ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  caucvgprlemnbj Unicode version

Theorem caucvgprlemnbj 6746
Description: Lemma for caucvgpr 6761. 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 3764 . . . . . . . . . 10  |-  ( n  =  B  ->  (
n  <N  k  <->  B  <N  k ) )
5 fveq2 5165 . . . . . . . . . . . 12  |-  ( n  =  B  ->  ( F `  n )  =  ( F `  B ) )
6 opeq1 3546 . . . . . . . . . . . . . . 15  |-  ( n  =  B  ->  <. n ,  1o >.  =  <. B ,  1o >. )
76eceq1d 6129 . . . . . . . . . . . . . 14  |-  ( n  =  B  ->  [ <. n ,  1o >. ]  ~Q  =  [ <. B ,  1o >. ]  ~Q  )
87fveq2d 5169 . . . . . . . . . . . . 13  |-  ( n  =  B  ->  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  =  ( *Q `  [ <. B ,  1o >. ]  ~Q  ) )
98oveq2d 5515 . . . . . . . . . . . 12  |-  ( n  =  B  ->  (
( F `  k
)  +Q  ( *Q
`  [ <. n ,  1o >. ]  ~Q  )
)  =  ( ( F `  k )  +Q  ( *Q `  [ <. B ,  1o >. ]  ~Q  ) ) )
105, 9breq12d 3774 . . . . . . . . . . 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 5517 . . . . . . . . . . . 12  |-  ( n  =  B  ->  (
( F `  n
)  +Q  ( *Q
`  [ <. n ,  1o >. ]  ~Q  )
)  =  ( ( F `  B )  +Q  ( *Q `  [ <. B ,  1o >. ]  ~Q  ) ) )
1211breq2d 3773 . . . . . . . . . . 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 442 . . . . . . . . . 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 223 . . . . . . . . 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 3765 . . . . . . . . . 10  |-  ( k  =  J  ->  ( B  <N  k  <->  B  <N  J ) )
16 fveq2 5165 . . . . . . . . . . . . 13  |-  ( k  =  J  ->  ( F `  k )  =  ( F `  J ) )
1716oveq1d 5514 . . . . . . . . . . . 12  |-  ( k  =  J  ->  (
( F `  k
)  +Q  ( *Q
`  [ <. B ,  1o >. ]  ~Q  )
)  =  ( ( F `  J )  +Q  ( *Q `  [ <. B ,  1o >. ]  ~Q  ) ) )
1817breq2d 3773 . . . . . . . . . . 11  |-  ( k  =  J  ->  (
( F `  B
)  <Q  ( ( F `
 k )  +Q  ( *Q `  [ <. B ,  1o >. ]  ~Q  ) )  <->  ( F `  B )  <Q  (
( F `  J
)  +Q  ( *Q
`  [ <. B ,  1o >. ]  ~Q  )
) ) )
1916breq1d 3771 . . . . . . . . . . 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 442 . . . . . . . . . 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 223 . . . . . . . . 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 2659 . . . . . . . 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 391 . . . . . . 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 115 . . . . 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 107 . . . 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 5290 . . . . . . 7  |-  ( ph  ->  ( F `  B
)  e.  Q. )
29 nnnq 6501 . . . . . . . 8  |-  ( B  e.  N.  ->  [ <. B ,  1o >. ]  ~Q  e.  Q. )
30 recclnq 6471 . . . . . . . 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 6454 . . . . . . 7  |-  ( ( ( F `  B
)  e.  Q.  /\  ( *Q `  [ <. B ,  1o >. ]  ~Q  )  e.  Q. )  ->  ( ( F `  B )  +Q  ( *Q `  [ <. B ,  1o >. ]  ~Q  )
)  e.  Q. )
3328, 31, 32syl2anc 391 . . . . . 6  |-  ( ph  ->  ( ( F `  B )  +Q  ( *Q `  [ <. B ,  1o >. ]  ~Q  )
)  e.  Q. )
34 nnnq 6501 . . . . . . 7  |-  ( J  e.  N.  ->  [ <. J ,  1o >. ]  ~Q  e.  Q. )
35 recclnq 6471 . . . . . . 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 6486 . . . . . 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 391 . . . . 5  |-  ( ph  ->  ( ( F `  B )  +Q  ( *Q `  [ <. B ,  1o >. ]  ~Q  )
)  <Q  ( ( ( F `  B )  +Q  ( *Q `  [ <. B ,  1o >. ]  ~Q  ) )  +Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  ) ) )
3938adantr 261 . . . 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 6477 . . . . 5  |-  <Q  Or  Q.
41 ltrelnq 6444 . . . . 5  |-  <Q  C_  ( Q.  X.  Q. )
4240, 41sotri 4707 . . . 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 391 . . 3  |-  ( (
ph  /\  B  <N  J )  ->  ( F `  J )  <Q  (
( ( F `  B )  +Q  ( *Q `  [ <. B ,  1o >. ]  ~Q  )
)  +Q  ( *Q
`  [ <. J ,  1o >. ]  ~Q  )
) )
44 ltaddnq 6486 . . . . . . 7  |-  ( ( ( F `  B
)  e.  Q.  /\  ( *Q `  [ <. B ,  1o >. ]  ~Q  )  e.  Q. )  ->  ( F `  B
)  <Q  ( ( F `
 B )  +Q  ( *Q `  [ <. B ,  1o >. ]  ~Q  ) ) )
4528, 31, 44syl2anc 391 . . . . . 6  |-  ( ph  ->  ( F `  B
)  <Q  ( ( F `
 B )  +Q  ( *Q `  [ <. B ,  1o >. ]  ~Q  ) ) )
4645adantr 261 . . . . 5  |-  ( (
ph  /\  B  =  J )  ->  ( F `  B )  <Q  ( ( F `  B )  +Q  ( *Q `  [ <. B ,  1o >. ]  ~Q  )
) )
47 fveq2 5165 . . . . . . 7  |-  ( B  =  J  ->  ( F `  B )  =  ( F `  J ) )
4847breq1d 3771 . . . . . 6  |-  ( B  =  J  ->  (
( F `  B
)  <Q  ( ( F `
 B )  +Q  ( *Q `  [ <. B ,  1o >. ]  ~Q  ) )  <->  ( F `  J )  <Q  (
( F `  B
)  +Q  ( *Q
`  [ <. B ,  1o >. ]  ~Q  )
) ) )
4948adantl 262 . . . . 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 135 . . . 4  |-  ( (
ph  /\  B  =  J )  ->  ( F `  J )  <Q  ( ( F `  B )  +Q  ( *Q `  [ <. B ,  1o >. ]  ~Q  )
) )
5138adantr 261 . . . 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 391 . . 3  |-  ( (
ph  /\  B  =  J )  ->  ( F `  J )  <Q  ( ( ( F `
 B )  +Q  ( *Q `  [ <. B ,  1o >. ]  ~Q  ) )  +Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  ) ) )
53 breq1 3764 . . . . . . . . . 10  |-  ( n  =  J  ->  (
n  <N  k  <->  J  <N  k ) )
54 fveq2 5165 . . . . . . . . . . . 12  |-  ( n  =  J  ->  ( F `  n )  =  ( F `  J ) )
55 opeq1 3546 . . . . . . . . . . . . . . 15  |-  ( n  =  J  ->  <. n ,  1o >.  =  <. J ,  1o >. )
5655eceq1d 6129 . . . . . . . . . . . . . 14  |-  ( n  =  J  ->  [ <. n ,  1o >. ]  ~Q  =  [ <. J ,  1o >. ]  ~Q  )
5756fveq2d 5169 . . . . . . . . . . . . 13  |-  ( n  =  J  ->  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  =  ( *Q `  [ <. J ,  1o >. ]  ~Q  ) )
5857oveq2d 5515 . . . . . . . . . . . 12  |-  ( n  =  J  ->  (
( F `  k
)  +Q  ( *Q
`  [ <. n ,  1o >. ]  ~Q  )
)  =  ( ( F `  k )  +Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  ) ) )
5954, 58breq12d 3774 . . . . . . . . . . 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 5517 . . . . . . . . . . . 12  |-  ( n  =  J  ->  (
( F `  n
)  +Q  ( *Q
`  [ <. n ,  1o >. ]  ~Q  )
)  =  ( ( F `  J )  +Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  ) ) )
6160breq2d 3773 . . . . . . . . . . 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 442 . . . . . . . . . 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 223 . . . . . . . . 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 3765 . . . . . . . . . 10  |-  ( k  =  B  ->  ( J  <N  k  <->  J  <N  B ) )
65 fveq2 5165 . . . . . . . . . . . . 13  |-  ( k  =  B  ->  ( F `  k )  =  ( F `  B ) )
6665oveq1d 5514 . . . . . . . . . . . 12  |-  ( k  =  B  ->  (
( F `  k
)  +Q  ( *Q
`  [ <. J ,  1o >. ]  ~Q  )
)  =  ( ( F `  B )  +Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  ) ) )
6766breq2d 3773 . . . . . . . . . . 11  |-  ( k  =  B  ->  (
( F `  J
)  <Q  ( ( F `
 k )  +Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  ) )  <->  ( F `  J )  <Q  (
( F `  B
)  +Q  ( *Q
`  [ <. J ,  1o >. ]  ~Q  )
) ) )
6865breq1d 3771 . . . . . . . . . . 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 442 . . . . . . . . . 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 223 . . . . . . . . 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 2659 . . . . . . . 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 391 . . . . . . 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 115 . . . . 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 105 . . . 4  |-  ( (
ph  /\  J  <N  B )  ->  ( F `  J )  <Q  (
( F `  B
)  +Q  ( *Q
`  [ <. J ,  1o >. ]  ~Q  )
) )
76 ltanqg 6479 . . . . . . . 8  |-  ( ( f  e.  Q.  /\  g  e.  Q.  /\  h  e.  Q. )  ->  (
f  <Q  g  <->  ( h  +Q  f )  <Q  (
h  +Q  g ) ) )
7776adantl 262 . . . . . . 7  |-  ( (
ph  /\  ( f  e.  Q.  /\  g  e. 
Q.  /\  h  e.  Q. ) )  ->  (
f  <Q  g  <->  ( h  +Q  f )  <Q  (
h  +Q  g ) ) )
78 addcomnqg 6460 . . . . . . . 8  |-  ( ( f  e.  Q.  /\  g  e.  Q. )  ->  ( f  +Q  g
)  =  ( g  +Q  f ) )
7978adantl 262 . . . . . . 7  |-  ( (
ph  /\  ( f  e.  Q.  /\  g  e. 
Q. ) )  -> 
( f  +Q  g
)  =  ( g  +Q  f ) )
8077, 28, 33, 36, 79caovord2d 5657 . . . . . 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 135 . . . . 5  |-  ( ph  ->  ( ( F `  B )  +Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  )
)  <Q  ( ( ( F `  B )  +Q  ( *Q `  [ <. B ,  1o >. ]  ~Q  ) )  +Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  ) ) )
8281adantr 261 . . . 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 4707 . . . 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 391 . . 3  |-  ( (
ph  /\  J  <N  B )  ->  ( F `  J )  <Q  (
( ( F `  B )  +Q  ( *Q `  [ <. B ,  1o >. ]  ~Q  )
)  +Q  ( *Q
`  [ <. J ,  1o >. ]  ~Q  )
) )
85 pitri3or 6401 . . . 4  |-  ( ( B  e.  N.  /\  J  e.  N. )  ->  ( B  <N  J  \/  B  =  J  \/  J  <N  B ) )
862, 3, 85syl2anc 391 . . 3  |-  ( ph  ->  ( B  <N  J  \/  B  =  J  \/  J  <N  B ) )
8743, 52, 84, 86mpjao3dan 1202 . 2  |-  ( ph  ->  ( F `  J
)  <Q  ( ( ( F `  B )  +Q  ( *Q `  [ <. B ,  1o >. ]  ~Q  ) )  +Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  ) ) )
8827, 3ffvelrnd 5290 . . . 4  |-  ( ph  ->  ( F `  J
)  e.  Q. )
89 addclnq 6454 . . . . 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 391 . . . 4  |-  ( ph  ->  ( ( ( F `
 B )  +Q  ( *Q `  [ <. B ,  1o >. ]  ~Q  ) )  +Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  ) )  e. 
Q. )
91 so2nr 4055 . . . . 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 400 . . . 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 391 . . 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 624 . . 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 137 . 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 97    <-> wb 98    \/ w3o 884    /\ w3a 885    = wceq 1243    e. wcel 1393   A.wral 2303   <.cop 3375   class class class wbr 3761    Or wor 4029   -->wf 4885   ` cfv 4889  (class class class)co 5499   1oc1o 5981   [cec 6091   N.cnpi 6351    <N clti 6354    ~Q ceq 6358   Q.cnq 6359    +Q cplq 6361   *Qcrq 6363    <Q cltq 6364
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-coll 3869  ax-sep 3872  ax-nul 3880  ax-pow 3924  ax-pr 3941  ax-un 4166  ax-setind 4256  ax-iinf 4298
This theorem depends on definitions:  df-bi 110  df-dc 743  df-3or 886  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2308  df-rex 2309  df-reu 2310  df-rab 2312  df-v 2556  df-sbc 2762  df-csb 2850  df-dif 2917  df-un 2919  df-in 2921  df-ss 2928  df-nul 3222  df-pw 3358  df-sn 3378  df-pr 3379  df-op 3381  df-uni 3578  df-int 3613  df-iun 3656  df-br 3762  df-opab 3816  df-mpt 3817  df-tr 3852  df-eprel 4023  df-id 4027  df-po 4030  df-iso 4031  df-iord 4099  df-on 4101  df-suc 4104  df-iom 4301  df-xp 4338  df-rel 4339  df-cnv 4340  df-co 4341  df-dm 4342  df-rn 4343  df-res 4344  df-ima 4345  df-iota 4854  df-fun 4891  df-fn 4892  df-f 4893  df-f1 4894  df-fo 4895  df-f1o 4896  df-fv 4897  df-ov 5502  df-oprab 5503  df-mpt2 5504  df-1st 5754  df-2nd 5755  df-recs 5907  df-irdg 5944  df-1o 5988  df-oadd 5992  df-omul 5993  df-er 6093  df-ec 6095  df-qs 6099  df-ni 6383  df-pli 6384  df-mi 6385  df-lti 6386  df-plpq 6423  df-mpq 6424  df-enq 6426  df-nqqs 6427  df-plqqs 6428  df-mqqs 6429  df-1nqqs 6430  df-rq 6431  df-ltnqqs 6432
This theorem is referenced by:  caucvgprlemladdrl  6757
  Copyright terms: Public domain W3C validator