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Theorem caucvgsrlemoffcau 6882
Description: Lemma for caucvgsr 6886. 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 6880 . . . . . . . . . . 11  |-  ( (
ph  /\  n  e.  N. )  ->  ( ( G `  n )  +R  A )  =  ( ( F `  n )  +R  1R ) )
65adantr 261 . . . . . . . . . 10  |-  ( ( ( ph  /\  n  e.  N. )  /\  k  e.  N. )  ->  (
( G `  n
)  +R  A )  =  ( ( F `
 n )  +R 
1R ) )
76eqcomd 2045 . . . . . . . . 9  |-  ( ( ( ph  /\  n  e.  N. )  /\  k  e.  N. )  ->  (
( F `  n
)  +R  1R )  =  ( ( G `
 n )  +R  A ) )
82ad2antrr 457 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  n  e.  N. )  /\  k  e.  N. )  ->  F : N. --> R. )
9 simpr 103 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  n  e.  N. )  /\  k  e.  N. )  ->  k  e.  N. )
108, 9ffvelrnd 5303 . . . . . . . . . . 11  |-  ( ( ( ph  /\  n  e.  N. )  /\  k  e.  N. )  ->  ( F `  k )  e.  R. )
11 simplr 482 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  n  e.  N. )  /\  k  e.  N. )  ->  n  e.  N. )
12 recnnpr 6646 . . . . . . . . . . . 12  |-  ( n  e.  N.  ->  <. { l  |  l  <Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >.  e.  P. )
13 prsrcl 6868 . . . . . . . . . . . 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 6836 . . . . . . . . . . . 12  |-  1R  e.  R.
1615a1i 9 . . . . . . . . . . 11  |-  ( ( ( ph  /\  n  e.  N. )  /\  k  e.  N. )  ->  1R  e.  R. )
17 addcomsrg 6840 . . . . . . . . . . . 12  |-  ( ( f  e.  R.  /\  g  e.  R. )  ->  ( f  +R  g
)  =  ( g  +R  f ) )
1817adantl 262 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  n  e.  N. )  /\  k  e.  N. )  /\  ( f  e. 
R.  /\  g  e.  R. ) )  ->  (
f  +R  g )  =  ( g  +R  f ) )
19 addasssrg 6841 . . . . . . . . . . . 12  |-  ( ( f  e.  R.  /\  g  e.  R.  /\  h  e.  R. )  ->  (
( f  +R  g
)  +R  h )  =  ( f  +R  ( g  +R  h
) ) )
2019adantl 262 . . . . . . . . . . 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 5681 . . . . . . . . . 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 6880 . . . . . . . . . . . 12  |-  ( (
ph  /\  k  e.  N. )  ->  ( ( G `  k )  +R  A )  =  ( ( F `  k )  +R  1R ) )
2322adantlr 446 . . . . . . . . . . 11  |-  ( ( ( ph  /\  n  e.  N. )  /\  k  e.  N. )  ->  (
( G `  k
)  +R  A )  =  ( ( F `
 k )  +R 
1R ) )
2423oveq1d 5527 . . . . . . . . . 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 6881 . . . . . . . . . . . . 13  |-  ( ph  ->  G : N. --> R. )
2625ad2antrr 457 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  n  e.  N. )  /\  k  e.  N. )  ->  G : N. --> R. )
2726, 9ffvelrnd 5303 . . . . . . . . . . 11  |-  ( ( ( ph  /\  n  e.  N. )  /\  k  e.  N. )  ->  ( G `  k )  e.  R. )
283caucvgsrlemasr 6874 . . . . . . . . . . . 12  |-  ( ph  ->  A  e.  R. )
2928ad2antrr 457 . . . . . . . . . . 11  |-  ( ( ( ph  /\  n  e.  N. )  /\  k  e.  N. )  ->  A  e.  R. )
3027, 29, 14, 18, 20caov32d 5681 . . . . . . . . . 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 2078 . . . . . . . . 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 3777 . . . . . . . 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 6855 . . . . . . . . . 10  |-  ( ( f  e.  R.  /\  g  e.  R.  /\  h  e.  R. )  ->  (
f  <R  g  <->  ( h  +R  f )  <R  (
h  +R  g ) ) )
3433adantl 262 . . . . . . . . 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 5303 . . . . . . . . 9  |-  ( ( ( ph  /\  n  e.  N. )  /\  k  e.  N. )  ->  ( F `  n )  e.  R. )
36 addclsr 6838 . . . . . . . . . 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 391 . . . . . . . . 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 5670 . . . . . . . 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 5303 . . . . . . . . 9  |-  ( ( ( ph  /\  n  e.  N. )  /\  k  e.  N. )  ->  ( G `  n )  e.  R. )
40 addclsr 6838 . . . . . . . . . 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 391 . . . . . . . . 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 5670 . . . . . . . 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 209 . . . . . . 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 2045 . . . . . . . . 9  |-  ( ( ( ph  /\  n  e.  N. )  /\  k  e.  N. )  ->  (
( F `  k
)  +R  1R )  =  ( ( G `
 k )  +R  A ) )
4535, 14, 16, 18, 20caov32d 5681 . . . . . . . . . 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 5527 . . . . . . . . . 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 5681 . . . . . . . . . 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 2078 . . . . . . . . 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 3777 . . . . . . . 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 6838 . . . . . . . . . 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 391 . . . . . . . . 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 5670 . . . . . . . 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 6838 . . . . . . . . . 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 391 . . . . . . . . 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 5670 . . . . . . . 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 209 . . . . . . 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 442 . . . . . 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 132 . . . . 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 48 . . . 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 2387 . . 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 2387 . 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 97    <-> wb 98    /\ w3a 885    = wceq 1243    e. wcel 1393   {cab 2026   A.wral 2306   <.cop 3378   class class class wbr 3764    |-> cmpt 3818   -->wf 4898   ` cfv 4902  (class class class)co 5512   1oc1o 5994   [cec 6104   N.cnpi 6370    <N clti 6373    ~Q ceq 6377   *Qcrq 6382    <Q cltq 6383   P.cnp 6389   1Pc1p 6390    +P. cpp 6391    ~R cer 6394   R.cnr 6395   1Rc1r 6397   -1Rcm1r 6398    +R cplr 6399    .R cmr 6400    <R cltr 6401
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 3872  ax-sep 3875  ax-nul 3883  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-setind 4262  ax-iinf 4311
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 2311  df-rex 2312  df-reu 2313  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-int 3616  df-iun 3659  df-br 3765  df-opab 3819  df-mpt 3820  df-tr 3855  df-eprel 4026  df-id 4030  df-po 4033  df-iso 4034  df-iord 4103  df-on 4105  df-suc 4108  df-iom 4314  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909  df-fv 4910  df-ov 5515  df-oprab 5516  df-mpt2 5517  df-1st 5767  df-2nd 5768  df-recs 5920  df-irdg 5957  df-1o 6001  df-2o 6002  df-oadd 6005  df-omul 6006  df-er 6106  df-ec 6108  df-qs 6112  df-ni 6402  df-pli 6403  df-mi 6404  df-lti 6405  df-plpq 6442  df-mpq 6443  df-enq 6445  df-nqqs 6446  df-plqqs 6447  df-mqqs 6448  df-1nqqs 6449  df-rq 6450  df-ltnqqs 6451  df-enq0 6522  df-nq0 6523  df-0nq0 6524  df-plq0 6525  df-mq0 6526  df-inp 6564  df-i1p 6565  df-iplp 6566  df-imp 6567  df-iltp 6568  df-enr 6811  df-nr 6812  df-plr 6813  df-mr 6814  df-ltr 6815  df-0r 6816  df-1r 6817  df-m1r 6818
This theorem is referenced by:  caucvgsrlemoffres  6884
  Copyright terms: Public domain W3C validator