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Theorem caucvgprlemopu 7669
Description: Lemma for caucvgpr 7680. 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 4007 . . . . . 6  |-  ( u  =  r  ->  (
( ( F `  j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  u  <->  ( ( F `  j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) )  <Q 
r ) )
21rexbidv 2478 . . . . 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 5518 . . . . . 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 7361 . . . . . . . 8  |-  Q.  e.  _V
65rabex 4147 . . . . . . 7  |-  { l  e.  Q.  |  E. j  e.  N.  (
l  +Q  ( *Q
`  [ <. j ,  1o >. ]  ~Q  )
)  <Q  ( F `  j ) }  e.  _V
75rabex 4147 . . . . . . 7  |-  { u  e.  Q.  |  E. j  e.  N.  ( ( F `
 j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) )  <Q  u }  e.  _V
86, 7op2nd 6147 . . . . . 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 2198 . . . . 5  |-  ( 2nd `  L )  =  {
u  e.  Q.  |  E. j  e.  N.  ( ( F `  j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  u }
102, 9elrab2 2896 . . . 4  |-  ( r  e.  ( 2nd `  L
)  <->  ( r  e. 
Q.  /\  E. j  e.  N.  ( ( F `
 j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) )  <Q 
r ) )
1110simprbi 275 . . 3  |-  ( r  e.  ( 2nd `  L
)  ->  E. j  e.  N.  ( ( F `
 j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) )  <Q 
r )
1211adantl 277 . 2  |-  ( (
ph  /\  r  e.  ( 2nd `  L ) )  ->  E. j  e.  N.  ( ( F `
 j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) )  <Q 
r )
13 simprr 531 . . . 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 7413 . . . 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 122 . . 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 531 . . . . . 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 528 . . . . . . 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 535 . . . . . . . . 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 276 . . . . . . . 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 529 . . . . . . . 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 2526 . . . . . . . 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 411 . . . . . . 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 4007 . . . . . . . . 9  |-  ( u  =  s  ->  (
( ( F `  j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  u  <->  ( ( F `  j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) )  <Q 
s ) )
2423rexbidv 2478 . . . . . . . 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 2896 . . . . . . 7  |-  ( s  e.  ( 2nd `  L
)  <->  ( s  e. 
Q.  /\  E. j  e.  N.  ( ( F `
 j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) )  <Q 
s ) )
2617, 22, 25sylanbrc 417 . . . . . 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 306 . . . . 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 115 . . . 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 2579 . . 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 2599 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 104    = wceq 1353    e. wcel 2148   A.wral 2455   E.wrex 2456   {crab 2459   <.cop 3595   class class class wbr 4003   -->wf 5212   ` cfv 5216  (class class class)co 5874   2ndc2nd 6139   1oc1o 6409   [cec 6532   N.cnpi 7270    <N clti 7273    ~Q ceq 7277   Q.cnq 7278    +Q cplq 7280   *Qcrq 7282    <Q cltq 7283
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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4118  ax-sep 4121  ax-nul 4129  ax-pow 4174  ax-pr 4209  ax-un 4433  ax-setind 4536  ax-iinf 4587
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-iun 3888  df-br 4004  df-opab 4065  df-mpt 4066  df-tr 4102  df-eprel 4289  df-id 4293  df-po 4296  df-iso 4297  df-iord 4366  df-on 4368  df-suc 4371  df-iom 4590  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-res 4638  df-ima 4639  df-iota 5178  df-fun 5218  df-fn 5219  df-f 5220  df-f1 5221  df-fo 5222  df-f1o 5223  df-fv 5224  df-ov 5877  df-oprab 5878  df-mpo 5879  df-1st 6140  df-2nd 6141  df-recs 6305  df-irdg 6370  df-1o 6416  df-oadd 6420  df-omul 6421  df-er 6534  df-ec 6536  df-qs 6540  df-ni 7302  df-pli 7303  df-mi 7304  df-lti 7305  df-plpq 7342  df-mpq 7343  df-enq 7345  df-nqqs 7346  df-plqqs 7347  df-mqqs 7348  df-1nqqs 7349  df-rq 7350  df-ltnqqs 7351
This theorem is referenced by:  caucvgprlemrnd  7671
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