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Theorem caucvgprprlemrnd 7023
Description: Lemma for caucvgprpr 7034. The putative limit is rounded. (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
caucvgprprlemrnd  |-  ( ph  ->  ( A. s  e. 
Q.  ( s  e.  ( 1st `  L
)  <->  E. t  e.  Q.  ( s  <Q  t  /\  t  e.  ( 1st `  L ) ) )  /\  A. t  e.  Q.  ( t  e.  ( 2nd `  L
)  <->  E. s  e.  Q.  ( s  <Q  t  /\  s  e.  ( 2nd `  L ) ) ) ) )
Distinct variable groups:    A, m    m, F    F, l, t    u, F, t, r, s    L, s, t    p, l, q, r, s, t    u, p, q, r, s    ph, r,
s, t
Allowed substitution hints:    ph( u, k, m, n, q, p, l)    A( u, t, k, n, s, r, q, p, l)    F( k, n, q, p)    L( u, k, m, n, r, q, p, l)

Proof of Theorem caucvgprprlemrnd
StepHypRef Expression
1 caucvgprpr.f . . . . . 6  |-  ( ph  ->  F : N. --> P. )
2 caucvgprpr.cau . . . . . 6  |-  ( 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 } >. )
) ) )
3 caucvgprpr.bnd . . . . . 6  |-  ( ph  ->  A. m  e.  N.  A  <P  ( F `  m ) )
4 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 } >. } >.
51, 2, 3, 4caucvgprprlemopl 7019 . . . . 5  |-  ( (
ph  /\  s  e.  ( 1st `  L ) )  ->  E. t  e.  Q.  ( s  <Q 
t  /\  t  e.  ( 1st `  L ) ) )
65ex 113 . . . 4  |-  ( ph  ->  ( s  e.  ( 1st `  L )  ->  E. t  e.  Q.  ( s  <Q  t  /\  t  e.  ( 1st `  L ) ) ) )
71, 2, 3, 4caucvgprprlemlol 7020 . . . . . 6  |-  ( (
ph  /\  s  <Q  t  /\  t  e.  ( 1st `  L ) )  ->  s  e.  ( 1st `  L ) )
873expib 1142 . . . . 5  |-  ( ph  ->  ( ( s  <Q 
t  /\  t  e.  ( 1st `  L ) )  ->  s  e.  ( 1st `  L ) ) )
98rexlimdvw 2485 . . . 4  |-  ( ph  ->  ( E. t  e. 
Q.  ( s  <Q 
t  /\  t  e.  ( 1st `  L ) )  ->  s  e.  ( 1st `  L ) ) )
106, 9impbid 127 . . 3  |-  ( ph  ->  ( s  e.  ( 1st `  L )  <->  E. t  e.  Q.  ( s  <Q  t  /\  t  e.  ( 1st `  L ) ) ) )
1110ralrimivw 2440 . 2  |-  ( ph  ->  A. s  e.  Q.  ( s  e.  ( 1st `  L )  <->  E. t  e.  Q.  ( s  <Q  t  /\  t  e.  ( 1st `  L ) ) ) )
121, 2, 3, 4caucvgprprlemopu 7021 . . . . 5  |-  ( (
ph  /\  t  e.  ( 2nd `  L ) )  ->  E. s  e.  Q.  ( s  <Q 
t  /\  s  e.  ( 2nd `  L ) ) )
1312ex 113 . . . 4  |-  ( ph  ->  ( t  e.  ( 2nd `  L )  ->  E. s  e.  Q.  ( s  <Q  t  /\  s  e.  ( 2nd `  L ) ) ) )
141, 2, 3, 4caucvgprprlemupu 7022 . . . . . 6  |-  ( (
ph  /\  s  <Q  t  /\  s  e.  ( 2nd `  L ) )  ->  t  e.  ( 2nd `  L ) )
15143expib 1142 . . . . 5  |-  ( ph  ->  ( ( s  <Q 
t  /\  s  e.  ( 2nd `  L ) )  ->  t  e.  ( 2nd `  L ) ) )
1615rexlimdvw 2485 . . . 4  |-  ( ph  ->  ( E. s  e. 
Q.  ( s  <Q 
t  /\  s  e.  ( 2nd `  L ) )  ->  t  e.  ( 2nd `  L ) ) )
1713, 16impbid 127 . . 3  |-  ( ph  ->  ( t  e.  ( 2nd `  L )  <->  E. s  e.  Q.  ( s  <Q  t  /\  s  e.  ( 2nd `  L ) ) ) )
1817ralrimivw 2440 . 2  |-  ( ph  ->  A. t  e.  Q.  ( t  e.  ( 2nd `  L )  <->  E. s  e.  Q.  ( s  <Q  t  /\  s  e.  ( 2nd `  L ) ) ) )
1911, 18jca 300 1  |-  ( ph  ->  ( A. s  e. 
Q.  ( s  e.  ( 1st `  L
)  <->  E. t  e.  Q.  ( s  <Q  t  /\  t  e.  ( 1st `  L ) ) )  /\  A. t  e.  Q.  ( t  e.  ( 2nd `  L
)  <->  E. s  e.  Q.  ( s  <Q  t  /\  s  e.  ( 2nd `  L ) ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    = 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 5564   1stc1st 5817   2ndc2nd 5818   1oc1o 6079   [cec 6192   N.cnpi 6594    <N clti 6597    ~Q ceq 6601   Q.cnq 6602    +Q cplq 6604   *Qcrq 6606    <Q cltq 6607   P.cnp 6613    +P. cpp 6615    <P cltp 6617
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 5567  df-oprab 5568  df-mpt2 5569  df-1st 5819  df-2nd 5820  df-recs 5975  df-irdg 6040  df-1o 6086  df-2o 6087  df-oadd 6090  df-omul 6091  df-er 6194  df-ec 6196  df-qs 6200  df-ni 6626  df-pli 6627  df-mi 6628  df-lti 6629  df-plpq 6666  df-mpq 6667  df-enq 6669  df-nqqs 6670  df-plqqs 6671  df-mqqs 6672  df-1nqqs 6673  df-rq 6674  df-ltnqqs 6675  df-enq0 6746  df-nq0 6747  df-0nq0 6748  df-plq0 6749  df-mq0 6750  df-inp 6788  df-iplp 6790  df-iltp 6792
This theorem is referenced by:  caucvgprprlemcl  7026
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