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Theorem caucvgsrlemoffcau 7570
Description: Lemma for caucvgsr 7574. 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 7568 . . . . . . . . . . 11  |-  ( (
ph  /\  n  e.  N. )  ->  ( ( G `  n )  +R  A )  =  ( ( F `  n )  +R  1R ) )
65adantr 272 . . . . . . . . . 10  |-  ( ( ( ph  /\  n  e.  N. )  /\  k  e.  N. )  ->  (
( G `  n
)  +R  A )  =  ( ( F `
 n )  +R 
1R ) )
76eqcomd 2121 . . . . . . . . 9  |-  ( ( ( ph  /\  n  e.  N. )  /\  k  e.  N. )  ->  (
( F `  n
)  +R  1R )  =  ( ( G `
 n )  +R  A ) )
82ad2antrr 477 . . . . . . . . . . . 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 5522 . . . . . . . . . . 11  |-  ( ( ( ph  /\  n  e.  N. )  /\  k  e.  N. )  ->  ( F `  k )  e.  R. )
11 simplr 502 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  n  e.  N. )  /\  k  e.  N. )  ->  n  e.  N. )
12 recnnpr 7320 . . . . . . . . . . . 12  |-  ( n  e.  N.  ->  <. { l  |  l  <Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >.  e.  P. )
13 prsrcl 7556 . . . . . . . . . . . 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 7523 . . . . . . . . . . . 12  |-  1R  e.  R.
1615a1i 9 . . . . . . . . . . 11  |-  ( ( ( ph  /\  n  e.  N. )  /\  k  e.  N. )  ->  1R  e.  R. )
17 addcomsrg 7527 . . . . . . . . . . . 12  |-  ( ( f  e.  R.  /\  g  e.  R. )  ->  ( f  +R  g
)  =  ( g  +R  f ) )
1817adantl 273 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  n  e.  N. )  /\  k  e.  N. )  /\  ( f  e. 
R.  /\  g  e.  R. ) )  ->  (
f  +R  g )  =  ( g  +R  f ) )
19 addasssrg 7528 . . . . . . . . . . . 12  |-  ( ( f  e.  R.  /\  g  e.  R.  /\  h  e.  R. )  ->  (
( f  +R  g
)  +R  h )  =  ( f  +R  ( g  +R  h
) ) )
2019adantl 273 . . . . . . . . . . 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 5917 . . . . . . . . . 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 7568 . . . . . . . . . . . 12  |-  ( (
ph  /\  k  e.  N. )  ->  ( ( G `  k )  +R  A )  =  ( ( F `  k )  +R  1R ) )
2322adantlr 466 . . . . . . . . . . 11  |-  ( ( ( ph  /\  n  e.  N. )  /\  k  e.  N. )  ->  (
( G `  k
)  +R  A )  =  ( ( F `
 k )  +R 
1R ) )
2423oveq1d 5755 . . . . . . . . . 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 7569 . . . . . . . . . . . . 13  |-  ( ph  ->  G : N. --> R. )
2625ad2antrr 477 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  n  e.  N. )  /\  k  e.  N. )  ->  G : N. --> R. )
2726, 9ffvelrnd 5522 . . . . . . . . . . 11  |-  ( ( ( ph  /\  n  e.  N. )  /\  k  e.  N. )  ->  ( G `  k )  e.  R. )
283caucvgsrlemasr 7562 . . . . . . . . . . . 12  |-  ( ph  ->  A  e.  R. )
2928ad2antrr 477 . . . . . . . . . . 11  |-  ( ( ( ph  /\  n  e.  N. )  /\  k  e.  N. )  ->  A  e.  R. )
3027, 29, 14, 18, 20caov32d 5917 . . . . . . . . . 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 2154 . . . . . . . . 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 3910 . . . . . . . 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 7542 . . . . . . . . . 10  |-  ( ( f  e.  R.  /\  g  e.  R.  /\  h  e.  R. )  ->  (
f  <R  g  <->  ( h  +R  f )  <R  (
h  +R  g ) ) )
3433adantl 273 . . . . . . . . 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 5522 . . . . . . . . 9  |-  ( ( ( ph  /\  n  e.  N. )  /\  k  e.  N. )  ->  ( F `  n )  e.  R. )
36 addclsr 7525 . . . . . . . . . 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 406 . . . . . . . . 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 5906 . . . . . . . 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 5522 . . . . . . . . 9  |-  ( ( ( ph  /\  n  e.  N. )  /\  k  e.  N. )  ->  ( G `  n )  e.  R. )
40 addclsr 7525 . . . . . . . . . 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 406 . . . . . . . . 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 5906 . . . . . . . 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 2121 . . . . . . . . 9  |-  ( ( ( ph  /\  n  e.  N. )  /\  k  e.  N. )  ->  (
( F `  k
)  +R  1R )  =  ( ( G `
 k )  +R  A ) )
4535, 14, 16, 18, 20caov32d 5917 . . . . . . . . . 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 5755 . . . . . . . . . 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 5917 . . . . . . . . . 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 2154 . . . . . . . . 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 3910 . . . . . . . 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 7525 . . . . . . . . . 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 406 . . . . . . . . 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 5906 . . . . . . . 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 7525 . . . . . . . . . 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 406 . . . . . . . . 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 5906 . . . . . . . 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 462 . . . . . 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 2474 . . 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 2474 . 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 945    = wceq 1314    e. wcel 1463   {cab 2101   A.wral 2391   <.cop 3498   class class class wbr 3897    |-> cmpt 3957   -->wf 5087   ` cfv 5091  (class class class)co 5740   1oc1o 6272   [cec 6393   N.cnpi 7044    <N clti 7047    ~Q ceq 7051   *Qcrq 7056    <Q cltq 7057   P.cnp 7063   1Pc1p 7064    +P. cpp 7065    ~R cer 7068   R.cnr 7069   1Rc1r 7071   -1Rcm1r 7072    +R cplr 7073    .R cmr 7074    <R cltr 7075
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 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-coll 4011  ax-sep 4014  ax-nul 4022  ax-pow 4066  ax-pr 4099  ax-un 4323  ax-setind 4420  ax-iinf 4470
This theorem depends on definitions:  df-bi 116  df-dc 803  df-3or 946  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ne 2284  df-ral 2396  df-rex 2397  df-reu 2398  df-rab 2400  df-v 2660  df-sbc 2881  df-csb 2974  df-dif 3041  df-un 3043  df-in 3045  df-ss 3052  df-nul 3332  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-int 3740  df-iun 3783  df-br 3898  df-opab 3958  df-mpt 3959  df-tr 3995  df-eprel 4179  df-id 4183  df-po 4186  df-iso 4187  df-iord 4256  df-on 4258  df-suc 4261  df-iom 4473  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-rn 4518  df-res 4519  df-ima 4520  df-iota 5056  df-fun 5093  df-fn 5094  df-f 5095  df-f1 5096  df-fo 5097  df-f1o 5098  df-fv 5099  df-ov 5743  df-oprab 5744  df-mpo 5745  df-1st 6004  df-2nd 6005  df-recs 6168  df-irdg 6233  df-1o 6279  df-2o 6280  df-oadd 6283  df-omul 6284  df-er 6395  df-ec 6397  df-qs 6401  df-ni 7076  df-pli 7077  df-mi 7078  df-lti 7079  df-plpq 7116  df-mpq 7117  df-enq 7119  df-nqqs 7120  df-plqqs 7121  df-mqqs 7122  df-1nqqs 7123  df-rq 7124  df-ltnqqs 7125  df-enq0 7196  df-nq0 7197  df-0nq0 7198  df-plq0 7199  df-mq0 7200  df-inp 7238  df-i1p 7239  df-iplp 7240  df-imp 7241  df-iltp 7242  df-enr 7498  df-nr 7499  df-plr 7500  df-mr 7501  df-ltr 7502  df-0r 7503  df-1r 7504  df-m1r 7505
This theorem is referenced by:  caucvgsrlemoffres  7572
  Copyright terms: Public domain W3C validator