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Theorem caucvgprlemopu 7443
Description: Lemma for caucvgpr 7454. 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 3901 . . . . . 6  |-  ( u  =  r  ->  (
( ( F `  j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  u  <->  ( ( F `  j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) )  <Q 
r ) )
21rexbidv 2413 . . . . 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 5390 . . . . . 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 7135 . . . . . . . 8  |-  Q.  e.  _V
65rabex 4040 . . . . . . 7  |-  { l  e.  Q.  |  E. j  e.  N.  (
l  +Q  ( *Q
`  [ <. j ,  1o >. ]  ~Q  )
)  <Q  ( F `  j ) }  e.  _V
75rabex 4040 . . . . . . 7  |-  { u  e.  Q.  |  E. j  e.  N.  ( ( F `
 j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) )  <Q  u }  e.  _V
86, 7op2nd 6011 . . . . . 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 2136 . . . . 5  |-  ( 2nd `  L )  =  {
u  e.  Q.  |  E. j  e.  N.  ( ( F `  j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  u }
102, 9elrab2 2814 . . . 4  |-  ( r  e.  ( 2nd `  L
)  <->  ( r  e. 
Q.  /\  E. j  e.  N.  ( ( F `
 j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) )  <Q 
r ) )
1110simprbi 271 . . 3  |-  ( r  e.  ( 2nd `  L
)  ->  E. j  e.  N.  ( ( F `
 j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) )  <Q 
r )
1211adantl 273 . 2  |-  ( (
ph  /\  r  e.  ( 2nd `  L ) )  ->  E. j  e.  N.  ( ( F `
 j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) )  <Q 
r )
13 simprr 504 . . . 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 7187 . . . 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 121 . . 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 504 . . . . . 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 502 . . . . . . 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 507 . . . . . . . . 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 272 . . . . . . . 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 503 . . . . . . . 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 2456 . . . . . . . 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 406 . . . . . . 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 3901 . . . . . . . . 9  |-  ( u  =  s  ->  (
( ( F `  j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  u  <->  ( ( F `  j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) )  <Q 
s ) )
2423rexbidv 2413 . . . . . . . 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 2814 . . . . . . 7  |-  ( s  e.  ( 2nd `  L
)  <->  ( s  e. 
Q.  /\  E. j  e.  N.  ( ( F `
 j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) )  <Q 
s ) )
2617, 22, 25sylanbrc 411 . . . . . 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 302 . . . . 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 114 . . . 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 2509 . . 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 2529 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 103    = wceq 1314    e. wcel 1463   A.wral 2391   E.wrex 2392   {crab 2395   <.cop 3498   class class class wbr 3897   -->wf 5087   ` cfv 5091  (class class class)co 5740   2ndc2nd 6003   1oc1o 6272   [cec 6393   N.cnpi 7044    <N clti 7047    ~Q ceq 7051   Q.cnq 7052    +Q cplq 7054   *Qcrq 7056    <Q cltq 7057
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 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-coll 4011  ax-sep 4014  ax-nul 4022  ax-pow 4066  ax-pr 4099  ax-un 4323  ax-setind 4420  ax-iinf 4470
This theorem depends on definitions:  df-bi 116  df-dc 803  df-3or 946  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ne 2284  df-ral 2396  df-rex 2397  df-reu 2398  df-rab 2400  df-v 2660  df-sbc 2881  df-csb 2974  df-dif 3041  df-un 3043  df-in 3045  df-ss 3052  df-nul 3332  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-int 3740  df-iun 3783  df-br 3898  df-opab 3958  df-mpt 3959  df-tr 3995  df-eprel 4179  df-id 4183  df-po 4186  df-iso 4187  df-iord 4256  df-on 4258  df-suc 4261  df-iom 4473  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-rn 4518  df-res 4519  df-ima 4520  df-iota 5056  df-fun 5093  df-fn 5094  df-f 5095  df-f1 5096  df-fo 5097  df-f1o 5098  df-fv 5099  df-ov 5743  df-oprab 5744  df-mpo 5745  df-1st 6004  df-2nd 6005  df-recs 6168  df-irdg 6233  df-1o 6279  df-oadd 6283  df-omul 6284  df-er 6395  df-ec 6397  df-qs 6401  df-ni 7076  df-pli 7077  df-mi 7078  df-lti 7079  df-plpq 7116  df-mpq 7117  df-enq 7119  df-nqqs 7120  df-plqqs 7121  df-mqqs 7122  df-1nqqs 7123  df-rq 7124  df-ltnqqs 7125
This theorem is referenced by:  caucvgprlemrnd  7445
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