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Theorem cvmliftlem4 28930
Description: Lemma for cvmlift 28941. 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 5873 . . . 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 10896 . . . . 5  |-  0  e.  ZZ
4 seq1 12123 . . . . 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 5 . . . 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 2486 . . 3  |-  ( Q `
 0 )  =  ( ( (  _I  |`  NN )  u.  { <. 0 ,  { <. 0 ,  P >. }
>. } ) `  0
)
7 fnresi 5704 . . . 4  |-  (  _I  |`  NN )  Fn  NN
8 c0ex 9607 . . . . 5  |-  0  e.  _V
9 snex 4697 . . . . 5  |-  { <. 0 ,  P >. }  e.  _V
108, 9fnsn 5647 . . . 4  |-  { <. 0 ,  { <. 0 ,  P >. } >. }  Fn  { 0 }
11 0nnn 10588 . . . . . 6  |-  -.  0  e.  NN
12 disjsn 4092 . . . . . 6  |-  ( ( NN  i^i  { 0 } )  =  (/)  <->  -.  0  e.  NN )
1311, 12mpbir 209 . . . . 5  |-  ( NN 
i^i  { 0 } )  =  (/)
148snid 4060 . . . . 5  |-  0  e.  { 0 }
1513, 14pm3.2i 455 . . . 4  |-  ( ( NN  i^i  { 0 } )  =  (/)  /\  0  e.  { 0 } )
16 fvun2 5945 . . . 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 1324 . . 3  |-  ( ( (  _I  |`  NN )  u.  { <. 0 ,  { <. 0 ,  P >. } >. } ) ` 
0 )  =  ( { <. 0 ,  { <. 0 ,  P >. }
>. } `  0 )
186, 17eqtri 2486 . 2  |-  ( Q `
 0 )  =  ( { <. 0 ,  { <. 0 ,  P >. } >. } `  0
)
198, 9fvsn 6105 . 2  |-  ( {
<. 0 ,  { <. 0 ,  P >. }
>. } `  0 )  =  { <. 0 ,  P >. }
2018, 19eqtri 2486 1  |-  ( Q `
 0 )  =  { <. 0 ,  P >. }
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    = wceq 1395    e. wcel 1819   A.wral 2807   {crab 2811   _Vcvv 3109    \ cdif 3468    u. cun 3469    i^i cin 3470    C_ wss 3471   (/)c0 3793   ~Pcpw 4015   {csn 4032   <.cop 4038   U.cuni 4251   U_ciun 4332    |-> cmpt 4515    _I cid 4799    X. cxp 5006   `'ccnv 5007   ran crn 5009    |` cres 5010   "cima 5011    Fn wfn 5589   -->wf 5590   ` cfv 5594   iota_crio 6257  (class class class)co 6296    |-> cmpt2 6298   1stc1st 6797   2ndc2nd 6798   0cc0 9509   1c1 9510    - cmin 9824    / cdiv 10227   NNcn 10556   ZZcz 10885   (,)cioo 11554   [,]cicc 11557   ...cfz 11697    seqcseq 12110   ↾t crest 14838   topGenctg 14855    Cn ccn 19852   Homeochmeo 20380   IIcii 21505   CovMap ccvm 28897
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-2nd 6800  df-recs 7060  df-rdg 7094  df-er 7329  df-en 7536  df-dom 7537  df-sdom 7538  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-nn 10557  df-n0 10817  df-z 10886  df-uz 11107  df-seq 12111
This theorem is referenced by:  cvmliftlem7  28933  cvmliftlem13  28938
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