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Theorem caucvgprlem2 7670
Description: Lemma for caucvgpr 7672. 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 7655 . . . 4  |-  ( ph  ->  ( *Q `  [ <. K ,  1o >. ]  ~Q  )  <Q  Q )
4 caucvgpr.f . . . . 5  |-  ( ph  ->  F : N. --> Q. )
5 ltrelpi 7314 . . . . . . . 8  |-  <N  C_  ( N.  X.  N. )
65brel 4675 . . . . . . 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 5648 . . . 4  |-  ( ph  ->  ( F `  K
)  e.  Q. )
10 ltanqi 7392 . . . 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 7405 . . 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 5511 . . . . . . . . 9  |-  ( j  =  K  ->  ( F `  j )  =  ( F `  K ) )
18 opeq1 3776 . . . . . . . . . . 11  |-  ( j  =  K  ->  <. j ,  1o >.  =  <. K ,  1o >. )
1918eceq1d 6565 . . . . . . . . . 10  |-  ( j  =  K  ->  [ <. j ,  1o >. ]  ~Q  =  [ <. K ,  1o >. ]  ~Q  )
2019fveq2d 5515 . . . . . . . . 9  |-  ( j  =  K  ->  ( *Q `  [ <. j ,  1o >. ]  ~Q  )  =  ( *Q `  [ <. K ,  1o >. ]  ~Q  ) )
2117, 20oveq12d 5887 . . . . . . . 8  |-  ( j  =  K  ->  (
( F `  j
)  +Q  ( *Q
`  [ <. j ,  1o >. ]  ~Q  )
)  =  ( ( F `  K )  +Q  ( *Q `  [ <. K ,  1o >. ]  ~Q  ) ) )
2221breq1d 4010 . . . . . . 7  |-  ( j  =  K  ->  (
( ( F `  j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  x  <->  ( ( F `  K )  +Q  ( *Q `  [ <. K ,  1o >. ]  ~Q  ) )  <Q  x ) )
2322rspcev 2841 . . . . . 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 4004 . . . . . . 7  |-  ( u  =  x  ->  (
( ( F `  j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  u  <->  ( ( F `  j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) )  <Q  x ) )
2625rexbidv 2478 . . . . . 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 5514 . . . . . . 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 7353 . . . . . . . . 9  |-  Q.  e.  _V
3029rabex 4144 . . . . . . . 8  |-  { l  e.  Q.  |  E. j  e.  N.  (
l  +Q  ( *Q
`  [ <. j ,  1o >. ]  ~Q  )
)  <Q  ( F `  j ) }  e.  _V
3129rabex 4144 . . . . . . . 8  |-  { u  e.  Q.  |  E. j  e.  N.  ( ( F `
 j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) )  <Q  u }  e.  _V
3230, 31op2nd 6142 . . . . . . 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 2198 . . . . . 6  |-  ( 2nd `  L )  =  {
u  e.  Q.  |  E. j  e.  N.  ( ( F `  j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  u }
3426, 33elrab2 2896 . . . . 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 2740 . . . . . . 7  |-  x  e. 
_V
38 breq1 4003 . . . . . . 7  |-  ( l  =  x  ->  (
l  <Q  ( ( F `
 K )  +Q  Q )  <->  x  <Q  ( ( F `  K
)  +Q  Q ) ) )
3937, 38elab 2881 . . . . . 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 7539 . . . . . 6  |-  { l  |  l  <Q  (
( F `  K
)  +Q  Q ) }  e.  _V
42 gtnqex 7540 . . . . . 6  |-  { u  |  ( ( F `
 K )  +Q  Q )  <Q  u }  e.  _V
4341, 42op1st 6141 . . . . 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 2271 . . . 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 2526 . . . 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 1236 . . 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 7666 . . . . 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 7365 . . . . . . 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 7537 . . . . . 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 7496 . . . 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 2599 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 1353    e. wcel 2148   {cab 2163   A.wral 2455   E.wrex 2456   {crab 2459   <.cop 3594   class class class wbr 4000   -->wf 5208   ` cfv 5212  (class class class)co 5869   1stc1st 6133   2ndc2nd 6134   1oc1o 6404   [cec 6527   N.cnpi 7262    <N clti 7265    ~Q ceq 7269   Q.cnq 7270    +Q cplq 7272   *Qcrq 7274    <Q cltq 7275   P.cnp 7281    <P cltp 7285
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 4115  ax-sep 4118  ax-nul 4126  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-setind 4533  ax-iinf 4584
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 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-tr 4099  df-eprel 4286  df-id 4290  df-po 4293  df-iso 4294  df-iord 4363  df-on 4365  df-suc 4368  df-iom 4587  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-f1 5217  df-fo 5218  df-f1o 5219  df-fv 5220  df-ov 5872  df-oprab 5873  df-mpo 5874  df-1st 6135  df-2nd 6136  df-recs 6300  df-irdg 6365  df-1o 6411  df-oadd 6415  df-omul 6416  df-er 6529  df-ec 6531  df-qs 6535  df-ni 7294  df-pli 7295  df-mi 7296  df-lti 7297  df-plpq 7334  df-mpq 7335  df-enq 7337  df-nqqs 7338  df-plqqs 7339  df-mqqs 7340  df-1nqqs 7341  df-rq 7342  df-ltnqqs 7343  df-inp 7456  df-iltp 7460
This theorem is referenced by:  caucvgprlemlim  7671
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