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Theorem cvmliftlem5 27126
Description: Lemma for cvmlift 27136. Definition of  Q at a successor. This is a function defined on  W as  `' ( T  |`  I )  o.  G where  I is the unique covering set of  2nd `  ( T `  M ) that contains  Q ( M  -  1 ) evaluated at the last defined point, namely  ( M  - 
1 )  /  N (note that for  M  =  1 this is using the seed value  Q ( 0 ) ( 0 )  =  P). (Contributed by Mario Carneiro, 15-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
cvmliftlem5  |-  ( (
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 ) ) ) )
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 cvmliftlem5
StepHypRef Expression
1 0z 10649 . . . 4  |-  0  e.  ZZ
2 simpr 461 . . . . 5  |-  ( (
ph  /\  M  e.  NN )  ->  M  e.  NN )
3 nnuz 10888 . . . . . 6  |-  NN  =  ( ZZ>= `  1 )
4 1e0p1 10775 . . . . . . 7  |-  1  =  ( 0  +  1 )
54fveq2i 5687 . . . . . 6  |-  ( ZZ>= ` 
1 )  =  (
ZZ>= `  ( 0  +  1 ) )
63, 5eqtri 2457 . . . . 5  |-  NN  =  ( ZZ>= `  ( 0  +  1 ) )
72, 6syl6eleq 2527 . . . 4  |-  ( (
ph  /\  M  e.  NN )  ->  M  e.  ( ZZ>= `  ( 0  +  1 ) ) )
8 seqm1 11815 . . . 4  |-  ( ( 0  e.  ZZ  /\  M  e.  ( ZZ>= `  ( 0  +  1 ) ) )  -> 
(  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 >. } >. } ) ) `
 M )  =  ( (  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 >. } >. } ) ) `
 ( M  - 
1 ) ) ( 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 >. } >. } ) `  M ) ) )
91, 7, 8sylancr 663 . . 3  |-  ( (
ph  /\  M  e.  NN )  ->  (  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 >. } >. } ) ) `
 M )  =  ( (  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 >. } >. } ) ) `
 ( M  - 
1 ) ) ( 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 >. } >. } ) `  M ) ) )
10 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 >. } >. } ) )
1110fveq1i 5685 . . 3  |-  ( Q `
 M )  =  (  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 >. } >. } ) ) `
 M )
1210fveq1i 5685 . . . 4  |-  ( Q `
 ( M  - 
1 ) )  =  (  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 >. } >. } ) ) `
 ( M  - 
1 ) )
1312oveq1i 6096 . . 3  |-  ( ( Q `  ( M  -  1 ) ) ( 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 >. } >. } ) `  M ) )  =  ( (  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 >. } >. } ) ) `
 ( M  - 
1 ) ) ( 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 >. } >. } ) `  M ) )
149, 11, 133eqtr4g 2494 . 2  |-  ( (
ph  /\  M  e.  NN )  ->  ( Q `
 M )  =  ( ( Q `  ( M  -  1
) ) ( 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 >. } >. } ) `  M ) ) )
15 0nnn 10345 . . . . . 6  |-  -.  0  e.  NN
16 disjsn 3929 . . . . . 6  |-  ( ( NN  i^i  { 0 } )  =  (/)  <->  -.  0  e.  NN )
1715, 16mpbir 209 . . . . 5  |-  ( NN 
i^i  { 0 } )  =  (/)
18 fnresi 5521 . . . . . 6  |-  (  _I  |`  NN )  Fn  NN
19 c0ex 9372 . . . . . . 7  |-  0  e.  _V
20 snex 4526 . . . . . . 7  |-  { <. 0 ,  P >. }  e.  _V
2119, 20fnsn 5464 . . . . . 6  |-  { <. 0 ,  { <. 0 ,  P >. } >. }  Fn  { 0 }
22 fvun1 5755 . . . . . 6  |-  ( ( (  _I  |`  NN )  Fn  NN  /\  { <. 0 ,  { <. 0 ,  P >. }
>. }  Fn  { 0 }  /\  ( ( NN  i^i  { 0 } )  =  (/)  /\  M  e.  NN ) )  ->  ( (
(  _I  |`  NN )  u.  { <. 0 ,  { <. 0 ,  P >. } >. } ) `  M )  =  ( (  _I  |`  NN ) `
 M ) )
2318, 21, 22mp3an12 1304 . . . . 5  |-  ( ( ( NN  i^i  {
0 } )  =  (/)  /\  M  e.  NN )  ->  ( ( (  _I  |`  NN )  u.  { <. 0 ,  { <. 0 ,  P >. }
>. } ) `  M
)  =  ( (  _I  |`  NN ) `  M ) )
2417, 2, 23sylancr 663 . . . 4  |-  ( (
ph  /\  M  e.  NN )  ->  ( ( (  _I  |`  NN )  u.  { <. 0 ,  { <. 0 ,  P >. } >. } ) `  M )  =  ( (  _I  |`  NN ) `
 M ) )
25 fvresi 5897 . . . . 5  |-  ( M  e.  NN  ->  (
(  _I  |`  NN ) `
 M )  =  M )
2625adantl 466 . . . 4  |-  ( (
ph  /\  M  e.  NN )  ->  ( (  _I  |`  NN ) `  M )  =  M )
2724, 26eqtrd 2469 . . 3  |-  ( (
ph  /\  M  e.  NN )  ->  ( ( (  _I  |`  NN )  u.  { <. 0 ,  { <. 0 ,  P >. } >. } ) `  M )  =  M )
2827oveq2d 6102 . 2  |-  ( (
ph  /\  M  e.  NN )  ->  ( ( Q `  ( M  -  1 ) ) ( 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 >. } >. } ) `  M ) )  =  ( ( Q `  ( M  -  1
) ) ( 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 )
) ) ) M ) )
29 fvex 5694 . . . 4  |-  ( Q `
 ( M  - 
1 ) )  e. 
_V
3029a1i 11 . . 3  |-  ( ph  ->  ( Q `  ( M  -  1 ) )  e.  _V )
31 simpr 461 . . . . . . . . 9  |-  ( ( x  =  ( Q `
 ( M  - 
1 ) )  /\  m  =  M )  ->  m  =  M )
3231oveq1d 6101 . . . . . . . 8  |-  ( ( x  =  ( Q `
 ( M  - 
1 ) )  /\  m  =  M )  ->  ( m  -  1 )  =  ( M  -  1 ) )
3332oveq1d 6101 . . . . . . 7  |-  ( ( x  =  ( Q `
 ( M  - 
1 ) )  /\  m  =  M )  ->  ( ( m  - 
1 )  /  N
)  =  ( ( M  -  1 )  /  N ) )
3431oveq1d 6101 . . . . . . 7  |-  ( ( x  =  ( Q `
 ( M  - 
1 ) )  /\  m  =  M )  ->  ( m  /  N
)  =  ( M  /  N ) )
3533, 34oveq12d 6104 . . . . . 6  |-  ( ( x  =  ( Q `
 ( M  - 
1 ) )  /\  m  =  M )  ->  ( ( ( m  -  1 )  /  N ) [,] (
m  /  N ) )  =  ( ( ( M  -  1 )  /  N ) [,] ( M  /  N ) ) )
36 cvmliftlem5.3 . . . . . 6  |-  W  =  ( ( ( M  -  1 )  /  N ) [,] ( M  /  N ) )
3735, 36syl6eqr 2487 . . . . 5  |-  ( ( x  =  ( Q `
 ( M  - 
1 ) )  /\  m  =  M )  ->  ( ( ( m  -  1 )  /  N ) [,] (
m  /  N ) )  =  W )
3831fveq2d 5688 . . . . . . . . . 10  |-  ( ( x  =  ( Q `
 ( M  - 
1 ) )  /\  m  =  M )  ->  ( T `  m
)  =  ( T `
 M ) )
3938fveq2d 5688 . . . . . . . . 9  |-  ( ( x  =  ( Q `
 ( M  - 
1 ) )  /\  m  =  M )  ->  ( 2nd `  ( T `  m )
)  =  ( 2nd `  ( T `  M
) ) )
40 simpl 457 . . . . . . . . . . 11  |-  ( ( x  =  ( Q `
 ( M  - 
1 ) )  /\  m  =  M )  ->  x  =  ( Q `
 ( M  - 
1 ) ) )
4140, 33fveq12d 5690 . . . . . . . . . 10  |-  ( ( x  =  ( Q `
 ( M  - 
1 ) )  /\  m  =  M )  ->  ( x `  (
( m  -  1 )  /  N ) )  =  ( ( Q `  ( M  -  1 ) ) `
 ( ( M  -  1 )  /  N ) ) )
4241eleq1d 2503 . . . . . . . . 9  |-  ( ( x  =  ( Q `
 ( M  - 
1 ) )  /\  m  =  M )  ->  ( ( x `  ( ( m  - 
1 )  /  N
) )  e.  b  <-> 
( ( Q `  ( M  -  1
) ) `  (
( M  -  1 )  /  N ) )  e.  b ) )
4339, 42riotaeqbidv 6048 . . . . . . . 8  |-  ( ( x  =  ( Q `
 ( M  - 
1 ) )  /\  m  =  M )  ->  ( iota_ b  e.  ( 2nd `  ( T `
 m ) ) ( x `  (
( m  -  1 )  /  N ) )  e.  b )  =  ( iota_ b  e.  ( 2nd `  ( T `  M )
) ( ( Q `
 ( M  - 
1 ) ) `  ( ( M  - 
1 )  /  N
) )  e.  b ) )
4443reseq2d 5102 . . . . . . 7  |-  ( ( x  =  ( Q `
 ( M  - 
1 ) )  /\  m  =  M )  ->  ( F  |`  ( iota_ b  e.  ( 2nd `  ( T `  m
) ) ( x `
 ( ( m  -  1 )  /  N ) )  e.  b ) )  =  ( F  |`  ( iota_ b  e.  ( 2nd `  ( T `  M
) ) ( ( Q `  ( M  -  1 ) ) `
 ( ( M  -  1 )  /  N ) )  e.  b ) ) )
4544cnveqd 5007 . . . . . 6  |-  ( ( x  =  ( Q `
 ( M  - 
1 ) )  /\  m  =  M )  ->  `' ( F  |`  ( iota_ b  e.  ( 2nd `  ( T `
 m ) ) ( x `  (
( m  -  1 )  /  N ) )  e.  b ) )  =  `' ( F  |`  ( iota_ b  e.  ( 2nd `  ( T `  M )
) ( ( Q `
 ( M  - 
1 ) ) `  ( ( M  - 
1 )  /  N
) )  e.  b ) ) )
4645fveq1d 5686 . . . . 5  |-  ( ( x  =  ( Q `
 ( M  - 
1 ) )  /\  m  =  M )  ->  ( `' ( F  |`  ( iota_ b  e.  ( 2nd `  ( T `
 m ) ) ( x `  (
( m  -  1 )  /  N ) )  e.  b ) ) `  ( G `
 z ) )  =  ( `' ( F  |`  ( iota_ b  e.  ( 2nd `  ( T `  M )
) ( ( Q `
 ( M  - 
1 ) ) `  ( ( M  - 
1 )  /  N
) )  e.  b ) ) `  ( G `  z )
) )
4737, 46mpteq12dv 4363 . . . 4  |-  ( ( x  =  ( Q `
 ( M  - 
1 ) )  /\  m  =  M )  ->  ( z  e.  ( ( ( m  - 
1 )  /  N
) [,] ( m  /  N ) ) 
|->  ( `' ( F  |`  ( iota_ b  e.  ( 2nd `  ( T `
 m ) ) ( x `  (
( m  -  1 )  /  N ) )  e.  b ) ) `  ( G `
 z ) ) )  =  ( z  e.  W  |->  ( `' ( F  |`  ( iota_ b  e.  ( 2nd `  ( T `  M
) ) ( ( Q `  ( M  -  1 ) ) `
 ( ( M  -  1 )  /  N ) )  e.  b ) ) `  ( G `  z ) ) ) )
48 eqid 2437 . . . 4  |-  ( 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 )
) ) )  =  ( 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 ) ) ) )
49 ovex 6111 . . . . . 6  |-  ( ( ( M  -  1 )  /  N ) [,] ( M  /  N ) )  e. 
_V
5036, 49eqeltri 2507 . . . . 5  |-  W  e. 
_V
5150mptex 5941 . . . 4  |-  ( z  e.  W  |->  ( `' ( F  |`  ( iota_ b  e.  ( 2nd `  ( T `  M
) ) ( ( Q `  ( M  -  1 ) ) `
 ( ( M  -  1 )  /  N ) )  e.  b ) ) `  ( G `  z ) ) )  e.  _V
5247, 48, 51ovmpt2a 6216 . . 3  |-  ( ( ( Q `  ( M  -  1 ) )  e.  _V  /\  M  e.  NN )  ->  ( ( Q `  ( M  -  1
) ) ( 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 )
) ) ) M )  =  ( z  e.  W  |->  ( `' ( F  |`  ( iota_ b  e.  ( 2nd `  ( T `  M
) ) ( ( Q `  ( M  -  1 ) ) `
 ( ( M  -  1 )  /  N ) )  e.  b ) ) `  ( G `  z ) ) ) )
5330, 52sylan 471 . 2  |-  ( (
ph  /\  M  e.  NN )  ->  ( ( Q `  ( M  -  1 ) ) ( 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 ) ) ) ) M )  =  ( z  e.  W  |->  ( `' ( F  |`  ( iota_ b  e.  ( 2nd `  ( T `  M )
) ( ( Q `
 ( M  - 
1 ) ) `  ( ( M  - 
1 )  /  N
) )  e.  b ) ) `  ( G `  z )
) ) )
5414, 28, 533eqtrd 2473 1  |-  ( (
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 ) ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756   A.wral 2709   {crab 2713   _Vcvv 2966    \ cdif 3318    u. cun 3319    i^i cin 3320    C_ wss 3321   (/)c0 3630   ~Pcpw 3853   {csn 3870   <.cop 3876   U.cuni 4084   U_ciun 4164    e. cmpt 4343    _I cid 4623    X. cxp 4830   `'ccnv 4831   ran crn 4833    |` cres 4834   "cima 4835    Fn wfn 5406   -->wf 5407   ` cfv 5411   iota_crio 6044  (class class class)co 6086    e. cmpt2 6088   1stc1st 6570   2ndc2nd 6571   0cc0 9274   1c1 9275    + caddc 9277    - cmin 9587    / cdiv 9985   NNcn 10314   ZZcz 10638   ZZ>=cuz 10853   (,)cioo 11292   [,]cicc 11295   ...cfz 11429    seqcseq 11798   ↾t crest 14351   topGenctg 14368    Cn ccn 18797   Homeochmeo 19295   IIcii 20420   CovMap ccvm 27092
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2418  ax-rep 4396  ax-sep 4406  ax-nul 4414  ax-pow 4463  ax-pr 4524  ax-un 6367  ax-cnex 9330  ax-resscn 9331  ax-1cn 9332  ax-icn 9333  ax-addcl 9334  ax-addrcl 9335  ax-mulcl 9336  ax-mulrcl 9337  ax-mulcom 9338  ax-addass 9339  ax-mulass 9340  ax-distr 9341  ax-i2m1 9342  ax-1ne0 9343  ax-1rid 9344  ax-rnegex 9345  ax-rrecex 9346  ax-cnre 9347  ax-pre-lttri 9348  ax-pre-lttrn 9349  ax-pre-ltadd 9350  ax-pre-mulgt0 9351
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-nel 2603  df-ral 2714  df-rex 2715  df-reu 2716  df-rab 2718  df-v 2968  df-sbc 3180  df-csb 3282  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-pss 3337  df-nul 3631  df-if 3785  df-pw 3855  df-sn 3871  df-pr 3873  df-tp 3875  df-op 3877  df-uni 4085  df-iun 4166  df-br 4286  df-opab 4344  df-mpt 4345  df-tr 4379  df-eprel 4624  df-id 4628  df-po 4633  df-so 4634  df-fr 4671  df-we 4673  df-ord 4714  df-on 4715  df-lim 4716  df-suc 4717  df-xp 4838  df-rel 4839  df-cnv 4840  df-co 4841  df-dm 4842  df-rn 4843  df-res 4844  df-ima 4845  df-iota 5374  df-fun 5413  df-fn 5414  df-f 5415  df-f1 5416  df-fo 5417  df-f1o 5418  df-fv 5419  df-riota 6045  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-om 6472  df-2nd 6573  df-recs 6824  df-rdg 6858  df-er 7093  df-en 7303  df-dom 7304  df-sdom 7305  df-pnf 9412  df-mnf 9413  df-xr 9414  df-ltxr 9415  df-le 9416  df-sub 9589  df-neg 9590  df-nn 10315  df-n0 10572  df-z 10639  df-uz 10854  df-seq 11799
This theorem is referenced by:  cvmliftlem6  27127  cvmliftlem8  27129  cvmliftlem9  27130
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