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Theorem caucvgprprlemnkltj 7461
Description: Lemma for caucvgprpr 7484. 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 7368 . . . 4  |-  <P  Or  P.
2 ltrelpr 7277 . . . 4  |-  <P  C_  ( P.  X.  P. )
31, 2son2lpi 4903 . . 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 503 . . . . . . 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 7460 . . . . . . . . 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 111 . . . . . . . 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 272 . . . . . . 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 4902 . . . . . . 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 406 . . . . . 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 7391 . . . . . . . 8  |-  ( ( f  e.  P.  /\  g  e.  P.  /\  h  e.  P. )  ->  (
f  <P  g  <->  ( h  +P.  f )  <P  (
h  +P.  g )
) )
1312adantl 273 . . . . . . 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 477 . . . . . . . 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 7319 . . . . . . . 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 5522 . . . . . . . 8  |-  ( ph  ->  ( F `  J
)  e.  P. )
2019ad2antrr 477 . . . . . . 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 7320 . . . . . . . . 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 477 . . . . . . 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 7350 . . . . . . . 8  |-  ( ( f  e.  P.  /\  g  e.  P. )  ->  ( f  +P.  g
)  =  ( g  +P.  f ) )
2625adantl 273 . . . . . . 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 5906 . . . . . 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 166 . . . . 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 7320 . . . . . . . . 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 477 . . . . . . 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 7369 . . . . . . 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 406 . . . . . 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 504 . . . . . 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 4902 . . . . . 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 406 . . . . 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 302 . . . 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 114 . . 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 636 . 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 272 . . . . 5  |-  ( (
ph  /\  K  <N  J )  ->  S  e.  Q. )
41 nnnq 7194 . . . . . . 7  |-  ( K  e.  N.  ->  [ <. K ,  1o >. ]  ~Q  e.  Q. )
42 recclnq 7164 . . . . . . 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 272 . . . . 5  |-  ( (
ph  /\  K  <N  J )  ->  ( *Q `  [ <. K ,  1o >. ]  ~Q  )  e. 
Q. )
45 addnqpr 7333 . . . . 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 406 . . . 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 3907 . . 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 458 . 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 645 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 103    <-> wb 104    /\ w3a 945    = wceq 1314    e. wcel 1463   {cab 2101   A.wral 2391   <.cop 3498   class class class wbr 3897   -->wf 5087   ` cfv 5091  (class class class)co 5740   1oc1o 6272   [cec 6393   N.cnpi 7044    <N clti 7047    ~Q ceq 7051   Q.cnq 7052    +Q cplq 7054   *Qcrq 7056    <Q cltq 7057   P.cnp 7063    +P. cpp 7065    <P cltp 7067
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 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-coll 4011  ax-sep 4014  ax-nul 4022  ax-pow 4066  ax-pr 4099  ax-un 4323  ax-setind 4420  ax-iinf 4470
This theorem depends on definitions:  df-bi 116  df-dc 803  df-3or 946  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ne 2284  df-ral 2396  df-rex 2397  df-reu 2398  df-rab 2400  df-v 2660  df-sbc 2881  df-csb 2974  df-dif 3041  df-un 3043  df-in 3045  df-ss 3052  df-nul 3332  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-int 3740  df-iun 3783  df-br 3898  df-opab 3958  df-mpt 3959  df-tr 3995  df-eprel 4179  df-id 4183  df-po 4186  df-iso 4187  df-iord 4256  df-on 4258  df-suc 4261  df-iom 4473  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-rn 4518  df-res 4519  df-ima 4520  df-iota 5056  df-fun 5093  df-fn 5094  df-f 5095  df-f1 5096  df-fo 5097  df-f1o 5098  df-fv 5099  df-ov 5743  df-oprab 5744  df-mpo 5745  df-1st 6004  df-2nd 6005  df-recs 6168  df-irdg 6233  df-1o 6279  df-2o 6280  df-oadd 6283  df-omul 6284  df-er 6395  df-ec 6397  df-qs 6401  df-ni 7076  df-pli 7077  df-mi 7078  df-lti 7079  df-plpq 7116  df-mpq 7117  df-enq 7119  df-nqqs 7120  df-plqqs 7121  df-mqqs 7122  df-1nqqs 7123  df-rq 7124  df-ltnqqs 7125  df-enq0 7196  df-nq0 7197  df-0nq0 7198  df-plq0 7199  df-mq0 7200  df-inp 7238  df-iplp 7240  df-iltp 7242
This theorem is referenced by:  caucvgprprlemnkj  7464
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