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Theorem caucvgprprlemclphr 7533
Description: Lemma for caucvgprpr 7540. The putative limit is a positive real. Like caucvgprprlemcl 7532 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 3709 . . . . . . . . . . . . . 14  |-  ( r  =  s  ->  <. r ,  1o >.  =  <. s ,  1o >. )
65eceq1d 6469 . . . . . . . . . . . . 13  |-  ( r  =  s  ->  [ <. r ,  1o >. ]  ~Q  =  [ <. s ,  1o >. ]  ~Q  )
76fveq2d 5429 . . . . . . . . . . . 12  |-  ( r  =  s  ->  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  =  ( *Q `  [ <. s ,  1o >. ]  ~Q  ) )
87oveq2d 5794 . . . . . . . . . . 11  |-  ( r  =  s  ->  (
l  +Q  ( *Q
`  [ <. r ,  1o >. ]  ~Q  )
)  =  ( l  +Q  ( *Q `  [ <. s ,  1o >. ]  ~Q  ) ) )
98breq2d 3945 . . . . . . . . . 10  |-  ( r  =  s  ->  (
p  <Q  ( l  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) )  <->  p  <Q  ( l  +Q  ( *Q
`  [ <. s ,  1o >. ]  ~Q  )
) ) )
109abbidv 2258 . . . . . . . . 9  |-  ( r  =  s  ->  { p  |  p  <Q  ( l  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) ) }  =  { p  |  p  <Q  ( l  +Q  ( *Q `  [ <. s ,  1o >. ]  ~Q  ) ) } )
118breq1d 3943 . . . . . . . . . 10  |-  ( r  =  s  ->  (
( l  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  )
)  <Q  q  <->  ( l  +Q  ( *Q `  [ <. s ,  1o >. ]  ~Q  ) )  <Q 
q ) )
1211abbidv 2258 . . . . . . . . 9  |-  ( r  =  s  ->  { q  |  ( l  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) )  <Q 
q }  =  {
q  |  ( l  +Q  ( *Q `  [ <. s ,  1o >. ]  ~Q  ) ) 
<Q  q } )
1310, 12opeq12d 3717 . . . . . . . 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 5425 . . . . . . . 8  |-  ( r  =  s  ->  ( F `  r )  =  ( F `  s ) )
1513, 14breq12d 3946 . . . . . . 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 2656 . . . . . 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 2672 . . . 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 3945 . . . . . . . . . . 11  |-  ( r  =  s  ->  (
p  <Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <->  p  <Q  ( *Q `  [ <. s ,  1o >. ]  ~Q  ) ) )
2019abbidv 2258 . . . . . . . . . 10  |-  ( r  =  s  ->  { p  |  p  <Q  ( *Q
`  [ <. r ,  1o >. ]  ~Q  ) }  =  { p  |  p  <Q  ( *Q
`  [ <. s ,  1o >. ]  ~Q  ) } )
217breq1d 3943 . . . . . . . . . . 11  |-  ( r  =  s  ->  (
( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q  q  <->  ( *Q `  [ <. s ,  1o >. ]  ~Q  )  <Q  q ) )
2221abbidv 2258 . . . . . . . . . 10  |-  ( r  =  s  ->  { q  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q 
q }  =  {
q  |  ( *Q
`  [ <. s ,  1o >. ]  ~Q  )  <Q  q } )
2320, 22opeq12d 3717 . . . . . . . . 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 5796 . . . . . . . 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 3943 . . . . . . 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 2656 . . . . . 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 2672 . . . 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 3714 . . 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 2161 . 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 7532 1  |-  ( ph  ->  L  e.  P. )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1332    e. wcel 1481   {cab 2126   A.wral 2417   E.wrex 2418   {crab 2421   <.cop 3531   class class class wbr 3933   -->wf 5123   ` cfv 5127  (class class class)co 5778   1oc1o 6310   [cec 6431   N.cnpi 7100    <N clti 7103    ~Q ceq 7107   Q.cnq 7108    +Q cplq 7110   *Qcrq 7112    <Q cltq 7113   P.cnp 7119    +P. cpp 7121    <P cltp 7123
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 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-coll 4047  ax-sep 4050  ax-nul 4058  ax-pow 4102  ax-pr 4135  ax-un 4359  ax-setind 4456  ax-iinf 4506
This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-ral 2422  df-rex 2423  df-reu 2424  df-rab 2426  df-v 2689  df-sbc 2911  df-csb 3005  df-dif 3074  df-un 3076  df-in 3078  df-ss 3085  df-nul 3365  df-pw 3513  df-sn 3534  df-pr 3535  df-op 3537  df-uni 3741  df-int 3776  df-iun 3819  df-br 3934  df-opab 3994  df-mpt 3995  df-tr 4031  df-eprel 4215  df-id 4219  df-po 4222  df-iso 4223  df-iord 4292  df-on 4294  df-suc 4297  df-iom 4509  df-xp 4549  df-rel 4550  df-cnv 4551  df-co 4552  df-dm 4553  df-rn 4554  df-res 4555  df-ima 4556  df-iota 5092  df-fun 5129  df-fn 5130  df-f 5131  df-f1 5132  df-fo 5133  df-f1o 5134  df-fv 5135  df-ov 5781  df-oprab 5782  df-mpo 5783  df-1st 6042  df-2nd 6043  df-recs 6206  df-irdg 6271  df-1o 6317  df-2o 6318  df-oadd 6321  df-omul 6322  df-er 6433  df-ec 6435  df-qs 6439  df-ni 7132  df-pli 7133  df-mi 7134  df-lti 7135  df-plpq 7172  df-mpq 7173  df-enq 7175  df-nqqs 7176  df-plqqs 7177  df-mqqs 7178  df-1nqqs 7179  df-rq 7180  df-ltnqqs 7181  df-enq0 7252  df-nq0 7253  df-0nq0 7254  df-plq0 7255  df-mq0 7256  df-inp 7294  df-iplp 7296  df-iltp 7298
This theorem is referenced by:  caucvgprprlemexbt  7534  caucvgprprlemexb  7535
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