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Theorem caucvgprprlemml 6850
Description: Lemma for caucvgprpr 6868. 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 1pi 6471 . . . . 5  |-  1o  e.  N.
2 caucvgprpr.bnd . . . . 5  |-  ( ph  ->  A. m  e.  N.  A  <P  ( F `  m ) )
3 fveq2 5206 . . . . . . 7  |-  ( m  =  1o  ->  ( F `  m )  =  ( F `  1o ) )
43breq2d 3804 . . . . . 6  |-  ( m  =  1o  ->  ( A  <P  ( F `  m )  <->  A  <P  ( F `  1o ) ) )
54rspcv 2669 . . . . 5  |-  ( 1o  e.  N.  ->  ( A. m  e.  N.  A  <P  ( F `  m )  ->  A  <P  ( F `  1o ) ) )
61, 2, 5mpsyl 63 . . . 4  |-  ( ph  ->  A  <P  ( F `  1o ) )
7 ltrelpr 6661 . . . . . 6  |-  <P  C_  ( P.  X.  P. )
87brel 4420 . . . . 5  |-  ( A 
<P  ( F `  1o )  ->  ( A  e. 
P.  /\  ( F `  1o )  e.  P. ) )
98simpld 109 . . . 4  |-  ( A 
<P  ( F `  1o )  ->  A  e.  P. )
106, 9syl 14 . . 3  |-  ( ph  ->  A  e.  P. )
11 prop 6631 . . . 4  |-  ( A  e.  P.  ->  <. ( 1st `  A ) ,  ( 2nd `  A
) >.  e.  P. )
12 prml 6633 . . . 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 6570 . . . 4  |-  ( x  e.  Q.  ->  E. s  e.  Q.  ( s  +Q  s )  <Q  x
)
1615ad2antrl 467 . . 3  |-  ( (
ph  /\  ( x  e.  Q.  /\  x  e.  ( 1st `  A
) ) )  ->  E. s  e.  Q.  ( s  +Q  s
)  <Q  x )
17 simplr 490 . . . . . 6  |-  ( ( ( ( ph  /\  ( x  e.  Q.  /\  x  e.  ( 1st `  A ) ) )  /\  s  e.  Q. )  /\  ( s  +Q  s )  <Q  x
)  ->  s  e.  Q. )
18 archrecnq 6819 . . . . . . . 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 107 . . . . . . . . . . . . . 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 490 . . . . . . . . . . . . . . . 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 6578 . . . . . . . . . . . . . . . 16  |-  ( r  e.  N.  ->  [ <. r ,  1o >. ]  ~Q  e.  Q. )
23 recclnq 6548 . . . . . . . . . . . . . . . 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 465 . . . . . . . . . . . . . . 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 6556 . . . . . . . . . . . . . . 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 1146 . . . . . . . . . . . . . 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 139 . . . . . . . . . . . . 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 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
)  ->  ( s  +Q  s )  <Q  x
)
30 ltsonq 6554 . . . . . . . . . . . . . 14  |-  <Q  Or  Q.
31 ltrelnq 6521 . . . . . . . . . . . . . 14  |-  <Q  C_  ( Q.  X.  Q. )
3230, 31sotri 4748 . . . . . . . . . . . . 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 397 . . . . . . . . . . . 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 473 . . . . . . . . . . . . 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 492 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( x  e.  Q.  /\  x  e.  ( 1st `  A
) ) )  ->  x  e.  ( 1st `  A ) )
3635ad4antr 471 . . . . . . . . . . . . 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 6640 . . . . . . . . . . . . . 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 271 . . . . . . . . . . . . 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 397 . . . . . . . . . . . 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 6531 . . . . . . . . . . . . 13  |-  ( ( s  e.  Q.  /\  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  e.  Q. )  ->  ( s  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  )
)  e.  Q. )
4225, 24, 41syl2anc 397 . . . . . . . . . . . 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 6707 . . . . . . . . . . . 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 397 . . . . . . . . . . 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 139 . . . . . . . . . 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 )
462ad5antr 473 . . . . . . . . . . 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 )
)
47 fveq2 5206 . . . . . . . . . . . . 13  |-  ( m  =  r  ->  ( F `  m )  =  ( F `  r ) )
4847breq2d 3804 . . . . . . . . . . . 12  |-  ( m  =  r  ->  ( A  <P  ( F `  m )  <->  A  <P  ( F `  r ) ) )
4948rspcv 2669 . . . . . . . . . . 11  |-  ( r  e.  N.  ->  ( A. m  e.  N.  A  <P  ( F `  m )  ->  A  <P  ( F `  r
) ) )
5021, 46, 49sylc 60 . . . . . . . . . 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 ) )
51 ltsopr 6752 . . . . . . . . . . 11  |-  <P  Or  P.
5251, 7sotri 4748 . . . . . . . . . 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 )
)
5345, 50, 52syl2anc 397 . . . . . . . . 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 )
)
5453ex 112 . . . . . . . 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 )
) )
5554reximdva 2438 . . . . . . 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
) ) )
5619, 55mpd 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
) )
57 oveq1 5547 . . . . . . . . . . . 12  |-  ( l  =  s  ->  (
l  +Q  ( *Q
`  [ <. r ,  1o >. ]  ~Q  )
)  =  ( s  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) ) )
5857breq2d 3804 . . . . . . . . . . 11  |-  ( l  =  s  ->  (
p  <Q  ( l  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) )  <->  p  <Q  ( s  +Q  ( *Q
`  [ <. r ,  1o >. ]  ~Q  )
) ) )
5958abbidv 2171 . . . . . . . . . 10  |-  ( l  =  s  ->  { p  |  p  <Q  ( l  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) ) }  =  { p  |  p  <Q  ( s  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) ) } )
6057breq1d 3802 . . . . . . . . . . 11  |-  ( l  =  s  ->  (
( l  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  )
)  <Q  q  <->  ( s  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) )  <Q 
q ) )
6160abbidv 2171 . . . . . . . . . 10  |-  ( l  =  s  ->  { q  |  ( l  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) )  <Q 
q }  =  {
q  |  ( s  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) ) 
<Q  q } )
6259, 61opeq12d 3585 . . . . . . . . 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 } >. )
6362breq1d 3802 . . . . . . . 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 )
) )
6463rexbidv 2344 . . . . . . 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 )
) )
65 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 } >. } >.
6665fveq2i 5209 . . . . . . . 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 } >. } >. )
67 nqex 6519 . . . . . . . . . 10  |-  Q.  e.  _V
6867rabex 3929 . . . . . . . . 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
6967rabex 3929 . . . . . . . . 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
7068, 69op1st 5801 . . . . . . . 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 ) }
7166, 70eqtri 2076 . . . . . . 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 ) }
7264, 71elrab2 2723 . . . . . 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
) ) )
7317, 56, 72sylanbrc 402 . . . . 5  |-  ( ( ( ( ph  /\  ( x  e.  Q.  /\  x  e.  ( 1st `  A ) ) )  /\  s  e.  Q. )  /\  ( s  +Q  s )  <Q  x
)  ->  s  e.  ( 1st `  L ) )
7473ex 112 . . . 4  |-  ( ( ( ph  /\  (
x  e.  Q.  /\  x  e.  ( 1st `  A ) ) )  /\  s  e.  Q. )  ->  ( ( s  +Q  s )  <Q  x  ->  s  e.  ( 1st `  L ) ) )
7574reximdva 2438 . . 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 ) ) )
7616, 75mpd 13 . 2  |-  ( (
ph  /\  ( x  e.  Q.  /\  x  e.  ( 1st `  A
) ) )  ->  E. s  e.  Q.  s  e.  ( 1st `  L ) )
7714, 76rexlimddv 2454 1  |-  ( ph  ->  E. s  e.  Q.  s  e.  ( 1st `  L ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 101    <-> wb 102    = wceq 1259    e. wcel 1409   {cab 2042   A.wral 2323   E.wrex 2324   {crab 2327   <.cop 3406   class class class wbr 3792   -->wf 4926   ` cfv 4930  (class class class)co 5540   1stc1st 5793   2ndc2nd 5794   1oc1o 6025   [cec 6135   N.cnpi 6428    <N clti 6431    ~Q ceq 6435   Q.cnq 6436    +Q cplq 6438   *Qcrq 6440    <Q cltq 6441   P.cnp 6447    +P. cpp 6449    <P cltp 6451
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-coll 3900  ax-sep 3903  ax-nul 3911  ax-pow 3955  ax-pr 3972  ax-un 4198  ax-setind 4290  ax-iinf 4339
This theorem depends on definitions:  df-bi 114  df-dc 754  df-3or 897  df-3an 898  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ne 2221  df-ral 2328  df-rex 2329  df-reu 2330  df-rab 2332  df-v 2576  df-sbc 2788  df-csb 2881  df-dif 2948  df-un 2950  df-in 2952  df-ss 2959  df-nul 3253  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-int 3644  df-iun 3687  df-br 3793  df-opab 3847  df-mpt 3848  df-tr 3883  df-eprel 4054  df-id 4058  df-po 4061  df-iso 4062  df-iord 4131  df-on 4133  df-suc 4136  df-iom 4342  df-xp 4379  df-rel 4380  df-cnv 4381  df-co 4382  df-dm 4383  df-rn 4384  df-res 4385  df-ima 4386  df-iota 4895  df-fun 4932  df-fn 4933  df-f 4934  df-f1 4935  df-fo 4936  df-f1o 4937  df-fv 4938  df-ov 5543  df-oprab 5544  df-mpt2 5545  df-1st 5795  df-2nd 5796  df-recs 5951  df-irdg 5988  df-1o 6032  df-oadd 6036  df-omul 6037  df-er 6137  df-ec 6139  df-qs 6143  df-ni 6460  df-pli 6461  df-mi 6462  df-lti 6463  df-plpq 6500  df-mpq 6501  df-enq 6503  df-nqqs 6504  df-plqqs 6505  df-mqqs 6506  df-1nqqs 6507  df-rq 6508  df-ltnqqs 6509  df-inp 6622  df-iltp 6626
This theorem is referenced by:  caucvgprprlemm  6852
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