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Theorem caucvgsrlemcau 7783
Description: Lemma for caucvgsr 7792. 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 7782 . . . . . . . . . 10  |-  ( ph  ->  G : N. --> P. )
65ad2antrr 488 . . . . . . . . 9  |-  ( ( ( ph  /\  n  e.  N. )  /\  k  e.  N. )  ->  G : N. --> P. )
7 simplr 528 . . . . . . . . 9  |-  ( ( ( ph  /\  n  e.  N. )  /\  k  e.  N. )  ->  n  e.  N. )
86, 7ffvelcdmd 5648 . . . . . . . 8  |-  ( ( ( ph  /\  n  e.  N. )  /\  k  e.  N. )  ->  ( G `  n )  e.  P. )
95adantr 276 . . . . . . . . . 10  |-  ( (
ph  /\  n  e.  N. )  ->  G : N.
--> P. )
109ffvelcdmda 5647 . . . . . . . . 9  |-  ( ( ( ph  /\  n  e.  N. )  /\  k  e.  N. )  ->  ( G `  k )  e.  P. )
11 recnnpr 7538 . . . . . . . . . 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 7527 . . . . . . . . 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 411 . . . . . . . 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 7777 . . . . . . . 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 411 . . . . . . 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 7781 . . . . . . . . 9  |-  ( (
ph  /\  n  e.  N. )  ->  [ <. ( ( G `  n
)  +P.  1P ) ,  1P >. ]  ~R  =  ( F `  n ) )
1817adantr 276 . . . . . . . 8  |-  ( ( ( ph  /\  n  e.  N. )  /\  k  e.  N. )  ->  [ <. ( ( G `  n
)  +P.  1P ) ,  1P >. ]  ~R  =  ( F `  n ) )
19 prsradd 7776 . . . . . . . . . 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 411 . . . . . . . . 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 7781 . . . . . . . . . . 11  |-  ( (
ph  /\  k  e.  N. )  ->  [ <. ( ( G `  k
)  +P.  1P ) ,  1P >. ]  ~R  =  ( F `  k ) )
2221adantlr 477 . . . . . . . . . 10  |-  ( ( ( ph  /\  n  e.  N. )  /\  k  e.  N. )  ->  [ <. ( ( G `  k
)  +P.  1P ) ,  1P >. ]  ~R  =  ( F `  k ) )
2322oveq1d 5884 . . . . . . . . 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 2210 . . . . . . . 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 4013 . . . . . . 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 188 . . . . . 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 7527 . . . . . . . . 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 411 . . . . . . . 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 7777 . . . . . . . 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 411 . . . . . . 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 7776 . . . . . . . . . 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 411 . . . . . . . . 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 5884 . . . . . . . . 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 2210 . . . . . . . 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 4013 . . . . . . 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 188 . . . . . 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 473 . . . . 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 230 . . . 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 2473 . . 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 2473 . 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 167 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 104    <-> wb 105    = wceq 1353    e. wcel 2148   {cab 2163   A.wral 2455   <.cop 3594   class class class wbr 4000    |-> cmpt 4061   -->wf 5208   ` cfv 5212   iota_crio 5824  (class class class)co 5869   1oc1o 6404   [cec 6527   N.cnpi 7262    <N clti 7265    ~Q ceq 7269   *Qcrq 7274    <Q cltq 7275   P.cnp 7281   1Pc1p 7282    +P. cpp 7283    <P cltp 7285    ~R cer 7286   R.cnr 7287   1Rc1r 7289    +R cplr 7291    <R cltr 7293
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4115  ax-sep 4118  ax-nul 4126  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-setind 4533  ax-iinf 4584
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-tr 4099  df-eprel 4286  df-id 4290  df-po 4293  df-iso 4294  df-iord 4363  df-on 4365  df-suc 4368  df-iom 4587  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-f1 5217  df-fo 5218  df-f1o 5219  df-fv 5220  df-riota 5825  df-ov 5872  df-oprab 5873  df-mpo 5874  df-1st 6135  df-2nd 6136  df-recs 6300  df-irdg 6365  df-1o 6411  df-2o 6412  df-oadd 6415  df-omul 6416  df-er 6529  df-ec 6531  df-qs 6535  df-ni 7294  df-pli 7295  df-mi 7296  df-lti 7297  df-plpq 7334  df-mpq 7335  df-enq 7337  df-nqqs 7338  df-plqqs 7339  df-mqqs 7340  df-1nqqs 7341  df-rq 7342  df-ltnqqs 7343  df-enq0 7414  df-nq0 7415  df-0nq0 7416  df-plq0 7417  df-mq0 7418  df-inp 7456  df-i1p 7457  df-iplp 7458  df-iltp 7460  df-enr 7716  df-nr 7717  df-plr 7718  df-ltr 7720  df-0r 7721  df-1r 7722
This theorem is referenced by:  caucvgsrlemgt1  7785
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