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Theorem cvmliftlem9 27319
Description: Lemma for cvmlift 27325. 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 11588 . . . 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 2451 . . . . 5  |-  ( ( ( M  -  1 )  /  N ) [,] ( M  /  N ) )  =  ( ( ( M  -  1 )  /  N ) [,] ( M  /  N ) )
152, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14cvmliftlem5 27315 . . . 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 5796 . . . 4  |-  ( ( ( ph  /\  M  e.  ( 1 ... N
) )  /\  z  =  ( ( M  -  1 )  /  N ) )  -> 
( G `  z
)  =  ( G `
 ( ( M  -  1 )  /  N ) ) )
1918fveq2d 5796 . . 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 10441 . . . . . . 7  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  M  e.  RR )
22 peano2rem 9779 . . . . . . 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 10474 . . . . 5  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  (
( M  -  1 )  /  N )  e.  RR )
2625rexrd 9537 . . . 4  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  (
( M  -  1 )  /  N )  e.  RR* )
2721, 24nndivred 10474 . . . . 5  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  ( M  /  N )  e.  RR )
2827rexrd 9537 . . . 4  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  ( M  /  N )  e. 
RR* )
2921ltm1d 10369 . . . . . 6  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  ( M  -  1 )  <  M )
3024nnred 10441 . . . . . . 7  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  N  e.  RR )
3124nngt0d 10469 . . . . . . 7  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  0  <  N )
32 ltdiv1 10297 . . . . . . 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 1223 . . . . . 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 9626 . . . 4  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  (
( M  -  1 )  /  N )  <_  ( M  /  N ) )
36 lbicc2 11511 . . . 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 1219 . . 3  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  (
( M  -  1 )  /  N )  e.  ( ( ( M  -  1 )  /  N ) [,] ( M  /  N
) ) )
38 fvex 5802 . . . 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 5881 . 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 27311 . . . . . . 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 27317 . . . . . . . . 9  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  (
( Q `  ( M  -  1 ) ) `  ( ( M  -  1 )  /  N ) )  e.  ( `' F " { ( G `  ( ( M  - 
1 )  /  N
) ) } ) )
45 cvmcn 27288 . . . . . . . . . . 11  |-  ( F  e.  ( C CovMap  J
)  ->  F  e.  ( C  Cn  J
) )
463, 4cnf 18975 . . . . . . . . . . 11  |-  ( F  e.  ( C  Cn  J )  ->  F : B --> X )
4741, 45, 463syl 20 . . . . . . . . . 10  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  F : B --> X )
48 ffn 5660 . . . . . . . . . 10  |-  ( F : B --> X  ->  F  Fn  B )
49 fniniseg 5926 . . . . . . . . . 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 27313 . . . . . . . 8  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  ( G `  ( ( M  -  1 )  /  N ) )  e.  ( 1st `  ( T `  M )
) )
5553, 54eqeltrd 2539 . . . . . . 7  |-  ( (
ph  /\  M  e.  ( 1 ... N
) )  ->  ( F `  ( ( Q `  ( M  -  1 ) ) `
 ( ( M  -  1 )  /  N ) ) )  e.  ( 1st `  ( T `  M )
) )
56 eqid 2451 . . . . . . . 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 27303 . . . . . . 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 1221 . . . . . 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 5806 . . . . 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 2492 . . 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 27298 . . . . 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 1219 . . . 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 6087 . . . 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 2492 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 1370    e. wcel 1758   A.wral 2795   {crab 2799   _Vcvv 3071    \ cdif 3426    u. cun 3427    i^i cin 3428    C_ wss 3429   (/)c0 3738   ~Pcpw 3961   {csn 3978   <.cop 3984   U.cuni 4192   U_ciun 4272   class class class wbr 4393    |-> cmpt 4451    _I cid 4732    X. cxp 4939   `'ccnv 4940   ran crn 4942    |` cres 4943   "cima 4944    Fn wfn 5514   -->wf 5515   -1-1-onto->wf1o 5518   ` cfv 5519   iota_crio 6153  (class class class)co 6193    |-> cmpt2 6195   1stc1st 6678   2ndc2nd 6679   RRcr 9385   0cc0 9386   1c1 9387   RR*cxr 9521    < clt 9522    <_ cle 9523    - cmin 9699    / cdiv 10097   NNcn 10426   (,)cioo 11404   [,]cicc 11407   ...cfz 11547    seqcseq 11916   ↾t crest 14470   topGenctg 14487    Cn ccn 18953   Homeochmeo 19451   IIcii 20576   CovMap ccvm 27281
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4504  ax-sep 4514  ax-nul 4522  ax-pow 4571  ax-pr 4632  ax-un 6475  ax-cnex 9442  ax-resscn 9443  ax-1cn 9444  ax-icn 9445  ax-addcl 9446  ax-addrcl 9447  ax-mulcl 9448  ax-mulrcl 9449  ax-mulcom 9450  ax-addass 9451  ax-mulass 9452  ax-distr 9453  ax-i2m1 9454  ax-1ne0 9455  ax-1rid 9456  ax-rnegex 9457  ax-rrecex 9458  ax-cnre 9459  ax-pre-lttri 9460  ax-pre-lttrn 9461  ax-pre-ltadd 9462  ax-pre-mulgt0 9463  ax-pre-sup 9464
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3073  df-sbc 3288  df-csb 3390  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-pss 3445  df-nul 3739  df-if 3893  df-pw 3963  df-sn 3979  df-pr 3981  df-tp 3983  df-op 3985  df-uni 4193  df-int 4230  df-iun 4274  df-br 4394  df-opab 4452  df-mpt 4453  df-tr 4487  df-eprel 4733  df-id 4737  df-po 4742  df-so 4743  df-fr 4780  df-we 4782  df-ord 4823  df-on 4824  df-lim 4825  df-suc 4826  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-rn 4952  df-res 4953  df-ima 4954  df-iota 5482  df-fun 5521  df-fn 5522  df-f 5523  df-f1 5524  df-fo 5525  df-f1o 5526  df-fv 5527  df-riota 6154  df-ov 6196  df-oprab 6197  df-mpt2 6198  df-om 6580  df-1st 6680  df-2nd 6681  df-recs 6935  df-rdg 6969  df-oadd 7027  df-er 7204  df-map 7319  df-en 7414  df-dom 7415  df-sdom 7416  df-fin 7417  df-fi 7765  df-sup 7795  df-pnf 9524  df-mnf 9525  df-xr 9526  df-ltxr 9527  df-le 9528  df-sub 9701  df-neg 9702  df-div 10098  df-nn 10427  df-2 10484  df-3 10485  df-n0 10684  df-z 10751  df-uz 10966  df-q 11058  df-rp 11096  df-xneg 11193  df-xadd 11194  df-xmul 11195  df-icc 11411  df-fz 11548  df-seq 11917  df-exp 11976  df-cj 12699  df-re 12700  df-im 12701  df-sqr 12835  df-abs 12836  df-rest 14472  df-topgen 14493  df-psmet 17927  df-xmet 17928  df-met 17929  df-bl 17930  df-mopn 17931  df-top 18628  df-bases 18630  df-topon 18631  df-cn 18956  df-hmeo 19453  df-ii 20578  df-cvm 27282
This theorem is referenced by:  cvmliftlem10  27320  cvmliftlem13  27322
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