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Theorem caucvgprlem2 7708
Description: Lemma for caucvgpr 7710. Part of showing the putative limit to be a limit. (Contributed by Jim Kingdon, 3-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 } >.
caucvgprlemlim.q  |-  ( ph  ->  Q  e.  Q. )
caucvgprlemlim.jk  |-  ( ph  ->  J  <N  K )
caucvgprlemlim.jkq  |-  ( ph  ->  ( *Q `  [ <. J ,  1o >. ]  ~Q  )  <Q  Q )
Assertion
Ref Expression
caucvgprlem2  |-  ( ph  ->  L  <P  <. { l  |  l  <Q  (
( F `  K
)  +Q  Q ) } ,  { u  |  ( ( F `
 K )  +Q  Q )  <Q  u } >. )
Distinct variable groups:    A, j    j, F, u, l    n, F, k    j, K, u, l    j, L, k    Q, l, u    j, l   
j, k    k, n
Allowed substitution hints:    ph( u, j, k, n, l)    A( u, k, n, l)    Q( j, k, n)    J( u, j, k, n, l)    K( k, n)    L( u, n, l)

Proof of Theorem caucvgprlem2
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 caucvgprlemlim.jk . . . . 5  |-  ( ph  ->  J  <N  K )
2 caucvgprlemlim.jkq . . . . 5  |-  ( ph  ->  ( *Q `  [ <. J ,  1o >. ]  ~Q  )  <Q  Q )
31, 2caucvgprlemk 7693 . . . 4  |-  ( ph  ->  ( *Q `  [ <. K ,  1o >. ]  ~Q  )  <Q  Q )
4 caucvgpr.f . . . . 5  |-  ( ph  ->  F : N. --> Q. )
5 ltrelpi 7352 . . . . . . . 8  |-  <N  C_  ( N.  X.  N. )
65brel 4696 . . . . . . 7  |-  ( J 
<N  K  ->  ( J  e.  N.  /\  K  e.  N. ) )
71, 6syl 14 . . . . . 6  |-  ( ph  ->  ( J  e.  N.  /\  K  e.  N. )
)
87simprd 114 . . . . 5  |-  ( ph  ->  K  e.  N. )
94, 8ffvelcdmd 5672 . . . 4  |-  ( ph  ->  ( F `  K
)  e.  Q. )
10 ltanqi 7430 . . . 4  |-  ( ( ( *Q `  [ <. K ,  1o >. ]  ~Q  )  <Q  Q  /\  ( F `  K )  e.  Q. )  -> 
( ( F `  K )  +Q  ( *Q `  [ <. K ,  1o >. ]  ~Q  )
)  <Q  ( ( F `
 K )  +Q  Q ) )
113, 9, 10syl2anc 411 . . 3  |-  ( ph  ->  ( ( F `  K )  +Q  ( *Q `  [ <. K ,  1o >. ]  ~Q  )
)  <Q  ( ( F `
 K )  +Q  Q ) )
12 ltbtwnnqq 7443 . . 3  |-  ( ( ( F `  K
)  +Q  ( *Q
`  [ <. K ,  1o >. ]  ~Q  )
)  <Q  ( ( F `
 K )  +Q  Q )  <->  E. x  e.  Q.  ( ( ( F `  K )  +Q  ( *Q `  [ <. K ,  1o >. ]  ~Q  ) ) 
<Q  x  /\  x  <Q  ( ( F `  K )  +Q  Q
) ) )
1311, 12sylib 122 . 2  |-  ( ph  ->  E. x  e.  Q.  ( ( ( F `
 K )  +Q  ( *Q `  [ <. K ,  1o >. ]  ~Q  ) )  <Q  x  /\  x  <Q  (
( F `  K
)  +Q  Q ) ) )
14 simprl 529 . . . 4  |-  ( (
ph  /\  ( x  e.  Q.  /\  ( ( ( F `  K
)  +Q  ( *Q
`  [ <. K ,  1o >. ]  ~Q  )
)  <Q  x  /\  x  <Q  ( ( F `  K )  +Q  Q
) ) ) )  ->  x  e.  Q. )
158adantr 276 . . . . . 6  |-  ( (
ph  /\  ( x  e.  Q.  /\  ( ( ( F `  K
)  +Q  ( *Q
`  [ <. K ,  1o >. ]  ~Q  )
)  <Q  x  /\  x  <Q  ( ( F `  K )  +Q  Q
) ) ) )  ->  K  e.  N. )
16 simprrl 539 . . . . . 6  |-  ( (
ph  /\  ( x  e.  Q.  /\  ( ( ( F `  K
)  +Q  ( *Q
`  [ <. K ,  1o >. ]  ~Q  )
)  <Q  x  /\  x  <Q  ( ( F `  K )  +Q  Q
) ) ) )  ->  ( ( F `
 K )  +Q  ( *Q `  [ <. K ,  1o >. ]  ~Q  ) )  <Q  x )
17 fveq2 5534 . . . . . . . . 9  |-  ( j  =  K  ->  ( F `  j )  =  ( F `  K ) )
18 opeq1 3793 . . . . . . . . . . 11  |-  ( j  =  K  ->  <. j ,  1o >.  =  <. K ,  1o >. )
1918eceq1d 6594 . . . . . . . . . 10  |-  ( j  =  K  ->  [ <. j ,  1o >. ]  ~Q  =  [ <. K ,  1o >. ]  ~Q  )
2019fveq2d 5538 . . . . . . . . 9  |-  ( j  =  K  ->  ( *Q `  [ <. j ,  1o >. ]  ~Q  )  =  ( *Q `  [ <. K ,  1o >. ]  ~Q  ) )
2117, 20oveq12d 5913 . . . . . . . 8  |-  ( j  =  K  ->  (
( F `  j
)  +Q  ( *Q
`  [ <. j ,  1o >. ]  ~Q  )
)  =  ( ( F `  K )  +Q  ( *Q `  [ <. K ,  1o >. ]  ~Q  ) ) )
2221breq1d 4028 . . . . . . 7  |-  ( j  =  K  ->  (
( ( F `  j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  x  <->  ( ( F `  K )  +Q  ( *Q `  [ <. K ,  1o >. ]  ~Q  ) )  <Q  x ) )
2322rspcev 2856 . . . . . 6  |-  ( ( K  e.  N.  /\  ( ( F `  K )  +Q  ( *Q `  [ <. K ,  1o >. ]  ~Q  )
)  <Q  x )  ->  E. j  e.  N.  ( ( F `  j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  x )
2415, 16, 23syl2anc 411 . . . . 5  |-  ( (
ph  /\  ( x  e.  Q.  /\  ( ( ( F `  K
)  +Q  ( *Q
`  [ <. K ,  1o >. ]  ~Q  )
)  <Q  x  /\  x  <Q  ( ( F `  K )  +Q  Q
) ) ) )  ->  E. j  e.  N.  ( ( F `  j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  x )
25 breq2 4022 . . . . . . 7  |-  ( u  =  x  ->  (
( ( F `  j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  u  <->  ( ( F `  j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) )  <Q  x ) )
2625rexbidv 2491 . . . . . 6  |-  ( u  =  x  ->  ( 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  x ) )
27 caucvgpr.lim . . . . . . . 8  |-  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 } >.
2827fveq2i 5537 . . . . . . 7  |-  ( 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 } >. )
29 nqex 7391 . . . . . . . . 9  |-  Q.  e.  _V
3029rabex 4162 . . . . . . . 8  |-  { l  e.  Q.  |  E. j  e.  N.  (
l  +Q  ( *Q
`  [ <. j ,  1o >. ]  ~Q  )
)  <Q  ( F `  j ) }  e.  _V
3129rabex 4162 . . . . . . . 8  |-  { u  e.  Q.  |  E. j  e.  N.  ( ( F `
 j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) )  <Q  u }  e.  _V
3230, 31op2nd 6171 . . . . . . 7  |-  ( 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 }
3328, 32eqtri 2210 . . . . . 6  |-  ( 2nd `  L )  =  {
u  e.  Q.  |  E. j  e.  N.  ( ( F `  j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  u }
3426, 33elrab2 2911 . . . . 5  |-  ( x  e.  ( 2nd `  L
)  <->  ( x  e. 
Q.  /\  E. j  e.  N.  ( ( F `
 j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) )  <Q  x ) )
3514, 24, 34sylanbrc 417 . . . 4  |-  ( (
ph  /\  ( x  e.  Q.  /\  ( ( ( F `  K
)  +Q  ( *Q
`  [ <. K ,  1o >. ]  ~Q  )
)  <Q  x  /\  x  <Q  ( ( F `  K )  +Q  Q
) ) ) )  ->  x  e.  ( 2nd `  L ) )
36 simprrr 540 . . . . . 6  |-  ( (
ph  /\  ( x  e.  Q.  /\  ( ( ( F `  K
)  +Q  ( *Q
`  [ <. K ,  1o >. ]  ~Q  )
)  <Q  x  /\  x  <Q  ( ( F `  K )  +Q  Q
) ) ) )  ->  x  <Q  (
( F `  K
)  +Q  Q ) )
37 vex 2755 . . . . . . 7  |-  x  e. 
_V
38 breq1 4021 . . . . . . 7  |-  ( l  =  x  ->  (
l  <Q  ( ( F `
 K )  +Q  Q )  <->  x  <Q  ( ( F `  K
)  +Q  Q ) ) )
3937, 38elab 2896 . . . . . 6  |-  ( x  e.  { l  |  l  <Q  ( ( F `  K )  +Q  Q ) }  <->  x  <Q  ( ( F `  K
)  +Q  Q ) )
4036, 39sylibr 134 . . . . 5  |-  ( (
ph  /\  ( x  e.  Q.  /\  ( ( ( F `  K
)  +Q  ( *Q
`  [ <. K ,  1o >. ]  ~Q  )
)  <Q  x  /\  x  <Q  ( ( F `  K )  +Q  Q
) ) ) )  ->  x  e.  {
l  |  l  <Q 
( ( F `  K )  +Q  Q
) } )
41 ltnqex 7577 . . . . . 6  |-  { l  |  l  <Q  (
( F `  K
)  +Q  Q ) }  e.  _V
42 gtnqex 7578 . . . . . 6  |-  { u  |  ( ( F `
 K )  +Q  Q )  <Q  u }  e.  _V
4341, 42op1st 6170 . . . . 5  |-  ( 1st `  <. { l  |  l  <Q  ( ( F `  K )  +Q  Q ) } ,  { u  |  (
( F `  K
)  +Q  Q ) 
<Q  u } >. )  =  { l  |  l 
<Q  ( ( F `  K )  +Q  Q
) }
4440, 43eleqtrrdi 2283 . . . 4  |-  ( (
ph  /\  ( x  e.  Q.  /\  ( ( ( F `  K
)  +Q  ( *Q
`  [ <. K ,  1o >. ]  ~Q  )
)  <Q  x  /\  x  <Q  ( ( F `  K )  +Q  Q
) ) ) )  ->  x  e.  ( 1st `  <. { l  |  l  <Q  (
( F `  K
)  +Q  Q ) } ,  { u  |  ( ( F `
 K )  +Q  Q )  <Q  u } >. ) )
45 rspe 2539 . . . 4  |-  ( ( x  e.  Q.  /\  ( x  e.  ( 2nd `  L )  /\  x  e.  ( 1st ` 
<. { l  |  l 
<Q  ( ( F `  K )  +Q  Q
) } ,  {
u  |  ( ( F `  K )  +Q  Q )  <Q  u } >. ) ) )  ->  E. x  e.  Q.  ( x  e.  ( 2nd `  L )  /\  x  e.  ( 1st ` 
<. { l  |  l 
<Q  ( ( F `  K )  +Q  Q
) } ,  {
u  |  ( ( F `  K )  +Q  Q )  <Q  u } >. ) ) )
4614, 35, 44, 45syl12anc 1247 . . 3  |-  ( (
ph  /\  ( x  e.  Q.  /\  ( ( ( F `  K
)  +Q  ( *Q
`  [ <. K ,  1o >. ]  ~Q  )
)  <Q  x  /\  x  <Q  ( ( F `  K )  +Q  Q
) ) ) )  ->  E. x  e.  Q.  ( x  e.  ( 2nd `  L )  /\  x  e.  ( 1st ` 
<. { l  |  l 
<Q  ( ( F `  K )  +Q  Q
) } ,  {
u  |  ( ( F `  K )  +Q  Q )  <Q  u } >. ) ) )
47 caucvgpr.cau . . . . . 6  |-  ( 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  )
) ) ) )
48 caucvgpr.bnd . . . . . 6  |-  ( ph  ->  A. j  e.  N.  A  <Q  ( F `  j ) )
494, 47, 48, 27caucvgprlemcl 7704 . . . . 5  |-  ( ph  ->  L  e.  P. )
5049adantr 276 . . . 4  |-  ( (
ph  /\  ( x  e.  Q.  /\  ( ( ( F `  K
)  +Q  ( *Q
`  [ <. K ,  1o >. ]  ~Q  )
)  <Q  x  /\  x  <Q  ( ( F `  K )  +Q  Q
) ) ) )  ->  L  e.  P. )
51 caucvgprlemlim.q . . . . . . 7  |-  ( ph  ->  Q  e.  Q. )
52 addclnq 7403 . . . . . . 7  |-  ( ( ( F `  K
)  e.  Q.  /\  Q  e.  Q. )  ->  ( ( F `  K )  +Q  Q
)  e.  Q. )
539, 51, 52syl2anc 411 . . . . . 6  |-  ( ph  ->  ( ( F `  K )  +Q  Q
)  e.  Q. )
54 nqprlu 7575 . . . . . 6  |-  ( ( ( F `  K
)  +Q  Q )  e.  Q.  ->  <. { l  |  l  <Q  (
( F `  K
)  +Q  Q ) } ,  { u  |  ( ( F `
 K )  +Q  Q )  <Q  u } >.  e.  P. )
5553, 54syl 14 . . . . 5  |-  ( ph  -> 
<. { l  |  l 
<Q  ( ( F `  K )  +Q  Q
) } ,  {
u  |  ( ( F `  K )  +Q  Q )  <Q  u } >.  e.  P. )
5655adantr 276 . . . 4  |-  ( (
ph  /\  ( x  e.  Q.  /\  ( ( ( F `  K
)  +Q  ( *Q
`  [ <. K ,  1o >. ]  ~Q  )
)  <Q  x  /\  x  <Q  ( ( F `  K )  +Q  Q
) ) ) )  ->  <. { l  |  l  <Q  ( ( F `  K )  +Q  Q ) } ,  { u  |  (
( F `  K
)  +Q  Q ) 
<Q  u } >.  e.  P. )
57 ltdfpr 7534 . . . 4  |-  ( ( L  e.  P.  /\  <. { l  |  l 
<Q  ( ( F `  K )  +Q  Q
) } ,  {
u  |  ( ( F `  K )  +Q  Q )  <Q  u } >.  e.  P. )  ->  ( L  <P  <. { l  |  l 
<Q  ( ( F `  K )  +Q  Q
) } ,  {
u  |  ( ( F `  K )  +Q  Q )  <Q  u } >.  <->  E. x  e.  Q.  ( x  e.  ( 2nd `  L )  /\  x  e.  ( 1st ` 
<. { l  |  l 
<Q  ( ( F `  K )  +Q  Q
) } ,  {
u  |  ( ( F `  K )  +Q  Q )  <Q  u } >. ) ) ) )
5850, 56, 57syl2anc 411 . . 3  |-  ( (
ph  /\  ( x  e.  Q.  /\  ( ( ( F `  K
)  +Q  ( *Q
`  [ <. K ,  1o >. ]  ~Q  )
)  <Q  x  /\  x  <Q  ( ( F `  K )  +Q  Q
) ) ) )  ->  ( L  <P  <. { l  |  l 
<Q  ( ( F `  K )  +Q  Q
) } ,  {
u  |  ( ( F `  K )  +Q  Q )  <Q  u } >.  <->  E. x  e.  Q.  ( x  e.  ( 2nd `  L )  /\  x  e.  ( 1st ` 
<. { l  |  l 
<Q  ( ( F `  K )  +Q  Q
) } ,  {
u  |  ( ( F `  K )  +Q  Q )  <Q  u } >. ) ) ) )
5946, 58mpbird 167 . 2  |-  ( (
ph  /\  ( x  e.  Q.  /\  ( ( ( F `  K
)  +Q  ( *Q
`  [ <. K ,  1o >. ]  ~Q  )
)  <Q  x  /\  x  <Q  ( ( F `  K )  +Q  Q
) ) ) )  ->  L  <P  <. { l  |  l  <Q  (
( F `  K
)  +Q  Q ) } ,  { u  |  ( ( F `
 K )  +Q  Q )  <Q  u } >. )
6013, 59rexlimddv 2612 1  |-  ( ph  ->  L  <P  <. { l  |  l  <Q  (
( F `  K
)  +Q  Q ) } ,  { u  |  ( ( F `
 K )  +Q  Q )  <Q  u } >. )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105    = wceq 1364    e. wcel 2160   {cab 2175   A.wral 2468   E.wrex 2469   {crab 2472   <.cop 3610   class class class wbr 4018   -->wf 5231   ` cfv 5235  (class class class)co 5895   1stc1st 6162   2ndc2nd 6163   1oc1o 6433   [cec 6556   N.cnpi 7300    <N clti 7303    ~Q ceq 7307   Q.cnq 7308    +Q cplq 7310   *Qcrq 7312    <Q cltq 7313   P.cnp 7319    <P cltp 7323
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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-nul 4144  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554  ax-iinf 4605
This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-ral 2473  df-rex 2474  df-reu 2475  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-tr 4117  df-eprel 4307  df-id 4311  df-po 4314  df-iso 4315  df-iord 4384  df-on 4386  df-suc 4389  df-iom 4608  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-f1 5240  df-fo 5241  df-f1o 5242  df-fv 5243  df-ov 5898  df-oprab 5899  df-mpo 5900  df-1st 6164  df-2nd 6165  df-recs 6329  df-irdg 6394  df-1o 6440  df-oadd 6444  df-omul 6445  df-er 6558  df-ec 6560  df-qs 6564  df-ni 7332  df-pli 7333  df-mi 7334  df-lti 7335  df-plpq 7372  df-mpq 7373  df-enq 7375  df-nqqs 7376  df-plqqs 7377  df-mqqs 7378  df-1nqqs 7379  df-rq 7380  df-ltnqqs 7381  df-inp 7494  df-iltp 7498
This theorem is referenced by:  caucvgprlemlim  7709
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