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Theorem cvmliftlem4 24977
Description: Lemma for cvmlift 24988. 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 5731 . . . 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 10295 . . . . 5  |-  0  e.  ZZ
4 seq1 11338 . . . . 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 2458 . . 3  |-  ( Q `
 0 )  =  ( ( (  _I  |`  NN )  u.  { <. 0 ,  { <. 0 ,  P >. }
>. } ) `  0
)
7 fnresi 5564 . . . 4  |-  (  _I  |`  NN )  Fn  NN
8 c0ex 9087 . . . . 5  |-  0  e.  _V
9 snex 4407 . . . . 5  |-  { <. 0 ,  P >. }  e.  _V
108, 9fnsn 5506 . . . 4  |-  { <. 0 ,  { <. 0 ,  P >. } >. }  Fn  { 0 }
11 0nnn 10033 . . . . . 6  |-  -.  0  e.  NN
12 disjsn 3870 . . . . . 6  |-  ( ( NN  i^i  { 0 } )  =  (/)  <->  -.  0  e.  NN )
1311, 12mpbir 202 . . . . 5  |-  ( NN 
i^i  { 0 } )  =  (/)
148snid 3843 . . . . 5  |-  0  e.  { 0 }
1513, 14pm3.2i 443 . . . 4  |-  ( ( NN  i^i  { 0 } )  =  (/)  /\  0  e.  { 0 } )
16 fvun2 5797 . . . 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 1280 . . 3  |-  ( ( (  _I  |`  NN )  u.  { <. 0 ,  { <. 0 ,  P >. } >. } ) ` 
0 )  =  ( { <. 0 ,  { <. 0 ,  P >. }
>. } `  0 )
186, 17eqtri 2458 . 2  |-  ( Q `
 0 )  =  ( { <. 0 ,  { <. 0 ,  P >. } >. } `  0
)
198, 9fvsn 5928 . 2  |-  ( {
<. 0 ,  { <. 0 ,  P >. }
>. } `  0 )  =  { <. 0 ,  P >. }
2018, 19eqtri 2458 1  |-  ( Q `
 0 )  =  { <. 0 ,  P >. }
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 360    = wceq 1653    e. wcel 1726   A.wral 2707   {crab 2711   _Vcvv 2958    \ cdif 3319    u. cun 3320    i^i cin 3321    C_ wss 3322   (/)c0 3630   ~Pcpw 3801   {csn 3816   <.cop 3819   U.cuni 4017   U_ciun 4095    e. cmpt 4268    _I cid 4495    X. cxp 4878   `'ccnv 4879   ran crn 4881    |` cres 4882   "cima 4883    Fn wfn 5451   -->wf 5452   ` cfv 5456  (class class class)co 6083    e. cmpt2 6085   1stc1st 6349   2ndc2nd 6350   iota_crio 6544   0cc0 8992   1c1 8993    - cmin 9293    / cdiv 9679   NNcn 10002   ZZcz 10284   (,)cioo 10918   [,]cicc 10921   ...cfz 11045    seq cseq 11325   ↾t crest 13650   topGenctg 13667    Cn ccn 17290    Homeo chmeo 17787   IIcii 18907   CovMap ccvm 24944
This theorem is referenced by:  cvmliftlem7  24980  cvmliftlem13  24985
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703  ax-cnex 9048  ax-resscn 9049  ax-1cn 9050  ax-icn 9051  ax-addcl 9052  ax-addrcl 9053  ax-mulcl 9054  ax-mulrcl 9055  ax-mulcom 9056  ax-addass 9057  ax-mulass 9058  ax-distr 9059  ax-i2m1 9060  ax-1ne0 9061  ax-1rid 9062  ax-rnegex 9063  ax-rrecex 9064  ax-cnre 9065  ax-pre-lttri 9066  ax-pre-lttrn 9067  ax-pre-ltadd 9068  ax-pre-mulgt0 9069
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-2nd 6352  df-riota 6551  df-recs 6635  df-rdg 6670  df-er 6907  df-en 7112  df-dom 7113  df-sdom 7114  df-pnf 9124  df-mnf 9125  df-xr 9126  df-ltxr 9127  df-le 9128  df-sub 9295  df-neg 9296  df-nn 10003  df-n0 10224  df-z 10285  df-uz 10491  df-seq 11326
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