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Theorem caucvgsrlemoffcau 7760
Description: Lemma for caucvgsr 7764. Offsetting the values of the sequence so they are greater than one. (Contributed by Jim Kingdon, 3-Jul-2021.)
Hypotheses
Ref Expression
caucvgsr.f  |-  ( ph  ->  F : N. --> R. )
caucvgsr.cau  |-  ( ph  ->  A. n  e.  N.  A. k  e.  N.  (
n  <N  k  ->  (
( F `  n
)  <R  ( ( F `
 k )  +R 
[ <. ( <. { l  |  l  <Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  )  /\  ( F `  k )  <R  (
( F `  n
)  +R  [ <. (
<. { l  |  l 
<Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ) ) ) )
caucvgsrlembnd.bnd  |-  ( ph  ->  A. m  e.  N.  A  <R  ( F `  m ) )
caucvgsrlembnd.offset  |-  G  =  ( a  e.  N.  |->  ( ( ( F `
 a )  +R 
1R )  +R  ( A  .R  -1R ) ) )
Assertion
Ref Expression
caucvgsrlemoffcau  |-  ( ph  ->  A. n  e.  N.  A. k  e.  N.  (
n  <N  k  ->  (
( G `  n
)  <R  ( ( G `
 k )  +R 
[ <. ( <. { l  |  l  <Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  )  /\  ( G `  k )  <R  (
( G `  n
)  +R  [ <. (
<. { l  |  l 
<Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ) ) ) )
Distinct variable groups:    A, a    A, m    F, a    k, a, n, ph    n, l, u
Allowed substitution hints:    ph( u, m, l)    A( u, k, n, l)    F( u, k, m, n, l)    G( u, k, m, n, a, l)

Proof of Theorem caucvgsrlemoffcau
Dummy variables  f  g  h are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 caucvgsr.cau . 2  |-  ( ph  ->  A. n  e.  N.  A. k  e.  N.  (
n  <N  k  ->  (
( F `  n
)  <R  ( ( F `
 k )  +R 
[ <. ( <. { l  |  l  <Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  )  /\  ( F `  k )  <R  (
( F `  n
)  +R  [ <. (
<. { l  |  l 
<Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ) ) ) )
2 caucvgsr.f . . . . . . . . . . . 12  |-  ( ph  ->  F : N. --> R. )
3 caucvgsrlembnd.bnd . . . . . . . . . . . 12  |-  ( ph  ->  A. m  e.  N.  A  <R  ( F `  m ) )
4 caucvgsrlembnd.offset . . . . . . . . . . . 12  |-  G  =  ( a  e.  N.  |->  ( ( ( F `
 a )  +R 
1R )  +R  ( A  .R  -1R ) ) )
52, 1, 3, 4caucvgsrlemoffval 7758 . . . . . . . . . . 11  |-  ( (
ph  /\  n  e.  N. )  ->  ( ( G `  n )  +R  A )  =  ( ( F `  n )  +R  1R ) )
65adantr 274 . . . . . . . . . 10  |-  ( ( ( ph  /\  n  e.  N. )  /\  k  e.  N. )  ->  (
( G `  n
)  +R  A )  =  ( ( F `
 n )  +R 
1R ) )
76eqcomd 2176 . . . . . . . . 9  |-  ( ( ( ph  /\  n  e.  N. )  /\  k  e.  N. )  ->  (
( F `  n
)  +R  1R )  =  ( ( G `
 n )  +R  A ) )
82ad2antrr 485 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  n  e.  N. )  /\  k  e.  N. )  ->  F : N. --> R. )
9 simpr 109 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  n  e.  N. )  /\  k  e.  N. )  ->  k  e.  N. )
108, 9ffvelrnd 5632 . . . . . . . . . . 11  |-  ( ( ( ph  /\  n  e.  N. )  /\  k  e.  N. )  ->  ( F `  k )  e.  R. )
11 simplr 525 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  n  e.  N. )  /\  k  e.  N. )  ->  n  e.  N. )
12 recnnpr 7510 . . . . . . . . . . . 12  |-  ( n  e.  N.  ->  <. { l  |  l  <Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >.  e.  P. )
13 prsrcl 7746 . . . . . . . . . . . 12  |-  ( <. { l  |  l 
<Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >.  e.  P.  ->  [ <. ( <. { l  |  l  <Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  e.  R. )
1411, 12, 133syl 17 . . . . . . . . . . 11  |-  ( ( ( ph  /\  n  e.  N. )  /\  k  e.  N. )  ->  [ <. (
<. { l  |  l 
<Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  e.  R. )
15 1sr 7713 . . . . . . . . . . . 12  |-  1R  e.  R.
1615a1i 9 . . . . . . . . . . 11  |-  ( ( ( ph  /\  n  e.  N. )  /\  k  e.  N. )  ->  1R  e.  R. )
17 addcomsrg 7717 . . . . . . . . . . . 12  |-  ( ( f  e.  R.  /\  g  e.  R. )  ->  ( f  +R  g
)  =  ( g  +R  f ) )
1817adantl 275 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  n  e.  N. )  /\  k  e.  N. )  /\  ( f  e. 
R.  /\  g  e.  R. ) )  ->  (
f  +R  g )  =  ( g  +R  f ) )
19 addasssrg 7718 . . . . . . . . . . . 12  |-  ( ( f  e.  R.  /\  g  e.  R.  /\  h  e.  R. )  ->  (
( f  +R  g
)  +R  h )  =  ( f  +R  ( g  +R  h
) ) )
2019adantl 275 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  n  e.  N. )  /\  k  e.  N. )  /\  ( f  e. 
R.  /\  g  e.  R.  /\  h  e.  R. ) )  ->  (
( f  +R  g
)  +R  h )  =  ( f  +R  ( g  +R  h
) ) )
2110, 14, 16, 18, 20caov32d 6033 . . . . . . . . . 10  |-  ( ( ( ph  /\  n  e.  N. )  /\  k  e.  N. )  ->  (
( ( F `  k )  +R  [ <. ( <. { l  |  l  <Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  )  +R  1R )  =  ( ( ( F `  k )  +R  1R )  +R  [ <. ( <. { l  |  l 
<Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ) )
222, 1, 3, 4caucvgsrlemoffval 7758 . . . . . . . . . . . 12  |-  ( (
ph  /\  k  e.  N. )  ->  ( ( G `  k )  +R  A )  =  ( ( F `  k )  +R  1R ) )
2322adantlr 474 . . . . . . . . . . 11  |-  ( ( ( ph  /\  n  e.  N. )  /\  k  e.  N. )  ->  (
( G `  k
)  +R  A )  =  ( ( F `
 k )  +R 
1R ) )
2423oveq1d 5868 . . . . . . . . . 10  |-  ( ( ( ph  /\  n  e.  N. )  /\  k  e.  N. )  ->  (
( ( G `  k )  +R  A
)  +R  [ <. (
<. { l  |  l 
<Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  )  =  ( (
( F `  k
)  +R  1R )  +R  [ <. ( <. { l  |  l  <Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ) )
252, 1, 3, 4caucvgsrlemofff 7759 . . . . . . . . . . . . 13  |-  ( ph  ->  G : N. --> R. )
2625ad2antrr 485 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  n  e.  N. )  /\  k  e.  N. )  ->  G : N. --> R. )
2726, 9ffvelrnd 5632 . . . . . . . . . . 11  |-  ( ( ( ph  /\  n  e.  N. )  /\  k  e.  N. )  ->  ( G `  k )  e.  R. )
283caucvgsrlemasr 7752 . . . . . . . . . . . 12  |-  ( ph  ->  A  e.  R. )
2928ad2antrr 485 . . . . . . . . . . 11  |-  ( ( ( ph  /\  n  e.  N. )  /\  k  e.  N. )  ->  A  e.  R. )
3027, 29, 14, 18, 20caov32d 6033 . . . . . . . . . 10  |-  ( ( ( ph  /\  n  e.  N. )  /\  k  e.  N. )  ->  (
( ( G `  k )  +R  A
)  +R  [ <. (
<. { l  |  l 
<Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  )  =  ( (
( G `  k
)  +R  [ <. (
<. { l  |  l 
<Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  )  +R  A ) )
3121, 24, 303eqtr2d 2209 . . . . . . . . 9  |-  ( ( ( ph  /\  n  e.  N. )  /\  k  e.  N. )  ->  (
( ( F `  k )  +R  [ <. ( <. { l  |  l  <Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  )  +R  1R )  =  ( ( ( G `  k )  +R  [ <. ( <. { l  |  l  <Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  )  +R  A ) )
327, 31breq12d 4002 . . . . . . . 8  |-  ( ( ( ph  /\  n  e.  N. )  /\  k  e.  N. )  ->  (
( ( F `  n )  +R  1R )  <R  ( ( ( F `  k )  +R  [ <. ( <. { l  |  l 
<Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  )  +R  1R )  <->  ( ( G `  n )  +R  A )  <R  (
( ( G `  k )  +R  [ <. ( <. { l  |  l  <Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  )  +R  A ) ) )
33 ltasrg 7732 . . . . . . . . . 10  |-  ( ( f  e.  R.  /\  g  e.  R.  /\  h  e.  R. )  ->  (
f  <R  g  <->  ( h  +R  f )  <R  (
h  +R  g ) ) )
3433adantl 275 . . . . . . . . 9  |-  ( ( ( ( ph  /\  n  e.  N. )  /\  k  e.  N. )  /\  ( f  e. 
R.  /\  g  e.  R.  /\  h  e.  R. ) )  ->  (
f  <R  g  <->  ( h  +R  f )  <R  (
h  +R  g ) ) )
358, 11ffvelrnd 5632 . . . . . . . . 9  |-  ( ( ( ph  /\  n  e.  N. )  /\  k  e.  N. )  ->  ( F `  n )  e.  R. )
36 addclsr 7715 . . . . . . . . . 10  |-  ( ( ( F `  k
)  e.  R.  /\  [
<. ( <. { l  |  l  <Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  e.  R. )  ->  ( ( F `  k )  +R  [ <. ( <. { l  |  l 
<Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  )  e.  R. )
3710, 14, 36syl2anc 409 . . . . . . . . 9  |-  ( ( ( ph  /\  n  e.  N. )  /\  k  e.  N. )  ->  (
( F `  k
)  +R  [ <. (
<. { l  |  l 
<Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  )  e.  R. )
3834, 35, 37, 16, 18caovord2d 6022 . . . . . . . 8  |-  ( ( ( ph  /\  n  e.  N. )  /\  k  e.  N. )  ->  (
( F `  n
)  <R  ( ( F `
 k )  +R 
[ <. ( <. { l  |  l  <Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ) 
<->  ( ( F `  n )  +R  1R )  <R  ( ( ( F `  k )  +R  [ <. ( <. { l  |  l 
<Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  )  +R  1R ) ) )
3926, 11ffvelrnd 5632 . . . . . . . . 9  |-  ( ( ( ph  /\  n  e.  N. )  /\  k  e.  N. )  ->  ( G `  n )  e.  R. )
40 addclsr 7715 . . . . . . . . . 10  |-  ( ( ( G `  k
)  e.  R.  /\  [
<. ( <. { l  |  l  <Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  e.  R. )  ->  ( ( G `  k )  +R  [ <. ( <. { l  |  l 
<Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  )  e.  R. )
4127, 14, 40syl2anc 409 . . . . . . . . 9  |-  ( ( ( ph  /\  n  e.  N. )  /\  k  e.  N. )  ->  (
( G `  k
)  +R  [ <. (
<. { l  |  l 
<Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  )  e.  R. )
4234, 39, 41, 29, 18caovord2d 6022 . . . . . . . 8  |-  ( ( ( ph  /\  n  e.  N. )  /\  k  e.  N. )  ->  (
( G `  n
)  <R  ( ( G `
 k )  +R 
[ <. ( <. { l  |  l  <Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ) 
<->  ( ( G `  n )  +R  A
)  <R  ( ( ( G `  k )  +R  [ <. ( <. { l  |  l 
<Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  )  +R  A ) ) )
4332, 38, 423bitr4d 219 . . . . . . 7  |-  ( ( ( ph  /\  n  e.  N. )  /\  k  e.  N. )  ->  (
( F `  n
)  <R  ( ( F `
 k )  +R 
[ <. ( <. { l  |  l  <Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ) 
<->  ( G `  n
)  <R  ( ( G `
 k )  +R 
[ <. ( <. { l  |  l  <Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ) ) )
4423eqcomd 2176 . . . . . . . . 9  |-  ( ( ( ph  /\  n  e.  N. )  /\  k  e.  N. )  ->  (
( F `  k
)  +R  1R )  =  ( ( G `
 k )  +R  A ) )
4535, 14, 16, 18, 20caov32d 6033 . . . . . . . . . 10  |-  ( ( ( ph  /\  n  e.  N. )  /\  k  e.  N. )  ->  (
( ( F `  n )  +R  [ <. ( <. { l  |  l  <Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  )  +R  1R )  =  ( ( ( F `  n )  +R  1R )  +R  [ <. ( <. { l  |  l 
<Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ) )
466oveq1d 5868 . . . . . . . . . 10  |-  ( ( ( ph  /\  n  e.  N. )  /\  k  e.  N. )  ->  (
( ( G `  n )  +R  A
)  +R  [ <. (
<. { l  |  l 
<Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  )  =  ( (
( F `  n
)  +R  1R )  +R  [ <. ( <. { l  |  l  <Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ) )
4739, 29, 14, 18, 20caov32d 6033 . . . . . . . . . 10  |-  ( ( ( ph  /\  n  e.  N. )  /\  k  e.  N. )  ->  (
( ( G `  n )  +R  A
)  +R  [ <. (
<. { l  |  l 
<Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  )  =  ( (
( G `  n
)  +R  [ <. (
<. { l  |  l 
<Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  )  +R  A ) )
4845, 46, 473eqtr2d 2209 . . . . . . . . 9  |-  ( ( ( ph  /\  n  e.  N. )  /\  k  e.  N. )  ->  (
( ( F `  n )  +R  [ <. ( <. { l  |  l  <Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  )  +R  1R )  =  ( ( ( G `  n )  +R  [ <. ( <. { l  |  l  <Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  )  +R  A ) )
4944, 48breq12d 4002 . . . . . . . 8  |-  ( ( ( ph  /\  n  e.  N. )  /\  k  e.  N. )  ->  (
( ( F `  k )  +R  1R )  <R  ( ( ( F `  n )  +R  [ <. ( <. { l  |  l 
<Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  )  +R  1R )  <->  ( ( G `  k )  +R  A )  <R  (
( ( G `  n )  +R  [ <. ( <. { l  |  l  <Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  )  +R  A ) ) )
50 addclsr 7715 . . . . . . . . . 10  |-  ( ( ( F `  n
)  e.  R.  /\  [
<. ( <. { l  |  l  <Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  e.  R. )  ->  ( ( F `  n )  +R  [ <. ( <. { l  |  l 
<Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  )  e.  R. )
5135, 14, 50syl2anc 409 . . . . . . . . 9  |-  ( ( ( ph  /\  n  e.  N. )  /\  k  e.  N. )  ->  (
( F `  n
)  +R  [ <. (
<. { l  |  l 
<Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  )  e.  R. )
5234, 10, 51, 16, 18caovord2d 6022 . . . . . . . 8  |-  ( ( ( ph  /\  n  e.  N. )  /\  k  e.  N. )  ->  (
( F `  k
)  <R  ( ( F `
 n )  +R 
[ <. ( <. { l  |  l  <Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ) 
<->  ( ( F `  k )  +R  1R )  <R  ( ( ( F `  n )  +R  [ <. ( <. { l  |  l 
<Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  )  +R  1R ) ) )
53 addclsr 7715 . . . . . . . . . 10  |-  ( ( ( G `  n
)  e.  R.  /\  [
<. ( <. { l  |  l  <Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  e.  R. )  ->  ( ( G `  n )  +R  [ <. ( <. { l  |  l 
<Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  )  e.  R. )
5439, 14, 53syl2anc 409 . . . . . . . . 9  |-  ( ( ( ph  /\  n  e.  N. )  /\  k  e.  N. )  ->  (
( G `  n
)  +R  [ <. (
<. { l  |  l 
<Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  )  e.  R. )
5534, 27, 54, 29, 18caovord2d 6022 . . . . . . . 8  |-  ( ( ( ph  /\  n  e.  N. )  /\  k  e.  N. )  ->  (
( G `  k
)  <R  ( ( G `
 n )  +R 
[ <. ( <. { l  |  l  <Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ) 
<->  ( ( G `  k )  +R  A
)  <R  ( ( ( G `  n )  +R  [ <. ( <. { l  |  l 
<Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  )  +R  A ) ) )
5649, 52, 553bitr4d 219 . . . . . . 7  |-  ( ( ( ph  /\  n  e.  N. )  /\  k  e.  N. )  ->  (
( F `  k
)  <R  ( ( F `
 n )  +R 
[ <. ( <. { l  |  l  <Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ) 
<->  ( G `  k
)  <R  ( ( G `
 n )  +R 
[ <. ( <. { l  |  l  <Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ) ) )
5743, 56anbi12d 470 . . . . . 6  |-  ( ( ( ph  /\  n  e.  N. )  /\  k  e.  N. )  ->  (
( ( F `  n )  <R  (
( F `  k
)  +R  [ <. (
<. { l  |  l 
<Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  )  /\  ( F `  k )  <R  (
( F `  n
)  +R  [ <. (
<. { l  |  l 
<Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ) )  <->  ( ( G `  n )  <R  ( ( G `  k )  +R  [ <. ( <. { l  |  l  <Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  )  /\  ( G `  k
)  <R  ( ( G `
 n )  +R 
[ <. ( <. { l  |  l  <Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ) ) ) )
5857biimpd 143 . . . . 5  |-  ( ( ( ph  /\  n  e.  N. )  /\  k  e.  N. )  ->  (
( ( F `  n )  <R  (
( F `  k
)  +R  [ <. (
<. { l  |  l 
<Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  )  /\  ( F `  k )  <R  (
( F `  n
)  +R  [ <. (
<. { l  |  l 
<Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ) )  ->  (
( G `  n
)  <R  ( ( G `
 k )  +R 
[ <. ( <. { l  |  l  <Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  )  /\  ( G `  k )  <R  (
( G `  n
)  +R  [ <. (
<. { l  |  l 
<Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ) ) ) )
5958imim2d 54 . . . 4  |-  ( ( ( ph  /\  n  e.  N. )  /\  k  e.  N. )  ->  (
( n  <N  k  ->  ( ( F `  n )  <R  (
( F `  k
)  +R  [ <. (
<. { l  |  l 
<Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  )  /\  ( F `  k )  <R  (
( F `  n
)  +R  [ <. (
<. { l  |  l 
<Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ) ) )  -> 
( n  <N  k  ->  ( ( G `  n )  <R  (
( G `  k
)  +R  [ <. (
<. { l  |  l 
<Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  )  /\  ( G `  k )  <R  (
( G `  n
)  +R  [ <. (
<. { l  |  l 
<Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ) ) ) ) )
6059ralimdva 2537 . . 3  |-  ( (
ph  /\  n  e.  N. )  ->  ( A. k  e.  N.  (
n  <N  k  ->  (
( F `  n
)  <R  ( ( F `
 k )  +R 
[ <. ( <. { l  |  l  <Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  )  /\  ( F `  k )  <R  (
( F `  n
)  +R  [ <. (
<. { l  |  l 
<Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ) ) )  ->  A. k  e.  N.  ( n  <N  k  -> 
( ( G `  n )  <R  (
( G `  k
)  +R  [ <. (
<. { l  |  l 
<Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  )  /\  ( G `  k )  <R  (
( G `  n
)  +R  [ <. (
<. { l  |  l 
<Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ) ) ) ) )
6160ralimdva 2537 . 2  |-  ( ph  ->  ( A. n  e. 
N.  A. k  e.  N.  ( n  <N  k  -> 
( ( F `  n )  <R  (
( F `  k
)  +R  [ <. (
<. { l  |  l 
<Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  )  /\  ( F `  k )  <R  (
( F `  n
)  +R  [ <. (
<. { l  |  l 
<Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ) ) )  ->  A. n  e.  N.  A. k  e.  N.  (
n  <N  k  ->  (
( G `  n
)  <R  ( ( G `
 k )  +R 
[ <. ( <. { l  |  l  <Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  )  /\  ( G `  k )  <R  (
( G `  n
)  +R  [ <. (
<. { l  |  l 
<Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ) ) ) ) )
621, 61mpd 13 1  |-  ( ph  ->  A. n  e.  N.  A. k  e.  N.  (
n  <N  k  ->  (
( G `  n
)  <R  ( ( G `
 k )  +R 
[ <. ( <. { l  |  l  <Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  )  /\  ( G `  k )  <R  (
( G `  n
)  +R  [ <. (
<. { l  |  l 
<Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    /\ w3a 973    = wceq 1348    e. wcel 2141   {cab 2156   A.wral 2448   <.cop 3586   class class class wbr 3989    |-> cmpt 4050   -->wf 5194   ` cfv 5198  (class class class)co 5853   1oc1o 6388   [cec 6511   N.cnpi 7234    <N clti 7237    ~Q ceq 7241   *Qcrq 7246    <Q cltq 7247   P.cnp 7253   1Pc1p 7254    +P. cpp 7255    ~R cer 7258   R.cnr 7259   1Rc1r 7261   -1Rcm1r 7262    +R cplr 7263    .R cmr 7264    <R cltr 7265
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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-iinf 4572
This theorem depends on definitions:  df-bi 116  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-eprel 4274  df-id 4278  df-po 4281  df-iso 4282  df-iord 4351  df-on 4353  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-recs 6284  df-irdg 6349  df-1o 6395  df-2o 6396  df-oadd 6399  df-omul 6400  df-er 6513  df-ec 6515  df-qs 6519  df-ni 7266  df-pli 7267  df-mi 7268  df-lti 7269  df-plpq 7306  df-mpq 7307  df-enq 7309  df-nqqs 7310  df-plqqs 7311  df-mqqs 7312  df-1nqqs 7313  df-rq 7314  df-ltnqqs 7315  df-enq0 7386  df-nq0 7387  df-0nq0 7388  df-plq0 7389  df-mq0 7390  df-inp 7428  df-i1p 7429  df-iplp 7430  df-imp 7431  df-iltp 7432  df-enr 7688  df-nr 7689  df-plr 7690  df-mr 7691  df-ltr 7692  df-0r 7693  df-1r 7694  df-m1r 7695
This theorem is referenced by:  caucvgsrlemoffres  7762
  Copyright terms: Public domain W3C validator