftlem.q . . . 4  |-  Q  =  seq 0 ( ( x  e.  _V ,  m  e.  NN  |->  ( z  e.  ( ( ( m  -  1 )  /  N ) [,] ( m  /  N
) )  |->  ( `' ( F  |`  ( iota_ b  e.  ( 2nd `  ( T `  m
) ) ( x `
 ( ( m  -  1 )  /  N ) )  e.  b ) ) `  ( G `  z ) ) ) ) ,  ( (  _I  |`  NN )  u.  { <. 0 ,  { <. 0 ,  P >. } >. } ) )
14 cvmliftlem5.3 . . . 4  |-  W  =  ( ( ( M  -  1 )  /  N ) [,] ( M  /  N ) )
152, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14cvmliftlem5 28224 . . 3  |-  ( (
ph  /\  M  e.  NN )  ->  ( Q `
 M )  =  ( z  e.  W  |->  ( `' ( F  |`  ( iota_ b  e.  ( 2nd `  ( T `
 M ) ) ( ( Q `  ( M  -  1
) ) `  (
( M  -  1 )  /  N ) )  e.  b ) ) `  ( G `
 z ) ) ) )
161, 15sylan2 474 . 2  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  ( Q `  M )  =  ( z  e.  W  |->  ( `' ( F  |`  ( iota_ b  e.  ( 2nd `  ( T `  M )
) ( ( Q `
 ( M  - 
1 ) ) `  ( ( M  - 
1 )  /  N
) )  e.  b ) ) `  ( G `  z )
) ) )
175adantr 465 . . . 4  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  F  e.  ( C CovMap  J ) )
18 cvmtop1 28195 . . . 4  |-  ( F  e.  ( C CovMap  J
)  ->  C  e.  Top )
19 cnrest2r 19547 . . . 4  |-  ( C  e.  Top  ->  (
( Lt  W )  Cn  ( Ct  ( iota_ b  e.  ( 2nd `  ( T `
 M ) ) ( ( Q `  ( M  -  1
) ) `  (
( M  -  1 )  /  N ) )  e.  b ) ) )  C_  (
( Lt  W )  Cn  C
) )
2017, 18, 193syl 20 . . 3  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  (
( Lt  W )  Cn  ( Ct  ( iota_ b  e.  ( 2nd `  ( T `
 M ) ) ( ( Q `  ( M  -  1
) ) `  (
( M  -  1 )  /  N ) )  e.  b ) ) )  C_  (
( Lt  W )  Cn  C
) )
21 retopon 20998 . . . . . 6  |-  ( topGen ` 
ran  (,) )  e.  (TopOn `  RR )
2212, 21eqeltri 2544 . . . . 5  |-  L  e.  (TopOn `  RR )
23 simpr 461 . . . . . . 7  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  M  e.  ( 1 ... N
) )
242, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 23, 14cvmliftlem2 28221 . . . . . 6  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  W  C_  ( 0 [,] 1
) )
25 unitssre 11656 . . . . . 6  |-  ( 0 [,] 1 )  C_  RR
2624, 25syl6ss 3509 . . . . 5  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  W  C_  RR )
27 resttopon 19421 . . . . 5  |-  ( ( L  e.  (TopOn `  RR )  /\  W  C_  RR )  ->  ( Lt  W )  e.  (TopOn `  W ) )
2822, 26, 27sylancr 663 . . . 4  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  ( Lt  W )  e.  (TopOn `  W ) )
29 eqid 2460 . . . . . . 7  |-  ( IIt  W )  =  ( IIt  W )
30 iitopon 21111 . . . . . . . 8  |-  II  e.  (TopOn `  ( 0 [,] 1 ) )
3130a1i 11 . . . . . . 7  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  II  e.  (TopOn `  ( 0 [,] 1 ) ) )
326adantr 465 . . . . . . . . . 10  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  G  e.  ( II  Cn  J
) )
33 iiuni 21113 . . . . . . . . . . 11  |-  ( 0 [,] 1 )  = 
U. II
3433, 4cnf 19506 . . . . . . . . . 10  |-  ( G  e.  ( II  Cn  J )  ->  G : ( 0 [,] 1 ) --> X )
3532, 34syl 16 . . . . . . . . 9  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  G : ( 0 [,] 1 ) --> X )
3635feqmptd 5911 . . . . . . . 8  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  G  =  ( z  e.  ( 0 [,] 1
)  |->  ( G `  z ) ) )
3736, 32eqeltrrd 2549 . . . . . . 7  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  (
z  e.  ( 0 [,] 1 )  |->  ( G `  z ) )  e.  ( II 
Cn  J ) )
3829, 31, 24, 37cnmpt1res 19905 . . . . . 6  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  (
z  e.  W  |->  ( G `  z ) )  e.  ( ( IIt  W )  Cn  J
) )
39 dfii2 21114 . . . . . . . . . 10  |-  II  =  ( ( topGen `  ran  (,) )t  ( 0 [,] 1
) )
4012oveq1i 6285 . . . . . . . . . 10  |-  ( Lt  ( 0 [,] 1 ) )  =  ( (
topGen `  ran  (,) )t  (
0 [,] 1 ) )
4139, 40eqtr4i 2492 . . . . . . . . 9  |-  II  =  ( Lt  ( 0 [,] 1 ) )
4241oveq1i 6285 . . . . . . . 8  |-  ( IIt  W )  =  ( ( Lt  ( 0 [,] 1
) )t  W )
43 retop 20996 . . . . . . . . . . 11  |-  ( topGen ` 
ran  (,) )  e.  Top
4412, 43eqeltri 2544 . . . . . . . . . 10  |-  L  e. 
Top
4544a1i 11 . . . . . . . . 9  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  L  e.  Top )
46 ovex 6300 . . . . . . . . . 10  |-  ( 0 [,] 1 )  e. 
_V
4746a1i 11 . . . . . . . . 9  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  (
0 [,] 1 )  e.  _V )
48 restabs 19425 . . . . . . . . 9  |-  ( ( L  e.  Top  /\  W  C_  ( 0 [,] 1 )  /\  (
0 [,] 1 )  e.  _V )  -> 
( ( Lt  ( 0 [,] 1 ) )t  W )  =  ( Lt  W ) )
4945, 24, 47, 48syl3anc 1223 . . . . . . . 8  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  (
( Lt  ( 0 [,] 1 ) )t  W )  =  ( Lt  W ) )
5042, 49syl5eq 2513 . . . . . . 7  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  (
IIt 
W )  =  ( Lt  W ) )
5150oveq1d 6290 . . . . . 6  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  (
( IIt  W )  Cn  J
)  =  ( ( Lt  W )  Cn  J
) )
5238, 51eleqtrd 2550 . . . . 5  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  (
z  e.  W  |->  ( G `  z ) )  e.  ( ( Lt  W )  Cn  J
) )
53 cvmtop2 28196 . . . . . . . 8  |-  ( F  e.  ( C CovMap  J
)  ->  J  e.  Top )
5417, 53syl 16 . . . . . . 7  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  J  e.  Top )
554toptopon 19194 . . . . . . 7  |-  ( J  e.  Top  <->  J  e.  (TopOn `  X ) )
5654, 55sylib 196 . . . . . 6  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  J  e.  (TopOn `  X )
)
57 simprl 755 . . . . . . . . . 10  |-  ( (
ph  /\  ( M  e.  ( 1 ... N
)  /\  z  e.  W ) )  ->  M  e.  ( 1 ... N ) )
58 simprr 756 . . . . . . . . . 10  |-  ( (
ph  /\  ( M  e.  ( 1 ... N
)  /\  z  e.  W ) )  -> 
z  e.  W )
592, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 57, 14, 58cvmliftlem3 28222 . . . . . . . . 9  |-  ( (
ph  /\  ( M  e.  ( 1 ... N
)  /\  z  e.  W ) )  -> 
( G `  z
)  e.  ( 1st `  ( T `  M
) ) )
6059anassrs 648 . . . . . . . 8  |-  ( ( ( ph  /\  M  e.  ( 1 ... N
) )  /\  z  e.  W )  ->  ( G `  z )  e.  ( 1st `  ( T `  M )
) )
61 eqid 2460 . . . . . . . 8  |-  ( z  e.  W  |->  ( G `
 z ) )  =  ( z  e.  W  |->  ( G `  z ) )
6260, 61fmptd 6036 . . . . . . 7  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  (
z  e.  W  |->  ( G `  z ) ) : W --> ( 1st `  ( T `  M
) ) )
63 frn 5728 . . . . . . 7  |-  ( ( z  e.  W  |->  ( G `  z ) ) : W --> ( 1st `  ( T `  M
) )  ->  ran  ( z  e.  W  |->  ( G `  z
) )  C_  ( 1st `  ( T `  M ) ) )
6462, 63syl 16 . . . . . 6  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  ran  ( z  e.  W  |->  ( G `  z
) )  C_  ( 1st `  ( T `  M ) ) )
652, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 23cvmliftlem1 28220 . . . . . . . 8  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  ( 2nd `  ( T `  M ) )  e.  ( S `  ( 1st `  ( T `  M ) ) ) )
662cvmsrcl 28199 . . . . . . . 8  |-  ( ( 2nd `  ( T `
 M ) )  e.  ( S `  ( 1st `  ( T `
 M ) ) )  ->  ( 1st `  ( T `  M
) )  e.  J
)
67 elssuni 4268 . . . . . . . 8  |-  ( ( 1st `  ( T `
 M ) )  e.  J  ->  ( 1st `  ( T `  M ) )  C_  U. J )
6865, 66, 673syl 20 . . . . . . 7  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  ( 1st `  ( T `  M ) )  C_  U. J )
6968, 4syl6sseqr 3544 . . . . . 6  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  ( 1st `  ( T `  M ) )  C_  X )
70 cnrest2 19546 . . . . . 6  |-  ( ( J  e.  (TopOn `  X )  /\  ran  ( z  e.  W  |->  ( G `  z
) )  C_  ( 1st `  ( T `  M ) )  /\  ( 1st `  ( T `
 M ) ) 
C_  X )  -> 
( ( z  e.  W  |->  ( G `  z ) )  e.  ( ( Lt  W )  Cn  J )  <->  ( z  e.  W  |->  ( G `
 z ) )  e.  ( ( Lt  W )  Cn  ( Jt  ( 1st `  ( T `
 M ) ) ) ) ) )
7156, 64, 69, 70syl3anc 1223 . . . . 5  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  (
( z  e.  W  |->  ( G `  z
) )  e.  ( ( Lt  W )  Cn  J
)  <->  ( z  e.  W  |->  ( G `  z ) )  e.  ( ( Lt  W )  Cn  ( Jt  ( 1st `  ( T `  M
) ) ) ) ) )
7252, 71mpbid 210 . . . 4  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  (
z  e.  W  |->  ( G `  z ) )  e.  ( ( Lt  W )  Cn  ( Jt  ( 1st `  ( T `
 M ) ) ) ) )
732, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14cvmliftlem7 28226 . . . . . . . . . 10  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  (
( Q `  ( M  -  1 ) ) `  ( ( M  -  1 )  /  N ) )  e.  ( `' F " { ( G `  ( ( M  - 
1 )  /  N
) ) } ) )
74 cvmcn 28197 . . . . . . . . . . . 12  |-  ( F  e.  ( C CovMap  J
)  ->  F  e.  ( C  Cn  J
) )
753, 4cnf 19506 . . . . . . . . . . . 12  |-  ( F  e.  ( C  Cn  J )  ->  F : B --> X )
7617, 74, 753syl 20 . . . . . . . . . . 11  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  F : B --> X )
77 ffn 5722 . . . . . . . . . . 11  |-  ( F : B --> X  ->  F  Fn  B )
78 fniniseg 5993 . . . . . . . . . . 11  |-  ( F  Fn  B  ->  (
( ( Q `  ( M  -  1
) ) `  (
( M  -  1 )  /  N ) )  e.  ( `' F " { ( G `  ( ( M  -  1 )  /  N ) ) } )  <->  ( (
( Q `  ( M  -  1 ) ) `  ( ( M  -  1 )  /  N ) )  e.  B  /\  ( F `  ( ( Q `  ( M  -  1 ) ) `
 ( ( M  -  1 )  /  N ) ) )  =  ( G `  ( ( M  - 
1 )  /  N
) ) ) ) )
7976, 77, 783syl 20 . . . . . . . . . 10  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  (
( ( Q `  ( M  -  1
) ) `  (
( M  -  1 )  /  N ) )  e.  ( `' F " { ( G `  ( ( M  -  1 )  /  N ) ) } )  <->  ( (
( Q `  ( M  -  1 ) ) `  ( ( M  -  1 )  /  N ) )  e.  B  /\  ( F `  ( ( Q `  ( M  -  1 ) ) `
 ( ( M  -  1 )  /  N ) ) )  =  ( G `  ( ( M  - 
1 )  /  N
) ) ) ) )
8073, 79mpbid 210 . . . . . . . . 9  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  (
( ( Q `  ( M  -  1
) ) `  (
( M  -  1 )  /  N ) )  e.  B  /\  ( F `  ( ( Q `  ( M  -  1 ) ) `
 ( ( M  -  1 )  /  N ) ) )  =  ( G `  ( ( M  - 
1 )  /  N
) ) ) )
8180simpld 459 . . . . . . . 8  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  (
( Q `  ( M  -  1 ) ) `  ( ( M  -  1 )  /  N ) )  e.  B )
8280simprd 463 . . . . . . . . 9  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  ( F `  ( ( Q `  ( M  -  1 ) ) `
 ( ( M  -  1 )  /  N ) ) )  =  ( G `  ( ( M  - 
1 )  /  N
) ) )
831adantl 466 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  M  e.  NN )
8483nnred 10540 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  M  e.  RR )
85 peano2rem 9875 . . . . . . . . . . . . . . 15  |-  ( M  e.  RR  ->  ( M  -  1 )  e.  RR )
8684, 85syl 16 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  ( M  -  1 )  e.  RR )
879adantr 465 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  N  e.  NN )
8886, 87nndivred 10573 . . . . . . . . . . . . 13  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  (
( M  -  1 )  /  N )  e.  RR )
8988rexrd 9632 . . . . . . . . . . . 12  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  (
( M  -  1 )  /  N )  e.  RR* )
9084, 87nndivred 10573 . . . . . . . . . . . . 13  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  ( M  /  N )  e.  RR )
9190rexrd 9632 . . . . . . . . . . . 12  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  ( M  /  N )  e. 
RR* )
9284ltm1d 10467 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  ( M  -  1 )  <  M )
9387nnred 10540 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  N  e.  RR )
9487nngt0d 10568 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  0  <  N )
95 ltdiv1 10395 . . . . . . . . . . . . . . 15  |-  ( ( ( M  -  1 )  e.  RR  /\  M  e.  RR  /\  ( N  e.  RR  /\  0  <  N ) )  -> 
( ( M  - 
1 )  <  M  <->  ( ( M  -  1 )  /  N )  <  ( M  /  N ) ) )
9686, 84, 93, 94, 95syl112anc 1227 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  (
( M  -  1 )  <  M  <->  ( ( M  -  1 )  /  N )  < 
( M  /  N
) ) )
9792, 96mpbid 210 . . . . . . . . . . . . 13  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  (
( M  -  1 )  /  N )  <  ( M  /  N ) )
9888, 90, 97ltled 9721 . . . . . . . . . . . 12  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  (
( M  -  1 )  /  N )  <_  ( M  /  N ) )
99 lbicc2 11625 . . . . . . . . . . . 12  |-  ( ( ( ( M  - 
1 )  /  N
)  e.  RR*  /\  ( M  /  N )  e. 
RR*  /\  ( ( M  -  1 )  /  N )  <_ 
( M  /  N
) )  ->  (
( M  -  1 )  /  N )  e.  ( ( ( M  -  1 )  /  N ) [,] ( M  /  N
) ) )
10089, 91, 98, 99syl3anc 1223 . . . . . . . . . . 11  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  (
( M  -  1 )  /  N )  e.  ( ( ( M  -  1 )  /  N ) [,] ( M  /  N
) ) )
101100, 14syl6eleqr 2559 . . . . . . . . . 10  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  (
( M  -  1 )  /  N )  e.  W )
1022, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 23, 14, 101cvmliftlem3 28222 . . . . . . . . 9  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  ( G `  ( ( M  -  1 )  /  N ) )  e.  ( 1st `  ( T `  M )
) )
10382, 102eqeltrd 2548 . . . . . . . 8  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  ( F `  ( ( Q `  ( M  -  1 ) ) `
 ( ( M  -  1 )  /  N ) ) )  e.  ( 1st `  ( T `  M )
) )
104 eqid 2460 . . . . . . . . 9  |-  ( iota_ b  e.  ( 2nd `  ( T `  M )
) ( ( Q `
 ( M  - 
1 ) ) `  ( ( M  - 
1 )  /  N
) )  e.  b )  =  ( iota_ b  e.  ( 2nd `  ( T `  M )
) ( ( Q `
 ( M  - 
1 ) ) `  ( ( M  - 
1 )  /  N
) )  e.  b )
1052, 3, 104cvmsiota 28212 . . . . . . . 8  |-  ( ( F  e.  ( C CovMap  J )  /\  (
( 2nd `  ( T `  M )
)  e.  ( S `
 ( 1st `  ( T `  M )
) )  /\  (
( Q `  ( M  -  1 ) ) `  ( ( M  -  1 )  /  N ) )  e.  B  /\  ( F `  ( ( Q `  ( M  -  1 ) ) `
 ( ( M  -  1 )  /  N ) ) )  e.  ( 1st `  ( T `  M )
) ) )  -> 
( ( iota_ b  e.  ( 2nd `  ( T `  M )
) ( ( Q `
 ( M  - 
1 ) ) `  ( ( M  - 
1 )  /  N
) )  e.  b )  e.  ( 2nd `  ( T `  M
) )  /\  (
( Q `  ( M  -  1 ) ) `  ( ( M  -  1 )  /  N ) )  e.  ( iota_ b  e.  ( 2nd `  ( T `  M )
) ( ( Q `
 ( M  - 
1 ) ) `  ( ( M  - 
1 )  /  N
) )  e.  b ) ) )
10617, 65, 81, 103, 105syl13anc 1225 . . . . . . 7  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  (
( iota_ b  e.  ( 2nd `  ( T `
 M ) ) ( ( Q `  ( M  -  1
) ) `  (
( M  -  1 )  /  N ) )  e.  b )  e.  ( 2nd `  ( T `  M )
)  /\  ( ( Q `  ( M  -  1 ) ) `
 ( ( M  -  1 )  /  N ) )  e.  ( iota_ b  e.  ( 2nd `  ( T `
 M ) ) ( ( Q `  ( M  -  1
) ) `  (
( M  -  1 )  /  N ) )  e.  b ) ) )
107106simpld 459 . . . . . 6  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  ( iota_ b  e.  ( 2nd `  ( T `  M
) ) ( ( Q `  ( M  -  1 ) ) `
 ( ( M  -  1 )  /  N ) )  e.  b )  e.  ( 2nd `  ( T `
 M ) ) )
1082cvmshmeo 28206 . . . . . 6  |-  ( ( ( 2nd `  ( T `  M )
)  e.  ( S `
 ( 1st `  ( T `  M )
) )  /\  ( iota_ b  e.  ( 2nd `  ( T `  M
) ) ( ( Q `  ( M  -  1 ) ) `
 ( ( M  -  1 )  /  N ) )  e.  b )  e.  ( 2nd `  ( T `
 M ) ) )  ->  ( F  |`  ( iota_ b  e.  ( 2nd `  ( T `
 M ) ) ( ( Q `  ( M  -  1
) ) `  (
( M  -  1 )  /  N ) )  e.  b ) )  e.  ( ( Ct  ( iota_ b  e.  ( 2nd `  ( T `
 M ) ) ( ( Q `  ( M  -  1
) ) `  (
( M  -  1 )  /  N ) )  e.  b ) ) Homeo ( Jt  ( 1st `  ( T `  M
) ) ) ) )
10965, 107, 108syl2anc 661 . . . . 5  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  ( F  |`  ( iota_ b  e.  ( 2nd `  ( T `  M )
) ( ( Q `
 ( M  - 
1 ) ) `  ( ( M  - 
1 )  /  N
) )  e.  b ) )  e.  ( ( Ct  ( iota_ b  e.  ( 2nd `  ( T `  M )
) ( ( Q `
 ( M  - 
1 ) ) `  ( ( M  - 
1 )  /  N
) )  e.  b ) ) Homeo ( Jt  ( 1st `  ( T `
 M ) ) ) ) )
110 hmeocnvcn 19990 . . . . 5  |-  ( ( F  |`  ( iota_ b  e.  ( 2nd `  ( T `  M )
) ( ( Q `
 ( M  - 
1 ) ) `  ( ( M  - 
1 )  /  N
) )  e.  b ) )  e.  ( ( Ct  ( iota_ b  e.  ( 2nd `  ( T `  M )
) ( ( Q `
 ( M  - 
1 ) ) `  ( ( M  - 
1 )  /  N
) )  e.  b ) ) Homeo ( Jt  ( 1st `  ( T `
 M ) ) ) )  ->  `' ( F  |`  ( iota_ b  e.  ( 2nd `  ( T `  M )
) ( ( Q `
 ( M  - 
1 ) ) `  ( ( M  - 
1 )  /  N
) )  e.  b ) )  e.  ( ( Jt  ( 1st `  ( T `  M )
) )  Cn  ( Ct  ( iota_ b  e.  ( 2nd `  ( T `
 M ) ) ( ( Q `  ( M  -  1
) ) `  (
( M  -  1 )  /  N ) )  e.  b ) ) ) )
111109, 110syl 16 . . . 4  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  `' ( F  |`  ( iota_ b  e.  ( 2nd `  ( T `  M )
) ( ( Q `
 ( M  - 
1 ) ) `  ( ( M  - 
1 )  /  N
) )  e.  b ) )  e.  ( ( Jt  ( 1st `  ( T `  M )
) )  Cn  ( Ct  ( iota_ b  e.  ( 2nd `  ( T `
 M ) ) ( ( Q `  ( M  -  1
) ) `  (
( M  -  1 )  /  N ) )  e.  b ) ) ) )
11228, 72, 111cnmpt11f 19893 . . 3  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  (
z  e.  W  |->  ( `' ( F  |`  ( iota_ b  e.  ( 2nd `  ( T `
 M ) ) ( ( Q `  ( M  -  1
) ) `  (
( M  -  1 )  /  N ) )  e.  b ) ) `  ( G `
 z ) ) )  e.  ( ( Lt  W )  Cn  ( Ct  ( iota_ b  e.  ( 2nd `  ( T `
 M ) ) ( ( Q `  ( M  -  1
) ) `  (
( M  -  1 )  /  N ) )  e.  b ) ) ) )
11320, 112sseldd 3498 . 2  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  (
z  e.  W  |->  ( `' ( F  |`  ( iota_ b  e.  ( 2nd `  ( T `
 M ) ) ( ( Q `  ( M  -  1
) ) `  (
( M  -  1 )  /  N ) )  e.  b ) ) `  ( G `
 z ) ) )  e.  ( ( Lt  W )  Cn  C
) )
11416, 113eqeltrd 2548 1  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  ( Q `  M )  e.  ( ( Lt  W )  Cn  C ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1374    e. wcel 1762   A.wral 2807   {crab 2811   _Vcvv 3106    \ cdif 3466    u. cun 3467    i^i cin 3468    C_ wss 3469   (/)c0 3778   ~Pcpw 4003   {csn 4020   <.cop 4026   U.cuni 4238   U_ciun 4318   class class class wbr 4440    |-> cmpt 4498    _I cid 4783    X. cxp 4990   `'ccnv 4991   ran crn 4993    |` cres 4994   "cima 4995    Fn wfn 5574   -->wf 5575   ` cfv 5579   iota_crio 6235  (class class class)co 6275    |-> cmpt2 6277   1stc1st 6772   2ndc2nd 6773   RRcr 9480   0cc0 9481   1c1 9482   RR*cxr 9616    < clt 9617    <_ cle 9618    - cmin 9794    / cdiv 10195   NNcn 10525   (,)cioo 11518   [,]cicc 11521   ...cfz 11661    seqcseq 12063   ↾t crest 14665   topGenctg 14682   Topctop 19154  TopOnctopon 19155    Cn ccn 19484   Homeochmeo 19982   IIcii 21107   CovMap ccvm 28190
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-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
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  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-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-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-riota 6236  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-om 6672  df-1st 6774  df-2nd 6775  df-recs 7032  df-rdg 7066  df-oadd 7124  df-er 7301  df-map 7412  df-en 7507  df-dom 7508  df-sdom 7509  df-fin 7510  df-fi 7860  df-sup 7890  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-n0 10785  df-z 10854  df-uz 11072  df-q 11172  df-rp 11210  df-xneg 11307  df-xadd 11308  df-xmul 11309  df-ioo 11522  df-icc 11525  df-fz 11662  df-seq 12064  df-exp 12123  df-cj 12882  df-re 12883  df-im 12884  df-sqr 13018  df-abs 13019  df-rest 14667  df-topgen 14688  df-psmet 18175  df-xmet 18176  df-met 18177  df-bl 18178  df-mopn 18179  df-top 19159  df-bases 19161  df-topon 19162  df-cn 19487  df-hmeo 19984  df-ii 21109  df-cvm 28191
This theorem is referenced by:  cvmliftlem10  28229
  Copyright terms: Public domain W3C validator