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Theorem caucvgprlemnkj 6745
Description: Lemma for caucvgpr 6761. Part of disjointness. (Contributed by Jim Kingdon, 23-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  )
) ) ) )
caucvgprlemnkj.k  |-  ( ph  ->  K  e.  N. )
caucvgprlemnkj.j  |-  ( ph  ->  J  e.  N. )
caucvgprlemnkj.s  |-  ( ph  ->  S  e.  Q. )
Assertion
Ref Expression
caucvgprlemnkj  |-  ( ph  ->  -.  ( ( S  +Q  ( *Q `  [ <. K ,  1o >. ]  ~Q  ) ) 
<Q  ( F `  K
)  /\  ( ( F `  J )  +Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  ) )  <Q  S ) )
Distinct variable group:    k, F, n
Allowed substitution hints:    ph( k, n)    S( k, n)    J( k, n)    K( k, n)

Proof of Theorem caucvgprlemnkj
Dummy variables  a  b  f  g  h are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ltsonq 6477 . . . 4  |-  <Q  Or  Q.
2 ltrelnq 6444 . . . 4  |-  <Q  C_  ( Q.  X.  Q. )
31, 2son2lpi 4708 . . 3  |-  -.  ( S  <Q  ( F `  J )  /\  ( F `  J )  <Q  S )
4 simprl 483 . . . . . . 7  |-  ( ( ( ph  /\  K  <N  J )  /\  (
( S  +Q  ( *Q `  [ <. K ,  1o >. ]  ~Q  )
)  <Q  ( F `  K )  /\  (
( F `  J
)  +Q  ( *Q
`  [ <. J ,  1o >. ]  ~Q  )
)  <Q  S ) )  ->  ( S  +Q  ( *Q `  [ <. K ,  1o >. ]  ~Q  ) )  <Q  ( F `  K )
)
5 caucvgpr.cau . . . . . . . . . . . 12  |-  ( 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  )
) ) ) )
6 breq1 3764 . . . . . . . . . . . . . 14  |-  ( n  =  a  ->  (
n  <N  k  <->  a  <N  k ) )
7 fveq2 5165 . . . . . . . . . . . . . . . 16  |-  ( n  =  a  ->  ( F `  n )  =  ( F `  a ) )
8 opeq1 3546 . . . . . . . . . . . . . . . . . . 19  |-  ( n  =  a  ->  <. n ,  1o >.  =  <. a ,  1o >. )
98eceq1d 6129 . . . . . . . . . . . . . . . . . 18  |-  ( n  =  a  ->  [ <. n ,  1o >. ]  ~Q  =  [ <. a ,  1o >. ]  ~Q  )
109fveq2d 5169 . . . . . . . . . . . . . . . . 17  |-  ( n  =  a  ->  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  =  ( *Q `  [ <. a ,  1o >. ]  ~Q  ) )
1110oveq2d 5515 . . . . . . . . . . . . . . . 16  |-  ( n  =  a  ->  (
( F `  k
)  +Q  ( *Q
`  [ <. n ,  1o >. ]  ~Q  )
)  =  ( ( F `  k )  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  ) ) )
127, 11breq12d 3774 . . . . . . . . . . . . . . 15  |-  ( n  =  a  ->  (
( F `  n
)  <Q  ( ( F `
 k )  +Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) )  <->  ( F `  a )  <Q  (
( F `  k
)  +Q  ( *Q
`  [ <. a ,  1o >. ]  ~Q  )
) ) )
137, 10oveq12d 5517 . . . . . . . . . . . . . . . 16  |-  ( n  =  a  ->  (
( F `  n
)  +Q  ( *Q
`  [ <. n ,  1o >. ]  ~Q  )
)  =  ( ( F `  a )  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  ) ) )
1413breq2d 3773 . . . . . . . . . . . . . . 15  |-  ( n  =  a  ->  (
( F `  k
)  <Q  ( ( F `
 n )  +Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) )  <->  ( F `  k )  <Q  (
( F `  a
)  +Q  ( *Q
`  [ <. a ,  1o >. ]  ~Q  )
) ) )
1512, 14anbi12d 442 . . . . . . . . . . . . . 14  |-  ( n  =  a  ->  (
( ( F `  n )  <Q  (
( F `  k
)  +Q  ( *Q
`  [ <. n ,  1o >. ]  ~Q  )
)  /\  ( F `  k )  <Q  (
( F `  n
)  +Q  ( *Q
`  [ <. n ,  1o >. ]  ~Q  )
) )  <->  ( ( F `  a )  <Q  ( ( F `  k )  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  )
)  /\  ( F `  k )  <Q  (
( F `  a
)  +Q  ( *Q
`  [ <. a ,  1o >. ]  ~Q  )
) ) ) )
166, 15imbi12d 223 . . . . . . . . . . . . 13  |-  ( n  =  a  ->  (
( n  <N  k  ->  ( ( F `  n )  <Q  (
( F `  k
)  +Q  ( *Q
`  [ <. n ,  1o >. ]  ~Q  )
)  /\  ( F `  k )  <Q  (
( F `  n
)  +Q  ( *Q
`  [ <. n ,  1o >. ]  ~Q  )
) ) )  <->  ( a  <N  k  ->  ( ( F `  a )  <Q  ( ( F `  k )  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  )
)  /\  ( F `  k )  <Q  (
( F `  a
)  +Q  ( *Q
`  [ <. a ,  1o >. ]  ~Q  )
) ) ) ) )
17 breq2 3765 . . . . . . . . . . . . . 14  |-  ( k  =  b  ->  (
a  <N  k  <->  a  <N  b ) )
18 fveq2 5165 . . . . . . . . . . . . . . . . 17  |-  ( k  =  b  ->  ( F `  k )  =  ( F `  b ) )
1918oveq1d 5514 . . . . . . . . . . . . . . . 16  |-  ( k  =  b  ->  (
( F `  k
)  +Q  ( *Q
`  [ <. a ,  1o >. ]  ~Q  )
)  =  ( ( F `  b )  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  ) ) )
2019breq2d 3773 . . . . . . . . . . . . . . 15  |-  ( k  =  b  ->  (
( F `  a
)  <Q  ( ( F `
 k )  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  ) )  <->  ( F `  a )  <Q  (
( F `  b
)  +Q  ( *Q
`  [ <. a ,  1o >. ]  ~Q  )
) ) )
2118breq1d 3771 . . . . . . . . . . . . . . 15  |-  ( k  =  b  ->  (
( F `  k
)  <Q  ( ( F `
 a )  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  ) )  <->  ( F `  b )  <Q  (
( F `  a
)  +Q  ( *Q
`  [ <. a ,  1o >. ]  ~Q  )
) ) )
2220, 21anbi12d 442 . . . . . . . . . . . . . 14  |-  ( k  =  b  ->  (
( ( F `  a )  <Q  (
( F `  k
)  +Q  ( *Q
`  [ <. a ,  1o >. ]  ~Q  )
)  /\  ( F `  k )  <Q  (
( F `  a
)  +Q  ( *Q
`  [ <. a ,  1o >. ]  ~Q  )
) )  <->  ( ( F `  a )  <Q  ( ( F `  b )  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  )
)  /\  ( F `  b )  <Q  (
( F `  a
)  +Q  ( *Q
`  [ <. a ,  1o >. ]  ~Q  )
) ) ) )
2317, 22imbi12d 223 . . . . . . . . . . . . 13  |-  ( k  =  b  ->  (
( a  <N  k  ->  ( ( F `  a )  <Q  (
( F `  k
)  +Q  ( *Q
`  [ <. a ,  1o >. ]  ~Q  )
)  /\  ( F `  k )  <Q  (
( F `  a
)  +Q  ( *Q
`  [ <. a ,  1o >. ]  ~Q  )
) ) )  <->  ( a  <N  b  ->  ( ( F `  a )  <Q  ( ( F `  b )  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  )
)  /\  ( F `  b )  <Q  (
( F `  a
)  +Q  ( *Q
`  [ <. a ,  1o >. ]  ~Q  )
) ) ) ) )
2416, 23cbvral2v 2538 . . . . . . . . . . . 12  |-  ( 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  )
) ) )  <->  A. a  e.  N.  A. b  e. 
N.  ( a  <N 
b  ->  ( ( F `  a )  <Q  ( ( F `  b )  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  )
)  /\  ( F `  b )  <Q  (
( F `  a
)  +Q  ( *Q
`  [ <. a ,  1o >. ]  ~Q  )
) ) ) )
255, 24sylib 127 . . . . . . . . . . 11  |-  ( ph  ->  A. a  e.  N.  A. b  e.  N.  (
a  <N  b  ->  (
( F `  a
)  <Q  ( ( F `
 b )  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  ) )  /\  ( F `  b ) 
<Q  ( ( F `  a )  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  )
) ) ) )
26 caucvgprlemnkj.k . . . . . . . . . . . 12  |-  ( ph  ->  K  e.  N. )
27 caucvgprlemnkj.j . . . . . . . . . . . 12  |-  ( ph  ->  J  e.  N. )
28 breq1 3764 . . . . . . . . . . . . . 14  |-  ( a  =  K  ->  (
a  <N  b  <->  K  <N  b ) )
29 fveq2 5165 . . . . . . . . . . . . . . . 16  |-  ( a  =  K  ->  ( F `  a )  =  ( F `  K ) )
30 opeq1 3546 . . . . . . . . . . . . . . . . . . 19  |-  ( a  =  K  ->  <. a ,  1o >.  =  <. K ,  1o >. )
3130eceq1d 6129 . . . . . . . . . . . . . . . . . 18  |-  ( a  =  K  ->  [ <. a ,  1o >. ]  ~Q  =  [ <. K ,  1o >. ]  ~Q  )
3231fveq2d 5169 . . . . . . . . . . . . . . . . 17  |-  ( a  =  K  ->  ( *Q `  [ <. a ,  1o >. ]  ~Q  )  =  ( *Q `  [ <. K ,  1o >. ]  ~Q  ) )
3332oveq2d 5515 . . . . . . . . . . . . . . . 16  |-  ( a  =  K  ->  (
( F `  b
)  +Q  ( *Q
`  [ <. a ,  1o >. ]  ~Q  )
)  =  ( ( F `  b )  +Q  ( *Q `  [ <. K ,  1o >. ]  ~Q  ) ) )
3429, 33breq12d 3774 . . . . . . . . . . . . . . 15  |-  ( a  =  K  ->  (
( F `  a
)  <Q  ( ( F `
 b )  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  ) )  <->  ( F `  K )  <Q  (
( F `  b
)  +Q  ( *Q
`  [ <. K ,  1o >. ]  ~Q  )
) ) )
3529, 32oveq12d 5517 . . . . . . . . . . . . . . . 16  |-  ( a  =  K  ->  (
( F `  a
)  +Q  ( *Q
`  [ <. a ,  1o >. ]  ~Q  )
)  =  ( ( F `  K )  +Q  ( *Q `  [ <. K ,  1o >. ]  ~Q  ) ) )
3635breq2d 3773 . . . . . . . . . . . . . . 15  |-  ( a  =  K  ->  (
( F `  b
)  <Q  ( ( F `
 a )  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  ) )  <->  ( F `  b )  <Q  (
( F `  K
)  +Q  ( *Q
`  [ <. K ,  1o >. ]  ~Q  )
) ) )
3734, 36anbi12d 442 . . . . . . . . . . . . . 14  |-  ( a  =  K  ->  (
( ( F `  a )  <Q  (
( F `  b
)  +Q  ( *Q
`  [ <. a ,  1o >. ]  ~Q  )
)  /\  ( F `  b )  <Q  (
( F `  a
)  +Q  ( *Q
`  [ <. a ,  1o >. ]  ~Q  )
) )  <->  ( ( F `  K )  <Q  ( ( F `  b )  +Q  ( *Q `  [ <. K ,  1o >. ]  ~Q  )
)  /\  ( F `  b )  <Q  (
( F `  K
)  +Q  ( *Q
`  [ <. K ,  1o >. ]  ~Q  )
) ) ) )
3828, 37imbi12d 223 . . . . . . . . . . . . 13  |-  ( a  =  K  ->  (
( a  <N  b  ->  ( ( F `  a )  <Q  (
( F `  b
)  +Q  ( *Q
`  [ <. a ,  1o >. ]  ~Q  )
)  /\  ( F `  b )  <Q  (
( F `  a
)  +Q  ( *Q
`  [ <. a ,  1o >. ]  ~Q  )
) ) )  <->  ( K  <N  b  ->  ( ( F `  K )  <Q  ( ( F `  b )  +Q  ( *Q `  [ <. K ,  1o >. ]  ~Q  )
)  /\  ( F `  b )  <Q  (
( F `  K
)  +Q  ( *Q
`  [ <. K ,  1o >. ]  ~Q  )
) ) ) ) )
39 breq2 3765 . . . . . . . . . . . . . 14  |-  ( b  =  J  ->  ( K  <N  b  <->  K  <N  J ) )
40 fveq2 5165 . . . . . . . . . . . . . . . . 17  |-  ( b  =  J  ->  ( F `  b )  =  ( F `  J ) )
4140oveq1d 5514 . . . . . . . . . . . . . . . 16  |-  ( b  =  J  ->  (
( F `  b
)  +Q  ( *Q
`  [ <. K ,  1o >. ]  ~Q  )
)  =  ( ( F `  J )  +Q  ( *Q `  [ <. K ,  1o >. ]  ~Q  ) ) )
4241breq2d 3773 . . . . . . . . . . . . . . 15  |-  ( b  =  J  ->  (
( F `  K
)  <Q  ( ( F `
 b )  +Q  ( *Q `  [ <. K ,  1o >. ]  ~Q  ) )  <->  ( F `  K )  <Q  (
( F `  J
)  +Q  ( *Q
`  [ <. K ,  1o >. ]  ~Q  )
) ) )
4340breq1d 3771 . . . . . . . . . . . . . . 15  |-  ( b  =  J  ->  (
( F `  b
)  <Q  ( ( F `
 K )  +Q  ( *Q `  [ <. K ,  1o >. ]  ~Q  ) )  <->  ( F `  J )  <Q  (
( F `  K
)  +Q  ( *Q
`  [ <. K ,  1o >. ]  ~Q  )
) ) )
4442, 43anbi12d 442 . . . . . . . . . . . . . 14  |-  ( b  =  J  ->  (
( ( F `  K )  <Q  (
( F `  b
)  +Q  ( *Q
`  [ <. K ,  1o >. ]  ~Q  )
)  /\  ( F `  b )  <Q  (
( F `  K
)  +Q  ( *Q
`  [ <. K ,  1o >. ]  ~Q  )
) )  <->  ( ( F `  K )  <Q  ( ( F `  J )  +Q  ( *Q `  [ <. K ,  1o >. ]  ~Q  )
)  /\  ( F `  J )  <Q  (
( F `  K
)  +Q  ( *Q
`  [ <. K ,  1o >. ]  ~Q  )
) ) ) )
4539, 44imbi12d 223 . . . . . . . . . . . . 13  |-  ( b  =  J  ->  (
( K  <N  b  ->  ( ( F `  K )  <Q  (
( F `  b
)  +Q  ( *Q
`  [ <. K ,  1o >. ]  ~Q  )
)  /\  ( F `  b )  <Q  (
( F `  K
)  +Q  ( *Q
`  [ <. K ,  1o >. ]  ~Q  )
) ) )  <->  ( K  <N  J  ->  ( ( F `  K )  <Q  ( ( F `  J )  +Q  ( *Q `  [ <. K ,  1o >. ]  ~Q  )
)  /\  ( F `  J )  <Q  (
( F `  K
)  +Q  ( *Q
`  [ <. K ,  1o >. ]  ~Q  )
) ) ) ) )
4638, 45rspc2v 2659 . . . . . . . . . . . 12  |-  ( ( K  e.  N.  /\  J  e.  N. )  ->  ( A. a  e. 
N.  A. b  e.  N.  ( a  <N  b  ->  ( ( F `  a )  <Q  (
( F `  b
)  +Q  ( *Q
`  [ <. a ,  1o >. ]  ~Q  )
)  /\  ( F `  b )  <Q  (
( F `  a
)  +Q  ( *Q
`  [ <. a ,  1o >. ]  ~Q  )
) ) )  -> 
( K  <N  J  -> 
( ( F `  K )  <Q  (
( F `  J
)  +Q  ( *Q
`  [ <. K ,  1o >. ]  ~Q  )
)  /\  ( F `  J )  <Q  (
( F `  K
)  +Q  ( *Q
`  [ <. K ,  1o >. ]  ~Q  )
) ) ) ) )
4726, 27, 46syl2anc 391 . . . . . . . . . . 11  |-  ( ph  ->  ( A. a  e. 
N.  A. b  e.  N.  ( a  <N  b  ->  ( ( F `  a )  <Q  (
( F `  b
)  +Q  ( *Q
`  [ <. a ,  1o >. ]  ~Q  )
)  /\  ( F `  b )  <Q  (
( F `  a
)  +Q  ( *Q
`  [ <. a ,  1o >. ]  ~Q  )
) ) )  -> 
( K  <N  J  -> 
( ( F `  K )  <Q  (
( F `  J
)  +Q  ( *Q
`  [ <. K ,  1o >. ]  ~Q  )
)  /\  ( F `  J )  <Q  (
( F `  K
)  +Q  ( *Q
`  [ <. K ,  1o >. ]  ~Q  )
) ) ) ) )
4825, 47mpd 13 . . . . . . . . . 10  |-  ( ph  ->  ( K  <N  J  -> 
( ( F `  K )  <Q  (
( F `  J
)  +Q  ( *Q
`  [ <. K ,  1o >. ]  ~Q  )
)  /\  ( F `  J )  <Q  (
( F `  K
)  +Q  ( *Q
`  [ <. K ,  1o >. ]  ~Q  )
) ) ) )
4948imp 115 . . . . . . . . 9  |-  ( (
ph  /\  K  <N  J )  ->  ( ( F `  K )  <Q  ( ( F `  J )  +Q  ( *Q `  [ <. K ,  1o >. ]  ~Q  )
)  /\  ( F `  J )  <Q  (
( F `  K
)  +Q  ( *Q
`  [ <. K ,  1o >. ]  ~Q  )
) ) )
5049simpld 105 . . . . . . . 8  |-  ( (
ph  /\  K  <N  J )  ->  ( F `  K )  <Q  (
( F `  J
)  +Q  ( *Q
`  [ <. K ,  1o >. ]  ~Q  )
) )
5150adantr 261 . . . . . . 7  |-  ( ( ( ph  /\  K  <N  J )  /\  (
( S  +Q  ( *Q `  [ <. K ,  1o >. ]  ~Q  )
)  <Q  ( F `  K )  /\  (
( F `  J
)  +Q  ( *Q
`  [ <. J ,  1o >. ]  ~Q  )
)  <Q  S ) )  ->  ( F `  K )  <Q  (
( F `  J
)  +Q  ( *Q
`  [ <. K ,  1o >. ]  ~Q  )
) )
521, 2sotri 4707 . . . . . . 7  |-  ( ( ( S  +Q  ( *Q `  [ <. K ,  1o >. ]  ~Q  )
)  <Q  ( F `  K )  /\  ( F `  K )  <Q  ( ( F `  J )  +Q  ( *Q `  [ <. K ,  1o >. ]  ~Q  )
) )  ->  ( S  +Q  ( *Q `  [ <. K ,  1o >. ]  ~Q  ) ) 
<Q  ( ( F `  J )  +Q  ( *Q `  [ <. K ,  1o >. ]  ~Q  )
) )
534, 51, 52syl2anc 391 . . . . . 6  |-  ( ( ( ph  /\  K  <N  J )  /\  (
( S  +Q  ( *Q `  [ <. K ,  1o >. ]  ~Q  )
)  <Q  ( F `  K )  /\  (
( F `  J
)  +Q  ( *Q
`  [ <. J ,  1o >. ]  ~Q  )
)  <Q  S ) )  ->  ( S  +Q  ( *Q `  [ <. K ,  1o >. ]  ~Q  ) )  <Q  (
( F `  J
)  +Q  ( *Q
`  [ <. K ,  1o >. ]  ~Q  )
) )
54 ltanqg 6479 . . . . . . . 8  |-  ( ( f  e.  Q.  /\  g  e.  Q.  /\  h  e.  Q. )  ->  (
f  <Q  g  <->  ( h  +Q  f )  <Q  (
h  +Q  g ) ) )
5554adantl 262 . . . . . . 7  |-  ( ( ( ( ph  /\  K  <N  J )  /\  ( ( S  +Q  ( *Q `  [ <. K ,  1o >. ]  ~Q  ) )  <Q  ( F `  K )  /\  ( ( F `  J )  +Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  )
)  <Q  S ) )  /\  ( f  e. 
Q.  /\  g  e.  Q.  /\  h  e.  Q. ) )  ->  (
f  <Q  g  <->  ( h  +Q  f )  <Q  (
h  +Q  g ) ) )
56 caucvgprlemnkj.s . . . . . . . 8  |-  ( ph  ->  S  e.  Q. )
5756ad2antrr 457 . . . . . . 7  |-  ( ( ( ph  /\  K  <N  J )  /\  (
( S  +Q  ( *Q `  [ <. K ,  1o >. ]  ~Q  )
)  <Q  ( F `  K )  /\  (
( F `  J
)  +Q  ( *Q
`  [ <. J ,  1o >. ]  ~Q  )
)  <Q  S ) )  ->  S  e.  Q. )
58 caucvgpr.f . . . . . . . . 9  |-  ( ph  ->  F : N. --> Q. )
5958, 27ffvelrnd 5290 . . . . . . . 8  |-  ( ph  ->  ( F `  J
)  e.  Q. )
6059ad2antrr 457 . . . . . . 7  |-  ( ( ( ph  /\  K  <N  J )  /\  (
( S  +Q  ( *Q `  [ <. K ,  1o >. ]  ~Q  )
)  <Q  ( F `  K )  /\  (
( F `  J
)  +Q  ( *Q
`  [ <. J ,  1o >. ]  ~Q  )
)  <Q  S ) )  ->  ( F `  J )  e.  Q. )
61 nnnq 6501 . . . . . . . . 9  |-  ( K  e.  N.  ->  [ <. K ,  1o >. ]  ~Q  e.  Q. )
62 recclnq 6471 . . . . . . . . 9  |-  ( [
<. K ,  1o >. ]  ~Q  e.  Q.  ->  ( *Q `  [ <. K ,  1o >. ]  ~Q  )  e.  Q. )
6326, 61, 623syl 17 . . . . . . . 8  |-  ( ph  ->  ( *Q `  [ <. K ,  1o >. ]  ~Q  )  e.  Q. )
6463ad2antrr 457 . . . . . . 7  |-  ( ( ( ph  /\  K  <N  J )  /\  (
( S  +Q  ( *Q `  [ <. K ,  1o >. ]  ~Q  )
)  <Q  ( F `  K )  /\  (
( F `  J
)  +Q  ( *Q
`  [ <. J ,  1o >. ]  ~Q  )
)  <Q  S ) )  ->  ( *Q `  [ <. K ,  1o >. ]  ~Q  )  e. 
Q. )
65 addcomnqg 6460 . . . . . . . 8  |-  ( ( f  e.  Q.  /\  g  e.  Q. )  ->  ( f  +Q  g
)  =  ( g  +Q  f ) )
6665adantl 262 . . . . . . 7  |-  ( ( ( ( ph  /\  K  <N  J )  /\  ( ( S  +Q  ( *Q `  [ <. K ,  1o >. ]  ~Q  ) )  <Q  ( F `  K )  /\  ( ( F `  J )  +Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  )
)  <Q  S ) )  /\  ( f  e. 
Q.  /\  g  e.  Q. ) )  ->  (
f  +Q  g )  =  ( g  +Q  f ) )
6755, 57, 60, 64, 66caovord2d 5657 . . . . . 6  |-  ( ( ( ph  /\  K  <N  J )  /\  (
( S  +Q  ( *Q `  [ <. K ,  1o >. ]  ~Q  )
)  <Q  ( F `  K )  /\  (
( F `  J
)  +Q  ( *Q
`  [ <. J ,  1o >. ]  ~Q  )
)  <Q  S ) )  ->  ( S  <Q  ( F `  J )  <-> 
( S  +Q  ( *Q `  [ <. K ,  1o >. ]  ~Q  )
)  <Q  ( ( F `
 J )  +Q  ( *Q `  [ <. K ,  1o >. ]  ~Q  ) ) ) )
6853, 67mpbird 156 . . . . 5  |-  ( ( ( ph  /\  K  <N  J )  /\  (
( S  +Q  ( *Q `  [ <. K ,  1o >. ]  ~Q  )
)  <Q  ( F `  K )  /\  (
( F `  J
)  +Q  ( *Q
`  [ <. J ,  1o >. ]  ~Q  )
)  <Q  S ) )  ->  S  <Q  ( F `  J )
)
69 nnnq 6501 . . . . . . . . 9  |-  ( J  e.  N.  ->  [ <. J ,  1o >. ]  ~Q  e.  Q. )
70 recclnq 6471 . . . . . . . . 9  |-  ( [
<. J ,  1o >. ]  ~Q  e.  Q.  ->  ( *Q `  [ <. J ,  1o >. ]  ~Q  )  e.  Q. )
7127, 69, 703syl 17 . . . . . . . 8  |-  ( ph  ->  ( *Q `  [ <. J ,  1o >. ]  ~Q  )  e.  Q. )
7271ad2antrr 457 . . . . . . 7  |-  ( ( ( ph  /\  K  <N  J )  /\  (
( S  +Q  ( *Q `  [ <. K ,  1o >. ]  ~Q  )
)  <Q  ( F `  K )  /\  (
( F `  J
)  +Q  ( *Q
`  [ <. J ,  1o >. ]  ~Q  )
)  <Q  S ) )  ->  ( *Q `  [ <. J ,  1o >. ]  ~Q  )  e. 
Q. )
73 ltaddnq 6486 . . . . . . 7  |-  ( ( ( F `  J
)  e.  Q.  /\  ( *Q `  [ <. J ,  1o >. ]  ~Q  )  e.  Q. )  ->  ( F `  J
)  <Q  ( ( F `
 J )  +Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  ) ) )
7460, 72, 73syl2anc 391 . . . . . 6  |-  ( ( ( ph  /\  K  <N  J )  /\  (
( S  +Q  ( *Q `  [ <. K ,  1o >. ]  ~Q  )
)  <Q  ( F `  K )  /\  (
( F `  J
)  +Q  ( *Q
`  [ <. J ,  1o >. ]  ~Q  )
)  <Q  S ) )  ->  ( F `  J )  <Q  (
( F `  J
)  +Q  ( *Q
`  [ <. J ,  1o >. ]  ~Q  )
) )
75 simprr 484 . . . . . 6  |-  ( ( ( ph  /\  K  <N  J )  /\  (
( S  +Q  ( *Q `  [ <. K ,  1o >. ]  ~Q  )
)  <Q  ( F `  K )  /\  (
( F `  J
)  +Q  ( *Q
`  [ <. J ,  1o >. ]  ~Q  )
)  <Q  S ) )  ->  ( ( F `
 J )  +Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  ) )  <Q  S )
761, 2sotri 4707 . . . . . 6  |-  ( ( ( F `  J
)  <Q  ( ( F `
 J )  +Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  ) )  /\  ( ( F `  J )  +Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  )
)  <Q  S )  -> 
( F `  J
)  <Q  S )
7774, 75, 76syl2anc 391 . . . . 5  |-  ( ( ( ph  /\  K  <N  J )  /\  (
( S  +Q  ( *Q `  [ <. K ,  1o >. ]  ~Q  )
)  <Q  ( F `  K )  /\  (
( F `  J
)  +Q  ( *Q
`  [ <. J ,  1o >. ]  ~Q  )
)  <Q  S ) )  ->  ( F `  J )  <Q  S )
7868, 77jca 290 . . . 4  |-  ( ( ( ph  /\  K  <N  J )  /\  (
( S  +Q  ( *Q `  [ <. K ,  1o >. ]  ~Q  )
)  <Q  ( F `  K )  /\  (
( F `  J
)  +Q  ( *Q
`  [ <. J ,  1o >. ]  ~Q  )
)  <Q  S ) )  ->  ( S  <Q  ( F `  J )  /\  ( F `  J )  <Q  S ) )
7978ex 108 . . 3  |-  ( (
ph  /\  K  <N  J )  ->  ( (
( S  +Q  ( *Q `  [ <. K ,  1o >. ]  ~Q  )
)  <Q  ( F `  K )  /\  (
( F `  J
)  +Q  ( *Q
`  [ <. J ,  1o >. ]  ~Q  )
)  <Q  S )  -> 
( S  <Q  ( F `  J )  /\  ( F `  J
)  <Q  S ) ) )
803, 79mtoi 590 . 2  |-  ( (
ph  /\  K  <N  J )  ->  -.  (
( S  +Q  ( *Q `  [ <. K ,  1o >. ]  ~Q  )
)  <Q  ( F `  K )  /\  (
( F `  J
)  +Q  ( *Q
`  [ <. J ,  1o >. ]  ~Q  )
)  <Q  S ) )
811, 2son2lpi 4708 . . 3  |-  -.  (
( ( S  +Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  ) )  +Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  )
)  <Q  S  /\  S  <Q  ( ( S  +Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  ) )  +Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  )
) )
82 opeq1 3546 . . . . . . . . . . . 12  |-  ( K  =  J  ->  <. K ,  1o >.  =  <. J ,  1o >. )
8382eceq1d 6129 . . . . . . . . . . 11  |-  ( K  =  J  ->  [ <. K ,  1o >. ]  ~Q  =  [ <. J ,  1o >. ]  ~Q  )
8483fveq2d 5169 . . . . . . . . . 10  |-  ( K  =  J  ->  ( *Q `  [ <. K ,  1o >. ]  ~Q  )  =  ( *Q `  [ <. J ,  1o >. ]  ~Q  ) )
8584oveq2d 5515 . . . . . . . . 9  |-  ( K  =  J  ->  ( S  +Q  ( *Q `  [ <. K ,  1o >. ]  ~Q  ) )  =  ( S  +Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  ) ) )
86 fveq2 5165 . . . . . . . . 9  |-  ( K  =  J  ->  ( F `  K )  =  ( F `  J ) )
8785, 86breq12d 3774 . . . . . . . 8  |-  ( K  =  J  ->  (
( S  +Q  ( *Q `  [ <. K ,  1o >. ]  ~Q  )
)  <Q  ( F `  K )  <->  ( S  +Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  ) )  <Q 
( F `  J
) ) )
8887anbi1d 438 . . . . . . 7  |-  ( K  =  J  ->  (
( ( S  +Q  ( *Q `  [ <. K ,  1o >. ]  ~Q  ) )  <Q  ( F `  K )  /\  ( ( F `  J )  +Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  )
)  <Q  S )  <->  ( ( S  +Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  ) ) 
<Q  ( F `  J
)  /\  ( ( F `  J )  +Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  ) )  <Q  S ) ) )
8988adantl 262 . . . . . 6  |-  ( (
ph  /\  K  =  J )  ->  (
( ( S  +Q  ( *Q `  [ <. K ,  1o >. ]  ~Q  ) )  <Q  ( F `  K )  /\  ( ( F `  J )  +Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  )
)  <Q  S )  <->  ( ( S  +Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  ) ) 
<Q  ( F `  J
)  /\  ( ( F `  J )  +Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  ) )  <Q  S ) ) )
9054adantl 262 . . . . . . . . 9  |-  ( (
ph  /\  ( f  e.  Q.  /\  g  e. 
Q.  /\  h  e.  Q. ) )  ->  (
f  <Q  g  <->  ( h  +Q  f )  <Q  (
h  +Q  g ) ) )
91 addclnq 6454 . . . . . . . . . 10  |-  ( ( S  e.  Q.  /\  ( *Q `  [ <. J ,  1o >. ]  ~Q  )  e.  Q. )  ->  ( S  +Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  )
)  e.  Q. )
9256, 71, 91syl2anc 391 . . . . . . . . 9  |-  ( ph  ->  ( S  +Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  )
)  e.  Q. )
9365adantl 262 . . . . . . . . 9  |-  ( (
ph  /\  ( f  e.  Q.  /\  g  e. 
Q. ) )  -> 
( f  +Q  g
)  =  ( g  +Q  f ) )
9490, 92, 59, 71, 93caovord2d 5657 . . . . . . . 8  |-  ( ph  ->  ( ( S  +Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  ) )  <Q  ( F `  J )  <->  ( ( S  +Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  )
)  +Q  ( *Q
`  [ <. J ,  1o >. ]  ~Q  )
)  <Q  ( ( F `
 J )  +Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  ) ) ) )
9594adantr 261 . . . . . . 7  |-  ( (
ph  /\  K  =  J )  ->  (
( S  +Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  )
)  <Q  ( F `  J )  <->  ( ( S  +Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  ) )  +Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  ) ) 
<Q  ( ( F `  J )  +Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  )
) ) )
9695anbi1d 438 . . . . . 6  |-  ( (
ph  /\  K  =  J )  ->  (
( ( S  +Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  ) )  <Q  ( F `  J )  /\  ( ( F `  J )  +Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  )
)  <Q  S )  <->  ( (
( S  +Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  )
)  +Q  ( *Q
`  [ <. J ,  1o >. ]  ~Q  )
)  <Q  ( ( F `
 J )  +Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  ) )  /\  ( ( F `  J )  +Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  )
)  <Q  S ) ) )
9789, 96bitrd 177 . . . . 5  |-  ( (
ph  /\  K  =  J )  ->  (
( ( S  +Q  ( *Q `  [ <. K ,  1o >. ]  ~Q  ) )  <Q  ( F `  K )  /\  ( ( F `  J )  +Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  )
)  <Q  S )  <->  ( (
( S  +Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  )
)  +Q  ( *Q
`  [ <. J ,  1o >. ]  ~Q  )
)  <Q  ( ( F `
 J )  +Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  ) )  /\  ( ( F `  J )  +Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  )
)  <Q  S ) ) )
981, 2sotri 4707 . . . . 5  |-  ( ( ( ( S  +Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  ) )  +Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  )
)  <Q  ( ( F `
 J )  +Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  ) )  /\  ( ( F `  J )  +Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  )
)  <Q  S )  -> 
( ( S  +Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  ) )  +Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  )
)  <Q  S )
9997, 98syl6bi 152 . . . 4  |-  ( (
ph  /\  K  =  J )  ->  (
( ( S  +Q  ( *Q `  [ <. K ,  1o >. ]  ~Q  ) )  <Q  ( F `  K )  /\  ( ( F `  J )  +Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  )
)  <Q  S )  -> 
( ( S  +Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  ) )  +Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  )
)  <Q  S ) )
100 ltaddnq 6486 . . . . . . 7  |-  ( ( S  e.  Q.  /\  ( *Q `  [ <. J ,  1o >. ]  ~Q  )  e.  Q. )  ->  S  <Q  ( S  +Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  ) ) )
10156, 71, 100syl2anc 391 . . . . . 6  |-  ( ph  ->  S  <Q  ( S  +Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  ) ) )
102 ltaddnq 6486 . . . . . . 7  |-  ( ( ( S  +Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  )
)  e.  Q.  /\  ( *Q `  [ <. J ,  1o >. ]  ~Q  )  e.  Q. )  ->  ( S  +Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  )
)  <Q  ( ( S  +Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  ) )  +Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  ) ) )
10392, 71, 102syl2anc 391 . . . . . 6  |-  ( ph  ->  ( S  +Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  )
)  <Q  ( ( S  +Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  ) )  +Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  ) ) )
1041, 2sotri 4707 . . . . . 6  |-  ( ( S  <Q  ( S  +Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  ) )  /\  ( S  +Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  )
)  <Q  ( ( S  +Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  ) )  +Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  ) ) )  ->  S  <Q  ( ( S  +Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  )
)  +Q  ( *Q
`  [ <. J ,  1o >. ]  ~Q  )
) )
105101, 103, 104syl2anc 391 . . . . 5  |-  ( ph  ->  S  <Q  ( ( S  +Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  ) )  +Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  ) ) )
106105adantr 261 . . . 4  |-  ( (
ph  /\  K  =  J )  ->  S  <Q  ( ( S  +Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  ) )  +Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  )
) )
10799, 106jctird 300 . . 3  |-  ( (
ph  /\  K  =  J )  ->  (
( ( S  +Q  ( *Q `  [ <. K ,  1o >. ]  ~Q  ) )  <Q  ( F `  K )  /\  ( ( F `  J )  +Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  )
)  <Q  S )  -> 
( ( ( S  +Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  ) )  +Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  ) ) 
<Q  S  /\  S  <Q  ( ( S  +Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  )
)  +Q  ( *Q
`  [ <. J ,  1o >. ]  ~Q  )
) ) ) )
10881, 107mtoi 590 . 2  |-  ( (
ph  /\  K  =  J )  ->  -.  ( ( S  +Q  ( *Q `  [ <. K ,  1o >. ]  ~Q  ) )  <Q  ( F `  K )  /\  ( ( F `  J )  +Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  )
)  <Q  S ) )
1091, 2son2lpi 4708 . . 3  |-  -.  ( S  <Q  ( ( F `
 J )  +Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  ) )  /\  ( ( F `  J )  +Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  )
)  <Q  S )
11056ad2antrr 457 . . . . . . 7  |-  ( ( ( ph  /\  J  <N  K )  /\  (
( S  +Q  ( *Q `  [ <. K ,  1o >. ]  ~Q  )
)  <Q  ( F `  K )  /\  (
( F `  J
)  +Q  ( *Q
`  [ <. J ,  1o >. ]  ~Q  )
)  <Q  S ) )  ->  S  e.  Q. )
11163ad2antrr 457 . . . . . . 7  |-  ( ( ( ph  /\  J  <N  K )  /\  (
( S  +Q  ( *Q `  [ <. K ,  1o >. ]  ~Q  )
)  <Q  ( F `  K )  /\  (
( F `  J
)  +Q  ( *Q
`  [ <. J ,  1o >. ]  ~Q  )
)  <Q  S ) )  ->  ( *Q `  [ <. K ,  1o >. ]  ~Q  )  e. 
Q. )
112 ltaddnq 6486 . . . . . . 7  |-  ( ( S  e.  Q.  /\  ( *Q `  [ <. K ,  1o >. ]  ~Q  )  e.  Q. )  ->  S  <Q  ( S  +Q  ( *Q `  [ <. K ,  1o >. ]  ~Q  ) ) )
113110, 111, 112syl2anc 391 . . . . . 6  |-  ( ( ( ph  /\  J  <N  K )  /\  (
( S  +Q  ( *Q `  [ <. K ,  1o >. ]  ~Q  )
)  <Q  ( F `  K )  /\  (
( F `  J
)  +Q  ( *Q
`  [ <. J ,  1o >. ]  ~Q  )
)  <Q  S ) )  ->  S  <Q  ( S  +Q  ( *Q `  [ <. K ,  1o >. ]  ~Q  ) ) )
114 simprl 483 . . . . . . 7  |-  ( ( ( ph  /\  J  <N  K )  /\  (
( S  +Q  ( *Q `  [ <. K ,  1o >. ]  ~Q  )
)  <Q  ( F `  K )  /\  (
( F `  J
)  +Q  ( *Q
`  [ <. J ,  1o >. ]  ~Q  )
)  <Q  S ) )  ->  ( S  +Q  ( *Q `  [ <. K ,  1o >. ]  ~Q  ) )  <Q  ( F `  K )
)
115 breq1 3764 . . . . . . . . . . . . . 14  |-  ( a  =  J  ->  (
a  <N  b  <->  J  <N  b ) )
116 fveq2 5165 . . . . . . . . . . . . . . . 16  |-  ( a  =  J  ->  ( F `  a )  =  ( F `  J ) )
117 opeq1 3546 . . . . . . . . . . . . . . . . . . 19  |-  ( a  =  J  ->  <. a ,  1o >.  =  <. J ,  1o >. )
118117eceq1d 6129 . . . . . . . . . . . . . . . . . 18  |-  ( a  =  J  ->  [ <. a ,  1o >. ]  ~Q  =  [ <. J ,  1o >. ]  ~Q  )
119118fveq2d 5169 . . . . . . . . . . . . . . . . 17  |-  ( a  =  J  ->  ( *Q `  [ <. a ,  1o >. ]  ~Q  )  =  ( *Q `  [ <. J ,  1o >. ]  ~Q  ) )
120119oveq2d 5515 . . . . . . . . . . . . . . . 16  |-  ( a  =  J  ->  (
( F `  b
)  +Q  ( *Q
`  [ <. a ,  1o >. ]  ~Q  )
)  =  ( ( F `  b )  +Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  ) ) )
121116, 120breq12d 3774 . . . . . . . . . . . . . . 15  |-  ( a  =  J  ->  (
( F `  a
)  <Q  ( ( F `
 b )  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  ) )  <->  ( F `  J )  <Q  (
( F `  b
)  +Q  ( *Q
`  [ <. J ,  1o >. ]  ~Q  )
) ) )
122116, 119oveq12d 5517 . . . . . . . . . . . . . . . 16  |-  ( a  =  J  ->  (
( F `  a
)  +Q  ( *Q
`  [ <. a ,  1o >. ]  ~Q  )
)  =  ( ( F `  J )  +Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  ) ) )
123122breq2d 3773 . . . . . . . . . . . . . . 15  |-  ( a  =  J  ->  (
( F `  b
)  <Q  ( ( F `
 a )  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  ) )  <->  ( F `  b )  <Q  (
( F `  J
)  +Q  ( *Q
`  [ <. J ,  1o >. ]  ~Q  )
) ) )
124121, 123anbi12d 442 . . . . . . . . . . . . . 14  |-  ( a  =  J  ->  (
( ( F `  a )  <Q  (
( F `  b
)  +Q  ( *Q
`  [ <. a ,  1o >. ]  ~Q  )
)  /\  ( F `  b )  <Q  (
( F `  a
)  +Q  ( *Q
`  [ <. a ,  1o >. ]  ~Q  )
) )  <->  ( ( F `  J )  <Q  ( ( F `  b )  +Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  )
)  /\  ( F `  b )  <Q  (
( F `  J
)  +Q  ( *Q
`  [ <. J ,  1o >. ]  ~Q  )
) ) ) )
125115, 124imbi12d 223 . . . . . . . . . . . . 13  |-  ( a  =  J  ->  (
( a  <N  b  ->  ( ( F `  a )  <Q  (
( F `  b
)  +Q  ( *Q
`  [ <. a ,  1o >. ]  ~Q  )
)  /\  ( F `  b )  <Q  (
( F `  a
)  +Q  ( *Q
`  [ <. a ,  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  )
) ) ) ) )
126 breq2 3765 . . . . . . . . . . . . . 14  |-  ( b  =  K  ->  ( J  <N  b  <->  J  <N  K ) )
127 fveq2 5165 . . . . . . . . . . . . . . . . 17  |-  ( b  =  K  ->  ( F `  b )  =  ( F `  K ) )
128127oveq1d 5514 . . . . . . . . . . . . . . . 16  |-  ( b  =  K  ->  (
( F `  b
)  +Q  ( *Q
`  [ <. J ,  1o >. ]  ~Q  )
)  =  ( ( F `  K )  +Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  ) ) )
129128breq2d 3773 . . . . . . . . . . . . . . 15  |-  ( b  =  K  ->  (
( F `  J
)  <Q  ( ( F `
 b )  +Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  ) )  <->  ( F `  J )  <Q  (
( F `  K
)  +Q  ( *Q
`  [ <. J ,  1o >. ]  ~Q  )
) ) )
130127breq1d 3771 . . . . . . . . . . . . . . 15  |-  ( b  =  K  ->  (
( F `  b
)  <Q  ( ( F `
 J )  +Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  ) )  <->  ( F `  K )  <Q  (
( F `  J
)  +Q  ( *Q
`  [ <. J ,  1o >. ]  ~Q  )
) ) )
131129, 130anbi12d 442 . . . . . . . . . . . . . 14  |-  ( b  =  K  ->  (
( ( F `  J )  <Q  (
( F `  b
)  +Q  ( *Q
`  [ <. J ,  1o >. ]  ~Q  )
)  /\  ( F `  b )  <Q  (
( F `  J
)  +Q  ( *Q
`  [ <. J ,  1o >. ]  ~Q  )
) )  <->  ( ( F `  J )  <Q  ( ( F `  K )  +Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  )
)  /\  ( F `  K )  <Q  (
( F `  J
)  +Q  ( *Q
`  [ <. J ,  1o >. ]  ~Q  )
) ) ) )
132126, 131imbi12d 223 . . . . . . . . . . . . 13  |-  ( b  =  K  ->  (
( J  <N  b  ->  ( ( F `  J )  <Q  (
( F `  b
)  +Q  ( *Q
`  [ <. J ,  1o >. ]  ~Q  )
)  /\  ( F `  b )  <Q  (
( F `  J
)  +Q  ( *Q
`  [ <. J ,  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  )
) ) ) ) )
133125, 132rspc2v 2659 . . . . . . . . . . . 12  |-  ( ( J  e.  N.  /\  K  e.  N. )  ->  ( A. a  e. 
N.  A. b  e.  N.  ( a  <N  b  ->  ( ( F `  a )  <Q  (
( F `  b
)  +Q  ( *Q
`  [ <. a ,  1o >. ]  ~Q  )
)  /\  ( F `  b )  <Q  (
( F `  a
)  +Q  ( *Q
`  [ <. a ,  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  )
) ) ) ) )
13427, 26, 133syl2anc 391 . . . . . . . . . . 11  |-  ( ph  ->  ( A. a  e. 
N.  A. b  e.  N.  ( a  <N  b  ->  ( ( F `  a )  <Q  (
( F `  b
)  +Q  ( *Q
`  [ <. a ,  1o >. ]  ~Q  )
)  /\  ( F `  b )  <Q  (
( F `  a
)  +Q  ( *Q
`  [ <. a ,  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  )
) ) ) ) )
13525, 134mpd 13 . . . . . . . . . 10  |-  ( ph  ->  ( J  <N  K  -> 
( ( F `  J )  <Q  (
( F `  K
)  +Q  ( *Q
`  [ <. J ,  1o >. ]  ~Q  )
)  /\  ( F `  K )  <Q  (
( F `  J
)  +Q  ( *Q
`  [ <. J ,  1o >. ]  ~Q  )
) ) ) )
136135imp 115 . . . . . . . . 9  |-  ( (
ph  /\  J  <N  K )  ->  ( ( F `  J )  <Q  ( ( F `  K )  +Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  )
)  /\  ( F `  K )  <Q  (
( F `  J
)  +Q  ( *Q
`  [ <. J ,  1o >. ]  ~Q  )
) ) )
137136simprd 107 . . . . . . . 8  |-  ( (
ph  /\  J  <N  K )  ->  ( F `  K )  <Q  (
( F `  J
)  +Q  ( *Q
`  [ <. J ,  1o >. ]  ~Q  )
) )
138137adantr 261 . . . . . . 7  |-  ( ( ( ph  /\  J  <N  K )  /\  (
( S  +Q  ( *Q `  [ <. K ,  1o >. ]  ~Q  )
)  <Q  ( F `  K )  /\  (
( F `  J
)  +Q  ( *Q
`  [ <. J ,  1o >. ]  ~Q  )
)  <Q  S ) )  ->  ( F `  K )  <Q  (
( F `  J
)  +Q  ( *Q
`  [ <. J ,  1o >. ]  ~Q  )
) )
1391, 2sotri 4707 . . . . . . 7  |-  ( ( ( S  +Q  ( *Q `  [ <. K ,  1o >. ]  ~Q  )
)  <Q  ( F `  K )  /\  ( F `  K )  <Q  ( ( F `  J )  +Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  )
) )  ->  ( S  +Q  ( *Q `  [ <. K ,  1o >. ]  ~Q  ) ) 
<Q  ( ( F `  J )  +Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  )
) )
140114, 138, 139syl2anc 391 . . . . . 6  |-  ( ( ( ph  /\  J  <N  K )  /\  (
( S  +Q  ( *Q `  [ <. K ,  1o >. ]  ~Q  )
)  <Q  ( F `  K )  /\  (
( F `  J
)  +Q  ( *Q
`  [ <. J ,  1o >. ]  ~Q  )
)  <Q  S ) )  ->  ( S  +Q  ( *Q `  [ <. K ,  1o >. ]  ~Q  ) )  <Q  (
( F `  J
)  +Q  ( *Q
`  [ <. J ,  1o >. ]  ~Q  )
) )
1411, 2sotri 4707 . . . . . 6  |-  ( ( S  <Q  ( S  +Q  ( *Q `  [ <. K ,  1o >. ]  ~Q  ) )  /\  ( S  +Q  ( *Q `  [ <. K ,  1o >. ]  ~Q  )
)  <Q  ( ( F `
 J )  +Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  ) ) )  ->  S  <Q  (
( F `  J
)  +Q  ( *Q
`  [ <. J ,  1o >. ]  ~Q  )
) )
142113, 140, 141syl2anc 391 . . . . 5  |-  ( ( ( ph  /\  J  <N  K )  /\  (
( S  +Q  ( *Q `  [ <. K ,  1o >. ]  ~Q  )
)  <Q  ( F `  K )  /\  (
( F `  J
)  +Q  ( *Q
`  [ <. J ,  1o >. ]  ~Q  )
)  <Q  S ) )  ->  S  <Q  (
( F `  J
)  +Q  ( *Q
`  [ <. J ,  1o >. ]  ~Q  )
) )
143 simprr 484 . . . . 5  |-  ( ( ( ph  /\  J  <N  K )  /\  (
( S  +Q  ( *Q `  [ <. K ,  1o >. ]  ~Q  )
)  <Q  ( F `  K )  /\  (
( F `  J
)  +Q  ( *Q
`  [ <. J ,  1o >. ]  ~Q  )
)  <Q  S ) )  ->  ( ( F `
 J )  +Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  ) )  <Q  S )
144142, 143jca 290 . . . 4  |-  ( ( ( ph  /\  J  <N  K )  /\  (
( S  +Q  ( *Q `  [ <. K ,  1o >. ]  ~Q  )
)  <Q  ( F `  K )  /\  (
( F `  J
)  +Q  ( *Q
`  [ <. J ,  1o >. ]  ~Q  )
)  <Q  S ) )  ->  ( S  <Q  ( ( F `  J
)  +Q  ( *Q
`  [ <. J ,  1o >. ]  ~Q  )
)  /\  ( ( F `  J )  +Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  ) )  <Q  S ) )
145144ex 108 . . 3  |-  ( (
ph  /\  J  <N  K )  ->  ( (
( S  +Q  ( *Q `  [ <. K ,  1o >. ]  ~Q  )
)  <Q  ( F `  K )  /\  (
( F `  J
)  +Q  ( *Q
`  [ <. J ,  1o >. ]  ~Q  )
)  <Q  S )  -> 
( S  <Q  (
( F `  J
)  +Q  ( *Q
`  [ <. J ,  1o >. ]  ~Q  )
)  /\  ( ( F `  J )  +Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  ) )  <Q  S ) ) )
146109, 145mtoi 590 . 2  |-  ( (
ph  /\  J  <N  K )  ->  -.  (
( S  +Q  ( *Q `  [ <. K ,  1o >. ]  ~Q  )
)  <Q  ( F `  K )  /\  (
( F `  J
)  +Q  ( *Q
`  [ <. J ,  1o >. ]  ~Q  )
)  <Q  S ) )
147 pitri3or 6401 . . 3  |-  ( ( K  e.  N.  /\  J  e.  N. )  ->  ( K  <N  J  \/  K  =  J  \/  J  <N  K ) )
14826, 27, 147syl2anc 391 . 2  |-  ( ph  ->  ( K  <N  J  \/  K  =  J  \/  J  <N  K ) )
14980, 108, 146, 148mpjao3dan 1202 1  |-  ( ph  ->  -.  ( ( S  +Q  ( *Q `  [ <. K ,  1o >. ]  ~Q  ) ) 
<Q  ( F `  K
)  /\  ( ( F `  J )  +Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  ) )  <Q  S ) )
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   -->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:  caucvgprlemdisj  6753
  Copyright terms: Public domain W3C validator