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Theorem caucvgprprlemlim 6963
Description: Lemma for caucvgprpr 6964. The putative limit is a limit. (Contributed by Jim Kingdon, 21-Nov-2020.)
Hypotheses
Ref Expression
caucvgprpr.f  |-  ( ph  ->  F : N. --> P. )
caucvgprpr.cau  |-  ( ph  ->  A. n  e.  N.  A. k  e.  N.  (
n  <N  k  ->  (
( F `  n
)  <P  ( ( F `
 k )  +P. 
<. { l  |  l 
<Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >. )  /\  ( F `  k
)  <P  ( ( F `
 n )  +P. 
<. { l  |  l 
<Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >. )
) ) )
caucvgprpr.bnd  |-  ( ph  ->  A. m  e.  N.  A  <P  ( F `  m ) )
caucvgprpr.lim  |-  L  = 
<. { l  e.  Q.  |  E. r  e.  N.  <. { p  |  p  <Q  ( l  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  )
) } ,  {
q  |  ( l  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) ) 
<Q  q } >.  <P  ( F `  r ) } ,  { u  e.  Q.  |  E. r  e.  N.  ( ( F `
 r )  +P. 
<. { p  |  p 
<Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q  q } >. ) 
<P  <. { p  |  p  <Q  u } ,  { q  |  u 
<Q  q } >. } >.
Assertion
Ref Expression
caucvgprprlemlim  |-  ( ph  ->  A. x  e.  P.  E. j  e.  N.  A. k  e.  N.  (
j  <N  k  ->  (
( F `  k
)  <P  ( L  +P.  x )  /\  L  <P  ( ( F `  k )  +P.  x
) ) ) )
Distinct variable groups:    A, m    m, F    A, r, j    u, F, r, l, k, n    ph, k, r    k, L   
j, k, ph, x    k, l, u, p, q, r    j, r, x   
q, l, r    u, p, q, r    m, r   
k, n, u, l   
j, l, u    n, r
Allowed substitution hints:    ph( u, m, n, q, p, l)    A( x, u, k, n, q, p, l)    F( x, j, q, p)    L( x, u, j, m, n, r, q, p, l)

Proof of Theorem caucvgprprlemlim
StepHypRef Expression
1 archrecpr 6916 . . . 4  |-  ( x  e.  P.  ->  E. j  e.  N.  <. { l  |  l  <Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. j ,  1o >. ]  ~Q  )  <Q  u } >.  <P  x )
21adantl 271 . . 3  |-  ( (
ph  /\  x  e.  P. )  ->  E. j  e.  N.  <. { l  |  l  <Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. j ,  1o >. ]  ~Q  )  <Q  u } >.  <P  x )
3 caucvgprpr.f . . . . . . . . . 10  |-  ( ph  ->  F : N. --> P. )
43ad5antr 480 . . . . . . . . 9  |-  ( ( ( ( ( (
ph  /\  x  e.  P. )  /\  j  e.  N. )  /\  <. { l  |  l  <Q 
( *Q `  [ <. j ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. j ,  1o >. ]  ~Q  )  <Q  u } >.  <P  x
)  /\  k  e.  N. )  /\  j  <N  k )  ->  F : N. --> P. )
5 caucvgprpr.cau . . . . . . . . . 10  |-  ( ph  ->  A. n  e.  N.  A. k  e.  N.  (
n  <N  k  ->  (
( F `  n
)  <P  ( ( F `
 k )  +P. 
<. { l  |  l 
<Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >. )  /\  ( F `  k
)  <P  ( ( F `
 n )  +P. 
<. { l  |  l 
<Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >. )
) ) )
65ad5antr 480 . . . . . . . . 9  |-  ( ( ( ( ( (
ph  /\  x  e.  P. )  /\  j  e.  N. )  /\  <. { l  |  l  <Q 
( *Q `  [ <. j ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. j ,  1o >. ]  ~Q  )  <Q  u } >.  <P  x
)  /\  k  e.  N. )  /\  j  <N  k )  ->  A. n  e.  N.  A. k  e. 
N.  ( n  <N  k  ->  ( ( F `
 n )  <P 
( ( F `  k )  +P.  <. { l  |  l  <Q 
( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >. )  /\  ( F `  k
)  <P  ( ( F `
 n )  +P. 
<. { l  |  l 
<Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >. )
) ) )
7 caucvgprpr.bnd . . . . . . . . . 10  |-  ( ph  ->  A. m  e.  N.  A  <P  ( F `  m ) )
87ad5antr 480 . . . . . . . . 9  |-  ( ( ( ( ( (
ph  /\  x  e.  P. )  /\  j  e.  N. )  /\  <. { l  |  l  <Q 
( *Q `  [ <. j ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. j ,  1o >. ]  ~Q  )  <Q  u } >.  <P  x
)  /\  k  e.  N. )  /\  j  <N  k )  ->  A. m  e.  N.  A  <P  ( F `  m )
)
9 caucvgprpr.lim . . . . . . . . 9  |-  L  = 
<. { l  e.  Q.  |  E. r  e.  N.  <. { p  |  p  <Q  ( l  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  )
) } ,  {
q  |  ( l  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) ) 
<Q  q } >.  <P  ( F `  r ) } ,  { u  e.  Q.  |  E. r  e.  N.  ( ( F `
 r )  +P. 
<. { p  |  p 
<Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q  q } >. ) 
<P  <. { p  |  p  <Q  u } ,  { q  |  u 
<Q  q } >. } >.
10 simpr 108 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  P. )  ->  x  e. 
P. )
1110ad4antr 478 . . . . . . . . 9  |-  ( ( ( ( ( (
ph  /\  x  e.  P. )  /\  j  e.  N. )  /\  <. { l  |  l  <Q 
( *Q `  [ <. j ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. j ,  1o >. ]  ~Q  )  <Q  u } >.  <P  x
)  /\  k  e.  N. )  /\  j  <N  k )  ->  x  e.  P. )
12 simpr 108 . . . . . . . . 9  |-  ( ( ( ( ( (
ph  /\  x  e.  P. )  /\  j  e.  N. )  /\  <. { l  |  l  <Q 
( *Q `  [ <. j ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. j ,  1o >. ]  ~Q  )  <Q  u } >.  <P  x
)  /\  k  e.  N. )  /\  j  <N  k )  ->  j  <N  k )
13 simpllr 501 . . . . . . . . 9  |-  ( ( ( ( ( (
ph  /\  x  e.  P. )  /\  j  e.  N. )  /\  <. { l  |  l  <Q 
( *Q `  [ <. j ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. j ,  1o >. ]  ~Q  )  <Q  u } >.  <P  x
)  /\  k  e.  N. )  /\  j  <N  k )  ->  <. { l  |  l  <Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. j ,  1o >. ]  ~Q  )  <Q  u } >.  <P  x )
144, 6, 8, 9, 11, 12, 13caucvgprprlem1 6961 . . . . . . . 8  |-  ( ( ( ( ( (
ph  /\  x  e.  P. )  /\  j  e.  N. )  /\  <. { l  |  l  <Q 
( *Q `  [ <. j ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. j ,  1o >. ]  ~Q  )  <Q  u } >.  <P  x
)  /\  k  e.  N. )  /\  j  <N  k )  ->  ( F `  k )  <P  ( L  +P.  x
) )
154, 6, 8, 9, 11, 12, 13caucvgprprlem2 6962 . . . . . . . 8  |-  ( ( ( ( ( (
ph  /\  x  e.  P. )  /\  j  e.  N. )  /\  <. { l  |  l  <Q 
( *Q `  [ <. j ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. j ,  1o >. ]  ~Q  )  <Q  u } >.  <P  x
)  /\  k  e.  N. )  /\  j  <N  k )  ->  L  <P  ( ( F `  k )  +P.  x
) )
1614, 15jca 300 . . . . . . 7  |-  ( ( ( ( ( (
ph  /\  x  e.  P. )  /\  j  e.  N. )  /\  <. { l  |  l  <Q 
( *Q `  [ <. j ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. j ,  1o >. ]  ~Q  )  <Q  u } >.  <P  x
)  /\  k  e.  N. )  /\  j  <N  k )  ->  (
( F `  k
)  <P  ( L  +P.  x )  /\  L  <P  ( ( F `  k )  +P.  x
) ) )
1716ex 113 . . . . . 6  |-  ( ( ( ( ( ph  /\  x  e.  P. )  /\  j  e.  N. )  /\  <. { l  |  l  <Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. j ,  1o >. ]  ~Q  )  <Q  u } >.  <P  x )  /\  k  e.  N. )  ->  ( j  <N  k  ->  ( ( F `  k )  <P  ( L  +P.  x )  /\  L  <P  ( ( F `
 k )  +P.  x ) ) ) )
1817ralrimiva 2435 . . . . 5  |-  ( ( ( ( ph  /\  x  e.  P. )  /\  j  e.  N. )  /\  <. { l  |  l  <Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. j ,  1o >. ]  ~Q  )  <Q  u } >.  <P  x )  ->  A. k  e.  N.  ( j  <N  k  ->  ( ( F `  k )  <P  ( L  +P.  x )  /\  L  <P  ( ( F `
 k )  +P.  x ) ) ) )
1918ex 113 . . . 4  |-  ( ( ( ph  /\  x  e.  P. )  /\  j  e.  N. )  ->  ( <. { l  |  l 
<Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. j ,  1o >. ]  ~Q  )  <Q  u } >.  <P  x  ->  A. k  e.  N.  ( j  <N  k  ->  ( ( F `  k )  <P  ( L  +P.  x )  /\  L  <P  ( ( F `
 k )  +P.  x ) ) ) ) )
2019reximdva 2464 . . 3  |-  ( (
ph  /\  x  e.  P. )  ->  ( E. j  e.  N.  <. { l  |  l  <Q 
( *Q `  [ <. j ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. j ,  1o >. ]  ~Q  )  <Q  u } >.  <P  x  ->  E. j  e.  N.  A. k  e.  N.  (
j  <N  k  ->  (
( F `  k
)  <P  ( L  +P.  x )  /\  L  <P  ( ( F `  k )  +P.  x
) ) ) ) )
212, 20mpd 13 . 2  |-  ( (
ph  /\  x  e.  P. )  ->  E. j  e.  N.  A. k  e. 
N.  ( j  <N 
k  ->  ( ( F `  k )  <P  ( L  +P.  x
)  /\  L  <P  ( ( F `  k
)  +P.  x )
) ) )
2221ralrimiva 2435 1  |-  ( ph  ->  A. x  e.  P.  E. j  e.  N.  A. k  e.  N.  (
j  <N  k  ->  (
( F `  k
)  <P  ( L  +P.  x )  /\  L  <P  ( ( F `  k )  +P.  x
) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    = wceq 1285    e. wcel 1434   {cab 2068   A.wral 2349   E.wrex 2350   {crab 2353   <.cop 3409   class class class wbr 3793   -->wf 4928   ` cfv 4932  (class class class)co 5543   1oc1o 6058   [cec 6170   N.cnpi 6524    <N clti 6527    ~Q ceq 6531   Q.cnq 6532    +Q cplq 6534   *Qcrq 6536    <Q cltq 6537   P.cnp 6543    +P. cpp 6545    <P cltp 6547
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-coll 3901  ax-sep 3904  ax-nul 3912  ax-pow 3956  ax-pr 3972  ax-un 4196  ax-setind 4288  ax-iinf 4337
This theorem depends on definitions:  df-bi 115  df-dc 777  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ne 2247  df-ral 2354  df-rex 2355  df-reu 2356  df-rab 2358  df-v 2604  df-sbc 2817  df-csb 2910  df-dif 2976  df-un 2978  df-in 2980  df-ss 2987  df-nul 3259  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-uni 3610  df-int 3645  df-iun 3688  df-br 3794  df-opab 3848  df-mpt 3849  df-tr 3884  df-eprel 4052  df-id 4056  df-po 4059  df-iso 4060  df-iord 4129  df-on 4131  df-suc 4134  df-iom 4340  df-xp 4377  df-rel 4378  df-cnv 4379  df-co 4380  df-dm 4381  df-rn 4382  df-res 4383  df-ima 4384  df-iota 4897  df-fun 4934  df-fn 4935  df-f 4936  df-f1 4937  df-fo 4938  df-f1o 4939  df-fv 4940  df-ov 5546  df-oprab 5547  df-mpt2 5548  df-1st 5798  df-2nd 5799  df-recs 5954  df-irdg 6019  df-1o 6065  df-2o 6066  df-oadd 6069  df-omul 6070  df-er 6172  df-ec 6174  df-qs 6178  df-ni 6556  df-pli 6557  df-mi 6558  df-lti 6559  df-plpq 6596  df-mpq 6597  df-enq 6599  df-nqqs 6600  df-plqqs 6601  df-mqqs 6602  df-1nqqs 6603  df-rq 6604  df-ltnqqs 6605  df-enq0 6676  df-nq0 6677  df-0nq0 6678  df-plq0 6679  df-mq0 6680  df-inp 6718  df-iplp 6720  df-iltp 6722
This theorem is referenced by:  caucvgprpr  6964
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