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Theorem caucvgprprlem2 7525
Description: Lemma for caucvgprpr 7527. 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 7498 . . . 4  |-  ( ph  -> 
<. { l  |  l 
<Q  ( *Q `  [ <. K ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. K ,  1o >. ]  ~Q  )  <Q  u } >.  <P  Q )
4 ltrelpi 7139 . . . . . . . . . 10  |-  <N  C_  ( N.  X.  N. )
54brel 4591 . . . . . . . . 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 7237 . . . . . . . 8  |-  ( K  e.  N.  ->  [ <. K ,  1o >. ]  ~Q  e.  Q. )
9 recclnq 7207 . . . . . . . 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 7362 . . . . . 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 5556 . . . . 5  |-  ( ph  ->  ( F `  K
)  e.  P. )
17 ltaprg 7434 . . . . 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 1216 . . . 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 7352 . . . . 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 408 . . . 4  |-  ( ph  ->  ( ( F `  K )  +P.  <. { l  |  l  <Q 
( *Q `  [ <. K ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. K ,  1o >. ]  ~Q  )  <Q  u } >. )  e.  P. )
22 addclpr 7352 . . . . 5  |-  ( ( ( F `  K
)  e.  P.  /\  Q  e.  P. )  ->  ( ( F `  K )  +P.  Q
)  e.  P. )
2316, 14, 22syl2anc 408 . . . 4  |-  ( ph  ->  ( ( F `  K )  +P.  Q
)  e.  P. )
24 ltdfpr 7321 . . . 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 408 . . 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 520 . . . 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 528 . . . . . . . 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 3932 . . . . . . . . . . . 12  |-  ( l  =  p  ->  (
l  <Q  ( *Q `  [ <. K ,  1o >. ]  ~Q  )  <->  p  <Q  ( *Q `  [ <. K ,  1o >. ]  ~Q  ) ) )
3130cbvabv 2264 . . . . . . . . . . 11  |-  { l  |  l  <Q  ( *Q `  [ <. K ,  1o >. ]  ~Q  ) }  =  { p  |  p  <Q  ( *Q
`  [ <. K ,  1o >. ]  ~Q  ) }
32 breq2 3933 . . . . . . . . . . . 12  |-  ( u  =  q  ->  (
( *Q `  [ <. K ,  1o >. ]  ~Q  )  <Q  u  <->  ( *Q `  [ <. K ,  1o >. ]  ~Q  )  <Q  q ) )
3332cbvabv 2264 . . . . . . . . . . 11  |-  { u  |  ( *Q `  [ <. K ,  1o >. ]  ~Q  )  <Q  u }  =  {
q  |  ( *Q
`  [ <. K ,  1o >. ]  ~Q  )  <Q  q }
3431, 33opeq12i 3710 . . . . . . . . . 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 5785 . . . . . . . . 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 5424 . . . . . . . 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 2232 . . . . . . 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 7362 . . . . . . . . . . 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 7352 . . . . . . . . . 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 408 . . . . . . . . 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 7367 . . . . . . . 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 408 . . . . . . 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 5421 . . . . . . . . 9  |-  ( r  =  K  ->  ( F `  r )  =  ( F `  K ) )
47 opeq1 3705 . . . . . . . . . . . . . 14  |-  ( r  =  K  ->  <. r ,  1o >.  =  <. K ,  1o >. )
4847eceq1d 6465 . . . . . . . . . . . . 13  |-  ( r  =  K  ->  [ <. r ,  1o >. ]  ~Q  =  [ <. K ,  1o >. ]  ~Q  )
4948fveq2d 5425 . . . . . . . . . . . 12  |-  ( r  =  K  ->  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  =  ( *Q `  [ <. K ,  1o >. ]  ~Q  ) )
5049breq2d 3941 . . . . . . . . . . 11  |-  ( r  =  K  ->  (
p  <Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <->  p  <Q  ( *Q `  [ <. K ,  1o >. ]  ~Q  ) ) )
5150abbidv 2257 . . . . . . . . . 10  |-  ( r  =  K  ->  { p  |  p  <Q  ( *Q
`  [ <. r ,  1o >. ]  ~Q  ) }  =  { p  |  p  <Q  ( *Q
`  [ <. K ,  1o >. ]  ~Q  ) } )
5249breq1d 3939 . . . . . . . . . . 11  |-  ( r  =  K  ->  (
( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q  q  <->  ( *Q `  [ <. K ,  1o >. ]  ~Q  )  <Q  q ) )
5352abbidv 2257 . . . . . . . . . 10  |-  ( r  =  K  ->  { q  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q 
q }  =  {
q  |  ( *Q
`  [ <. K ,  1o >. ]  ~Q  )  <Q  q } )
5451, 53opeq12d 3713 . . . . . . . . 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 5792 . . . . . . . 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 3939 . . . . . . 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 2789 . . . . . 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 408 . . . . 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 3933 . . . . . . . . . 10  |-  ( u  =  x  ->  (
p  <Q  u  <->  p  <Q  x ) )
6059abbidv 2257 . . . . . . . . 9  |-  ( u  =  x  ->  { p  |  p  <Q  u }  =  { p  |  p 
<Q  x } )
61 breq1 3932 . . . . . . . . . 10  |-  ( u  =  x  ->  (
u  <Q  q  <->  x  <Q  q ) )
6261abbidv 2257 . . . . . . . . 9  |-  ( u  =  x  ->  { q  |  u  <Q  q }  =  { q  |  x  <Q  q } )
6360, 62opeq12d 3713 . . . . . . . 8  |-  ( u  =  x  ->  <. { p  |  p  <Q  u } ,  { q  |  u 
<Q  q } >.  =  <. { p  |  p  <Q  x } ,  { q  |  x  <Q  q } >. )
6463breq2d 3941 . . . . . . 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 2438 . . . . . 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 5424 . . . . . . 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 7178 . . . . . . . . 9  |-  Q.  e.  _V
6968rabex 4072 . . . . . . . 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 4072 . . . . . . . 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 6045 . . . . . . 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 2160 . . . . . 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 2843 . . . . 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 413 . . . 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 529 . . . 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 2481 . . . 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 1214 . . 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 7519 . . . . 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 7321 . . . 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 408 . . 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 2554 1  |-  ( ph  ->  L  <P  ( ( F `  K )  +P.  Q ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1331    e. wcel 1480   {cab 2125   A.wral 2416   E.wrex 2417   {crab 2420   <.cop 3530   class class class wbr 3929   -->wf 5119   ` cfv 5123  (class class class)co 5774   1stc1st 6036   2ndc2nd 6037   1oc1o 6306   [cec 6427   N.cnpi 7087    <N clti 7090    ~Q ceq 7094   Q.cnq 7095    +Q cplq 7097   *Qcrq 7099    <Q cltq 7100   P.cnp 7106    +P. cpp 7108    <P cltp 7110
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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502
This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-eprel 4211  df-id 4215  df-po 4218  df-iso 4219  df-iord 4288  df-on 4290  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-recs 6202  df-irdg 6267  df-1o 6313  df-2o 6314  df-oadd 6317  df-omul 6318  df-er 6429  df-ec 6431  df-qs 6435  df-ni 7119  df-pli 7120  df-mi 7121  df-lti 7122  df-plpq 7159  df-mpq 7160  df-enq 7162  df-nqqs 7163  df-plqqs 7164  df-mqqs 7165  df-1nqqs 7166  df-rq 7167  df-ltnqqs 7168  df-enq0 7239  df-nq0 7240  df-0nq0 7241  df-plq0 7242  df-mq0 7243  df-inp 7281  df-iplp 7283  df-iltp 7285
This theorem is referenced by:  caucvgprprlemlim  7526
  Copyright terms: Public domain W3C validator