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Theorem caucvgprprlem2 7672
Description: Lemma for caucvgprpr 7674. Part of showing the putative limit to be a limit. (Contributed by Jim Kingdon, 25-Nov-2020.)
Hypotheses
Ref Expression
caucvgprpr.f  |-  ( ph  ->  F : N. --> P. )
caucvgprpr.cau  |-  ( ph  ->  A. n  e.  N.  A. k  e.  N.  (
n  <N  k  ->  (
( F `  n
)  <P  ( ( F `
 k )  +P. 
<. { l  |  l 
<Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >. )  /\  ( F `  k
)  <P  ( ( F `
 n )  +P. 
<. { l  |  l 
<Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >. )
) ) )
caucvgprpr.bnd  |-  ( ph  ->  A. m  e.  N.  A  <P  ( F `  m ) )
caucvgprpr.lim  |-  L  = 
<. { l  e.  Q.  |  E. r  e.  N.  <. { p  |  p  <Q  ( l  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  )
) } ,  {
q  |  ( l  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) ) 
<Q  q } >.  <P  ( F `  r ) } ,  { u  e.  Q.  |  E. r  e.  N.  ( ( F `
 r )  +P. 
<. { p  |  p 
<Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q  q } >. ) 
<P  <. { p  |  p  <Q  u } ,  { q  |  u 
<Q  q } >. } >.
caucvgprprlemlim.q  |-  ( ph  ->  Q  e.  P. )
caucvgprprlemlim.jk  |-  ( ph  ->  J  <N  K )
caucvgprprlemlim.jkq  |-  ( ph  -> 
<. { l  |  l 
<Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. J ,  1o >. ]  ~Q  )  <Q  u } >.  <P  Q )
Assertion
Ref Expression
caucvgprprlem2  |-  ( ph  ->  L  <P  ( ( F `  K )  +P.  Q ) )
Distinct variable groups:    A, m    m, F    A, r    F, r, u, l, k    n, F    K, l, p, u, q, r    J, l, u    k, L    ph, r    k, n    k, r    q,
l, r    m, r    k, p, q    u, n, l, k
Allowed substitution hints:    ph( u, k, m, n, q, p, l)    A( u, k, n, q, p, l)    Q( u, k, m, n, r, q, p, l)    F( q, p)    J( k, m, n, r, q, p)    K( k, m, n)    L( u, m, n, r, q, p, l)

Proof of Theorem caucvgprprlem2
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 caucvgprprlemlim.jk . . . . 5  |-  ( ph  ->  J  <N  K )
2 caucvgprprlemlim.jkq . . . . 5  |-  ( ph  -> 
<. { l  |  l 
<Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. J ,  1o >. ]  ~Q  )  <Q  u } >.  <P  Q )
31, 2caucvgprprlemk 7645 . . . 4  |-  ( ph  -> 
<. { l  |  l 
<Q  ( *Q `  [ <. K ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. K ,  1o >. ]  ~Q  )  <Q  u } >.  <P  Q )
4 ltrelpi 7286 . . . . . . . . . 10  |-  <N  C_  ( N.  X.  N. )
54brel 4663 . . . . . . . . 9  |-  ( J 
<N  K  ->  ( J  e.  N.  /\  K  e.  N. ) )
61, 5syl 14 . . . . . . . 8  |-  ( ph  ->  ( J  e.  N.  /\  K  e.  N. )
)
76simprd 113 . . . . . . 7  |-  ( ph  ->  K  e.  N. )
8 nnnq 7384 . . . . . . . 8  |-  ( K  e.  N.  ->  [ <. K ,  1o >. ]  ~Q  e.  Q. )
9 recclnq 7354 . . . . . . . 8  |-  ( [
<. K ,  1o >. ]  ~Q  e.  Q.  ->  ( *Q `  [ <. K ,  1o >. ]  ~Q  )  e.  Q. )
108, 9syl 14 . . . . . . 7  |-  ( K  e.  N.  ->  ( *Q `  [ <. K ,  1o >. ]  ~Q  )  e.  Q. )
117, 10syl 14 . . . . . 6  |-  ( ph  ->  ( *Q `  [ <. K ,  1o >. ]  ~Q  )  e.  Q. )
12 nqprlu 7509 . . . . . 6  |-  ( ( *Q `  [ <. K ,  1o >. ]  ~Q  )  e.  Q.  ->  <. { l  |  l 
<Q  ( *Q `  [ <. K ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. K ,  1o >. ]  ~Q  )  <Q  u } >.  e.  P. )
1311, 12syl 14 . . . . 5  |-  ( ph  -> 
<. { l  |  l 
<Q  ( *Q `  [ <. K ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. K ,  1o >. ]  ~Q  )  <Q  u } >.  e.  P. )
14 caucvgprprlemlim.q . . . . 5  |-  ( ph  ->  Q  e.  P. )
15 caucvgprpr.f . . . . . 6  |-  ( ph  ->  F : N. --> P. )
1615, 7ffvelrnd 5632 . . . . 5  |-  ( ph  ->  ( F `  K
)  e.  P. )
17 ltaprg 7581 . . . . 5  |-  ( (
<. { l  |  l 
<Q  ( *Q `  [ <. K ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. K ,  1o >. ]  ~Q  )  <Q  u } >.  e.  P.  /\  Q  e.  P.  /\  ( F `  K )  e.  P. )  -> 
( <. { l  |  l  <Q  ( *Q `  [ <. K ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. K ,  1o >. ]  ~Q  )  <Q  u } >.  <P  Q  <->  ( ( F `  K )  +P.  <. { l  |  l  <Q  ( *Q `  [ <. K ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. K ,  1o >. ]  ~Q  )  <Q  u } >. )  <P  (
( F `  K
)  +P.  Q )
) )
1813, 14, 16, 17syl3anc 1233 . . . 4  |-  ( ph  ->  ( <. { l  |  l  <Q  ( *Q `  [ <. K ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. K ,  1o >. ]  ~Q  )  <Q  u } >.  <P  Q  <->  ( ( F `  K )  +P.  <. { l  |  l  <Q  ( *Q `  [ <. K ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. K ,  1o >. ]  ~Q  )  <Q  u } >. )  <P  (
( F `  K
)  +P.  Q )
) )
193, 18mpbid 146 . . 3  |-  ( ph  ->  ( ( F `  K )  +P.  <. { l  |  l  <Q 
( *Q `  [ <. K ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. K ,  1o >. ]  ~Q  )  <Q  u } >. )  <P  ( ( F `  K )  +P.  Q
) )
20 addclpr 7499 . . . . 5  |-  ( ( ( F `  K
)  e.  P.  /\  <. { l  |  l 
<Q  ( *Q `  [ <. K ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. K ,  1o >. ]  ~Q  )  <Q  u } >.  e.  P. )  ->  ( ( F `
 K )  +P. 
<. { l  |  l 
<Q  ( *Q `  [ <. K ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. K ,  1o >. ]  ~Q  )  <Q  u } >. )  e.  P. )
2116, 13, 20syl2anc 409 . . . 4  |-  ( ph  ->  ( ( F `  K )  +P.  <. { l  |  l  <Q 
( *Q `  [ <. K ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. K ,  1o >. ]  ~Q  )  <Q  u } >. )  e.  P. )
22 addclpr 7499 . . . . 5  |-  ( ( ( F `  K
)  e.  P.  /\  Q  e.  P. )  ->  ( ( F `  K )  +P.  Q
)  e.  P. )
2316, 14, 22syl2anc 409 . . . 4  |-  ( ph  ->  ( ( F `  K )  +P.  Q
)  e.  P. )
24 ltdfpr 7468 . . . 4  |-  ( ( ( ( F `  K )  +P.  <. { l  |  l  <Q 
( *Q `  [ <. K ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. K ,  1o >. ]  ~Q  )  <Q  u } >. )  e.  P.  /\  ( ( F `  K )  +P.  Q )  e. 
P. )  ->  (
( ( F `  K )  +P.  <. { l  |  l  <Q 
( *Q `  [ <. K ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. K ,  1o >. ]  ~Q  )  <Q  u } >. )  <P  ( ( F `  K )  +P.  Q
)  <->  E. x  e.  Q.  ( x  e.  ( 2nd `  ( ( F `
 K )  +P. 
<. { l  |  l 
<Q  ( *Q `  [ <. K ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. K ,  1o >. ]  ~Q  )  <Q  u } >. )
)  /\  x  e.  ( 1st `  ( ( F `  K )  +P.  Q ) ) ) ) )
2521, 23, 24syl2anc 409 . . 3  |-  ( ph  ->  ( ( ( F `
 K )  +P. 
<. { l  |  l 
<Q  ( *Q `  [ <. K ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. K ,  1o >. ]  ~Q  )  <Q  u } >. )  <P  ( ( F `  K )  +P.  Q
)  <->  E. x  e.  Q.  ( x  e.  ( 2nd `  ( ( F `
 K )  +P. 
<. { l  |  l 
<Q  ( *Q `  [ <. K ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. K ,  1o >. ]  ~Q  )  <Q  u } >. )
)  /\  x  e.  ( 1st `  ( ( F `  K )  +P.  Q ) ) ) ) )
2619, 25mpbid 146 . 2  |-  ( ph  ->  E. x  e.  Q.  ( x  e.  ( 2nd `  ( ( F `
 K )  +P. 
<. { l  |  l 
<Q  ( *Q `  [ <. K ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. K ,  1o >. ]  ~Q  )  <Q  u } >. )
)  /\  x  e.  ( 1st `  ( ( F `  K )  +P.  Q ) ) ) )
27 simprl 526 . . . 4  |-  ( (
ph  /\  ( x  e.  Q.  /\  ( x  e.  ( 2nd `  (
( F `  K
)  +P.  <. { l  |  l  <Q  ( *Q `  [ <. K ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. K ,  1o >. ]  ~Q  )  <Q  u } >. ) )  /\  x  e.  ( 1st `  ( ( F `  K )  +P.  Q
) ) ) ) )  ->  x  e.  Q. )
287adantr 274 . . . . . 6  |-  ( (
ph  /\  ( x  e.  Q.  /\  ( x  e.  ( 2nd `  (
( F `  K
)  +P.  <. { l  |  l  <Q  ( *Q `  [ <. K ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. K ,  1o >. ]  ~Q  )  <Q  u } >. ) )  /\  x  e.  ( 1st `  ( ( F `  K )  +P.  Q
) ) ) ) )  ->  K  e.  N. )
29 simprrl 534 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  Q.  /\  ( x  e.  ( 2nd `  (
( F `  K
)  +P.  <. { l  |  l  <Q  ( *Q `  [ <. K ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. K ,  1o >. ]  ~Q  )  <Q  u } >. ) )  /\  x  e.  ( 1st `  ( ( F `  K )  +P.  Q
) ) ) ) )  ->  x  e.  ( 2nd `  ( ( F `  K )  +P.  <. { l  |  l  <Q  ( *Q `  [ <. K ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. K ,  1o >. ]  ~Q  )  <Q  u } >. ) ) )
30 breq1 3992 . . . . . . . . . . . 12  |-  ( l  =  p  ->  (
l  <Q  ( *Q `  [ <. K ,  1o >. ]  ~Q  )  <->  p  <Q  ( *Q `  [ <. K ,  1o >. ]  ~Q  ) ) )
3130cbvabv 2295 . . . . . . . . . . 11  |-  { l  |  l  <Q  ( *Q `  [ <. K ,  1o >. ]  ~Q  ) }  =  { p  |  p  <Q  ( *Q
`  [ <. K ,  1o >. ]  ~Q  ) }
32 breq2 3993 . . . . . . . . . . . 12  |-  ( u  =  q  ->  (
( *Q `  [ <. K ,  1o >. ]  ~Q  )  <Q  u  <->  ( *Q `  [ <. K ,  1o >. ]  ~Q  )  <Q  q ) )
3332cbvabv 2295 . . . . . . . . . . 11  |-  { u  |  ( *Q `  [ <. K ,  1o >. ]  ~Q  )  <Q  u }  =  {
q  |  ( *Q
`  [ <. K ,  1o >. ]  ~Q  )  <Q  q }
3431, 33opeq12i 3770 . . . . . . . . . 10  |-  <. { l  |  l  <Q  ( *Q `  [ <. K ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. K ,  1o >. ]  ~Q  )  <Q  u } >.  =  <. { p  |  p  <Q  ( *Q `  [ <. K ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. K ,  1o >. ]  ~Q  )  <Q 
q } >.
3534oveq2i 5864 . . . . . . . . 9  |-  ( ( F `  K )  +P.  <. { l  |  l  <Q  ( *Q `  [ <. K ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. K ,  1o >. ]  ~Q  )  <Q  u } >. )  =  ( ( F `  K
)  +P.  <. { p  |  p  <Q  ( *Q
`  [ <. K ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. K ,  1o >. ]  ~Q  )  <Q 
q } >. )
3635fveq2i 5499 . . . . . . . 8  |-  ( 2nd `  ( ( F `  K )  +P.  <. { l  |  l  <Q 
( *Q `  [ <. K ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. K ,  1o >. ]  ~Q  )  <Q  u } >. )
)  =  ( 2nd `  ( ( F `  K )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. K ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. K ,  1o >. ]  ~Q  )  <Q 
q } >. )
)
3729, 36eleqtrdi 2263 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  Q.  /\  ( x  e.  ( 2nd `  (
( F `  K
)  +P.  <. { l  |  l  <Q  ( *Q `  [ <. K ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. K ,  1o >. ]  ~Q  )  <Q  u } >. ) )  /\  x  e.  ( 1st `  ( ( F `  K )  +P.  Q
) ) ) ) )  ->  x  e.  ( 2nd `  ( ( F `  K )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. K ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. K ,  1o >. ]  ~Q  )  <Q  q } >. ) ) )
38 nqprlu 7509 . . . . . . . . . . 11  |-  ( ( *Q `  [ <. K ,  1o >. ]  ~Q  )  e.  Q.  ->  <. { p  |  p  <Q  ( *Q `  [ <. K ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. K ,  1o >. ]  ~Q  )  <Q  q } >.  e. 
P. )
3911, 38syl 14 . . . . . . . . . 10  |-  ( ph  -> 
<. { p  |  p 
<Q  ( *Q `  [ <. K ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. K ,  1o >. ]  ~Q  )  <Q  q } >.  e. 
P. )
40 addclpr 7499 . . . . . . . . . 10  |-  ( ( ( F `  K
)  e.  P.  /\  <. { p  |  p  <Q  ( *Q `  [ <. K ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. K ,  1o >. ]  ~Q  )  <Q  q } >.  e. 
P. )  ->  (
( F `  K
)  +P.  <. { p  |  p  <Q  ( *Q
`  [ <. K ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. K ,  1o >. ]  ~Q  )  <Q 
q } >. )  e.  P. )
4116, 39, 40syl2anc 409 . . . . . . . . 9  |-  ( ph  ->  ( ( F `  K )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. K ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. K ,  1o >. ]  ~Q  )  <Q 
q } >. )  e.  P. )
4241adantr 274 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  Q.  /\  ( x  e.  ( 2nd `  (
( F `  K
)  +P.  <. { l  |  l  <Q  ( *Q `  [ <. K ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. K ,  1o >. ]  ~Q  )  <Q  u } >. ) )  /\  x  e.  ( 1st `  ( ( F `  K )  +P.  Q
) ) ) ) )  ->  ( ( F `  K )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. K ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. K ,  1o >. ]  ~Q  )  <Q  q } >. )  e.  P. )
43 nqpru 7514 . . . . . . . 8  |-  ( ( x  e.  Q.  /\  ( ( F `  K )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. K ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. K ,  1o >. ]  ~Q  )  <Q 
q } >. )  e.  P. )  ->  (
x  e.  ( 2nd `  ( ( F `  K )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. K ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. K ,  1o >. ]  ~Q  )  <Q 
q } >. )
)  <->  ( ( F `
 K )  +P. 
<. { p  |  p 
<Q  ( *Q `  [ <. K ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. K ,  1o >. ]  ~Q  )  <Q  q } >. ) 
<P  <. { p  |  p  <Q  x } ,  { q  |  x 
<Q  q } >. )
)
4427, 42, 43syl2anc 409 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  Q.  /\  ( x  e.  ( 2nd `  (
( F `  K
)  +P.  <. { l  |  l  <Q  ( *Q `  [ <. K ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. K ,  1o >. ]  ~Q  )  <Q  u } >. ) )  /\  x  e.  ( 1st `  ( ( F `  K )  +P.  Q
) ) ) ) )  ->  ( x  e.  ( 2nd `  (
( F `  K
)  +P.  <. { p  |  p  <Q  ( *Q
`  [ <. K ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. K ,  1o >. ]  ~Q  )  <Q 
q } >. )
)  <->  ( ( F `
 K )  +P. 
<. { p  |  p 
<Q  ( *Q `  [ <. K ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. K ,  1o >. ]  ~Q  )  <Q  q } >. ) 
<P  <. { p  |  p  <Q  x } ,  { q  |  x 
<Q  q } >. )
)
4537, 44mpbid 146 . . . . . 6  |-  ( (
ph  /\  ( x  e.  Q.  /\  ( x  e.  ( 2nd `  (
( F `  K
)  +P.  <. { l  |  l  <Q  ( *Q `  [ <. K ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. K ,  1o >. ]  ~Q  )  <Q  u } >. ) )  /\  x  e.  ( 1st `  ( ( F `  K )  +P.  Q
) ) ) ) )  ->  ( ( F `  K )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. K ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. K ,  1o >. ]  ~Q  )  <Q  q } >. )  <P  <. { p  |  p  <Q  x } ,  { q  |  x 
<Q  q } >. )
46 fveq2 5496 . . . . . . . . 9  |-  ( r  =  K  ->  ( F `  r )  =  ( F `  K ) )
47 opeq1 3765 . . . . . . . . . . . . . 14  |-  ( r  =  K  ->  <. r ,  1o >.  =  <. K ,  1o >. )
4847eceq1d 6549 . . . . . . . . . . . . 13  |-  ( r  =  K  ->  [ <. r ,  1o >. ]  ~Q  =  [ <. K ,  1o >. ]  ~Q  )
4948fveq2d 5500 . . . . . . . . . . . 12  |-  ( r  =  K  ->  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  =  ( *Q `  [ <. K ,  1o >. ]  ~Q  ) )
5049breq2d 4001 . . . . . . . . . . 11  |-  ( r  =  K  ->  (
p  <Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <->  p  <Q  ( *Q `  [ <. K ,  1o >. ]  ~Q  ) ) )
5150abbidv 2288 . . . . . . . . . 10  |-  ( r  =  K  ->  { p  |  p  <Q  ( *Q
`  [ <. r ,  1o >. ]  ~Q  ) }  =  { p  |  p  <Q  ( *Q
`  [ <. K ,  1o >. ]  ~Q  ) } )
5249breq1d 3999 . . . . . . . . . . 11  |-  ( r  =  K  ->  (
( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q  q  <->  ( *Q `  [ <. K ,  1o >. ]  ~Q  )  <Q  q ) )
5352abbidv 2288 . . . . . . . . . 10  |-  ( r  =  K  ->  { q  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q 
q }  =  {
q  |  ( *Q
`  [ <. K ,  1o >. ]  ~Q  )  <Q  q } )
5451, 53opeq12d 3773 . . . . . . . . 9  |-  ( r  =  K  ->  <. { p  |  p  <Q  ( *Q
`  [ <. r ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q 
q } >.  =  <. { p  |  p  <Q  ( *Q `  [ <. K ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. K ,  1o >. ]  ~Q  )  <Q 
q } >. )
5546, 54oveq12d 5871 . . . . . . . 8  |-  ( r  =  K  ->  (
( F `  r
)  +P.  <. { p  |  p  <Q  ( *Q
`  [ <. r ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q 
q } >. )  =  ( ( F `
 K )  +P. 
<. { p  |  p 
<Q  ( *Q `  [ <. K ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. K ,  1o >. ]  ~Q  )  <Q  q } >. ) )
5655breq1d 3999 . . . . . . 7  |-  ( r  =  K  ->  (
( ( F `  r )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q 
q } >. )  <P 
<. { p  |  p 
<Q  x } ,  {
q  |  x  <Q  q } >.  <->  ( ( F `
 K )  +P. 
<. { p  |  p 
<Q  ( *Q `  [ <. K ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. K ,  1o >. ]  ~Q  )  <Q  q } >. ) 
<P  <. { p  |  p  <Q  x } ,  { q  |  x 
<Q  q } >. )
)
5756rspcev 2834 . . . . . 6  |-  ( ( K  e.  N.  /\  ( ( F `  K )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. K ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. K ,  1o >. ]  ~Q  )  <Q 
q } >. )  <P 
<. { p  |  p 
<Q  x } ,  {
q  |  x  <Q  q } >. )  ->  E. r  e.  N.  ( ( F `
 r )  +P. 
<. { p  |  p 
<Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q  q } >. ) 
<P  <. { p  |  p  <Q  x } ,  { q  |  x 
<Q  q } >. )
5828, 45, 57syl2anc 409 . . . . 5  |-  ( (
ph  /\  ( x  e.  Q.  /\  ( x  e.  ( 2nd `  (
( F `  K
)  +P.  <. { l  |  l  <Q  ( *Q `  [ <. K ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. K ,  1o >. ]  ~Q  )  <Q  u } >. ) )  /\  x  e.  ( 1st `  ( ( F `  K )  +P.  Q
) ) ) ) )  ->  E. r  e.  N.  ( ( F `
 r )  +P. 
<. { p  |  p 
<Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q  q } >. ) 
<P  <. { p  |  p  <Q  x } ,  { q  |  x 
<Q  q } >. )
59 breq2 3993 . . . . . . . . . 10  |-  ( u  =  x  ->  (
p  <Q  u  <->  p  <Q  x ) )
6059abbidv 2288 . . . . . . . . 9  |-  ( u  =  x  ->  { p  |  p  <Q  u }  =  { p  |  p 
<Q  x } )
61 breq1 3992 . . . . . . . . . 10  |-  ( u  =  x  ->  (
u  <Q  q  <->  x  <Q  q ) )
6261abbidv 2288 . . . . . . . . 9  |-  ( u  =  x  ->  { q  |  u  <Q  q }  =  { q  |  x  <Q  q } )
6360, 62opeq12d 3773 . . . . . . . 8  |-  ( u  =  x  ->  <. { p  |  p  <Q  u } ,  { q  |  u 
<Q  q } >.  =  <. { p  |  p  <Q  x } ,  { q  |  x  <Q  q } >. )
6463breq2d 4001 . . . . . . 7  |-  ( u  =  x  ->  (
( ( F `  r )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q 
q } >. )  <P 
<. { p  |  p 
<Q  u } ,  {
q  |  u  <Q  q } >.  <->  ( ( F `
 r )  +P. 
<. { p  |  p 
<Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q  q } >. ) 
<P  <. { p  |  p  <Q  x } ,  { q  |  x 
<Q  q } >. )
)
6564rexbidv 2471 . . . . . 6  |-  ( u  =  x  ->  ( E. r  e.  N.  ( ( F `  r )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q 
q } >. )  <P 
<. { p  |  p 
<Q  u } ,  {
q  |  u  <Q  q } >.  <->  E. r  e.  N.  ( ( F `  r )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q 
q } >. )  <P 
<. { p  |  p 
<Q  x } ,  {
q  |  x  <Q  q } >. ) )
66 caucvgprpr.lim . . . . . . . 8  |-  L  = 
<. { l  e.  Q.  |  E. r  e.  N.  <. { p  |  p  <Q  ( l  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  )
) } ,  {
q  |  ( l  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) ) 
<Q  q } >.  <P  ( F `  r ) } ,  { u  e.  Q.  |  E. r  e.  N.  ( ( F `
 r )  +P. 
<. { p  |  p 
<Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q  q } >. ) 
<P  <. { p  |  p  <Q  u } ,  { q  |  u 
<Q  q } >. } >.
6766fveq2i 5499 . . . . . . 7  |-  ( 2nd `  L )  =  ( 2nd `  <. { l  e.  Q.  |  E. r  e.  N.  <. { p  |  p  <Q  ( l  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) ) } ,  { q  |  ( l  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) )  <Q 
q } >.  <P  ( F `  r ) } ,  { u  e.  Q.  |  E. r  e.  N.  ( ( F `
 r )  +P. 
<. { p  |  p 
<Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q  q } >. ) 
<P  <. { p  |  p  <Q  u } ,  { q  |  u 
<Q  q } >. } >. )
68 nqex 7325 . . . . . . . . 9  |-  Q.  e.  _V
6968rabex 4133 . . . . . . . 8  |-  { l  e.  Q.  |  E. r  e.  N.  <. { p  |  p  <Q  ( l  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) ) } ,  { q  |  ( l  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) )  <Q 
q } >.  <P  ( F `  r ) }  e.  _V
7068rabex 4133 . . . . . . . 8  |-  { u  e.  Q.  |  E. r  e.  N.  ( ( F `
 r )  +P. 
<. { p  |  p 
<Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q  q } >. ) 
<P  <. { p  |  p  <Q  u } ,  { q  |  u 
<Q  q } >. }  e.  _V
7169, 70op2nd 6126 . . . . . . 7  |-  ( 2nd `  <. { l  e. 
Q.  |  E. r  e.  N.  <. { p  |  p  <Q  ( l  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) ) } ,  { q  |  ( l  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  )
)  <Q  q } >.  <P 
( F `  r
) } ,  {
u  e.  Q.  |  E. r  e.  N.  ( ( F `  r )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q 
q } >. )  <P 
<. { p  |  p 
<Q  u } ,  {
q  |  u  <Q  q } >. } >. )  =  { u  e.  Q.  |  E. r  e.  N.  ( ( F `  r )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q 
q } >. )  <P 
<. { p  |  p 
<Q  u } ,  {
q  |  u  <Q  q } >. }
7267, 71eqtri 2191 . . . . . 6  |-  ( 2nd `  L )  =  {
u  e.  Q.  |  E. r  e.  N.  ( ( F `  r )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q 
q } >. )  <P 
<. { p  |  p 
<Q  u } ,  {
q  |  u  <Q  q } >. }
7365, 72elrab2 2889 . . . . 5  |-  ( x  e.  ( 2nd `  L
)  <->  ( x  e. 
Q.  /\  E. r  e.  N.  ( ( F `
 r )  +P. 
<. { p  |  p 
<Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q  q } >. ) 
<P  <. { p  |  p  <Q  x } ,  { q  |  x 
<Q  q } >. )
)
7427, 58, 73sylanbrc 415 . . . 4  |-  ( (
ph  /\  ( x  e.  Q.  /\  ( x  e.  ( 2nd `  (
( F `  K
)  +P.  <. { l  |  l  <Q  ( *Q `  [ <. K ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. K ,  1o >. ]  ~Q  )  <Q  u } >. ) )  /\  x  e.  ( 1st `  ( ( F `  K )  +P.  Q
) ) ) ) )  ->  x  e.  ( 2nd `  L ) )
75 simprrr 535 . . . 4  |-  ( (
ph  /\  ( x  e.  Q.  /\  ( x  e.  ( 2nd `  (
( F `  K
)  +P.  <. { l  |  l  <Q  ( *Q `  [ <. K ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. K ,  1o >. ]  ~Q  )  <Q  u } >. ) )  /\  x  e.  ( 1st `  ( ( F `  K )  +P.  Q
) ) ) ) )  ->  x  e.  ( 1st `  ( ( F `  K )  +P.  Q ) ) )
76 rspe 2519 . . . 4  |-  ( ( x  e.  Q.  /\  ( x  e.  ( 2nd `  L )  /\  x  e.  ( 1st `  ( ( F `  K )  +P.  Q
) ) ) )  ->  E. x  e.  Q.  ( x  e.  ( 2nd `  L )  /\  x  e.  ( 1st `  ( ( F `  K )  +P.  Q
) ) ) )
7727, 74, 75, 76syl12anc 1231 . . 3  |-  ( (
ph  /\  ( x  e.  Q.  /\  ( x  e.  ( 2nd `  (
( F `  K
)  +P.  <. { l  |  l  <Q  ( *Q `  [ <. K ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. K ,  1o >. ]  ~Q  )  <Q  u } >. ) )  /\  x  e.  ( 1st `  ( ( F `  K )  +P.  Q
) ) ) ) )  ->  E. x  e.  Q.  ( x  e.  ( 2nd `  L
)  /\  x  e.  ( 1st `  ( ( F `  K )  +P.  Q ) ) ) )
78 caucvgprpr.cau . . . . . 6  |-  ( ph  ->  A. n  e.  N.  A. k  e.  N.  (
n  <N  k  ->  (
( F `  n
)  <P  ( ( F `
 k )  +P. 
<. { l  |  l 
<Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >. )  /\  ( F `  k
)  <P  ( ( F `
 n )  +P. 
<. { l  |  l 
<Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >. )
) ) )
79 caucvgprpr.bnd . . . . . 6  |-  ( ph  ->  A. m  e.  N.  A  <P  ( F `  m ) )
8015, 78, 79, 66caucvgprprlemcl 7666 . . . . 5  |-  ( ph  ->  L  e.  P. )
8180adantr 274 . . . 4  |-  ( (
ph  /\  ( x  e.  Q.  /\  ( x  e.  ( 2nd `  (
( F `  K
)  +P.  <. { l  |  l  <Q  ( *Q `  [ <. K ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. K ,  1o >. ]  ~Q  )  <Q  u } >. ) )  /\  x  e.  ( 1st `  ( ( F `  K )  +P.  Q
) ) ) ) )  ->  L  e.  P. )
8223adantr 274 . . . 4  |-  ( (
ph  /\  ( x  e.  Q.  /\  ( x  e.  ( 2nd `  (
( F `  K
)  +P.  <. { l  |  l  <Q  ( *Q `  [ <. K ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. K ,  1o >. ]  ~Q  )  <Q  u } >. ) )  /\  x  e.  ( 1st `  ( ( F `  K )  +P.  Q
) ) ) ) )  ->  ( ( F `  K )  +P.  Q )  e.  P. )
83 ltdfpr 7468 . . . 4  |-  ( ( L  e.  P.  /\  ( ( F `  K )  +P.  Q
)  e.  P. )  ->  ( L  <P  (
( F `  K
)  +P.  Q )  <->  E. x  e.  Q.  (
x  e.  ( 2nd `  L )  /\  x  e.  ( 1st `  (
( F `  K
)  +P.  Q )
) ) ) )
8481, 82, 83syl2anc 409 . . 3  |-  ( (
ph  /\  ( x  e.  Q.  /\  ( x  e.  ( 2nd `  (
( F `  K
)  +P.  <. { l  |  l  <Q  ( *Q `  [ <. K ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. K ,  1o >. ]  ~Q  )  <Q  u } >. ) )  /\  x  e.  ( 1st `  ( ( F `  K )  +P.  Q
) ) ) ) )  ->  ( L  <P  ( ( F `  K )  +P.  Q
)  <->  E. x  e.  Q.  ( x  e.  ( 2nd `  L )  /\  x  e.  ( 1st `  ( ( F `  K )  +P.  Q
) ) ) ) )
8577, 84mpbird 166 . 2  |-  ( (
ph  /\  ( x  e.  Q.  /\  ( x  e.  ( 2nd `  (
( F `  K
)  +P.  <. { l  |  l  <Q  ( *Q `  [ <. K ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. K ,  1o >. ]  ~Q  )  <Q  u } >. ) )  /\  x  e.  ( 1st `  ( ( F `  K )  +P.  Q
) ) ) ) )  ->  L  <P  ( ( F `  K
)  +P.  Q )
)
8626, 85rexlimddv 2592 1  |-  ( ph  ->  L  <P  ( ( F `  K )  +P.  Q ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1348    e. wcel 2141   {cab 2156   A.wral 2448   E.wrex 2449   {crab 2452   <.cop 3586   class class class wbr 3989   -->wf 5194   ` cfv 5198  (class class class)co 5853   1stc1st 6117   2ndc2nd 6118   1oc1o 6388   [cec 6511   N.cnpi 7234    <N clti 7237    ~Q ceq 7241   Q.cnq 7242    +Q cplq 7244   *Qcrq 7246    <Q cltq 7247   P.cnp 7253    +P. cpp 7255    <P cltp 7257
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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-iinf 4572
This theorem depends on definitions:  df-bi 116  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-eprel 4274  df-id 4278  df-po 4281  df-iso 4282  df-iord 4351  df-on 4353  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-recs 6284  df-irdg 6349  df-1o 6395  df-2o 6396  df-oadd 6399  df-omul 6400  df-er 6513  df-ec 6515  df-qs 6519  df-ni 7266  df-pli 7267  df-mi 7268  df-lti 7269  df-plpq 7306  df-mpq 7307  df-enq 7309  df-nqqs 7310  df-plqqs 7311  df-mqqs 7312  df-1nqqs 7313  df-rq 7314  df-ltnqqs 7315  df-enq0 7386  df-nq0 7387  df-0nq0 7388  df-plq0 7389  df-mq0 7390  df-inp 7428  df-iplp 7430  df-iltp 7432
This theorem is referenced by:  caucvgprprlemlim  7673
  Copyright terms: Public domain W3C validator