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Theorem caucvgprprlem1 7171
Description: Lemma for caucvgprpr 7174. Part of showing the putative limit to be a limit. (Contributed by Jim Kingdon, 25-Nov-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 } >. } >.
caucvgprprlemlim.q  |-  ( ph  ->  Q  e.  P. )
caucvgprprlemlim.jk  |-  ( ph  ->  J  <N  K )
caucvgprprlemlim.jkq  |-  ( ph  -> 
<. { l  |  l 
<Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. J ,  1o >. ]  ~Q  )  <Q  u } >.  <P  Q )
Assertion
Ref Expression
caucvgprprlem1  |-  ( ph  ->  ( F `  K
)  <P  ( L  +P.  Q ) )
Distinct variable groups:    A, m    m, F    A, r    F, r, l, u, n, k    J, l, u    K, l, r, u    Q, r   
k, L    ph, r    q, p, r, l, u    m, r    k, l, u, r, p, q    n, l, u, r
Allowed substitution hints:    ph( u, k, m, n, q, p, l)    A( u, k, n, q, p, l)    Q( u, k, m, n, q, p, l)    F( q, p)    J( k, m, n, r, q, p)    K( k, m, n, q, p)    L( u, m, n, r, q, p, l)

Proof of Theorem caucvgprprlem1
StepHypRef Expression
1 caucvgprpr.f . 2  |-  ( ph  ->  F : N. --> P. )
2 caucvgprpr.cau . 2  |-  ( 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 . 2  |-  ( ph  ->  A. m  e.  N.  A  <P  ( F `  m ) )
4 caucvgprpr.lim . 2  |-  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 } >. } >.
5 caucvgprprlemlim.jk . . . . 5  |-  ( ph  ->  J  <N  K )
6 ltrelpi 6786 . . . . . 6  |-  <N  C_  ( N.  X.  N. )
76brel 4448 . . . . 5  |-  ( J 
<N  K  ->  ( J  e.  N.  /\  K  e.  N. ) )
85, 7syl 14 . . . 4  |-  ( ph  ->  ( J  e.  N.  /\  K  e.  N. )
)
98simprd 112 . . 3  |-  ( ph  ->  K  e.  N. )
101, 9ffvelrnd 5380 . 2  |-  ( ph  ->  ( F `  K
)  e.  P. )
11 caucvgprprlemlim.q . 2  |-  ( ph  ->  Q  e.  P. )
12 caucvgprprlemlim.jkq . . . . . 6  |-  ( ph  -> 
<. { l  |  l 
<Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. J ,  1o >. ]  ~Q  )  <Q  u } >.  <P  Q )
135, 12caucvgprprlemk 7145 . . . . 5  |-  ( ph  -> 
<. { l  |  l 
<Q  ( *Q `  [ <. K ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. K ,  1o >. ]  ~Q  )  <Q  u } >.  <P  Q )
14 nnnq 6884 . . . . . . . 8  |-  ( K  e.  N.  ->  [ <. K ,  1o >. ]  ~Q  e.  Q. )
159, 14syl 14 . . . . . . 7  |-  ( ph  ->  [ <. K ,  1o >. ]  ~Q  e.  Q. )
16 recclnq 6854 . . . . . . 7  |-  ( [
<. K ,  1o >. ]  ~Q  e.  Q.  ->  ( *Q `  [ <. K ,  1o >. ]  ~Q  )  e.  Q. )
17 nqprlu 7009 . . . . . . 7  |-  ( ( *Q `  [ <. K ,  1o >. ]  ~Q  )  e.  Q.  ->  <. { l  |  l 
<Q  ( *Q `  [ <. K ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. K ,  1o >. ]  ~Q  )  <Q  u } >.  e.  P. )
1815, 16, 173syl 17 . . . . . 6  |-  ( ph  -> 
<. { l  |  l 
<Q  ( *Q `  [ <. K ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. K ,  1o >. ]  ~Q  )  <Q  u } >.  e.  P. )
19 ltaprg 7081 . . . . . 6  |-  ( (
<. { l  |  l 
<Q  ( *Q `  [ <. K ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. K ,  1o >. ]  ~Q  )  <Q  u } >.  e.  P.  /\  Q  e.  P.  /\  ( F `  K )  e.  P. )  -> 
( <. { l  |  l  <Q  ( *Q `  [ <. K ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. K ,  1o >. ]  ~Q  )  <Q  u } >.  <P  Q  <->  ( ( F `  K )  +P.  <. { l  |  l  <Q  ( *Q `  [ <. K ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. K ,  1o >. ]  ~Q  )  <Q  u } >. )  <P  (
( F `  K
)  +P.  Q )
) )
2018, 11, 10, 19syl3anc 1170 . . . . 5  |-  ( ph  ->  ( <. { l  |  l  <Q  ( *Q `  [ <. K ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. K ,  1o >. ]  ~Q  )  <Q  u } >.  <P  Q  <->  ( ( F `  K )  +P.  <. { l  |  l  <Q  ( *Q `  [ <. K ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. K ,  1o >. ]  ~Q  )  <Q  u } >. )  <P  (
( F `  K
)  +P.  Q )
) )
2113, 20mpbid 145 . . . 4  |-  ( ph  ->  ( ( F `  K )  +P.  <. { l  |  l  <Q 
( *Q `  [ <. K ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. K ,  1o >. ]  ~Q  )  <Q  u } >. )  <P  ( ( F `  K )  +P.  Q
) )
22 opeq1 3596 . . . . . . . . . . . 12  |-  ( r  =  K  ->  <. r ,  1o >.  =  <. K ,  1o >. )
2322eceq1d 6258 . . . . . . . . . . 11  |-  ( r  =  K  ->  [ <. r ,  1o >. ]  ~Q  =  [ <. K ,  1o >. ]  ~Q  )
2423fveq2d 5257 . . . . . . . . . 10  |-  ( r  =  K  ->  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  =  ( *Q `  [ <. K ,  1o >. ]  ~Q  ) )
2524breq2d 3823 . . . . . . . . 9  |-  ( r  =  K  ->  (
l  <Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <->  l  <Q  ( *Q `  [ <. K ,  1o >. ]  ~Q  ) ) )
2625abbidv 2200 . . . . . . . 8  |-  ( r  =  K  ->  { l  |  l  <Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) }  =  { l  |  l  <Q  ( *Q
`  [ <. K ,  1o >. ]  ~Q  ) } )
2724breq1d 3821 . . . . . . . . 9  |-  ( r  =  K  ->  (
( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q  u  <->  ( *Q `  [ <. K ,  1o >. ]  ~Q  )  <Q  u ) )
2827abbidv 2200 . . . . . . . 8  |-  ( r  =  K  ->  { u  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q  u }  =  {
u  |  ( *Q
`  [ <. K ,  1o >. ]  ~Q  )  <Q  u } )
2926, 28opeq12d 3604 . . . . . . 7  |-  ( r  =  K  ->  <. { l  |  l  <Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q  u } >.  =  <. { l  |  l  <Q 
( *Q `  [ <. K ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. K ,  1o >. ]  ~Q  )  <Q  u } >. )
3029oveq2d 5607 . . . . . 6  |-  ( r  =  K  ->  (
( F `  K
)  +P.  <. { l  |  l  <Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q  u } >. )  =  ( ( F `  K
)  +P.  <. { l  |  l  <Q  ( *Q `  [ <. K ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. K ,  1o >. ]  ~Q  )  <Q  u } >. ) )
31 fveq2 5253 . . . . . . 7  |-  ( r  =  K  ->  ( F `  r )  =  ( F `  K ) )
3231oveq1d 5606 . . . . . 6  |-  ( r  =  K  ->  (
( F `  r
)  +P.  Q )  =  ( ( F `
 K )  +P. 
Q ) )
3330, 32breq12d 3824 . . . . 5  |-  ( r  =  K  ->  (
( ( F `  K )  +P.  <. { l  |  l  <Q 
( *Q `  [ <. r ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q  u } >. )  <P  ( ( F `  r )  +P.  Q
)  <->  ( ( F `
 K )  +P. 
<. { l  |  l 
<Q  ( *Q `  [ <. K ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. K ,  1o >. ]  ~Q  )  <Q  u } >. )  <P  ( ( F `  K )  +P.  Q
) ) )
3433rspcev 2712 . . . 4  |-  ( ( K  e.  N.  /\  ( ( F `  K )  +P.  <. { l  |  l  <Q 
( *Q `  [ <. K ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. K ,  1o >. ]  ~Q  )  <Q  u } >. )  <P  ( ( F `  K )  +P.  Q
) )  ->  E. r  e.  N.  ( ( F `
 K )  +P. 
<. { l  |  l 
<Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q  u } >. )  <P  ( ( F `  r )  +P.  Q
) )
359, 21, 34syl2anc 403 . . 3  |-  ( ph  ->  E. r  e.  N.  ( ( F `  K )  +P.  <. { l  |  l  <Q 
( *Q `  [ <. r ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q  u } >. )  <P  ( ( F `  r )  +P.  Q
) )
36 breq1 3814 . . . . . . . 8  |-  ( l  =  p  ->  (
l  <Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <->  p  <Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) ) )
3736cbvabv 2206 . . . . . . 7  |-  { l  |  l  <Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) }  =  { p  |  p  <Q  ( *Q
`  [ <. r ,  1o >. ]  ~Q  ) }
38 breq2 3815 . . . . . . . 8  |-  ( u  =  q  ->  (
( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q  u  <->  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q  q ) )
3938cbvabv 2206 . . . . . . 7  |-  { u  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q  u }  =  {
q  |  ( *Q
`  [ <. r ,  1o >. ]  ~Q  )  <Q  q }
4037, 39opeq12i 3601 . . . . . 6  |-  <. { l  |  l  <Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q  u } >.  =  <. { p  |  p  <Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q 
q } >.
4140oveq2i 5602 . . . . 5  |-  ( ( F `  K )  +P.  <. { l  |  l  <Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q  u } >. )  =  ( ( F `  K
)  +P.  <. { p  |  p  <Q  ( *Q
`  [ <. r ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q 
q } >. )
4241breq1i 3818 . . . 4  |-  ( ( ( F `  K
)  +P.  <. { l  |  l  <Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q  u } >. )  <P  (
( F `  r
)  +P.  Q )  <->  ( ( F `  K
)  +P.  <. { p  |  p  <Q  ( *Q
`  [ <. r ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q 
q } >. )  <P  ( ( F `  r )  +P.  Q
) )
4342rexbii 2379 . . 3  |-  ( E. r  e.  N.  (
( F `  K
)  +P.  <. { l  |  l  <Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q  u } >. )  <P  (
( F `  r
)  +P.  Q )  <->  E. r  e.  N.  (
( F `  K
)  +P.  <. { p  |  p  <Q  ( *Q
`  [ <. r ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q 
q } >. )  <P  ( ( F `  r )  +P.  Q
) )
4435, 43sylib 120 . 2  |-  ( ph  ->  E. r  e.  N.  ( ( F `  K )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q 
q } >. )  <P  ( ( F `  r )  +P.  Q
) )
451, 2, 3, 4, 10, 11, 44caucvgprprlemaddq 7170 1  |-  ( ph  ->  ( F `  K
)  <P  ( L  +P.  Q ) )
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 3425   class class class wbr 3811   -->wf 4965   ` cfv 4969  (class class class)co 5591   1oc1o 6106   [cec 6220   N.cnpi 6734    <N clti 6737    ~Q ceq 6741   Q.cnq 6742    +Q cplq 6744   *Qcrq 6746    <Q cltq 6747   P.cnp 6753    +P. cpp 6755    <P cltp 6757
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 3919  ax-sep 3922  ax-nul 3930  ax-pow 3974  ax-pr 4000  ax-un 4224  ax-setind 4316  ax-iinf 4366
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 2614  df-sbc 2827  df-csb 2920  df-dif 2986  df-un 2988  df-in 2990  df-ss 2997  df-nul 3270  df-pw 3408  df-sn 3428  df-pr 3429  df-op 3431  df-uni 3628  df-int 3663  df-iun 3706  df-br 3812  df-opab 3866  df-mpt 3867  df-tr 3902  df-eprel 4080  df-id 4084  df-po 4087  df-iso 4088  df-iord 4157  df-on 4159  df-suc 4162  df-iom 4369  df-xp 4407  df-rel 4408  df-cnv 4409  df-co 4410  df-dm 4411  df-rn 4412  df-res 4413  df-ima 4414  df-iota 4934  df-fun 4971  df-fn 4972  df-f 4973  df-f1 4974  df-fo 4975  df-f1o 4976  df-fv 4977  df-ov 5594  df-oprab 5595  df-mpt2 5596  df-1st 5846  df-2nd 5847  df-recs 6002  df-irdg 6067  df-1o 6113  df-2o 6114  df-oadd 6117  df-omul 6118  df-er 6222  df-ec 6224  df-qs 6228  df-ni 6766  df-pli 6767  df-mi 6768  df-lti 6769  df-plpq 6806  df-mpq 6807  df-enq 6809  df-nqqs 6810  df-plqqs 6811  df-mqqs 6812  df-1nqqs 6813  df-rq 6814  df-ltnqqs 6815  df-enq0 6886  df-nq0 6887  df-0nq0 6888  df-plq0 6889  df-mq0 6890  df-inp 6928  df-iplp 6930  df-iltp 6932
This theorem is referenced by:  caucvgprprlemlim  7173
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