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Theorem caucvgprlemlim 7238
Description: Lemma for caucvgpr 7239. 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 7220 . . . 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 7236 . . . . . . . 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 7237 . . . . . . . 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 2446 . . . . 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 2475 . . 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 2446 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 1289    e. wcel 1438   {cab 2074   A.wral 2359   E.wrex 2360   {crab 2363   <.cop 3449   class class class wbr 3845   -->wf 5011   ` cfv 5015  (class class class)co 5652   1oc1o 6174   [cec 6288   N.cnpi 6829    <N clti 6832    ~Q ceq 6836   Q.cnq 6837    +Q cplq 6839   *Qcrq 6841    <Q cltq 6842    +P. cpp 6850    <P cltp 6852
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 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-coll 3954  ax-sep 3957  ax-nul 3965  ax-pow 4009  ax-pr 4036  ax-un 4260  ax-setind 4353  ax-iinf 4403
This theorem depends on definitions:  df-bi 115  df-dc 781  df-3or 925  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-ral 2364  df-rex 2365  df-reu 2366  df-rab 2368  df-v 2621  df-sbc 2841  df-csb 2934  df-dif 3001  df-un 3003  df-in 3005  df-ss 3012  df-nul 3287  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-int 3689  df-iun 3732  df-br 3846  df-opab 3900  df-mpt 3901  df-tr 3937  df-eprel 4116  df-id 4120  df-po 4123  df-iso 4124  df-iord 4193  df-on 4195  df-suc 4198  df-iom 4406  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-rn 4449  df-res 4450  df-ima 4451  df-iota 4980  df-fun 5017  df-fn 5018  df-f 5019  df-f1 5020  df-fo 5021  df-f1o 5022  df-fv 5023  df-ov 5655  df-oprab 5656  df-mpt2 5657  df-1st 5911  df-2nd 5912  df-recs 6070  df-irdg 6135  df-1o 6181  df-2o 6182  df-oadd 6185  df-omul 6186  df-er 6290  df-ec 6292  df-qs 6296  df-ni 6861  df-pli 6862  df-mi 6863  df-lti 6864  df-plpq 6901  df-mpq 6902  df-enq 6904  df-nqqs 6905  df-plqqs 6906  df-mqqs 6907  df-1nqqs 6908  df-rq 6909  df-ltnqqs 6910  df-enq0 6981  df-nq0 6982  df-0nq0 6983  df-plq0 6984  df-mq0 6985  df-inp 7023  df-iplp 7025  df-iltp 7027
This theorem is referenced by:  caucvgpr  7239
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