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Theorem cvmliftlem9 28378
Description: Lemma for cvmlift 28384. The  Q ( M ) functions are defined on almost disjoint intervals, but they overlap at the edges. Here we show that at these points the  Q functions agree on their common domain. (Contributed by Mario Carneiro, 14-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 >. } >. } ) )
Assertion
Ref Expression
cvmliftlem9  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  (
( Q `  M
) `  ( ( M  -  1 )  /  N ) )  =  ( ( Q `
 ( M  - 
1 ) ) `  ( ( M  - 
1 )  /  N
) ) )
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
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)    X( x, z, v, u, k, m, s, b)

Proof of Theorem cvmliftlem9
StepHypRef Expression
1 elfznn 11710 . . . 4  |-  ( M  e.  ( 1 ... N )  ->  M  e.  NN )
2 cvmliftlem.1 . . . . 5  |-  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 . . . . 5  |-  B  = 
U. C
4 cvmliftlem.x . . . . 5  |-  X  = 
U. J
5 cvmliftlem.f . . . . 5  |-  ( ph  ->  F  e.  ( C CovMap  J ) )
6 cvmliftlem.g . . . . 5  |-  ( ph  ->  G  e.  ( II 
Cn  J ) )
7 cvmliftlem.p . . . . 5  |-  ( ph  ->  P  e.  B )
8 cvmliftlem.e . . . . 5  |-  ( ph  ->  ( F `  P
)  =  ( G `
 0 ) )
9 cvmliftlem.n . . . . 5  |-  ( ph  ->  N  e.  NN )
10 cvmliftlem.t . . . . 5  |-  ( ph  ->  T : ( 1 ... N ) --> U_ j  e.  J  ( { j }  X.  ( S `  j ) ) )
11 cvmliftlem.a . . . . 5  |-  ( ph  ->  A. k  e.  ( 1 ... N ) ( G " (
( ( k  - 
1 )  /  N
) [,] ( k  /  N ) ) )  C_  ( 1st `  ( T `  k
) ) )
12 cvmliftlem.l . . . . 5  |-  L  =  ( topGen `  ran  (,) )
13 cvmliftlem.q . . . . 5  |-  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 eqid 2467 . . . . 5  |-  ( ( ( M  -  1 )  /  N ) [,] ( M  /  N ) )  =  ( ( ( M  -  1 )  /  N ) [,] ( M  /  N ) )
152, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14cvmliftlem5 28374 . . . 4  |-  ( (
ph  /\  M  e.  NN )  ->  ( Q `
 M )  =  ( z  e.  ( ( ( M  - 
1 )  /  N
) [,] ( M  /  N ) ) 
|->  ( `' ( F  |`  ( iota_ b  e.  ( 2nd `  ( T `
 M ) ) ( ( Q `  ( M  -  1
) ) `  (
( M  -  1 )  /  N ) )  e.  b ) ) `  ( G `
 z ) ) ) )
161, 15sylan2 474 . . 3  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  ( Q `  M )  =  ( z  e.  ( ( ( M  -  1 )  /  N ) [,] ( M  /  N ) ) 
|->  ( `' ( F  |`  ( iota_ b  e.  ( 2nd `  ( T `
 M ) ) ( ( Q `  ( M  -  1
) ) `  (
( M  -  1 )  /  N ) )  e.  b ) ) `  ( G `
 z ) ) ) )
17 simpr 461 . . . . 5  |-  ( ( ( ph  /\  M  e.  ( 1 ... N
) )  /\  z  =  ( ( M  -  1 )  /  N ) )  -> 
z  =  ( ( M  -  1 )  /  N ) )
1817fveq2d 5868 . . . 4  |-  ( ( ( ph  /\  M  e.  ( 1 ... N
) )  /\  z  =  ( ( M  -  1 )  /  N ) )  -> 
( G `  z
)  =  ( G `
 ( ( M  -  1 )  /  N ) ) )
1918fveq2d 5868 . . 3  |-  ( ( ( ph  /\  M  e.  ( 1 ... N
) )  /\  z  =  ( ( M  -  1 )  /  N ) )  -> 
( `' ( F  |`  ( iota_ b  e.  ( 2nd `  ( T `
 M ) ) ( ( Q `  ( M  -  1
) ) `  (
( M  -  1 )  /  N ) )  e.  b ) ) `  ( G `
 z ) )  =  ( `' ( F  |`  ( iota_ b  e.  ( 2nd `  ( T `  M )
) ( ( Q `
 ( M  - 
1 ) ) `  ( ( M  - 
1 )  /  N
) )  e.  b ) ) `  ( G `  ( ( M  -  1 )  /  N ) ) ) )
201adantl 466 . . . . . . . 8  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  M  e.  NN )
2120nnred 10547 . . . . . . 7  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  M  e.  RR )
22 peano2rem 9882 . . . . . . 7  |-  ( M  e.  RR  ->  ( M  -  1 )  e.  RR )
2321, 22syl 16 . . . . . 6  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  ( M  -  1 )  e.  RR )
249adantr 465 . . . . . 6  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  N  e.  NN )
2523, 24nndivred 10580 . . . . 5  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  (
( M  -  1 )  /  N )  e.  RR )
2625rexrd 9639 . . . 4  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  (
( M  -  1 )  /  N )  e.  RR* )
2721, 24nndivred 10580 . . . . 5  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  ( M  /  N )  e.  RR )
2827rexrd 9639 . . . 4  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  ( M  /  N )  e. 
RR* )
2921ltm1d 10474 . . . . . 6  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  ( M  -  1 )  <  M )
3024nnred 10547 . . . . . . 7  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  N  e.  RR )
3124nngt0d 10575 . . . . . . 7  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  0  <  N )
32 ltdiv1 10402 . . . . . . 7  |-  ( ( ( M  -  1 )  e.  RR  /\  M  e.  RR  /\  ( N  e.  RR  /\  0  <  N ) )  -> 
( ( M  - 
1 )  <  M  <->  ( ( M  -  1 )  /  N )  <  ( M  /  N ) ) )
3323, 21, 30, 31, 32syl112anc 1232 . . . . . 6  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  (
( M  -  1 )  <  M  <->  ( ( M  -  1 )  /  N )  < 
( M  /  N
) ) )
3429, 33mpbid 210 . . . . 5  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  (
( M  -  1 )  /  N )  <  ( M  /  N ) )
3525, 27, 34ltled 9728 . . . 4  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  (
( M  -  1 )  /  N )  <_  ( M  /  N ) )
36 lbicc2 11632 . . . 4  |-  ( ( ( ( M  - 
1 )  /  N
)  e.  RR*  /\  ( M  /  N )  e. 
RR*  /\  ( ( M  -  1 )  /  N )  <_ 
( M  /  N
) )  ->  (
( M  -  1 )  /  N )  e.  ( ( ( M  -  1 )  /  N ) [,] ( M  /  N
) ) )
3726, 28, 35, 36syl3anc 1228 . . 3  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  (
( M  -  1 )  /  N )  e.  ( ( ( M  -  1 )  /  N ) [,] ( M  /  N
) ) )
38 fvex 5874 . . . 4  |-  ( `' ( F  |`  ( iota_ b  e.  ( 2nd `  ( T `  M
) ) ( ( Q `  ( M  -  1 ) ) `
 ( ( M  -  1 )  /  N ) )  e.  b ) ) `  ( G `  ( ( M  -  1 )  /  N ) ) )  e.  _V
3938a1i 11 . . 3  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  ( `' ( F  |`  ( iota_ b  e.  ( 2nd `  ( T `
 M ) ) ( ( Q `  ( M  -  1
) ) `  (
( M  -  1 )  /  N ) )  e.  b ) ) `  ( G `
 ( ( M  -  1 )  /  N ) ) )  e.  _V )
4016, 19, 37, 39fvmptd 5953 . 2  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  (
( Q `  M
) `  ( ( M  -  1 )  /  N ) )  =  ( `' ( F  |`  ( iota_ b  e.  ( 2nd `  ( T `  M )
) ( ( Q `
 ( M  - 
1 ) ) `  ( ( M  - 
1 )  /  N
) )  e.  b ) ) `  ( G `  ( ( M  -  1 )  /  N ) ) ) )
415adantr 465 . . . . . . 7  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  F  e.  ( C CovMap  J ) )
42 simpr 461 . . . . . . . 8  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  M  e.  ( 1 ... N
) )
432, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 42cvmliftlem1 28370 . . . . . . 7  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  ( 2nd `  ( T `  M ) )  e.  ( S `  ( 1st `  ( T `  M ) ) ) )
442, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14cvmliftlem7 28376 . . . . . . . . 9  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  (
( Q `  ( M  -  1 ) ) `  ( ( M  -  1 )  /  N ) )  e.  ( `' F " { ( G `  ( ( M  - 
1 )  /  N
) ) } ) )
45 cvmcn 28347 . . . . . . . . . . 11  |-  ( F  e.  ( C CovMap  J
)  ->  F  e.  ( C  Cn  J
) )
463, 4cnf 19513 . . . . . . . . . . 11  |-  ( F  e.  ( C  Cn  J )  ->  F : B --> X )
4741, 45, 463syl 20 . . . . . . . . . 10  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  F : B --> X )
48 ffn 5729 . . . . . . . . . 10  |-  ( F : B --> X  ->  F  Fn  B )
49 fniniseg 6000 . . . . . . . . . 10  |-  ( 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
) ) ) ) )
5047, 48, 493syl 20 . . . . . . . . 9  |-  ( (
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
) ) ) ) )
5144, 50mpbid 210 . . . . . . . 8  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  (
( ( Q `  ( M  -  1
) ) `  (
( M  -  1 )  /  N ) )  e.  B  /\  ( F `  ( ( Q `  ( M  -  1 ) ) `
 ( ( M  -  1 )  /  N ) ) )  =  ( G `  ( ( M  - 
1 )  /  N
) ) ) )
5251simpld 459 . . . . . . 7  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  (
( Q `  ( M  -  1 ) ) `  ( ( M  -  1 )  /  N ) )  e.  B )
5351simprd 463 . . . . . . . 8  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  ( F `  ( ( Q `  ( M  -  1 ) ) `
 ( ( M  -  1 )  /  N ) ) )  =  ( G `  ( ( M  - 
1 )  /  N
) ) )
542, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 42, 14, 37cvmliftlem3 28372 . . . . . . . 8  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  ( G `  ( ( M  -  1 )  /  N ) )  e.  ( 1st `  ( T `  M )
) )
5553, 54eqeltrd 2555 . . . . . . 7  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  ( F `  ( ( Q `  ( M  -  1 ) ) `
 ( ( M  -  1 )  /  N ) ) )  e.  ( 1st `  ( T `  M )
) )
56 eqid 2467 . . . . . . . 8  |-  ( 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 )
572, 3, 56cvmsiota 28362 . . . . . . 7  |-  ( ( 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 ) ) )
5841, 43, 52, 55, 57syl13anc 1230 . . . . . 6  |-  ( (
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 ) ) )
5958simprd 463 . . . . 5  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  (
( Q `  ( M  -  1 ) ) `  ( ( M  -  1 )  /  N ) )  e.  ( iota_ b  e.  ( 2nd `  ( T `  M )
) ( ( Q `
 ( M  - 
1 ) ) `  ( ( M  - 
1 )  /  N
) )  e.  b ) )
60 fvres 5878 . . . . 5  |-  ( ( ( Q `  ( M  -  1 ) ) `  ( ( M  -  1 )  /  N ) )  e.  ( iota_ b  e.  ( 2nd `  ( T `  M )
) ( ( Q `
 ( M  - 
1 ) ) `  ( ( M  - 
1 )  /  N
) )  e.  b )  ->  ( ( F  |`  ( iota_ b  e.  ( 2nd `  ( T `  M )
) ( ( Q `
 ( M  - 
1 ) ) `  ( ( M  - 
1 )  /  N
) )  e.  b ) ) `  (
( Q `  ( M  -  1 ) ) `  ( ( M  -  1 )  /  N ) ) )  =  ( F `
 ( ( Q `
 ( M  - 
1 ) ) `  ( ( M  - 
1 )  /  N
) ) ) )
6159, 60syl 16 . . . 4  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  (
( F  |`  ( iota_ b  e.  ( 2nd `  ( T `  M
) ) ( ( Q `  ( M  -  1 ) ) `
 ( ( M  -  1 )  /  N ) )  e.  b ) ) `  ( ( Q `  ( M  -  1
) ) `  (
( M  -  1 )  /  N ) ) )  =  ( F `  ( ( Q `  ( M  -  1 ) ) `
 ( ( M  -  1 )  /  N ) ) ) )
6261, 53eqtrd 2508 . . 3  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  (
( F  |`  ( iota_ b  e.  ( 2nd `  ( T `  M
) ) ( ( Q `  ( M  -  1 ) ) `
 ( ( M  -  1 )  /  N ) )  e.  b ) ) `  ( ( Q `  ( M  -  1
) ) `  (
( M  -  1 )  /  N ) ) )  =  ( G `  ( ( M  -  1 )  /  N ) ) )
6358simpld 459 . . . . 5  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  ( iota_ b  e.  ( 2nd `  ( T `  M
) ) ( ( Q `  ( M  -  1 ) ) `
 ( ( M  -  1 )  /  N ) )  e.  b )  e.  ( 2nd `  ( T `
 M ) ) )
642cvmsf1o 28357 . . . . 5  |-  ( ( F  e.  ( C CovMap  J )  /\  ( 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 ) ) : ( iota_ b  e.  ( 2nd `  ( T `
 M ) ) ( ( Q `  ( M  -  1
) ) `  (
( M  -  1 )  /  N ) )  e.  b ) -1-1-onto-> ( 1st `  ( T `
 M ) ) )
6541, 43, 63, 64syl3anc 1228 . . . 4  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  ( F  |`  ( 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 ) -1-1-onto-> ( 1st `  ( T `  M
) ) )
66 f1ocnvfv 6170 . . . 4  |-  ( ( ( F  |`  ( 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 ) -1-1-onto-> ( 1st `  ( T `
 M ) )  /\  ( ( Q `
 ( M  - 
1 ) ) `  ( ( M  - 
1 )  /  N
) )  e.  (
iota_ b  e.  ( 2nd `  ( T `  M ) ) ( ( Q `  ( M  -  1 ) ) `  ( ( M  -  1 )  /  N ) )  e.  b ) )  ->  ( ( ( F  |`  ( iota_ b  e.  ( 2nd `  ( T `  M )
) ( ( Q `
 ( M  - 
1 ) ) `  ( ( M  - 
1 )  /  N
) )  e.  b ) ) `  (
( Q `  ( M  -  1 ) ) `  ( ( M  -  1 )  /  N ) ) )  =  ( G `
 ( ( M  -  1 )  /  N ) )  -> 
( `' ( F  |`  ( iota_ b  e.  ( 2nd `  ( T `
 M ) ) ( ( Q `  ( M  -  1
) ) `  (
( M  -  1 )  /  N ) )  e.  b ) ) `  ( G `
 ( ( M  -  1 )  /  N ) ) )  =  ( ( Q `
 ( M  - 
1 ) ) `  ( ( M  - 
1 )  /  N
) ) ) )
6765, 59, 66syl2anc 661 . . 3  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  (
( ( F  |`  ( iota_ b  e.  ( 2nd `  ( T `
 M ) ) ( ( Q `  ( M  -  1
) ) `  (
( M  -  1 )  /  N ) )  e.  b ) ) `  ( ( Q `  ( M  -  1 ) ) `
 ( ( M  -  1 )  /  N ) ) )  =  ( G `  ( ( M  - 
1 )  /  N
) )  ->  ( `' ( F  |`  ( iota_ b  e.  ( 2nd `  ( T `
 M ) ) ( ( Q `  ( M  -  1
) ) `  (
( M  -  1 )  /  N ) )  e.  b ) ) `  ( G `
 ( ( M  -  1 )  /  N ) ) )  =  ( ( Q `
 ( M  - 
1 ) ) `  ( ( M  - 
1 )  /  N
) ) ) )
6862, 67mpd 15 . 2  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  ( `' ( F  |`  ( iota_ b  e.  ( 2nd `  ( T `
 M ) ) ( ( Q `  ( M  -  1
) ) `  (
( M  -  1 )  /  N ) )  e.  b ) ) `  ( G `
 ( ( M  -  1 )  /  N ) ) )  =  ( ( Q `
 ( M  - 
1 ) ) `  ( ( M  - 
1 )  /  N
) ) )
6940, 68eqtrd 2508 1  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  (
( Q `  M
) `  ( ( M  -  1 )  /  N ) )  =  ( ( Q `
 ( M  - 
1 ) ) `  ( ( M  - 
1 )  /  N
) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767   A.wral 2814   {crab 2818   _Vcvv 3113    \ cdif 3473    u. cun 3474    i^i cin 3475    C_ wss 3476   (/)c0 3785   ~Pcpw 4010   {csn 4027   <.cop 4033   U.cuni 4245   U_ciun 4325   class class class wbr 4447    |-> cmpt 4505    _I cid 4790    X. cxp 4997   `'ccnv 4998   ran crn 5000    |` cres 5001   "cima 5002    Fn wfn 5581   -->wf 5582   -1-1-onto->wf1o 5585   ` cfv 5586   iota_crio 6242  (class class class)co 6282    |-> cmpt2 6284   1stc1st 6779   2ndc2nd 6780   RRcr 9487   0cc0 9488   1c1 9489   RR*cxr 9623    < clt 9624    <_ cle 9625    - cmin 9801    / cdiv 10202   NNcn 10532   (,)cioo 11525   [,]cicc 11528   ...cfz 11668    seqcseq 12071   ↾t crest 14672   topGenctg 14689    Cn ccn 19491   Homeochmeo 19989   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-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
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  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-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-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-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-1st 6781  df-2nd 6782  df-recs 7039  df-rdg 7073  df-oadd 7131  df-er 7308  df-map 7419  df-en 7514  df-dom 7515  df-sdom 7516  df-fin 7517  df-fi 7867  df-sup 7897  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-n0 10792  df-z 10861  df-uz 11079  df-q 11179  df-rp 11217  df-xneg 11314  df-xadd 11315  df-xmul 11316  df-icc 11532  df-fz 11669  df-seq 12072  df-exp 12131  df-cj 12891  df-re 12892  df-im 12893  df-sqrt 13027  df-abs 13028  df-rest 14674  df-topgen 14695  df-psmet 18182  df-xmet 18183  df-met 18184  df-bl 18185  df-mopn 18186  df-top 19166  df-bases 19168  df-topon 19169  df-cn 19494  df-hmeo 19991  df-ii 21116  df-cvm 28341
This theorem is referenced by:  cvmliftlem10  28379  cvmliftlem13  28381
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