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Theorem caucvgprprlemopu 7619
Description: Lemma for caucvgprpr 7632. The upper cut of the putative limit is open. (Contributed by Jim Kingdon, 21-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
caucvgprprlemopu  |-  ( (
ph  /\  t  e.  ( 2nd `  L ) )  ->  E. s  e.  Q.  ( s  <Q 
t  /\  s  e.  ( 2nd `  L ) ) )
Distinct variable groups:    A, m    m, F    F, l, r, s   
u, F, r, s    L, s    p, l, q, t, r, s    u, p, q, t    ph, r,
s
Allowed substitution hints:    ph( u, t, k, m, n, q, p, l)    A( u, t, k, n, s, r, q, p, l)    F( t, k, n, q, p)    L( u, t, k, m, n, r, q, p, l)

Proof of Theorem caucvgprprlemopu
Dummy variable  b is distinct from all other variables.
StepHypRef Expression
1 caucvgprpr.lim . . . . 5  |-  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 } >. } >.
21caucvgprprlemelu 7606 . . . 4  |-  ( t  e.  ( 2nd `  L
)  <->  ( t  e. 
Q.  /\  E. b  e.  N.  ( ( F `
 b )  +P. 
<. { p  |  p 
<Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q  q } >. ) 
<P  <. { p  |  p  <Q  t } ,  { q  |  t 
<Q  q } >. )
)
32simprbi 273 . . 3  |-  ( t  e.  ( 2nd `  L
)  ->  E. b  e.  N.  ( ( F `
 b )  +P. 
<. { p  |  p 
<Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q  q } >. ) 
<P  <. { p  |  p  <Q  t } ,  { q  |  t 
<Q  q } >. )
43adantl 275 . 2  |-  ( (
ph  /\  t  e.  ( 2nd `  L ) )  ->  E. b  e.  N.  ( ( F `
 b )  +P. 
<. { p  |  p 
<Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q  q } >. ) 
<P  <. { p  |  p  <Q  t } ,  { q  |  t 
<Q  q } >. )
5 simprr 522 . . . . 5  |-  ( ( ( ph  /\  t  e.  ( 2nd `  L
) )  /\  (
b  e.  N.  /\  ( ( F `  b )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q 
q } >. )  <P 
<. { p  |  p 
<Q  t } ,  {
q  |  t  <Q 
q } >. )
)  ->  ( ( F `  b )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q  q } >. )  <P  <. { p  |  p  <Q  t } ,  { q  |  t  <Q  q } >. )
6 caucvgprpr.f . . . . . . . . 9  |-  ( ph  ->  F : N. --> P. )
76ffvelrnda 5602 . . . . . . . 8  |-  ( (
ph  /\  b  e.  N. )  ->  ( F `
 b )  e. 
P. )
8 recnnpr 7468 . . . . . . . . 9  |-  ( b  e.  N.  ->  <. { p  |  p  <Q  ( *Q
`  [ <. b ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q 
q } >.  e.  P. )
98adantl 275 . . . . . . . 8  |-  ( (
ph  /\  b  e.  N. )  ->  <. { p  |  p  <Q  ( *Q
`  [ <. b ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q 
q } >.  e.  P. )
10 addclpr 7457 . . . . . . . 8  |-  ( ( ( F `  b
)  e.  P.  /\  <. { p  |  p  <Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q  q } >.  e. 
P. )  ->  (
( F `  b
)  +P.  <. { p  |  p  <Q  ( *Q
`  [ <. b ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q 
q } >. )  e.  P. )
117, 9, 10syl2anc 409 . . . . . . 7  |-  ( (
ph  /\  b  e.  N. )  ->  ( ( F `  b )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q  q } >. )  e.  P. )
1211ad2ant2r 501 . . . . . 6  |-  ( ( ( ph  /\  t  e.  ( 2nd `  L
) )  /\  (
b  e.  N.  /\  ( ( F `  b )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q 
q } >. )  <P 
<. { p  |  p 
<Q  t } ,  {
q  |  t  <Q 
q } >. )
)  ->  ( ( F `  b )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q  q } >. )  e.  P. )
132simplbi 272 . . . . . . . 8  |-  ( t  e.  ( 2nd `  L
)  ->  t  e.  Q. )
1413ad2antlr 481 . . . . . . 7  |-  ( ( ( ph  /\  t  e.  ( 2nd `  L
) )  /\  (
b  e.  N.  /\  ( ( F `  b )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q 
q } >. )  <P 
<. { p  |  p 
<Q  t } ,  {
q  |  t  <Q 
q } >. )
)  ->  t  e.  Q. )
15 nqprlu 7467 . . . . . . 7  |-  ( t  e.  Q.  ->  <. { p  |  p  <Q  t } ,  { q  |  t  <Q  q } >.  e.  P. )
1614, 15syl 14 . . . . . 6  |-  ( ( ( ph  /\  t  e.  ( 2nd `  L
) )  /\  (
b  e.  N.  /\  ( ( F `  b )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q 
q } >. )  <P 
<. { p  |  p 
<Q  t } ,  {
q  |  t  <Q 
q } >. )
)  ->  <. { p  |  p  <Q  t } ,  { q  |  t  <Q  q } >.  e.  P. )
17 ltdfpr 7426 . . . . . 6  |-  ( ( ( ( F `  b )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q 
q } >. )  e.  P.  /\  <. { p  |  p  <Q  t } ,  { q  |  t  <Q  q } >.  e.  P. )  -> 
( ( ( F `
 b )  +P. 
<. { p  |  p 
<Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q  q } >. ) 
<P  <. { p  |  p  <Q  t } ,  { q  |  t 
<Q  q } >.  <->  E. s  e.  Q.  ( s  e.  ( 2nd `  (
( F `  b
)  +P.  <. { p  |  p  <Q  ( *Q
`  [ <. b ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q 
q } >. )
)  /\  s  e.  ( 1st `  <. { p  |  p  <Q  t } ,  { q  |  t  <Q  q } >. ) ) ) )
1812, 16, 17syl2anc 409 . . . . 5  |-  ( ( ( ph  /\  t  e.  ( 2nd `  L
) )  /\  (
b  e.  N.  /\  ( ( F `  b )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q 
q } >. )  <P 
<. { p  |  p 
<Q  t } ,  {
q  |  t  <Q 
q } >. )
)  ->  ( (
( F `  b
)  +P.  <. { p  |  p  <Q  ( *Q
`  [ <. b ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q 
q } >. )  <P 
<. { p  |  p 
<Q  t } ,  {
q  |  t  <Q 
q } >.  <->  E. s  e.  Q.  ( s  e.  ( 2nd `  (
( F `  b
)  +P.  <. { p  |  p  <Q  ( *Q
`  [ <. b ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q 
q } >. )
)  /\  s  e.  ( 1st `  <. { p  |  p  <Q  t } ,  { q  |  t  <Q  q } >. ) ) ) )
195, 18mpbid 146 . . . 4  |-  ( ( ( ph  /\  t  e.  ( 2nd `  L
) )  /\  (
b  e.  N.  /\  ( ( F `  b )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q 
q } >. )  <P 
<. { p  |  p 
<Q  t } ,  {
q  |  t  <Q 
q } >. )
)  ->  E. s  e.  Q.  ( s  e.  ( 2nd `  (
( F `  b
)  +P.  <. { p  |  p  <Q  ( *Q
`  [ <. b ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q 
q } >. )
)  /\  s  e.  ( 1st `  <. { p  |  p  <Q  t } ,  { q  |  t  <Q  q } >. ) ) )
20 simpr 109 . . . . . . . 8  |-  ( ( ( ( ph  /\  t  e.  ( 2nd `  L ) )  /\  ( b  e.  N.  /\  ( ( F `  b )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q 
q } >. )  <P 
<. { p  |  p 
<Q  t } ,  {
q  |  t  <Q 
q } >. )
)  /\  s  e.  Q. )  ->  s  e. 
Q. )
2112adantr 274 . . . . . . . 8  |-  ( ( ( ( ph  /\  t  e.  ( 2nd `  L ) )  /\  ( b  e.  N.  /\  ( ( F `  b )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q 
q } >. )  <P 
<. { p  |  p 
<Q  t } ,  {
q  |  t  <Q 
q } >. )
)  /\  s  e.  Q. )  ->  ( ( F `  b )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q  q } >. )  e.  P. )
22 nqpru 7472 . . . . . . . 8  |-  ( ( s  e.  Q.  /\  ( ( F `  b )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q 
q } >. )  e.  P. )  ->  (
s  e.  ( 2nd `  ( ( F `  b )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q 
q } >. )
)  <->  ( ( F `
 b )  +P. 
<. { p  |  p 
<Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q  q } >. ) 
<P  <. { p  |  p  <Q  s } ,  { q  |  s 
<Q  q } >. )
)
2320, 21, 22syl2anc 409 . . . . . . 7  |-  ( ( ( ( ph  /\  t  e.  ( 2nd `  L ) )  /\  ( b  e.  N.  /\  ( ( F `  b )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q 
q } >. )  <P 
<. { p  |  p 
<Q  t } ,  {
q  |  t  <Q 
q } >. )
)  /\  s  e.  Q. )  ->  ( s  e.  ( 2nd `  (
( F `  b
)  +P.  <. { p  |  p  <Q  ( *Q
`  [ <. b ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q 
q } >. )
)  <->  ( ( F `
 b )  +P. 
<. { p  |  p 
<Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q  q } >. ) 
<P  <. { p  |  p  <Q  s } ,  { q  |  s 
<Q  q } >. )
)
24 vex 2715 . . . . . . . . 9  |-  s  e. 
_V
25 breq1 3968 . . . . . . . . 9  |-  ( p  =  s  ->  (
p  <Q  t  <->  s  <Q  t ) )
26 ltnqex 7469 . . . . . . . . . 10  |-  { p  |  p  <Q  t }  e.  _V
27 gtnqex 7470 . . . . . . . . . 10  |-  { q  |  t  <Q  q }  e.  _V
2826, 27op1st 6094 . . . . . . . . 9  |-  ( 1st `  <. { p  |  p  <Q  t } ,  { q  |  t 
<Q  q } >. )  =  { p  |  p 
<Q  t }
2924, 25, 28elab2 2860 . . . . . . . 8  |-  ( s  e.  ( 1st `  <. { p  |  p  <Q  t } ,  { q  |  t  <Q  q } >. )  <->  s  <Q  t )
3029a1i 9 . . . . . . 7  |-  ( ( ( ( ph  /\  t  e.  ( 2nd `  L ) )  /\  ( b  e.  N.  /\  ( ( F `  b )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q 
q } >. )  <P 
<. { p  |  p 
<Q  t } ,  {
q  |  t  <Q 
q } >. )
)  /\  s  e.  Q. )  ->  ( s  e.  ( 1st `  <. { p  |  p  <Q  t } ,  { q  |  t  <Q  q } >. )  <->  s  <Q  t ) )
3123, 30anbi12d 465 . . . . . 6  |-  ( ( ( ( ph  /\  t  e.  ( 2nd `  L ) )  /\  ( b  e.  N.  /\  ( ( F `  b )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q 
q } >. )  <P 
<. { p  |  p 
<Q  t } ,  {
q  |  t  <Q 
q } >. )
)  /\  s  e.  Q. )  ->  ( ( s  e.  ( 2nd `  ( ( F `  b )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q 
q } >. )
)  /\  s  e.  ( 1st `  <. { p  |  p  <Q  t } ,  { q  |  t  <Q  q } >. ) )  <->  ( (
( F `  b
)  +P.  <. { p  |  p  <Q  ( *Q
`  [ <. b ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q 
q } >. )  <P 
<. { p  |  p 
<Q  s } ,  {
q  |  s  <Q 
q } >.  /\  s  <Q  t ) ) )
3231biimpd 143 . . . . 5  |-  ( ( ( ( ph  /\  t  e.  ( 2nd `  L ) )  /\  ( b  e.  N.  /\  ( ( F `  b )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q 
q } >. )  <P 
<. { p  |  p 
<Q  t } ,  {
q  |  t  <Q 
q } >. )
)  /\  s  e.  Q. )  ->  ( ( s  e.  ( 2nd `  ( ( F `  b )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q 
q } >. )
)  /\  s  e.  ( 1st `  <. { p  |  p  <Q  t } ,  { q  |  t  <Q  q } >. ) )  ->  (
( ( F `  b )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q 
q } >. )  <P 
<. { p  |  p 
<Q  s } ,  {
q  |  s  <Q 
q } >.  /\  s  <Q  t ) ) )
3332reximdva 2559 . . . 4  |-  ( ( ( ph  /\  t  e.  ( 2nd `  L
) )  /\  (
b  e.  N.  /\  ( ( F `  b )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q 
q } >. )  <P 
<. { p  |  p 
<Q  t } ,  {
q  |  t  <Q 
q } >. )
)  ->  ( E. s  e.  Q.  (
s  e.  ( 2nd `  ( ( F `  b )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q 
q } >. )
)  /\  s  e.  ( 1st `  <. { p  |  p  <Q  t } ,  { q  |  t  <Q  q } >. ) )  ->  E. s  e.  Q.  ( ( ( F `  b )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q  q } >. )  <P  <. { p  |  p  <Q  s } ,  { q  |  s  <Q  q } >.  /\  s  <Q  t
) ) )
3419, 33mpd 13 . . 3  |-  ( ( ( ph  /\  t  e.  ( 2nd `  L
) )  /\  (
b  e.  N.  /\  ( ( F `  b )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q 
q } >. )  <P 
<. { p  |  p 
<Q  t } ,  {
q  |  t  <Q 
q } >. )
)  ->  E. s  e.  Q.  ( ( ( F `  b )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q  q } >. )  <P  <. { p  |  p  <Q  s } ,  { q  |  s  <Q  q } >.  /\  s  <Q  t
) )
35 simprr 522 . . . . . 6  |-  ( ( ( ( ( ph  /\  t  e.  ( 2nd `  L ) )  /\  ( b  e.  N.  /\  ( ( F `  b )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q 
q } >. )  <P 
<. { p  |  p 
<Q  t } ,  {
q  |  t  <Q 
q } >. )
)  /\  s  e.  Q. )  /\  (
( ( F `  b )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q 
q } >. )  <P 
<. { p  |  p 
<Q  s } ,  {
q  |  s  <Q 
q } >.  /\  s  <Q  t ) )  -> 
s  <Q  t )
36 simplr 520 . . . . . . 7  |-  ( ( ( ( ( ph  /\  t  e.  ( 2nd `  L ) )  /\  ( b  e.  N.  /\  ( ( F `  b )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q 
q } >. )  <P 
<. { p  |  p 
<Q  t } ,  {
q  |  t  <Q 
q } >. )
)  /\  s  e.  Q. )  /\  (
( ( F `  b )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q 
q } >. )  <P 
<. { p  |  p 
<Q  s } ,  {
q  |  s  <Q 
q } >.  /\  s  <Q  t ) )  -> 
s  e.  Q. )
37 simplrl 525 . . . . . . . . 9  |-  ( ( ( ( ph  /\  t  e.  ( 2nd `  L ) )  /\  ( b  e.  N.  /\  ( ( F `  b )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q 
q } >. )  <P 
<. { p  |  p 
<Q  t } ,  {
q  |  t  <Q 
q } >. )
)  /\  s  e.  Q. )  ->  b  e. 
N. )
3837adantr 274 . . . . . . . 8  |-  ( ( ( ( ( ph  /\  t  e.  ( 2nd `  L ) )  /\  ( b  e.  N.  /\  ( ( F `  b )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q 
q } >. )  <P 
<. { p  |  p 
<Q  t } ,  {
q  |  t  <Q 
q } >. )
)  /\  s  e.  Q. )  /\  (
( ( F `  b )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q 
q } >. )  <P 
<. { p  |  p 
<Q  s } ,  {
q  |  s  <Q 
q } >.  /\  s  <Q  t ) )  -> 
b  e.  N. )
39 simprl 521 . . . . . . . 8  |-  ( ( ( ( ( ph  /\  t  e.  ( 2nd `  L ) )  /\  ( b  e.  N.  /\  ( ( F `  b )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q 
q } >. )  <P 
<. { p  |  p 
<Q  t } ,  {
q  |  t  <Q 
q } >. )
)  /\  s  e.  Q. )  /\  (
( ( F `  b )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q 
q } >. )  <P 
<. { p  |  p 
<Q  s } ,  {
q  |  s  <Q 
q } >.  /\  s  <Q  t ) )  -> 
( ( F `  b )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q 
q } >. )  <P 
<. { p  |  p 
<Q  s } ,  {
q  |  s  <Q 
q } >. )
40 fveq2 5468 . . . . . . . . . . 11  |-  ( r  =  b  ->  ( F `  r )  =  ( F `  b ) )
41 opeq1 3741 . . . . . . . . . . . . . . . 16  |-  ( r  =  b  ->  <. r ,  1o >.  =  <. b ,  1o >. )
4241eceq1d 6516 . . . . . . . . . . . . . . 15  |-  ( r  =  b  ->  [ <. r ,  1o >. ]  ~Q  =  [ <. b ,  1o >. ]  ~Q  )
4342fveq2d 5472 . . . . . . . . . . . . . 14  |-  ( r  =  b  ->  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  =  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) )
4443breq2d 3977 . . . . . . . . . . . . 13  |-  ( r  =  b  ->  (
p  <Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <->  p  <Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) ) )
4544abbidv 2275 . . . . . . . . . . . 12  |-  ( r  =  b  ->  { p  |  p  <Q  ( *Q
`  [ <. r ,  1o >. ]  ~Q  ) }  =  { p  |  p  <Q  ( *Q
`  [ <. b ,  1o >. ]  ~Q  ) } )
4643breq1d 3975 . . . . . . . . . . . . 13  |-  ( r  =  b  ->  (
( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q  q  <->  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q  q ) )
4746abbidv 2275 . . . . . . . . . . . 12  |-  ( r  =  b  ->  { q  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q 
q }  =  {
q  |  ( *Q
`  [ <. b ,  1o >. ]  ~Q  )  <Q  q } )
4845, 47opeq12d 3749 . . . . . . . . . . 11  |-  ( r  =  b  ->  <. { p  |  p  <Q  ( *Q
`  [ <. r ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q 
q } >.  =  <. { p  |  p  <Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q 
q } >. )
4940, 48oveq12d 5842 . . . . . . . . . 10  |-  ( r  =  b  ->  (
( F `  r
)  +P.  <. { p  |  p  <Q  ( *Q
`  [ <. r ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q 
q } >. )  =  ( ( F `
 b )  +P. 
<. { p  |  p 
<Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q  q } >. ) )
5049breq1d 3975 . . . . . . . . 9  |-  ( r  =  b  ->  (
( ( F `  r )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q 
q } >. )  <P 
<. { p  |  p 
<Q  s } ,  {
q  |  s  <Q 
q } >.  <->  ( ( F `  b )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q  q } >. )  <P  <. { p  |  p  <Q  s } ,  { q  |  s  <Q  q } >. ) )
5150rspcev 2816 . . . . . . . 8  |-  ( ( b  e.  N.  /\  ( ( F `  b )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q 
q } >. )  <P 
<. { p  |  p 
<Q  s } ,  {
q  |  s  <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  s } ,  {
q  |  s  <Q 
q } >. )
5238, 39, 51syl2anc 409 . . . . . . 7  |-  ( ( ( ( ( ph  /\  t  e.  ( 2nd `  L ) )  /\  ( b  e.  N.  /\  ( ( F `  b )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q 
q } >. )  <P 
<. { p  |  p 
<Q  t } ,  {
q  |  t  <Q 
q } >. )
)  /\  s  e.  Q. )  /\  (
( ( F `  b )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q 
q } >. )  <P 
<. { p  |  p 
<Q  s } ,  {
q  |  s  <Q 
q } >.  /\  s  <Q  t ) )  ->  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  s } ,  {
q  |  s  <Q 
q } >. )
531caucvgprprlemelu 7606 . . . . . . 7  |-  ( s  e.  ( 2nd `  L
)  <->  ( s  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  s } ,  { q  |  s 
<Q  q } >. )
)
5436, 52, 53sylanbrc 414 . . . . . 6  |-  ( ( ( ( ( ph  /\  t  e.  ( 2nd `  L ) )  /\  ( b  e.  N.  /\  ( ( F `  b )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q 
q } >. )  <P 
<. { p  |  p 
<Q  t } ,  {
q  |  t  <Q 
q } >. )
)  /\  s  e.  Q. )  /\  (
( ( F `  b )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q 
q } >. )  <P 
<. { p  |  p 
<Q  s } ,  {
q  |  s  <Q 
q } >.  /\  s  <Q  t ) )  -> 
s  e.  ( 2nd `  L ) )
5535, 54jca 304 . . . . 5  |-  ( ( ( ( ( ph  /\  t  e.  ( 2nd `  L ) )  /\  ( b  e.  N.  /\  ( ( F `  b )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q 
q } >. )  <P 
<. { p  |  p 
<Q  t } ,  {
q  |  t  <Q 
q } >. )
)  /\  s  e.  Q. )  /\  (
( ( F `  b )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q 
q } >. )  <P 
<. { p  |  p 
<Q  s } ,  {
q  |  s  <Q 
q } >.  /\  s  <Q  t ) )  -> 
( s  <Q  t  /\  s  e.  ( 2nd `  L ) ) )
5655ex 114 . . . 4  |-  ( ( ( ( ph  /\  t  e.  ( 2nd `  L ) )  /\  ( b  e.  N.  /\  ( ( F `  b )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q 
q } >. )  <P 
<. { p  |  p 
<Q  t } ,  {
q  |  t  <Q 
q } >. )
)  /\  s  e.  Q. )  ->  ( ( ( ( F `  b )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q 
q } >. )  <P 
<. { p  |  p 
<Q  s } ,  {
q  |  s  <Q 
q } >.  /\  s  <Q  t )  ->  (
s  <Q  t  /\  s  e.  ( 2nd `  L
) ) ) )
5756reximdva 2559 . . 3  |-  ( ( ( ph  /\  t  e.  ( 2nd `  L
) )  /\  (
b  e.  N.  /\  ( ( F `  b )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q 
q } >. )  <P 
<. { p  |  p 
<Q  t } ,  {
q  |  t  <Q 
q } >. )
)  ->  ( E. s  e.  Q.  (
( ( F `  b )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q 
q } >. )  <P 
<. { p  |  p 
<Q  s } ,  {
q  |  s  <Q 
q } >.  /\  s  <Q  t )  ->  E. s  e.  Q.  ( s  <Q 
t  /\  s  e.  ( 2nd `  L ) ) ) )
5834, 57mpd 13 . 2  |-  ( ( ( ph  /\  t  e.  ( 2nd `  L
) )  /\  (
b  e.  N.  /\  ( ( F `  b )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q 
q } >. )  <P 
<. { p  |  p 
<Q  t } ,  {
q  |  t  <Q 
q } >. )
)  ->  E. s  e.  Q.  ( s  <Q 
t  /\  s  e.  ( 2nd `  L ) ) )
594, 58rexlimddv 2579 1  |-  ( (
ph  /\  t  e.  ( 2nd `  L ) )  ->  E. s  e.  Q.  ( s  <Q 
t  /\  s  e.  ( 2nd `  L ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1335    e. wcel 2128   {cab 2143   A.wral 2435   E.wrex 2436   {crab 2439   <.cop 3563   class class class wbr 3965   -->wf 5166   ` cfv 5170  (class class class)co 5824   1stc1st 6086   2ndc2nd 6087   1oc1o 6356   [cec 6478   N.cnpi 7192    <N clti 7195    ~Q ceq 7199   Q.cnq 7200    +Q cplq 7202   *Qcrq 7204    <Q cltq 7205   P.cnp 7211    +P. cpp 7213    <P cltp 7215
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 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-coll 4079  ax-sep 4082  ax-nul 4090  ax-pow 4135  ax-pr 4169  ax-un 4393  ax-setind 4496  ax-iinf 4547
This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-ral 2440  df-rex 2441  df-reu 2442  df-rab 2444  df-v 2714  df-sbc 2938  df-csb 3032  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3395  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-int 3808  df-iun 3851  df-br 3966  df-opab 4026  df-mpt 4027  df-tr 4063  df-eprel 4249  df-id 4253  df-po 4256  df-iso 4257  df-iord 4326  df-on 4328  df-suc 4331  df-iom 4550  df-xp 4592  df-rel 4593  df-cnv 4594  df-co 4595  df-dm 4596  df-rn 4597  df-res 4598  df-ima 4599  df-iota 5135  df-fun 5172  df-fn 5173  df-f 5174  df-f1 5175  df-fo 5176  df-f1o 5177  df-fv 5178  df-ov 5827  df-oprab 5828  df-mpo 5829  df-1st 6088  df-2nd 6089  df-recs 6252  df-irdg 6317  df-1o 6363  df-2o 6364  df-oadd 6367  df-omul 6368  df-er 6480  df-ec 6482  df-qs 6486  df-ni 7224  df-pli 7225  df-mi 7226  df-lti 7227  df-plpq 7264  df-mpq 7265  df-enq 7267  df-nqqs 7268  df-plqqs 7269  df-mqqs 7270  df-1nqqs 7271  df-rq 7272  df-ltnqqs 7273  df-enq0 7344  df-nq0 7345  df-0nq0 7346  df-plq0 7347  df-mq0 7348  df-inp 7386  df-iplp 7388  df-iltp 7390
This theorem is referenced by:  caucvgprprlemrnd  7621
  Copyright terms: Public domain W3C validator