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Theorem caucvgprlemopu 6769
Description: Lemma for caucvgpr 6780. The upper cut of the putative limit is open. (Contributed by Jim Kingdon, 20-Oct-2020.)
Hypotheses
Ref Expression
caucvgpr.f  |-  ( ph  ->  F : N. --> Q. )
caucvgpr.cau  |-  ( ph  ->  A. n  e.  N.  A. k  e.  N.  (
n  <N  k  ->  (
( F `  n
)  <Q  ( ( F `
 k )  +Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) )  /\  ( F `  k ) 
<Q  ( ( F `  n )  +Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  )
) ) ) )
caucvgpr.bnd  |-  ( ph  ->  A. j  e.  N.  A  <Q  ( F `  j ) )
caucvgpr.lim  |-  L  = 
<. { l  e.  Q.  |  E. j  e.  N.  ( l  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  ( F `  j ) } ,  { u  e.  Q.  |  E. j  e.  N.  ( ( F `  j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  u } >.
Assertion
Ref Expression
caucvgprlemopu  |-  ( (
ph  /\  r  e.  ( 2nd `  L ) )  ->  E. s  e.  Q.  ( s  <Q 
r  /\  s  e.  ( 2nd `  L ) ) )
Distinct variable groups:    A, j    F, l, r, s    u, F   
j, L, r, s   
j, l, s    ph, j,
r, s    u, j,
r, s
Allowed substitution hints:    ph( u, k, n, l)    A( u, k, n, s, r, l)    F( j, k, n)    L( u, k, n, l)

Proof of Theorem caucvgprlemopu
StepHypRef Expression
1 breq2 3768 . . . . . 6  |-  ( u  =  r  ->  (
( ( F `  j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  u  <->  ( ( F `  j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) )  <Q 
r ) )
21rexbidv 2327 . . . . 5  |-  ( u  =  r  ->  ( E. j  e.  N.  ( ( F `  j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  u  <->  E. j  e.  N.  ( ( F `
 j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) )  <Q 
r ) )
3 caucvgpr.lim . . . . . . 7  |-  L  = 
<. { l  e.  Q.  |  E. j  e.  N.  ( l  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  ( F `  j ) } ,  { u  e.  Q.  |  E. j  e.  N.  ( ( F `  j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  u } >.
43fveq2i 5181 . . . . . 6  |-  ( 2nd `  L )  =  ( 2nd `  <. { l  e.  Q.  |  E. j  e.  N.  (
l  +Q  ( *Q
`  [ <. j ,  1o >. ]  ~Q  )
)  <Q  ( F `  j ) } ,  { u  e.  Q.  |  E. j  e.  N.  ( ( F `  j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  u } >. )
5 nqex 6461 . . . . . . . 8  |-  Q.  e.  _V
65rabex 3901 . . . . . . 7  |-  { l  e.  Q.  |  E. j  e.  N.  (
l  +Q  ( *Q
`  [ <. j ,  1o >. ]  ~Q  )
)  <Q  ( F `  j ) }  e.  _V
75rabex 3901 . . . . . . 7  |-  { u  e.  Q.  |  E. j  e.  N.  ( ( F `
 j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) )  <Q  u }  e.  _V
86, 7op2nd 5774 . . . . . 6  |-  ( 2nd `  <. { l  e. 
Q.  |  E. j  e.  N.  ( l  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) )  <Q 
( F `  j
) } ,  {
u  e.  Q.  |  E. j  e.  N.  ( ( F `  j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  u } >. )  =  { u  e. 
Q.  |  E. j  e.  N.  ( ( F `
 j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) )  <Q  u }
94, 8eqtri 2060 . . . . 5  |-  ( 2nd `  L )  =  {
u  e.  Q.  |  E. j  e.  N.  ( ( F `  j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  u }
102, 9elrab2 2700 . . . 4  |-  ( r  e.  ( 2nd `  L
)  <->  ( r  e. 
Q.  /\  E. j  e.  N.  ( ( F `
 j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) )  <Q 
r ) )
1110simprbi 260 . . 3  |-  ( r  e.  ( 2nd `  L
)  ->  E. j  e.  N.  ( ( F `
 j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) )  <Q 
r )
1211adantl 262 . 2  |-  ( (
ph  /\  r  e.  ( 2nd `  L ) )  ->  E. j  e.  N.  ( ( F `
 j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) )  <Q 
r )
13 simprr 484 . . . 4  |-  ( ( ( ph  /\  r  e.  ( 2nd `  L
) )  /\  (
j  e.  N.  /\  ( ( F `  j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  r ) )  ->  ( ( F `
 j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) )  <Q 
r )
14 ltbtwnnqq 6513 . . . 4  |-  ( ( ( F `  j
)  +Q  ( *Q
`  [ <. j ,  1o >. ]  ~Q  )
)  <Q  r  <->  E. s  e.  Q.  ( ( ( F `  j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) ) 
<Q  s  /\  s  <Q  r ) )
1513, 14sylib 127 . . 3  |-  ( ( ( ph  /\  r  e.  ( 2nd `  L
) )  /\  (
j  e.  N.  /\  ( ( F `  j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  r ) )  ->  E. s  e.  Q.  ( ( ( F `
 j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) )  <Q 
s  /\  s  <Q  r ) )
16 simprr 484 . . . . . 6  |-  ( ( ( ( ( ph  /\  r  e.  ( 2nd `  L ) )  /\  ( j  e.  N.  /\  ( ( F `  j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  r ) )  /\  s  e.  Q. )  /\  ( ( ( F `  j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) ) 
<Q  s  /\  s  <Q  r ) )  -> 
s  <Q  r )
17 simplr 482 . . . . . . 7  |-  ( ( ( ( ( ph  /\  r  e.  ( 2nd `  L ) )  /\  ( j  e.  N.  /\  ( ( F `  j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  r ) )  /\  s  e.  Q. )  /\  ( ( ( F `  j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) ) 
<Q  s  /\  s  <Q  r ) )  -> 
s  e.  Q. )
18 simplrl 487 . . . . . . . . 9  |-  ( ( ( ( ph  /\  r  e.  ( 2nd `  L ) )  /\  ( j  e.  N.  /\  ( ( F `  j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  r ) )  /\  s  e.  Q. )  ->  j  e.  N. )
1918adantr 261 . . . . . . . 8  |-  ( ( ( ( ( ph  /\  r  e.  ( 2nd `  L ) )  /\  ( j  e.  N.  /\  ( ( F `  j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  r ) )  /\  s  e.  Q. )  /\  ( ( ( F `  j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) ) 
<Q  s  /\  s  <Q  r ) )  -> 
j  e.  N. )
20 simprl 483 . . . . . . . 8  |-  ( ( ( ( ( ph  /\  r  e.  ( 2nd `  L ) )  /\  ( j  e.  N.  /\  ( ( F `  j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  r ) )  /\  s  e.  Q. )  /\  ( ( ( F `  j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) ) 
<Q  s  /\  s  <Q  r ) )  -> 
( ( F `  j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  s )
21 rspe 2370 . . . . . . . 8  |-  ( ( j  e.  N.  /\  ( ( F `  j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  s )  ->  E. j  e.  N.  ( ( F `  j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  s )
2219, 20, 21syl2anc 391 . . . . . . 7  |-  ( ( ( ( ( ph  /\  r  e.  ( 2nd `  L ) )  /\  ( j  e.  N.  /\  ( ( F `  j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  r ) )  /\  s  e.  Q. )  /\  ( ( ( F `  j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) ) 
<Q  s  /\  s  <Q  r ) )  ->  E. j  e.  N.  ( ( F `  j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  s )
23 breq2 3768 . . . . . . . . 9  |-  ( u  =  s  ->  (
( ( F `  j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  u  <->  ( ( F `  j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) )  <Q 
s ) )
2423rexbidv 2327 . . . . . . . 8  |-  ( u  =  s  ->  ( E. j  e.  N.  ( ( F `  j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  u  <->  E. j  e.  N.  ( ( F `
 j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) )  <Q 
s ) )
2524, 9elrab2 2700 . . . . . . 7  |-  ( s  e.  ( 2nd `  L
)  <->  ( s  e. 
Q.  /\  E. j  e.  N.  ( ( F `
 j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) )  <Q 
s ) )
2617, 22, 25sylanbrc 394 . . . . . 6  |-  ( ( ( ( ( ph  /\  r  e.  ( 2nd `  L ) )  /\  ( j  e.  N.  /\  ( ( F `  j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  r ) )  /\  s  e.  Q. )  /\  ( ( ( F `  j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) ) 
<Q  s  /\  s  <Q  r ) )  -> 
s  e.  ( 2nd `  L ) )
2716, 26jca 290 . . . . 5  |-  ( ( ( ( ( ph  /\  r  e.  ( 2nd `  L ) )  /\  ( j  e.  N.  /\  ( ( F `  j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  r ) )  /\  s  e.  Q. )  /\  ( ( ( F `  j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) ) 
<Q  s  /\  s  <Q  r ) )  -> 
( s  <Q  r  /\  s  e.  ( 2nd `  L ) ) )
2827ex 108 . . . 4  |-  ( ( ( ( ph  /\  r  e.  ( 2nd `  L ) )  /\  ( j  e.  N.  /\  ( ( F `  j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  r ) )  /\  s  e.  Q. )  ->  ( ( ( ( F `  j
)  +Q  ( *Q
`  [ <. j ,  1o >. ]  ~Q  )
)  <Q  s  /\  s  <Q  r )  ->  (
s  <Q  r  /\  s  e.  ( 2nd `  L
) ) ) )
2928reximdva 2421 . . 3  |-  ( ( ( ph  /\  r  e.  ( 2nd `  L
) )  /\  (
j  e.  N.  /\  ( ( F `  j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  r ) )  ->  ( E. s  e.  Q.  ( ( ( F `  j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) ) 
<Q  s  /\  s  <Q  r )  ->  E. s  e.  Q.  ( s  <Q 
r  /\  s  e.  ( 2nd `  L ) ) ) )
3015, 29mpd 13 . 2  |-  ( ( ( ph  /\  r  e.  ( 2nd `  L
) )  /\  (
j  e.  N.  /\  ( ( F `  j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  r ) )  ->  E. s  e.  Q.  ( s  <Q  r  /\  s  e.  ( 2nd `  L ) ) )
3112, 30rexlimddv 2437 1  |-  ( (
ph  /\  r  e.  ( 2nd `  L ) )  ->  E. s  e.  Q.  ( s  <Q 
r  /\  s  e.  ( 2nd `  L ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 97    = wceq 1243    e. wcel 1393   A.wral 2306   E.wrex 2307   {crab 2310   <.cop 3378   class class class wbr 3764   -->wf 4898   ` cfv 4902  (class class class)co 5512   2ndc2nd 5766   1oc1o 5994   [cec 6104   N.cnpi 6370    <N clti 6373    ~Q ceq 6377   Q.cnq 6378    +Q cplq 6380   *Qcrq 6382    <Q cltq 6383
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-coll 3872  ax-sep 3875  ax-nul 3883  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-setind 4262  ax-iinf 4311
This theorem depends on definitions:  df-bi 110  df-dc 743  df-3or 886  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2311  df-rex 2312  df-reu 2313  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-int 3616  df-iun 3659  df-br 3765  df-opab 3819  df-mpt 3820  df-tr 3855  df-eprel 4026  df-id 4030  df-po 4033  df-iso 4034  df-iord 4103  df-on 4105  df-suc 4108  df-iom 4314  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909  df-fv 4910  df-ov 5515  df-oprab 5516  df-mpt2 5517  df-1st 5767  df-2nd 5768  df-recs 5920  df-irdg 5957  df-1o 6001  df-oadd 6005  df-omul 6006  df-er 6106  df-ec 6108  df-qs 6112  df-ni 6402  df-pli 6403  df-mi 6404  df-lti 6405  df-plpq 6442  df-mpq 6443  df-enq 6445  df-nqqs 6446  df-plqqs 6447  df-mqqs 6448  df-1nqqs 6449  df-rq 6450  df-ltnqqs 6451
This theorem is referenced by:  caucvgprlemrnd  6771
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