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Theorem caucvgprprlemupu 7728
Description: Lemma for caucvgprpr 7740. The upper cut of the putative limit is upper. (Contributed by Jim Kingdon, 21-Dec-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
caucvgprprlemupu  |-  ( (
ph  /\  s  <Q  t  /\  s  e.  ( 2nd `  L ) )  ->  t  e.  ( 2nd `  L ) )
Distinct variable groups:    A, m    m, F    F, l, r, s   
u, F, r, s    L, s    p, l, q, t, r, s    u, p, q, t    ph, r,
s
Allowed substitution hints:    ph( u, t, k, m, n, q, p, l)    A( u, t, k, n, s, r, q, p, l)    F( t, k, n, q, p)    L( u, t, k, m, n, r, q, p, l)

Proof of Theorem caucvgprprlemupu
Dummy variable  b is distinct from all other variables.
StepHypRef Expression
1 ltrelnq 7393 . . . . 5  |-  <Q  C_  ( Q.  X.  Q. )
21brel 4696 . . . 4  |-  ( s 
<Q  t  ->  ( s  e.  Q.  /\  t  e.  Q. ) )
32simprd 114 . . 3  |-  ( s 
<Q  t  ->  t  e. 
Q. )
433ad2ant2 1021 . 2  |-  ( (
ph  /\  s  <Q  t  /\  s  e.  ( 2nd `  L ) )  ->  t  e.  Q. )
5 caucvgprpr.lim . . . . . 6  |-  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 } >. } >.
65caucvgprprlemelu 7714 . . . . 5  |-  ( s  e.  ( 2nd `  L
)  <->  ( s  e. 
Q.  /\  E. b  e.  N.  ( ( F `
 b )  +P. 
<. { p  |  p 
<Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q  q } >. ) 
<P  <. { p  |  p  <Q  s } ,  { q  |  s 
<Q  q } >. )
)
76simprbi 275 . . . 4  |-  ( s  e.  ( 2nd `  L
)  ->  E. b  e.  N.  ( ( F `
 b )  +P. 
<. { p  |  p 
<Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q  q } >. ) 
<P  <. { p  |  p  <Q  s } ,  { q  |  s 
<Q  q } >. )
873ad2ant3 1022 . . 3  |-  ( (
ph  /\  s  <Q  t  /\  s  e.  ( 2nd `  L ) )  ->  E. b  e.  N.  ( ( F `
 b )  +P. 
<. { p  |  p 
<Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q  q } >. ) 
<P  <. { p  |  p  <Q  s } ,  { q  |  s 
<Q  q } >. )
9 ltnqpri 7622 . . . . . 6  |-  ( s 
<Q  t  ->  <. { p  |  p  <Q  s } ,  { q  |  s  <Q  q } >.  <P  <. { p  |  p  <Q  t } ,  { q  |  t 
<Q  q } >. )
1093ad2ant2 1021 . . . . 5  |-  ( (
ph  /\  s  <Q  t  /\  s  e.  ( 2nd `  L ) )  ->  <. { p  |  p  <Q  s } ,  { q  |  s  <Q  q } >.  <P  <. { p  |  p  <Q  t } ,  { q  |  t 
<Q  q } >. )
11 ltsopr 7624 . . . . . . 7  |-  <P  Or  P.
12 ltrelpr 7533 . . . . . . 7  |-  <P  C_  ( P.  X.  P. )
1311, 12sotri 5042 . . . . . 6  |-  ( ( ( ( F `  b )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q 
q } >. )  <P 
<. { p  |  p 
<Q  s } ,  {
q  |  s  <Q 
q } >.  /\  <. { p  |  p  <Q  s } ,  { q  |  s  <Q  q } >.  <P  <. { p  |  p  <Q  t } ,  { q  |  t 
<Q  q } >. )  ->  ( ( F `  b )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q 
q } >. )  <P 
<. { p  |  p 
<Q  t } ,  {
q  |  t  <Q 
q } >. )
1413expcom 116 . . . . 5  |-  ( <. { p  |  p  <Q  s } ,  {
q  |  s  <Q 
q } >.  <P  <. { p  |  p  <Q  t } ,  { q  |  t  <Q  q } >.  ->  ( ( ( F `  b )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q  q } >. )  <P  <. { p  |  p  <Q  s } ,  { q  |  s  <Q  q } >.  ->  ( ( F `
 b )  +P. 
<. { p  |  p 
<Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q  q } >. ) 
<P  <. { p  |  p  <Q  t } ,  { q  |  t 
<Q  q } >. )
)
1510, 14syl 14 . . . 4  |-  ( (
ph  /\  s  <Q  t  /\  s  e.  ( 2nd `  L ) )  ->  ( (
( F `  b
)  +P.  <. { p  |  p  <Q  ( *Q
`  [ <. b ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q 
q } >. )  <P 
<. { p  |  p 
<Q  s } ,  {
q  |  s  <Q 
q } >.  ->  (
( F `  b
)  +P.  <. { p  |  p  <Q  ( *Q
`  [ <. b ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q 
q } >. )  <P 
<. { p  |  p 
<Q  t } ,  {
q  |  t  <Q 
q } >. )
)
1615reximdv 2591 . . 3  |-  ( (
ph  /\  s  <Q  t  /\  s  e.  ( 2nd `  L ) )  ->  ( E. b  e.  N.  (
( F `  b
)  +P.  <. { p  |  p  <Q  ( *Q
`  [ <. b ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q 
q } >. )  <P 
<. { p  |  p 
<Q  s } ,  {
q  |  s  <Q 
q } >.  ->  E. b  e.  N.  ( ( F `
 b )  +P. 
<. { p  |  p 
<Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q  q } >. ) 
<P  <. { p  |  p  <Q  t } ,  { q  |  t 
<Q  q } >. )
)
178, 16mpd 13 . 2  |-  ( (
ph  /\  s  <Q  t  /\  s  e.  ( 2nd `  L ) )  ->  E. b  e.  N.  ( ( F `
 b )  +P. 
<. { p  |  p 
<Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q  q } >. ) 
<P  <. { p  |  p  <Q  t } ,  { q  |  t 
<Q  q } >. )
185caucvgprprlemelu 7714 . 2  |-  ( t  e.  ( 2nd `  L
)  <->  ( t  e. 
Q.  /\  E. b  e.  N.  ( ( F `
 b )  +P. 
<. { p  |  p 
<Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q  q } >. ) 
<P  <. { p  |  p  <Q  t } ,  { q  |  t 
<Q  q } >. )
)
194, 17, 18sylanbrc 417 1  |-  ( (
ph  /\  s  <Q  t  /\  s  e.  ( 2nd `  L ) )  ->  t  e.  ( 2nd `  L ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    /\ w3a 980    = wceq 1364    e. wcel 2160   {cab 2175   A.wral 2468   E.wrex 2469   {crab 2472   <.cop 3610   class class class wbr 4018   -->wf 5231   ` cfv 5235  (class class class)co 5895   2ndc2nd 6163   1oc1o 6433   [cec 6556   N.cnpi 7300    <N clti 7303    ~Q ceq 7307   Q.cnq 7308    +Q cplq 7310   *Qcrq 7312    <Q cltq 7313   P.cnp 7319    +P. cpp 7321    <P cltp 7323
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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-nul 4144  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554  ax-iinf 4605
This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-ral 2473  df-rex 2474  df-reu 2475  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-tr 4117  df-eprel 4307  df-id 4311  df-po 4314  df-iso 4315  df-iord 4384  df-on 4386  df-suc 4389  df-iom 4608  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-f1 5240  df-fo 5241  df-f1o 5242  df-fv 5243  df-ov 5898  df-oprab 5899  df-mpo 5900  df-1st 6164  df-2nd 6165  df-recs 6329  df-irdg 6394  df-1o 6440  df-oadd 6444  df-omul 6445  df-er 6558  df-ec 6560  df-qs 6564  df-ni 7332  df-pli 7333  df-mi 7334  df-lti 7335  df-plpq 7372  df-mpq 7373  df-enq 7375  df-nqqs 7376  df-plqqs 7377  df-mqqs 7378  df-1nqqs 7379  df-rq 7380  df-ltnqqs 7381  df-inp 7494  df-iltp 7498
This theorem is referenced by:  caucvgprprlemrnd  7729
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