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

Theorem cvmliftlem8 27029
Description: Lemma for cvmlift 27036. 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 11465 . . 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 27026 . . 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 471 . 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 462 . . . 4  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  F  e.  ( C CovMap  J ) )
18 cvmtop1 26997 . . . 4  |-  ( F  e.  ( C CovMap  J
)  ->  C  e.  Top )
19 cnrest2r 18733 . . . 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 20184 . . . . . 6  |-  ( topGen ` 
ran  (,) )  e.  (TopOn `  RR )
2212, 21eqeltri 2503 . . . . 5  |-  L  e.  (TopOn `  RR )
23 simpr 458 . . . . . . 7  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  M  e.  ( 1 ... N
) )
242, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 23, 14cvmliftlem2 27023 . . . . . 6  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  W  C_  ( 0 [,] 1
) )
25 unitssre 11419 . . . . . 6  |-  ( 0 [,] 1 )  C_  RR
2624, 25syl6ss 3356 . . . . 5  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  W  C_  RR )
27 resttopon 18607 . . . . 5  |-  ( ( L  e.  (TopOn `  RR )  /\  W  C_  RR )  ->  ( Lt  W )  e.  (TopOn `  W ) )
2822, 26, 27sylancr 656 . . . 4  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  ( Lt  W )  e.  (TopOn `  W ) )
29 eqid 2433 . . . . . . 7  |-  ( IIt  W )  =  ( IIt  W )
30 iitopon 20297 . . . . . . . 8  |-  II  e.  (TopOn `  ( 0 [,] 1 ) )
3130a1i 11 . . . . . . 7  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  II  e.  (TopOn `  ( 0 [,] 1 ) ) )
326adantr 462 . . . . . . . . . 10  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  G  e.  ( II  Cn  J
) )
33 iiuni 20299 . . . . . . . . . . 11  |-  ( 0 [,] 1 )  = 
U. II
3433, 4cnf 18692 . . . . . . . . . 10  |-  ( G  e.  ( II  Cn  J )  ->  G : ( 0 [,] 1 ) --> X )
3532, 34syl 16 . . . . . . . . 9  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  G : ( 0 [,] 1 ) --> X )
3635feqmptd 5732 . . . . . . . 8  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  G  =  ( z  e.  ( 0 [,] 1
)  |->  ( G `  z ) ) )
3736, 32eqeltrrd 2508 . . . . . . 7  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  (
z  e.  ( 0 [,] 1 )  |->  ( G `  z ) )  e.  ( II 
Cn  J ) )
3829, 31, 24, 37cnmpt1res 19091 . . . . . 6  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  (
z  e.  W  |->  ( G `  z ) )  e.  ( ( IIt  W )  Cn  J
) )
39 dfii2 20300 . . . . . . . . . 10  |-  II  =  ( ( topGen `  ran  (,) )t  ( 0 [,] 1
) )
4012oveq1i 6090 . . . . . . . . . 10  |-  ( Lt  ( 0 [,] 1 ) )  =  ( (
topGen `  ran  (,) )t  (
0 [,] 1 ) )
4139, 40eqtr4i 2456 . . . . . . . . 9  |-  II  =  ( Lt  ( 0 [,] 1 ) )
4241oveq1i 6090 . . . . . . . 8  |-  ( IIt  W )  =  ( ( Lt  ( 0 [,] 1
) )t  W )
43 retop 20182 . . . . . . . . . . 11  |-  ( topGen ` 
ran  (,) )  e.  Top
4412, 43eqeltri 2503 . . . . . . . . . 10  |-  L  e. 
Top
4544a1i 11 . . . . . . . . 9  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  L  e.  Top )
46 ovex 6105 . . . . . . . . . 10  |-  ( 0 [,] 1 )  e. 
_V
4746a1i 11 . . . . . . . . 9  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  (
0 [,] 1 )  e.  _V )
48 restabs 18611 . . . . . . . . 9  |-  ( ( L  e.  Top  /\  W  C_  ( 0 [,] 1 )  /\  (
0 [,] 1 )  e.  _V )  -> 
( ( Lt  ( 0 [,] 1 ) )t  W )  =  ( Lt  W ) )
4945, 24, 47, 48syl3anc 1211 . . . . . . . 8  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  (
( Lt  ( 0 [,] 1 ) )t  W )  =  ( Lt  W ) )
5042, 49syl5eq 2477 . . . . . . 7  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  (
IIt 
W )  =  ( Lt  W ) )
5150oveq1d 6095 . . . . . 6  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  (
( IIt  W )  Cn  J
)  =  ( ( Lt  W )  Cn  J
) )
5238, 51eleqtrd 2509 . . . . 5  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  (
z  e.  W  |->  ( G `  z ) )  e.  ( ( Lt  W )  Cn  J
) )
53 cvmtop2 26998 . . . . . . . 8  |-  ( F  e.  ( C CovMap  J
)  ->  J  e.  Top )
5417, 53syl 16 . . . . . . 7  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  J  e.  Top )
554toptopon 18380 . . . . . . 7  |-  ( J  e.  Top  <->  J  e.  (TopOn `  X ) )
5654, 55sylib 196 . . . . . 6  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  J  e.  (TopOn `  X )
)
57 simprl 748 . . . . . . . . . 10  |-  ( (
ph  /\  ( M  e.  ( 1 ... N
)  /\  z  e.  W ) )  ->  M  e.  ( 1 ... N ) )
58 simprr 749 . . . . . . . . . 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 27024 . . . . . . . . 9  |-  ( (
ph  /\  ( M  e.  ( 1 ... N
)  /\  z  e.  W ) )  -> 
( G `  z
)  e.  ( 1st `  ( T `  M
) ) )
6059anassrs 641 . . . . . . . 8  |-  ( ( ( ph  /\  M  e.  ( 1 ... N
) )  /\  z  e.  W )  ->  ( G `  z )  e.  ( 1st `  ( T `  M )
) )
61 eqid 2433 . . . . . . . 8  |-  ( z  e.  W  |->  ( G `
 z ) )  =  ( z  e.  W  |->  ( G `  z ) )
6260, 61fmptd 5855 . . . . . . 7  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  (
z  e.  W  |->  ( G `  z ) ) : W --> ( 1st `  ( T `  M
) ) )
63 frn 5553 . . . . . . 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 27022 . . . . . . . 8  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  ( 2nd `  ( T `  M ) )  e.  ( S `  ( 1st `  ( T `  M ) ) ) )
662cvmsrcl 27001 . . . . . . . 8  |-  ( ( 2nd `  ( T `
 M ) )  e.  ( S `  ( 1st `  ( T `
 M ) ) )  ->  ( 1st `  ( T `  M
) )  e.  J
)
67 elssuni 4109 . . . . . . . 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 3391 . . . . . 6  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  ( 1st `  ( T `  M ) )  C_  X )
70 cnrest2 18732 . . . . . 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 1211 . . . . 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 27028 . . . . . . . . . 10  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  (
( Q `  ( M  -  1 ) ) `  ( ( M  -  1 )  /  N ) )  e.  ( `' F " { ( G `  ( ( M  - 
1 )  /  N
) ) } ) )
74 cvmcn 26999 . . . . . . . . . . . 12  |-  ( F  e.  ( C CovMap  J
)  ->  F  e.  ( C  Cn  J
) )
753, 4cnf 18692 . . . . . . . . . . . 12  |-  ( F  e.  ( C  Cn  J )  ->  F : B --> X )
7617, 74, 753syl 20 . . . . . . . . . . 11  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  F : B --> X )
77 ffn 5547 . . . . . . . . . . 11  |-  ( F : B --> X  ->  F  Fn  B )
78 fniniseg 5812 . . . . . . . . . . 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 456 . . . . . . . 8  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  (
( Q `  ( M  -  1 ) ) `  ( ( M  -  1 )  /  N ) )  e.  B )
8280simprd 460 . . . . . . . . 9  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  ( F `  ( ( Q `  ( M  -  1 ) ) `
 ( ( M  -  1 )  /  N ) ) )  =  ( G `  ( ( M  - 
1 )  /  N
) ) )
831adantl 463 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  M  e.  NN )
8483nnred 10325 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  M  e.  RR )
85 peano2rem 9663 . . . . . . . . . . . . . . 15  |-  ( M  e.  RR  ->  ( M  -  1 )  e.  RR )
8684, 85syl 16 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  ( M  -  1 )  e.  RR )
879adantr 462 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  N  e.  NN )
8886, 87nndivred 10358 . . . . . . . . . . . . 13  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  (
( M  -  1 )  /  N )  e.  RR )
8988rexrd 9421 . . . . . . . . . . . 12  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  (
( M  -  1 )  /  N )  e.  RR* )
9084, 87nndivred 10358 . . . . . . . . . . . . 13  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  ( M  /  N )  e.  RR )
9190rexrd 9421 . . . . . . . . . . . 12  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  ( M  /  N )  e. 
RR* )
9284ltm1d 10253 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  ( M  -  1 )  <  M )
9387nnred 10325 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  N  e.  RR )
9487nngt0d 10353 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  0  <  N )
95 ltdiv1 10181 . . . . . . . . . . . . . . 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 1215 . . . . . . . . . . . . . 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 9510 . . . . . . . . . . . 12  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  (
( M  -  1 )  /  N )  <_  ( M  /  N ) )
99 lbicc2 11388 . . . . . . . . . . . 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 1211 . . . . . . . . . . 11  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  (
( M  -  1 )  /  N )  e.  ( ( ( M  -  1 )  /  N ) [,] ( M  /  N
) ) )
101100, 14syl6eleqr 2524 . . . . . . . . . 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 27024 . . . . . . . . 9  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  ( G `  ( ( M  -  1 )  /  N ) )  e.  ( 1st `  ( T `  M )
) )
10382, 102eqeltrd 2507 . . . . . . . 8  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  ( F `  ( ( Q `  ( M  -  1 ) ) `
 ( ( M  -  1 )  /  N ) ) )  e.  ( 1st `  ( T `  M )
) )
104 eqid 2433 . . . . . . . . 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 27014 . . . . . . . 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 1213 . . . . . . 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 456 . . . . . 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 27008 . . . . . 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 654 . . . . 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 19176 . . . . 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 19079 . . 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 3345 . 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 2507 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 1362    e. wcel 1755   A.wral 2705   {crab 2709   _Vcvv 2962    \ cdif 3313    u. cun 3314    i^i cin 3315    C_ wss 3316   (/)c0 3625   ~Pcpw 3848   {csn 3865   <.cop 3871   U.cuni 4079   U_ciun 4159   class class class wbr 4280    e. cmpt 4338    _I cid 4618    X. cxp 4825   `'ccnv 4826   ran crn 4828    |` cres 4829   "cima 4830    Fn wfn 5401   -->wf 5402   ` cfv 5406   iota_crio 6038  (class class class)co 6080    e. cmpt2 6082   1stc1st 6564   2ndc2nd 6565   RRcr 9269   0cc0 9270   1c1 9271   RR*cxr 9405    < clt 9406    <_ cle 9407    - cmin 9583    / cdiv 9981   NNcn 10310   (,)cioo 11288   [,]cicc 11291   ...cfz 11424    seqcseq 11790   ↾t crest 14342   topGenctg 14359   Topctop 18340  TopOnctopon 18341    Cn ccn 18670   Homeochmeo 19168   IIcii 20293   CovMap ccvm 26992
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-8 1757  ax-9 1759  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414  ax-rep 4391  ax-sep 4401  ax-nul 4409  ax-pow 4458  ax-pr 4519  ax-un 6361  ax-cnex 9326  ax-resscn 9327  ax-1cn 9328  ax-icn 9329  ax-addcl 9330  ax-addrcl 9331  ax-mulcl 9332  ax-mulrcl 9333  ax-mulcom 9334  ax-addass 9335  ax-mulass 9336  ax-distr 9337  ax-i2m1 9338  ax-1ne0 9339  ax-1rid 9340  ax-rnegex 9341  ax-rrecex 9342  ax-cnre 9343  ax-pre-lttri 9344  ax-pre-lttrn 9345  ax-pre-ltadd 9346  ax-pre-mulgt0 9347  ax-pre-sup 9348
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 959  df-3an 960  df-tru 1365  df-ex 1590  df-nf 1593  df-sb 1700  df-eu 2258  df-mo 2259  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-ne 2598  df-nel 2599  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2964  df-sbc 3176  df-csb 3277  df-dif 3319  df-un 3321  df-in 3323  df-ss 3330  df-pss 3332  df-nul 3626  df-if 3780  df-pw 3850  df-sn 3866  df-pr 3868  df-tp 3870  df-op 3872  df-uni 4080  df-int 4117  df-iun 4161  df-br 4281  df-opab 4339  df-mpt 4340  df-tr 4374  df-eprel 4619  df-id 4623  df-po 4628  df-so 4629  df-fr 4666  df-we 4668  df-ord 4709  df-on 4710  df-lim 4711  df-suc 4712  df-xp 4833  df-rel 4834  df-cnv 4835  df-co 4836  df-dm 4837  df-rn 4838  df-res 4839  df-ima 4840  df-iota 5369  df-fun 5408  df-fn 5409  df-f 5410  df-f1 5411  df-fo 5412  df-f1o 5413  df-fv 5414  df-riota 6039  df-ov 6083  df-oprab 6084  df-mpt2 6085  df-om 6466  df-1st 6566  df-2nd 6567  df-recs 6818  df-rdg 6852  df-oadd 6912  df-er 7089  df-map 7204  df-en 7299  df-dom 7300  df-sdom 7301  df-fin 7302  df-fi 7649  df-sup 7679  df-pnf 9408  df-mnf 9409  df-xr 9410  df-ltxr 9411  df-le 9412  df-sub 9585  df-neg 9586  df-div 9982  df-nn 10311  df-2 10368  df-3 10369  df-n0 10568  df-z 10635  df-uz 10850  df-q 10942  df-rp 10980  df-xneg 11077  df-xadd 11078  df-xmul 11079  df-ioo 11292  df-icc 11295  df-fz 11425  df-seq 11791  df-exp 11850  df-cj 12572  df-re 12573  df-im 12574  df-sqr 12708  df-abs 12709  df-rest 14344  df-topgen 14365  df-psmet 17653  df-xmet 17654  df-met 17655  df-bl 17656  df-mopn 17657  df-top 18345  df-bases 18347  df-topon 18348  df-cn 18673  df-hmeo 19170  df-ii 20295  df-cvm 26993
This theorem is referenced by:  cvmliftlem10  27031
  Copyright terms: Public domain W3C validator