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Theorem caucvgprprlemnkeqj 7803
Description: Lemma for caucvgprpr 7825. 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 7709 . . . 4  |-  <P  Or  P.
2 ltrelpr 7618 . . . 4  |-  <P  C_  ( P.  X.  P. )
31, 2son2lpi 5079 . . 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, 5ffvelcdmd 5716 . . . . . . . 8  |-  ( ph  ->  ( F `  J
)  e.  P. )
76ad2antrr 488 . . . . . . 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 276 . . . . . . . . . . 11  |-  ( (
ph  /\  K  =  J )  ->  J  e.  N. )
9 nnnq 7535 . . . . . . . . . . 11  |-  ( J  e.  N.  ->  [ <. J ,  1o >. ]  ~Q  e.  Q. )
108, 9syl 14 . . . . . . . . . 10  |-  ( (
ph  /\  K  =  J )  ->  [ <. J ,  1o >. ]  ~Q  e.  Q. )
11 recclnq 7505 . . . . . . . . . 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 7660 . . . . . . . . 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 276 . . . . . . 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 7710 . . . . . . 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 411 . . . . . 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 531 . . . . . 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 5078 . . . . . 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 411 . . . . 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 276 . . . . . . . . 9  |-  ( (
ph  /\  K  =  J )  ->  S  e.  Q. )
23 nqprlu 7660 . . . . . . . . 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 7710 . . . . . . . 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 411 . . . . . . 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 276 . . . . . 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 529 . . . . . . 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 7674 . . . . . . . . . 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 411 . . . . . . . . 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 4054 . . . . . . . 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 276 . . . . . . 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 147 . . . . . 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 5078 . . . . . 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 411 . . . . 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 306 . . . 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 115 . . 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 666 . 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 3819 . . . . . . . . . . 11  |-  ( K  =  J  ->  <. K ,  1o >.  =  <. J ,  1o >. )
4039eceq1d 6656 . . . . . . . . . 10  |-  ( K  =  J  ->  [ <. K ,  1o >. ]  ~Q  =  [ <. J ,  1o >. ]  ~Q  )
4140fveq2d 5580 . . . . . . . . 9  |-  ( K  =  J  ->  ( *Q `  [ <. K ,  1o >. ]  ~Q  )  =  ( *Q `  [ <. J ,  1o >. ]  ~Q  ) )
4241oveq2d 5960 . . . . . . . 8  |-  ( K  =  J  ->  ( S  +Q  ( *Q `  [ <. K ,  1o >. ]  ~Q  ) )  =  ( S  +Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  ) ) )
4342breq2d 4056 . . . . . . 7  |-  ( K  =  J  ->  (
p  <Q  ( S  +Q  ( *Q `  [ <. K ,  1o >. ]  ~Q  ) )  <->  p  <Q  ( S  +Q  ( *Q
`  [ <. J ,  1o >. ]  ~Q  )
) ) )
4443abbidv 2323 . . . . . 6  |-  ( K  =  J  ->  { p  |  p  <Q  ( S  +Q  ( *Q `  [ <. K ,  1o >. ]  ~Q  ) ) }  =  { p  |  p  <Q  ( S  +Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  ) ) } )
4542breq1d 4054 . . . . . . 7  |-  ( K  =  J  ->  (
( S  +Q  ( *Q `  [ <. K ,  1o >. ]  ~Q  )
)  <Q  q  <->  ( S  +Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  ) )  <Q 
q ) )
4645abbidv 2323 . . . . . 6  |-  ( K  =  J  ->  { q  |  ( S  +Q  ( *Q `  [ <. K ,  1o >. ]  ~Q  ) )  <Q  q }  =  { q  |  ( S  +Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  ) )  <Q  q } )
4744, 46opeq12d 3827 . . . . 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 5576 . . . . 5  |-  ( K  =  J  ->  ( F `  K )  =  ( F `  J ) )
4947, 48breq12d 4057 . . . 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 465 . . 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 277 . 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 675 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 104    <-> wb 105    = wceq 1373    e. wcel 2176   {cab 2191   A.wral 2484   <.cop 3636   class class class wbr 4044   -->wf 5267   ` cfv 5271  (class class class)co 5944   1oc1o 6495   [cec 6618   N.cnpi 7385    <N clti 7388    ~Q ceq 7392   Q.cnq 7393    +Q cplq 7395   *Qcrq 7397    <Q cltq 7398   P.cnp 7404    +P. cpp 7406    <P cltp 7408
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-coll 4159  ax-sep 4162  ax-nul 4170  ax-pow 4218  ax-pr 4253  ax-un 4480  ax-setind 4585  ax-iinf 4636
This theorem depends on definitions:  df-bi 117  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-ral 2489  df-rex 2490  df-reu 2491  df-rab 2493  df-v 2774  df-sbc 2999  df-csb 3094  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3461  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-int 3886  df-iun 3929  df-br 4045  df-opab 4106  df-mpt 4107  df-tr 4143  df-eprel 4336  df-id 4340  df-po 4343  df-iso 4344  df-iord 4413  df-on 4415  df-suc 4418  df-iom 4639  df-xp 4681  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-rn 4686  df-res 4687  df-ima 4688  df-iota 5232  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5947  df-oprab 5948  df-mpo 5949  df-1st 6226  df-2nd 6227  df-recs 6391  df-irdg 6456  df-1o 6502  df-2o 6503  df-oadd 6506  df-omul 6507  df-er 6620  df-ec 6622  df-qs 6626  df-ni 7417  df-pli 7418  df-mi 7419  df-lti 7420  df-plpq 7457  df-mpq 7458  df-enq 7460  df-nqqs 7461  df-plqqs 7462  df-mqqs 7463  df-1nqqs 7464  df-rq 7465  df-ltnqqs 7466  df-enq0 7537  df-nq0 7538  df-0nq0 7539  df-plq0 7540  df-mq0 7541  df-inp 7579  df-iplp 7581  df-iltp 7583
This theorem is referenced by:  caucvgprprlemnkj  7805
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