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Theorem caucvgprlemlim 7677
Description: Lemma for caucvgpr 7678. 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 7659 . . . 4  |-  ( x  e.  Q.  ->  E. j  e.  N.  ( *Q `  [ <. j ,  1o >. ]  ~Q  )  <Q  x )
21adantl 277 . . 3  |-  ( (
ph  /\  x  e.  Q. )  ->  E. j  e.  N.  ( *Q `  [ <. j ,  1o >. ]  ~Q  )  <Q  x )
3 caucvgpr.f . . . . . . . . . 10  |-  ( ph  ->  F : N. --> Q. )
43ad5antr 496 . . . . . . . . 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 496 . . . . . . . . 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 496 . . . . . . . . 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 110 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  Q. )  ->  x  e. 
Q. )
1110ad4antr 494 . . . . . . . . 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 110 . . . . . . . . 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 534 . . . . . . . . 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 7675 . . . . . . . 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 7676 . . . . . . . 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 306 . . . . . . 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 115 . . . . . 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 2550 . . . . 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 115 . . . 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 2579 . . 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 2550 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 104    = wceq 1353    e. wcel 2148   {cab 2163   A.wral 2455   E.wrex 2456   {crab 2459   <.cop 3595   class class class wbr 4002   -->wf 5211   ` cfv 5215  (class class class)co 5872   1oc1o 6407   [cec 6530   N.cnpi 7268    <N clti 7271    ~Q ceq 7275   Q.cnq 7276    +Q cplq 7278   *Qcrq 7280    <Q cltq 7281    +P. cpp 7289    <P cltp 7291
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4117  ax-sep 4120  ax-nul 4128  ax-pow 4173  ax-pr 4208  ax-un 4432  ax-setind 4535  ax-iinf 4586
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-iun 3888  df-br 4003  df-opab 4064  df-mpt 4065  df-tr 4101  df-eprel 4288  df-id 4292  df-po 4295  df-iso 4296  df-iord 4365  df-on 4367  df-suc 4370  df-iom 4589  df-xp 4631  df-rel 4632  df-cnv 4633  df-co 4634  df-dm 4635  df-rn 4636  df-res 4637  df-ima 4638  df-iota 5177  df-fun 5217  df-fn 5218  df-f 5219  df-f1 5220  df-fo 5221  df-f1o 5222  df-fv 5223  df-ov 5875  df-oprab 5876  df-mpo 5877  df-1st 6138  df-2nd 6139  df-recs 6303  df-irdg 6368  df-1o 6414  df-2o 6415  df-oadd 6418  df-omul 6419  df-er 6532  df-ec 6534  df-qs 6538  df-ni 7300  df-pli 7301  df-mi 7302  df-lti 7303  df-plpq 7340  df-mpq 7341  df-enq 7343  df-nqqs 7344  df-plqqs 7345  df-mqqs 7346  df-1nqqs 7347  df-rq 7348  df-ltnqqs 7349  df-enq0 7420  df-nq0 7421  df-0nq0 7422  df-plq0 7423  df-mq0 7424  df-inp 7462  df-iplp 7464  df-iltp 7466
This theorem is referenced by:  caucvgpr  7678
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