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Theorem caucvgprprlemnkltj 6844
Description: Lemma for caucvgprpr 6867. Part of disjointness. (Contributed by Jim Kingdon, 12-Feb-2021.)
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 } >. )
) ) )
caucvgprprlemnkj.k  |-  ( ph  ->  K  e.  N. )
caucvgprprlemnkj.j  |-  ( ph  ->  J  e.  N. )
caucvgprprlemnkj.s  |-  ( ph  ->  S  e.  Q. )
Assertion
Ref Expression
caucvgprprlemnkltj  |-  ( (
ph  /\  K  <N  J )  ->  -.  ( <. { p  |  p 
<Q  ( S  +Q  ( *Q `  [ <. K ,  1o >. ]  ~Q  )
) } ,  {
q  |  ( S  +Q  ( *Q `  [ <. K ,  1o >. ]  ~Q  ) ) 
<Q  q } >.  <P  ( F `  K )  /\  ( ( F `  J )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. J ,  1o >. ]  ~Q  )  <Q 
q } >. )  <P 
<. { p  |  p 
<Q  S } ,  {
q  |  S  <Q  q } >. ) )
Distinct variable groups:    k, F, n    J, p, q    K, p, q    K, l, p    u, K, q    S, p, q   
k, l, n    u, k, n
Allowed substitution hints:    ph( u, k, n, q, p, l)    S( u, k, n, l)    F( u, q, p, l)    J( u, k, n, l)    K( k, n)

Proof of Theorem caucvgprprlemnkltj
Dummy variables  f  g  h are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ltsopr 6751 . . . 4  |-  <P  Or  P.
2 ltrelpr 6660 . . . 4  |-  <P  C_  ( P.  X.  P. )
31, 2son2lpi 4748 . . 3  |-  -.  ( <. { p  |  p 
<Q  S } ,  {
q  |  S  <Q  q } >.  <P  ( F `
 J )  /\  ( F `  J ) 
<P  <. { p  |  p  <Q  S } ,  { q  |  S  <Q  q } >. )
4 simprl 491 . . . . . . 7  |-  ( ( ( ph  /\  K  <N  J )  /\  (
( <. { p  |  p  <Q  S } ,  { q  |  S  <Q  q } >.  +P.  <. { p  |  p  <Q  ( *Q `  [ <. K ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. K ,  1o >. ]  ~Q  )  <Q 
q } >. )  <P  ( F `  K
)  /\  ( ( F `  J )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. J ,  1o >. ]  ~Q  )  <Q  q } >. )  <P  <. { p  |  p  <Q  S } ,  { q  |  S  <Q  q } >. )
)  ->  ( <. { p  |  p  <Q  S } ,  { q  |  S  <Q  q } >.  +P.  <. { p  |  p  <Q  ( *Q
`  [ <. K ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. K ,  1o >. ]  ~Q  )  <Q 
q } >. )  <P  ( F `  K
) )
5 caucvgprpr.f . . . . . . . . . 10  |-  ( ph  ->  F : N. --> P. )
6 caucvgprpr.cau . . . . . . . . . 10  |-  ( 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 } >. )
) ) )
75, 6caucvgprprlemval 6843 . . . . . . . . 9  |-  ( (
ph  /\  K  <N  J )  ->  ( ( F `  K )  <P  ( ( F `  J )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. K ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. K ,  1o >. ]  ~Q  )  <Q 
q } >. )  /\  ( F `  J
)  <P  ( ( F `
 K )  +P. 
<. { p  |  p 
<Q  ( *Q `  [ <. K ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. K ,  1o >. ]  ~Q  )  <Q  q } >. ) ) )
87simpld 109 . . . . . . . 8  |-  ( (
ph  /\  K  <N  J )  ->  ( F `  K )  <P  (
( F `  J
)  +P.  <. { p  |  p  <Q  ( *Q
`  [ <. K ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. K ,  1o >. ]  ~Q  )  <Q 
q } >. )
)
98adantr 265 . . . . . . 7  |-  ( ( ( ph  /\  K  <N  J )  /\  (
( <. { p  |  p  <Q  S } ,  { q  |  S  <Q  q } >.  +P.  <. { p  |  p  <Q  ( *Q `  [ <. K ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. K ,  1o >. ]  ~Q  )  <Q 
q } >. )  <P  ( F `  K
)  /\  ( ( F `  J )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. J ,  1o >. ]  ~Q  )  <Q  q } >. )  <P  <. { p  |  p  <Q  S } ,  { q  |  S  <Q  q } >. )
)  ->  ( F `  K )  <P  (
( F `  J
)  +P.  <. { p  |  p  <Q  ( *Q
`  [ <. K ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. K ,  1o >. ]  ~Q  )  <Q 
q } >. )
)
101, 2sotri 4747 . . . . . . 7  |-  ( ( ( <. { p  |  p  <Q  S } ,  { q  |  S  <Q  q } >.  +P.  <. { p  |  p  <Q  ( *Q `  [ <. K ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. K ,  1o >. ]  ~Q  )  <Q 
q } >. )  <P  ( F `  K
)  /\  ( F `  K )  <P  (
( F `  J
)  +P.  <. { p  |  p  <Q  ( *Q
`  [ <. K ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. K ,  1o >. ]  ~Q  )  <Q 
q } >. )
)  ->  ( <. { p  |  p  <Q  S } ,  { q  |  S  <Q  q } >.  +P.  <. { p  |  p  <Q  ( *Q
`  [ <. K ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. K ,  1o >. ]  ~Q  )  <Q 
q } >. )  <P  ( ( F `  J )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. K ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. K ,  1o >. ]  ~Q  )  <Q 
q } >. )
)
114, 9, 10syl2anc 397 . . . . . 6  |-  ( ( ( ph  /\  K  <N  J )  /\  (
( <. { p  |  p  <Q  S } ,  { q  |  S  <Q  q } >.  +P.  <. { p  |  p  <Q  ( *Q `  [ <. K ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. K ,  1o >. ]  ~Q  )  <Q 
q } >. )  <P  ( F `  K
)  /\  ( ( F `  J )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. J ,  1o >. ]  ~Q  )  <Q  q } >. )  <P  <. { p  |  p  <Q  S } ,  { q  |  S  <Q  q } >. )
)  ->  ( <. { p  |  p  <Q  S } ,  { q  |  S  <Q  q } >.  +P.  <. { p  |  p  <Q  ( *Q
`  [ <. K ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. K ,  1o >. ]  ~Q  )  <Q 
q } >. )  <P  ( ( F `  J )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. K ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. K ,  1o >. ]  ~Q  )  <Q 
q } >. )
)
12 ltaprg 6774 . . . . . . . 8  |-  ( ( f  e.  P.  /\  g  e.  P.  /\  h  e.  P. )  ->  (
f  <P  g  <->  ( h  +P.  f )  <P  (
h  +P.  g )
) )
1312adantl 266 . . . . . . 7  |-  ( ( ( ( ph  /\  K  <N  J )  /\  ( ( <. { p  |  p  <Q  S } ,  { q  |  S  <Q  q } >.  +P.  <. { p  |  p  <Q  ( *Q `  [ <. K ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. K ,  1o >. ]  ~Q  )  <Q 
q } >. )  <P  ( F `  K
)  /\  ( ( F `  J )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. J ,  1o >. ]  ~Q  )  <Q  q } >. )  <P  <. { p  |  p  <Q  S } ,  { q  |  S  <Q  q } >. )
)  /\  ( f  e.  P.  /\  g  e. 
P.  /\  h  e.  P. ) )  ->  (
f  <P  g  <->  ( h  +P.  f )  <P  (
h  +P.  g )
) )
14 caucvgprprlemnkj.s . . . . . . . . 9  |-  ( ph  ->  S  e.  Q. )
1514ad2antrr 465 . . . . . . . 8  |-  ( ( ( ph  /\  K  <N  J )  /\  (
( <. { p  |  p  <Q  S } ,  { q  |  S  <Q  q } >.  +P.  <. { p  |  p  <Q  ( *Q `  [ <. K ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. K ,  1o >. ]  ~Q  )  <Q 
q } >. )  <P  ( F `  K
)  /\  ( ( F `  J )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. J ,  1o >. ]  ~Q  )  <Q  q } >. )  <P  <. { p  |  p  <Q  S } ,  { q  |  S  <Q  q } >. )
)  ->  S  e.  Q. )
16 nqprlu 6702 . . . . . . . 8  |-  ( S  e.  Q.  ->  <. { p  |  p  <Q  S } ,  { q  |  S  <Q  q } >.  e.  P. )
1715, 16syl 14 . . . . . . 7  |-  ( ( ( ph  /\  K  <N  J )  /\  (
( <. { p  |  p  <Q  S } ,  { q  |  S  <Q  q } >.  +P.  <. { p  |  p  <Q  ( *Q `  [ <. K ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. K ,  1o >. ]  ~Q  )  <Q 
q } >. )  <P  ( F `  K
)  /\  ( ( F `  J )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. J ,  1o >. ]  ~Q  )  <Q  q } >. )  <P  <. { p  |  p  <Q  S } ,  { q  |  S  <Q  q } >. )
)  ->  <. { p  |  p  <Q  S } ,  { q  |  S  <Q  q } >.  e.  P. )
18 caucvgprprlemnkj.j . . . . . . . . 9  |-  ( ph  ->  J  e.  N. )
195, 18ffvelrnd 5330 . . . . . . . 8  |-  ( ph  ->  ( F `  J
)  e.  P. )
2019ad2antrr 465 . . . . . . 7  |-  ( ( ( ph  /\  K  <N  J )  /\  (
( <. { p  |  p  <Q  S } ,  { q  |  S  <Q  q } >.  +P.  <. { p  |  p  <Q  ( *Q `  [ <. K ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. K ,  1o >. ]  ~Q  )  <Q 
q } >. )  <P  ( F `  K
)  /\  ( ( F `  J )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. J ,  1o >. ]  ~Q  )  <Q  q } >. )  <P  <. { p  |  p  <Q  S } ,  { q  |  S  <Q  q } >. )
)  ->  ( F `  J )  e.  P. )
21 caucvgprprlemnkj.k . . . . . . . . 9  |-  ( ph  ->  K  e.  N. )
22 recnnpr 6703 . . . . . . . . 9  |-  ( K  e.  N.  ->  <. { p  |  p  <Q  ( *Q
`  [ <. K ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. K ,  1o >. ]  ~Q  )  <Q 
q } >.  e.  P. )
2321, 22syl 14 . . . . . . . 8  |-  ( ph  -> 
<. { p  |  p 
<Q  ( *Q `  [ <. K ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. K ,  1o >. ]  ~Q  )  <Q  q } >.  e. 
P. )
2423ad2antrr 465 . . . . . . 7  |-  ( ( ( ph  /\  K  <N  J )  /\  (
( <. { p  |  p  <Q  S } ,  { q  |  S  <Q  q } >.  +P.  <. { p  |  p  <Q  ( *Q `  [ <. K ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. K ,  1o >. ]  ~Q  )  <Q 
q } >. )  <P  ( F `  K
)  /\  ( ( F `  J )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. J ,  1o >. ]  ~Q  )  <Q  q } >. )  <P  <. { p  |  p  <Q  S } ,  { q  |  S  <Q  q } >. )
)  ->  <. { p  |  p  <Q  ( *Q
`  [ <. K ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. K ,  1o >. ]  ~Q  )  <Q 
q } >.  e.  P. )
25 addcomprg 6733 . . . . . . . 8  |-  ( ( f  e.  P.  /\  g  e.  P. )  ->  ( f  +P.  g
)  =  ( g  +P.  f ) )
2625adantl 266 . . . . . . 7  |-  ( ( ( ( ph  /\  K  <N  J )  /\  ( ( <. { p  |  p  <Q  S } ,  { q  |  S  <Q  q } >.  +P.  <. { p  |  p  <Q  ( *Q `  [ <. K ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. K ,  1o >. ]  ~Q  )  <Q 
q } >. )  <P  ( F `  K
)  /\  ( ( F `  J )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. J ,  1o >. ]  ~Q  )  <Q  q } >. )  <P  <. { p  |  p  <Q  S } ,  { q  |  S  <Q  q } >. )
)  /\  ( f  e.  P.  /\  g  e. 
P. ) )  -> 
( f  +P.  g
)  =  ( g  +P.  f ) )
2713, 17, 20, 24, 26caovord2d 5697 . . . . . 6  |-  ( ( ( ph  /\  K  <N  J )  /\  (
( <. { p  |  p  <Q  S } ,  { q  |  S  <Q  q } >.  +P.  <. { p  |  p  <Q  ( *Q `  [ <. K ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. K ,  1o >. ]  ~Q  )  <Q 
q } >. )  <P  ( F `  K
)  /\  ( ( F `  J )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. J ,  1o >. ]  ~Q  )  <Q  q } >. )  <P  <. { p  |  p  <Q  S } ,  { q  |  S  <Q  q } >. )
)  ->  ( <. { p  |  p  <Q  S } ,  { q  |  S  <Q  q } >.  <P  ( F `  J )  <->  ( <. { p  |  p  <Q  S } ,  { q  |  S  <Q  q } >.  +P.  <. { p  |  p  <Q  ( *Q
`  [ <. K ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. K ,  1o >. ]  ~Q  )  <Q 
q } >. )  <P  ( ( F `  J )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. K ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. K ,  1o >. ]  ~Q  )  <Q 
q } >. )
) )
2811, 27mpbird 160 . . . . 5  |-  ( ( ( ph  /\  K  <N  J )  /\  (
( <. { p  |  p  <Q  S } ,  { q  |  S  <Q  q } >.  +P.  <. { p  |  p  <Q  ( *Q `  [ <. K ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. K ,  1o >. ]  ~Q  )  <Q 
q } >. )  <P  ( F `  K
)  /\  ( ( F `  J )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. J ,  1o >. ]  ~Q  )  <Q  q } >. )  <P  <. { p  |  p  <Q  S } ,  { q  |  S  <Q  q } >. )
)  ->  <. { p  |  p  <Q  S } ,  { q  |  S  <Q  q } >.  <P  ( F `  J )
)
29 recnnpr 6703 . . . . . . . . 9  |-  ( J  e.  N.  ->  <. { p  |  p  <Q  ( *Q
`  [ <. J ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. J ,  1o >. ]  ~Q  )  <Q 
q } >.  e.  P. )
3018, 29syl 14 . . . . . . . 8  |-  ( ph  -> 
<. { p  |  p 
<Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. J ,  1o >. ]  ~Q  )  <Q  q } >.  e. 
P. )
3130ad2antrr 465 . . . . . . 7  |-  ( ( ( ph  /\  K  <N  J )  /\  (
( <. { p  |  p  <Q  S } ,  { q  |  S  <Q  q } >.  +P.  <. { p  |  p  <Q  ( *Q `  [ <. K ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. K ,  1o >. ]  ~Q  )  <Q 
q } >. )  <P  ( F `  K
)  /\  ( ( F `  J )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. J ,  1o >. ]  ~Q  )  <Q  q } >. )  <P  <. { p  |  p  <Q  S } ,  { q  |  S  <Q  q } >. )
)  ->  <. { p  |  p  <Q  ( *Q
`  [ <. J ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. J ,  1o >. ]  ~Q  )  <Q 
q } >.  e.  P. )
32 ltaddpr 6752 . . . . . . 7  |-  ( ( ( F `  J
)  e.  P.  /\  <. { p  |  p  <Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. J ,  1o >. ]  ~Q  )  <Q  q } >.  e. 
P. )  ->  ( F `  J )  <P  ( ( F `  J )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. J ,  1o >. ]  ~Q  )  <Q 
q } >. )
)
3320, 31, 32syl2anc 397 . . . . . 6  |-  ( ( ( ph  /\  K  <N  J )  /\  (
( <. { p  |  p  <Q  S } ,  { q  |  S  <Q  q } >.  +P.  <. { p  |  p  <Q  ( *Q `  [ <. K ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. K ,  1o >. ]  ~Q  )  <Q 
q } >. )  <P  ( F `  K
)  /\  ( ( F `  J )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. J ,  1o >. ]  ~Q  )  <Q  q } >. )  <P  <. { p  |  p  <Q  S } ,  { q  |  S  <Q  q } >. )
)  ->  ( F `  J )  <P  (
( F `  J
)  +P.  <. { p  |  p  <Q  ( *Q
`  [ <. J ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. J ,  1o >. ]  ~Q  )  <Q 
q } >. )
)
34 simprr 492 . . . . . 6  |-  ( ( ( ph  /\  K  <N  J )  /\  (
( <. { p  |  p  <Q  S } ,  { q  |  S  <Q  q } >.  +P.  <. { p  |  p  <Q  ( *Q `  [ <. K ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. K ,  1o >. ]  ~Q  )  <Q 
q } >. )  <P  ( F `  K
)  /\  ( ( F `  J )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. J ,  1o >. ]  ~Q  )  <Q  q } >. )  <P  <. { p  |  p  <Q  S } ,  { q  |  S  <Q  q } >. )
)  ->  ( ( F `  J )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. J ,  1o >. ]  ~Q  )  <Q  q } >. )  <P  <. { p  |  p  <Q  S } ,  { q  |  S  <Q  q } >. )
351, 2sotri 4747 . . . . . 6  |-  ( ( ( F `  J
)  <P  ( ( F `
 J )  +P. 
<. { p  |  p 
<Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. J ,  1o >. ]  ~Q  )  <Q  q } >. )  /\  ( ( F `
 J )  +P. 
<. { p  |  p 
<Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. J ,  1o >. ]  ~Q  )  <Q  q } >. ) 
<P  <. { p  |  p  <Q  S } ,  { q  |  S  <Q  q } >. )  ->  ( F `  J
)  <P  <. { p  |  p  <Q  S } ,  { q  |  S  <Q  q } >. )
3633, 34, 35syl2anc 397 . . . . 5  |-  ( ( ( ph  /\  K  <N  J )  /\  (
( <. { p  |  p  <Q  S } ,  { q  |  S  <Q  q } >.  +P.  <. { p  |  p  <Q  ( *Q `  [ <. K ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. K ,  1o >. ]  ~Q  )  <Q 
q } >. )  <P  ( F `  K
)  /\  ( ( F `  J )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. J ,  1o >. ]  ~Q  )  <Q  q } >. )  <P  <. { p  |  p  <Q  S } ,  { q  |  S  <Q  q } >. )
)  ->  ( F `  J )  <P  <. { p  |  p  <Q  S } ,  { q  |  S  <Q  q } >. )
3728, 36jca 294 . . . 4  |-  ( ( ( ph  /\  K  <N  J )  /\  (
( <. { p  |  p  <Q  S } ,  { q  |  S  <Q  q } >.  +P.  <. { p  |  p  <Q  ( *Q `  [ <. K ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. K ,  1o >. ]  ~Q  )  <Q 
q } >. )  <P  ( F `  K
)  /\  ( ( F `  J )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. J ,  1o >. ]  ~Q  )  <Q  q } >. )  <P  <. { p  |  p  <Q  S } ,  { q  |  S  <Q  q } >. )
)  ->  ( <. { p  |  p  <Q  S } ,  { q  |  S  <Q  q } >.  <P  ( F `  J )  /\  ( F `  J )  <P 
<. { p  |  p 
<Q  S } ,  {
q  |  S  <Q  q } >. ) )
3837ex 112 . . 3  |-  ( (
ph  /\  K  <N  J )  ->  ( (
( <. { p  |  p  <Q  S } ,  { q  |  S  <Q  q } >.  +P.  <. { p  |  p  <Q  ( *Q `  [ <. K ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. K ,  1o >. ]  ~Q  )  <Q 
q } >. )  <P  ( F `  K
)  /\  ( ( F `  J )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. J ,  1o >. ]  ~Q  )  <Q  q } >. )  <P  <. { p  |  p  <Q  S } ,  { q  |  S  <Q  q } >. )  ->  ( <. { p  |  p  <Q  S } ,  { q  |  S  <Q  q } >.  <P  ( F `  J )  /\  ( F `  J
)  <P  <. { p  |  p  <Q  S } ,  { q  |  S  <Q  q } >. )
) )
393, 38mtoi 600 . 2  |-  ( (
ph  /\  K  <N  J )  ->  -.  (
( <. { p  |  p  <Q  S } ,  { q  |  S  <Q  q } >.  +P.  <. { p  |  p  <Q  ( *Q `  [ <. K ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. K ,  1o >. ]  ~Q  )  <Q 
q } >. )  <P  ( F `  K
)  /\  ( ( F `  J )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. J ,  1o >. ]  ~Q  )  <Q  q } >. )  <P  <. { p  |  p  <Q  S } ,  { q  |  S  <Q  q } >. )
)
4014adantr 265 . . . . 5  |-  ( (
ph  /\  K  <N  J )  ->  S  e.  Q. )
41 nnnq 6577 . . . . . . 7  |-  ( K  e.  N.  ->  [ <. K ,  1o >. ]  ~Q  e.  Q. )
42 recclnq 6547 . . . . . . 7  |-  ( [
<. K ,  1o >. ]  ~Q  e.  Q.  ->  ( *Q `  [ <. K ,  1o >. ]  ~Q  )  e.  Q. )
4321, 41, 423syl 17 . . . . . 6  |-  ( ph  ->  ( *Q `  [ <. K ,  1o >. ]  ~Q  )  e.  Q. )
4443adantr 265 . . . . 5  |-  ( (
ph  /\  K  <N  J )  ->  ( *Q `  [ <. K ,  1o >. ]  ~Q  )  e. 
Q. )
45 addnqpr 6716 . . . . 5  |-  ( ( S  e.  Q.  /\  ( *Q `  [ <. K ,  1o >. ]  ~Q  )  e.  Q. )  -> 
<. { p  |  p 
<Q  ( S  +Q  ( *Q `  [ <. K ,  1o >. ]  ~Q  )
) } ,  {
q  |  ( S  +Q  ( *Q `  [ <. K ,  1o >. ]  ~Q  ) ) 
<Q  q } >.  =  (
<. { p  |  p 
<Q  S } ,  {
q  |  S  <Q  q } >.  +P.  <. { p  |  p  <Q  ( *Q
`  [ <. K ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. K ,  1o >. ]  ~Q  )  <Q 
q } >. )
)
4640, 44, 45syl2anc 397 . . . 4  |-  ( (
ph  /\  K  <N  J )  ->  <. { p  |  p  <Q  ( S  +Q  ( *Q `  [ <. K ,  1o >. ]  ~Q  ) ) } ,  { q  |  ( S  +Q  ( *Q `  [ <. K ,  1o >. ]  ~Q  ) )  <Q  q } >.  =  ( <. { p  |  p  <Q  S } ,  {
q  |  S  <Q  q } >.  +P.  <. { p  |  p  <Q  ( *Q
`  [ <. K ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. K ,  1o >. ]  ~Q  )  <Q 
q } >. )
)
4746breq1d 3801 . . 3  |-  ( (
ph  /\  K  <N  J )  ->  ( <. { p  |  p  <Q  ( S  +Q  ( *Q
`  [ <. K ,  1o >. ]  ~Q  )
) } ,  {
q  |  ( S  +Q  ( *Q `  [ <. K ,  1o >. ]  ~Q  ) ) 
<Q  q } >.  <P  ( F `  K )  <->  (
<. { p  |  p 
<Q  S } ,  {
q  |  S  <Q  q } >.  +P.  <. { p  |  p  <Q  ( *Q
`  [ <. K ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. K ,  1o >. ]  ~Q  )  <Q 
q } >. )  <P  ( F `  K
) ) )
4847anbi1d 446 . 2  |-  ( (
ph  /\  K  <N  J )  ->  ( ( <. { p  |  p 
<Q  ( S  +Q  ( *Q `  [ <. K ,  1o >. ]  ~Q  )
) } ,  {
q  |  ( S  +Q  ( *Q `  [ <. K ,  1o >. ]  ~Q  ) ) 
<Q  q } >.  <P  ( F `  K )  /\  ( ( F `  J )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. J ,  1o >. ]  ~Q  )  <Q 
q } >. )  <P 
<. { p  |  p 
<Q  S } ,  {
q  |  S  <Q  q } >. )  <->  ( ( <. { p  |  p 
<Q  S } ,  {
q  |  S  <Q  q } >.  +P.  <. { p  |  p  <Q  ( *Q
`  [ <. K ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. K ,  1o >. ]  ~Q  )  <Q 
q } >. )  <P  ( F `  K
)  /\  ( ( F `  J )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. J ,  1o >. ]  ~Q  )  <Q  q } >. )  <P  <. { p  |  p  <Q  S } ,  { q  |  S  <Q  q } >. )
) )
4939, 48mtbird 608 1  |-  ( (
ph  /\  K  <N  J )  ->  -.  ( <. { p  |  p 
<Q  ( S  +Q  ( *Q `  [ <. K ,  1o >. ]  ~Q  )
) } ,  {
q  |  ( S  +Q  ( *Q `  [ <. K ,  1o >. ]  ~Q  ) ) 
<Q  q } >.  <P  ( F `  K )  /\  ( ( F `  J )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. J ,  1o >. ]  ~Q  )  <Q 
q } >. )  <P 
<. { p  |  p 
<Q  S } ,  {
q  |  S  <Q  q } >. ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 101    <-> wb 102    /\ w3a 896    = wceq 1259    e. wcel 1409   {cab 2042   A.wral 2323   <.cop 3405   class class class wbr 3791   -->wf 4925   ` cfv 4929  (class class class)co 5539   1oc1o 6024   [cec 6134   N.cnpi 6427    <N clti 6430    ~Q ceq 6434   Q.cnq 6435    +Q cplq 6437   *Qcrq 6439    <Q cltq 6440   P.cnp 6446    +P. cpp 6448    <P cltp 6450
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 3899  ax-sep 3902  ax-nul 3910  ax-pow 3954  ax-pr 3971  ax-un 4197  ax-setind 4289  ax-iinf 4338
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 2787  df-csb 2880  df-dif 2947  df-un 2949  df-in 2951  df-ss 2958  df-nul 3252  df-pw 3388  df-sn 3408  df-pr 3409  df-op 3411  df-uni 3608  df-int 3643  df-iun 3686  df-br 3792  df-opab 3846  df-mpt 3847  df-tr 3882  df-eprel 4053  df-id 4057  df-po 4060  df-iso 4061  df-iord 4130  df-on 4132  df-suc 4135  df-iom 4341  df-xp 4378  df-rel 4379  df-cnv 4380  df-co 4381  df-dm 4382  df-rn 4383  df-res 4384  df-ima 4385  df-iota 4894  df-fun 4931  df-fn 4932  df-f 4933  df-f1 4934  df-fo 4935  df-f1o 4936  df-fv 4937  df-ov 5542  df-oprab 5543  df-mpt2 5544  df-1st 5794  df-2nd 5795  df-recs 5950  df-irdg 5987  df-1o 6031  df-2o 6032  df-oadd 6035  df-omul 6036  df-er 6136  df-ec 6138  df-qs 6142  df-ni 6459  df-pli 6460  df-mi 6461  df-lti 6462  df-plpq 6499  df-mpq 6500  df-enq 6502  df-nqqs 6503  df-plqqs 6504  df-mqqs 6505  df-1nqqs 6506  df-rq 6507  df-ltnqqs 6508  df-enq0 6579  df-nq0 6580  df-0nq0 6581  df-plq0 6582  df-mq0 6583  df-inp 6621  df-iplp 6623  df-iltp 6625
This theorem is referenced by:  caucvgprprlemnkj  6847
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