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Theorem caucvgprprlem1 7268
Description: Lemma for caucvgprpr 7271. 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 6883 . . . . . 6  |-  <N  C_  ( N.  X.  N. )
76brel 4490 . . . . 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 5435 . 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 7242 . . . . 5  |-  ( ph  -> 
<. { l  |  l 
<Q  ( *Q `  [ <. K ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. K ,  1o >. ]  ~Q  )  <Q  u } >.  <P  Q )
14 nnnq 6981 . . . . . . . 8  |-  ( K  e.  N.  ->  [ <. K ,  1o >. ]  ~Q  e.  Q. )
159, 14syl 14 . . . . . . 7  |-  ( ph  ->  [ <. K ,  1o >. ]  ~Q  e.  Q. )
16 recclnq 6951 . . . . . . 7  |-  ( [
<. K ,  1o >. ]  ~Q  e.  Q.  ->  ( *Q `  [ <. K ,  1o >. ]  ~Q  )  e.  Q. )
17 nqprlu 7106 . . . . . . 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 7178 . . . . . 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 1174 . . . . 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 3622 . . . . . . . . . . . 12  |-  ( r  =  K  ->  <. r ,  1o >.  =  <. K ,  1o >. )
2322eceq1d 6328 . . . . . . . . . . 11  |-  ( r  =  K  ->  [ <. r ,  1o >. ]  ~Q  =  [ <. K ,  1o >. ]  ~Q  )
2423fveq2d 5309 . . . . . . . . . 10  |-  ( r  =  K  ->  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  =  ( *Q `  [ <. K ,  1o >. ]  ~Q  ) )
2524breq2d 3857 . . . . . . . . 9  |-  ( r  =  K  ->  (
l  <Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <->  l  <Q  ( *Q `  [ <. K ,  1o >. ]  ~Q  ) ) )
2625abbidv 2205 . . . . . . . 8  |-  ( r  =  K  ->  { l  |  l  <Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) }  =  { l  |  l  <Q  ( *Q
`  [ <. K ,  1o >. ]  ~Q  ) } )
2724breq1d 3855 . . . . . . . . 9  |-  ( r  =  K  ->  (
( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q  u  <->  ( *Q `  [ <. K ,  1o >. ]  ~Q  )  <Q  u ) )
2827abbidv 2205 . . . . . . . 8  |-  ( r  =  K  ->  { u  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q  u }  =  {
u  |  ( *Q
`  [ <. K ,  1o >. ]  ~Q  )  <Q  u } )
2926, 28opeq12d 3630 . . . . . . 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 5668 . . . . . 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 5305 . . . . . . 7  |-  ( r  =  K  ->  ( F `  r )  =  ( F `  K ) )
3231oveq1d 5667 . . . . . 6  |-  ( r  =  K  ->  (
( F `  r
)  +P.  Q )  =  ( ( F `
 K )  +P. 
Q ) )
3330, 32breq12d 3858 . . . . 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 2722 . . . 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 3848 . . . . . . . 8  |-  ( l  =  p  ->  (
l  <Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <->  p  <Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) ) )
3736cbvabv 2211 . . . . . . 7  |-  { l  |  l  <Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) }  =  { p  |  p  <Q  ( *Q
`  [ <. r ,  1o >. ]  ~Q  ) }
38 breq2 3849 . . . . . . . 8  |-  ( u  =  q  ->  (
( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q  u  <->  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q  q ) )
3938cbvabv 2211 . . . . . . 7  |-  { u  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q  u }  =  {
q  |  ( *Q
`  [ <. r ,  1o >. ]  ~Q  )  <Q  q }
4037, 39opeq12i 3627 . . . . . 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 5663 . . . . 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 3852 . . . 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 2385 . . 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 7267 1  |-  ( ph  ->  ( F `  K
)  <P  ( L  +P.  Q ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    = wceq 1289    e. wcel 1438   {cab 2074   A.wral 2359   E.wrex 2360   {crab 2363   <.cop 3449   class class class wbr 3845   -->wf 5011   ` cfv 5015  (class class class)co 5652   1oc1o 6174   [cec 6290   N.cnpi 6831    <N clti 6834    ~Q ceq 6838   Q.cnq 6839    +Q cplq 6841   *Qcrq 6843    <Q cltq 6844   P.cnp 6850    +P. cpp 6852    <P cltp 6854
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 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-coll 3954  ax-sep 3957  ax-nul 3965  ax-pow 4009  ax-pr 4036  ax-un 4260  ax-setind 4353  ax-iinf 4403
This theorem depends on definitions:  df-bi 115  df-dc 781  df-3or 925  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-ral 2364  df-rex 2365  df-reu 2366  df-rab 2368  df-v 2621  df-sbc 2841  df-csb 2934  df-dif 3001  df-un 3003  df-in 3005  df-ss 3012  df-nul 3287  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-int 3689  df-iun 3732  df-br 3846  df-opab 3900  df-mpt 3901  df-tr 3937  df-eprel 4116  df-id 4120  df-po 4123  df-iso 4124  df-iord 4193  df-on 4195  df-suc 4198  df-iom 4406  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-rn 4449  df-res 4450  df-ima 4451  df-iota 4980  df-fun 5017  df-fn 5018  df-f 5019  df-f1 5020  df-fo 5021  df-f1o 5022  df-fv 5023  df-ov 5655  df-oprab 5656  df-mpt2 5657  df-1st 5911  df-2nd 5912  df-recs 6070  df-irdg 6135  df-1o 6181  df-2o 6182  df-oadd 6185  df-omul 6186  df-er 6292  df-ec 6294  df-qs 6298  df-ni 6863  df-pli 6864  df-mi 6865  df-lti 6866  df-plpq 6903  df-mpq 6904  df-enq 6906  df-nqqs 6907  df-plqqs 6908  df-mqqs 6909  df-1nqqs 6910  df-rq 6911  df-ltnqqs 6912  df-enq0 6983  df-nq0 6984  df-0nq0 6985  df-plq0 6986  df-mq0 6987  df-inp 7025  df-iplp 7027  df-iltp 7029
This theorem is referenced by:  caucvgprprlemlim  7270
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