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Theorem caucvgprlemlim 6985
Description: Lemma for caucvgpr 6986. The putative limit is a limit. (Contributed by Jim Kingdon, 1-Oct-2020.)
Hypotheses
Ref Expression
caucvgpr.f  |-  ( ph  ->  F : N. --> Q. )
caucvgpr.cau  |-  ( ph  ->  A. n  e.  N.  A. k  e.  N.  (
n  <N  k  ->  (
( F `  n
)  <Q  ( ( F `
 k )  +Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) )  /\  ( F `  k ) 
<Q  ( ( F `  n )  +Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  )
) ) ) )
caucvgpr.bnd  |-  ( ph  ->  A. j  e.  N.  A  <Q  ( F `  j ) )
caucvgpr.lim  |-  L  = 
<. { l  e.  Q.  |  E. j  e.  N.  ( l  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  ( F `  j ) } ,  { u  e.  Q.  |  E. j  e.  N.  ( ( F `  j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  u } >.
Assertion
Ref Expression
caucvgprlemlim  |-  ( ph  ->  A. x  e.  Q.  E. j  e.  N.  A. k  e.  N.  (
j  <N  k  ->  ( <. { l  |  l 
<Q  ( F `  k
) } ,  {
u  |  ( F `
 k )  <Q  u } >.  <P  ( L  +P.  <. { l  |  l  <Q  x } ,  { u  |  x 
<Q  u } >. )  /\  L  <P  <. { l  |  l  <Q  (
( F `  k
)  +Q  x ) } ,  { u  |  ( ( F `
 k )  +Q  x )  <Q  u } >. ) ) )
Distinct variable groups:    A, j    j, F, u, l, k    n, F, k    j, k, ph, x    k, l, u, x, j    j, L, k
Allowed substitution hints:    ph( u, n, l)    A( x, u, k, n, l)    F( x)    L( x, u, n, l)

Proof of Theorem caucvgprlemlim
StepHypRef Expression
1 archrecnq 6967 . . . 4  |-  ( x  e.  Q.  ->  E. j  e.  N.  ( *Q `  [ <. j ,  1o >. ]  ~Q  )  <Q  x )
21adantl 271 . . 3  |-  ( (
ph  /\  x  e.  Q. )  ->  E. j  e.  N.  ( *Q `  [ <. j ,  1o >. ]  ~Q  )  <Q  x )
3 caucvgpr.f . . . . . . . . . 10  |-  ( ph  ->  F : N. --> Q. )
43ad5antr 480 . . . . . . . . 9  |-  ( ( ( ( ( (
ph  /\  x  e.  Q. )  /\  j  e.  N. )  /\  ( *Q `  [ <. j ,  1o >. ]  ~Q  )  <Q  x )  /\  k  e.  N. )  /\  j  <N  k )  ->  F : N. --> Q. )
5 caucvgpr.cau . . . . . . . . . 10  |-  ( ph  ->  A. n  e.  N.  A. k  e.  N.  (
n  <N  k  ->  (
( F `  n
)  <Q  ( ( F `
 k )  +Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) )  /\  ( F `  k ) 
<Q  ( ( F `  n )  +Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  )
) ) ) )
65ad5antr 480 . . . . . . . . 9  |-  ( ( ( ( ( (
ph  /\  x  e.  Q. )  /\  j  e.  N. )  /\  ( *Q `  [ <. j ,  1o >. ]  ~Q  )  <Q  x )  /\  k  e.  N. )  /\  j  <N  k )  ->  A. n  e.  N.  A. k  e. 
N.  ( n  <N  k  ->  ( ( F `
 n )  <Q 
( ( F `  k )  +Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  )
)  /\  ( F `  k )  <Q  (
( F `  n
)  +Q  ( *Q
`  [ <. n ,  1o >. ]  ~Q  )
) ) ) )
7 caucvgpr.bnd . . . . . . . . . 10  |-  ( ph  ->  A. j  e.  N.  A  <Q  ( F `  j ) )
87ad5antr 480 . . . . . . . . 9  |-  ( ( ( ( ( (
ph  /\  x  e.  Q. )  /\  j  e.  N. )  /\  ( *Q `  [ <. j ,  1o >. ]  ~Q  )  <Q  x )  /\  k  e.  N. )  /\  j  <N  k )  ->  A. j  e.  N.  A  <Q  ( F `  j )
)
9 caucvgpr.lim . . . . . . . . 9  |-  L  = 
<. { l  e.  Q.  |  E. j  e.  N.  ( l  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  ( F `  j ) } ,  { u  e.  Q.  |  E. j  e.  N.  ( ( F `  j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  u } >.
10 simpr 108 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  Q. )  ->  x  e. 
Q. )
1110ad4antr 478 . . . . . . . . 9  |-  ( ( ( ( ( (
ph  /\  x  e.  Q. )  /\  j  e.  N. )  /\  ( *Q `  [ <. j ,  1o >. ]  ~Q  )  <Q  x )  /\  k  e.  N. )  /\  j  <N  k )  ->  x  e.  Q. )
12 simpr 108 . . . . . . . . 9  |-  ( ( ( ( ( (
ph  /\  x  e.  Q. )  /\  j  e.  N. )  /\  ( *Q `  [ <. j ,  1o >. ]  ~Q  )  <Q  x )  /\  k  e.  N. )  /\  j  <N  k )  ->  j  <N  k )
13 simpllr 501 . . . . . . . . 9  |-  ( ( ( ( ( (
ph  /\  x  e.  Q. )  /\  j  e.  N. )  /\  ( *Q `  [ <. j ,  1o >. ]  ~Q  )  <Q  x )  /\  k  e.  N. )  /\  j  <N  k )  ->  ( *Q `  [ <. j ,  1o >. ]  ~Q  )  <Q  x )
144, 6, 8, 9, 11, 12, 13caucvgprlem1 6983 . . . . . . . 8  |-  ( ( ( ( ( (
ph  /\  x  e.  Q. )  /\  j  e.  N. )  /\  ( *Q `  [ <. j ,  1o >. ]  ~Q  )  <Q  x )  /\  k  e.  N. )  /\  j  <N  k )  ->  <. { l  |  l  <Q  ( F `  k ) } ,  { u  |  ( F `  k )  <Q  u } >.  <P  ( L  +P.  <. { l  |  l 
<Q  x } ,  {
u  |  x  <Q  u } >. ) )
154, 6, 8, 9, 11, 12, 13caucvgprlem2 6984 . . . . . . . 8  |-  ( ( ( ( ( (
ph  /\  x  e.  Q. )  /\  j  e.  N. )  /\  ( *Q `  [ <. j ,  1o >. ]  ~Q  )  <Q  x )  /\  k  e.  N. )  /\  j  <N  k )  ->  L  <P 
<. { l  |  l 
<Q  ( ( F `  k )  +Q  x
) } ,  {
u  |  ( ( F `  k )  +Q  x )  <Q  u } >. )
1614, 15jca 300 . . . . . . 7  |-  ( ( ( ( ( (
ph  /\  x  e.  Q. )  /\  j  e.  N. )  /\  ( *Q `  [ <. j ,  1o >. ]  ~Q  )  <Q  x )  /\  k  e.  N. )  /\  j  <N  k )  ->  ( <. { l  |  l 
<Q  ( F `  k
) } ,  {
u  |  ( F `
 k )  <Q  u } >.  <P  ( L  +P.  <. { l  |  l  <Q  x } ,  { u  |  x 
<Q  u } >. )  /\  L  <P  <. { l  |  l  <Q  (
( F `  k
)  +Q  x ) } ,  { u  |  ( ( F `
 k )  +Q  x )  <Q  u } >. ) )
1716ex 113 . . . . . 6  |-  ( ( ( ( ( ph  /\  x  e.  Q. )  /\  j  e.  N. )  /\  ( *Q `  [ <. j ,  1o >. ]  ~Q  )  <Q  x )  /\  k  e.  N. )  ->  (
j  <N  k  ->  ( <. { l  |  l 
<Q  ( F `  k
) } ,  {
u  |  ( F `
 k )  <Q  u } >.  <P  ( L  +P.  <. { l  |  l  <Q  x } ,  { u  |  x 
<Q  u } >. )  /\  L  <P  <. { l  |  l  <Q  (
( F `  k
)  +Q  x ) } ,  { u  |  ( ( F `
 k )  +Q  x )  <Q  u } >. ) ) )
1817ralrimiva 2439 . . . . 5  |-  ( ( ( ( ph  /\  x  e.  Q. )  /\  j  e.  N. )  /\  ( *Q `  [ <. j ,  1o >. ]  ~Q  )  <Q  x )  ->  A. k  e.  N.  ( j  <N 
k  ->  ( <. { l  |  l  <Q 
( F `  k
) } ,  {
u  |  ( F `
 k )  <Q  u } >.  <P  ( L  +P.  <. { l  |  l  <Q  x } ,  { u  |  x 
<Q  u } >. )  /\  L  <P  <. { l  |  l  <Q  (
( F `  k
)  +Q  x ) } ,  { u  |  ( ( F `
 k )  +Q  x )  <Q  u } >. ) ) )
1918ex 113 . . . 4  |-  ( ( ( ph  /\  x  e.  Q. )  /\  j  e.  N. )  ->  (
( *Q `  [ <. j ,  1o >. ]  ~Q  )  <Q  x  ->  A. k  e.  N.  ( j  <N  k  ->  ( <. { l  |  l  <Q  ( F `  k ) } ,  { u  |  ( F `  k )  <Q  u } >.  <P  ( L  +P.  <. { l  |  l  <Q  x } ,  { u  |  x 
<Q  u } >. )  /\  L  <P  <. { l  |  l  <Q  (
( F `  k
)  +Q  x ) } ,  { u  |  ( ( F `
 k )  +Q  x )  <Q  u } >. ) ) ) )
2019reximdva 2468 . . 3  |-  ( (
ph  /\  x  e.  Q. )  ->  ( E. j  e.  N.  ( *Q `  [ <. j ,  1o >. ]  ~Q  )  <Q  x  ->  E. j  e.  N.  A. k  e. 
N.  ( j  <N 
k  ->  ( <. { l  |  l  <Q 
( F `  k
) } ,  {
u  |  ( F `
 k )  <Q  u } >.  <P  ( L  +P.  <. { l  |  l  <Q  x } ,  { u  |  x 
<Q  u } >. )  /\  L  <P  <. { l  |  l  <Q  (
( F `  k
)  +Q  x ) } ,  { u  |  ( ( F `
 k )  +Q  x )  <Q  u } >. ) ) ) )
212, 20mpd 13 . 2  |-  ( (
ph  /\  x  e.  Q. )  ->  E. j  e.  N.  A. k  e. 
N.  ( j  <N 
k  ->  ( <. { l  |  l  <Q 
( F `  k
) } ,  {
u  |  ( F `
 k )  <Q  u } >.  <P  ( L  +P.  <. { l  |  l  <Q  x } ,  { u  |  x 
<Q  u } >. )  /\  L  <P  <. { l  |  l  <Q  (
( F `  k
)  +Q  x ) } ,  { u  |  ( ( F `
 k )  +Q  x )  <Q  u } >. ) ) )
2221ralrimiva 2439 1  |-  ( ph  ->  A. x  e.  Q.  E. j  e.  N.  A. k  e.  N.  (
j  <N  k  ->  ( <. { l  |  l 
<Q  ( F `  k
) } ,  {
u  |  ( F `
 k )  <Q  u } >.  <P  ( L  +P.  <. { l  |  l  <Q  x } ,  { u  |  x 
<Q  u } >. )  /\  L  <P  <. { l  |  l  <Q  (
( F `  k
)  +Q  x ) } ,  { u  |  ( ( F `
 k )  +Q  x )  <Q  u } >. ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    = wceq 1285    e. wcel 1434   {cab 2069   A.wral 2353   E.wrex 2354   {crab 2357   <.cop 3419   class class class wbr 3805   -->wf 4948   ` cfv 4952  (class class class)co 5563   1oc1o 6078   [cec 6191   N.cnpi 6576    <N clti 6579    ~Q ceq 6583   Q.cnq 6584    +Q cplq 6586   *Qcrq 6588    <Q cltq 6589    +P. cpp 6597    <P cltp 6599
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 2065  ax-coll 3913  ax-sep 3916  ax-nul 3924  ax-pow 3968  ax-pr 3992  ax-un 4216  ax-setind 4308  ax-iinf 4357
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 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-ral 2358  df-rex 2359  df-reu 2360  df-rab 2362  df-v 2612  df-sbc 2825  df-csb 2918  df-dif 2984  df-un 2986  df-in 2988  df-ss 2995  df-nul 3268  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-uni 3622  df-int 3657  df-iun 3700  df-br 3806  df-opab 3860  df-mpt 3861  df-tr 3896  df-eprel 4072  df-id 4076  df-po 4079  df-iso 4080  df-iord 4149  df-on 4151  df-suc 4154  df-iom 4360  df-xp 4397  df-rel 4398  df-cnv 4399  df-co 4400  df-dm 4401  df-rn 4402  df-res 4403  df-ima 4404  df-iota 4917  df-fun 4954  df-fn 4955  df-f 4956  df-f1 4957  df-fo 4958  df-f1o 4959  df-fv 4960  df-ov 5566  df-oprab 5567  df-mpt2 5568  df-1st 5818  df-2nd 5819  df-recs 5974  df-irdg 6039  df-1o 6085  df-2o 6086  df-oadd 6089  df-omul 6090  df-er 6193  df-ec 6195  df-qs 6199  df-ni 6608  df-pli 6609  df-mi 6610  df-lti 6611  df-plpq 6648  df-mpq 6649  df-enq 6651  df-nqqs 6652  df-plqqs 6653  df-mqqs 6654  df-1nqqs 6655  df-rq 6656  df-ltnqqs 6657  df-enq0 6728  df-nq0 6729  df-0nq0 6730  df-plq0 6731  df-mq0 6732  df-inp 6770  df-iplp 6772  df-iltp 6774
This theorem is referenced by:  caucvgpr  6986
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