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Theorem cvmliftlem4 23819
Description: Lemma for cvmlift 23830. The function  Q will be our lifted path, defined piecewise on each section  [ ( M  -  1 )  /  N ,  M  /  N ] for  M  e.  ( 1 ... N ). For 
M  =  0, it is a "seed" value which makes the rest of the recursion work, a singleton function mapping  0 to  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 >. } >. } ) )
Assertion
Ref Expression
cvmliftlem4  |-  ( Q `
 0 )  =  { <. 0 ,  P >. }
Distinct variable groups:    v, b,
z, B    j, b,
k, m, s, u, x, F, v, z   
z, L    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 cvmliftlem4
StepHypRef Expression
1 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 >. } >. } ) )
21fveq1i 5526 . . . 4  |-  ( Q `
 0 )  =  (  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 >. } >. } ) ) `
 0 )
3 0z 10035 . . . . 5  |-  0  e.  ZZ
4 seq1 11059 . . . . 5  |-  ( 0  e.  ZZ  ->  (  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 >. } >. } ) ) `
 0 )  =  ( ( (  _I  |`  NN )  u.  { <. 0 ,  { <. 0 ,  P >. }
>. } ) `  0
) )
53, 4ax-mp 8 . . . 4  |-  (  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 >. } >. } ) ) `
 0 )  =  ( ( (  _I  |`  NN )  u.  { <. 0 ,  { <. 0 ,  P >. }
>. } ) `  0
)
62, 5eqtri 2303 . . 3  |-  ( Q `
 0 )  =  ( ( (  _I  |`  NN )  u.  { <. 0 ,  { <. 0 ,  P >. }
>. } ) `  0
)
7 fnresi 5361 . . . 4  |-  (  _I  |`  NN )  Fn  NN
8 c0ex 8832 . . . . 5  |-  0  e.  _V
9 snex 4216 . . . . 5  |-  { <. 0 ,  P >. }  e.  _V
108, 9fnsn 5304 . . . 4  |-  { <. 0 ,  { <. 0 ,  P >. } >. }  Fn  { 0 }
11 0nnn 9777 . . . . . 6  |-  -.  0  e.  NN
12 disjsn 3693 . . . . . 6  |-  ( ( NN  i^i  { 0 } )  =  (/)  <->  -.  0  e.  NN )
1311, 12mpbir 200 . . . . 5  |-  ( NN 
i^i  { 0 } )  =  (/)
148snid 3667 . . . . 5  |-  0  e.  { 0 }
1513, 14pm3.2i 441 . . . 4  |-  ( ( NN  i^i  { 0 } )  =  (/)  /\  0  e.  { 0 } )
16 fvun2 5591 . . . 4  |-  ( ( (  _I  |`  NN )  Fn  NN  /\  { <. 0 ,  { <. 0 ,  P >. }
>. }  Fn  { 0 }  /\  ( ( NN  i^i  { 0 } )  =  (/)  /\  0  e.  { 0 } ) )  -> 
( ( (  _I  |`  NN )  u.  { <. 0 ,  { <. 0 ,  P >. }
>. } ) `  0
)  =  ( {
<. 0 ,  { <. 0 ,  P >. }
>. } `  0 ) )
177, 10, 15, 16mp3an 1277 . . 3  |-  ( ( (  _I  |`  NN )  u.  { <. 0 ,  { <. 0 ,  P >. } >. } ) ` 
0 )  =  ( { <. 0 ,  { <. 0 ,  P >. }
>. } `  0 )
186, 17eqtri 2303 . 2  |-  ( Q `
 0 )  =  ( { <. 0 ,  { <. 0 ,  P >. } >. } `  0
)
198, 9fvsn 5713 . 2  |-  ( {
<. 0 ,  { <. 0 ,  P >. }
>. } `  0 )  =  { <. 0 ,  P >. }
2018, 19eqtri 2303 1  |-  ( Q `
 0 )  =  { <. 0 ,  P >. }
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543   {crab 2547   _Vcvv 2788    \ cdif 3149    u. cun 3150    i^i cin 3151    C_ wss 3152   (/)c0 3455   ~Pcpw 3625   {csn 3640   <.cop 3643   U.cuni 3827   U_ciun 3905    e. cmpt 4077    _I cid 4304    X. cxp 4687   `'ccnv 4688   ran crn 4690    |` cres 4691   "cima 4692    Fn wfn 5250   -->wf 5251   ` cfv 5255  (class class class)co 5858    e. cmpt2 5860   1stc1st 6120   2ndc2nd 6121   iota_crio 6297   0cc0 8737   1c1 8738    - cmin 9037    / cdiv 9423   NNcn 9746   ZZcz 10024   (,)cioo 10656   [,]cicc 10659   ...cfz 10782    seq cseq 11046   ↾t crest 13325   topGenctg 13342    Cn ccn 16954    Homeo chmeo 17444   IIcii 18379   CovMap ccvm 23786
This theorem is referenced by:  cvmliftlem7  23822  cvmliftlem13  23827
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-n0 9966  df-z 10025  df-uz 10231  df-seq 11047
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