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Theorem caucvgprlem2 6759
Description: Lemma for caucvgpr 6761. 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 6744 . . . 4  |-  ( ph  ->  ( *Q `  [ <. K ,  1o >. ]  ~Q  )  <Q  Q )
4 caucvgpr.f . . . . 5  |-  ( ph  ->  F : N. --> Q. )
5 ltrelpi 6403 . . . . . . . 8  |-  <N  C_  ( N.  X.  N. )
65brel 4379 . . . . . . 7  |-  ( J 
<N  K  ->  ( J  e.  N.  /\  K  e.  N. ) )
71, 6syl 14 . . . . . 6  |-  ( ph  ->  ( J  e.  N.  /\  K  e.  N. )
)
87simprd 107 . . . . 5  |-  ( ph  ->  K  e.  N. )
94, 8ffvelrnd 5290 . . . 4  |-  ( ph  ->  ( F `  K
)  e.  Q. )
10 ltanqi 6481 . . . 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 391 . . 3  |-  ( ph  ->  ( ( F `  K )  +Q  ( *Q `  [ <. K ,  1o >. ]  ~Q  )
)  <Q  ( ( F `
 K )  +Q  Q ) )
12 ltbtwnnqq 6494 . . 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 127 . 2  |-  ( ph  ->  E. x  e.  Q.  ( ( ( F `
 K )  +Q  ( *Q `  [ <. K ,  1o >. ]  ~Q  ) )  <Q  x  /\  x  <Q  (
( F `  K
)  +Q  Q ) ) )
14 simprl 483 . . . 4  |-  ( (
ph  /\  ( x  e.  Q.  /\  ( ( ( F `  K
)  +Q  ( *Q
`  [ <. K ,  1o >. ]  ~Q  )
)  <Q  x  /\  x  <Q  ( ( F `  K )  +Q  Q
) ) ) )  ->  x  e.  Q. )
158adantr 261 . . . . . 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 491 . . . . . 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 5165 . . . . . . . . 9  |-  ( j  =  K  ->  ( F `  j )  =  ( F `  K ) )
18 opeq1 3546 . . . . . . . . . . 11  |-  ( j  =  K  ->  <. j ,  1o >.  =  <. K ,  1o >. )
1918eceq1d 6129 . . . . . . . . . 10  |-  ( j  =  K  ->  [ <. j ,  1o >. ]  ~Q  =  [ <. K ,  1o >. ]  ~Q  )
2019fveq2d 5169 . . . . . . . . 9  |-  ( j  =  K  ->  ( *Q `  [ <. j ,  1o >. ]  ~Q  )  =  ( *Q `  [ <. K ,  1o >. ]  ~Q  ) )
2117, 20oveq12d 5517 . . . . . . . 8  |-  ( j  =  K  ->  (
( F `  j
)  +Q  ( *Q
`  [ <. j ,  1o >. ]  ~Q  )
)  =  ( ( F `  K )  +Q  ( *Q `  [ <. K ,  1o >. ]  ~Q  ) ) )
2221breq1d 3771 . . . . . . 7  |-  ( j  =  K  ->  (
( ( F `  j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  x  <->  ( ( F `  K )  +Q  ( *Q `  [ <. K ,  1o >. ]  ~Q  ) )  <Q  x ) )
2322rspcev 2653 . . . . . 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 391 . . . . 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 3765 . . . . . . 7  |-  ( u  =  x  ->  (
( ( F `  j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  u  <->  ( ( F `  j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) )  <Q  x ) )
2625rexbidv 2324 . . . . . 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 5168 . . . . . . 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 6442 . . . . . . . . 9  |-  Q.  e.  _V
3029rabex 3898 . . . . . . . 8  |-  { l  e.  Q.  |  E. j  e.  N.  (
l  +Q  ( *Q
`  [ <. j ,  1o >. ]  ~Q  )
)  <Q  ( F `  j ) }  e.  _V
3129rabex 3898 . . . . . . . 8  |-  { u  e.  Q.  |  E. j  e.  N.  ( ( F `
 j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) )  <Q  u }  e.  _V
3230, 31op2nd 5761 . . . . . . 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 2060 . . . . . 6  |-  ( 2nd `  L )  =  {
u  e.  Q.  |  E. j  e.  N.  ( ( F `  j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  u }
3426, 33elrab2 2697 . . . . 5  |-  ( x  e.  ( 2nd `  L
)  <->  ( x  e. 
Q.  /\  E. j  e.  N.  ( ( F `
 j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) )  <Q  x ) )
3514, 24, 34sylanbrc 394 . . . 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 492 . . . . . 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 2557 . . . . . . 7  |-  x  e. 
_V
38 breq1 3764 . . . . . . 7  |-  ( l  =  x  ->  (
l  <Q  ( ( F `
 K )  +Q  Q )  <->  x  <Q  ( ( F `  K
)  +Q  Q ) ) )
3937, 38elab 2684 . . . . . 6  |-  ( x  e.  { l  |  l  <Q  ( ( F `  K )  +Q  Q ) }  <->  x  <Q  ( ( F `  K
)  +Q  Q ) )
4036, 39sylibr 137 . . . . 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 6628 . . . . . 6  |-  { l  |  l  <Q  (
( F `  K
)  +Q  Q ) }  e.  _V
42 gtnqex 6629 . . . . . 6  |-  { u  |  ( ( F `
 K )  +Q  Q )  <Q  u }  e.  _V
4341, 42op1st 5760 . . . . 5  |-  ( 1st `  <. { l  |  l  <Q  ( ( F `  K )  +Q  Q ) } ,  { u  |  (
( F `  K
)  +Q  Q ) 
<Q  u } >. )  =  { l  |  l 
<Q  ( ( F `  K )  +Q  Q
) }
4440, 43syl6eleqr 2131 . . . 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 2367 . . . 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 1133 . . 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 6755 . . . . 5  |-  ( ph  ->  L  e.  P. )
5049adantr 261 . . . 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 6454 . . . . . . 7  |-  ( ( ( F `  K
)  e.  Q.  /\  Q  e.  Q. )  ->  ( ( F `  K )  +Q  Q
)  e.  Q. )
539, 51, 52syl2anc 391 . . . . . 6  |-  ( ph  ->  ( ( F `  K )  +Q  Q
)  e.  Q. )
54 nqprlu 6626 . . . . . 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 261 . . . 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 6585 . . . 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 391 . . 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 156 . 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 2434 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 97    <-> wb 98    = wceq 1243    e. wcel 1393   {cab 2026   A.wral 2303   E.wrex 2304   {crab 2307   <.cop 3375   class class class wbr 3761   -->wf 4885   ` cfv 4889  (class class class)co 5499   1stc1st 5752   2ndc2nd 5753   1oc1o 5981   [cec 6091   N.cnpi 6351    <N clti 6354    ~Q ceq 6358   Q.cnq 6359    +Q cplq 6361   *Qcrq 6363    <Q cltq 6364   P.cnp 6370    <P cltp 6374
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-coll 3869  ax-sep 3872  ax-nul 3880  ax-pow 3924  ax-pr 3941  ax-un 4166  ax-setind 4256  ax-iinf 4298
This theorem depends on definitions:  df-bi 110  df-dc 743  df-3or 886  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2308  df-rex 2309  df-reu 2310  df-rab 2312  df-v 2556  df-sbc 2762  df-csb 2850  df-dif 2917  df-un 2919  df-in 2921  df-ss 2928  df-nul 3222  df-pw 3358  df-sn 3378  df-pr 3379  df-op 3381  df-uni 3578  df-int 3613  df-iun 3656  df-br 3762  df-opab 3816  df-mpt 3817  df-tr 3852  df-eprel 4023  df-id 4027  df-po 4030  df-iso 4031  df-iord 4099  df-on 4101  df-suc 4104  df-iom 4301  df-xp 4338  df-rel 4339  df-cnv 4340  df-co 4341  df-dm 4342  df-rn 4343  df-res 4344  df-ima 4345  df-iota 4854  df-fun 4891  df-fn 4892  df-f 4893  df-f1 4894  df-fo 4895  df-f1o 4896  df-fv 4897  df-ov 5502  df-oprab 5503  df-mpt2 5504  df-1st 5754  df-2nd 5755  df-recs 5907  df-irdg 5944  df-1o 5988  df-oadd 5992  df-omul 5993  df-er 6093  df-ec 6095  df-qs 6099  df-ni 6383  df-pli 6384  df-mi 6385  df-lti 6386  df-plpq 6423  df-mpq 6424  df-enq 6426  df-nqqs 6427  df-plqqs 6428  df-mqqs 6429  df-1nqqs 6430  df-rq 6431  df-ltnqqs 6432  df-inp 6545  df-iltp 6549
This theorem is referenced by:  caucvgprlemlim  6760
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