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Theorem caucvgprprlemclphr 7262
Description: Lemma for caucvgprpr 7269. The putative limit is a positive real. Like caucvgprprlemcl 7261 but without a distinct variable constraint between  ph and  r. (Contributed by Jim Kingdon, 19-Jun-2021.)
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
caucvgprprlemclphr  |-  ( ph  ->  L  e.  P. )
Distinct variable groups:    A, m    m, F    A, r    F, l, u, r, k    n, F, k    k, L    u, l, p, q, r    m, r    k, p, q, r   
u, n, l, k
Allowed substitution hints:    ph( u, k, m, n, r, q, p, l)    A( u, k, n, q, p, l)    F( q, p)    L( u, m, n, r, q, p, l)

Proof of Theorem caucvgprprlemclphr
Dummy variable  s is distinct from all other variables.
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 . . 3  |-  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 opeq1 3622 . . . . . . . . . . . . . 14  |-  ( r  =  s  ->  <. r ,  1o >.  =  <. s ,  1o >. )
65eceq1d 6326 . . . . . . . . . . . . 13  |-  ( r  =  s  ->  [ <. r ,  1o >. ]  ~Q  =  [ <. s ,  1o >. ]  ~Q  )
76fveq2d 5309 . . . . . . . . . . . 12  |-  ( r  =  s  ->  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  =  ( *Q `  [ <. s ,  1o >. ]  ~Q  ) )
87oveq2d 5668 . . . . . . . . . . 11  |-  ( r  =  s  ->  (
l  +Q  ( *Q
`  [ <. r ,  1o >. ]  ~Q  )
)  =  ( l  +Q  ( *Q `  [ <. s ,  1o >. ]  ~Q  ) ) )
98breq2d 3857 . . . . . . . . . 10  |-  ( r  =  s  ->  (
p  <Q  ( l  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) )  <->  p  <Q  ( l  +Q  ( *Q
`  [ <. s ,  1o >. ]  ~Q  )
) ) )
109abbidv 2205 . . . . . . . . 9  |-  ( r  =  s  ->  { p  |  p  <Q  ( l  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) ) }  =  { p  |  p  <Q  ( l  +Q  ( *Q `  [ <. s ,  1o >. ]  ~Q  ) ) } )
118breq1d 3855 . . . . . . . . . 10  |-  ( r  =  s  ->  (
( l  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  )
)  <Q  q  <->  ( l  +Q  ( *Q `  [ <. s ,  1o >. ]  ~Q  ) )  <Q 
q ) )
1211abbidv 2205 . . . . . . . . 9  |-  ( r  =  s  ->  { q  |  ( l  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) )  <Q 
q }  =  {
q  |  ( l  +Q  ( *Q `  [ <. s ,  1o >. ]  ~Q  ) ) 
<Q  q } )
1310, 12opeq12d 3630 . . . . . . . 8  |-  ( r  =  s  ->  <. { p  |  p  <Q  ( l  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) ) } ,  { q  |  ( l  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) )  <Q 
q } >.  =  <. { p  |  p  <Q  ( l  +Q  ( *Q
`  [ <. s ,  1o >. ]  ~Q  )
) } ,  {
q  |  ( l  +Q  ( *Q `  [ <. s ,  1o >. ]  ~Q  ) ) 
<Q  q } >. )
14 fveq2 5305 . . . . . . . 8  |-  ( r  =  s  ->  ( F `  r )  =  ( F `  s ) )
1513, 14breq12d 3858 . . . . . . 7  |-  ( r  =  s  ->  ( <. { p  |  p 
<Q  ( l  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  )
) } ,  {
q  |  ( l  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) ) 
<Q  q } >.  <P  ( F `  r )  <->  <. { p  |  p  <Q  ( l  +Q  ( *Q `  [ <. s ,  1o >. ]  ~Q  )
) } ,  {
q  |  ( l  +Q  ( *Q `  [ <. s ,  1o >. ]  ~Q  ) ) 
<Q  q } >.  <P  ( F `  s )
) )
1615cbvrexv 2591 . . . . . 6  |-  ( 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 )  <->  E. s  e.  N.  <. { p  |  p  <Q  ( l  +Q  ( *Q
`  [ <. s ,  1o >. ]  ~Q  )
) } ,  {
q  |  ( l  +Q  ( *Q `  [ <. s ,  1o >. ]  ~Q  ) ) 
<Q  q } >.  <P  ( F `  s )
)
1716a1i 9 . . . . 5  |-  ( 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 )  <->  E. s  e.  N.  <. { p  |  p  <Q  ( l  +Q  ( *Q
`  [ <. s ,  1o >. ]  ~Q  )
) } ,  {
q  |  ( l  +Q  ( *Q `  [ <. s ,  1o >. ]  ~Q  ) ) 
<Q  q } >.  <P  ( F `  s )
) )
1817rabbiia 2604 . . . 4  |-  { 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 ) }  =  { l  e.  Q.  |  E. s  e.  N.  <. { p  |  p  <Q  ( l  +Q  ( *Q `  [ <. s ,  1o >. ]  ~Q  ) ) } ,  { q  |  ( l  +Q  ( *Q `  [ <. s ,  1o >. ]  ~Q  )
)  <Q  q } >.  <P 
( F `  s
) }
197breq2d 3857 . . . . . . . . . . 11  |-  ( r  =  s  ->  (
p  <Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <->  p  <Q  ( *Q `  [ <. s ,  1o >. ]  ~Q  ) ) )
2019abbidv 2205 . . . . . . . . . 10  |-  ( r  =  s  ->  { p  |  p  <Q  ( *Q
`  [ <. r ,  1o >. ]  ~Q  ) }  =  { p  |  p  <Q  ( *Q
`  [ <. s ,  1o >. ]  ~Q  ) } )
217breq1d 3855 . . . . . . . . . . 11  |-  ( r  =  s  ->  (
( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q  q  <->  ( *Q `  [ <. s ,  1o >. ]  ~Q  )  <Q  q ) )
2221abbidv 2205 . . . . . . . . . 10  |-  ( r  =  s  ->  { q  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q 
q }  =  {
q  |  ( *Q
`  [ <. s ,  1o >. ]  ~Q  )  <Q  q } )
2320, 22opeq12d 3630 . . . . . . . . 9  |-  ( r  =  s  ->  <. { p  |  p  <Q  ( *Q
`  [ <. r ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q 
q } >.  =  <. { p  |  p  <Q  ( *Q `  [ <. s ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. s ,  1o >. ]  ~Q  )  <Q 
q } >. )
2414, 23oveq12d 5670 . . . . . . . 8  |-  ( r  =  s  ->  (
( F `  r
)  +P.  <. { p  |  p  <Q  ( *Q
`  [ <. r ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q 
q } >. )  =  ( ( F `
 s )  +P. 
<. { p  |  p 
<Q  ( *Q `  [ <. s ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. s ,  1o >. ]  ~Q  )  <Q  q } >. ) )
2524breq1d 3855 . . . . . . 7  |-  ( r  =  s  ->  (
( ( 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 } >.  <->  ( ( F `
 s )  +P. 
<. { p  |  p 
<Q  ( *Q `  [ <. s ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. s ,  1o >. ]  ~Q  )  <Q  q } >. ) 
<P  <. { p  |  p  <Q  u } ,  { q  |  u 
<Q  q } >. )
)
2625cbvrexv 2591 . . . . . 6  |-  ( 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 } >.  <->  E. s  e.  N.  ( ( F `  s )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. s ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. s ,  1o >. ]  ~Q  )  <Q 
q } >. )  <P 
<. { p  |  p 
<Q  u } ,  {
q  |  u  <Q  q } >. )
2726a1i 9 . . . . 5  |-  ( 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 } >.  <->  E. s  e.  N.  ( ( F `  s )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. s ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. s ,  1o >. ]  ~Q  )  <Q 
q } >. )  <P 
<. { p  |  p 
<Q  u } ,  {
q  |  u  <Q  q } >. ) )
2827rabbiia 2604 . . . 4  |-  { 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 } >. }  =  { u  e.  Q.  |  E. s  e.  N.  ( ( F `  s )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. s ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. s ,  1o >. ]  ~Q  )  <Q 
q } >. )  <P 
<. { p  |  p 
<Q  u } ,  {
q  |  u  <Q  q } >. }
2918, 28opeq12i 3627 . . 3  |-  <. { 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 } >. } >.  = 
<. { l  e.  Q.  |  E. s  e.  N.  <. { p  |  p  <Q  ( l  +Q  ( *Q `  [ <. s ,  1o >. ]  ~Q  )
) } ,  {
q  |  ( l  +Q  ( *Q `  [ <. s ,  1o >. ]  ~Q  ) ) 
<Q  q } >.  <P  ( F `  s ) } ,  { u  e.  Q.  |  E. s  e.  N.  ( ( F `
 s )  +P. 
<. { p  |  p 
<Q  ( *Q `  [ <. s ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. s ,  1o >. ]  ~Q  )  <Q  q } >. ) 
<P  <. { p  |  p  <Q  u } ,  { q  |  u 
<Q  q } >. } >.
304, 29eqtri 2108 . 2  |-  L  = 
<. { l  e.  Q.  |  E. s  e.  N.  <. { p  |  p  <Q  ( l  +Q  ( *Q `  [ <. s ,  1o >. ]  ~Q  )
) } ,  {
q  |  ( l  +Q  ( *Q `  [ <. s ,  1o >. ]  ~Q  ) ) 
<Q  q } >.  <P  ( F `  s ) } ,  { u  e.  Q.  |  E. s  e.  N.  ( ( F `
 s )  +P. 
<. { p  |  p 
<Q  ( *Q `  [ <. s ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. s ,  1o >. ]  ~Q  )  <Q  q } >. ) 
<P  <. { p  |  p  <Q  u } ,  { q  |  u 
<Q  q } >. } >.
311, 2, 3, 30caucvgprprlemcl 7261 1  |-  ( ph  ->  L  e.  P. )
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 6288   N.cnpi 6829    <N clti 6832    ~Q ceq 6836   Q.cnq 6837    +Q cplq 6839   *Qcrq 6841    <Q cltq 6842   P.cnp 6848    +P. cpp 6850    <P cltp 6852
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 6290  df-ec 6292  df-qs 6296  df-ni 6861  df-pli 6862  df-mi 6863  df-lti 6864  df-plpq 6901  df-mpq 6902  df-enq 6904  df-nqqs 6905  df-plqqs 6906  df-mqqs 6907  df-1nqqs 6908  df-rq 6909  df-ltnqqs 6910  df-enq0 6981  df-nq0 6982  df-0nq0 6983  df-plq0 6984  df-mq0 6985  df-inp 7023  df-iplp 7025  df-iltp 7027
This theorem is referenced by:  caucvgprprlemexbt  7263  caucvgprprlemexb  7264
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