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Theorem cvmliftlem4 27297
Description: Lemma for cvmlift 27308. 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 5776 . . . 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 10744 . . . . 5  |-  0  e.  ZZ
4 seq1 11906 . . . . 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 2478 . . 3  |-  ( Q `
 0 )  =  ( ( (  _I  |`  NN )  u.  { <. 0 ,  { <. 0 ,  P >. }
>. } ) `  0
)
7 fnresi 5612 . . . 4  |-  (  _I  |`  NN )  Fn  NN
8 c0ex 9467 . . . . 5  |-  0  e.  _V
9 snex 4617 . . . . 5  |-  { <. 0 ,  P >. }  e.  _V
108, 9fnsn 5555 . . . 4  |-  { <. 0 ,  { <. 0 ,  P >. } >. }  Fn  { 0 }
11 0nnn 10440 . . . . . 6  |-  -.  0  e.  NN
12 disjsn 4020 . . . . . 6  |-  ( ( NN  i^i  { 0 } )  =  (/)  <->  -.  0  e.  NN )
1311, 12mpbir 209 . . . . 5  |-  ( NN 
i^i  { 0 } )  =  (/)
148snid 3989 . . . . 5  |-  0  e.  { 0 }
1513, 14pm3.2i 455 . . . 4  |-  ( ( NN  i^i  { 0 } )  =  (/)  /\  0  e.  { 0 } )
16 fvun2 5848 . . . 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 1315 . . 3  |-  ( ( (  _I  |`  NN )  u.  { <. 0 ,  { <. 0 ,  P >. } >. } ) ` 
0 )  =  ( { <. 0 ,  { <. 0 ,  P >. }
>. } `  0 )
186, 17eqtri 2478 . 2  |-  ( Q `
 0 )  =  ( { <. 0 ,  { <. 0 ,  P >. } >. } `  0
)
198, 9fvsn 5996 . 2  |-  ( {
<. 0 ,  { <. 0 ,  P >. }
>. } `  0 )  =  { <. 0 ,  P >. }
2018, 19eqtri 2478 1  |-  ( Q `
 0 )  =  { <. 0 ,  P >. }
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1757   A.wral 2792   {crab 2796   _Vcvv 3054    \ cdif 3409    u. cun 3410    i^i cin 3411    C_ wss 3412   (/)c0 3721   ~Pcpw 3944   {csn 3961   <.cop 3967   U.cuni 4175   U_ciun 4255    |-> cmpt 4434    _I cid 4715    X. cxp 4922   `'ccnv 4923   ran crn 4925    |` cres 4926   "cima 4927    Fn wfn 5497   -->wf 5498   ` cfv 5502   iota_crio 6136  (class class class)co 6176    |-> cmpt2 6178   1stc1st 6661   2ndc2nd 6662   0cc0 9369   1c1 9370    - cmin 9682    / cdiv 10080   NNcn 10409   ZZcz 10733   (,)cioo 11387   [,]cicc 11390   ...cfz 11524    seqcseq 11893   ↾t crest 14447   topGenctg 14464    Cn ccn 18930   Homeochmeo 19428   IIcii 20553   CovMap ccvm 27264
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1709  ax-7 1729  ax-8 1759  ax-9 1761  ax-10 1776  ax-11 1781  ax-12 1793  ax-13 1944  ax-ext 2429  ax-sep 4497  ax-nul 4505  ax-pow 4554  ax-pr 4615  ax-un 6458  ax-cnex 9425  ax-resscn 9426  ax-1cn 9427  ax-icn 9428  ax-addcl 9429  ax-addrcl 9430  ax-mulcl 9431  ax-mulrcl 9432  ax-mulcom 9433  ax-addass 9434  ax-mulass 9435  ax-distr 9436  ax-i2m1 9437  ax-1ne0 9438  ax-1rid 9439  ax-rnegex 9440  ax-rrecex 9441  ax-cnre 9442  ax-pre-lttri 9443  ax-pre-lttrn 9444  ax-pre-ltadd 9445  ax-pre-mulgt0 9446
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1702  df-eu 2263  df-mo 2264  df-clab 2436  df-cleq 2442  df-clel 2445  df-nfc 2598  df-ne 2643  df-nel 2644  df-ral 2797  df-rex 2798  df-reu 2799  df-rab 2801  df-v 3056  df-sbc 3271  df-csb 3373  df-dif 3415  df-un 3417  df-in 3419  df-ss 3426  df-pss 3428  df-nul 3722  df-if 3876  df-pw 3946  df-sn 3962  df-pr 3964  df-tp 3966  df-op 3968  df-uni 4176  df-iun 4257  df-br 4377  df-opab 4435  df-mpt 4436  df-tr 4470  df-eprel 4716  df-id 4720  df-po 4725  df-so 4726  df-fr 4763  df-we 4765  df-ord 4806  df-on 4807  df-lim 4808  df-suc 4809  df-xp 4930  df-rel 4931  df-cnv 4932  df-co 4933  df-dm 4934  df-rn 4935  df-res 4936  df-ima 4937  df-iota 5465  df-fun 5504  df-fn 5505  df-f 5506  df-f1 5507  df-fo 5508  df-f1o 5509  df-fv 5510  df-riota 6137  df-ov 6179  df-oprab 6180  df-mpt2 6181  df-om 6563  df-2nd 6664  df-recs 6918  df-rdg 6952  df-er 7187  df-en 7397  df-dom 7398  df-sdom 7399  df-pnf 9507  df-mnf 9508  df-xr 9509  df-ltxr 9510  df-le 9511  df-sub 9684  df-neg 9685  df-nn 10410  df-n0 10667  df-z 10734  df-uz 10949  df-seq 11894
This theorem is referenced by:  cvmliftlem7  27300  cvmliftlem13  27305
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