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Theorem caucvgprprlemnbj 7231
Description: Lemma for caucvgprpr 7250. 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 7226 . . . . . 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 112 . . . . 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 5419 . . . . . . . 8  |-  ( ph  ->  ( F `  B
)  e.  P. )
7 recnnpr 7086 . . . . . . . . 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 7075 . . . . . . . 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 403 . . . . . . 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 7086 . . . . . . . 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 7135 . . . . . . 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 403 . . . . . 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 270 . . . . 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 7134 . . . . . 6  |-  <P  Or  P.
18 ltrelpr 7043 . . . . . 6  |-  <P  C_  ( P.  X.  P. )
1917, 18sotri 4814 . . . . 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 403 . . . 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 7135 . . . . . . . 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 403 . . . . . . 7  |-  ( ph  ->  ( F `  B
)  <P  ( ( F `
 B )  +P. 
<. { p  |  p 
<Q  ( *Q `  [ <. B ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. B ,  1o >. ]  ~Q  )  <Q  q } >. ) )
2322adantr 270 . . . . . 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 5289 . . . . . . . 8  |-  ( B  =  J  ->  ( F `  B )  =  ( F `  J ) )
2524breq1d 3847 . . . . . . 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 271 . . . . . 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 145 . . . . 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 270 . . . . 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 403 . . . 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 7226 . . . . . 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 110 . . . . 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 7157 . . . . . . . . 9  |-  ( ( x  e.  P.  /\  y  e.  P.  /\  z  e.  P. )  ->  (
x  <P  y  <->  ( z  +P.  x )  <P  (
z  +P.  y )
) )
3332adantl 271 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  P.  /\  y  e. 
P.  /\  z  e.  P. ) )  ->  (
x  <P  y  <->  ( z  +P.  x )  <P  (
z  +P.  y )
) )
34 addcomprg 7116 . . . . . . . . 9  |-  ( ( x  e.  P.  /\  y  e.  P. )  ->  ( x  +P.  y
)  =  ( y  +P.  x ) )
3534adantl 271 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  P.  /\  y  e. 
P. ) )  -> 
( x  +P.  y
)  =  ( y  +P.  x ) )
3633, 6, 10, 13, 35caovord2d 5796 . . . . . . 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 145 . . . . . 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 270 . . . . 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 4814 . . . . 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 403 . . . 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 6860 . . . . 5  |-  ( ( B  e.  N.  /\  J  e.  N. )  ->  ( B  <N  J  \/  B  =  J  \/  J  <N  B ) )
425, 11, 41syl2anc 403 . . . 4  |-  ( ph  ->  ( B  <N  J  \/  B  =  J  \/  J  <N  B ) )
4320, 29, 40, 42mpjao3dan 1243 . . 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 5419 . . . . 5  |-  ( ph  ->  ( F `  J
)  e.  P. )
45 addclpr 7075 . . . . . 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 403 . . . . 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 4139 . . . . . 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 415 . . . . 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 403 . . . 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 659 . . . 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 132 . . 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 3840 . . . . . . 7  |-  ( p  =  l  ->  (
p  <Q  ( *Q `  [ <. B ,  1o >. ]  ~Q  )  <->  l  <Q  ( *Q `  [ <. B ,  1o >. ]  ~Q  ) ) )
5453cbvabv 2211 . . . . . 6  |-  { p  |  p  <Q  ( *Q
`  [ <. B ,  1o >. ]  ~Q  ) }  =  { l  |  l  <Q  ( *Q
`  [ <. B ,  1o >. ]  ~Q  ) }
55 breq2 3841 . . . . . . 7  |-  ( q  =  u  ->  (
( *Q `  [ <. B ,  1o >. ]  ~Q  )  <Q  q  <->  ( *Q `  [ <. B ,  1o >. ]  ~Q  )  <Q  u ) )
5655cbvabv 2211 . . . . . 6  |-  { q  |  ( *Q `  [ <. B ,  1o >. ]  ~Q  )  <Q 
q }  =  {
u  |  ( *Q
`  [ <. B ,  1o >. ]  ~Q  )  <Q  u }
5754, 56opeq12i 3622 . . . . 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 5645 . . . 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 3840 . . . . . 6  |-  ( p  =  l  ->  (
p  <Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  )  <->  l  <Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  ) ) )
6059cbvabv 2211 . . . . 5  |-  { p  |  p  <Q  ( *Q
`  [ <. J ,  1o >. ]  ~Q  ) }  =  { l  |  l  <Q  ( *Q
`  [ <. J ,  1o >. ]  ~Q  ) }
61 breq2 3841 . . . . . 6  |-  ( q  =  u  ->  (
( *Q `  [ <. J ,  1o >. ]  ~Q  )  <Q  q  <->  ( *Q `  [ <. J ,  1o >. ]  ~Q  )  <Q  u ) )
6261cbvabv 2211 . . . . 5  |-  { q  |  ( *Q `  [ <. J ,  1o >. ]  ~Q  )  <Q 
q }  =  {
u  |  ( *Q
`  [ <. J ,  1o >. ]  ~Q  )  <Q  u }
6360, 62opeq12i 3622 . . . 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 5646 . . 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 3844 . 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 636 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 102    <-> wb 103    \/ w3o 923    /\ w3a 924    = wceq 1289    e. wcel 1438   {cab 2074   A.wral 2359   <.cop 3444   class class class wbr 3837    Or wor 4113   -->wf 4998   ` cfv 5002  (class class class)co 5634   1oc1o 6156   [cec 6270   N.cnpi 6810    <N clti 6813    ~Q ceq 6817   *Qcrq 6822    <Q cltq 6823   P.cnp 6829    +P. cpp 6831    <P cltp 6833
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-coll 3946  ax-sep 3949  ax-nul 3957  ax-pow 4001  ax-pr 4027  ax-un 4251  ax-setind 4343  ax-iinf 4393
This theorem depends on definitions:  df-bi 115  df-dc 781  df-3or 925  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-ral 2364  df-rex 2365  df-reu 2366  df-rab 2368  df-v 2621  df-sbc 2839  df-csb 2932  df-dif 2999  df-un 3001  df-in 3003  df-ss 3010  df-nul 3285  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-int 3684  df-iun 3727  df-br 3838  df-opab 3892  df-mpt 3893  df-tr 3929  df-eprel 4107  df-id 4111  df-po 4114  df-iso 4115  df-iord 4184  df-on 4186  df-suc 4189  df-iom 4396  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-rn 4439  df-res 4440  df-ima 4441  df-iota 4967  df-fun 5004  df-fn 5005  df-f 5006  df-f1 5007  df-fo 5008  df-f1o 5009  df-fv 5010  df-ov 5637  df-oprab 5638  df-mpt2 5639  df-1st 5893  df-2nd 5894  df-recs 6052  df-irdg 6117  df-1o 6163  df-2o 6164  df-oadd 6167  df-omul 6168  df-er 6272  df-ec 6274  df-qs 6278  df-ni 6842  df-pli 6843  df-mi 6844  df-lti 6845  df-plpq 6882  df-mpq 6883  df-enq 6885  df-nqqs 6886  df-plqqs 6887  df-mqqs 6888  df-1nqqs 6889  df-rq 6890  df-ltnqqs 6891  df-enq0 6962  df-nq0 6963  df-0nq0 6964  df-plq0 6965  df-mq0 6966  df-inp 7004  df-iplp 7006  df-iltp 7008
This theorem is referenced by:  caucvgprprlemaddq  7246
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