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Theorem caucvgprlemnbj 6908
Description: Lemma for caucvgpr 6923. 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 3790 . . . . . . . . . 10  |-  ( n  =  B  ->  (
n  <N  k  <->  B  <N  k ) )
5 fveq2 5203 . . . . . . . . . . . 12  |-  ( n  =  B  ->  ( F `  n )  =  ( F `  B ) )
6 opeq1 3572 . . . . . . . . . . . . . . 15  |-  ( n  =  B  ->  <. n ,  1o >.  =  <. B ,  1o >. )
76eceq1d 6201 . . . . . . . . . . . . . 14  |-  ( n  =  B  ->  [ <. n ,  1o >. ]  ~Q  =  [ <. B ,  1o >. ]  ~Q  )
87fveq2d 5207 . . . . . . . . . . . . 13  |-  ( n  =  B  ->  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  =  ( *Q `  [ <. B ,  1o >. ]  ~Q  ) )
98oveq2d 5553 . . . . . . . . . . . 12  |-  ( n  =  B  ->  (
( F `  k
)  +Q  ( *Q
`  [ <. n ,  1o >. ]  ~Q  )
)  =  ( ( F `  k )  +Q  ( *Q `  [ <. B ,  1o >. ]  ~Q  ) ) )
105, 9breq12d 3800 . . . . . . . . . . 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 5555 . . . . . . . . . . . 12  |-  ( n  =  B  ->  (
( F `  n
)  +Q  ( *Q
`  [ <. n ,  1o >. ]  ~Q  )
)  =  ( ( F `  B )  +Q  ( *Q `  [ <. B ,  1o >. ]  ~Q  ) ) )
1211breq2d 3799 . . . . . . . . . . 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 457 . . . . . . . . . 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 232 . . . . . . . . 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 3791 . . . . . . . . . 10  |-  ( k  =  J  ->  ( B  <N  k  <->  B  <N  J ) )
16 fveq2 5203 . . . . . . . . . . . . 13  |-  ( k  =  J  ->  ( F `  k )  =  ( F `  J ) )
1716oveq1d 5552 . . . . . . . . . . . 12  |-  ( k  =  J  ->  (
( F `  k
)  +Q  ( *Q
`  [ <. B ,  1o >. ]  ~Q  )
)  =  ( ( F `  J )  +Q  ( *Q `  [ <. B ,  1o >. ]  ~Q  ) ) )
1817breq2d 3799 . . . . . . . . . . 11  |-  ( k  =  J  ->  (
( F `  B
)  <Q  ( ( F `
 k )  +Q  ( *Q `  [ <. B ,  1o >. ]  ~Q  ) )  <->  ( F `  B )  <Q  (
( F `  J
)  +Q  ( *Q
`  [ <. B ,  1o >. ]  ~Q  )
) ) )
1916breq1d 3797 . . . . . . . . . . 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 457 . . . . . . . . . 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 232 . . . . . . . . 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 2714 . . . . . . . 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 403 . . . . . . 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 122 . . . . 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 112 . . . 4  |-  ( (
ph  /\  B  <N  J )  ->  ( F `  J )  <Q  (
( F `  B
)  +Q  ( *Q
`  [ <. B ,  1o >. ]  ~Q  )
) )
27 caucvgpr.f . . . . . . . 8  |-  ( ph  ->  F : N. --> Q. )
2827, 2ffvelrnd 5329 . . . . . . 7  |-  ( ph  ->  ( F `  B
)  e.  Q. )
29 nnnq 6663 . . . . . . . 8  |-  ( B  e.  N.  ->  [ <. B ,  1o >. ]  ~Q  e.  Q. )
30 recclnq 6633 . . . . . . . 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 6616 . . . . . . 7  |-  ( ( ( F `  B
)  e.  Q.  /\  ( *Q `  [ <. B ,  1o >. ]  ~Q  )  e.  Q. )  ->  ( ( F `  B )  +Q  ( *Q `  [ <. B ,  1o >. ]  ~Q  )
)  e.  Q. )
3328, 31, 32syl2anc 403 . . . . . 6  |-  ( ph  ->  ( ( F `  B )  +Q  ( *Q `  [ <. B ,  1o >. ]  ~Q  )
)  e.  Q. )
34 nnnq 6663 . . . . . . 7  |-  ( J  e.  N.  ->  [ <. J ,  1o >. ]  ~Q  e.  Q. )
35 recclnq 6633 . . . . . . 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 6648 . . . . . 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 403 . . . . 5  |-  ( ph  ->  ( ( F `  B )  +Q  ( *Q `  [ <. B ,  1o >. ]  ~Q  )
)  <Q  ( ( ( F `  B )  +Q  ( *Q `  [ <. B ,  1o >. ]  ~Q  ) )  +Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  ) ) )
3938adantr 270 . . . 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 6639 . . . . 5  |-  <Q  Or  Q.
41 ltrelnq 6606 . . . . 5  |-  <Q  C_  ( Q.  X.  Q. )
4240, 41sotri 4744 . . . 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 403 . . 3  |-  ( (
ph  /\  B  <N  J )  ->  ( F `  J )  <Q  (
( ( F `  B )  +Q  ( *Q `  [ <. B ,  1o >. ]  ~Q  )
)  +Q  ( *Q
`  [ <. J ,  1o >. ]  ~Q  )
) )
44 ltaddnq 6648 . . . . . . 7  |-  ( ( ( F `  B
)  e.  Q.  /\  ( *Q `  [ <. B ,  1o >. ]  ~Q  )  e.  Q. )  ->  ( F `  B
)  <Q  ( ( F `
 B )  +Q  ( *Q `  [ <. B ,  1o >. ]  ~Q  ) ) )
4528, 31, 44syl2anc 403 . . . . . 6  |-  ( ph  ->  ( F `  B
)  <Q  ( ( F `
 B )  +Q  ( *Q `  [ <. B ,  1o >. ]  ~Q  ) ) )
4645adantr 270 . . . . 5  |-  ( (
ph  /\  B  =  J )  ->  ( F `  B )  <Q  ( ( F `  B )  +Q  ( *Q `  [ <. B ,  1o >. ]  ~Q  )
) )
47 fveq2 5203 . . . . . . 7  |-  ( B  =  J  ->  ( F `  B )  =  ( F `  J ) )
4847breq1d 3797 . . . . . 6  |-  ( B  =  J  ->  (
( F `  B
)  <Q  ( ( F `
 B )  +Q  ( *Q `  [ <. B ,  1o >. ]  ~Q  ) )  <->  ( F `  J )  <Q  (
( F `  B
)  +Q  ( *Q
`  [ <. B ,  1o >. ]  ~Q  )
) ) )
4948adantl 271 . . . . 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 145 . . . 4  |-  ( (
ph  /\  B  =  J )  ->  ( F `  J )  <Q  ( ( F `  B )  +Q  ( *Q `  [ <. B ,  1o >. ]  ~Q  )
) )
5138adantr 270 . . . 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 403 . . 3  |-  ( (
ph  /\  B  =  J )  ->  ( F `  J )  <Q  ( ( ( F `
 B )  +Q  ( *Q `  [ <. B ,  1o >. ]  ~Q  ) )  +Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  ) ) )
53 breq1 3790 . . . . . . . . . 10  |-  ( n  =  J  ->  (
n  <N  k  <->  J  <N  k ) )
54 fveq2 5203 . . . . . . . . . . . 12  |-  ( n  =  J  ->  ( F `  n )  =  ( F `  J ) )
55 opeq1 3572 . . . . . . . . . . . . . . 15  |-  ( n  =  J  ->  <. n ,  1o >.  =  <. J ,  1o >. )
5655eceq1d 6201 . . . . . . . . . . . . . 14  |-  ( n  =  J  ->  [ <. n ,  1o >. ]  ~Q  =  [ <. J ,  1o >. ]  ~Q  )
5756fveq2d 5207 . . . . . . . . . . . . 13  |-  ( n  =  J  ->  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  =  ( *Q `  [ <. J ,  1o >. ]  ~Q  ) )
5857oveq2d 5553 . . . . . . . . . . . 12  |-  ( n  =  J  ->  (
( F `  k
)  +Q  ( *Q
`  [ <. n ,  1o >. ]  ~Q  )
)  =  ( ( F `  k )  +Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  ) ) )
5954, 58breq12d 3800 . . . . . . . . . . 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 5555 . . . . . . . . . . . 12  |-  ( n  =  J  ->  (
( F `  n
)  +Q  ( *Q
`  [ <. n ,  1o >. ]  ~Q  )
)  =  ( ( F `  J )  +Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  ) ) )
6160breq2d 3799 . . . . . . . . . . 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 457 . . . . . . . . . 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 232 . . . . . . . . 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 3791 . . . . . . . . . 10  |-  ( k  =  B  ->  ( J  <N  k  <->  J  <N  B ) )
65 fveq2 5203 . . . . . . . . . . . . 13  |-  ( k  =  B  ->  ( F `  k )  =  ( F `  B ) )
6665oveq1d 5552 . . . . . . . . . . . 12  |-  ( k  =  B  ->  (
( F `  k
)  +Q  ( *Q
`  [ <. J ,  1o >. ]  ~Q  )
)  =  ( ( F `  B )  +Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  ) ) )
6766breq2d 3799 . . . . . . . . . . 11  |-  ( k  =  B  ->  (
( F `  J
)  <Q  ( ( F `
 k )  +Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  ) )  <->  ( F `  J )  <Q  (
( F `  B
)  +Q  ( *Q
`  [ <. J ,  1o >. ]  ~Q  )
) ) )
6865breq1d 3797 . . . . . . . . . . 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 457 . . . . . . . . . 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 232 . . . . . . . . 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 2714 . . . . . . . 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 403 . . . . . . 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 122 . . . . 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 110 . . . 4  |-  ( (
ph  /\  J  <N  B )  ->  ( F `  J )  <Q  (
( F `  B
)  +Q  ( *Q
`  [ <. J ,  1o >. ]  ~Q  )
) )
76 ltanqg 6641 . . . . . . . 8  |-  ( ( f  e.  Q.  /\  g  e.  Q.  /\  h  e.  Q. )  ->  (
f  <Q  g  <->  ( h  +Q  f )  <Q  (
h  +Q  g ) ) )
7776adantl 271 . . . . . . 7  |-  ( (
ph  /\  ( f  e.  Q.  /\  g  e. 
Q.  /\  h  e.  Q. ) )  ->  (
f  <Q  g  <->  ( h  +Q  f )  <Q  (
h  +Q  g ) ) )
78 addcomnqg 6622 . . . . . . . 8  |-  ( ( f  e.  Q.  /\  g  e.  Q. )  ->  ( f  +Q  g
)  =  ( g  +Q  f ) )
7978adantl 271 . . . . . . 7  |-  ( (
ph  /\  ( f  e.  Q.  /\  g  e. 
Q. ) )  -> 
( f  +Q  g
)  =  ( g  +Q  f ) )
8077, 28, 33, 36, 79caovord2d 5695 . . . . . 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 145 . . . . 5  |-  ( ph  ->  ( ( F `  B )  +Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  )
)  <Q  ( ( ( F `  B )  +Q  ( *Q `  [ <. B ,  1o >. ]  ~Q  ) )  +Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  ) ) )
8281adantr 270 . . . 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 4744 . . . 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 403 . . 3  |-  ( (
ph  /\  J  <N  B )  ->  ( F `  J )  <Q  (
( ( F `  B )  +Q  ( *Q `  [ <. B ,  1o >. ]  ~Q  )
)  +Q  ( *Q
`  [ <. J ,  1o >. ]  ~Q  )
) )
85 pitri3or 6563 . . . 4  |-  ( ( B  e.  N.  /\  J  e.  N. )  ->  ( B  <N  J  \/  B  =  J  \/  J  <N  B ) )
862, 3, 85syl2anc 403 . . 3  |-  ( ph  ->  ( B  <N  J  \/  B  =  J  \/  J  <N  B ) )
8743, 52, 84, 86mpjao3dan 1239 . 2  |-  ( ph  ->  ( F `  J
)  <Q  ( ( ( F `  B )  +Q  ( *Q `  [ <. B ,  1o >. ]  ~Q  ) )  +Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  ) ) )
8827, 3ffvelrnd 5329 . . . 4  |-  ( ph  ->  ( F `  J
)  e.  Q. )
89 addclnq 6616 . . . . 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 403 . . . 4  |-  ( ph  ->  ( ( ( F `
 B )  +Q  ( *Q `  [ <. B ,  1o >. ]  ~Q  ) )  +Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  ) )  e. 
Q. )
91 so2nr 4078 . . . . 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 415 . . . 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 403 . . 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 657 . . 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 132 . 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 102    <-> wb 103    \/ w3o 919    /\ w3a 920    = wceq 1285    e. wcel 1434   A.wral 2349   <.cop 3403   class class class wbr 3787    Or wor 4052   -->wf 4922   ` cfv 4926  (class class class)co 5537   1oc1o 6052   [cec 6163   N.cnpi 6513    <N clti 6516    ~Q ceq 6520   Q.cnq 6521    +Q cplq 6523   *Qcrq 6525    <Q cltq 6526
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 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-coll 3895  ax-sep 3898  ax-nul 3906  ax-pow 3950  ax-pr 3966  ax-un 4190  ax-setind 4282  ax-iinf 4331
This theorem depends on definitions:  df-bi 115  df-dc 777  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ne 2247  df-ral 2354  df-rex 2355  df-reu 2356  df-rab 2358  df-v 2604  df-sbc 2817  df-csb 2910  df-dif 2976  df-un 2978  df-in 2980  df-ss 2987  df-nul 3253  df-pw 3386  df-sn 3406  df-pr 3407  df-op 3409  df-uni 3604  df-int 3639  df-iun 3682  df-br 3788  df-opab 3842  df-mpt 3843  df-tr 3878  df-eprel 4046  df-id 4050  df-po 4053  df-iso 4054  df-iord 4123  df-on 4125  df-suc 4128  df-iom 4334  df-xp 4371  df-rel 4372  df-cnv 4373  df-co 4374  df-dm 4375  df-rn 4376  df-res 4377  df-ima 4378  df-iota 4891  df-fun 4928  df-fn 4929  df-f 4930  df-f1 4931  df-fo 4932  df-f1o 4933  df-fv 4934  df-ov 5540  df-oprab 5541  df-mpt2 5542  df-1st 5792  df-2nd 5793  df-recs 5948  df-irdg 6013  df-1o 6059  df-oadd 6063  df-omul 6064  df-er 6165  df-ec 6167  df-qs 6171  df-ni 6545  df-pli 6546  df-mi 6547  df-lti 6548  df-plpq 6585  df-mpq 6586  df-enq 6588  df-nqqs 6589  df-plqqs 6590  df-mqqs 6591  df-1nqqs 6592  df-rq 6593  df-ltnqqs 6594
This theorem is referenced by:  caucvgprlemladdrl  6919
  Copyright terms: Public domain W3C validator