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Theorem caucvgprlemnbj 7666
Description: Lemma for caucvgpr 7681. Non-existence of two elements of the sequence which are too far from each other. (Contributed by Jim Kingdon, 18-Oct-2020.)
Hypotheses
Ref Expression
caucvgpr.f  |-  ( ph  ->  F : N. --> Q. )
caucvgpr.cau  |-  ( ph  ->  A. n  e.  N.  A. k  e.  N.  (
n  <N  k  ->  (
( F `  n
)  <Q  ( ( F `
 k )  +Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) )  /\  ( F `  k ) 
<Q  ( ( F `  n )  +Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  )
) ) ) )
caucvgprlemnbj.b  |-  ( ph  ->  B  e.  N. )
caucvgprlemnbj.j  |-  ( ph  ->  J  e.  N. )
Assertion
Ref Expression
caucvgprlemnbj  |-  ( ph  ->  -.  ( ( ( F `  B )  +Q  ( *Q `  [ <. B ,  1o >. ]  ~Q  ) )  +Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  ) ) 
<Q  ( F `  J
) )
Distinct variable groups:    B, k, n   
k, F, n    k, J, n
Allowed substitution hints:    ph( k, n)

Proof of Theorem caucvgprlemnbj
Dummy variables  f  g  h are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 caucvgpr.cau . . . . . . 7  |-  ( ph  ->  A. n  e.  N.  A. k  e.  N.  (
n  <N  k  ->  (
( F `  n
)  <Q  ( ( F `
 k )  +Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) )  /\  ( F `  k ) 
<Q  ( ( F `  n )  +Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  )
) ) ) )
2 caucvgprlemnbj.b . . . . . . . 8  |-  ( ph  ->  B  e.  N. )
3 caucvgprlemnbj.j . . . . . . . 8  |-  ( ph  ->  J  e.  N. )
4 breq1 4007 . . . . . . . . . 10  |-  ( n  =  B  ->  (
n  <N  k  <->  B  <N  k ) )
5 fveq2 5516 . . . . . . . . . . . 12  |-  ( n  =  B  ->  ( F `  n )  =  ( F `  B ) )
6 opeq1 3779 . . . . . . . . . . . . . . 15  |-  ( n  =  B  ->  <. n ,  1o >.  =  <. B ,  1o >. )
76eceq1d 6571 . . . . . . . . . . . . . 14  |-  ( n  =  B  ->  [ <. n ,  1o >. ]  ~Q  =  [ <. B ,  1o >. ]  ~Q  )
87fveq2d 5520 . . . . . . . . . . . . 13  |-  ( n  =  B  ->  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  =  ( *Q `  [ <. B ,  1o >. ]  ~Q  ) )
98oveq2d 5891 . . . . . . . . . . . 12  |-  ( n  =  B  ->  (
( F `  k
)  +Q  ( *Q
`  [ <. n ,  1o >. ]  ~Q  )
)  =  ( ( F `  k )  +Q  ( *Q `  [ <. B ,  1o >. ]  ~Q  ) ) )
105, 9breq12d 4017 . . . . . . . . . . 11  |-  ( n  =  B  ->  (
( F `  n
)  <Q  ( ( F `
 k )  +Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) )  <->  ( F `  B )  <Q  (
( F `  k
)  +Q  ( *Q
`  [ <. B ,  1o >. ]  ~Q  )
) ) )
115, 8oveq12d 5893 . . . . . . . . . . . 12  |-  ( n  =  B  ->  (
( F `  n
)  +Q  ( *Q
`  [ <. n ,  1o >. ]  ~Q  )
)  =  ( ( F `  B )  +Q  ( *Q `  [ <. B ,  1o >. ]  ~Q  ) ) )
1211breq2d 4016 . . . . . . . . . . 11  |-  ( n  =  B  ->  (
( F `  k
)  <Q  ( ( F `
 n )  +Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) )  <->  ( F `  k )  <Q  (
( F `  B
)  +Q  ( *Q
`  [ <. B ,  1o >. ]  ~Q  )
) ) )
1310, 12anbi12d 473 . . . . . . . . . 10  |-  ( n  =  B  ->  (
( ( F `  n )  <Q  (
( F `  k
)  +Q  ( *Q
`  [ <. n ,  1o >. ]  ~Q  )
)  /\  ( F `  k )  <Q  (
( F `  n
)  +Q  ( *Q
`  [ <. n ,  1o >. ]  ~Q  )
) )  <->  ( ( F `  B )  <Q  ( ( F `  k )  +Q  ( *Q `  [ <. B ,  1o >. ]  ~Q  )
)  /\  ( F `  k )  <Q  (
( F `  B
)  +Q  ( *Q
`  [ <. B ,  1o >. ]  ~Q  )
) ) ) )
144, 13imbi12d 234 . . . . . . . . 9  |-  ( n  =  B  ->  (
( n  <N  k  ->  ( ( F `  n )  <Q  (
( F `  k
)  +Q  ( *Q
`  [ <. n ,  1o >. ]  ~Q  )
)  /\  ( F `  k )  <Q  (
( F `  n
)  +Q  ( *Q
`  [ <. n ,  1o >. ]  ~Q  )
) ) )  <->  ( B  <N  k  ->  ( ( F `  B )  <Q  ( ( F `  k )  +Q  ( *Q `  [ <. B ,  1o >. ]  ~Q  )
)  /\  ( F `  k )  <Q  (
( F `  B
)  +Q  ( *Q
`  [ <. B ,  1o >. ]  ~Q  )
) ) ) ) )
15 breq2 4008 . . . . . . . . . 10  |-  ( k  =  J  ->  ( B  <N  k  <->  B  <N  J ) )
16 fveq2 5516 . . . . . . . . . . . . 13  |-  ( k  =  J  ->  ( F `  k )  =  ( F `  J ) )
1716oveq1d 5890 . . . . . . . . . . . 12  |-  ( k  =  J  ->  (
( F `  k
)  +Q  ( *Q
`  [ <. B ,  1o >. ]  ~Q  )
)  =  ( ( F `  J )  +Q  ( *Q `  [ <. B ,  1o >. ]  ~Q  ) ) )
1817breq2d 4016 . . . . . . . . . . 11  |-  ( k  =  J  ->  (
( F `  B
)  <Q  ( ( F `
 k )  +Q  ( *Q `  [ <. B ,  1o >. ]  ~Q  ) )  <->  ( F `  B )  <Q  (
( F `  J
)  +Q  ( *Q
`  [ <. B ,  1o >. ]  ~Q  )
) ) )
1916breq1d 4014 . . . . . . . . . . 11  |-  ( k  =  J  ->  (
( F `  k
)  <Q  ( ( F `
 B )  +Q  ( *Q `  [ <. B ,  1o >. ]  ~Q  ) )  <->  ( F `  J )  <Q  (
( F `  B
)  +Q  ( *Q
`  [ <. B ,  1o >. ]  ~Q  )
) ) )
2018, 19anbi12d 473 . . . . . . . . . 10  |-  ( k  =  J  ->  (
( ( F `  B )  <Q  (
( F `  k
)  +Q  ( *Q
`  [ <. B ,  1o >. ]  ~Q  )
)  /\  ( F `  k )  <Q  (
( F `  B
)  +Q  ( *Q
`  [ <. B ,  1o >. ]  ~Q  )
) )  <->  ( ( F `  B )  <Q  ( ( F `  J )  +Q  ( *Q `  [ <. B ,  1o >. ]  ~Q  )
)  /\  ( F `  J )  <Q  (
( F `  B
)  +Q  ( *Q
`  [ <. B ,  1o >. ]  ~Q  )
) ) ) )
2115, 20imbi12d 234 . . . . . . . . 9  |-  ( k  =  J  ->  (
( B  <N  k  ->  ( ( F `  B )  <Q  (
( F `  k
)  +Q  ( *Q
`  [ <. B ,  1o >. ]  ~Q  )
)  /\  ( F `  k )  <Q  (
( F `  B
)  +Q  ( *Q
`  [ <. B ,  1o >. ]  ~Q  )
) ) )  <->  ( B  <N  J  ->  ( ( F `  B )  <Q  ( ( F `  J )  +Q  ( *Q `  [ <. B ,  1o >. ]  ~Q  )
)  /\  ( F `  J )  <Q  (
( F `  B
)  +Q  ( *Q
`  [ <. B ,  1o >. ]  ~Q  )
) ) ) ) )
2214, 21rspc2v 2855 . . . . . . . 8  |-  ( ( B  e.  N.  /\  J  e.  N. )  ->  ( A. n  e. 
N.  A. k  e.  N.  ( n  <N  k  -> 
( ( F `  n )  <Q  (
( F `  k
)  +Q  ( *Q
`  [ <. n ,  1o >. ]  ~Q  )
)  /\  ( F `  k )  <Q  (
( F `  n
)  +Q  ( *Q
`  [ <. n ,  1o >. ]  ~Q  )
) ) )  -> 
( B  <N  J  -> 
( ( F `  B )  <Q  (
( F `  J
)  +Q  ( *Q
`  [ <. B ,  1o >. ]  ~Q  )
)  /\  ( F `  J )  <Q  (
( F `  B
)  +Q  ( *Q
`  [ <. B ,  1o >. ]  ~Q  )
) ) ) ) )
232, 3, 22syl2anc 411 . . . . . . 7  |-  ( ph  ->  ( A. n  e. 
N.  A. k  e.  N.  ( n  <N  k  -> 
( ( F `  n )  <Q  (
( F `  k
)  +Q  ( *Q
`  [ <. n ,  1o >. ]  ~Q  )
)  /\  ( F `  k )  <Q  (
( F `  n
)  +Q  ( *Q
`  [ <. n ,  1o >. ]  ~Q  )
) ) )  -> 
( B  <N  J  -> 
( ( F `  B )  <Q  (
( F `  J
)  +Q  ( *Q
`  [ <. B ,  1o >. ]  ~Q  )
)  /\  ( F `  J )  <Q  (
( F `  B
)  +Q  ( *Q
`  [ <. B ,  1o >. ]  ~Q  )
) ) ) ) )
241, 23mpd 13 . . . . . 6  |-  ( ph  ->  ( B  <N  J  -> 
( ( F `  B )  <Q  (
( F `  J
)  +Q  ( *Q
`  [ <. B ,  1o >. ]  ~Q  )
)  /\  ( F `  J )  <Q  (
( F `  B
)  +Q  ( *Q
`  [ <. B ,  1o >. ]  ~Q  )
) ) ) )
2524imp 124 . . . . 5  |-  ( (
ph  /\  B  <N  J )  ->  ( ( F `  B )  <Q  ( ( F `  J )  +Q  ( *Q `  [ <. B ,  1o >. ]  ~Q  )
)  /\  ( F `  J )  <Q  (
( F `  B
)  +Q  ( *Q
`  [ <. B ,  1o >. ]  ~Q  )
) ) )
2625simprd 114 . . . 4  |-  ( (
ph  /\  B  <N  J )  ->  ( F `  J )  <Q  (
( F `  B
)  +Q  ( *Q
`  [ <. B ,  1o >. ]  ~Q  )
) )
27 caucvgpr.f . . . . . . . 8  |-  ( ph  ->  F : N. --> Q. )
2827, 2ffvelcdmd 5653 . . . . . . 7  |-  ( ph  ->  ( F `  B
)  e.  Q. )
29 nnnq 7421 . . . . . . . 8  |-  ( B  e.  N.  ->  [ <. B ,  1o >. ]  ~Q  e.  Q. )
30 recclnq 7391 . . . . . . . 8  |-  ( [
<. B ,  1o >. ]  ~Q  e.  Q.  ->  ( *Q `  [ <. B ,  1o >. ]  ~Q  )  e.  Q. )
312, 29, 303syl 17 . . . . . . 7  |-  ( ph  ->  ( *Q `  [ <. B ,  1o >. ]  ~Q  )  e.  Q. )
32 addclnq 7374 . . . . . . 7  |-  ( ( ( F `  B
)  e.  Q.  /\  ( *Q `  [ <. B ,  1o >. ]  ~Q  )  e.  Q. )  ->  ( ( F `  B )  +Q  ( *Q `  [ <. B ,  1o >. ]  ~Q  )
)  e.  Q. )
3328, 31, 32syl2anc 411 . . . . . 6  |-  ( ph  ->  ( ( F `  B )  +Q  ( *Q `  [ <. B ,  1o >. ]  ~Q  )
)  e.  Q. )
34 nnnq 7421 . . . . . . 7  |-  ( J  e.  N.  ->  [ <. J ,  1o >. ]  ~Q  e.  Q. )
35 recclnq 7391 . . . . . . 7  |-  ( [
<. J ,  1o >. ]  ~Q  e.  Q.  ->  ( *Q `  [ <. J ,  1o >. ]  ~Q  )  e.  Q. )
363, 34, 353syl 17 . . . . . 6  |-  ( ph  ->  ( *Q `  [ <. J ,  1o >. ]  ~Q  )  e.  Q. )
37 ltaddnq 7406 . . . . . 6  |-  ( ( ( ( F `  B )  +Q  ( *Q `  [ <. B ,  1o >. ]  ~Q  )
)  e.  Q.  /\  ( *Q `  [ <. J ,  1o >. ]  ~Q  )  e.  Q. )  ->  ( ( F `  B )  +Q  ( *Q `  [ <. B ,  1o >. ]  ~Q  )
)  <Q  ( ( ( F `  B )  +Q  ( *Q `  [ <. B ,  1o >. ]  ~Q  ) )  +Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  ) ) )
3833, 36, 37syl2anc 411 . . . . 5  |-  ( ph  ->  ( ( F `  B )  +Q  ( *Q `  [ <. B ,  1o >. ]  ~Q  )
)  <Q  ( ( ( F `  B )  +Q  ( *Q `  [ <. B ,  1o >. ]  ~Q  ) )  +Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  ) ) )
3938adantr 276 . . . 4  |-  ( (
ph  /\  B  <N  J )  ->  ( ( F `  B )  +Q  ( *Q `  [ <. B ,  1o >. ]  ~Q  ) )  <Q 
( ( ( F `
 B )  +Q  ( *Q `  [ <. B ,  1o >. ]  ~Q  ) )  +Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  ) ) )
40 ltsonq 7397 . . . . 5  |-  <Q  Or  Q.
41 ltrelnq 7364 . . . . 5  |-  <Q  C_  ( Q.  X.  Q. )
4240, 41sotri 5025 . . . 4  |-  ( ( ( F `  J
)  <Q  ( ( F `
 B )  +Q  ( *Q `  [ <. B ,  1o >. ]  ~Q  ) )  /\  ( ( F `  B )  +Q  ( *Q `  [ <. B ,  1o >. ]  ~Q  )
)  <Q  ( ( ( F `  B )  +Q  ( *Q `  [ <. B ,  1o >. ]  ~Q  ) )  +Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  ) ) )  ->  ( F `  J )  <Q  (
( ( F `  B )  +Q  ( *Q `  [ <. B ,  1o >. ]  ~Q  )
)  +Q  ( *Q
`  [ <. J ,  1o >. ]  ~Q  )
) )
4326, 39, 42syl2anc 411 . . 3  |-  ( (
ph  /\  B  <N  J )  ->  ( F `  J )  <Q  (
( ( F `  B )  +Q  ( *Q `  [ <. B ,  1o >. ]  ~Q  )
)  +Q  ( *Q
`  [ <. J ,  1o >. ]  ~Q  )
) )
44 ltaddnq 7406 . . . . . . 7  |-  ( ( ( F `  B
)  e.  Q.  /\  ( *Q `  [ <. B ,  1o >. ]  ~Q  )  e.  Q. )  ->  ( F `  B
)  <Q  ( ( F `
 B )  +Q  ( *Q `  [ <. B ,  1o >. ]  ~Q  ) ) )
4528, 31, 44syl2anc 411 . . . . . 6  |-  ( ph  ->  ( F `  B
)  <Q  ( ( F `
 B )  +Q  ( *Q `  [ <. B ,  1o >. ]  ~Q  ) ) )
4645adantr 276 . . . . 5  |-  ( (
ph  /\  B  =  J )  ->  ( F `  B )  <Q  ( ( F `  B )  +Q  ( *Q `  [ <. B ,  1o >. ]  ~Q  )
) )
47 fveq2 5516 . . . . . . 7  |-  ( B  =  J  ->  ( F `  B )  =  ( F `  J ) )
4847breq1d 4014 . . . . . 6  |-  ( B  =  J  ->  (
( F `  B
)  <Q  ( ( F `
 B )  +Q  ( *Q `  [ <. B ,  1o >. ]  ~Q  ) )  <->  ( F `  J )  <Q  (
( F `  B
)  +Q  ( *Q
`  [ <. B ,  1o >. ]  ~Q  )
) ) )
4948adantl 277 . . . . 5  |-  ( (
ph  /\  B  =  J )  ->  (
( F `  B
)  <Q  ( ( F `
 B )  +Q  ( *Q `  [ <. B ,  1o >. ]  ~Q  ) )  <->  ( F `  J )  <Q  (
( F `  B
)  +Q  ( *Q
`  [ <. B ,  1o >. ]  ~Q  )
) ) )
5046, 49mpbid 147 . . . 4  |-  ( (
ph  /\  B  =  J )  ->  ( F `  J )  <Q  ( ( F `  B )  +Q  ( *Q `  [ <. B ,  1o >. ]  ~Q  )
) )
5138adantr 276 . . . 4  |-  ( (
ph  /\  B  =  J )  ->  (
( F `  B
)  +Q  ( *Q
`  [ <. B ,  1o >. ]  ~Q  )
)  <Q  ( ( ( F `  B )  +Q  ( *Q `  [ <. B ,  1o >. ]  ~Q  ) )  +Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  ) ) )
5250, 51, 42syl2anc 411 . . 3  |-  ( (
ph  /\  B  =  J )  ->  ( F `  J )  <Q  ( ( ( F `
 B )  +Q  ( *Q `  [ <. B ,  1o >. ]  ~Q  ) )  +Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  ) ) )
53 breq1 4007 . . . . . . . . . 10  |-  ( n  =  J  ->  (
n  <N  k  <->  J  <N  k ) )
54 fveq2 5516 . . . . . . . . . . . 12  |-  ( n  =  J  ->  ( F `  n )  =  ( F `  J ) )
55 opeq1 3779 . . . . . . . . . . . . . . 15  |-  ( n  =  J  ->  <. n ,  1o >.  =  <. J ,  1o >. )
5655eceq1d 6571 . . . . . . . . . . . . . 14  |-  ( n  =  J  ->  [ <. n ,  1o >. ]  ~Q  =  [ <. J ,  1o >. ]  ~Q  )
5756fveq2d 5520 . . . . . . . . . . . . 13  |-  ( n  =  J  ->  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  =  ( *Q `  [ <. J ,  1o >. ]  ~Q  ) )
5857oveq2d 5891 . . . . . . . . . . . 12  |-  ( n  =  J  ->  (
( F `  k
)  +Q  ( *Q
`  [ <. n ,  1o >. ]  ~Q  )
)  =  ( ( F `  k )  +Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  ) ) )
5954, 58breq12d 4017 . . . . . . . . . . 11  |-  ( n  =  J  ->  (
( F `  n
)  <Q  ( ( F `
 k )  +Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) )  <->  ( F `  J )  <Q  (
( F `  k
)  +Q  ( *Q
`  [ <. J ,  1o >. ]  ~Q  )
) ) )
6054, 57oveq12d 5893 . . . . . . . . . . . 12  |-  ( n  =  J  ->  (
( F `  n
)  +Q  ( *Q
`  [ <. n ,  1o >. ]  ~Q  )
)  =  ( ( F `  J )  +Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  ) ) )
6160breq2d 4016 . . . . . . . . . . 11  |-  ( n  =  J  ->  (
( F `  k
)  <Q  ( ( F `
 n )  +Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) )  <->  ( F `  k )  <Q  (
( F `  J
)  +Q  ( *Q
`  [ <. J ,  1o >. ]  ~Q  )
) ) )
6259, 61anbi12d 473 . . . . . . . . . 10  |-  ( n  =  J  ->  (
( ( F `  n )  <Q  (
( F `  k
)  +Q  ( *Q
`  [ <. n ,  1o >. ]  ~Q  )
)  /\  ( F `  k )  <Q  (
( F `  n
)  +Q  ( *Q
`  [ <. n ,  1o >. ]  ~Q  )
) )  <->  ( ( F `  J )  <Q  ( ( F `  k )  +Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  )
)  /\  ( F `  k )  <Q  (
( F `  J
)  +Q  ( *Q
`  [ <. J ,  1o >. ]  ~Q  )
) ) ) )
6353, 62imbi12d 234 . . . . . . . . 9  |-  ( n  =  J  ->  (
( n  <N  k  ->  ( ( F `  n )  <Q  (
( F `  k
)  +Q  ( *Q
`  [ <. n ,  1o >. ]  ~Q  )
)  /\  ( F `  k )  <Q  (
( F `  n
)  +Q  ( *Q
`  [ <. n ,  1o >. ]  ~Q  )
) ) )  <->  ( J  <N  k  ->  ( ( F `  J )  <Q  ( ( F `  k )  +Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  )
)  /\  ( F `  k )  <Q  (
( F `  J
)  +Q  ( *Q
`  [ <. J ,  1o >. ]  ~Q  )
) ) ) ) )
64 breq2 4008 . . . . . . . . . 10  |-  ( k  =  B  ->  ( J  <N  k  <->  J  <N  B ) )
65 fveq2 5516 . . . . . . . . . . . . 13  |-  ( k  =  B  ->  ( F `  k )  =  ( F `  B ) )
6665oveq1d 5890 . . . . . . . . . . . 12  |-  ( k  =  B  ->  (
( F `  k
)  +Q  ( *Q
`  [ <. J ,  1o >. ]  ~Q  )
)  =  ( ( F `  B )  +Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  ) ) )
6766breq2d 4016 . . . . . . . . . . 11  |-  ( k  =  B  ->  (
( F `  J
)  <Q  ( ( F `
 k )  +Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  ) )  <->  ( F `  J )  <Q  (
( F `  B
)  +Q  ( *Q
`  [ <. J ,  1o >. ]  ~Q  )
) ) )
6865breq1d 4014 . . . . . . . . . . 11  |-  ( k  =  B  ->  (
( F `  k
)  <Q  ( ( F `
 J )  +Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  ) )  <->  ( F `  B )  <Q  (
( F `  J
)  +Q  ( *Q
`  [ <. J ,  1o >. ]  ~Q  )
) ) )
6967, 68anbi12d 473 . . . . . . . . . 10  |-  ( k  =  B  ->  (
( ( F `  J )  <Q  (
( F `  k
)  +Q  ( *Q
`  [ <. J ,  1o >. ]  ~Q  )
)  /\  ( F `  k )  <Q  (
( F `  J
)  +Q  ( *Q
`  [ <. J ,  1o >. ]  ~Q  )
) )  <->  ( ( F `  J )  <Q  ( ( F `  B )  +Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  )
)  /\  ( F `  B )  <Q  (
( F `  J
)  +Q  ( *Q
`  [ <. J ,  1o >. ]  ~Q  )
) ) ) )
7064, 69imbi12d 234 . . . . . . . . 9  |-  ( k  =  B  ->  (
( J  <N  k  ->  ( ( F `  J )  <Q  (
( F `  k
)  +Q  ( *Q
`  [ <. J ,  1o >. ]  ~Q  )
)  /\  ( F `  k )  <Q  (
( F `  J
)  +Q  ( *Q
`  [ <. J ,  1o >. ]  ~Q  )
) ) )  <->  ( J  <N  B  ->  ( ( F `  J )  <Q  ( ( F `  B )  +Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  )
)  /\  ( F `  B )  <Q  (
( F `  J
)  +Q  ( *Q
`  [ <. J ,  1o >. ]  ~Q  )
) ) ) ) )
7163, 70rspc2v 2855 . . . . . . . 8  |-  ( ( J  e.  N.  /\  B  e.  N. )  ->  ( A. n  e. 
N.  A. k  e.  N.  ( n  <N  k  -> 
( ( F `  n )  <Q  (
( F `  k
)  +Q  ( *Q
`  [ <. n ,  1o >. ]  ~Q  )
)  /\  ( F `  k )  <Q  (
( F `  n
)  +Q  ( *Q
`  [ <. n ,  1o >. ]  ~Q  )
) ) )  -> 
( J  <N  B  -> 
( ( F `  J )  <Q  (
( F `  B
)  +Q  ( *Q
`  [ <. J ,  1o >. ]  ~Q  )
)  /\  ( F `  B )  <Q  (
( F `  J
)  +Q  ( *Q
`  [ <. J ,  1o >. ]  ~Q  )
) ) ) ) )
723, 2, 71syl2anc 411 . . . . . . 7  |-  ( ph  ->  ( A. n  e. 
N.  A. k  e.  N.  ( n  <N  k  -> 
( ( F `  n )  <Q  (
( F `  k
)  +Q  ( *Q
`  [ <. n ,  1o >. ]  ~Q  )
)  /\  ( F `  k )  <Q  (
( F `  n
)  +Q  ( *Q
`  [ <. n ,  1o >. ]  ~Q  )
) ) )  -> 
( J  <N  B  -> 
( ( F `  J )  <Q  (
( F `  B
)  +Q  ( *Q
`  [ <. J ,  1o >. ]  ~Q  )
)  /\  ( F `  B )  <Q  (
( F `  J
)  +Q  ( *Q
`  [ <. J ,  1o >. ]  ~Q  )
) ) ) ) )
731, 72mpd 13 . . . . . 6  |-  ( ph  ->  ( J  <N  B  -> 
( ( F `  J )  <Q  (
( F `  B
)  +Q  ( *Q
`  [ <. J ,  1o >. ]  ~Q  )
)  /\  ( F `  B )  <Q  (
( F `  J
)  +Q  ( *Q
`  [ <. J ,  1o >. ]  ~Q  )
) ) ) )
7473imp 124 . . . . 5  |-  ( (
ph  /\  J  <N  B )  ->  ( ( F `  J )  <Q  ( ( F `  B )  +Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  )
)  /\  ( F `  B )  <Q  (
( F `  J
)  +Q  ( *Q
`  [ <. J ,  1o >. ]  ~Q  )
) ) )
7574simpld 112 . . . 4  |-  ( (
ph  /\  J  <N  B )  ->  ( F `  J )  <Q  (
( F `  B
)  +Q  ( *Q
`  [ <. J ,  1o >. ]  ~Q  )
) )
76 ltanqg 7399 . . . . . . . 8  |-  ( ( f  e.  Q.  /\  g  e.  Q.  /\  h  e.  Q. )  ->  (
f  <Q  g  <->  ( h  +Q  f )  <Q  (
h  +Q  g ) ) )
7776adantl 277 . . . . . . 7  |-  ( (
ph  /\  ( f  e.  Q.  /\  g  e. 
Q.  /\  h  e.  Q. ) )  ->  (
f  <Q  g  <->  ( h  +Q  f )  <Q  (
h  +Q  g ) ) )
78 addcomnqg 7380 . . . . . . . 8  |-  ( ( f  e.  Q.  /\  g  e.  Q. )  ->  ( f  +Q  g
)  =  ( g  +Q  f ) )
7978adantl 277 . . . . . . 7  |-  ( (
ph  /\  ( f  e.  Q.  /\  g  e. 
Q. ) )  -> 
( f  +Q  g
)  =  ( g  +Q  f ) )
8077, 28, 33, 36, 79caovord2d 6044 . . . . . 6  |-  ( ph  ->  ( ( F `  B )  <Q  (
( F `  B
)  +Q  ( *Q
`  [ <. B ,  1o >. ]  ~Q  )
)  <->  ( ( F `
 B )  +Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  ) )  <Q 
( ( ( F `
 B )  +Q  ( *Q `  [ <. B ,  1o >. ]  ~Q  ) )  +Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  ) ) ) )
8145, 80mpbid 147 . . . . 5  |-  ( ph  ->  ( ( F `  B )  +Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  )
)  <Q  ( ( ( F `  B )  +Q  ( *Q `  [ <. B ,  1o >. ]  ~Q  ) )  +Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  ) ) )
8281adantr 276 . . . 4  |-  ( (
ph  /\  J  <N  B )  ->  ( ( F `  B )  +Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  ) )  <Q 
( ( ( F `
 B )  +Q  ( *Q `  [ <. B ,  1o >. ]  ~Q  ) )  +Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  ) ) )
8340, 41sotri 5025 . . . 4  |-  ( ( ( F `  J
)  <Q  ( ( F `
 B )  +Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  ) )  /\  ( ( F `  B )  +Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  )
)  <Q  ( ( ( F `  B )  +Q  ( *Q `  [ <. B ,  1o >. ]  ~Q  ) )  +Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  ) ) )  ->  ( F `  J )  <Q  (
( ( F `  B )  +Q  ( *Q `  [ <. B ,  1o >. ]  ~Q  )
)  +Q  ( *Q
`  [ <. J ,  1o >. ]  ~Q  )
) )
8475, 82, 83syl2anc 411 . . 3  |-  ( (
ph  /\  J  <N  B )  ->  ( F `  J )  <Q  (
( ( F `  B )  +Q  ( *Q `  [ <. B ,  1o >. ]  ~Q  )
)  +Q  ( *Q
`  [ <. J ,  1o >. ]  ~Q  )
) )
85 pitri3or 7321 . . . 4  |-  ( ( B  e.  N.  /\  J  e.  N. )  ->  ( B  <N  J  \/  B  =  J  \/  J  <N  B ) )
862, 3, 85syl2anc 411 . . 3  |-  ( ph  ->  ( B  <N  J  \/  B  =  J  \/  J  <N  B ) )
8743, 52, 84, 86mpjao3dan 1307 . 2  |-  ( ph  ->  ( F `  J
)  <Q  ( ( ( F `  B )  +Q  ( *Q `  [ <. B ,  1o >. ]  ~Q  ) )  +Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  ) ) )
8827, 3ffvelcdmd 5653 . . . 4  |-  ( ph  ->  ( F `  J
)  e.  Q. )
89 addclnq 7374 . . . . 5  |-  ( ( ( ( F `  B )  +Q  ( *Q `  [ <. B ,  1o >. ]  ~Q  )
)  e.  Q.  /\  ( *Q `  [ <. J ,  1o >. ]  ~Q  )  e.  Q. )  ->  ( ( ( F `
 B )  +Q  ( *Q `  [ <. B ,  1o >. ]  ~Q  ) )  +Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  ) )  e. 
Q. )
9033, 36, 89syl2anc 411 . . . 4  |-  ( ph  ->  ( ( ( F `
 B )  +Q  ( *Q `  [ <. B ,  1o >. ]  ~Q  ) )  +Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  ) )  e. 
Q. )
91 so2nr 4322 . . . . 5  |-  ( ( 
<Q  Or  Q.  /\  (
( F `  J
)  e.  Q.  /\  ( ( ( F `
 B )  +Q  ( *Q `  [ <. B ,  1o >. ]  ~Q  ) )  +Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  ) )  e. 
Q. ) )  ->  -.  ( ( F `  J )  <Q  (
( ( F `  B )  +Q  ( *Q `  [ <. B ,  1o >. ]  ~Q  )
)  +Q  ( *Q
`  [ <. J ,  1o >. ]  ~Q  )
)  /\  ( (
( F `  B
)  +Q  ( *Q
`  [ <. B ,  1o >. ]  ~Q  )
)  +Q  ( *Q
`  [ <. J ,  1o >. ]  ~Q  )
)  <Q  ( F `  J ) ) )
9240, 91mpan 424 . . . 4  |-  ( ( ( F `  J
)  e.  Q.  /\  ( ( ( F `
 B )  +Q  ( *Q `  [ <. B ,  1o >. ]  ~Q  ) )  +Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  ) )  e. 
Q. )  ->  -.  ( ( F `  J )  <Q  (
( ( F `  B )  +Q  ( *Q `  [ <. B ,  1o >. ]  ~Q  )
)  +Q  ( *Q
`  [ <. J ,  1o >. ]  ~Q  )
)  /\  ( (
( F `  B
)  +Q  ( *Q
`  [ <. B ,  1o >. ]  ~Q  )
)  +Q  ( *Q
`  [ <. J ,  1o >. ]  ~Q  )
)  <Q  ( F `  J ) ) )
9388, 90, 92syl2anc 411 . . 3  |-  ( ph  ->  -.  ( ( F `
 J )  <Q 
( ( ( F `
 B )  +Q  ( *Q `  [ <. B ,  1o >. ]  ~Q  ) )  +Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  ) )  /\  ( ( ( F `
 B )  +Q  ( *Q `  [ <. B ,  1o >. ]  ~Q  ) )  +Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  ) )  <Q 
( F `  J
) ) )
94 imnan 690 . . 3  |-  ( ( ( F `  J
)  <Q  ( ( ( F `  B )  +Q  ( *Q `  [ <. B ,  1o >. ]  ~Q  ) )  +Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  ) )  ->  -.  ( (
( F `  B
)  +Q  ( *Q
`  [ <. B ,  1o >. ]  ~Q  )
)  +Q  ( *Q
`  [ <. J ,  1o >. ]  ~Q  )
)  <Q  ( F `  J ) )  <->  -.  (
( F `  J
)  <Q  ( ( ( F `  B )  +Q  ( *Q `  [ <. B ,  1o >. ]  ~Q  ) )  +Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  ) )  /\  ( ( ( F `  B )  +Q  ( *Q `  [ <. B ,  1o >. ]  ~Q  ) )  +Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  ) ) 
<Q  ( F `  J
) ) )
9593, 94sylibr 134 . 2  |-  ( ph  ->  ( ( F `  J )  <Q  (
( ( F `  B )  +Q  ( *Q `  [ <. B ,  1o >. ]  ~Q  )
)  +Q  ( *Q
`  [ <. J ,  1o >. ]  ~Q  )
)  ->  -.  (
( ( F `  B )  +Q  ( *Q `  [ <. B ,  1o >. ]  ~Q  )
)  +Q  ( *Q
`  [ <. J ,  1o >. ]  ~Q  )
)  <Q  ( F `  J ) ) )
9687, 95mpd 13 1  |-  ( ph  ->  -.  ( ( ( F `  B )  +Q  ( *Q `  [ <. B ,  1o >. ]  ~Q  ) )  +Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  ) ) 
<Q  ( F `  J
) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 104    <-> wb 105    \/ w3o 977    /\ w3a 978    = wceq 1353    e. wcel 2148   A.wral 2455   <.cop 3596   class class class wbr 4004    Or wor 4296   -->wf 5213   ` cfv 5217  (class class class)co 5875   1oc1o 6410   [cec 6533   N.cnpi 7271    <N clti 7274    ~Q ceq 7278   Q.cnq 7279    +Q cplq 7281   *Qcrq 7283    <Q cltq 7284
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4119  ax-sep 4122  ax-nul 4130  ax-pow 4175  ax-pr 4210  ax-un 4434  ax-setind 4537  ax-iinf 4588
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2740  df-sbc 2964  df-csb 3059  df-dif 3132  df-un 3134  df-in 3136  df-ss 3143  df-nul 3424  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-int 3846  df-iun 3889  df-br 4005  df-opab 4066  df-mpt 4067  df-tr 4103  df-eprel 4290  df-id 4294  df-po 4297  df-iso 4298  df-iord 4367  df-on 4369  df-suc 4372  df-iom 4591  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-rn 4638  df-res 4639  df-ima 4640  df-iota 5179  df-fun 5219  df-fn 5220  df-f 5221  df-f1 5222  df-fo 5223  df-f1o 5224  df-fv 5225  df-ov 5878  df-oprab 5879  df-mpo 5880  df-1st 6141  df-2nd 6142  df-recs 6306  df-irdg 6371  df-1o 6417  df-oadd 6421  df-omul 6422  df-er 6535  df-ec 6537  df-qs 6541  df-ni 7303  df-pli 7304  df-mi 7305  df-lti 7306  df-plpq 7343  df-mpq 7344  df-enq 7346  df-nqqs 7347  df-plqqs 7348  df-mqqs 7349  df-1nqqs 7350  df-rq 7351  df-ltnqqs 7352
This theorem is referenced by:  caucvgprlemladdrl  7677
  Copyright terms: Public domain W3C validator