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Theorem caucvgprprlem1 7738
Description: Lemma for caucvgprpr 7741. 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 7353 . . . . . 6  |-  <N  C_  ( N.  X.  N. )
76brel 4696 . . . . 5  |-  ( J 
<N  K  ->  ( J  e.  N.  /\  K  e.  N. ) )
85, 7syl 14 . . . 4  |-  ( ph  ->  ( J  e.  N.  /\  K  e.  N. )
)
98simprd 114 . . 3  |-  ( ph  ->  K  e.  N. )
101, 9ffvelcdmd 5673 . 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 7712 . . . . 5  |-  ( ph  -> 
<. { l  |  l 
<Q  ( *Q `  [ <. K ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. K ,  1o >. ]  ~Q  )  <Q  u } >.  <P  Q )
14 nnnq 7451 . . . . . . . 8  |-  ( K  e.  N.  ->  [ <. K ,  1o >. ]  ~Q  e.  Q. )
159, 14syl 14 . . . . . . 7  |-  ( ph  ->  [ <. K ,  1o >. ]  ~Q  e.  Q. )
16 recclnq 7421 . . . . . . 7  |-  ( [
<. K ,  1o >. ]  ~Q  e.  Q.  ->  ( *Q `  [ <. K ,  1o >. ]  ~Q  )  e.  Q. )
17 nqprlu 7576 . . . . . . 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 7648 . . . . . 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 1249 . . . . 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 147 . . . 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 3793 . . . . . . . . . . . 12  |-  ( r  =  K  ->  <. r ,  1o >.  =  <. K ,  1o >. )
2322eceq1d 6595 . . . . . . . . . . 11  |-  ( r  =  K  ->  [ <. r ,  1o >. ]  ~Q  =  [ <. K ,  1o >. ]  ~Q  )
2423fveq2d 5538 . . . . . . . . . 10  |-  ( r  =  K  ->  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  =  ( *Q `  [ <. K ,  1o >. ]  ~Q  ) )
2524breq2d 4030 . . . . . . . . 9  |-  ( r  =  K  ->  (
l  <Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <->  l  <Q  ( *Q `  [ <. K ,  1o >. ]  ~Q  ) ) )
2625abbidv 2307 . . . . . . . 8  |-  ( r  =  K  ->  { l  |  l  <Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) }  =  { l  |  l  <Q  ( *Q
`  [ <. K ,  1o >. ]  ~Q  ) } )
2724breq1d 4028 . . . . . . . . 9  |-  ( r  =  K  ->  (
( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q  u  <->  ( *Q `  [ <. K ,  1o >. ]  ~Q  )  <Q  u ) )
2827abbidv 2307 . . . . . . . 8  |-  ( r  =  K  ->  { u  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q  u }  =  {
u  |  ( *Q
`  [ <. K ,  1o >. ]  ~Q  )  <Q  u } )
2926, 28opeq12d 3801 . . . . . . 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 5912 . . . . . 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 5534 . . . . . . 7  |-  ( r  =  K  ->  ( F `  r )  =  ( F `  K ) )
3231oveq1d 5911 . . . . . 6  |-  ( r  =  K  ->  (
( F `  r
)  +P.  Q )  =  ( ( F `
 K )  +P. 
Q ) )
3330, 32breq12d 4031 . . . . 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 2856 . . . 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 411 . . 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 4021 . . . . . . . 8  |-  ( l  =  p  ->  (
l  <Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <->  p  <Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) ) )
3736cbvabv 2314 . . . . . . 7  |-  { l  |  l  <Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) }  =  { p  |  p  <Q  ( *Q
`  [ <. r ,  1o >. ]  ~Q  ) }
38 breq2 4022 . . . . . . . 8  |-  ( u  =  q  ->  (
( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q  u  <->  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q  q ) )
3938cbvabv 2314 . . . . . . 7  |-  { u  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q  u }  =  {
q  |  ( *Q
`  [ <. r ,  1o >. ]  ~Q  )  <Q  q }
4037, 39opeq12i 3798 . . . . . 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 5907 . . . . 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 4025 . . . 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 2497 . . 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 122 . 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 7737 1  |-  ( ph  ->  ( F `  K
)  <P  ( L  +P.  Q ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105    = 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 5896   1oc1o 6434   [cec 6557   N.cnpi 7301    <N clti 7304    ~Q ceq 7308   Q.cnq 7309    +Q cplq 7311   *Qcrq 7313    <Q cltq 7314   P.cnp 7320    +P. cpp 7322    <P cltp 7324
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 5899  df-oprab 5900  df-mpo 5901  df-1st 6165  df-2nd 6166  df-recs 6330  df-irdg 6395  df-1o 6441  df-2o 6442  df-oadd 6445  df-omul 6446  df-er 6559  df-ec 6561  df-qs 6565  df-ni 7333  df-pli 7334  df-mi 7335  df-lti 7336  df-plpq 7373  df-mpq 7374  df-enq 7376  df-nqqs 7377  df-plqqs 7378  df-mqqs 7379  df-1nqqs 7380  df-rq 7381  df-ltnqqs 7382  df-enq0 7453  df-nq0 7454  df-0nq0 7455  df-plq0 7456  df-mq0 7457  df-inp 7495  df-iplp 7497  df-iltp 7499
This theorem is referenced by:  caucvgprprlemlim  7740
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