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Theorem caucvgprlem2 7654
Description: Lemma for caucvgpr 7656. 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 7639 . . . 4  |-  ( ph  ->  ( *Q `  [ <. K ,  1o >. ]  ~Q  )  <Q  Q )
4 caucvgpr.f . . . . 5  |-  ( ph  ->  F : N. --> Q. )
5 ltrelpi 7298 . . . . . . . 8  |-  <N  C_  ( N.  X.  N. )
65brel 4672 . . . . . . 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 5644 . . . 4  |-  ( ph  ->  ( F `  K
)  e.  Q. )
10 ltanqi 7376 . . . 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 7389 . . 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 5507 . . . . . . . . 9  |-  ( j  =  K  ->  ( F `  j )  =  ( F `  K ) )
18 opeq1 3774 . . . . . . . . . . 11  |-  ( j  =  K  ->  <. j ,  1o >.  =  <. K ,  1o >. )
1918eceq1d 6561 . . . . . . . . . 10  |-  ( j  =  K  ->  [ <. j ,  1o >. ]  ~Q  =  [ <. K ,  1o >. ]  ~Q  )
2019fveq2d 5511 . . . . . . . . 9  |-  ( j  =  K  ->  ( *Q `  [ <. j ,  1o >. ]  ~Q  )  =  ( *Q `  [ <. K ,  1o >. ]  ~Q  ) )
2117, 20oveq12d 5883 . . . . . . . 8  |-  ( j  =  K  ->  (
( F `  j
)  +Q  ( *Q
`  [ <. j ,  1o >. ]  ~Q  )
)  =  ( ( F `  K )  +Q  ( *Q `  [ <. K ,  1o >. ]  ~Q  ) ) )
2221breq1d 4008 . . . . . . 7  |-  ( j  =  K  ->  (
( ( F `  j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  x  <->  ( ( F `  K )  +Q  ( *Q `  [ <. K ,  1o >. ]  ~Q  ) )  <Q  x ) )
2322rspcev 2839 . . . . . 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 4002 . . . . . . 7  |-  ( u  =  x  ->  (
( ( F `  j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  u  <->  ( ( F `  j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) )  <Q  x ) )
2625rexbidv 2476 . . . . . 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 5510 . . . . . . 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 7337 . . . . . . . . 9  |-  Q.  e.  _V
3029rabex 4142 . . . . . . . 8  |-  { l  e.  Q.  |  E. j  e.  N.  (
l  +Q  ( *Q
`  [ <. j ,  1o >. ]  ~Q  )
)  <Q  ( F `  j ) }  e.  _V
3129rabex 4142 . . . . . . . 8  |-  { u  e.  Q.  |  E. j  e.  N.  ( ( F `
 j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) )  <Q  u }  e.  _V
3230, 31op2nd 6138 . . . . . . 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 2196 . . . . . 6  |-  ( 2nd `  L )  =  {
u  e.  Q.  |  E. j  e.  N.  ( ( F `  j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  u }
3426, 33elrab2 2894 . . . . 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 2738 . . . . . . 7  |-  x  e. 
_V
38 breq1 4001 . . . . . . 7  |-  ( l  =  x  ->  (
l  <Q  ( ( F `
 K )  +Q  Q )  <->  x  <Q  ( ( F `  K
)  +Q  Q ) ) )
3937, 38elab 2879 . . . . . 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 7523 . . . . . 6  |-  { l  |  l  <Q  (
( F `  K
)  +Q  Q ) }  e.  _V
42 gtnqex 7524 . . . . . 6  |-  { u  |  ( ( F `
 K )  +Q  Q )  <Q  u }  e.  _V
4341, 42op1st 6137 . . . . 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 2269 . . . 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 2524 . . . 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 7650 . . . . 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 7349 . . . . . . 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 7521 . . . . . 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 7480 . . . 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 2597 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 2146   {cab 2161   A.wral 2453   E.wrex 2454   {crab 2457   <.cop 3592   class class class wbr 3998   -->wf 5204   ` cfv 5208  (class class class)co 5865   1stc1st 6129   2ndc2nd 6130   1oc1o 6400   [cec 6523   N.cnpi 7246    <N clti 7249    ~Q ceq 7253   Q.cnq 7254    +Q cplq 7256   *Qcrq 7258    <Q cltq 7259   P.cnp 7265    <P cltp 7269
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 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-13 2148  ax-14 2149  ax-ext 2157  ax-coll 4113  ax-sep 4116  ax-nul 4124  ax-pow 4169  ax-pr 4203  ax-un 4427  ax-setind 4530  ax-iinf 4581
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 1459  df-sb 1761  df-eu 2027  df-mo 2028  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ne 2346  df-ral 2458  df-rex 2459  df-reu 2460  df-rab 2462  df-v 2737  df-sbc 2961  df-csb 3056  df-dif 3129  df-un 3131  df-in 3133  df-ss 3140  df-nul 3421  df-pw 3574  df-sn 3595  df-pr 3596  df-op 3598  df-uni 3806  df-int 3841  df-iun 3884  df-br 3999  df-opab 4060  df-mpt 4061  df-tr 4097  df-eprel 4283  df-id 4287  df-po 4290  df-iso 4291  df-iord 4360  df-on 4362  df-suc 4365  df-iom 4584  df-xp 4626  df-rel 4627  df-cnv 4628  df-co 4629  df-dm 4630  df-rn 4631  df-res 4632  df-ima 4633  df-iota 5170  df-fun 5210  df-fn 5211  df-f 5212  df-f1 5213  df-fo 5214  df-f1o 5215  df-fv 5216  df-ov 5868  df-oprab 5869  df-mpo 5870  df-1st 6131  df-2nd 6132  df-recs 6296  df-irdg 6361  df-1o 6407  df-oadd 6411  df-omul 6412  df-er 6525  df-ec 6527  df-qs 6531  df-ni 7278  df-pli 7279  df-mi 7280  df-lti 7281  df-plpq 7318  df-mpq 7319  df-enq 7321  df-nqqs 7322  df-plqqs 7323  df-mqqs 7324  df-1nqqs 7325  df-rq 7326  df-ltnqqs 7327  df-inp 7440  df-iltp 7444
This theorem is referenced by:  caucvgprlemlim  7655
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