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Theorem caucvgprprlemnbj 6848
Description: Lemma for caucvgprpr 6867. 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 6843 . . . . . 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 111 . . . . 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 5330 . . . . . . . 8  |-  ( ph  ->  ( F `  B
)  e.  P. )
7 recnnpr 6703 . . . . . . . . 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 6692 . . . . . . . 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 397 . . . . . . 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 6703 . . . . . . . 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 6752 . . . . . . 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 397 . . . . . 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 265 . . . . 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 6751 . . . . . 6  |-  <P  Or  P.
18 ltrelpr 6660 . . . . . 6  |-  <P  C_  ( P.  X.  P. )
1917, 18sotri 4747 . . . . 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 397 . . . 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 6752 . . . . . . . 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 397 . . . . . . 7  |-  ( ph  ->  ( F `  B
)  <P  ( ( F `
 B )  +P. 
<. { p  |  p 
<Q  ( *Q `  [ <. B ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. B ,  1o >. ]  ~Q  )  <Q  q } >. ) )
2322adantr 265 . . . . . 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 5205 . . . . . . . 8  |-  ( B  =  J  ->  ( F `  B )  =  ( F `  J ) )
2524breq1d 3801 . . . . . . 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 266 . . . . . 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 139 . . . . 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 265 . . . . 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 397 . . . 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 6843 . . . . . 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 109 . . . . 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 6774 . . . . . . . . 9  |-  ( ( x  e.  P.  /\  y  e.  P.  /\  z  e.  P. )  ->  (
x  <P  y  <->  ( z  +P.  x )  <P  (
z  +P.  y )
) )
3332adantl 266 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  P.  /\  y  e. 
P.  /\  z  e.  P. ) )  ->  (
x  <P  y  <->  ( z  +P.  x )  <P  (
z  +P.  y )
) )
34 addcomprg 6733 . . . . . . . . 9  |-  ( ( x  e.  P.  /\  y  e.  P. )  ->  ( x  +P.  y
)  =  ( y  +P.  x ) )
3534adantl 266 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  P.  /\  y  e. 
P. ) )  -> 
( x  +P.  y
)  =  ( y  +P.  x ) )
3633, 6, 10, 13, 35caovord2d 5697 . . . . . . 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 139 . . . . . 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 265 . . . . 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 4747 . . . . 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 397 . . . 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 6477 . . . . 5  |-  ( ( B  e.  N.  /\  J  e.  N. )  ->  ( B  <N  J  \/  B  =  J  \/  J  <N  B ) )
425, 11, 41syl2anc 397 . . . 4  |-  ( ph  ->  ( B  <N  J  \/  B  =  J  \/  J  <N  B ) )
4320, 29, 40, 42mpjao3dan 1213 . . 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 5330 . . . . 5  |-  ( ph  ->  ( F `  J
)  e.  P. )
45 addclpr 6692 . . . . . 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 397 . . . . 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 4085 . . . . . 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 408 . . . . 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 397 . . . 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 634 . . . 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 141 . . 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 3794 . . . . . . 7  |-  ( p  =  l  ->  (
p  <Q  ( *Q `  [ <. B ,  1o >. ]  ~Q  )  <->  l  <Q  ( *Q `  [ <. B ,  1o >. ]  ~Q  ) ) )
5453cbvabv 2177 . . . . . 6  |-  { p  |  p  <Q  ( *Q
`  [ <. B ,  1o >. ]  ~Q  ) }  =  { l  |  l  <Q  ( *Q
`  [ <. B ,  1o >. ]  ~Q  ) }
55 breq2 3795 . . . . . . 7  |-  ( q  =  u  ->  (
( *Q `  [ <. B ,  1o >. ]  ~Q  )  <Q  q  <->  ( *Q `  [ <. B ,  1o >. ]  ~Q  )  <Q  u ) )
5655cbvabv 2177 . . . . . 6  |-  { q  |  ( *Q `  [ <. B ,  1o >. ]  ~Q  )  <Q 
q }  =  {
u  |  ( *Q
`  [ <. B ,  1o >. ]  ~Q  )  <Q  u }
5754, 56opeq12i 3581 . . . . 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 5550 . . . 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 3794 . . . . . 6  |-  ( p  =  l  ->  (
p  <Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  )  <->  l  <Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  ) ) )
6059cbvabv 2177 . . . . 5  |-  { p  |  p  <Q  ( *Q
`  [ <. J ,  1o >. ]  ~Q  ) }  =  { l  |  l  <Q  ( *Q
`  [ <. J ,  1o >. ]  ~Q  ) }
61 breq2 3795 . . . . . 6  |-  ( q  =  u  ->  (
( *Q `  [ <. J ,  1o >. ]  ~Q  )  <Q  q  <->  ( *Q `  [ <. J ,  1o >. ]  ~Q  )  <Q  u ) )
6261cbvabv 2177 . . . . 5  |-  { q  |  ( *Q `  [ <. J ,  1o >. ]  ~Q  )  <Q 
q }  =  {
u  |  ( *Q
`  [ <. J ,  1o >. ]  ~Q  )  <Q  u }
6360, 62opeq12i 3581 . . . 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 5551 . . 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 3798 . 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 611 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 101    <-> wb 102    \/ w3o 895    /\ w3a 896    = wceq 1259    e. wcel 1409   {cab 2042   A.wral 2323   <.cop 3405   class class class wbr 3791    Or wor 4059   -->wf 4925   ` cfv 4929  (class class class)co 5539   1oc1o 6024   [cec 6134   N.cnpi 6427    <N clti 6430    ~Q ceq 6434   *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:  caucvgprprlemaddq  6863
  Copyright terms: Public domain W3C validator