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Theorem caucvgprprlemnbj 7469
Description: Lemma for caucvgprpr 7488. Non-existence of two elements of the sequence which are too far from each other. (Contributed by Jim Kingdon, 17-Jun-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 } >. )
) ) )
caucvgprprlemnbj.b  |-  ( ph  ->  B  e.  N. )
caucvgprprlemnbj.j  |-  ( ph  ->  J  e.  N. )
Assertion
Ref Expression
caucvgprprlemnbj  |-  ( ph  ->  -.  ( ( ( F `  B )  +P.  <. { l  |  l  <Q  ( *Q `  [ <. B ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. B ,  1o >. ]  ~Q  )  <Q  u } >. )  +P.  <. { l  |  l  <Q 
( *Q `  [ <. J ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. J ,  1o >. ]  ~Q  )  <Q  u } >. )  <P  ( F `  J
) )
Distinct variable groups:    B, k, l, n    u, B, k, n    k, F, n   
k, J, l, n   
u, J
Allowed substitution hints:    ph( u, k, n, l)    F( u, l)

Proof of Theorem caucvgprprlemnbj
Dummy variables  p  q  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 caucvgprpr.f . . . . . . 7  |-  ( ph  ->  F : N. --> P. )
2 caucvgprpr.cau . . . . . . 7  |-  ( 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 } >. )
) ) )
31, 2caucvgprprlemval 7464 . . . . . 6  |-  ( (
ph  /\  B  <N  J )  ->  ( ( F `  B )  <P  ( ( F `  J )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. B ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. B ,  1o >. ]  ~Q  )  <Q 
q } >. )  /\  ( F `  J
)  <P  ( ( F `
 B )  +P. 
<. { p  |  p 
<Q  ( *Q `  [ <. B ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. B ,  1o >. ]  ~Q  )  <Q  q } >. ) ) )
43simprd 113 . . . . 5  |-  ( (
ph  /\  B  <N  J )  ->  ( F `  J )  <P  (
( F `  B
)  +P.  <. { p  |  p  <Q  ( *Q
`  [ <. B ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. B ,  1o >. ]  ~Q  )  <Q 
q } >. )
)
5 caucvgprprlemnbj.b . . . . . . . . 9  |-  ( ph  ->  B  e.  N. )
61, 5ffvelrnd 5524 . . . . . . . 8  |-  ( ph  ->  ( F `  B
)  e.  P. )
7 recnnpr 7324 . . . . . . . . 9  |-  ( B  e.  N.  ->  <. { p  |  p  <Q  ( *Q
`  [ <. B ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. B ,  1o >. ]  ~Q  )  <Q 
q } >.  e.  P. )
85, 7syl 14 . . . . . . . 8  |-  ( ph  -> 
<. { p  |  p 
<Q  ( *Q `  [ <. B ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. B ,  1o >. ]  ~Q  )  <Q  q } >.  e. 
P. )
9 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. )
106, 8, 9syl2anc 408 . . . . . . 7  |-  ( ph  ->  ( ( F `  B )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. B ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. B ,  1o >. ]  ~Q  )  <Q 
q } >. )  e.  P. )
11 caucvgprprlemnbj.j . . . . . . . 8  |-  ( ph  ->  J  e.  N. )
12 recnnpr 7324 . . . . . . . 8  |-  ( J  e.  N.  ->  <. { p  |  p  <Q  ( *Q
`  [ <. J ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. J ,  1o >. ]  ~Q  )  <Q 
q } >.  e.  P. )
1311, 12syl 14 . . . . . . 7  |-  ( ph  -> 
<. { p  |  p 
<Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. J ,  1o >. ]  ~Q  )  <Q  q } >.  e. 
P. )
14 ltaddpr 7373 . . . . . . 7  |-  ( ( ( ( F `  B )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. B ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. B ,  1o >. ]  ~Q  )  <Q 
q } >. )  e.  P.  /\  <. { p  |  p  <Q  ( *Q
`  [ <. J ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. J ,  1o >. ]  ~Q  )  <Q 
q } >.  e.  P. )  ->  ( ( F `
 B )  +P. 
<. { p  |  p 
<Q  ( *Q `  [ <. B ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. B ,  1o >. ]  ~Q  )  <Q  q } >. ) 
<P  ( ( ( F `
 B )  +P. 
<. { p  |  p 
<Q  ( *Q `  [ <. B ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. B ,  1o >. ]  ~Q  )  <Q  q } >. )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. J ,  1o >. ]  ~Q  )  <Q  q } >. ) )
1510, 13, 14syl2anc 408 . . . . . 6  |-  ( ph  ->  ( ( F `  B )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. B ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. B ,  1o >. ]  ~Q  )  <Q 
q } >. )  <P  ( ( ( F `
 B )  +P. 
<. { p  |  p 
<Q  ( *Q `  [ <. B ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. B ,  1o >. ]  ~Q  )  <Q  q } >. )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. J ,  1o >. ]  ~Q  )  <Q  q } >. ) )
1615adantr 274 . . . . 5  |-  ( (
ph  /\  B  <N  J )  ->  ( ( F `  B )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. B ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. B ,  1o >. ]  ~Q  )  <Q  q } >. )  <P  (
( ( F `  B )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. B ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. B ,  1o >. ]  ~Q  )  <Q 
q } >. )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. J ,  1o >. ]  ~Q  )  <Q  q } >. ) )
17 ltsopr 7372 . . . . . 6  |-  <P  Or  P.
18 ltrelpr 7281 . . . . . 6  |-  <P  C_  ( P.  X.  P. )
1917, 18sotri 4904 . . . . 5  |-  ( ( ( F `  J
)  <P  ( ( 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  ( ( ( F `
 B )  +P. 
<. { p  |  p 
<Q  ( *Q `  [ <. B ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. B ,  1o >. ]  ~Q  )  <Q  q } >. )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. J ,  1o >. ]  ~Q  )  <Q  q } >. ) )  -> 
( F `  J
)  <P  ( ( ( F `  B )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. B ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. B ,  1o >. ]  ~Q  )  <Q  q } >. )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. J ,  1o >. ]  ~Q  )  <Q 
q } >. )
)
204, 16, 19syl2anc 408 . . . 4  |-  ( (
ph  /\  B  <N  J )  ->  ( F `  J )  <P  (
( ( F `  B )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. B ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. B ,  1o >. ]  ~Q  )  <Q 
q } >. )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. J ,  1o >. ]  ~Q  )  <Q  q } >. ) )
21 ltaddpr 7373 . . . . . . . 8  |-  ( ( ( F `  B
)  e.  P.  /\  <. { p  |  p  <Q  ( *Q `  [ <. B ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. B ,  1o >. ]  ~Q  )  <Q  q } >.  e. 
P. )  ->  ( F `  B )  <P  ( ( F `  B )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. B ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. B ,  1o >. ]  ~Q  )  <Q 
q } >. )
)
226, 8, 21syl2anc 408 . . . . . . 7  |-  ( ph  ->  ( F `  B
)  <P  ( ( F `
 B )  +P. 
<. { p  |  p 
<Q  ( *Q `  [ <. B ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. B ,  1o >. ]  ~Q  )  <Q  q } >. ) )
2322adantr 274 . . . . . 6  |-  ( (
ph  /\  B  =  J )  ->  ( F `  B )  <P  ( ( F `  B )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. B ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. B ,  1o >. ]  ~Q  )  <Q 
q } >. )
)
24 fveq2 5389 . . . . . . . 8  |-  ( B  =  J  ->  ( F `  B )  =  ( F `  J ) )
2524breq1d 3909 . . . . . . 7  |-  ( B  =  J  ->  (
( F `  B
)  <P  ( ( F `
 B )  +P. 
<. { p  |  p 
<Q  ( *Q `  [ <. B ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. B ,  1o >. ]  ~Q  )  <Q  q } >. )  <-> 
( F `  J
)  <P  ( ( F `
 B )  +P. 
<. { p  |  p 
<Q  ( *Q `  [ <. B ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. B ,  1o >. ]  ~Q  )  <Q  q } >. ) ) )
2625adantl 275 . . . . . 6  |-  ( (
ph  /\  B  =  J )  ->  (
( F `  B
)  <P  ( ( F `
 B )  +P. 
<. { p  |  p 
<Q  ( *Q `  [ <. B ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. B ,  1o >. ]  ~Q  )  <Q  q } >. )  <-> 
( F `  J
)  <P  ( ( F `
 B )  +P. 
<. { p  |  p 
<Q  ( *Q `  [ <. B ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. B ,  1o >. ]  ~Q  )  <Q  q } >. ) ) )
2723, 26mpbid 146 . . . . 5  |-  ( (
ph  /\  B  =  J )  ->  ( F `  J )  <P  ( ( F `  B )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. B ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. B ,  1o >. ]  ~Q  )  <Q 
q } >. )
)
2815adantr 274 . . . . 5  |-  ( (
ph  /\  B  =  J )  ->  (
( F `  B
)  +P.  <. { p  |  p  <Q  ( *Q
`  [ <. B ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. B ,  1o >. ]  ~Q  )  <Q 
q } >. )  <P  ( ( ( F `
 B )  +P. 
<. { p  |  p 
<Q  ( *Q `  [ <. B ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. B ,  1o >. ]  ~Q  )  <Q  q } >. )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. J ,  1o >. ]  ~Q  )  <Q  q } >. ) )
2927, 28, 19syl2anc 408 . . . 4  |-  ( (
ph  /\  B  =  J )  ->  ( F `  J )  <P  ( ( ( F `
 B )  +P. 
<. { p  |  p 
<Q  ( *Q `  [ <. B ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. B ,  1o >. ]  ~Q  )  <Q  q } >. )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. J ,  1o >. ]  ~Q  )  <Q  q } >. ) )
301, 2caucvgprprlemval 7464 . . . . . 6  |-  ( (
ph  /\  J  <N  B )  ->  ( ( F `  J )  <P  ( ( F `  B )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. J ,  1o >. ]  ~Q  )  <Q 
q } >. )  /\  ( F `  B
)  <P  ( ( F `
 J )  +P. 
<. { p  |  p 
<Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. J ,  1o >. ]  ~Q  )  <Q  q } >. ) ) )
3130simpld 111 . . . . 5  |-  ( (
ph  /\  J  <N  B )  ->  ( F `  J )  <P  (
( F `  B
)  +P.  <. { p  |  p  <Q  ( *Q
`  [ <. J ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. J ,  1o >. ]  ~Q  )  <Q 
q } >. )
)
32 ltaprg 7395 . . . . . . . . 9  |-  ( ( x  e.  P.  /\  y  e.  P.  /\  z  e.  P. )  ->  (
x  <P  y  <->  ( z  +P.  x )  <P  (
z  +P.  y )
) )
3332adantl 275 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  P.  /\  y  e. 
P.  /\  z  e.  P. ) )  ->  (
x  <P  y  <->  ( z  +P.  x )  <P  (
z  +P.  y )
) )
34 addcomprg 7354 . . . . . . . . 9  |-  ( ( x  e.  P.  /\  y  e.  P. )  ->  ( x  +P.  y
)  =  ( y  +P.  x ) )
3534adantl 275 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  P.  /\  y  e. 
P. ) )  -> 
( x  +P.  y
)  =  ( y  +P.  x ) )
3633, 6, 10, 13, 35caovord2d 5908 . . . . . . 7  |-  ( ph  ->  ( ( F `  B )  <P  (
( F `  B
)  +P.  <. { p  |  p  <Q  ( *Q
`  [ <. B ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. B ,  1o >. ]  ~Q  )  <Q 
q } >. )  <->  ( ( F `  B
)  +P.  <. { p  |  p  <Q  ( *Q
`  [ <. J ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. J ,  1o >. ]  ~Q  )  <Q 
q } >. )  <P  ( ( ( F `
 B )  +P. 
<. { p  |  p 
<Q  ( *Q `  [ <. B ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. B ,  1o >. ]  ~Q  )  <Q  q } >. )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. J ,  1o >. ]  ~Q  )  <Q  q } >. ) ) )
3722, 36mpbid 146 . . . . . 6  |-  ( ph  ->  ( ( F `  B )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. J ,  1o >. ]  ~Q  )  <Q 
q } >. )  <P  ( ( ( F `
 B )  +P. 
<. { p  |  p 
<Q  ( *Q `  [ <. B ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. B ,  1o >. ]  ~Q  )  <Q  q } >. )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. J ,  1o >. ]  ~Q  )  <Q  q } >. ) )
3837adantr 274 . . . . 5  |-  ( (
ph  /\  J  <N  B )  ->  ( ( F `  B )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. J ,  1o >. ]  ~Q  )  <Q  q } >. )  <P  (
( ( F `  B )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. B ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. B ,  1o >. ]  ~Q  )  <Q 
q } >. )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. J ,  1o >. ]  ~Q  )  <Q  q } >. ) )
3917, 18sotri 4904 . . . . 5  |-  ( ( ( F `  J
)  <P  ( ( F `
 B )  +P. 
<. { p  |  p 
<Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. J ,  1o >. ]  ~Q  )  <Q  q } >. )  /\  ( ( F `
 B )  +P. 
<. { p  |  p 
<Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. J ,  1o >. ]  ~Q  )  <Q  q } >. ) 
<P  ( ( ( F `
 B )  +P. 
<. { p  |  p 
<Q  ( *Q `  [ <. B ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. B ,  1o >. ]  ~Q  )  <Q  q } >. )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. J ,  1o >. ]  ~Q  )  <Q  q } >. ) )  -> 
( F `  J
)  <P  ( ( ( F `  B )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. B ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. B ,  1o >. ]  ~Q  )  <Q  q } >. )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. J ,  1o >. ]  ~Q  )  <Q 
q } >. )
)
4031, 38, 39syl2anc 408 . . . 4  |-  ( (
ph  /\  J  <N  B )  ->  ( F `  J )  <P  (
( ( F `  B )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. B ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. B ,  1o >. ]  ~Q  )  <Q 
q } >. )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. J ,  1o >. ]  ~Q  )  <Q  q } >. ) )
41 pitri3or 7098 . . . . 5  |-  ( ( B  e.  N.  /\  J  e.  N. )  ->  ( B  <N  J  \/  B  =  J  \/  J  <N  B ) )
425, 11, 41syl2anc 408 . . . 4  |-  ( ph  ->  ( B  <N  J  \/  B  =  J  \/  J  <N  B ) )
4320, 29, 40, 42mpjao3dan 1270 . . 3  |-  ( ph  ->  ( F `  J
)  <P  ( ( ( F `  B )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. B ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. B ,  1o >. ]  ~Q  )  <Q  q } >. )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. J ,  1o >. ]  ~Q  )  <Q 
q } >. )
)
441, 11ffvelrnd 5524 . . . . 5  |-  ( ph  ->  ( F `  J
)  e.  P. )
45 addclpr 7313 . . . . . 6  |-  ( ( ( ( F `  B )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. B ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. B ,  1o >. ]  ~Q  )  <Q 
q } >. )  e.  P.  /\  <. { p  |  p  <Q  ( *Q
`  [ <. J ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. J ,  1o >. ]  ~Q  )  <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  ( *Q `  [ <. J ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. J ,  1o >. ]  ~Q  )  <Q 
q } >. )  e.  P. )
4610, 13, 45syl2anc 408 . . . . 5  |-  ( ph  ->  ( ( ( F `
 B )  +P. 
<. { p  |  p 
<Q  ( *Q `  [ <. B ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. B ,  1o >. ]  ~Q  )  <Q  q } >. )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. J ,  1o >. ]  ~Q  )  <Q  q } >. )  e.  P. )
47 so2nr 4213 . . . . . 6  |-  ( ( 
<P  Or  P.  /\  (
( F `  J
)  e.  P.  /\  ( ( ( F `
 B )  +P. 
<. { p  |  p 
<Q  ( *Q `  [ <. B ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. B ,  1o >. ]  ~Q  )  <Q  q } >. )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. J ,  1o >. ]  ~Q  )  <Q  q } >. )  e.  P. ) )  ->  -.  ( ( F `  J )  <P  (
( ( F `  B )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. B ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. B ,  1o >. ]  ~Q  )  <Q 
q } >. )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. J ,  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  ( *Q `  [ <. J ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. J ,  1o >. ]  ~Q  )  <Q  q } >. )  <P  ( F `  J )
) )
4817, 47mpan 420 . . . . 5  |-  ( ( ( F `  J
)  e.  P.  /\  ( ( ( F `
 B )  +P. 
<. { p  |  p 
<Q  ( *Q `  [ <. B ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. B ,  1o >. ]  ~Q  )  <Q  q } >. )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. J ,  1o >. ]  ~Q  )  <Q  q } >. )  e.  P. )  ->  -.  ( ( F `  J )  <P  ( ( ( F `
 B )  +P. 
<. { p  |  p 
<Q  ( *Q `  [ <. B ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. B ,  1o >. ]  ~Q  )  <Q  q } >. )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. J ,  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  ( *Q `  [ <. J ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. J ,  1o >. ]  ~Q  )  <Q  q } >. )  <P  ( F `  J )
) )
4944, 46, 48syl2anc 408 . . . 4  |-  ( ph  ->  -.  ( ( F `
 J )  <P 
( ( ( F `
 B )  +P. 
<. { p  |  p 
<Q  ( *Q `  [ <. B ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. B ,  1o >. ]  ~Q  )  <Q  q } >. )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. J ,  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  ( *Q `  [ <. J ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. J ,  1o >. ]  ~Q  )  <Q  q } >. )  <P  ( F `  J )
) )
50 imnan 664 . . . 4  |-  ( ( ( F `  J
)  <P  ( ( ( F `  B )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. B ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. B ,  1o >. ]  ~Q  )  <Q  q } >. )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. J ,  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  ( *Q `  [ <. J ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. J ,  1o >. ]  ~Q  )  <Q 
q } >. )  <P  ( F `  J
) )  <->  -.  (
( F `  J
)  <P  ( ( ( F `  B )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. B ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. B ,  1o >. ]  ~Q  )  <Q  q } >. )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. J ,  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  ( *Q `  [ <. J ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. J ,  1o >. ]  ~Q  )  <Q  q } >. )  <P  ( F `  J )
) )
5149, 50sylibr 133 . . 3  |-  ( ph  ->  ( ( F `  J )  <P  (
( ( F `  B )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. B ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. B ,  1o >. ]  ~Q  )  <Q 
q } >. )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. J ,  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  ( *Q `  [ <. J ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. J ,  1o >. ]  ~Q  )  <Q  q } >. )  <P  ( F `  J )
) )
5243, 51mpd 13 . 2  |-  ( ph  ->  -.  ( ( ( F `  B )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. B ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. B ,  1o >. ]  ~Q  )  <Q  q } >. )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. J ,  1o >. ]  ~Q  )  <Q 
q } >. )  <P  ( F `  J
) )
53 breq1 3902 . . . . . . 7  |-  ( p  =  l  ->  (
p  <Q  ( *Q `  [ <. B ,  1o >. ]  ~Q  )  <->  l  <Q  ( *Q `  [ <. B ,  1o >. ]  ~Q  ) ) )
5453cbvabv 2241 . . . . . 6  |-  { p  |  p  <Q  ( *Q
`  [ <. B ,  1o >. ]  ~Q  ) }  =  { l  |  l  <Q  ( *Q
`  [ <. B ,  1o >. ]  ~Q  ) }
55 breq2 3903 . . . . . . 7  |-  ( q  =  u  ->  (
( *Q `  [ <. B ,  1o >. ]  ~Q  )  <Q  q  <->  ( *Q `  [ <. B ,  1o >. ]  ~Q  )  <Q  u ) )
5655cbvabv 2241 . . . . . 6  |-  { q  |  ( *Q `  [ <. B ,  1o >. ]  ~Q  )  <Q 
q }  =  {
u  |  ( *Q
`  [ <. B ,  1o >. ]  ~Q  )  <Q  u }
5754, 56opeq12i 3680 . . . . 5  |-  <. { p  |  p  <Q  ( *Q
`  [ <. B ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. B ,  1o >. ]  ~Q  )  <Q 
q } >.  =  <. { l  |  l  <Q 
( *Q `  [ <. B ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. B ,  1o >. ]  ~Q  )  <Q  u } >.
5857oveq2i 5753 . . . 4  |-  ( ( F `  B )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. B ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. B ,  1o >. ]  ~Q  )  <Q  q } >. )  =  ( ( F `  B
)  +P.  <. { l  |  l  <Q  ( *Q `  [ <. B ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. B ,  1o >. ]  ~Q  )  <Q  u } >. )
59 breq1 3902 . . . . . 6  |-  ( p  =  l  ->  (
p  <Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  )  <->  l  <Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  ) ) )
6059cbvabv 2241 . . . . 5  |-  { p  |  p  <Q  ( *Q
`  [ <. J ,  1o >. ]  ~Q  ) }  =  { l  |  l  <Q  ( *Q
`  [ <. J ,  1o >. ]  ~Q  ) }
61 breq2 3903 . . . . . 6  |-  ( q  =  u  ->  (
( *Q `  [ <. J ,  1o >. ]  ~Q  )  <Q  q  <->  ( *Q `  [ <. J ,  1o >. ]  ~Q  )  <Q  u ) )
6261cbvabv 2241 . . . . 5  |-  { q  |  ( *Q `  [ <. J ,  1o >. ]  ~Q  )  <Q 
q }  =  {
u  |  ( *Q
`  [ <. J ,  1o >. ]  ~Q  )  <Q  u }
6360, 62opeq12i 3680 . . . 4  |-  <. { p  |  p  <Q  ( *Q
`  [ <. J ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. J ,  1o >. ]  ~Q  )  <Q 
q } >.  =  <. { l  |  l  <Q 
( *Q `  [ <. J ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. J ,  1o >. ]  ~Q  )  <Q  u } >.
6458, 63oveq12i 5754 . . 3  |-  ( ( ( F `  B
)  +P.  <. { p  |  p  <Q  ( *Q
`  [ <. B ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. B ,  1o >. ]  ~Q  )  <Q 
q } >. )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. J ,  1o >. ]  ~Q  )  <Q  q } >. )  =  ( ( ( F `  B )  +P.  <. { l  |  l  <Q 
( *Q `  [ <. B ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. B ,  1o >. ]  ~Q  )  <Q  u } >. )  +P.  <. { l  |  l  <Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. J ,  1o >. ]  ~Q  )  <Q  u } >. )
6564breq1i 3906 . 2  |-  ( ( ( ( F `  B )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. B ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. B ,  1o >. ]  ~Q  )  <Q 
q } >. )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. J ,  1o >. ]  ~Q  )  <Q  q } >. )  <P  ( F `  J )  <->  ( ( ( F `  B )  +P.  <. { l  |  l  <Q 
( *Q `  [ <. B ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. B ,  1o >. ]  ~Q  )  <Q  u } >. )  +P.  <. { l  |  l  <Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. J ,  1o >. ]  ~Q  )  <Q  u } >. )  <P  ( F `  J )
)
6652, 65sylnib 650 1  |-  ( ph  ->  -.  ( ( ( F `  B )  +P.  <. { l  |  l  <Q  ( *Q `  [ <. B ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. B ,  1o >. ]  ~Q  )  <Q  u } >. )  +P.  <. { l  |  l  <Q 
( *Q `  [ <. J ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. J ,  1o >. ]  ~Q  )  <Q  u } >. )  <P  ( F `  J
) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 103    <-> wb 104    \/ w3o 946    /\ w3a 947    = wceq 1316    e. wcel 1465   {cab 2103   A.wral 2393   <.cop 3500   class class class wbr 3899    Or wor 4187   -->wf 5089   ` cfv 5093  (class class class)co 5742   1oc1o 6274   [cec 6395   N.cnpi 7048    <N clti 7051    ~Q ceq 7055   *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:  caucvgprprlemaddq  7484
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