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Theorem cvmliftlem8 23823
Description: Lemma for cvmlift 23830. 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 10819 . . 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 23820 . . 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 460 . 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 451 . . . 4  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  F  e.  ( C CovMap  J ) )
18 cvmtop1 23791 . . . 4  |-  ( F  e.  ( C CovMap  J
)  ->  C  e.  Top )
19 cnrest2r 17015 . . . 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 18 . . 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 18272 . . . . . 6  |-  ( topGen ` 
ran  (,) )  e.  (TopOn `  RR )
2212, 21eqeltri 2353 . . . . 5  |-  L  e.  (TopOn `  RR )
23 simpr 447 . . . . . . 7  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  M  e.  ( 1 ... N
) )
242, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 23, 14cvmliftlem2 23817 . . . . . 6  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  W  C_  ( 0 [,] 1
) )
25 0re 8838 . . . . . . 7  |-  0  e.  RR
26 1re 8837 . . . . . . 7  |-  1  e.  RR
27 iccssre 10731 . . . . . . 7  |-  ( ( 0  e.  RR  /\  1  e.  RR )  ->  ( 0 [,] 1
)  C_  RR )
2825, 26, 27mp2an 653 . . . . . 6  |-  ( 0 [,] 1 )  C_  RR
2924, 28syl6ss 3191 . . . . 5  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  W  C_  RR )
30 resttopon 16892 . . . . 5  |-  ( ( L  e.  (TopOn `  RR )  /\  W  C_  RR )  ->  ( Lt  W )  e.  (TopOn `  W ) )
3122, 29, 30sylancr 644 . . . 4  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  ( Lt  W )  e.  (TopOn `  W ) )
32 eqid 2283 . . . . . . 7  |-  ( IIt  W )  =  ( IIt  W )
33 iitopon 18383 . . . . . . . 8  |-  II  e.  (TopOn `  ( 0 [,] 1 ) )
3433a1i 10 . . . . . . 7  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  II  e.  (TopOn `  ( 0 [,] 1 ) ) )
356adantr 451 . . . . . . . . . 10  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  G  e.  ( II  Cn  J
) )
36 iiuni 18385 . . . . . . . . . . 11  |-  ( 0 [,] 1 )  = 
U. II
3736, 4cnf 16976 . . . . . . . . . 10  |-  ( G  e.  ( II  Cn  J )  ->  G : ( 0 [,] 1 ) --> X )
3835, 37syl 15 . . . . . . . . 9  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  G : ( 0 [,] 1 ) --> X )
3938feqmptd 5575 . . . . . . . 8  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  G  =  ( z  e.  ( 0 [,] 1
)  |->  ( G `  z ) ) )
4039, 35eqeltrrd 2358 . . . . . . 7  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  (
z  e.  ( 0 [,] 1 )  |->  ( G `  z ) )  e.  ( II 
Cn  J ) )
4132, 34, 24, 40cnmpt1res 17370 . . . . . 6  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  (
z  e.  W  |->  ( G `  z ) )  e.  ( ( IIt  W )  Cn  J
) )
42 dfii2 18386 . . . . . . . . . 10  |-  II  =  ( ( topGen `  ran  (,) )t  ( 0 [,] 1
) )
4312oveq1i 5868 . . . . . . . . . 10  |-  ( Lt  ( 0 [,] 1 ) )  =  ( (
topGen `  ran  (,) )t  (
0 [,] 1 ) )
4442, 43eqtr4i 2306 . . . . . . . . 9  |-  II  =  ( Lt  ( 0 [,] 1 ) )
4544oveq1i 5868 . . . . . . . 8  |-  ( IIt  W )  =  ( ( Lt  ( 0 [,] 1
) )t  W )
46 retop 18270 . . . . . . . . . . 11  |-  ( topGen ` 
ran  (,) )  e.  Top
4712, 46eqeltri 2353 . . . . . . . . . 10  |-  L  e. 
Top
4847a1i 10 . . . . . . . . 9  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  L  e.  Top )
49 ovex 5883 . . . . . . . . . 10  |-  ( 0 [,] 1 )  e. 
_V
5049a1i 10 . . . . . . . . 9  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  (
0 [,] 1 )  e.  _V )
51 restabs 16896 . . . . . . . . 9  |-  ( ( L  e.  Top  /\  W  C_  ( 0 [,] 1 )  /\  (
0 [,] 1 )  e.  _V )  -> 
( ( Lt  ( 0 [,] 1 ) )t  W )  =  ( Lt  W ) )
5248, 24, 50, 51syl3anc 1182 . . . . . . . 8  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  (
( Lt  ( 0 [,] 1 ) )t  W )  =  ( Lt  W ) )
5345, 52syl5eq 2327 . . . . . . 7  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  (
IIt 
W )  =  ( Lt  W ) )
5453oveq1d 5873 . . . . . 6  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  (
( IIt  W )  Cn  J
)  =  ( ( Lt  W )  Cn  J
) )
5541, 54eleqtrd 2359 . . . . 5  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  (
z  e.  W  |->  ( G `  z ) )  e.  ( ( Lt  W )  Cn  J
) )
56 cvmtop2 23792 . . . . . . . 8  |-  ( F  e.  ( C CovMap  J
)  ->  J  e.  Top )
5717, 56syl 15 . . . . . . 7  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  J  e.  Top )
584toptopon 16671 . . . . . . 7  |-  ( J  e.  Top  <->  J  e.  (TopOn `  X ) )
5957, 58sylib 188 . . . . . 6  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  J  e.  (TopOn `  X )
)
60 simprl 732 . . . . . . . . . 10  |-  ( (
ph  /\  ( M  e.  ( 1 ... N
)  /\  z  e.  W ) )  ->  M  e.  ( 1 ... N ) )
61 simprr 733 . . . . . . . . . 10  |-  ( (
ph  /\  ( M  e.  ( 1 ... N
)  /\  z  e.  W ) )  -> 
z  e.  W )
622, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 60, 14, 61cvmliftlem3 23818 . . . . . . . . 9  |-  ( (
ph  /\  ( M  e.  ( 1 ... N
)  /\  z  e.  W ) )  -> 
( G `  z
)  e.  ( 1st `  ( T `  M
) ) )
6362anassrs 629 . . . . . . . 8  |-  ( ( ( ph  /\  M  e.  ( 1 ... N
) )  /\  z  e.  W )  ->  ( G `  z )  e.  ( 1st `  ( T `  M )
) )
64 eqid 2283 . . . . . . . 8  |-  ( z  e.  W  |->  ( G `
 z ) )  =  ( z  e.  W  |->  ( G `  z ) )
6563, 64fmptd 5684 . . . . . . 7  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  (
z  e.  W  |->  ( G `  z ) ) : W --> ( 1st `  ( T `  M
) ) )
66 frn 5395 . . . . . . 7  |-  ( ( z  e.  W  |->  ( G `  z ) ) : W --> ( 1st `  ( T `  M
) )  ->  ran  ( z  e.  W  |->  ( G `  z
) )  C_  ( 1st `  ( T `  M ) ) )
6765, 66syl 15 . . . . . 6  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  ran  ( z  e.  W  |->  ( G `  z
) )  C_  ( 1st `  ( T `  M ) ) )
682, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 23cvmliftlem1 23816 . . . . . . . 8  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  ( 2nd `  ( T `  M ) )  e.  ( S `  ( 1st `  ( T `  M ) ) ) )
692cvmsrcl 23795 . . . . . . . 8  |-  ( ( 2nd `  ( T `
 M ) )  e.  ( S `  ( 1st `  ( T `
 M ) ) )  ->  ( 1st `  ( T `  M
) )  e.  J
)
70 elssuni 3855 . . . . . . . 8  |-  ( ( 1st `  ( T `
 M ) )  e.  J  ->  ( 1st `  ( T `  M ) )  C_  U. J )
7168, 69, 703syl 18 . . . . . . 7  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  ( 1st `  ( T `  M ) )  C_  U. J )
7271, 4syl6sseqr 3225 . . . . . 6  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  ( 1st `  ( T `  M ) )  C_  X )
73 cnrest2 17014 . . . . . 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 ) ) ) ) ) )
7459, 67, 72, 73syl3anc 1182 . . . . 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
) ) ) ) ) )
7555, 74mpbid 201 . . . 4  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  (
z  e.  W  |->  ( G `  z ) )  e.  ( ( Lt  W )  Cn  ( Jt  ( 1st `  ( T `
 M ) ) ) ) )
762, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14cvmliftlem7 23822 . . . . . . . . . 10  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  (
( Q `  ( M  -  1 ) ) `  ( ( M  -  1 )  /  N ) )  e.  ( `' F " { ( G `  ( ( M  - 
1 )  /  N
) ) } ) )
77 cvmcn 23793 . . . . . . . . . . . 12  |-  ( F  e.  ( C CovMap  J
)  ->  F  e.  ( C  Cn  J
) )
783, 4cnf 16976 . . . . . . . . . . . 12  |-  ( F  e.  ( C  Cn  J )  ->  F : B --> X )
7917, 77, 783syl 18 . . . . . . . . . . 11  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  F : B --> X )
80 ffn 5389 . . . . . . . . . . 11  |-  ( F : B --> X  ->  F  Fn  B )
81 fniniseg 5646 . . . . . . . . . . 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
) ) ) ) )
8279, 80, 813syl 18 . . . . . . . . . 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
) ) ) ) )
8376, 82mpbid 201 . . . . . . . . 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
) ) ) )
8483simpld 445 . . . . . . . 8  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  (
( Q `  ( M  -  1 ) ) `  ( ( M  -  1 )  /  N ) )  e.  B )
8583simprd 449 . . . . . . . . 9  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  ( F `  ( ( Q `  ( M  -  1 ) ) `
 ( ( M  -  1 )  /  N ) ) )  =  ( G `  ( ( M  - 
1 )  /  N
) ) )
861adantl 452 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  M  e.  NN )
8786nnred 9761 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  M  e.  RR )
88 peano2rem 9113 . . . . . . . . . . . . . . 15  |-  ( M  e.  RR  ->  ( M  -  1 )  e.  RR )
8987, 88syl 15 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  ( M  -  1 )  e.  RR )
909adantr 451 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  N  e.  NN )
9189, 90nndivred 9794 . . . . . . . . . . . . 13  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  (
( M  -  1 )  /  N )  e.  RR )
9291rexrd 8881 . . . . . . . . . . . 12  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  (
( M  -  1 )  /  N )  e.  RR* )
9387, 90nndivred 9794 . . . . . . . . . . . . 13  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  ( M  /  N )  e.  RR )
9493rexrd 8881 . . . . . . . . . . . 12  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  ( M  /  N )  e. 
RR* )
9587ltm1d 9689 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  ( M  -  1 )  <  M )
9690nnred 9761 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  N  e.  RR )
9790nngt0d 9789 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  0  <  N )
98 ltdiv1 9620 . . . . . . . . . . . . . . 15  |-  ( ( ( M  -  1 )  e.  RR  /\  M  e.  RR  /\  ( N  e.  RR  /\  0  <  N ) )  -> 
( ( M  - 
1 )  <  M  <->  ( ( M  -  1 )  /  N )  <  ( M  /  N ) ) )
9989, 87, 96, 97, 98syl112anc 1186 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  (
( M  -  1 )  <  M  <->  ( ( M  -  1 )  /  N )  < 
( M  /  N
) ) )
10095, 99mpbid 201 . . . . . . . . . . . . 13  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  (
( M  -  1 )  /  N )  <  ( M  /  N ) )
10191, 93, 100ltled 8967 . . . . . . . . . . . 12  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  (
( M  -  1 )  /  N )  <_  ( M  /  N ) )
102 lbicc2 10752 . . . . . . . . . . . 12  |-  ( ( ( ( M  - 
1 )  /  N
)  e.  RR*  /\  ( M  /  N )  e. 
RR*  /\  ( ( M  -  1 )  /  N )  <_ 
( M  /  N
) )  ->  (
( M  -  1 )  /  N )  e.  ( ( ( M  -  1 )  /  N ) [,] ( M  /  N
) ) )
10392, 94, 101, 102syl3anc 1182 . . . . . . . . . . 11  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  (
( M  -  1 )  /  N )  e.  ( ( ( M  -  1 )  /  N ) [,] ( M  /  N
) ) )
104103, 14syl6eleqr 2374 . . . . . . . . . 10  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  (
( M  -  1 )  /  N )  e.  W )
1052, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 23, 14, 104cvmliftlem3 23818 . . . . . . . . 9  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  ( G `  ( ( M  -  1 )  /  N ) )  e.  ( 1st `  ( T `  M )
) )
10685, 105eqeltrd 2357 . . . . . . . 8  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  ( F `  ( ( Q `  ( M  -  1 ) ) `
 ( ( M  -  1 )  /  N ) ) )  e.  ( 1st `  ( T `  M )
) )
107 eqid 2283 . . . . . . . . 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 )
1082, 3, 107cvmsiota 23808 . . . . . . . 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 ) ) )
10917, 68, 84, 106, 108syl13anc 1184 . . . . . . 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 ) ) )
110109simpld 445 . . . . . 6  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  ( iota_ b  e.  ( 2nd `  ( T `  M
) ) ( ( Q `  ( M  -  1 ) ) `
 ( ( M  -  1 )  /  N ) )  e.  b )  e.  ( 2nd `  ( T `
 M ) ) )
1112cvmshmeo 23802 . . . . . 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 ) ) ) ) )
11268, 110, 111syl2anc 642 . . . . 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 ) ) ) ) )
113 hmeocnvcn 17452 . . . . 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 ) ) ) )
114112, 113syl 15 . . . 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 ) ) ) )
11531, 75, 114cnmpt11f 17358 . . 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 ) ) ) )
11620, 115sseldd 3181 . 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
) )
11716, 116eqeltrd 2357 1  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  ( Q `  M )  e.  ( ( Lt  W )  Cn  C ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543   {crab 2547   _Vcvv 2788    \ cdif 3149    u. cun 3150    i^i cin 3151    C_ wss 3152   (/)c0 3455   ~Pcpw 3625   {csn 3640   <.cop 3643   U.cuni 3827   U_ciun 3905   class class class wbr 4023    e. cmpt 4077    _I cid 4304    X. cxp 4687   `'ccnv 4688   ran crn 4690    |` cres 4691   "cima 4692    Fn wfn 5250   -->wf 5251   ` cfv 5255  (class class class)co 5858    e. cmpt2 5860   1stc1st 6120   2ndc2nd 6121   iota_crio 6297   RRcr 8736   0cc0 8737   1c1 8738   RR*cxr 8866    < clt 8867    <_ cle 8868    - cmin 9037    / cdiv 9423   NNcn 9746   (,)cioo 10656   [,]cicc 10659   ...cfz 10782    seq cseq 11046   ↾t crest 13325   topGenctg 13342   Topctop 16631  TopOnctopon 16632    Cn ccn 16954    Homeo chmeo 17444   IIcii 18379   CovMap ccvm 23786
This theorem is referenced by:  cvmliftlem10  23825
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-oadd 6483  df-er 6660  df-map 6774  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-fi 7165  df-sup 7194  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-n0 9966  df-z 10025  df-uz 10231  df-q 10317  df-rp 10355  df-xneg 10452  df-xadd 10453  df-xmul 10454  df-ioo 10660  df-icc 10663  df-fz 10783  df-seq 11047  df-exp 11105  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-rest 13327  df-topgen 13344  df-xmet 16373  df-met 16374  df-bl 16375  df-mopn 16376  df-top 16636  df-bases 16638  df-topon 16639  df-cn 16957  df-hmeo 17446  df-ii 18381  df-cvm 23787
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