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Theorem caucvgprprlemclphr 7481
Description: Lemma for caucvgprpr 7488. The putative limit is a positive real. Like caucvgprprlemcl 7480 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 3675 . . . . . . . . . . . . . 14  |-  ( r  =  s  ->  <. r ,  1o >.  =  <. s ,  1o >. )
65eceq1d 6433 . . . . . . . . . . . . 13  |-  ( r  =  s  ->  [ <. r ,  1o >. ]  ~Q  =  [ <. s ,  1o >. ]  ~Q  )
76fveq2d 5393 . . . . . . . . . . . 12  |-  ( r  =  s  ->  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  =  ( *Q `  [ <. s ,  1o >. ]  ~Q  ) )
87oveq2d 5758 . . . . . . . . . . 11  |-  ( r  =  s  ->  (
l  +Q  ( *Q
`  [ <. r ,  1o >. ]  ~Q  )
)  =  ( l  +Q  ( *Q `  [ <. s ,  1o >. ]  ~Q  ) ) )
98breq2d 3911 . . . . . . . . . 10  |-  ( r  =  s  ->  (
p  <Q  ( l  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) )  <->  p  <Q  ( l  +Q  ( *Q
`  [ <. s ,  1o >. ]  ~Q  )
) ) )
109abbidv 2235 . . . . . . . . 9  |-  ( r  =  s  ->  { p  |  p  <Q  ( l  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) ) }  =  { p  |  p  <Q  ( l  +Q  ( *Q `  [ <. s ,  1o >. ]  ~Q  ) ) } )
118breq1d 3909 . . . . . . . . . 10  |-  ( r  =  s  ->  (
( l  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  )
)  <Q  q  <->  ( l  +Q  ( *Q `  [ <. s ,  1o >. ]  ~Q  ) )  <Q 
q ) )
1211abbidv 2235 . . . . . . . . 9  |-  ( r  =  s  ->  { q  |  ( l  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) )  <Q 
q }  =  {
q  |  ( l  +Q  ( *Q `  [ <. s ,  1o >. ]  ~Q  ) ) 
<Q  q } )
1310, 12opeq12d 3683 . . . . . . . 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 5389 . . . . . . . 8  |-  ( r  =  s  ->  ( F `  r )  =  ( F `  s ) )
1513, 14breq12d 3912 . . . . . . 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 2632 . . . . . 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 2645 . . . 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 3911 . . . . . . . . . . 11  |-  ( r  =  s  ->  (
p  <Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <->  p  <Q  ( *Q `  [ <. s ,  1o >. ]  ~Q  ) ) )
2019abbidv 2235 . . . . . . . . . 10  |-  ( r  =  s  ->  { p  |  p  <Q  ( *Q
`  [ <. r ,  1o >. ]  ~Q  ) }  =  { p  |  p  <Q  ( *Q
`  [ <. s ,  1o >. ]  ~Q  ) } )
217breq1d 3909 . . . . . . . . . . 11  |-  ( r  =  s  ->  (
( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q  q  <->  ( *Q `  [ <. s ,  1o >. ]  ~Q  )  <Q  q ) )
2221abbidv 2235 . . . . . . . . . 10  |-  ( r  =  s  ->  { q  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q 
q }  =  {
q  |  ( *Q
`  [ <. s ,  1o >. ]  ~Q  )  <Q  q } )
2320, 22opeq12d 3683 . . . . . . . . 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 5760 . . . . . . . 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 3909 . . . . . . 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 2632 . . . . . 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 2645 . . . 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 3680 . . 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 2138 . 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 7480 1  |-  ( ph  ->  L  e.  P. )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1316    e. wcel 1465   {cab 2103   A.wral 2393   E.wrex 2394   {crab 2397   <.cop 3500   class class class wbr 3899   -->wf 5089   ` cfv 5093  (class class class)co 5742   1oc1o 6274   [cec 6395   N.cnpi 7048    <N clti 7051    ~Q ceq 7055   Q.cnq 7056    +Q cplq 7058   *Qcrq 7060    <Q cltq 7061   P.cnp 7067    +P. cpp 7069    <P cltp 7071
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-coll 4013  ax-sep 4016  ax-nul 4024  ax-pow 4068  ax-pr 4101  ax-un 4325  ax-setind 4422  ax-iinf 4472
This theorem depends on definitions:  df-bi 116  df-dc 805  df-3or 948  df-3an 949  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ne 2286  df-ral 2398  df-rex 2399  df-reu 2400  df-rab 2402  df-v 2662  df-sbc 2883  df-csb 2976  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-nul 3334  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-int 3742  df-iun 3785  df-br 3900  df-opab 3960  df-mpt 3961  df-tr 3997  df-eprel 4181  df-id 4185  df-po 4188  df-iso 4189  df-iord 4258  df-on 4260  df-suc 4263  df-iom 4475  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522  df-iota 5058  df-fun 5095  df-fn 5096  df-f 5097  df-f1 5098  df-fo 5099  df-f1o 5100  df-fv 5101  df-ov 5745  df-oprab 5746  df-mpo 5747  df-1st 6006  df-2nd 6007  df-recs 6170  df-irdg 6235  df-1o 6281  df-2o 6282  df-oadd 6285  df-omul 6286  df-er 6397  df-ec 6399  df-qs 6403  df-ni 7080  df-pli 7081  df-mi 7082  df-lti 7083  df-plpq 7120  df-mpq 7121  df-enq 7123  df-nqqs 7124  df-plqqs 7125  df-mqqs 7126  df-1nqqs 7127  df-rq 7128  df-ltnqqs 7129  df-enq0 7200  df-nq0 7201  df-0nq0 7202  df-plq0 7203  df-mq0 7204  df-inp 7242  df-iplp 7244  df-iltp 7246
This theorem is referenced by:  caucvgprprlemexbt  7482  caucvgprprlemexb  7483
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