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Theorem caucvgprprlemml 7495
Description: Lemma for caucvgprpr 7513. 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 5414 . . . . . 6  |-  ( m  =  1o  ->  ( F `  m )  =  ( F `  1o ) )
21breq2d 3936 . . . . 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 7116 . . . . . 6  |-  1o  e.  N.
54a1i 9 . . . . 5  |-  ( ph  ->  1o  e.  N. )
62, 3, 5rspcdva 2789 . . . 4  |-  ( ph  ->  A  <P  ( F `  1o ) )
7 ltrelpr 7306 . . . . . 6  |-  <P  C_  ( P.  X.  P. )
87brel 4586 . . . . 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 7276 . . . 4  |-  ( A  e.  P.  ->  <. ( 1st `  A ) ,  ( 2nd `  A
) >.  e.  P. )
12 prml 7278 . . . 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 7215 . . . 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 519 . . . . . 6  |-  ( ( ( ( ph  /\  ( x  e.  Q.  /\  x  e.  ( 1st `  A ) ) )  /\  s  e.  Q. )  /\  ( s  +Q  s )  <Q  x
)  ->  s  e.  Q. )
18 archrecnq 7464 . . . . . . . 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 519 . . . . . . . . . . . . . . . 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 7223 . . . . . . . . . . . . . . . 16  |-  ( r  e.  N.  ->  [ <. r ,  1o >. ]  ~Q  e.  Q. )
23 recclnq 7193 . . . . . . . . . . . . . . . 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 7201 . . . . . . . . . . . . . . 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 1216 . . . . . . . . . . . . . 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 523 . . . . . . . . . . . . 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 7199 . . . . . . . . . . . . . 14  |-  <Q  Or  Q.
31 ltrelnq 7166 . . . . . . . . . . . . . 14  |-  <Q  C_  ( Q.  X.  Q. )
3230, 31sotri 4929 . . . . . . . . . . . . 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 521 . . . . . . . . . . . . . 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 7285 . . . . . . . . . . . . . 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 7176 . . . . . . . . . . . . 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 7352 . . . . . . . . . . . 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 5414 . . . . . . . . . . . 12  |-  ( m  =  r  ->  ( F `  m )  =  ( F `  r ) )
4746breq2d 3936 . . . . . . . . . . 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 2789 . . . . . . . . . 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 7397 . . . . . . . . . . 11  |-  <P  Or  P.
5150, 7sotri 4929 . . . . . . . . . 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 2532 . . . . . . 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 5774 . . . . . . . . . . . 12  |-  ( l  =  s  ->  (
l  +Q  ( *Q
`  [ <. r ,  1o >. ]  ~Q  )
)  =  ( s  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) ) )
5756breq2d 3936 . . . . . . . . . . 11  |-  ( l  =  s  ->  (
p  <Q  ( l  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) )  <->  p  <Q  ( s  +Q  ( *Q
`  [ <. r ,  1o >. ]  ~Q  )
) ) )
5857abbidv 2255 . . . . . . . . . 10  |-  ( l  =  s  ->  { p  |  p  <Q  ( l  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) ) }  =  { p  |  p  <Q  ( s  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) ) } )
5956breq1d 3934 . . . . . . . . . . 11  |-  ( l  =  s  ->  (
( l  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  )
)  <Q  q  <->  ( s  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) )  <Q 
q ) )
6059abbidv 2255 . . . . . . . . . 10  |-  ( l  =  s  ->  { q  |  ( l  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) )  <Q 
q }  =  {
q  |  ( s  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) ) 
<Q  q } )
6158, 60opeq12d 3708 . . . . . . . . 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 3934 . . . . . . . 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 2436 . . . . . . 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 5417 . . . . . . . 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 7164 . . . . . . . . . 10  |-  Q.  e.  _V
6766rabex 4067 . . . . . . . . 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 4067 . . . . . . . . 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 6037 . . . . . . . 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 2158 . . . . . . 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 2838 . . . . . 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 2532 . . 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 2552 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 1331    e. wcel 1480   {cab 2123   A.wral 2414   E.wrex 2415   {crab 2418   <.cop 3525   class class class wbr 3924   -->wf 5114   ` cfv 5118  (class class class)co 5767   1stc1st 6029   2ndc2nd 6030   1oc1o 6299   [cec 6420   N.cnpi 7073    <N clti 7076    ~Q ceq 7080   Q.cnq 7081    +Q cplq 7083   *Qcrq 7085    <Q cltq 7086   P.cnp 7092    +P. cpp 7094    <P cltp 7096
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 2119  ax-coll 4038  ax-sep 4041  ax-nul 4049  ax-pow 4093  ax-pr 4126  ax-un 4350  ax-setind 4447  ax-iinf 4497
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 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ne 2307  df-ral 2419  df-rex 2420  df-reu 2421  df-rab 2423  df-v 2683  df-sbc 2905  df-csb 2999  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-nul 3359  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-int 3767  df-iun 3810  df-br 3925  df-opab 3985  df-mpt 3986  df-tr 4022  df-eprel 4206  df-id 4210  df-po 4213  df-iso 4214  df-iord 4283  df-on 4285  df-suc 4288  df-iom 4500  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-res 4546  df-ima 4547  df-iota 5083  df-fun 5120  df-fn 5121  df-f 5122  df-f1 5123  df-fo 5124  df-f1o 5125  df-fv 5126  df-ov 5770  df-oprab 5771  df-mpo 5772  df-1st 6031  df-2nd 6032  df-recs 6195  df-irdg 6260  df-1o 6306  df-oadd 6310  df-omul 6311  df-er 6422  df-ec 6424  df-qs 6428  df-ni 7105  df-pli 7106  df-mi 7107  df-lti 7108  df-plpq 7145  df-mpq 7146  df-enq 7148  df-nqqs 7149  df-plqqs 7150  df-mqqs 7151  df-1nqqs 7152  df-rq 7153  df-ltnqqs 7154  df-inp 7267  df-iltp 7271
This theorem is referenced by:  caucvgprprlemm  7497
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