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Theorem caucvgprprlemopu 7475
Description: Lemma for caucvgprpr 7488. 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 7462 . . . 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 506 . . . . 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 5523 . . . . . . . 8  |-  ( (
ph  /\  b  e.  N. )  ->  ( F `
 b )  e. 
P. )
8 recnnpr 7324 . . . . . . . . 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 7313 . . . . . . . 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 408 . . . . . . 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 500 . . . . . 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 480 . . . . . . 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 7323 . . . . . . 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 7282 . . . . . 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 408 . . . . 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 7328 . . . . . . . 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 408 . . . . . . 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 2663 . . . . . . . . 9  |-  s  e. 
_V
25 breq1 3902 . . . . . . . . 9  |-  ( p  =  s  ->  (
p  <Q  t  <->  s  <Q  t ) )
26 ltnqex 7325 . . . . . . . . . 10  |-  { p  |  p  <Q  t }  e.  _V
27 gtnqex 7326 . . . . . . . . . 10  |-  { q  |  t  <Q  q }  e.  _V
2826, 27op1st 6012 . . . . . . . . 9  |-  ( 1st `  <. { p  |  p  <Q  t } ,  { q  |  t 
<Q  q } >. )  =  { p  |  p 
<Q  t }
2924, 25, 28elab2 2805 . . . . . . . 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 464 . . . . . 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 2511 . . . 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 506 . . . . . 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 504 . . . . . . 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 509 . . . . . . . . 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 505 . . . . . . . 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 5389 . . . . . . . . . . 11  |-  ( r  =  b  ->  ( F `  r )  =  ( F `  b ) )
41 opeq1 3675 . . . . . . . . . . . . . . . 16  |-  ( r  =  b  ->  <. r ,  1o >.  =  <. b ,  1o >. )
4241eceq1d 6433 . . . . . . . . . . . . . . 15  |-  ( r  =  b  ->  [ <. r ,  1o >. ]  ~Q  =  [ <. b ,  1o >. ]  ~Q  )
4342fveq2d 5393 . . . . . . . . . . . . . 14  |-  ( r  =  b  ->  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  =  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) )
4443breq2d 3911 . . . . . . . . . . . . 13  |-  ( r  =  b  ->  (
p  <Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <->  p  <Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) ) )
4544abbidv 2235 . . . . . . . . . . . 12  |-  ( r  =  b  ->  { p  |  p  <Q  ( *Q
`  [ <. r ,  1o >. ]  ~Q  ) }  =  { p  |  p  <Q  ( *Q
`  [ <. b ,  1o >. ]  ~Q  ) } )
4643breq1d 3909 . . . . . . . . . . . . 13  |-  ( r  =  b  ->  (
( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q  q  <->  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q  q ) )
4746abbidv 2235 . . . . . . . . . . . 12  |-  ( r  =  b  ->  { q  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q 
q }  =  {
q  |  ( *Q
`  [ <. b ,  1o >. ]  ~Q  )  <Q  q } )
4845, 47opeq12d 3683 . . . . . . . . . . 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 5760 . . . . . . . . . 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 3909 . . . . . . . . 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 2763 . . . . . . . 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 408 . . . . . . 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 7462 . . . . . . 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 413 . . . . . 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 2511 . . 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 2531 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 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-2o 6282  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-enq0 7200  df-nq0 7201  df-0nq0 7202  df-plq0 7203  df-mq0 7204  df-inp 7242  df-iplp 7244  df-iltp 7246
This theorem is referenced by:  caucvgprprlemrnd  7477
  Copyright terms: Public domain W3C validator