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Theorem caucvgprprlemupu 7515
Description: Lemma for caucvgprpr 7527. 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 7180 . . . . 5  |-  <Q  C_  ( Q.  X.  Q. )
21brel 4591 . . . 4  |-  ( s 
<Q  t  ->  ( s  e.  Q.  /\  t  e.  Q. ) )
32simprd 113 . . 3  |-  ( s 
<Q  t  ->  t  e. 
Q. )
433ad2ant2 1003 . 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 7501 . . . . 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 273 . . . 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 1004 . . 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 7409 . . . . . 6  |-  ( s 
<Q  t  ->  <. { p  |  p  <Q  s } ,  { q  |  s  <Q  q } >.  <P  <. { p  |  p  <Q  t } ,  { q  |  t 
<Q  q } >. )
1093ad2ant2 1003 . . . . 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 7411 . . . . . . 7  |-  <P  Or  P.
12 ltrelpr 7320 . . . . . . 7  |-  <P  C_  ( P.  X.  P. )
1311, 12sotri 4934 . . . . . 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 115 . . . . 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 2533 . . 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 7501 . 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 413 1  |-  ( (
ph  /\  s  <Q  t  /\  s  e.  ( 2nd `  L ) )  ->  t  e.  ( 2nd `  L ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    /\ w3a 962    = wceq 1331    e. wcel 1480   {cab 2125   A.wral 2416   E.wrex 2417   {crab 2420   <.cop 3530   class class class wbr 3929   -->wf 5119   ` cfv 5123  (class class class)co 5774   2ndc2nd 6037   1oc1o 6306   [cec 6427   N.cnpi 7087    <N clti 7090    ~Q ceq 7094   Q.cnq 7095    +Q cplq 7097   *Qcrq 7099    <Q cltq 7100   P.cnp 7106    +P. cpp 7108    <P cltp 7110
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502
This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-eprel 4211  df-id 4215  df-po 4218  df-iso 4219  df-iord 4288  df-on 4290  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-recs 6202  df-irdg 6267  df-1o 6313  df-oadd 6317  df-omul 6318  df-er 6429  df-ec 6431  df-qs 6435  df-ni 7119  df-pli 7120  df-mi 7121  df-lti 7122  df-plpq 7159  df-mpq 7160  df-enq 7162  df-nqqs 7163  df-plqqs 7164  df-mqqs 7165  df-1nqqs 7166  df-rq 7167  df-ltnqqs 7168  df-inp 7281  df-iltp 7285
This theorem is referenced by:  caucvgprprlemrnd  7516
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