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Theorem caucvgprprlem2 6789
Description: Lemma for caucvgprpr 6791. 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 6762 . . . 4  |-  ( ph  -> 
<. { l  |  l 
<Q  ( *Q `  [ <. K ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. K ,  1o >. ]  ~Q  )  <Q  u } >.  <P  Q )
4 ltrelpi 6403 . . . . . . . . . 10  |-  <N  C_  ( N.  X.  N. )
54brel 4379 . . . . . . . . 9  |-  ( J 
<N  K  ->  ( J  e.  N.  /\  K  e.  N. ) )
61, 5syl 14 . . . . . . . 8  |-  ( ph  ->  ( J  e.  N.  /\  K  e.  N. )
)
76simprd 107 . . . . . . 7  |-  ( ph  ->  K  e.  N. )
8 nnnq 6501 . . . . . . . 8  |-  ( K  e.  N.  ->  [ <. K ,  1o >. ]  ~Q  e.  Q. )
9 recclnq 6471 . . . . . . . 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 6626 . . . . . 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 5290 . . . . 5  |-  ( ph  ->  ( F `  K
)  e.  P. )
17 ltaprg 6698 . . . . 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 1135 . . . 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 135 . . 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 6616 . . . . 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 391 . . . 4  |-  ( ph  ->  ( ( F `  K )  +P.  <. { l  |  l  <Q 
( *Q `  [ <. K ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. K ,  1o >. ]  ~Q  )  <Q  u } >. )  e.  P. )
22 addclpr 6616 . . . . 5  |-  ( ( ( F `  K
)  e.  P.  /\  Q  e.  P. )  ->  ( ( F `  K )  +P.  Q
)  e.  P. )
2316, 14, 22syl2anc 391 . . . 4  |-  ( ph  ->  ( ( F `  K )  +P.  Q
)  e.  P. )
24 ltdfpr 6585 . . . 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 391 . . 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 135 . 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 483 . . . 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 261 . . . . . 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 491 . . . . . . . 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 3764 . . . . . . . . . . . 12  |-  ( l  =  p  ->  (
l  <Q  ( *Q `  [ <. K ,  1o >. ]  ~Q  )  <->  p  <Q  ( *Q `  [ <. K ,  1o >. ]  ~Q  ) ) )
3130cbvabv 2161 . . . . . . . . . . 11  |-  { l  |  l  <Q  ( *Q `  [ <. K ,  1o >. ]  ~Q  ) }  =  { p  |  p  <Q  ( *Q
`  [ <. K ,  1o >. ]  ~Q  ) }
32 breq2 3765 . . . . . . . . . . . 12  |-  ( u  =  q  ->  (
( *Q `  [ <. K ,  1o >. ]  ~Q  )  <Q  u  <->  ( *Q `  [ <. K ,  1o >. ]  ~Q  )  <Q  q ) )
3332cbvabv 2161 . . . . . . . . . . 11  |-  { u  |  ( *Q `  [ <. K ,  1o >. ]  ~Q  )  <Q  u }  =  {
q  |  ( *Q
`  [ <. K ,  1o >. ]  ~Q  )  <Q  q }
3431, 33opeq12i 3551 . . . . . . . . . 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 5510 . . . . . . . . 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 5168 . . . . . . . 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, 36syl6eleq 2130 . . . . . . 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 6626 . . . . . . . . . . 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 6616 . . . . . . . . . 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 391 . . . . . . . . 9  |-  ( ph  ->  ( ( F `  K )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. K ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. K ,  1o >. ]  ~Q  )  <Q 
q } >. )  e.  P. )
4241adantr 261 . . . . . . . 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 6631 . . . . . . . 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 391 . . . . . . 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 135 . . . . . 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 5165 . . . . . . . . 9  |-  ( r  =  K  ->  ( F `  r )  =  ( F `  K ) )
47 opeq1 3546 . . . . . . . . . . . . . 14  |-  ( r  =  K  ->  <. r ,  1o >.  =  <. K ,  1o >. )
4847eceq1d 6129 . . . . . . . . . . . . 13  |-  ( r  =  K  ->  [ <. r ,  1o >. ]  ~Q  =  [ <. K ,  1o >. ]  ~Q  )
4948fveq2d 5169 . . . . . . . . . . . 12  |-  ( r  =  K  ->  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  =  ( *Q `  [ <. K ,  1o >. ]  ~Q  ) )
5049breq2d 3773 . . . . . . . . . . 11  |-  ( r  =  K  ->  (
p  <Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <->  p  <Q  ( *Q `  [ <. K ,  1o >. ]  ~Q  ) ) )
5150abbidv 2155 . . . . . . . . . 10  |-  ( r  =  K  ->  { p  |  p  <Q  ( *Q
`  [ <. r ,  1o >. ]  ~Q  ) }  =  { p  |  p  <Q  ( *Q
`  [ <. K ,  1o >. ]  ~Q  ) } )
5249breq1d 3771 . . . . . . . . . . 11  |-  ( r  =  K  ->  (
( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q  q  <->  ( *Q `  [ <. K ,  1o >. ]  ~Q  )  <Q  q ) )
5352abbidv 2155 . . . . . . . . . 10  |-  ( r  =  K  ->  { q  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q 
q }  =  {
q  |  ( *Q
`  [ <. K ,  1o >. ]  ~Q  )  <Q  q } )
5451, 53opeq12d 3554 . . . . . . . . 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 5517 . . . . . . . 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 3771 . . . . . . 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 2653 . . . . . 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 391 . . . . 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 3765 . . . . . . . . . 10  |-  ( u  =  x  ->  (
p  <Q  u  <->  p  <Q  x ) )
6059abbidv 2155 . . . . . . . . 9  |-  ( u  =  x  ->  { p  |  p  <Q  u }  =  { p  |  p 
<Q  x } )
61 breq1 3764 . . . . . . . . . 10  |-  ( u  =  x  ->  (
u  <Q  q  <->  x  <Q  q ) )
6261abbidv 2155 . . . . . . . . 9  |-  ( u  =  x  ->  { q  |  u  <Q  q }  =  { q  |  x  <Q  q } )
6360, 62opeq12d 3554 . . . . . . . 8  |-  ( u  =  x  ->  <. { p  |  p  <Q  u } ,  { q  |  u 
<Q  q } >.  =  <. { p  |  p  <Q  x } ,  { q  |  x  <Q  q } >. )
6463breq2d 3773 . . . . . . 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 2324 . . . . . 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 5168 . . . . . . 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 6442 . . . . . . . . 9  |-  Q.  e.  _V
6968rabex 3898 . . . . . . . 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 3898 . . . . . . . 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 5761 . . . . . . 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 2060 . . . . . 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 2697 . . . . 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 394 . . . 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 492 . . . 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 2367 . . . 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 1133 . . 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 6783 . . . . 5  |-  ( ph  ->  L  e.  P. )
8180adantr 261 . . . 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 261 . . . 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 6585 . . . 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 391 . . 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 156 . 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 2434 1  |-  ( ph  ->  L  <P  ( ( F `  K )  +P.  Q ) )
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. cpp 6372    <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-2o 5989  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-enq0 6503  df-nq0 6504  df-0nq0 6505  df-plq0 6506  df-mq0 6507  df-inp 6545  df-iplp 6547  df-iltp 6549
This theorem is referenced by:  caucvgprprlemlim  6790
  Copyright terms: Public domain W3C validator