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Theorem caucvgsrlemcau 7067
Description: Lemma for caucvgsr 7076. Defining the Cauchy condition in terms of positive reals. (Contributed by Jim Kingdon, 23-Jun-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  ) ) ) )
caucvgsrlemgt1.gt1  |-  ( ph  ->  A. m  e.  N.  1R  <R  ( F `  m ) )
caucvgsrlemf.xfr  |-  G  =  ( x  e.  N.  |->  ( iota_ y  e.  P.  ( F `  x )  =  [ <. (
y  +P.  1P ) ,  1P >. ]  ~R  )
)
Assertion
Ref Expression
caucvgsrlemcau  |-  ( ph  ->  A. n  e.  N.  A. k  e.  N.  (
n  <N  k  ->  (
( G `  n
)  <P  ( ( G `
 k )  +P. 
<. { l  |  l 
<Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >. )  /\  ( G `  k
)  <P  ( ( G `
 n )  +P. 
<. { l  |  l 
<Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >. )
) ) )
Distinct variable groups:    m, F, x   
y, F, x    k, m, n, x    ph, k, n, x    y, k, n   
n, l, u
Allowed substitution hints:    ph( y, u, m, l)    F( u, k, n, l)    G( x, y, u, k, m, n, l)

Proof of Theorem caucvgsrlemcau
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 . . . . . . . . . . 11  |-  ( ph  ->  F : N. --> R. )
3 caucvgsrlemgt1.gt1 . . . . . . . . . . 11  |-  ( ph  ->  A. m  e.  N.  1R  <R  ( F `  m ) )
4 caucvgsrlemf.xfr . . . . . . . . . . 11  |-  G  =  ( x  e.  N.  |->  ( iota_ y  e.  P.  ( F `  x )  =  [ <. (
y  +P.  1P ) ,  1P >. ]  ~R  )
)
52, 1, 3, 4caucvgsrlemf 7066 . . . . . . . . . 10  |-  ( ph  ->  G : N. --> P. )
65ad2antrr 472 . . . . . . . . 9  |-  ( ( ( ph  /\  n  e.  N. )  /\  k  e.  N. )  ->  G : N. --> P. )
7 simplr 497 . . . . . . . . 9  |-  ( ( ( ph  /\  n  e.  N. )  /\  k  e.  N. )  ->  n  e.  N. )
86, 7ffvelrnd 5356 . . . . . . . 8  |-  ( ( ( ph  /\  n  e.  N. )  /\  k  e.  N. )  ->  ( G `  n )  e.  P. )
95adantr 270 . . . . . . . . . 10  |-  ( (
ph  /\  n  e.  N. )  ->  G : N.
--> P. )
109ffvelrnda 5355 . . . . . . . . 9  |-  ( ( ( ph  /\  n  e.  N. )  /\  k  e.  N. )  ->  ( G `  k )  e.  P. )
11 recnnpr 6836 . . . . . . . . . 10  |-  ( n  e.  N.  ->  <. { l  |  l  <Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >.  e.  P. )
127, 11syl 14 . . . . . . . . 9  |-  ( ( ( ph  /\  n  e.  N. )  /\  k  e.  N. )  ->  <. { l  |  l  <Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >.  e.  P. )
13 addclpr 6825 . . . . . . . . 9  |-  ( ( ( G `  k
)  e.  P.  /\  <. { l  |  l 
<Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >.  e.  P. )  ->  ( ( G `
 k )  +P. 
<. { l  |  l 
<Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >. )  e.  P. )
1410, 12, 13syl2anc 403 . . . . . . . 8  |-  ( ( ( ph  /\  n  e.  N. )  /\  k  e.  N. )  ->  (
( G `  k
)  +P.  <. { l  |  l  <Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >. )  e.  P. )
15 prsrlt 7061 . . . . . . . 8  |-  ( ( ( G `  n
)  e.  P.  /\  ( ( G `  k )  +P.  <. { l  |  l  <Q 
( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >. )  e.  P. )  ->  (
( G `  n
)  <P  ( ( G `
 k )  +P. 
<. { l  |  l 
<Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >. )  <->  [
<. ( ( G `  n )  +P.  1P ) ,  1P >. ]  ~R  <R  [ <. ( ( ( G `  k )  +P.  <. { l  |  l  <Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >. )  +P.  1P ) ,  1P >. ]  ~R  ) )
168, 14, 15syl2anc 403 . . . . . . 7  |-  ( ( ( ph  /\  n  e.  N. )  /\  k  e.  N. )  ->  (
( G `  n
)  <P  ( ( G `
 k )  +P. 
<. { l  |  l 
<Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >. )  <->  [
<. ( ( G `  n )  +P.  1P ) ,  1P >. ]  ~R  <R  [ <. ( ( ( G `  k )  +P.  <. { l  |  l  <Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >. )  +P.  1P ) ,  1P >. ]  ~R  ) )
172, 1, 3, 4caucvgsrlemfv 7065 . . . . . . . . 9  |-  ( (
ph  /\  n  e.  N. )  ->  [ <. ( ( G `  n
)  +P.  1P ) ,  1P >. ]  ~R  =  ( F `  n ) )
1817adantr 270 . . . . . . . 8  |-  ( ( ( ph  /\  n  e.  N. )  /\  k  e.  N. )  ->  [ <. ( ( G `  n
)  +P.  1P ) ,  1P >. ]  ~R  =  ( F `  n ) )
19 prsradd 7060 . . . . . . . . . 10  |-  ( ( ( G `  k
)  e.  P.  /\  <. { l  |  l 
<Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >.  e.  P. )  ->  [ <. (
( ( G `  k )  +P.  <. { l  |  l  <Q 
( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >. )  +P.  1P ) ,  1P >. ]  ~R  =  ( [ <. ( ( G `
 k )  +P. 
1P ) ,  1P >. ]  ~R  +R  [ <. ( <. { l  |  l  <Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  )
)
2010, 12, 19syl2anc 403 . . . . . . . . 9  |-  ( ( ( ph  /\  n  e.  N. )  /\  k  e.  N. )  ->  [ <. ( ( ( G `  k )  +P.  <. { l  |  l  <Q 
( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >. )  +P.  1P ) ,  1P >. ]  ~R  =  ( [ <. ( ( G `
 k )  +P. 
1P ) ,  1P >. ]  ~R  +R  [ <. ( <. { l  |  l  <Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  )
)
212, 1, 3, 4caucvgsrlemfv 7065 . . . . . . . . . . 11  |-  ( (
ph  /\  k  e.  N. )  ->  [ <. ( ( G `  k
)  +P.  1P ) ,  1P >. ]  ~R  =  ( F `  k ) )
2221adantlr 461 . . . . . . . . . 10  |-  ( ( ( ph  /\  n  e.  N. )  /\  k  e.  N. )  ->  [ <. ( ( G `  k
)  +P.  1P ) ,  1P >. ]  ~R  =  ( F `  k ) )
2322oveq1d 5579 . . . . . . . . 9  |-  ( ( ( ph  /\  n  e.  N. )  /\  k  e.  N. )  ->  ( [ <. ( ( G `
 k )  +P. 
1P ) ,  1P >. ]  ~R  +R  [ <. ( <. { l  |  l  <Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  )  =  ( ( F `
 k )  +R 
[ <. ( <. { l  |  l  <Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ) )
2420, 23eqtrd 2115 . . . . . . . 8  |-  ( ( ( ph  /\  n  e.  N. )  /\  k  e.  N. )  ->  [ <. ( ( ( G `  k )  +P.  <. { l  |  l  <Q 
( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >. )  +P.  1P ) ,  1P >. ]  ~R  =  ( ( F `  k
)  +R  [ <. (
<. { l  |  l 
<Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ) )
2518, 24breq12d 3819 . . . . . . 7  |-  ( ( ( ph  /\  n  e.  N. )  /\  k  e.  N. )  ->  ( [ <. ( ( G `
 n )  +P. 
1P ) ,  1P >. ]  ~R  <R  [ <. ( ( ( G `  k )  +P.  <. { l  |  l  <Q 
( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >. )  +P.  1P ) ,  1P >. ]  ~R  <->  ( F `  n )  <R  (
( F `  k
)  +R  [ <. (
<. { l  |  l 
<Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ) ) )
2616, 25bitrd 186 . . . . . 6  |-  ( ( ( ph  /\  n  e.  N. )  /\  k  e.  N. )  ->  (
( G `  n
)  <P  ( ( G `
 k )  +P. 
<. { l  |  l 
<Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >. )  <->  ( F `  n ) 
<R  ( ( F `  k )  +R  [ <. ( <. { l  |  l  <Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  )
) )
27 addclpr 6825 . . . . . . . . 9  |-  ( ( ( G `  n
)  e.  P.  /\  <. { l  |  l 
<Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >.  e.  P. )  ->  ( ( G `
 n )  +P. 
<. { l  |  l 
<Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >. )  e.  P. )
288, 12, 27syl2anc 403 . . . . . . . 8  |-  ( ( ( ph  /\  n  e.  N. )  /\  k  e.  N. )  ->  (
( G `  n
)  +P.  <. { l  |  l  <Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >. )  e.  P. )
29 prsrlt 7061 . . . . . . . 8  |-  ( ( ( G `  k
)  e.  P.  /\  ( ( G `  n )  +P.  <. { l  |  l  <Q 
( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >. )  e.  P. )  ->  (
( G `  k
)  <P  ( ( G `
 n )  +P. 
<. { l  |  l 
<Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >. )  <->  [
<. ( ( G `  k )  +P.  1P ) ,  1P >. ]  ~R  <R  [ <. ( ( ( G `  n )  +P.  <. { l  |  l  <Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >. )  +P.  1P ) ,  1P >. ]  ~R  ) )
3010, 28, 29syl2anc 403 . . . . . . 7  |-  ( ( ( ph  /\  n  e.  N. )  /\  k  e.  N. )  ->  (
( G `  k
)  <P  ( ( G `
 n )  +P. 
<. { l  |  l 
<Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >. )  <->  [
<. ( ( G `  k )  +P.  1P ) ,  1P >. ]  ~R  <R  [ <. ( ( ( G `  n )  +P.  <. { l  |  l  <Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >. )  +P.  1P ) ,  1P >. ]  ~R  ) )
31 prsradd 7060 . . . . . . . . . 10  |-  ( ( ( G `  n
)  e.  P.  /\  <. { l  |  l 
<Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >.  e.  P. )  ->  [ <. (
( ( G `  n )  +P.  <. { l  |  l  <Q 
( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >. )  +P.  1P ) ,  1P >. ]  ~R  =  ( [ <. ( ( G `
 n )  +P. 
1P ) ,  1P >. ]  ~R  +R  [ <. ( <. { l  |  l  <Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  )
)
328, 12, 31syl2anc 403 . . . . . . . . 9  |-  ( ( ( ph  /\  n  e.  N. )  /\  k  e.  N. )  ->  [ <. ( ( ( G `  n )  +P.  <. { l  |  l  <Q 
( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >. )  +P.  1P ) ,  1P >. ]  ~R  =  ( [ <. ( ( G `
 n )  +P. 
1P ) ,  1P >. ]  ~R  +R  [ <. ( <. { l  |  l  <Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  )
)
3318oveq1d 5579 . . . . . . . . 9  |-  ( ( ( ph  /\  n  e.  N. )  /\  k  e.  N. )  ->  ( [ <. ( ( G `
 n )  +P. 
1P ) ,  1P >. ]  ~R  +R  [ <. ( <. { l  |  l  <Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  )  =  ( ( F `
 n )  +R 
[ <. ( <. { l  |  l  <Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ) )
3432, 33eqtrd 2115 . . . . . . . 8  |-  ( ( ( ph  /\  n  e.  N. )  /\  k  e.  N. )  ->  [ <. ( ( ( G `  n )  +P.  <. { l  |  l  <Q 
( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >. )  +P.  1P ) ,  1P >. ]  ~R  =  ( ( F `  n
)  +R  [ <. (
<. { l  |  l 
<Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ) )
3522, 34breq12d 3819 . . . . . . 7  |-  ( ( ( ph  /\  n  e.  N. )  /\  k  e.  N. )  ->  ( [ <. ( ( G `
 k )  +P. 
1P ) ,  1P >. ]  ~R  <R  [ <. ( ( ( G `  n )  +P.  <. { 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  ) ) )
3630, 35bitrd 186 . . . . . 6  |-  ( ( ( ph  /\  n  e.  N. )  /\  k  e.  N. )  ->  (
( G `  k
)  <P  ( ( G `
 n )  +P. 
<. { l  |  l 
<Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >. )  <->  ( F `  k ) 
<R  ( ( F `  n )  +R  [ <. ( <. { l  |  l  <Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  )
) )
3726, 36anbi12d 457 . . . . 5  |-  ( ( ( ph  /\  n  e.  N. )  /\  k  e.  N. )  ->  (
( ( G `  n )  <P  (
( G `  k
)  +P.  <. { l  |  l  <Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >. )  /\  ( G `  k )  <P  ( ( G `  n )  +P.  <. { l  |  l  <Q 
( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >. )
)  <->  ( ( 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  ) ) ) )
3837imbi2d 228 . . . 4  |-  ( ( ( ph  /\  n  e.  N. )  /\  k  e.  N. )  ->  (
( n  <N  k  ->  ( ( G `  n )  <P  (
( G `  k
)  +P.  <. { l  |  l  <Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >. )  /\  ( G `  k )  <P  ( ( G `  n )  +P.  <. { l  |  l  <Q 
( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >. )
) )  <->  ( 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  ) ) ) ) )
3938ralbidva 2369 . . 3  |-  ( (
ph  /\  n  e.  N. )  ->  ( A. k  e.  N.  (
n  <N  k  ->  (
( G `  n
)  <P  ( ( G `
 k )  +P. 
<. { l  |  l 
<Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >. )  /\  ( G `  k
)  <P  ( ( G `
 n )  +P. 
<. { l  |  l 
<Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >. )
) )  <->  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  ) ) ) ) )
4039ralbidva 2369 . 2  |-  ( ph  ->  ( A. n  e. 
N.  A. k  e.  N.  ( n  <N  k  -> 
( ( G `  n )  <P  (
( G `  k
)  +P.  <. { l  |  l  <Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >. )  /\  ( G `  k )  <P  ( ( G `  n )  +P.  <. { l  |  l  <Q 
( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >. )
) )  <->  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  ) ) ) ) )
411, 40mpbird 165 1  |-  ( ph  ->  A. n  e.  N.  A. k  e.  N.  (
n  <N  k  ->  (
( G `  n
)  <P  ( ( G `
 k )  +P. 
<. { l  |  l 
<Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >. )  /\  ( G `  k
)  <P  ( ( G `
 n )  +P. 
<. { l  |  l 
<Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >. )
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    = wceq 1285    e. wcel 1434   {cab 2069   A.wral 2353   <.cop 3420   class class class wbr 3806    |-> cmpt 3860   -->wf 4949   ` cfv 4953   iota_crio 5519  (class class class)co 5564   1oc1o 6079   [cec 6192   N.cnpi 6560    <N clti 6563    ~Q ceq 6567   *Qcrq 6572    <Q cltq 6573   P.cnp 6579   1Pc1p 6580    +P. cpp 6581    <P cltp 6583    ~R cer 6584   R.cnr 6585   1Rc1r 6587    +R cplr 6589    <R cltr 6591
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-coll 3914  ax-sep 3917  ax-nul 3925  ax-pow 3969  ax-pr 3993  ax-un 4217  ax-setind 4309  ax-iinf 4358
This theorem depends on definitions:  df-bi 115  df-dc 777  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-ral 2358  df-rex 2359  df-reu 2360  df-rmo 2361  df-rab 2362  df-v 2612  df-sbc 2826  df-csb 2919  df-dif 2985  df-un 2987  df-in 2989  df-ss 2996  df-nul 3269  df-pw 3403  df-sn 3423  df-pr 3424  df-op 3426  df-uni 3623  df-int 3658  df-iun 3701  df-br 3807  df-opab 3861  df-mpt 3862  df-tr 3897  df-eprel 4073  df-id 4077  df-po 4080  df-iso 4081  df-iord 4150  df-on 4152  df-suc 4155  df-iom 4361  df-xp 4398  df-rel 4399  df-cnv 4400  df-co 4401  df-dm 4402  df-rn 4403  df-res 4404  df-ima 4405  df-iota 4918  df-fun 4955  df-fn 4956  df-f 4957  df-f1 4958  df-fo 4959  df-f1o 4960  df-fv 4961  df-riota 5520  df-ov 5567  df-oprab 5568  df-mpt2 5569  df-1st 5819  df-2nd 5820  df-recs 5975  df-irdg 6040  df-1o 6086  df-2o 6087  df-oadd 6090  df-omul 6091  df-er 6194  df-ec 6196  df-qs 6200  df-ni 6592  df-pli 6593  df-mi 6594  df-lti 6595  df-plpq 6632  df-mpq 6633  df-enq 6635  df-nqqs 6636  df-plqqs 6637  df-mqqs 6638  df-1nqqs 6639  df-rq 6640  df-ltnqqs 6641  df-enq0 6712  df-nq0 6713  df-0nq0 6714  df-plq0 6715  df-mq0 6716  df-inp 6754  df-i1p 6755  df-iplp 6756  df-iltp 6758  df-enr 7001  df-nr 7002  df-plr 7003  df-ltr 7005  df-0r 7006  df-1r 7007
This theorem is referenced by:  caucvgsrlemgt1  7069
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