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Theorem caucvgprlemopu 6827
Description: Lemma for caucvgpr 6838. 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 3796 . . . . . 6  |-  ( u  =  r  ->  (
( ( F `  j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  u  <->  ( ( F `  j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) )  <Q 
r ) )
21rexbidv 2344 . . . . 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 5209 . . . . . 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 6519 . . . . . . . 8  |-  Q.  e.  _V
65rabex 3929 . . . . . . 7  |-  { l  e.  Q.  |  E. j  e.  N.  (
l  +Q  ( *Q
`  [ <. j ,  1o >. ]  ~Q  )
)  <Q  ( F `  j ) }  e.  _V
75rabex 3929 . . . . . . 7  |-  { u  e.  Q.  |  E. j  e.  N.  ( ( F `
 j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) )  <Q  u }  e.  _V
86, 7op2nd 5802 . . . . . 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 2076 . . . . 5  |-  ( 2nd `  L )  =  {
u  e.  Q.  |  E. j  e.  N.  ( ( F `  j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  u }
102, 9elrab2 2723 . . . 4  |-  ( r  e.  ( 2nd `  L
)  <->  ( r  e. 
Q.  /\  E. j  e.  N.  ( ( F `
 j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) )  <Q 
r ) )
1110simprbi 264 . . 3  |-  ( r  e.  ( 2nd `  L
)  ->  E. j  e.  N.  ( ( F `
 j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) )  <Q 
r )
1211adantl 266 . 2  |-  ( (
ph  /\  r  e.  ( 2nd `  L ) )  ->  E. j  e.  N.  ( ( F `
 j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) )  <Q 
r )
13 simprr 492 . . . 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 6571 . . . 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 131 . . 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 492 . . . . . 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 490 . . . . . . 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 495 . . . . . . . . 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 265 . . . . . . . 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 491 . . . . . . . 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 2387 . . . . . . . 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 397 . . . . . . 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 3796 . . . . . . . . 9  |-  ( u  =  s  ->  (
( ( F `  j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  u  <->  ( ( F `  j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) )  <Q 
s ) )
2423rexbidv 2344 . . . . . . . 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 2723 . . . . . . 7  |-  ( s  e.  ( 2nd `  L
)  <->  ( s  e. 
Q.  /\  E. j  e.  N.  ( ( F `
 j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) )  <Q 
s ) )
2617, 22, 25sylanbrc 402 . . . . . 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 294 . . . . 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 112 . . . 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 2438 . . 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 2454 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 101    = wceq 1259    e. wcel 1409   A.wral 2323   E.wrex 2324   {crab 2327   <.cop 3406   class class class wbr 3792   -->wf 4926   ` cfv 4930  (class class class)co 5540   2ndc2nd 5794   1oc1o 6025   [cec 6135   N.cnpi 6428    <N clti 6431    ~Q ceq 6435   Q.cnq 6436    +Q cplq 6438   *Qcrq 6440    <Q cltq 6441
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-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
This theorem is referenced by:  caucvgprlemrnd  6829
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