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Theorem caucvgprprlemnkeqj 6769
Description: Lemma for caucvgprpr 6791. 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
caucvgprprlemnkeqj  |-  ( (
ph  /\  K  =  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    S, p, q
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( u, k, n, l)

Proof of Theorem caucvgprprlemnkeqj
StepHypRef Expression
1 ltsopr 6675 . . . 4  |-  <P  Or  P.
2 ltrelpr 6584 . . . 4  |-  <P  C_  ( P.  X.  P. )
31, 2son2lpi 4708 . . 3  |-  -.  (
( F `  J
)  <P  <. { p  |  p  <Q  S } ,  { q  |  S  <Q  q } >.  /\  <. { p  |  p  <Q  S } ,  { q  |  S  <Q  q } >.  <P  ( F `  J ) )
4 caucvgprpr.f . . . . . . . . 9  |-  ( ph  ->  F : N. --> P. )
5 caucvgprprlemnkj.j . . . . . . . . 9  |-  ( ph  ->  J  e.  N. )
64, 5ffvelrnd 5290 . . . . . . . 8  |-  ( ph  ->  ( F `  J
)  e.  P. )
76ad2antrr 457 . . . . . . 7  |-  ( ( ( ph  /\  K  =  J )  /\  ( <. { p  |  p 
<Q  ( S  +Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  )
) } ,  {
q  |  ( S  +Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  ) ) 
<Q  q } >.  <P  ( F `  J )  /\  ( ( 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. )
85adantr 261 . . . . . . . . . . 11  |-  ( (
ph  /\  K  =  J )  ->  J  e.  N. )
9 nnnq 6501 . . . . . . . . . . 11  |-  ( J  e.  N.  ->  [ <. J ,  1o >. ]  ~Q  e.  Q. )
108, 9syl 14 . . . . . . . . . 10  |-  ( (
ph  /\  K  =  J )  ->  [ <. J ,  1o >. ]  ~Q  e.  Q. )
11 recclnq 6471 . . . . . . . . . 10  |-  ( [
<. J ,  1o >. ]  ~Q  e.  Q.  ->  ( *Q `  [ <. J ,  1o >. ]  ~Q  )  e.  Q. )
1210, 11syl 14 . . . . . . . . 9  |-  ( (
ph  /\  K  =  J )  ->  ( *Q `  [ <. J ,  1o >. ]  ~Q  )  e.  Q. )
13 nqprlu 6626 . . . . . . . . 9  |-  ( ( *Q `  [ <. J ,  1o >. ]  ~Q  )  e.  Q.  ->  <. { p  |  p  <Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. J ,  1o >. ]  ~Q  )  <Q  q } >.  e. 
P. )
1412, 13syl 14 . . . . . . . 8  |-  ( (
ph  /\  K  =  J )  ->  <. { p  |  p  <Q  ( *Q
`  [ <. J ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. J ,  1o >. ]  ~Q  )  <Q 
q } >.  e.  P. )
1514adantr 261 . . . . . . 7  |-  ( ( ( ph  /\  K  =  J )  /\  ( <. { p  |  p 
<Q  ( S  +Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  )
) } ,  {
q  |  ( S  +Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  ) ) 
<Q  q } >.  <P  ( F `  J )  /\  ( ( 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. )
16 ltaddpr 6676 . . . . . . 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 } >. )
)
177, 15, 16syl2anc 391 . . . . . 6  |-  ( ( ( ph  /\  K  =  J )  /\  ( <. { p  |  p 
<Q  ( S  +Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  )
) } ,  {
q  |  ( S  +Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  ) ) 
<Q  q } >.  <P  ( F `  J )  /\  ( ( 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 } >. ) )
18 simprr 484 . . . . . 6  |-  ( ( ( ph  /\  K  =  J )  /\  ( <. { p  |  p 
<Q  ( S  +Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  )
) } ,  {
q  |  ( S  +Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  ) ) 
<Q  q } >.  <P  ( F `  J )  /\  ( ( 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 } >. )
191, 2sotri 4707 . . . . . 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 } >. )
2017, 18, 19syl2anc 391 . . . . 5  |-  ( ( ( ph  /\  K  =  J )  /\  ( <. { p  |  p 
<Q  ( S  +Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  )
) } ,  {
q  |  ( S  +Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  ) ) 
<Q  q } >.  <P  ( F `  J )  /\  ( ( 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 } >. )
21 caucvgprprlemnkj.s . . . . . . . . . 10  |-  ( ph  ->  S  e.  Q. )
2221adantr 261 . . . . . . . . 9  |-  ( (
ph  /\  K  =  J )  ->  S  e.  Q. )
23 nqprlu 6626 . . . . . . . . 9  |-  ( S  e.  Q.  ->  <. { p  |  p  <Q  S } ,  { q  |  S  <Q  q } >.  e.  P. )
2422, 23syl 14 . . . . . . . 8  |-  ( (
ph  /\  K  =  J )  ->  <. { p  |  p  <Q  S } ,  { q  |  S  <Q  q } >.  e.  P. )
25 ltaddpr 6676 . . . . . . . 8  |-  ( (
<. { p  |  p 
<Q  S } ,  {
q  |  S  <Q  q } >.  e.  P.  /\ 
<. { p  |  p 
<Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. J ,  1o >. ]  ~Q  )  <Q  q } >.  e. 
P. )  ->  <. { p  |  p  <Q  S } ,  { q  |  S  <Q  q } >.  <P  ( <. { p  |  p 
<Q  S } ,  {
q  |  S  <Q  q } >.  +P.  <. { p  |  p  <Q  ( *Q
`  [ <. J ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. J ,  1o >. ]  ~Q  )  <Q 
q } >. )
)
2624, 14, 25syl2anc 391 . . . . . . 7  |-  ( (
ph  /\  K  =  J )  ->  <. { p  |  p  <Q  S } ,  { q  |  S  <Q  q } >.  <P  ( <. { p  |  p 
<Q  S } ,  {
q  |  S  <Q  q } >.  +P.  <. { p  |  p  <Q  ( *Q
`  [ <. J ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. J ,  1o >. ]  ~Q  )  <Q 
q } >. )
)
2726adantr 261 . . . . . 6  |-  ( ( ( ph  /\  K  =  J )  /\  ( <. { p  |  p 
<Q  ( S  +Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  )
) } ,  {
q  |  ( S  +Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  ) ) 
<Q  q } >.  <P  ( F `  J )  /\  ( ( 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  S } ,  {
q  |  S  <Q  q } >.  +P.  <. { p  |  p  <Q  ( *Q
`  [ <. J ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. J ,  1o >. ]  ~Q  )  <Q 
q } >. )
)
28 simprl 483 . . . . . . 7  |-  ( ( ( ph  /\  K  =  J )  /\  ( <. { p  |  p 
<Q  ( S  +Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  )
) } ,  {
q  |  ( S  +Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  ) ) 
<Q  q } >.  <P  ( F `  J )  /\  ( ( 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  ( *Q `  [ <. J ,  1o >. ]  ~Q  )
) } ,  {
q  |  ( S  +Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  ) ) 
<Q  q } >.  <P  ( F `  J )
)
29 addnqpr 6640 . . . . . . . . . 10  |-  ( ( S  e.  Q.  /\  ( *Q `  [ <. J ,  1o >. ]  ~Q  )  e.  Q. )  -> 
<. { p  |  p 
<Q  ( S  +Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  )
) } ,  {
q  |  ( S  +Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  ) ) 
<Q  q } >.  =  (
<. { p  |  p 
<Q  S } ,  {
q  |  S  <Q  q } >.  +P.  <. { p  |  p  <Q  ( *Q
`  [ <. J ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. J ,  1o >. ]  ~Q  )  <Q 
q } >. )
)
3022, 12, 29syl2anc 391 . . . . . . . . 9  |-  ( (
ph  /\  K  =  J )  ->  <. { p  |  p  <Q  ( S  +Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  ) ) } ,  { q  |  ( S  +Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  ) )  <Q  q } >.  =  ( <. { p  |  p  <Q  S } ,  {
q  |  S  <Q  q } >.  +P.  <. { p  |  p  <Q  ( *Q
`  [ <. J ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. J ,  1o >. ]  ~Q  )  <Q 
q } >. )
)
3130breq1d 3771 . . . . . . . 8  |-  ( (
ph  /\  K  =  J )  ->  ( <. { p  |  p 
<Q  ( S  +Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  )
) } ,  {
q  |  ( S  +Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  ) ) 
<Q  q } >.  <P  ( F `  J )  <->  (
<. { p  |  p 
<Q  S } ,  {
q  |  S  <Q  q } >.  +P.  <. { p  |  p  <Q  ( *Q
`  [ <. J ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. J ,  1o >. ]  ~Q  )  <Q 
q } >. )  <P  ( F `  J
) ) )
3231adantr 261 . . . . . . 7  |-  ( ( ( ph  /\  K  =  J )  /\  ( <. { p  |  p 
<Q  ( S  +Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  )
) } ,  {
q  |  ( S  +Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  ) ) 
<Q  q } >.  <P  ( F `  J )  /\  ( ( 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  ( *Q `  [ <. J ,  1o >. ]  ~Q  ) ) } ,  { q  |  ( S  +Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  )
)  <Q  q } >.  <P 
( F `  J
)  <->  ( <. { p  |  p  <Q  S } ,  { q  |  S  <Q  q } >.  +P.  <. { p  |  p  <Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. J ,  1o >. ]  ~Q  )  <Q 
q } >. )  <P  ( F `  J
) ) )
3328, 32mpbid 135 . . . . . 6  |-  ( ( ( ph  /\  K  =  J )  /\  ( <. { p  |  p 
<Q  ( S  +Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  )
) } ,  {
q  |  ( S  +Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  ) ) 
<Q  q } >.  <P  ( F `  J )  /\  ( ( 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 `  [ <. J ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. J ,  1o >. ]  ~Q  )  <Q 
q } >. )  <P  ( F `  J
) )
341, 2sotri 4707 . . . . . 6  |-  ( (
<. { p  |  p 
<Q  S } ,  {
q  |  S  <Q  q } >.  <P  ( <. { p  |  p  <Q  S } ,  {
q  |  S  <Q  q } >.  +P.  <. { p  |  p  <Q  ( *Q
`  [ <. J ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. J ,  1o >. ]  ~Q  )  <Q 
q } >. )  /\  ( <. { p  |  p  <Q  S } ,  { q  |  S  <Q  q } >.  +P.  <. { p  |  p  <Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. J ,  1o >. ]  ~Q  )  <Q 
q } >. )  <P  ( F `  J
) )  ->  <. { p  |  p  <Q  S } ,  { q  |  S  <Q  q } >.  <P  ( F `  J )
)
3527, 33, 34syl2anc 391 . . . . 5  |-  ( ( ( ph  /\  K  =  J )  /\  ( <. { p  |  p 
<Q  ( S  +Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  )
) } ,  {
q  |  ( S  +Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  ) ) 
<Q  q } >.  <P  ( F `  J )  /\  ( ( 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 ) )
3620, 35jca 290 . . . 4  |-  ( ( ( ph  /\  K  =  J )  /\  ( <. { p  |  p 
<Q  ( S  +Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  )
) } ,  {
q  |  ( S  +Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  ) ) 
<Q  q } >.  <P  ( F `  J )  /\  ( ( 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 } >.  /\  <. { p  |  p  <Q  S } ,  { q  |  S  <Q  q } >.  <P  ( F `  J ) ) )
3736ex 108 . . 3  |-  ( (
ph  /\  K  =  J )  ->  (
( <. { p  |  p  <Q  ( S  +Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  ) ) } ,  { q  |  ( S  +Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  )
)  <Q  q } >.  <P 
( F `  J
)  /\  ( ( 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 } >.  /\  <. { p  |  p  <Q  S } ,  { q  |  S  <Q  q } >.  <P  ( F `  J ) ) ) )
383, 37mtoi 590 . 2  |-  ( (
ph  /\  K  =  J )  ->  -.  ( <. { p  |  p  <Q  ( S  +Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  ) ) } ,  { q  |  ( S  +Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  )
)  <Q  q } >.  <P 
( F `  J
)  /\  ( ( 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 } >. )
)
39 opeq1 3546 . . . . . . . . . . 11  |-  ( K  =  J  ->  <. K ,  1o >.  =  <. J ,  1o >. )
4039eceq1d 6129 . . . . . . . . . 10  |-  ( K  =  J  ->  [ <. K ,  1o >. ]  ~Q  =  [ <. J ,  1o >. ]  ~Q  )
4140fveq2d 5169 . . . . . . . . 9  |-  ( K  =  J  ->  ( *Q `  [ <. K ,  1o >. ]  ~Q  )  =  ( *Q `  [ <. J ,  1o >. ]  ~Q  ) )
4241oveq2d 5515 . . . . . . . 8  |-  ( K  =  J  ->  ( S  +Q  ( *Q `  [ <. K ,  1o >. ]  ~Q  ) )  =  ( S  +Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  ) ) )
4342breq2d 3773 . . . . . . 7  |-  ( K  =  J  ->  (
p  <Q  ( S  +Q  ( *Q `  [ <. K ,  1o >. ]  ~Q  ) )  <->  p  <Q  ( S  +Q  ( *Q
`  [ <. J ,  1o >. ]  ~Q  )
) ) )
4443abbidv 2155 . . . . . 6  |-  ( K  =  J  ->  { p  |  p  <Q  ( S  +Q  ( *Q `  [ <. K ,  1o >. ]  ~Q  ) ) }  =  { p  |  p  <Q  ( S  +Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  ) ) } )
4542breq1d 3771 . . . . . . 7  |-  ( K  =  J  ->  (
( S  +Q  ( *Q `  [ <. K ,  1o >. ]  ~Q  )
)  <Q  q  <->  ( S  +Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  ) )  <Q 
q ) )
4645abbidv 2155 . . . . . 6  |-  ( K  =  J  ->  { q  |  ( S  +Q  ( *Q `  [ <. K ,  1o >. ]  ~Q  ) )  <Q  q }  =  { q  |  ( S  +Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  ) )  <Q  q } )
4744, 46opeq12d 3554 . . . . 5  |-  ( K  =  J  ->  <. { p  |  p  <Q  ( S  +Q  ( *Q `  [ <. K ,  1o >. ]  ~Q  ) ) } ,  { q  |  ( S  +Q  ( *Q `  [ <. K ,  1o >. ]  ~Q  ) )  <Q  q } >.  =  <. { p  |  p  <Q  ( S  +Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  ) ) } ,  { q  |  ( S  +Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  ) )  <Q  q } >. )
48 fveq2 5165 . . . . 5  |-  ( K  =  J  ->  ( F `  K )  =  ( F `  J ) )
4947, 48breq12d 3774 . . . 4  |-  ( K  =  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  ( *Q `  [ <. J ,  1o >. ]  ~Q  )
) } ,  {
q  |  ( S  +Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  ) ) 
<Q  q } >.  <P  ( F `  J )
) )
5049anbi1d 438 . . 3  |-  ( K  =  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  ( *Q `  [ <. J ,  1o >. ]  ~Q  )
) } ,  {
q  |  ( S  +Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  ) ) 
<Q  q } >.  <P  ( F `  J )  /\  ( ( 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 } >. ) ) )
5150adantl 262 . 2  |-  ( (
ph  /\  K  =  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  ( *Q `  [ <. J ,  1o >. ]  ~Q  )
) } ,  {
q  |  ( S  +Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  ) ) 
<Q  q } >.  <P  ( F `  J )  /\  ( ( 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 } >. ) ) )
5238, 51mtbird 598 1  |-  ( (
ph  /\  K  =  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 97    <-> wb 98    = wceq 1243    e. wcel 1393   {cab 2026   A.wral 2303   <.cop 3375   class class class wbr 3761   -->wf 4885   ` cfv 4889  (class class class)co 5499   1oc1o 5981   [cec 6091   N.cnpi 6351    <N clti 6354    ~Q ceq 6358   Q.cnq 6359    +Q cplq 6361   *Qcrq 6363    <Q cltq 6364   P.cnp 6370    +P. cpp 6372    <P cltp 6374
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-coll 3869  ax-sep 3872  ax-nul 3880  ax-pow 3924  ax-pr 3941  ax-un 4166  ax-setind 4256  ax-iinf 4298
This theorem depends on definitions:  df-bi 110  df-dc 743  df-3or 886  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2308  df-rex 2309  df-reu 2310  df-rab 2312  df-v 2556  df-sbc 2762  df-csb 2850  df-dif 2917  df-un 2919  df-in 2921  df-ss 2928  df-nul 3222  df-pw 3358  df-sn 3378  df-pr 3379  df-op 3381  df-uni 3578  df-int 3613  df-iun 3656  df-br 3762  df-opab 3816  df-mpt 3817  df-tr 3852  df-eprel 4023  df-id 4027  df-po 4030  df-iso 4031  df-iord 4099  df-on 4101  df-suc 4104  df-iom 4301  df-xp 4338  df-rel 4339  df-cnv 4340  df-co 4341  df-dm 4342  df-rn 4343  df-res 4344  df-ima 4345  df-iota 4854  df-fun 4891  df-fn 4892  df-f 4893  df-f1 4894  df-fo 4895  df-f1o 4896  df-fv 4897  df-ov 5502  df-oprab 5503  df-mpt2 5504  df-1st 5754  df-2nd 5755  df-recs 5907  df-irdg 5944  df-1o 5988  df-2o 5989  df-oadd 5992  df-omul 5993  df-er 6093  df-ec 6095  df-qs 6099  df-ni 6383  df-pli 6384  df-mi 6385  df-lti 6386  df-plpq 6423  df-mpq 6424  df-enq 6426  df-nqqs 6427  df-plqqs 6428  df-mqqs 6429  df-1nqqs 6430  df-rq 6431  df-ltnqqs 6432  df-enq0 6503  df-nq0 6504  df-0nq0 6505  df-plq0 6506  df-mq0 6507  df-inp 6545  df-iplp 6547  df-iltp 6549
This theorem is referenced by:  caucvgprprlemnkj  6771
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