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Theorem caucvgprprlemml 7470
Description: Lemma for caucvgprpr 7488. 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 5389 . . . . . 6  |-  ( m  =  1o  ->  ( F `  m )  =  ( F `  1o ) )
21breq2d 3911 . . . . 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 7091 . . . . . 6  |-  1o  e.  N.
54a1i 9 . . . . 5  |-  ( ph  ->  1o  e.  N. )
62, 3, 5rspcdva 2768 . . . 4  |-  ( ph  ->  A  <P  ( F `  1o ) )
7 ltrelpr 7281 . . . . . 6  |-  <P  C_  ( P.  X.  P. )
87brel 4561 . . . . 5  |-  ( A 
<P  ( F `  1o )  ->  ( A  e. 
P.  /\  ( F `  1o )  e.  P. ) )
98simpld 111 . . . 4  |-  ( A 
<P  ( F `  1o )  ->  A  e.  P. )
106, 9syl 14 . . 3  |-  ( ph  ->  A  e.  P. )
11 prop 7251 . . . 4  |-  ( A  e.  P.  ->  <. ( 1st `  A ) ,  ( 2nd `  A
) >.  e.  P. )
12 prml 7253 . . . 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 7190 . . . 4  |-  ( x  e.  Q.  ->  E. s  e.  Q.  ( s  +Q  s )  <Q  x
)
1615ad2antrl 481 . . 3  |-  ( (
ph  /\  ( x  e.  Q.  /\  x  e.  ( 1st `  A
) ) )  ->  E. s  e.  Q.  ( s  +Q  s
)  <Q  x )
17 simplr 504 . . . . . 6  |-  ( ( ( ( ph  /\  ( x  e.  Q.  /\  x  e.  ( 1st `  A ) ) )  /\  s  e.  Q. )  /\  ( s  +Q  s )  <Q  x
)  ->  s  e.  Q. )
18 archrecnq 7439 . . . . . . . 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 109 . . . . . . . . . . . . . 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 504 . . . . . . . . . . . . . . . 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 7198 . . . . . . . . . . . . . . . 16  |-  ( r  e.  N.  ->  [ <. r ,  1o >. ]  ~Q  e.  Q. )
23 recclnq 7168 . . . . . . . . . . . . . . . 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 479 . . . . . . . . . . . . . . 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 7176 . . . . . . . . . . . . . . 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 1201 . . . . . . . . . . . . . 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 146 . . . . . . . . . . . . 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 508 . . . . . . . . . . . . 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 7174 . . . . . . . . . . . . . 14  |-  <Q  Or  Q.
31 ltrelnq 7141 . . . . . . . . . . . . . 14  |-  <Q  C_  ( Q.  X.  Q. )
3230, 31sotri 4904 . . . . . . . . . . . . 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 408 . . . . . . . . . . . 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 487 . . . . . . . . . . . . 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 506 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( x  e.  Q.  /\  x  e.  ( 1st `  A
) ) )  ->  x  e.  ( 1st `  A ) )
3635ad4antr 485 . . . . . . . . . . . . 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 7260 . . . . . . . . . . . . . 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 281 . . . . . . . . . . . . 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 408 . . . . . . . . . . . 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 7151 . . . . . . . . . . . . 13  |-  ( ( s  e.  Q.  /\  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  e.  Q. )  ->  ( s  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  )
)  e.  Q. )
4225, 24, 41syl2anc 408 . . . . . . . . . . . 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 7327 . . . . . . . . . . . 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 408 . . . . . . . . . . 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 146 . . . . . . . . . 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 5389 . . . . . . . . . . . 12  |-  ( m  =  r  ->  ( F `  m )  =  ( F `  r ) )
4746breq2d 3911 . . . . . . . . . . 11  |-  ( m  =  r  ->  ( A  <P  ( F `  m )  <->  A  <P  ( F `  r ) ) )
483ad5antr 487 . . . . . . . . . . 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 2768 . . . . . . . . . 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 7372 . . . . . . . . . . 11  |-  <P  Or  P.
5150, 7sotri 4904 . . . . . . . . . 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 408 . . . . . . . . 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 114 . . . . . . . 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 2511 . . . . . . 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 5749 . . . . . . . . . . . 12  |-  ( l  =  s  ->  (
l  +Q  ( *Q
`  [ <. r ,  1o >. ]  ~Q  )
)  =  ( s  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) ) )
5756breq2d 3911 . . . . . . . . . . 11  |-  ( l  =  s  ->  (
p  <Q  ( l  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) )  <->  p  <Q  ( s  +Q  ( *Q
`  [ <. r ,  1o >. ]  ~Q  )
) ) )
5857abbidv 2235 . . . . . . . . . 10  |-  ( l  =  s  ->  { p  |  p  <Q  ( l  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) ) }  =  { p  |  p  <Q  ( s  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) ) } )
5956breq1d 3909 . . . . . . . . . . 11  |-  ( l  =  s  ->  (
( l  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  )
)  <Q  q  <->  ( s  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) )  <Q 
q ) )
6059abbidv 2235 . . . . . . . . . 10  |-  ( l  =  s  ->  { q  |  ( l  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) )  <Q 
q }  =  {
q  |  ( s  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) ) 
<Q  q } )
6158, 60opeq12d 3683 . . . . . . . . 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 3909 . . . . . . . 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 2415 . . . . . . 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 5392 . . . . . . . 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 7139 . . . . . . . . . 10  |-  Q.  e.  _V
6766rabex 4042 . . . . . . . . 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 4042 . . . . . . . . 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 6012 . . . . . . . 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 2138 . . . . . . 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 2816 . . . . . 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 413 . . . . 5  |-  ( ( ( ( ph  /\  ( x  e.  Q.  /\  x  e.  ( 1st `  A ) ) )  /\  s  e.  Q. )  /\  ( s  +Q  s )  <Q  x
)  ->  s  e.  ( 1st `  L ) )
7372ex 114 . . . 4  |-  ( ( ( ph  /\  (
x  e.  Q.  /\  x  e.  ( 1st `  A ) ) )  /\  s  e.  Q. )  ->  ( ( s  +Q  s )  <Q  x  ->  s  e.  ( 1st `  L ) ) )
7473reximdva 2511 . . 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 2531 1  |-  ( ph  ->  E. s  e.  Q.  s  e.  ( 1st `  L ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1316    e. wcel 1465   {cab 2103   A.wral 2393   E.wrex 2394   {crab 2397   <.cop 3500   class class class wbr 3899   -->wf 5089   ` cfv 5093  (class class class)co 5742   1stc1st 6004   2ndc2nd 6005   1oc1o 6274   [cec 6395   N.cnpi 7048    <N clti 7051    ~Q ceq 7055   Q.cnq 7056    +Q cplq 7058   *Qcrq 7060    <Q cltq 7061   P.cnp 7067    +P. cpp 7069    <P cltp 7071
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 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-coll 4013  ax-sep 4016  ax-nul 4024  ax-pow 4068  ax-pr 4101  ax-un 4325  ax-setind 4422  ax-iinf 4472
This theorem depends on definitions:  df-bi 116  df-dc 805  df-3or 948  df-3an 949  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ne 2286  df-ral 2398  df-rex 2399  df-reu 2400  df-rab 2402  df-v 2662  df-sbc 2883  df-csb 2976  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-nul 3334  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-int 3742  df-iun 3785  df-br 3900  df-opab 3960  df-mpt 3961  df-tr 3997  df-eprel 4181  df-id 4185  df-po 4188  df-iso 4189  df-iord 4258  df-on 4260  df-suc 4263  df-iom 4475  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522  df-iota 5058  df-fun 5095  df-fn 5096  df-f 5097  df-f1 5098  df-fo 5099  df-f1o 5100  df-fv 5101  df-ov 5745  df-oprab 5746  df-mpo 5747  df-1st 6006  df-2nd 6007  df-recs 6170  df-irdg 6235  df-1o 6281  df-oadd 6285  df-omul 6286  df-er 6397  df-ec 6399  df-qs 6403  df-ni 7080  df-pli 7081  df-mi 7082  df-lti 7083  df-plpq 7120  df-mpq 7121  df-enq 7123  df-nqqs 7124  df-plqqs 7125  df-mqqs 7126  df-1nqqs 7127  df-rq 7128  df-ltnqqs 7129  df-inp 7242  df-iltp 7246
This theorem is referenced by:  caucvgprprlemm  7472
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