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Theorem cvmliftlem9 29795
Description: Lemma for cvmlift 29801. 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 11826 . . . 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 2429 . . . . 5  |-  ( ( ( M  -  1 )  /  N ) [,] ( M  /  N ) )  =  ( ( ( M  -  1 )  /  N ) [,] ( M  /  N ) )
152, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14cvmliftlem5 29791 . . . 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 476 . . 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 462 . . . . 5  |-  ( ( ( ph  /\  M  e.  ( 1 ... N
) )  /\  z  =  ( ( M  -  1 )  /  N ) )  -> 
z  =  ( ( M  -  1 )  /  N ) )
1817fveq2d 5885 . . . 4  |-  ( ( ( ph  /\  M  e.  ( 1 ... N
) )  /\  z  =  ( ( M  -  1 )  /  N ) )  -> 
( G `  z
)  =  ( G `
 ( ( M  -  1 )  /  N ) ) )
1918fveq2d 5885 . . 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 467 . . . . . . . 8  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  M  e.  NN )
2120nnred 10624 . . . . . . 7  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  M  e.  RR )
22 peano2rem 9940 . . . . . . 7  |-  ( M  e.  RR  ->  ( M  -  1 )  e.  RR )
2321, 22syl 17 . . . . . 6  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  ( M  -  1 )  e.  RR )
249adantr 466 . . . . . 6  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  N  e.  NN )
2523, 24nndivred 10658 . . . . 5  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  (
( M  -  1 )  /  N )  e.  RR )
2625rexrd 9689 . . . 4  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  (
( M  -  1 )  /  N )  e.  RR* )
2721, 24nndivred 10658 . . . . 5  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  ( M  /  N )  e.  RR )
2827rexrd 9689 . . . 4  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  ( M  /  N )  e. 
RR* )
2921ltm1d 10539 . . . . . 6  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  ( M  -  1 )  <  M )
3024nnred 10624 . . . . . . 7  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  N  e.  RR )
3124nngt0d 10653 . . . . . . 7  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  0  <  N )
32 ltdiv1 10468 . . . . . . 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 1268 . . . . . 6  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  (
( M  -  1 )  <  M  <->  ( ( M  -  1 )  /  N )  < 
( M  /  N
) ) )
3429, 33mpbid 213 . . . . 5  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  (
( M  -  1 )  /  N )  <  ( M  /  N ) )
3525, 27, 34ltled 9782 . . . 4  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  (
( M  -  1 )  /  N )  <_  ( M  /  N ) )
36 lbicc2 11746 . . . 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 1264 . . 3  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  (
( M  -  1 )  /  N )  e.  ( ( ( M  -  1 )  /  N ) [,] ( M  /  N
) ) )
38 fvex 5891 . . . 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 5970 . 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 466 . . . . . . 7  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  F  e.  ( C CovMap  J ) )
42 simpr 462 . . . . . . . 8  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  M  e.  ( 1 ... N
) )
432, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 42cvmliftlem1 29787 . . . . . . 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 29793 . . . . . . . . 9  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  (
( Q `  ( M  -  1 ) ) `  ( ( M  -  1 )  /  N ) )  e.  ( `' F " { ( G `  ( ( M  - 
1 )  /  N
) ) } ) )
45 cvmcn 29764 . . . . . . . . . . 11  |-  ( F  e.  ( C CovMap  J
)  ->  F  e.  ( C  Cn  J
) )
463, 4cnf 20184 . . . . . . . . . . 11  |-  ( F  e.  ( C  Cn  J )  ->  F : B --> X )
4741, 45, 463syl 18 . . . . . . . . . 10  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  F : B --> X )
48 ffn 5746 . . . . . . . . . 10  |-  ( F : B --> X  ->  F  Fn  B )
49 fniniseg 6018 . . . . . . . . . 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 18 . . . . . . . . 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 213 . . . . . . . 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 460 . . . . . . 7  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  (
( Q `  ( M  -  1 ) ) `  ( ( M  -  1 )  /  N ) )  e.  B )
5351simprd 464 . . . . . . . 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 29789 . . . . . . . 8  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  ( G `  ( ( M  -  1 )  /  N ) )  e.  ( 1st `  ( T `  M )
) )
5553, 54eqeltrd 2517 . . . . . . 7  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  ( F `  ( ( Q `  ( M  -  1 ) ) `
 ( ( M  -  1 )  /  N ) ) )  e.  ( 1st `  ( T `  M )
) )
56 eqid 2429 . . . . . . . 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 29779 . . . . . . 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 1266 . . . . . 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 464 . . . . 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 5895 . . . . 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 17 . . . 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 2470 . . 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 460 . . . . 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 29774 . . . . 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 1264 . . . 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 6192 . . . 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 665 . . 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 2470 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 187    /\ wa 370    = wceq 1437    e. wcel 1870   A.wral 2782   {crab 2786   _Vcvv 3087    \ cdif 3439    u. cun 3440    i^i cin 3441    C_ wss 3442   (/)c0 3767   ~Pcpw 3985   {csn 4002   <.cop 4008   U.cuni 4222   U_ciun 4302   class class class wbr 4426    |-> cmpt 4484    _I cid 4764    X. cxp 4852   `'ccnv 4853   ran crn 4855    |` cres 4856   "cima 4857    Fn wfn 5596   -->wf 5597   -1-1-onto->wf1o 5600   ` cfv 5601   iota_crio 6266  (class class class)co 6305    |-> cmpt2 6307   1stc1st 6805   2ndc2nd 6806   RRcr 9537   0cc0 9538   1c1 9539   RR*cxr 9673    < clt 9674    <_ cle 9675    - cmin 9859    / cdiv 10268   NNcn 10609   (,)cioo 11635   [,]cicc 11638   ...cfz 11782    seqcseq 12210   ↾t crest 15269   topGenctg 15286    Cn ccn 20162   Homeochmeo 20690   IIcii 21794   CovMap ccvm 29757
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-cnex 9594  ax-resscn 9595  ax-1cn 9596  ax-icn 9597  ax-addcl 9598  ax-addrcl 9599  ax-mulcl 9600  ax-mulrcl 9601  ax-mulcom 9602  ax-addass 9603  ax-mulass 9604  ax-distr 9605  ax-i2m1 9606  ax-1ne0 9607  ax-1rid 9608  ax-rnegex 9609  ax-rrecex 9610  ax-cnre 9611  ax-pre-lttri 9612  ax-pre-lttrn 9613  ax-pre-ltadd 9614  ax-pre-mulgt0 9615  ax-pre-sup 9616
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-nel 2628  df-ral 2787  df-rex 2788  df-reu 2789  df-rmo 2790  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-uni 4223  df-int 4259  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-tr 4521  df-eprel 4765  df-id 4769  df-po 4775  df-so 4776  df-fr 4813  df-we 4815  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-om 6707  df-1st 6807  df-2nd 6808  df-wrecs 7036  df-recs 7098  df-rdg 7136  df-oadd 7194  df-er 7371  df-map 7482  df-en 7578  df-dom 7579  df-sdom 7580  df-fin 7581  df-fi 7931  df-sup 7962  df-pnf 9676  df-mnf 9677  df-xr 9678  df-ltxr 9679  df-le 9680  df-sub 9861  df-neg 9862  df-div 10269  df-nn 10610  df-2 10668  df-3 10669  df-n0 10870  df-z 10938  df-uz 11160  df-q 11265  df-rp 11303  df-xneg 11409  df-xadd 11410  df-xmul 11411  df-icc 11642  df-fz 11783  df-seq 12211  df-exp 12270  df-cj 13141  df-re 13142  df-im 13143  df-sqrt 13277  df-abs 13278  df-rest 15271  df-topgen 15292  df-psmet 18888  df-xmet 18889  df-met 18890  df-bl 18891  df-mopn 18892  df-top 19843  df-bases 19844  df-topon 19845  df-cn 20165  df-hmeo 20692  df-ii 21796  df-cvm 29758
This theorem is referenced by:  cvmliftlem10  29796  cvmliftlem13  29798
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