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Theorem caucvgprprlemml 7706
Description: Lemma for caucvgprpr 7724. The lower cut of the putative limit is inhabited. (Contributed by Jim Kingdon, 29-Dec-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 } >. } >.
Assertion
Ref Expression
caucvgprprlemml  |-  ( ph  ->  E. s  e.  Q.  s  e.  ( 1st `  L ) )
Distinct variable groups:    A, m    m, F    A, r, m    A, s, r    F, l    p, l, q, r, s    u, l    ph, r, s
Allowed substitution hints:    ph( u, k, m, n, q, p, l)    A( u, k, n, q, p, l)    F( u, k, n, s, r, q, p)    L( u, k, m, n, s, r, q, p, l)

Proof of Theorem caucvgprprlemml
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 fveq2 5527 . . . . . 6  |-  ( m  =  1o  ->  ( F `  m )  =  ( F `  1o ) )
21breq2d 4027 . . . . 5  |-  ( m  =  1o  ->  ( A  <P  ( F `  m )  <->  A  <P  ( F `  1o ) ) )
3 caucvgprpr.bnd . . . . 5  |-  ( ph  ->  A. m  e.  N.  A  <P  ( F `  m ) )
4 1pi 7327 . . . . . 6  |-  1o  e.  N.
54a1i 9 . . . . 5  |-  ( ph  ->  1o  e.  N. )
62, 3, 5rspcdva 2858 . . . 4  |-  ( ph  ->  A  <P  ( F `  1o ) )
7 ltrelpr 7517 . . . . . 6  |-  <P  C_  ( P.  X.  P. )
87brel 4690 . . . . 5  |-  ( A 
<P  ( F `  1o )  ->  ( A  e. 
P.  /\  ( F `  1o )  e.  P. ) )
98simpld 112 . . . 4  |-  ( A 
<P  ( F `  1o )  ->  A  e.  P. )
106, 9syl 14 . . 3  |-  ( ph  ->  A  e.  P. )
11 prop 7487 . . . 4  |-  ( A  e.  P.  ->  <. ( 1st `  A ) ,  ( 2nd `  A
) >.  e.  P. )
12 prml 7489 . . . 4  |-  ( <.
( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P.  ->  E. x  e.  Q.  x  e.  ( 1st `  A ) )
1311, 12syl 14 . . 3  |-  ( A  e.  P.  ->  E. x  e.  Q.  x  e.  ( 1st `  A ) )
1410, 13syl 14 . 2  |-  ( ph  ->  E. x  e.  Q.  x  e.  ( 1st `  A ) )
15 subhalfnqq 7426 . . . 4  |-  ( x  e.  Q.  ->  E. s  e.  Q.  ( s  +Q  s )  <Q  x
)
1615ad2antrl 490 . . 3  |-  ( (
ph  /\  ( x  e.  Q.  /\  x  e.  ( 1st `  A
) ) )  ->  E. s  e.  Q.  ( s  +Q  s
)  <Q  x )
17 simplr 528 . . . . . 6  |-  ( ( ( ( ph  /\  ( x  e.  Q.  /\  x  e.  ( 1st `  A ) ) )  /\  s  e.  Q. )  /\  ( s  +Q  s )  <Q  x
)  ->  s  e.  Q. )
18 archrecnq 7675 . . . . . . . 8  |-  ( s  e.  Q.  ->  E. r  e.  N.  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q 
s )
1917, 18syl 14 . . . . . . 7  |-  ( ( ( ( ph  /\  ( x  e.  Q.  /\  x  e.  ( 1st `  A ) ) )  /\  s  e.  Q. )  /\  ( s  +Q  s )  <Q  x
)  ->  E. r  e.  N.  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q 
s )
20 simpr 110 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( (
ph  /\  ( x  e.  Q.  /\  x  e.  ( 1st `  A
) ) )  /\  s  e.  Q. )  /\  ( s  +Q  s
)  <Q  x )  /\  r  e.  N. )  /\  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q  s
)  ->  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q 
s )
21 simplr 528 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ( (
ph  /\  ( x  e.  Q.  /\  x  e.  ( 1st `  A
) ) )  /\  s  e.  Q. )  /\  ( s  +Q  s
)  <Q  x )  /\  r  e.  N. )  /\  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q  s
)  ->  r  e.  N. )
22 nnnq 7434 . . . . . . . . . . . . . . . 16  |-  ( r  e.  N.  ->  [ <. r ,  1o >. ]  ~Q  e.  Q. )
23 recclnq 7404 . . . . . . . . . . . . . . . 16  |-  ( [
<. r ,  1o >. ]  ~Q  e.  Q.  ->  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  e.  Q. )
2421, 22, 233syl 17 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( (
ph  /\  ( x  e.  Q.  /\  x  e.  ( 1st `  A
) ) )  /\  s  e.  Q. )  /\  ( s  +Q  s
)  <Q  x )  /\  r  e.  N. )  /\  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q  s
)  ->  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  e. 
Q. )
2517ad2antrr 488 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( (
ph  /\  ( x  e.  Q.  /\  x  e.  ( 1st `  A
) ) )  /\  s  e.  Q. )  /\  ( s  +Q  s
)  <Q  x )  /\  r  e.  N. )  /\  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q  s
)  ->  s  e.  Q. )
26 ltanqg 7412 . . . . . . . . . . . . . . 15  |-  ( ( ( *Q `  [ <. r ,  1o >. ]  ~Q  )  e.  Q.  /\  s  e.  Q.  /\  s  e.  Q. )  ->  ( ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q 
s  <->  ( s  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) )  <Q 
( s  +Q  s
) ) )
2724, 25, 25, 26syl3anc 1248 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( (
ph  /\  ( x  e.  Q.  /\  x  e.  ( 1st `  A
) ) )  /\  s  e.  Q. )  /\  ( s  +Q  s
)  <Q  x )  /\  r  e.  N. )  /\  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q  s
)  ->  ( ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q  s  <->  ( s  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) )  <Q 
( s  +Q  s
) ) )
2820, 27mpbid 147 . . . . . . . . . . . . 13  |-  ( ( ( ( ( (
ph  /\  ( x  e.  Q.  /\  x  e.  ( 1st `  A
) ) )  /\  s  e.  Q. )  /\  ( s  +Q  s
)  <Q  x )  /\  r  e.  N. )  /\  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q  s
)  ->  ( s  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) )  <Q 
( s  +Q  s
) )
29 simpllr 534 . . . . . . . . . . . . 13  |-  ( ( ( ( ( (
ph  /\  ( x  e.  Q.  /\  x  e.  ( 1st `  A
) ) )  /\  s  e.  Q. )  /\  ( s  +Q  s
)  <Q  x )  /\  r  e.  N. )  /\  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q  s
)  ->  ( s  +Q  s )  <Q  x
)
30 ltsonq 7410 . . . . . . . . . . . . . 14  |-  <Q  Or  Q.
31 ltrelnq 7377 . . . . . . . . . . . . . 14  |-  <Q  C_  ( Q.  X.  Q. )
3230, 31sotri 5036 . . . . . . . . . . . . 13  |-  ( ( ( s  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  )
)  <Q  ( s  +Q  s )  /\  (
s  +Q  s ) 
<Q  x )  ->  (
s  +Q  ( *Q
`  [ <. r ,  1o >. ]  ~Q  )
)  <Q  x )
3328, 29, 32syl2anc 411 . . . . . . . . . . . 12  |-  ( ( ( ( ( (
ph  /\  ( x  e.  Q.  /\  x  e.  ( 1st `  A
) ) )  /\  s  e.  Q. )  /\  ( s  +Q  s
)  <Q  x )  /\  r  e.  N. )  /\  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q  s
)  ->  ( s  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) )  <Q  x )
3410ad5antr 496 . . . . . . . . . . . . 13  |-  ( ( ( ( ( (
ph  /\  ( x  e.  Q.  /\  x  e.  ( 1st `  A
) ) )  /\  s  e.  Q. )  /\  ( s  +Q  s
)  <Q  x )  /\  r  e.  N. )  /\  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q  s
)  ->  A  e.  P. )
35 simprr 531 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( x  e.  Q.  /\  x  e.  ( 1st `  A
) ) )  ->  x  e.  ( 1st `  A ) )
3635ad4antr 494 . . . . . . . . . . . . 13  |-  ( ( ( ( ( (
ph  /\  ( x  e.  Q.  /\  x  e.  ( 1st `  A
) ) )  /\  s  e.  Q. )  /\  ( s  +Q  s
)  <Q  x )  /\  r  e.  N. )  /\  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q  s
)  ->  x  e.  ( 1st `  A ) )
37 prcdnql 7496 . . . . . . . . . . . . . 14  |-  ( (
<. ( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P.  /\  x  e.  ( 1st `  A ) )  -> 
( ( s  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) )  <Q  x  ->  ( s  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) )  e.  ( 1st `  A
) ) )
3811, 37sylan 283 . . . . . . . . . . . . 13  |-  ( ( A  e.  P.  /\  x  e.  ( 1st `  A ) )  -> 
( ( s  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) )  <Q  x  ->  ( s  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) )  e.  ( 1st `  A
) ) )
3934, 36, 38syl2anc 411 . . . . . . . . . . . 12  |-  ( ( ( ( ( (
ph  /\  ( x  e.  Q.  /\  x  e.  ( 1st `  A
) ) )  /\  s  e.  Q. )  /\  ( s  +Q  s
)  <Q  x )  /\  r  e.  N. )  /\  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q  s
)  ->  ( (
s  +Q  ( *Q
`  [ <. r ,  1o >. ]  ~Q  )
)  <Q  x  ->  (
s  +Q  ( *Q
`  [ <. r ,  1o >. ]  ~Q  )
)  e.  ( 1st `  A ) ) )
4033, 39mpd 13 . . . . . . . . . . 11  |-  ( ( ( ( ( (
ph  /\  ( x  e.  Q.  /\  x  e.  ( 1st `  A
) ) )  /\  s  e.  Q. )  /\  ( s  +Q  s
)  <Q  x )  /\  r  e.  N. )  /\  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q  s
)  ->  ( s  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) )  e.  ( 1st `  A
) )
41 addclnq 7387 . . . . . . . . . . . . 13  |-  ( ( s  e.  Q.  /\  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  e.  Q. )  ->  ( s  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  )
)  e.  Q. )
4225, 24, 41syl2anc 411 . . . . . . . . . . . 12  |-  ( ( ( ( ( (
ph  /\  ( x  e.  Q.  /\  x  e.  ( 1st `  A
) ) )  /\  s  e.  Q. )  /\  ( s  +Q  s
)  <Q  x )  /\  r  e.  N. )  /\  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q  s
)  ->  ( s  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) )  e. 
Q. )
43 nqprl 7563 . . . . . . . . . . . 12  |-  ( ( ( s  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  )
)  e.  Q.  /\  A  e.  P. )  ->  ( ( s  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) )  e.  ( 1st `  A
)  <->  <. { p  |  p  <Q  ( s  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) ) } ,  { q  |  ( s  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  )
)  <Q  q } >.  <P  A ) )
4442, 34, 43syl2anc 411 . . . . . . . . . . 11  |-  ( ( ( ( ( (
ph  /\  ( x  e.  Q.  /\  x  e.  ( 1st `  A
) ) )  /\  s  e.  Q. )  /\  ( s  +Q  s
)  <Q  x )  /\  r  e.  N. )  /\  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q  s
)  ->  ( (
s  +Q  ( *Q
`  [ <. r ,  1o >. ]  ~Q  )
)  e.  ( 1st `  A )  <->  <. { p  |  p  <Q  ( s  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) ) } ,  { q  |  ( s  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) )  <Q 
q } >.  <P  A ) )
4540, 44mpbid 147 . . . . . . . . . 10  |-  ( ( ( ( ( (
ph  /\  ( x  e.  Q.  /\  x  e.  ( 1st `  A
) ) )  /\  s  e.  Q. )  /\  ( s  +Q  s
)  <Q  x )  /\  r  e.  N. )  /\  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q  s
)  ->  <. { p  |  p  <Q  ( s  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) ) } ,  { q  |  ( s  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) )  <Q 
q } >.  <P  A )
46 fveq2 5527 . . . . . . . . . . . 12  |-  ( m  =  r  ->  ( F `  m )  =  ( F `  r ) )
4746breq2d 4027 . . . . . . . . . . 11  |-  ( m  =  r  ->  ( A  <P  ( F `  m )  <->  A  <P  ( F `  r ) ) )
483ad5antr 496 . . . . . . . . . . 11  |-  ( ( ( ( ( (
ph  /\  ( x  e.  Q.  /\  x  e.  ( 1st `  A
) ) )  /\  s  e.  Q. )  /\  ( s  +Q  s
)  <Q  x )  /\  r  e.  N. )  /\  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q  s
)  ->  A. m  e.  N.  A  <P  ( F `  m )
)
4947, 48, 21rspcdva 2858 . . . . . . . . . 10  |-  ( ( ( ( ( (
ph  /\  ( x  e.  Q.  /\  x  e.  ( 1st `  A
) ) )  /\  s  e.  Q. )  /\  ( s  +Q  s
)  <Q  x )  /\  r  e.  N. )  /\  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q  s
)  ->  A  <P  ( F `  r ) )
50 ltsopr 7608 . . . . . . . . . . 11  |-  <P  Or  P.
5150, 7sotri 5036 . . . . . . . . . 10  |-  ( (
<. { p  |  p 
<Q  ( s  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  )
) } ,  {
q  |  ( s  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) ) 
<Q  q } >.  <P  A  /\  A  <P  ( F `  r ) )  ->  <. { p  |  p 
<Q  ( s  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  )
) } ,  {
q  |  ( s  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) ) 
<Q  q } >.  <P  ( F `  r )
)
5245, 49, 51syl2anc 411 . . . . . . . . 9  |-  ( ( ( ( ( (
ph  /\  ( x  e.  Q.  /\  x  e.  ( 1st `  A
) ) )  /\  s  e.  Q. )  /\  ( s  +Q  s
)  <Q  x )  /\  r  e.  N. )  /\  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q  s
)  ->  <. { p  |  p  <Q  ( s  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) ) } ,  { q  |  ( s  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) )  <Q 
q } >.  <P  ( F `  r )
)
5352ex 115 . . . . . . . 8  |-  ( ( ( ( ( ph  /\  ( x  e.  Q.  /\  x  e.  ( 1st `  A ) ) )  /\  s  e.  Q. )  /\  ( s  +Q  s )  <Q  x
)  /\  r  e.  N. )  ->  ( ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q  s  ->  <. { p  |  p  <Q  ( s  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) ) } ,  { q  |  ( s  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) )  <Q 
q } >.  <P  ( F `  r )
) )
5453reximdva 2589 . . . . . . 7  |-  ( ( ( ( ph  /\  ( x  e.  Q.  /\  x  e.  ( 1st `  A ) ) )  /\  s  e.  Q. )  /\  ( s  +Q  s )  <Q  x
)  ->  ( E. r  e.  N.  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q  s  ->  E. r  e.  N.  <. { p  |  p  <Q  ( s  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) ) } ,  { q  |  ( s  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  )
)  <Q  q } >.  <P 
( F `  r
) ) )
5519, 54mpd 13 . . . . . 6  |-  ( ( ( ( ph  /\  ( x  e.  Q.  /\  x  e.  ( 1st `  A ) ) )  /\  s  e.  Q. )  /\  ( s  +Q  s )  <Q  x
)  ->  E. r  e.  N.  <. { p  |  p  <Q  ( s  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) ) } ,  { q  |  ( s  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  )
)  <Q  q } >.  <P 
( F `  r
) )
56 oveq1 5895 . . . . . . . . . . . 12  |-  ( l  =  s  ->  (
l  +Q  ( *Q
`  [ <. r ,  1o >. ]  ~Q  )
)  =  ( s  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) ) )
5756breq2d 4027 . . . . . . . . . . 11  |-  ( l  =  s  ->  (
p  <Q  ( l  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) )  <->  p  <Q  ( s  +Q  ( *Q
`  [ <. r ,  1o >. ]  ~Q  )
) ) )
5857abbidv 2305 . . . . . . . . . 10  |-  ( l  =  s  ->  { p  |  p  <Q  ( l  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) ) }  =  { p  |  p  <Q  ( s  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) ) } )
5956breq1d 4025 . . . . . . . . . . 11  |-  ( l  =  s  ->  (
( l  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  )
)  <Q  q  <->  ( s  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) )  <Q 
q ) )
6059abbidv 2305 . . . . . . . . . 10  |-  ( l  =  s  ->  { q  |  ( l  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) )  <Q 
q }  =  {
q  |  ( s  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) ) 
<Q  q } )
6158, 60opeq12d 3798 . . . . . . . . 9  |-  ( l  =  s  ->  <. { p  |  p  <Q  ( l  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) ) } ,  { q  |  ( l  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) )  <Q 
q } >.  =  <. { p  |  p  <Q  ( s  +Q  ( *Q
`  [ <. r ,  1o >. ]  ~Q  )
) } ,  {
q  |  ( s  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) ) 
<Q  q } >. )
6261breq1d 4025 . . . . . . . 8  |-  ( l  =  s  ->  ( <. { p  |  p 
<Q  ( l  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  )
) } ,  {
q  |  ( l  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) ) 
<Q  q } >.  <P  ( F `  r )  <->  <. { p  |  p  <Q  ( s  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  )
) } ,  {
q  |  ( s  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) ) 
<Q  q } >.  <P  ( F `  r )
) )
6362rexbidv 2488 . . . . . . 7  |-  ( l  =  s  ->  ( 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. r  e.  N.  <. { p  |  p  <Q  ( s  +Q  ( *Q
`  [ <. r ,  1o >. ]  ~Q  )
) } ,  {
q  |  ( s  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) ) 
<Q  q } >.  <P  ( F `  r )
) )
64 caucvgprpr.lim . . . . . . . . 9  |-  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 } >. } >.
6564fveq2i 5530 . . . . . . . 8  |-  ( 1st `  L )  =  ( 1st `  <. { 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 } >. } >. )
66 nqex 7375 . . . . . . . . . 10  |-  Q.  e.  _V
6766rabex 4159 . . . . . . . . 9  |-  { 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
6866rabex 4159 . . . . . . . . 9  |-  { 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
6967, 68op1st 6160 . . . . . . . 8  |-  ( 1st `  <. { 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 } >. } >. )  =  { 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 ) }
7065, 69eqtri 2208 . . . . . . 7  |-  ( 1st `  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 ) }
7163, 70elrab2 2908 . . . . . 6  |-  ( s  e.  ( 1st `  L
)  <->  ( s  e. 
Q.  /\  E. r  e.  N.  <. { p  |  p  <Q  ( s  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) ) } ,  { q  |  ( s  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  )
)  <Q  q } >.  <P 
( F `  r
) ) )
7217, 55, 71sylanbrc 417 . . . . 5  |-  ( ( ( ( ph  /\  ( x  e.  Q.  /\  x  e.  ( 1st `  A ) ) )  /\  s  e.  Q. )  /\  ( s  +Q  s )  <Q  x
)  ->  s  e.  ( 1st `  L ) )
7372ex 115 . . . 4  |-  ( ( ( ph  /\  (
x  e.  Q.  /\  x  e.  ( 1st `  A ) ) )  /\  s  e.  Q. )  ->  ( ( s  +Q  s )  <Q  x  ->  s  e.  ( 1st `  L ) ) )
7473reximdva 2589 . . 3  |-  ( (
ph  /\  ( x  e.  Q.  /\  x  e.  ( 1st `  A
) ) )  -> 
( E. s  e. 
Q.  ( s  +Q  s )  <Q  x  ->  E. s  e.  Q.  s  e.  ( 1st `  L ) ) )
7516, 74mpd 13 . 2  |-  ( (
ph  /\  ( x  e.  Q.  /\  x  e.  ( 1st `  A
) ) )  ->  E. s  e.  Q.  s  e.  ( 1st `  L ) )
7614, 75rexlimddv 2609 1  |-  ( ph  ->  E. s  e.  Q.  s  e.  ( 1st `  L ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105    = wceq 1363    e. wcel 2158   {cab 2173   A.wral 2465   E.wrex 2466   {crab 2469   <.cop 3607   class class class wbr 4015   -->wf 5224   ` cfv 5228  (class class class)co 5888   1stc1st 6152   2ndc2nd 6153   1oc1o 6423   [cec 6546   N.cnpi 7284    <N clti 7287    ~Q ceq 7291   Q.cnq 7292    +Q cplq 7294   *Qcrq 7296    <Q cltq 7297   P.cnp 7303    +P. cpp 7305    <P cltp 7307
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 615  ax-in2 616  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-coll 4130  ax-sep 4133  ax-nul 4141  ax-pow 4186  ax-pr 4221  ax-un 4445  ax-setind 4548  ax-iinf 4599
This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 980  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ne 2358  df-ral 2470  df-rex 2471  df-reu 2472  df-rab 2474  df-v 2751  df-sbc 2975  df-csb 3070  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-nul 3435  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-int 3857  df-iun 3900  df-br 4016  df-opab 4077  df-mpt 4078  df-tr 4114  df-eprel 4301  df-id 4305  df-po 4308  df-iso 4309  df-iord 4378  df-on 4380  df-suc 4383  df-iom 4602  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-rn 4649  df-res 4650  df-ima 4651  df-iota 5190  df-fun 5230  df-fn 5231  df-f 5232  df-f1 5233  df-fo 5234  df-f1o 5235  df-fv 5236  df-ov 5891  df-oprab 5892  df-mpo 5893  df-1st 6154  df-2nd 6155  df-recs 6319  df-irdg 6384  df-1o 6430  df-oadd 6434  df-omul 6435  df-er 6548  df-ec 6550  df-qs 6554  df-ni 7316  df-pli 7317  df-mi 7318  df-lti 7319  df-plpq 7356  df-mpq 7357  df-enq 7359  df-nqqs 7360  df-plqqs 7361  df-mqqs 7362  df-1nqqs 7363  df-rq 7364  df-ltnqqs 7365  df-inp 7478  df-iltp 7482
This theorem is referenced by:  caucvgprprlemm  7708
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