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Theorem cvmliftlem5 28998
Description: Lemma for cvmlift 29008. 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 10871 . . . 4  |-  0  e.  ZZ
2 simpr 459 . . . . 5  |-  ( (
ph  /\  M  e.  NN )  ->  M  e.  NN )
3 nnuz 11117 . . . . . 6  |-  NN  =  ( ZZ>= `  1 )
4 1e0p1 11004 . . . . . . 7  |-  1  =  ( 0  +  1 )
54fveq2i 5851 . . . . . 6  |-  ( ZZ>= ` 
1 )  =  (
ZZ>= `  ( 0  +  1 ) )
63, 5eqtri 2483 . . . . 5  |-  NN  =  ( ZZ>= `  ( 0  +  1 ) )
72, 6syl6eleq 2552 . . . 4  |-  ( (
ph  /\  M  e.  NN )  ->  M  e.  ( ZZ>= `  ( 0  +  1 ) ) )
8 seqm1 12106 . . . 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 661 . . 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 5849 . . 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 5849 . . . 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 6280 . . 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 2520 . 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 10563 . . . . . 6  |-  -.  0  e.  NN
16 disjsn 4076 . . . . . 6  |-  ( ( NN  i^i  { 0 } )  =  (/)  <->  -.  0  e.  NN )
1715, 16mpbir 209 . . . . 5  |-  ( NN 
i^i  { 0 } )  =  (/)
18 fnresi 5680 . . . . . 6  |-  (  _I  |`  NN )  Fn  NN
19 c0ex 9579 . . . . . . 7  |-  0  e.  _V
20 snex 4678 . . . . . . 7  |-  { <. 0 ,  P >. }  e.  _V
2119, 20fnsn 5623 . . . . . 6  |-  { <. 0 ,  { <. 0 ,  P >. } >. }  Fn  { 0 }
22 fvun1 5919 . . . . . 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 1312 . . . . 5  |-  ( ( ( NN  i^i  {
0 } )  =  (/)  /\  M  e.  NN )  ->  ( ( (  _I  |`  NN )  u.  { <. 0 ,  { <. 0 ,  P >. }
>. } ) `  M
)  =  ( (  _I  |`  NN ) `  M ) )
2417, 2, 23sylancr 661 . . . 4  |-  ( (
ph  /\  M  e.  NN )  ->  ( ( (  _I  |`  NN )  u.  { <. 0 ,  { <. 0 ,  P >. } >. } ) `  M )  =  ( (  _I  |`  NN ) `
 M ) )
25 fvresi 6073 . . . . 5  |-  ( M  e.  NN  ->  (
(  _I  |`  NN ) `
 M )  =  M )
2625adantl 464 . . . 4  |-  ( (
ph  /\  M  e.  NN )  ->  ( (  _I  |`  NN ) `  M )  =  M )
2724, 26eqtrd 2495 . . 3  |-  ( (
ph  /\  M  e.  NN )  ->  ( ( (  _I  |`  NN )  u.  { <. 0 ,  { <. 0 ,  P >. } >. } ) `  M )  =  M )
2827oveq2d 6286 . 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 5858 . . . 4  |-  ( Q `
 ( M  - 
1 ) )  e. 
_V
3029a1i 11 . . 3  |-  ( ph  ->  ( Q `  ( M  -  1 ) )  e.  _V )
31 simpr 459 . . . . . . . . 9  |-  ( ( x  =  ( Q `
 ( M  - 
1 ) )  /\  m  =  M )  ->  m  =  M )
3231oveq1d 6285 . . . . . . . 8  |-  ( ( x  =  ( Q `
 ( M  - 
1 ) )  /\  m  =  M )  ->  ( m  -  1 )  =  ( M  -  1 ) )
3332oveq1d 6285 . . . . . . 7  |-  ( ( x  =  ( Q `
 ( M  - 
1 ) )  /\  m  =  M )  ->  ( ( m  - 
1 )  /  N
)  =  ( ( M  -  1 )  /  N ) )
3431oveq1d 6285 . . . . . . 7  |-  ( ( x  =  ( Q `
 ( M  - 
1 ) )  /\  m  =  M )  ->  ( m  /  N
)  =  ( M  /  N ) )
3533, 34oveq12d 6288 . . . . . 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 2513 . . . . 5  |-  ( ( x  =  ( Q `
 ( M  - 
1 ) )  /\  m  =  M )  ->  ( ( ( m  -  1 )  /  N ) [,] (
m  /  N ) )  =  W )
3831fveq2d 5852 . . . . . . . . . 10  |-  ( ( x  =  ( Q `
 ( M  - 
1 ) )  /\  m  =  M )  ->  ( T `  m
)  =  ( T `
 M ) )
3938fveq2d 5852 . . . . . . . . 9  |-  ( ( x  =  ( Q `
 ( M  - 
1 ) )  /\  m  =  M )  ->  ( 2nd `  ( T `  m )
)  =  ( 2nd `  ( T `  M
) ) )
40 simpl 455 . . . . . . . . . . 11  |-  ( ( x  =  ( Q `
 ( M  - 
1 ) )  /\  m  =  M )  ->  x  =  ( Q `
 ( M  - 
1 ) ) )
4140, 33fveq12d 5854 . . . . . . . . . 10  |-  ( ( x  =  ( Q `
 ( M  - 
1 ) )  /\  m  =  M )  ->  ( x `  (
( m  -  1 )  /  N ) )  =  ( ( Q `  ( M  -  1 ) ) `
 ( ( M  -  1 )  /  N ) ) )
4241eleq1d 2523 . . . . . . . . 9  |-  ( ( x  =  ( Q `
 ( M  - 
1 ) )  /\  m  =  M )  ->  ( ( x `  ( ( m  - 
1 )  /  N
) )  e.  b  <-> 
( ( Q `  ( M  -  1
) ) `  (
( M  -  1 )  /  N ) )  e.  b ) )
4339, 42riotaeqbidv 6235 . . . . . . . 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 5262 . . . . . . 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 5167 . . . . . 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 5850 . . . . 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 4517 . . . 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 2454 . . . 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 6298 . . . . . 6  |-  ( ( ( M  -  1 )  /  N ) [,] ( M  /  N ) )  e. 
_V
5036, 49eqeltri 2538 . . . . 5  |-  W  e. 
_V
5150mptex 6118 . . . 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 6406 . . 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 469 . 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 2499 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 367    = wceq 1398    e. wcel 1823   A.wral 2804   {crab 2808   _Vcvv 3106    \ cdif 3458    u. cun 3459    i^i cin 3460    C_ wss 3461   (/)c0 3783   ~Pcpw 3999   {csn 4016   <.cop 4022   U.cuni 4235   U_ciun 4315    |-> cmpt 4497    _I cid 4779    X. cxp 4986   `'ccnv 4987   ran crn 4989    |` cres 4990   "cima 4991    Fn wfn 5565   -->wf 5566   ` cfv 5570   iota_crio 6231  (class class class)co 6270    |-> cmpt2 6272   1stc1st 6771   2ndc2nd 6772   0cc0 9481   1c1 9482    + caddc 9484    - cmin 9796    / cdiv 10202   NNcn 10531   ZZcz 10860   ZZ>=cuz 11082   (,)cioo 11532   [,]cicc 11535   ...cfz 11675    seqcseq 12089   ↾t crest 14910   topGenctg 14927    Cn ccn 19892   Homeochmeo 20420   IIcii 21545   CovMap ccvm 28964
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-2nd 6774  df-recs 7034  df-rdg 7068  df-er 7303  df-en 7510  df-dom 7511  df-sdom 7512  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-nn 10532  df-n0 10792  df-z 10861  df-uz 11083  df-seq 12090
This theorem is referenced by:  cvmliftlem6  28999  cvmliftlem8  29001  cvmliftlem9  29002
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