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Theorem caucvgprlem2 7456
Description: Lemma for caucvgpr 7458. 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 7441 . . . 4  |-  ( ph  ->  ( *Q `  [ <. K ,  1o >. ]  ~Q  )  <Q  Q )
4 caucvgpr.f . . . . 5  |-  ( ph  ->  F : N. --> Q. )
5 ltrelpi 7100 . . . . . . . 8  |-  <N  C_  ( N.  X.  N. )
65brel 4561 . . . . . . 7  |-  ( J 
<N  K  ->  ( J  e.  N.  /\  K  e.  N. ) )
71, 6syl 14 . . . . . 6  |-  ( ph  ->  ( J  e.  N.  /\  K  e.  N. )
)
87simprd 113 . . . . 5  |-  ( ph  ->  K  e.  N. )
94, 8ffvelrnd 5524 . . . 4  |-  ( ph  ->  ( F `  K
)  e.  Q. )
10 ltanqi 7178 . . . 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 408 . . 3  |-  ( ph  ->  ( ( F `  K )  +Q  ( *Q `  [ <. K ,  1o >. ]  ~Q  )
)  <Q  ( ( F `
 K )  +Q  Q ) )
12 ltbtwnnqq 7191 . . 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 121 . 2  |-  ( ph  ->  E. x  e.  Q.  ( ( ( F `
 K )  +Q  ( *Q `  [ <. K ,  1o >. ]  ~Q  ) )  <Q  x  /\  x  <Q  (
( F `  K
)  +Q  Q ) ) )
14 simprl 505 . . . 4  |-  ( (
ph  /\  ( x  e.  Q.  /\  ( ( ( F `  K
)  +Q  ( *Q
`  [ <. K ,  1o >. ]  ~Q  )
)  <Q  x  /\  x  <Q  ( ( F `  K )  +Q  Q
) ) ) )  ->  x  e.  Q. )
158adantr 274 . . . . . 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 513 . . . . . 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 5389 . . . . . . . . 9  |-  ( j  =  K  ->  ( F `  j )  =  ( F `  K ) )
18 opeq1 3675 . . . . . . . . . . 11  |-  ( j  =  K  ->  <. j ,  1o >.  =  <. K ,  1o >. )
1918eceq1d 6433 . . . . . . . . . 10  |-  ( j  =  K  ->  [ <. j ,  1o >. ]  ~Q  =  [ <. K ,  1o >. ]  ~Q  )
2019fveq2d 5393 . . . . . . . . 9  |-  ( j  =  K  ->  ( *Q `  [ <. j ,  1o >. ]  ~Q  )  =  ( *Q `  [ <. K ,  1o >. ]  ~Q  ) )
2117, 20oveq12d 5760 . . . . . . . 8  |-  ( j  =  K  ->  (
( F `  j
)  +Q  ( *Q
`  [ <. j ,  1o >. ]  ~Q  )
)  =  ( ( F `  K )  +Q  ( *Q `  [ <. K ,  1o >. ]  ~Q  ) ) )
2221breq1d 3909 . . . . . . 7  |-  ( j  =  K  ->  (
( ( F `  j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  x  <->  ( ( F `  K )  +Q  ( *Q `  [ <. K ,  1o >. ]  ~Q  ) )  <Q  x ) )
2322rspcev 2763 . . . . . 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 408 . . . . 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 3903 . . . . . . 7  |-  ( u  =  x  ->  (
( ( F `  j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  u  <->  ( ( F `  j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) )  <Q  x ) )
2625rexbidv 2415 . . . . . 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 5392 . . . . . . 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 7139 . . . . . . . . 9  |-  Q.  e.  _V
3029rabex 4042 . . . . . . . 8  |-  { l  e.  Q.  |  E. j  e.  N.  (
l  +Q  ( *Q
`  [ <. j ,  1o >. ]  ~Q  )
)  <Q  ( F `  j ) }  e.  _V
3129rabex 4042 . . . . . . . 8  |-  { u  e.  Q.  |  E. j  e.  N.  ( ( F `
 j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) )  <Q  u }  e.  _V
3230, 31op2nd 6013 . . . . . . 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 2138 . . . . . 6  |-  ( 2nd `  L )  =  {
u  e.  Q.  |  E. j  e.  N.  ( ( F `  j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  u }
3426, 33elrab2 2816 . . . . 5  |-  ( x  e.  ( 2nd `  L
)  <->  ( x  e. 
Q.  /\  E. j  e.  N.  ( ( F `
 j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) )  <Q  x ) )
3514, 24, 34sylanbrc 413 . . . 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 514 . . . . . 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 2663 . . . . . . 7  |-  x  e. 
_V
38 breq1 3902 . . . . . . 7  |-  ( l  =  x  ->  (
l  <Q  ( ( F `
 K )  +Q  Q )  <->  x  <Q  ( ( F `  K
)  +Q  Q ) ) )
3937, 38elab 2802 . . . . . 6  |-  ( x  e.  { l  |  l  <Q  ( ( F `  K )  +Q  Q ) }  <->  x  <Q  ( ( F `  K
)  +Q  Q ) )
4036, 39sylibr 133 . . . . 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 7325 . . . . . 6  |-  { l  |  l  <Q  (
( F `  K
)  +Q  Q ) }  e.  _V
42 gtnqex 7326 . . . . . 6  |-  { u  |  ( ( F `
 K )  +Q  Q )  <Q  u }  e.  _V
4341, 42op1st 6012 . . . . 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 2211 . . . 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 2458 . . . 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 1199 . . 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 7452 . . . . 5  |-  ( ph  ->  L  e.  P. )
5049adantr 274 . . . 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 7151 . . . . . . 7  |-  ( ( ( F `  K
)  e.  Q.  /\  Q  e.  Q. )  ->  ( ( F `  K )  +Q  Q
)  e.  Q. )
539, 51, 52syl2anc 408 . . . . . 6  |-  ( ph  ->  ( ( F `  K )  +Q  Q
)  e.  Q. )
54 nqprlu 7323 . . . . . 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 274 . . . 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 7282 . . . 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 408 . . 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 166 . 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 2531 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 103    <-> wb 104    = wceq 1316    e. wcel 1465   {cab 2103   A.wral 2393   E.wrex 2394   {crab 2397   <.cop 3500   class class class wbr 3899   -->wf 5089   ` cfv 5093  (class class class)co 5742   1stc1st 6004   2ndc2nd 6005   1oc1o 6274   [cec 6395   N.cnpi 7048    <N clti 7051    ~Q ceq 7055   Q.cnq 7056    +Q cplq 7058   *Qcrq 7060    <Q cltq 7061   P.cnp 7067    <P cltp 7071
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 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-coll 4013  ax-sep 4016  ax-nul 4024  ax-pow 4068  ax-pr 4101  ax-un 4325  ax-setind 4422  ax-iinf 4472
This theorem depends on definitions:  df-bi 116  df-dc 805  df-3or 948  df-3an 949  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ne 2286  df-ral 2398  df-rex 2399  df-reu 2400  df-rab 2402  df-v 2662  df-sbc 2883  df-csb 2976  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-nul 3334  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-int 3742  df-iun 3785  df-br 3900  df-opab 3960  df-mpt 3961  df-tr 3997  df-eprel 4181  df-id 4185  df-po 4188  df-iso 4189  df-iord 4258  df-on 4260  df-suc 4263  df-iom 4475  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522  df-iota 5058  df-fun 5095  df-fn 5096  df-f 5097  df-f1 5098  df-fo 5099  df-f1o 5100  df-fv 5101  df-ov 5745  df-oprab 5746  df-mpo 5747  df-1st 6006  df-2nd 6007  df-recs 6170  df-irdg 6235  df-1o 6281  df-oadd 6285  df-omul 6286  df-er 6397  df-ec 6399  df-qs 6403  df-ni 7080  df-pli 7081  df-mi 7082  df-lti 7083  df-plpq 7120  df-mpq 7121  df-enq 7123  df-nqqs 7124  df-plqqs 7125  df-mqqs 7126  df-1nqqs 7127  df-rq 7128  df-ltnqqs 7129  df-inp 7242  df-iltp 7246
This theorem is referenced by:  caucvgprlemlim  7457
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