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Theorem caucvgprlemnkj 7481
Description: Lemma for caucvgpr 7497. 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 7213 . . . 4  |-  <Q  Or  Q.
2 ltrelnq 7180 . . . 4  |-  <Q  C_  ( Q.  X.  Q. )
31, 2son2lpi 4935 . . 3  |-  -.  ( S  <Q  ( F `  J )  /\  ( F `  J )  <Q  S )
4 simprl 520 . . . . . . 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 3932 . . . . . . . . . . . . . 14  |-  ( n  =  a  ->  (
n  <N  k  <->  a  <N  k ) )
7 fveq2 5421 . . . . . . . . . . . . . . . 16  |-  ( n  =  a  ->  ( F `  n )  =  ( F `  a ) )
8 opeq1 3705 . . . . . . . . . . . . . . . . . . 19  |-  ( n  =  a  ->  <. n ,  1o >.  =  <. a ,  1o >. )
98eceq1d 6465 . . . . . . . . . . . . . . . . . 18  |-  ( n  =  a  ->  [ <. n ,  1o >. ]  ~Q  =  [ <. a ,  1o >. ]  ~Q  )
109fveq2d 5425 . . . . . . . . . . . . . . . . 17  |-  ( n  =  a  ->  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  =  ( *Q `  [ <. a ,  1o >. ]  ~Q  ) )
1110oveq2d 5790 . . . . . . . . . . . . . . . 16  |-  ( n  =  a  ->  (
( F `  k
)  +Q  ( *Q
`  [ <. n ,  1o >. ]  ~Q  )
)  =  ( ( F `  k )  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  ) ) )
127, 11breq12d 3942 . . . . . . . . . . . . . . 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 5792 . . . . . . . . . . . . . . . 16  |-  ( n  =  a  ->  (
( F `  n
)  +Q  ( *Q
`  [ <. n ,  1o >. ]  ~Q  )
)  =  ( ( F `  a )  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  ) ) )
1413breq2d 3941 . . . . . . . . . . . . . . 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 464 . . . . . . . . . . . . . 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 233 . . . . . . . . . . . . 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 3933 . . . . . . . . . . . . . 14  |-  ( k  =  b  ->  (
a  <N  k  <->  a  <N  b ) )
18 fveq2 5421 . . . . . . . . . . . . . . . . 17  |-  ( k  =  b  ->  ( F `  k )  =  ( F `  b ) )
1918oveq1d 5789 . . . . . . . . . . . . . . . 16  |-  ( k  =  b  ->  (
( F `  k
)  +Q  ( *Q
`  [ <. a ,  1o >. ]  ~Q  )
)  =  ( ( F `  b )  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  ) ) )
2019breq2d 3941 . . . . . . . . . . . . . . 15  |-  ( k  =  b  ->  (
( F `  a
)  <Q  ( ( F `
 k )  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  ) )  <->  ( F `  a )  <Q  (
( F `  b
)  +Q  ( *Q
`  [ <. a ,  1o >. ]  ~Q  )
) ) )
2118breq1d 3939 . . . . . . . . . . . . . . 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 464 . . . . . . . . . . . . . 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 233 . . . . . . . . . . . . 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 2665 . . . . . . . . . . . 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 121 . . . . . . . . . . 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 3932 . . . . . . . . . . . . . 14  |-  ( a  =  K  ->  (
a  <N  b  <->  K  <N  b ) )
29 fveq2 5421 . . . . . . . . . . . . . . . 16  |-  ( a  =  K  ->  ( F `  a )  =  ( F `  K ) )
30 opeq1 3705 . . . . . . . . . . . . . . . . . . 19  |-  ( a  =  K  ->  <. a ,  1o >.  =  <. K ,  1o >. )
3130eceq1d 6465 . . . . . . . . . . . . . . . . . 18  |-  ( a  =  K  ->  [ <. a ,  1o >. ]  ~Q  =  [ <. K ,  1o >. ]  ~Q  )
3231fveq2d 5425 . . . . . . . . . . . . . . . . 17  |-  ( a  =  K  ->  ( *Q `  [ <. a ,  1o >. ]  ~Q  )  =  ( *Q `  [ <. K ,  1o >. ]  ~Q  ) )
3332oveq2d 5790 . . . . . . . . . . . . . . . 16  |-  ( a  =  K  ->  (
( F `  b
)  +Q  ( *Q
`  [ <. a ,  1o >. ]  ~Q  )
)  =  ( ( F `  b )  +Q  ( *Q `  [ <. K ,  1o >. ]  ~Q  ) ) )
3429, 33breq12d 3942 . . . . . . . . . . . . . . 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 5792 . . . . . . . . . . . . . . . 16  |-  ( a  =  K  ->  (
( F `  a
)  +Q  ( *Q
`  [ <. a ,  1o >. ]  ~Q  )
)  =  ( ( F `  K )  +Q  ( *Q `  [ <. K ,  1o >. ]  ~Q  ) ) )
3635breq2d 3941 . . . . . . . . . . . . . . 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 464 . . . . . . . . . . . . . 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 233 . . . . . . . . . . . . 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 3933 . . . . . . . . . . . . . 14  |-  ( b  =  J  ->  ( K  <N  b  <->  K  <N  J ) )
40 fveq2 5421 . . . . . . . . . . . . . . . . 17  |-  ( b  =  J  ->  ( F `  b )  =  ( F `  J ) )
4140oveq1d 5789 . . . . . . . . . . . . . . . 16  |-  ( b  =  J  ->  (
( F `  b
)  +Q  ( *Q
`  [ <. K ,  1o >. ]  ~Q  )
)  =  ( ( F `  J )  +Q  ( *Q `  [ <. K ,  1o >. ]  ~Q  ) ) )
4241breq2d 3941 . . . . . . . . . . . . . . 15  |-  ( b  =  J  ->  (
( F `  K
)  <Q  ( ( F `
 b )  +Q  ( *Q `  [ <. K ,  1o >. ]  ~Q  ) )  <->  ( F `  K )  <Q  (
( F `  J
)  +Q  ( *Q
`  [ <. K ,  1o >. ]  ~Q  )
) ) )
4340breq1d 3939 . . . . . . . . . . . . . . 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 464 . . . . . . . . . . . . . 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 233 . . . . . . . . . . . . 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 2802 . . . . . . . . . . . 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 408 . . . . . . . . . . 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 123 . . . . . . . . 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 111 . . . . . . . 8  |-  ( (
ph  /\  K  <N  J )  ->  ( F `  K )  <Q  (
( F `  J
)  +Q  ( *Q
`  [ <. K ,  1o >. ]  ~Q  )
) )
5150adantr 274 . . . . . . 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 4934 . . . . . . 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 408 . . . . . 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 7215 . . . . . . . 8  |-  ( ( f  e.  Q.  /\  g  e.  Q.  /\  h  e.  Q. )  ->  (
f  <Q  g  <->  ( h  +Q  f )  <Q  (
h  +Q  g ) ) )
5554adantl 275 . . . . . . 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 479 . . . . . . 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 5556 . . . . . . . 8  |-  ( ph  ->  ( F `  J
)  e.  Q. )
6059ad2antrr 479 . . . . . . 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 7237 . . . . . . . . 9  |-  ( K  e.  N.  ->  [ <. K ,  1o >. ]  ~Q  e.  Q. )
62 recclnq 7207 . . . . . . . . 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 479 . . . . . . 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 7196 . . . . . . . 8  |-  ( ( f  e.  Q.  /\  g  e.  Q. )  ->  ( f  +Q  g
)  =  ( g  +Q  f ) )
6665adantl 275 . . . . . . 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 5940 . . . . . 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 166 . . . . 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 7237 . . . . . . . . 9  |-  ( J  e.  N.  ->  [ <. J ,  1o >. ]  ~Q  e.  Q. )
70 recclnq 7207 . . . . . . . . 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 479 . . . . . . 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 7222 . . . . . . 7  |-  ( ( ( F `  J
)  e.  Q.  /\  ( *Q `  [ <. J ,  1o >. ]  ~Q  )  e.  Q. )  ->  ( F `  J
)  <Q  ( ( F `
 J )  +Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  ) ) )
7460, 72, 73syl2anc 408 . . . . . 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 521 . . . . . 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 4934 . . . . . 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 408 . . . . 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 304 . . . 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 114 . . 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 653 . 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 4935 . . 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 3705 . . . . . . . . . . . 12  |-  ( K  =  J  ->  <. K ,  1o >.  =  <. J ,  1o >. )
8382eceq1d 6465 . . . . . . . . . . 11  |-  ( K  =  J  ->  [ <. K ,  1o >. ]  ~Q  =  [ <. J ,  1o >. ]  ~Q  )
8483fveq2d 5425 . . . . . . . . . 10  |-  ( K  =  J  ->  ( *Q `  [ <. K ,  1o >. ]  ~Q  )  =  ( *Q `  [ <. J ,  1o >. ]  ~Q  ) )
8584oveq2d 5790 . . . . . . . . 9  |-  ( K  =  J  ->  ( S  +Q  ( *Q `  [ <. K ,  1o >. ]  ~Q  ) )  =  ( S  +Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  ) ) )
86 fveq2 5421 . . . . . . . . 9  |-  ( K  =  J  ->  ( F `  K )  =  ( F `  J ) )
8785, 86breq12d 3942 . . . . . . . 8  |-  ( K  =  J  ->  (
( S  +Q  ( *Q `  [ <. K ,  1o >. ]  ~Q  )
)  <Q  ( F `  K )  <->  ( S  +Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  ) )  <Q 
( F `  J
) ) )
8887anbi1d 460 . . . . . . 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 275 . . . . . 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 275 . . . . . . . . 9  |-  ( (
ph  /\  ( f  e.  Q.  /\  g  e. 
Q.  /\  h  e.  Q. ) )  ->  (
f  <Q  g  <->  ( h  +Q  f )  <Q  (
h  +Q  g ) ) )
91 addclnq 7190 . . . . . . . . . 10  |-  ( ( S  e.  Q.  /\  ( *Q `  [ <. J ,  1o >. ]  ~Q  )  e.  Q. )  ->  ( S  +Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  )
)  e.  Q. )
9256, 71, 91syl2anc 408 . . . . . . . . 9  |-  ( ph  ->  ( S  +Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  )
)  e.  Q. )
9365adantl 275 . . . . . . . . 9  |-  ( (
ph  /\  ( f  e.  Q.  /\  g  e. 
Q. ) )  -> 
( f  +Q  g
)  =  ( g  +Q  f ) )
9490, 92, 59, 71, 93caovord2d 5940 . . . . . . . 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 274 . . . . . . 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 460 . . . . . 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 187 . . . . 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 4934 . . . . 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 162 . . . 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 7222 . . . . . . 7  |-  ( ( S  e.  Q.  /\  ( *Q `  [ <. J ,  1o >. ]  ~Q  )  e.  Q. )  ->  S  <Q  ( S  +Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  ) ) )
10156, 71, 100syl2anc 408 . . . . . 6  |-  ( ph  ->  S  <Q  ( S  +Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  ) ) )
102 ltaddnq 7222 . . . . . . 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 408 . . . . . 6  |-  ( ph  ->  ( S  +Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  )
)  <Q  ( ( S  +Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  ) )  +Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  ) ) )
1041, 2sotri 4934 . . . . . 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 408 . . . . 5  |-  ( ph  ->  S  <Q  ( ( S  +Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  ) )  +Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  ) ) )
106105adantr 274 . . . 4  |-  ( (
ph  /\  K  =  J )  ->  S  <Q  ( ( S  +Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  ) )  +Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  )
) )
10799, 106jctird 315 . . 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 653 . 2  |-  ( (
ph  /\  K  =  J )  ->  -.  ( ( S  +Q  ( *Q `  [ <. K ,  1o >. ]  ~Q  ) )  <Q  ( F `  K )  /\  ( ( F `  J )  +Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  )
)  <Q  S ) )
1091, 2son2lpi 4935 . . 3  |-  -.  ( S  <Q  ( ( F `
 J )  +Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  ) )  /\  ( ( F `  J )  +Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  )
)  <Q  S )
11056ad2antrr 479 . . . . . . 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 479 . . . . . . 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 7222 . . . . . . 7  |-  ( ( S  e.  Q.  /\  ( *Q `  [ <. K ,  1o >. ]  ~Q  )  e.  Q. )  ->  S  <Q  ( S  +Q  ( *Q `  [ <. K ,  1o >. ]  ~Q  ) ) )
113110, 111, 112syl2anc 408 . . . . . 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 520 . . . . . . 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 3932 . . . . . . . . . . . . . 14  |-  ( a  =  J  ->  (
a  <N  b  <->  J  <N  b ) )
116 fveq2 5421 . . . . . . . . . . . . . . . 16  |-  ( a  =  J  ->  ( F `  a )  =  ( F `  J ) )
117 opeq1 3705 . . . . . . . . . . . . . . . . . . 19  |-  ( a  =  J  ->  <. a ,  1o >.  =  <. J ,  1o >. )
118117eceq1d 6465 . . . . . . . . . . . . . . . . . 18  |-  ( a  =  J  ->  [ <. a ,  1o >. ]  ~Q  =  [ <. J ,  1o >. ]  ~Q  )
119118fveq2d 5425 . . . . . . . . . . . . . . . . 17  |-  ( a  =  J  ->  ( *Q `  [ <. a ,  1o >. ]  ~Q  )  =  ( *Q `  [ <. J ,  1o >. ]  ~Q  ) )
120119oveq2d 5790 . . . . . . . . . . . . . . . 16  |-  ( a  =  J  ->  (
( F `  b
)  +Q  ( *Q
`  [ <. a ,  1o >. ]  ~Q  )
)  =  ( ( F `  b )  +Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  ) ) )
121116, 120breq12d 3942 . . . . . . . . . . . . . . 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 5792 . . . . . . . . . . . . . . . 16  |-  ( a  =  J  ->  (
( F `  a
)  +Q  ( *Q
`  [ <. a ,  1o >. ]  ~Q  )
)  =  ( ( F `  J )  +Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  ) ) )
123122breq2d 3941 . . . . . . . . . . . . . . 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 464 . . . . . . . . . . . . . 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 233 . . . . . . . . . . . . 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 3933 . . . . . . . . . . . . . 14  |-  ( b  =  K  ->  ( J  <N  b  <->  J  <N  K ) )
127 fveq2 5421 . . . . . . . . . . . . . . . . 17  |-  ( b  =  K  ->  ( F `  b )  =  ( F `  K ) )
128127oveq1d 5789 . . . . . . . . . . . . . . . 16  |-  ( b  =  K  ->  (
( F `  b
)  +Q  ( *Q
`  [ <. J ,  1o >. ]  ~Q  )
)  =  ( ( F `  K )  +Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  ) ) )
129128breq2d 3941 . . . . . . . . . . . . . . 15  |-  ( b  =  K  ->  (
( F `  J
)  <Q  ( ( F `
 b )  +Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  ) )  <->  ( F `  J )  <Q  (
( F `  K
)  +Q  ( *Q
`  [ <. J ,  1o >. ]  ~Q  )
) ) )
130127breq1d 3939 . . . . . . . . . . . . . . 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 464 . . . . . . . . . . . . . 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 233 . . . . . . . . . . . . 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 2802 . . . . . . . . . . . 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 408 . . . . . . . . . . 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 123 . . . . . . . . 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 113 . . . . . . . 8  |-  ( (
ph  /\  J  <N  K )  ->  ( F `  K )  <Q  (
( F `  J
)  +Q  ( *Q
`  [ <. J ,  1o >. ]  ~Q  )
) )
138137adantr 274 . . . . . . 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 4934 . . . . . . 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 408 . . . . . 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 4934 . . . . . 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 408 . . . . 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 521 . . . . 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 304 . . . 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 114 . . 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 653 . 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 7137 . . 3  |-  ( ( K  e.  N.  /\  J  e.  N. )  ->  ( K  <N  J  \/  K  =  J  \/  J  <N  K ) )
14826, 27, 147syl2anc 408 . 2  |-  ( ph  ->  ( K  <N  J  \/  K  =  J  \/  J  <N  K ) )
14980, 108, 146, 148mpjao3dan 1285 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 103    <-> wb 104    \/ w3o 961    /\ w3a 962    = wceq 1331    e. wcel 1480   A.wral 2416   <.cop 3530   class class class wbr 3929   -->wf 5119   ` cfv 5123  (class class class)co 5774   1oc1o 6306   [cec 6427   N.cnpi 7087    <N clti 7090    ~Q ceq 7094   Q.cnq 7095    +Q cplq 7097   *Qcrq 7099    <Q cltq 7100
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502
This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-eprel 4211  df-id 4215  df-po 4218  df-iso 4219  df-iord 4288  df-on 4290  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-recs 6202  df-irdg 6267  df-1o 6313  df-oadd 6317  df-omul 6318  df-er 6429  df-ec 6431  df-qs 6435  df-ni 7119  df-pli 7120  df-mi 7121  df-lti 7122  df-plpq 7159  df-mpq 7160  df-enq 7162  df-nqqs 7163  df-plqqs 7164  df-mqqs 7165  df-1nqqs 7166  df-rq 7167  df-ltnqqs 7168
This theorem is referenced by:  caucvgprlemdisj  7489
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