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Theorem caucvgprprlemclphr 6861
Description: Lemma for caucvgprpr 6868. The putative limit is a positive real. Like caucvgprprlemcl 6860 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 3577 . . . . . . . . . . . . . 14  |-  ( r  =  s  ->  <. r ,  1o >.  =  <. s ,  1o >. )
65eceq1d 6173 . . . . . . . . . . . . 13  |-  ( r  =  s  ->  [ <. r ,  1o >. ]  ~Q  =  [ <. s ,  1o >. ]  ~Q  )
76fveq2d 5210 . . . . . . . . . . . 12  |-  ( r  =  s  ->  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  =  ( *Q `  [ <. s ,  1o >. ]  ~Q  ) )
87oveq2d 5556 . . . . . . . . . . 11  |-  ( r  =  s  ->  (
l  +Q  ( *Q
`  [ <. r ,  1o >. ]  ~Q  )
)  =  ( l  +Q  ( *Q `  [ <. s ,  1o >. ]  ~Q  ) ) )
98breq2d 3804 . . . . . . . . . 10  |-  ( r  =  s  ->  (
p  <Q  ( l  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) )  <->  p  <Q  ( l  +Q  ( *Q
`  [ <. s ,  1o >. ]  ~Q  )
) ) )
109abbidv 2171 . . . . . . . . 9  |-  ( r  =  s  ->  { p  |  p  <Q  ( l  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) ) }  =  { p  |  p  <Q  ( l  +Q  ( *Q `  [ <. s ,  1o >. ]  ~Q  ) ) } )
118breq1d 3802 . . . . . . . . . 10  |-  ( r  =  s  ->  (
( l  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  )
)  <Q  q  <->  ( l  +Q  ( *Q `  [ <. s ,  1o >. ]  ~Q  ) )  <Q 
q ) )
1211abbidv 2171 . . . . . . . . 9  |-  ( r  =  s  ->  { q  |  ( l  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) )  <Q 
q }  =  {
q  |  ( l  +Q  ( *Q `  [ <. s ,  1o >. ]  ~Q  ) ) 
<Q  q } )
1310, 12opeq12d 3585 . . . . . . . 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 5206 . . . . . . . 8  |-  ( r  =  s  ->  ( F `  r )  =  ( F `  s ) )
1513, 14breq12d 3805 . . . . . . 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 2551 . . . . . 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 2564 . . . 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 3804 . . . . . . . . . . 11  |-  ( r  =  s  ->  (
p  <Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <->  p  <Q  ( *Q `  [ <. s ,  1o >. ]  ~Q  ) ) )
2019abbidv 2171 . . . . . . . . . 10  |-  ( r  =  s  ->  { p  |  p  <Q  ( *Q
`  [ <. r ,  1o >. ]  ~Q  ) }  =  { p  |  p  <Q  ( *Q
`  [ <. s ,  1o >. ]  ~Q  ) } )
217breq1d 3802 . . . . . . . . . . 11  |-  ( r  =  s  ->  (
( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q  q  <->  ( *Q `  [ <. s ,  1o >. ]  ~Q  )  <Q  q ) )
2221abbidv 2171 . . . . . . . . . 10  |-  ( r  =  s  ->  { q  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q 
q }  =  {
q  |  ( *Q
`  [ <. s ,  1o >. ]  ~Q  )  <Q  q } )
2320, 22opeq12d 3585 . . . . . . . . 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 5558 . . . . . . . 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 3802 . . . . . . 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 2551 . . . . . 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 2564 . . . 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 3582 . . 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 2076 . 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 6860 1  |-  ( ph  ->  L  e.  P. )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 101    <-> wb 102    = wceq 1259    e. wcel 1409   {cab 2042   A.wral 2323   E.wrex 2324   {crab 2327   <.cop 3406   class class class wbr 3792   -->wf 4926   ` cfv 4930  (class class class)co 5540   1oc1o 6025   [cec 6135   N.cnpi 6428    <N clti 6431    ~Q ceq 6435   Q.cnq 6436    +Q cplq 6438   *Qcrq 6440    <Q cltq 6441   P.cnp 6447    +P. cpp 6449    <P cltp 6451
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-coll 3900  ax-sep 3903  ax-nul 3911  ax-pow 3955  ax-pr 3972  ax-un 4198  ax-setind 4290  ax-iinf 4339
This theorem depends on definitions:  df-bi 114  df-dc 754  df-3or 897  df-3an 898  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ne 2221  df-ral 2328  df-rex 2329  df-reu 2330  df-rab 2332  df-v 2576  df-sbc 2788  df-csb 2881  df-dif 2948  df-un 2950  df-in 2952  df-ss 2959  df-nul 3253  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-int 3644  df-iun 3687  df-br 3793  df-opab 3847  df-mpt 3848  df-tr 3883  df-eprel 4054  df-id 4058  df-po 4061  df-iso 4062  df-iord 4131  df-on 4133  df-suc 4136  df-iom 4342  df-xp 4379  df-rel 4380  df-cnv 4381  df-co 4382  df-dm 4383  df-rn 4384  df-res 4385  df-ima 4386  df-iota 4895  df-fun 4932  df-fn 4933  df-f 4934  df-f1 4935  df-fo 4936  df-f1o 4937  df-fv 4938  df-ov 5543  df-oprab 5544  df-mpt2 5545  df-1st 5795  df-2nd 5796  df-recs 5951  df-irdg 5988  df-1o 6032  df-2o 6033  df-oadd 6036  df-omul 6037  df-er 6137  df-ec 6139  df-qs 6143  df-ni 6460  df-pli 6461  df-mi 6462  df-lti 6463  df-plpq 6500  df-mpq 6501  df-enq 6503  df-nqqs 6504  df-plqqs 6505  df-mqqs 6506  df-1nqqs 6507  df-rq 6508  df-ltnqqs 6509  df-enq0 6580  df-nq0 6581  df-0nq0 6582  df-plq0 6583  df-mq0 6584  df-inp 6622  df-iplp 6624  df-iltp 6626
This theorem is referenced by:  caucvgprprlemexbt  6862  caucvgprprlemexb  6863
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