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Theorem caucvgprprlemclphr 7739
Description: Lemma for caucvgprpr 7746. The putative limit is a positive real. Like caucvgprprlemcl 7738 but without a disjoint variable condition 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 3796 . . . . . . . . . . . . . 14  |-  ( r  =  s  ->  <. r ,  1o >.  =  <. s ,  1o >. )
65eceq1d 6599 . . . . . . . . . . . . 13  |-  ( r  =  s  ->  [ <. r ,  1o >. ]  ~Q  =  [ <. s ,  1o >. ]  ~Q  )
76fveq2d 5541 . . . . . . . . . . . 12  |-  ( r  =  s  ->  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  =  ( *Q `  [ <. s ,  1o >. ]  ~Q  ) )
87oveq2d 5916 . . . . . . . . . . 11  |-  ( r  =  s  ->  (
l  +Q  ( *Q
`  [ <. r ,  1o >. ]  ~Q  )
)  =  ( l  +Q  ( *Q `  [ <. s ,  1o >. ]  ~Q  ) ) )
98breq2d 4033 . . . . . . . . . 10  |-  ( r  =  s  ->  (
p  <Q  ( l  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) )  <->  p  <Q  ( l  +Q  ( *Q
`  [ <. s ,  1o >. ]  ~Q  )
) ) )
109abbidv 2307 . . . . . . . . 9  |-  ( r  =  s  ->  { p  |  p  <Q  ( l  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) ) }  =  { p  |  p  <Q  ( l  +Q  ( *Q `  [ <. s ,  1o >. ]  ~Q  ) ) } )
118breq1d 4031 . . . . . . . . . 10  |-  ( r  =  s  ->  (
( l  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  )
)  <Q  q  <->  ( l  +Q  ( *Q `  [ <. s ,  1o >. ]  ~Q  ) )  <Q 
q ) )
1211abbidv 2307 . . . . . . . . 9  |-  ( r  =  s  ->  { q  |  ( l  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) )  <Q 
q }  =  {
q  |  ( l  +Q  ( *Q `  [ <. s ,  1o >. ]  ~Q  ) ) 
<Q  q } )
1310, 12opeq12d 3804 . . . . . . . 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 5537 . . . . . . . 8  |-  ( r  =  s  ->  ( F `  r )  =  ( F `  s ) )
1513, 14breq12d 4034 . . . . . . 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 2719 . . . . . 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 2737 . . . 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 4033 . . . . . . . . . . 11  |-  ( r  =  s  ->  (
p  <Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <->  p  <Q  ( *Q `  [ <. s ,  1o >. ]  ~Q  ) ) )
2019abbidv 2307 . . . . . . . . . 10  |-  ( r  =  s  ->  { p  |  p  <Q  ( *Q
`  [ <. r ,  1o >. ]  ~Q  ) }  =  { p  |  p  <Q  ( *Q
`  [ <. s ,  1o >. ]  ~Q  ) } )
217breq1d 4031 . . . . . . . . . . 11  |-  ( r  =  s  ->  (
( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q  q  <->  ( *Q `  [ <. s ,  1o >. ]  ~Q  )  <Q  q ) )
2221abbidv 2307 . . . . . . . . . 10  |-  ( r  =  s  ->  { q  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q 
q }  =  {
q  |  ( *Q
`  [ <. s ,  1o >. ]  ~Q  )  <Q  q } )
2320, 22opeq12d 3804 . . . . . . . . 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 5918 . . . . . . . 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 4031 . . . . . . 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 2719 . . . . . 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 2737 . . . 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 3801 . . 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 2210 . 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 7738 1  |-  ( ph  ->  L  e.  P. )
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 3613   class class class wbr 4021   -->wf 5234   ` cfv 5238  (class class class)co 5900   1oc1o 6438   [cec 6561   N.cnpi 7306    <N clti 7309    ~Q ceq 7313   Q.cnq 7314    +Q cplq 7316   *Qcrq 7318    <Q cltq 7319   P.cnp 7325    +P. cpp 7327    <P cltp 7329
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 4136  ax-sep 4139  ax-nul 4147  ax-pow 4195  ax-pr 4230  ax-un 4454  ax-setind 4557  ax-iinf 4608
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 3595  df-sn 3616  df-pr 3617  df-op 3619  df-uni 3828  df-int 3863  df-iun 3906  df-br 4022  df-opab 4083  df-mpt 4084  df-tr 4120  df-eprel 4310  df-id 4314  df-po 4317  df-iso 4318  df-iord 4387  df-on 4389  df-suc 4392  df-iom 4611  df-xp 4653  df-rel 4654  df-cnv 4655  df-co 4656  df-dm 4657  df-rn 4658  df-res 4659  df-ima 4660  df-iota 5199  df-fun 5240  df-fn 5241  df-f 5242  df-f1 5243  df-fo 5244  df-f1o 5245  df-fv 5246  df-ov 5903  df-oprab 5904  df-mpo 5905  df-1st 6169  df-2nd 6170  df-recs 6334  df-irdg 6399  df-1o 6445  df-2o 6446  df-oadd 6449  df-omul 6450  df-er 6563  df-ec 6565  df-qs 6569  df-ni 7338  df-pli 7339  df-mi 7340  df-lti 7341  df-plpq 7378  df-mpq 7379  df-enq 7381  df-nqqs 7382  df-plqqs 7383  df-mqqs 7384  df-1nqqs 7385  df-rq 7386  df-ltnqqs 7387  df-enq0 7458  df-nq0 7459  df-0nq0 7460  df-plq0 7461  df-mq0 7462  df-inp 7500  df-iplp 7502  df-iltp 7504
This theorem is referenced by:  caucvgprprlemexbt  7740  caucvgprprlemexb  7741
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