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Theorem caucvgprprlemnbj 7520
Description: Lemma for caucvgprpr 7539. 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 7515 . . . . . 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 5559 . . . . . . . 8  |-  ( ph  ->  ( F `  B
)  e.  P. )
7 recnnpr 7375 . . . . . . . . 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 7364 . . . . . . . 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 7375 . . . . . . . 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 7424 . . . . . . 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 7423 . . . . . 6  |-  <P  Or  P.
18 ltrelpr 7332 . . . . . 6  |-  <P  C_  ( P.  X.  P. )
1917, 18sotri 4937 . . . . 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 7424 . . . . . . . 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 5424 . . . . . . . 8  |-  ( B  =  J  ->  ( F `  B )  =  ( F `  J ) )
2524breq1d 3942 . . . . . . 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 7515 . . . . . 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 7446 . . . . . . . . 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 7405 . . . . . . . . 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 5943 . . . . . . 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 4937 . . . . 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 7149 . . . . 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 1285 . . 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 5559 . . . . 5  |-  ( ph  ->  ( F `  J
)  e.  P. )
45 addclpr 7364 . . . . . 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 4246 . . . . . 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 679 . . . 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 3935 . . . . . . 7  |-  ( p  =  l  ->  (
p  <Q  ( *Q `  [ <. B ,  1o >. ]  ~Q  )  <->  l  <Q  ( *Q `  [ <. B ,  1o >. ]  ~Q  ) ) )
5453cbvabv 2264 . . . . . 6  |-  { p  |  p  <Q  ( *Q
`  [ <. B ,  1o >. ]  ~Q  ) }  =  { l  |  l  <Q  ( *Q
`  [ <. B ,  1o >. ]  ~Q  ) }
55 breq2 3936 . . . . . . 7  |-  ( q  =  u  ->  (
( *Q `  [ <. B ,  1o >. ]  ~Q  )  <Q  q  <->  ( *Q `  [ <. B ,  1o >. ]  ~Q  )  <Q  u ) )
5655cbvabv 2264 . . . . . 6  |-  { q  |  ( *Q `  [ <. B ,  1o >. ]  ~Q  )  <Q 
q }  =  {
u  |  ( *Q
`  [ <. B ,  1o >. ]  ~Q  )  <Q  u }
5754, 56opeq12i 3713 . . . . 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 5788 . . . 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 3935 . . . . . 6  |-  ( p  =  l  ->  (
p  <Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  )  <->  l  <Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  ) ) )
6059cbvabv 2264 . . . . 5  |-  { p  |  p  <Q  ( *Q
`  [ <. J ,  1o >. ]  ~Q  ) }  =  { l  |  l  <Q  ( *Q
`  [ <. J ,  1o >. ]  ~Q  ) }
61 breq2 3936 . . . . . 6  |-  ( q  =  u  ->  (
( *Q `  [ <. J ,  1o >. ]  ~Q  )  <Q  q  <->  ( *Q `  [ <. J ,  1o >. ]  ~Q  )  <Q  u ) )
6261cbvabv 2264 . . . . 5  |-  { q  |  ( *Q `  [ <. J ,  1o >. ]  ~Q  )  <Q 
q }  =  {
u  |  ( *Q
`  [ <. J ,  1o >. ]  ~Q  )  <Q  u }
6360, 62opeq12i 3713 . . . 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 5789 . . 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 3939 . 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 665 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 961    /\ w3a 962    = wceq 1331    e. wcel 1480   {cab 2125   A.wral 2416   <.cop 3530   class class class wbr 3932    Or wor 4220   -->wf 5122   ` cfv 5126  (class class class)co 5777   1oc1o 6309   [cec 6430   N.cnpi 7099    <N clti 7102    ~Q ceq 7106   *Qcrq 7111    <Q cltq 7112   P.cnp 7118    +P. cpp 7120    <P cltp 7122
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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4046  ax-sep 4049  ax-nul 4057  ax-pow 4101  ax-pr 4134  ax-un 4358  ax-setind 4455  ax-iinf 4505
This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3740  df-int 3775  df-iun 3818  df-br 3933  df-opab 3993  df-mpt 3994  df-tr 4030  df-eprel 4214  df-id 4218  df-po 4221  df-iso 4222  df-iord 4291  df-on 4293  df-suc 4296  df-iom 4508  df-xp 4548  df-rel 4549  df-cnv 4550  df-co 4551  df-dm 4552  df-rn 4553  df-res 4554  df-ima 4555  df-iota 5091  df-fun 5128  df-fn 5129  df-f 5130  df-f1 5131  df-fo 5132  df-f1o 5133  df-fv 5134  df-ov 5780  df-oprab 5781  df-mpo 5782  df-1st 6041  df-2nd 6042  df-recs 6205  df-irdg 6270  df-1o 6316  df-2o 6317  df-oadd 6320  df-omul 6321  df-er 6432  df-ec 6434  df-qs 6438  df-ni 7131  df-pli 7132  df-mi 7133  df-lti 7134  df-plpq 7171  df-mpq 7172  df-enq 7174  df-nqqs 7175  df-plqqs 7176  df-mqqs 7177  df-1nqqs 7178  df-rq 7179  df-ltnqqs 7180  df-enq0 7251  df-nq0 7252  df-0nq0 7253  df-plq0 7254  df-mq0 7255  df-inp 7293  df-iplp 7295  df-iltp 7297
This theorem is referenced by:  caucvgprprlemaddq  7535
  Copyright terms: Public domain W3C validator