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Theorem cvmliftlem8 24980
Description: Lemma for cvmlift 24987. 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 11081 . . 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 24977 . . 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 462 . 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 453 . . . 4  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  F  e.  ( C CovMap  J ) )
18 cvmtop1 24948 . . . 4  |-  ( F  e.  ( C CovMap  J
)  ->  C  e.  Top )
19 cnrest2r 17352 . . . 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 19 . . 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 18798 . . . . . 6  |-  ( topGen ` 
ran  (,) )  e.  (TopOn `  RR )
2212, 21eqeltri 2507 . . . . 5  |-  L  e.  (TopOn `  RR )
23 simpr 449 . . . . . . 7  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  M  e.  ( 1 ... N
) )
242, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 23, 14cvmliftlem2 24974 . . . . . 6  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  W  C_  ( 0 [,] 1
) )
25 unitssre 11043 . . . . . 6  |-  ( 0 [,] 1 )  C_  RR
2624, 25syl6ss 3361 . . . . 5  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  W  C_  RR )
27 resttopon 17226 . . . . 5  |-  ( ( L  e.  (TopOn `  RR )  /\  W  C_  RR )  ->  ( Lt  W )  e.  (TopOn `  W ) )
2822, 26, 27sylancr 646 . . . 4  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  ( Lt  W )  e.  (TopOn `  W ) )
29 eqid 2437 . . . . . . 7  |-  ( IIt  W )  =  ( IIt  W )
30 iitopon 18910 . . . . . . . 8  |-  II  e.  (TopOn `  ( 0 [,] 1 ) )
3130a1i 11 . . . . . . 7  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  II  e.  (TopOn `  ( 0 [,] 1 ) ) )
326adantr 453 . . . . . . . . . 10  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  G  e.  ( II  Cn  J
) )
33 iiuni 18912 . . . . . . . . . . 11  |-  ( 0 [,] 1 )  = 
U. II
3433, 4cnf 17311 . . . . . . . . . 10  |-  ( G  e.  ( II  Cn  J )  ->  G : ( 0 [,] 1 ) --> X )
3532, 34syl 16 . . . . . . . . 9  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  G : ( 0 [,] 1 ) --> X )
3635feqmptd 5780 . . . . . . . 8  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  G  =  ( z  e.  ( 0 [,] 1
)  |->  ( G `  z ) ) )
3736, 32eqeltrrd 2512 . . . . . . 7  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  (
z  e.  ( 0 [,] 1 )  |->  ( G `  z ) )  e.  ( II 
Cn  J ) )
3829, 31, 24, 37cnmpt1res 17709 . . . . . 6  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  (
z  e.  W  |->  ( G `  z ) )  e.  ( ( IIt  W )  Cn  J
) )
39 dfii2 18913 . . . . . . . . . 10  |-  II  =  ( ( topGen `  ran  (,) )t  ( 0 [,] 1
) )
4012oveq1i 6092 . . . . . . . . . 10  |-  ( Lt  ( 0 [,] 1 ) )  =  ( (
topGen `  ran  (,) )t  (
0 [,] 1 ) )
4139, 40eqtr4i 2460 . . . . . . . . 9  |-  II  =  ( Lt  ( 0 [,] 1 ) )
4241oveq1i 6092 . . . . . . . 8  |-  ( IIt  W )  =  ( ( Lt  ( 0 [,] 1
) )t  W )
43 retop 18796 . . . . . . . . . . 11  |-  ( topGen ` 
ran  (,) )  e.  Top
4412, 43eqeltri 2507 . . . . . . . . . 10  |-  L  e. 
Top
4544a1i 11 . . . . . . . . 9  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  L  e.  Top )
46 ovex 6107 . . . . . . . . . 10  |-  ( 0 [,] 1 )  e. 
_V
4746a1i 11 . . . . . . . . 9  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  (
0 [,] 1 )  e.  _V )
48 restabs 17230 . . . . . . . . 9  |-  ( ( L  e.  Top  /\  W  C_  ( 0 [,] 1 )  /\  (
0 [,] 1 )  e.  _V )  -> 
( ( Lt  ( 0 [,] 1 ) )t  W )  =  ( Lt  W ) )
4945, 24, 47, 48syl3anc 1185 . . . . . . . 8  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  (
( Lt  ( 0 [,] 1 ) )t  W )  =  ( Lt  W ) )
5042, 49syl5eq 2481 . . . . . . 7  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  (
IIt 
W )  =  ( Lt  W ) )
5150oveq1d 6097 . . . . . 6  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  (
( IIt  W )  Cn  J
)  =  ( ( Lt  W )  Cn  J
) )
5238, 51eleqtrd 2513 . . . . 5  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  (
z  e.  W  |->  ( G `  z ) )  e.  ( ( Lt  W )  Cn  J
) )
53 cvmtop2 24949 . . . . . . . 8  |-  ( F  e.  ( C CovMap  J
)  ->  J  e.  Top )
5417, 53syl 16 . . . . . . 7  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  J  e.  Top )
554toptopon 16999 . . . . . . 7  |-  ( J  e.  Top  <->  J  e.  (TopOn `  X ) )
5654, 55sylib 190 . . . . . 6  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  J  e.  (TopOn `  X )
)
57 simprl 734 . . . . . . . . . 10  |-  ( (
ph  /\  ( M  e.  ( 1 ... N
)  /\  z  e.  W ) )  ->  M  e.  ( 1 ... N ) )
58 simprr 735 . . . . . . . . . 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 24975 . . . . . . . . 9  |-  ( (
ph  /\  ( M  e.  ( 1 ... N
)  /\  z  e.  W ) )  -> 
( G `  z
)  e.  ( 1st `  ( T `  M
) ) )
6059anassrs 631 . . . . . . . 8  |-  ( ( ( ph  /\  M  e.  ( 1 ... N
) )  /\  z  e.  W )  ->  ( G `  z )  e.  ( 1st `  ( T `  M )
) )
61 eqid 2437 . . . . . . . 8  |-  ( z  e.  W  |->  ( G `
 z ) )  =  ( z  e.  W  |->  ( G `  z ) )
6260, 61fmptd 5894 . . . . . . 7  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  (
z  e.  W  |->  ( G `  z ) ) : W --> ( 1st `  ( T `  M
) ) )
63 frn 5598 . . . . . . 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 24973 . . . . . . . 8  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  ( 2nd `  ( T `  M ) )  e.  ( S `  ( 1st `  ( T `  M ) ) ) )
662cvmsrcl 24952 . . . . . . . 8  |-  ( ( 2nd `  ( T `
 M ) )  e.  ( S `  ( 1st `  ( T `
 M ) ) )  ->  ( 1st `  ( T `  M
) )  e.  J
)
67 elssuni 4044 . . . . . . . 8  |-  ( ( 1st `  ( T `
 M ) )  e.  J  ->  ( 1st `  ( T `  M ) )  C_  U. J )
6865, 66, 673syl 19 . . . . . . 7  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  ( 1st `  ( T `  M ) )  C_  U. J )
6968, 4syl6sseqr 3396 . . . . . 6  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  ( 1st `  ( T `  M ) )  C_  X )
70 cnrest2 17351 . . . . . 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 1185 . . . . 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 203 . . . 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 24979 . . . . . . . . . 10  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  (
( Q `  ( M  -  1 ) ) `  ( ( M  -  1 )  /  N ) )  e.  ( `' F " { ( G `  ( ( M  - 
1 )  /  N
) ) } ) )
74 cvmcn 24950 . . . . . . . . . . . 12  |-  ( F  e.  ( C CovMap  J
)  ->  F  e.  ( C  Cn  J
) )
753, 4cnf 17311 . . . . . . . . . . . 12  |-  ( F  e.  ( C  Cn  J )  ->  F : B --> X )
7617, 74, 753syl 19 . . . . . . . . . . 11  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  F : B --> X )
77 ffn 5592 . . . . . . . . . . 11  |-  ( F : B --> X  ->  F  Fn  B )
78 fniniseg 5852 . . . . . . . . . . 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 19 . . . . . . . . . 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 203 . . . . . . . . 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 447 . . . . . . . 8  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  (
( Q `  ( M  -  1 ) ) `  ( ( M  -  1 )  /  N ) )  e.  B )
8280simprd 451 . . . . . . . . 9  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  ( F `  ( ( Q `  ( M  -  1 ) ) `
 ( ( M  -  1 )  /  N ) ) )  =  ( G `  ( ( M  - 
1 )  /  N
) ) )
831adantl 454 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  M  e.  NN )
8483nnred 10016 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  M  e.  RR )
85 peano2rem 9368 . . . . . . . . . . . . . . 15  |-  ( M  e.  RR  ->  ( M  -  1 )  e.  RR )
8684, 85syl 16 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  ( M  -  1 )  e.  RR )
879adantr 453 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  N  e.  NN )
8886, 87nndivred 10049 . . . . . . . . . . . . 13  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  (
( M  -  1 )  /  N )  e.  RR )
8988rexrd 9135 . . . . . . . . . . . 12  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  (
( M  -  1 )  /  N )  e.  RR* )
9084, 87nndivred 10049 . . . . . . . . . . . . 13  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  ( M  /  N )  e.  RR )
9190rexrd 9135 . . . . . . . . . . . 12  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  ( M  /  N )  e. 
RR* )
9284ltm1d 9944 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  ( M  -  1 )  <  M )
9387nnred 10016 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  N  e.  RR )
9487nngt0d 10044 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  0  <  N )
95 ltdiv1 9875 . . . . . . . . . . . . . . 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 1189 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  (
( M  -  1 )  <  M  <->  ( ( M  -  1 )  /  N )  < 
( M  /  N
) ) )
9792, 96mpbid 203 . . . . . . . . . . . . 13  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  (
( M  -  1 )  /  N )  <  ( M  /  N ) )
9888, 90, 97ltled 9222 . . . . . . . . . . . 12  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  (
( M  -  1 )  /  N )  <_  ( M  /  N ) )
99 lbicc2 11014 . . . . . . . . . . . 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 1185 . . . . . . . . . . 11  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  (
( M  -  1 )  /  N )  e.  ( ( ( M  -  1 )  /  N ) [,] ( M  /  N
) ) )
101100, 14syl6eleqr 2528 . . . . . . . . . 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 24975 . . . . . . . . 9  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  ( G `  ( ( M  -  1 )  /  N ) )  e.  ( 1st `  ( T `  M )
) )
10382, 102eqeltrd 2511 . . . . . . . 8  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  ( F `  ( ( Q `  ( M  -  1 ) ) `
 ( ( M  -  1 )  /  N ) ) )  e.  ( 1st `  ( T `  M )
) )
104 eqid 2437 . . . . . . . . 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 24965 . . . . . . . 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 1187 . . . . . . 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 447 . . . . . 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 24959 . . . . . 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 644 . . . . 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 17794 . . . . 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 17697 . . 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 3350 . 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 2511 1  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  ( Q `  M )  e.  ( ( Lt  W )  Cn  C ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    = wceq 1653    e. wcel 1726   A.wral 2706   {crab 2710   _Vcvv 2957    \ cdif 3318    u. cun 3319    i^i cin 3320    C_ wss 3321   (/)c0 3629   ~Pcpw 3800   {csn 3815   <.cop 3818   U.cuni 4016   U_ciun 4094   class class class wbr 4213    e. cmpt 4267    _I cid 4494    X. cxp 4877   `'ccnv 4878   ran crn 4880    |` cres 4881   "cima 4882    Fn wfn 5450   -->wf 5451   ` cfv 5455  (class class class)co 6082    e. cmpt2 6084   1stc1st 6348   2ndc2nd 6349   iota_crio 6543   RRcr 8990   0cc0 8991   1c1 8992   RR*cxr 9120    < clt 9121    <_ cle 9122    - cmin 9292    / cdiv 9678   NNcn 10001   (,)cioo 10917   [,]cicc 10920   ...cfz 11044    seq cseq 11324   ↾t crest 13649   topGenctg 13666   Topctop 16959  TopOnctopon 16960    Cn ccn 17289    Homeo chmeo 17786   IIcii 18906   CovMap ccvm 24943
This theorem is referenced by:  cvmliftlem10  24982
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-rep 4321  ax-sep 4331  ax-nul 4339  ax-pow 4378  ax-pr 4404  ax-un 4702  ax-cnex 9047  ax-resscn 9048  ax-1cn 9049  ax-icn 9050  ax-addcl 9051  ax-addrcl 9052  ax-mulcl 9053  ax-mulrcl 9054  ax-mulcom 9055  ax-addass 9056  ax-mulass 9057  ax-distr 9058  ax-i2m1 9059  ax-1ne0 9060  ax-1rid 9061  ax-rnegex 9062  ax-rrecex 9063  ax-cnre 9064  ax-pre-lttri 9065  ax-pre-lttrn 9066  ax-pre-ltadd 9067  ax-pre-mulgt0 9068  ax-pre-sup 9069
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-nel 2603  df-ral 2711  df-rex 2712  df-reu 2713  df-rmo 2714  df-rab 2715  df-v 2959  df-sbc 3163  df-csb 3253  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-pss 3337  df-nul 3630  df-if 3741  df-pw 3802  df-sn 3821  df-pr 3822  df-tp 3823  df-op 3824  df-uni 4017  df-int 4052  df-iun 4096  df-br 4214  df-opab 4268  df-mpt 4269  df-tr 4304  df-eprel 4495  df-id 4499  df-po 4504  df-so 4505  df-fr 4542  df-we 4544  df-ord 4585  df-on 4586  df-lim 4587  df-suc 4588  df-om 4847  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890  df-res 4891  df-ima 4892  df-iota 5419  df-fun 5457  df-fn 5458  df-f 5459  df-f1 5460  df-fo 5461  df-f1o 5462  df-fv 5463  df-ov 6085  df-oprab 6086  df-mpt2 6087  df-1st 6350  df-2nd 6351  df-riota 6550  df-recs 6634  df-rdg 6669  df-oadd 6729  df-er 6906  df-map 7021  df-en 7111  df-dom 7112  df-sdom 7113  df-fin 7114  df-fi 7417  df-sup 7447  df-pnf 9123  df-mnf 9124  df-xr 9125  df-ltxr 9126  df-le 9127  df-sub 9294  df-neg 9295  df-div 9679  df-nn 10002  df-2 10059  df-3 10060  df-n0 10223  df-z 10284  df-uz 10490  df-q 10576  df-rp 10614  df-xneg 10711  df-xadd 10712  df-xmul 10713  df-ioo 10921  df-icc 10924  df-fz 11045  df-seq 11325  df-exp 11384  df-cj 11905  df-re 11906  df-im 11907  df-sqr 12041  df-abs 12042  df-rest 13651  df-topgen 13668  df-psmet 16695  df-xmet 16696  df-met 16697  df-bl 16698  df-mopn 16699  df-top 16964  df-bases 16966  df-topon 16967  df-cn 17292  df-hmeo 17788  df-ii 18908  df-cvm 24944
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