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Theorem axcaucvglemcau 7126
Description: Lemma for axcaucvg 7128. The result of mapping to  N. and  R. satisfies the Cauchy condition. (Contributed by Jim Kingdon, 9-Jul-2021.)
Hypotheses
Ref Expression
axcaucvg.n  |-  N  = 
|^| { x  |  ( 1  e.  x  /\  A. y  e.  x  ( y  +  1 )  e.  x ) }
axcaucvg.f  |-  ( ph  ->  F : N --> RR )
axcaucvg.cau  |-  ( ph  ->  A. n  e.  N  A. k  e.  N  ( n  <RR  k  -> 
( ( F `  n )  <RR  ( ( F `  k )  +  ( iota_ r  e.  RR  ( n  x.  r )  =  1 ) )  /\  ( F `  k )  <RR  ( ( F `  n )  +  (
iota_ r  e.  RR  ( n  x.  r
)  =  1 ) ) ) ) )
axcaucvg.g  |-  G  =  ( j  e.  N.  |->  ( iota_ z  e.  R.  ( F `  <. [ <. (
<. { l  |  l 
<Q  [ <. j ,  1o >. ]  ~Q  } ,  { u  |  [ <. j ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >. )  =  <. z ,  0R >. )
)
Assertion
Ref Expression
axcaucvglemcau  |-  ( 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:    k, F, n, z, j    k, N, n    z, G    k,
l, r, u, n   
j, l, u, z    ph, j, k, n    y,
l, u    x, y    j, n, z, k
Allowed substitution hints:    ph( x, y, z, u, r, l)    F( x, y, u, r, l)    G( x, y, u, j, k, n, r, l)    N( x, y, z, u, j, r, l)

Proof of Theorem axcaucvglemcau
Dummy variables  a  b are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ltrenn 7085 . . . . . . . . . 10  |-  ( n 
<N  k  ->  <. [ <. (
<. { l  |  l 
<Q  [ <. n ,  1o >. ]  ~Q  } ,  { u  |  [ <. n ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  <RR  <. [ <. ( <. { l  |  l 
<Q  [ <. k ,  1o >. ]  ~Q  } ,  { u  |  [ <. k ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >. )
21adantl 271 . . . . . . . . 9  |-  ( ( ( ( ph  /\  n  e.  N. )  /\  k  e.  N. )  /\  n  <N  k
)  ->  <. [ <. (
<. { l  |  l 
<Q  [ <. n ,  1o >. ]  ~Q  } ,  { u  |  [ <. n ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  <RR  <. [ <. ( <. { l  |  l 
<Q  [ <. k ,  1o >. ]  ~Q  } ,  { u  |  [ <. k ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >. )
3 pitonn 7078 . . . . . . . . . . . 12  |-  ( k  e.  N.  ->  <. [ <. (
<. { l  |  l 
<Q  [ <. k ,  1o >. ]  ~Q  } ,  { u  |  [ <. k ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  e.  |^| { x  |  ( 1  e.  x  /\  A. y  e.  x  ( y  +  1 )  e.  x ) } )
4 axcaucvg.n . . . . . . . . . . . 12  |-  N  = 
|^| { x  |  ( 1  e.  x  /\  A. y  e.  x  ( y  +  1 )  e.  x ) }
53, 4syl6eleqr 2173 . . . . . . . . . . 11  |-  ( k  e.  N.  ->  <. [ <. (
<. { l  |  l 
<Q  [ <. k ,  1o >. ]  ~Q  } ,  { u  |  [ <. k ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  e.  N )
65ad2antlr 473 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  n  e.  N. )  /\  k  e.  N. )  /\  n  <N  k
)  ->  <. [ <. (
<. { l  |  l 
<Q  [ <. k ,  1o >. ]  ~Q  } ,  { u  |  [ <. k ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  e.  N )
7 pitonn 7078 . . . . . . . . . . . . 13  |-  ( n  e.  N.  ->  <. [ <. (
<. { l  |  l 
<Q  [ <. n ,  1o >. ]  ~Q  } ,  { u  |  [ <. n ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  e.  |^| { x  |  ( 1  e.  x  /\  A. y  e.  x  ( y  +  1 )  e.  x ) } )
87, 4syl6eleqr 2173 . . . . . . . . . . . 12  |-  ( n  e.  N.  ->  <. [ <. (
<. { l  |  l 
<Q  [ <. n ,  1o >. ]  ~Q  } ,  { u  |  [ <. n ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  e.  N )
98ad3antlr 477 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  n  e.  N. )  /\  k  e.  N. )  /\  n  <N  k
)  ->  <. [ <. (
<. { l  |  l 
<Q  [ <. n ,  1o >. ]  ~Q  } ,  { u  |  [ <. n ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  e.  N )
10 axcaucvg.cau . . . . . . . . . . . . 13  |-  ( ph  ->  A. n  e.  N  A. k  e.  N  ( n  <RR  k  -> 
( ( F `  n )  <RR  ( ( F `  k )  +  ( iota_ r  e.  RR  ( n  x.  r )  =  1 ) )  /\  ( F `  k )  <RR  ( ( F `  n )  +  (
iota_ r  e.  RR  ( n  x.  r
)  =  1 ) ) ) ) )
11 breq1 3796 . . . . . . . . . . . . . . 15  |-  ( n  =  a  ->  (
n  <RR  k  <->  a  <RR  k ) )
12 fveq2 5209 . . . . . . . . . . . . . . . . 17  |-  ( n  =  a  ->  ( F `  n )  =  ( F `  a ) )
13 oveq1 5550 . . . . . . . . . . . . . . . . . . . 20  |-  ( n  =  a  ->  (
n  x.  r )  =  ( a  x.  r ) )
1413eqeq1d 2090 . . . . . . . . . . . . . . . . . . 19  |-  ( n  =  a  ->  (
( n  x.  r
)  =  1  <->  (
a  x.  r )  =  1 ) )
1514riotabidv 5501 . . . . . . . . . . . . . . . . . 18  |-  ( n  =  a  ->  ( iota_ r  e.  RR  (
n  x.  r )  =  1 )  =  ( iota_ r  e.  RR  ( a  x.  r
)  =  1 ) )
1615oveq2d 5559 . . . . . . . . . . . . . . . . 17  |-  ( n  =  a  ->  (
( F `  k
)  +  ( iota_ r  e.  RR  ( n  x.  r )  =  1 ) )  =  ( ( F `  k )  +  (
iota_ r  e.  RR  ( a  x.  r
)  =  1 ) ) )
1712, 16breq12d 3806 . . . . . . . . . . . . . . . 16  |-  ( n  =  a  ->  (
( F `  n
)  <RR  ( ( F `
 k )  +  ( iota_ r  e.  RR  ( n  x.  r
)  =  1 ) )  <->  ( F `  a )  <RR  ( ( F `  k )  +  ( iota_ r  e.  RR  ( a  x.  r )  =  1 ) ) ) )
1812, 15oveq12d 5561 . . . . . . . . . . . . . . . . 17  |-  ( n  =  a  ->  (
( F `  n
)  +  ( iota_ r  e.  RR  ( n  x.  r )  =  1 ) )  =  ( ( F `  a )  +  (
iota_ r  e.  RR  ( a  x.  r
)  =  1 ) ) )
1918breq2d 3805 . . . . . . . . . . . . . . . 16  |-  ( n  =  a  ->  (
( F `  k
)  <RR  ( ( F `
 n )  +  ( iota_ r  e.  RR  ( n  x.  r
)  =  1 ) )  <->  ( F `  k )  <RR  ( ( F `  a )  +  ( iota_ r  e.  RR  ( a  x.  r )  =  1 ) ) ) )
2017, 19anbi12d 457 . . . . . . . . . . . . . . 15  |-  ( n  =  a  ->  (
( ( F `  n )  <RR  ( ( F `  k )  +  ( iota_ r  e.  RR  ( n  x.  r )  =  1 ) )  /\  ( F `  k )  <RR  ( ( F `  n )  +  (
iota_ r  e.  RR  ( n  x.  r
)  =  1 ) ) )  <->  ( ( F `  a )  <RR  ( ( F `  k )  +  (
iota_ r  e.  RR  ( a  x.  r
)  =  1 ) )  /\  ( F `
 k )  <RR  ( ( F `  a
)  +  ( iota_ r  e.  RR  ( a  x.  r )  =  1 ) ) ) ) )
2111, 20imbi12d 232 . . . . . . . . . . . . . 14  |-  ( n  =  a  ->  (
( n  <RR  k  -> 
( ( F `  n )  <RR  ( ( F `  k )  +  ( iota_ r  e.  RR  ( n  x.  r )  =  1 ) )  /\  ( F `  k )  <RR  ( ( F `  n )  +  (
iota_ r  e.  RR  ( n  x.  r
)  =  1 ) ) ) )  <->  ( a  <RR  k  ->  ( ( F `  a )  <RR  ( ( F `  k )  +  (
iota_ r  e.  RR  ( a  x.  r
)  =  1 ) )  /\  ( F `
 k )  <RR  ( ( F `  a
)  +  ( iota_ r  e.  RR  ( a  x.  r )  =  1 ) ) ) ) ) )
22 breq2 3797 . . . . . . . . . . . . . . 15  |-  ( k  =  b  ->  (
a  <RR  k  <->  a  <RR  b ) )
23 fveq2 5209 . . . . . . . . . . . . . . . . . 18  |-  ( k  =  b  ->  ( F `  k )  =  ( F `  b ) )
2423oveq1d 5558 . . . . . . . . . . . . . . . . 17  |-  ( k  =  b  ->  (
( F `  k
)  +  ( iota_ r  e.  RR  ( a  x.  r )  =  1 ) )  =  ( ( F `  b )  +  (
iota_ r  e.  RR  ( a  x.  r
)  =  1 ) ) )
2524breq2d 3805 . . . . . . . . . . . . . . . 16  |-  ( k  =  b  ->  (
( F `  a
)  <RR  ( ( F `
 k )  +  ( iota_ r  e.  RR  ( a  x.  r
)  =  1 ) )  <->  ( F `  a )  <RR  ( ( F `  b )  +  ( iota_ r  e.  RR  ( a  x.  r )  =  1 ) ) ) )
2623breq1d 3803 . . . . . . . . . . . . . . . 16  |-  ( k  =  b  ->  (
( F `  k
)  <RR  ( ( F `
 a )  +  ( iota_ r  e.  RR  ( a  x.  r
)  =  1 ) )  <->  ( F `  b )  <RR  ( ( F `  a )  +  ( iota_ r  e.  RR  ( a  x.  r )  =  1 ) ) ) )
2725, 26anbi12d 457 . . . . . . . . . . . . . . 15  |-  ( k  =  b  ->  (
( ( F `  a )  <RR  ( ( F `  k )  +  ( iota_ r  e.  RR  ( a  x.  r )  =  1 ) )  /\  ( F `  k )  <RR  ( ( F `  a )  +  (
iota_ r  e.  RR  ( a  x.  r
)  =  1 ) ) )  <->  ( ( F `  a )  <RR  ( ( F `  b )  +  (
iota_ r  e.  RR  ( a  x.  r
)  =  1 ) )  /\  ( F `
 b )  <RR  ( ( F `  a
)  +  ( iota_ r  e.  RR  ( a  x.  r )  =  1 ) ) ) ) )
2822, 27imbi12d 232 . . . . . . . . . . . . . 14  |-  ( k  =  b  ->  (
( a  <RR  k  -> 
( ( F `  a )  <RR  ( ( F `  k )  +  ( iota_ r  e.  RR  ( a  x.  r )  =  1 ) )  /\  ( F `  k )  <RR  ( ( F `  a )  +  (
iota_ r  e.  RR  ( a  x.  r
)  =  1 ) ) ) )  <->  ( a  <RR  b  ->  ( ( F `  a )  <RR  ( ( F `  b )  +  (
iota_ r  e.  RR  ( a  x.  r
)  =  1 ) )  /\  ( F `
 b )  <RR  ( ( F `  a
)  +  ( iota_ r  e.  RR  ( a  x.  r )  =  1 ) ) ) ) ) )
2921, 28cbvral2v 2586 . . . . . . . . . . . . 13  |-  ( A. n  e.  N  A. k  e.  N  (
n  <RR  k  ->  (
( F `  n
)  <RR  ( ( F `
 k )  +  ( iota_ r  e.  RR  ( n  x.  r
)  =  1 ) )  /\  ( F `
 k )  <RR  ( ( F `  n
)  +  ( iota_ r  e.  RR  ( n  x.  r )  =  1 ) ) ) )  <->  A. a  e.  N  A. b  e.  N  ( a  <RR  b  -> 
( ( F `  a )  <RR  ( ( F `  b )  +  ( iota_ r  e.  RR  ( a  x.  r )  =  1 ) )  /\  ( F `  b )  <RR  ( ( F `  a )  +  (
iota_ r  e.  RR  ( a  x.  r
)  =  1 ) ) ) ) )
3010, 29sylib 120 . . . . . . . . . . . 12  |-  ( ph  ->  A. a  e.  N  A. b  e.  N  ( a  <RR  b  -> 
( ( F `  a )  <RR  ( ( F `  b )  +  ( iota_ r  e.  RR  ( a  x.  r )  =  1 ) )  /\  ( F `  b )  <RR  ( ( F `  a )  +  (
iota_ r  e.  RR  ( a  x.  r
)  =  1 ) ) ) ) )
3130ad3antrrr 476 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  n  e.  N. )  /\  k  e.  N. )  /\  n  <N  k
)  ->  A. a  e.  N  A. b  e.  N  ( a  <RR  b  ->  ( ( F `  a )  <RR  ( ( F `  b )  +  (
iota_ r  e.  RR  ( a  x.  r
)  =  1 ) )  /\  ( F `
 b )  <RR  ( ( F `  a
)  +  ( iota_ r  e.  RR  ( a  x.  r )  =  1 ) ) ) ) )
32 breq1 3796 . . . . . . . . . . . . . 14  |-  ( a  =  <. [ <. ( <. { l  |  l 
<Q  [ <. n ,  1o >. ]  ~Q  } ,  { u  |  [ <. n ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  ->  ( a  <RR  b  <->  <. [ <. ( <. { l  |  l 
<Q  [ <. n ,  1o >. ]  ~Q  } ,  { u  |  [ <. n ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  <RR  b ) )
33 fveq2 5209 . . . . . . . . . . . . . . . 16  |-  ( a  =  <. [ <. ( <. { l  |  l 
<Q  [ <. n ,  1o >. ]  ~Q  } ,  { u  |  [ <. n ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  ->  ( F `  a )  =  ( F `  <. [ <. (
<. { l  |  l 
<Q  [ <. n ,  1o >. ]  ~Q  } ,  { u  |  [ <. n ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >. ) )
34 oveq1 5550 . . . . . . . . . . . . . . . . . . 19  |-  ( a  =  <. [ <. ( <. { l  |  l 
<Q  [ <. n ,  1o >. ]  ~Q  } ,  { u  |  [ <. n ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  ->  ( a  x.  r )  =  (
<. [ <. ( <. { l  |  l  <Q  [ <. n ,  1o >. ]  ~Q  } ,  { u  |  [ <. n ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  x.  r ) )
3534eqeq1d 2090 . . . . . . . . . . . . . . . . . 18  |-  ( a  =  <. [ <. ( <. { l  |  l 
<Q  [ <. n ,  1o >. ]  ~Q  } ,  { u  |  [ <. n ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  ->  ( (
a  x.  r )  =  1  <->  ( <. [
<. ( <. { l  |  l  <Q  [ <. n ,  1o >. ]  ~Q  } ,  { u  |  [ <. n ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  x.  r )  =  1 ) )
3635riotabidv 5501 . . . . . . . . . . . . . . . . 17  |-  ( a  =  <. [ <. ( <. { l  |  l 
<Q  [ <. n ,  1o >. ]  ~Q  } ,  { u  |  [ <. n ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  ->  ( iota_ r  e.  RR  ( a  x.  r )  =  1 )  =  (
iota_ r  e.  RR  ( <. [ <. ( <. { l  |  l 
<Q  [ <. n ,  1o >. ]  ~Q  } ,  { u  |  [ <. n ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  x.  r )  =  1 ) )
3736oveq2d 5559 . . . . . . . . . . . . . . . 16  |-  ( a  =  <. [ <. ( <. { l  |  l 
<Q  [ <. n ,  1o >. ]  ~Q  } ,  { u  |  [ <. n ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  ->  ( ( F `  b )  +  ( iota_ r  e.  RR  ( a  x.  r )  =  1 ) )  =  ( ( F `  b
)  +  ( iota_ r  e.  RR  ( <. [ <. ( <. { l  |  l  <Q  [ <. n ,  1o >. ]  ~Q  } ,  { u  |  [ <. n ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  x.  r )  =  1 ) ) )
3833, 37breq12d 3806 . . . . . . . . . . . . . . 15  |-  ( a  =  <. [ <. ( <. { l  |  l 
<Q  [ <. n ,  1o >. ]  ~Q  } ,  { u  |  [ <. n ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  ->  ( ( F `  a )  <RR  ( ( F `  b )  +  (
iota_ r  e.  RR  ( a  x.  r
)  =  1 ) )  <->  ( F `  <. [ <. ( <. { l  |  l  <Q  [ <. n ,  1o >. ]  ~Q  } ,  { u  |  [ <. n ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >. )  <RR  ( ( F `  b )  +  ( iota_ r  e.  RR  ( <. [ <. (
<. { l  |  l 
<Q  [ <. n ,  1o >. ]  ~Q  } ,  { u  |  [ <. n ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  x.  r )  =  1 ) ) ) )
3933, 36oveq12d 5561 . . . . . . . . . . . . . . . 16  |-  ( a  =  <. [ <. ( <. { l  |  l 
<Q  [ <. n ,  1o >. ]  ~Q  } ,  { u  |  [ <. n ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  ->  ( ( F `  a )  +  ( iota_ r  e.  RR  ( a  x.  r )  =  1 ) )  =  ( ( F `  <. [
<. ( <. { l  |  l  <Q  [ <. n ,  1o >. ]  ~Q  } ,  { u  |  [ <. n ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >. )  +  (
iota_ r  e.  RR  ( <. [ <. ( <. { l  |  l 
<Q  [ <. n ,  1o >. ]  ~Q  } ,  { u  |  [ <. n ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  x.  r )  =  1 ) ) )
4039breq2d 3805 . . . . . . . . . . . . . . 15  |-  ( a  =  <. [ <. ( <. { l  |  l 
<Q  [ <. n ,  1o >. ]  ~Q  } ,  { u  |  [ <. n ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  ->  ( ( F `  b )  <RR  ( ( F `  a )  +  (
iota_ r  e.  RR  ( a  x.  r
)  =  1 ) )  <->  ( F `  b )  <RR  ( ( F `  <. [ <. (
<. { l  |  l 
<Q  [ <. n ,  1o >. ]  ~Q  } ,  { u  |  [ <. n ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >. )  +  (
iota_ r  e.  RR  ( <. [ <. ( <. { l  |  l 
<Q  [ <. n ,  1o >. ]  ~Q  } ,  { u  |  [ <. n ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  x.  r )  =  1 ) ) ) )
4138, 40anbi12d 457 . . . . . . . . . . . . . 14  |-  ( a  =  <. [ <. ( <. { l  |  l 
<Q  [ <. n ,  1o >. ]  ~Q  } ,  { u  |  [ <. n ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  ->  ( (
( F `  a
)  <RR  ( ( F `
 b )  +  ( iota_ r  e.  RR  ( a  x.  r
)  =  1 ) )  /\  ( F `
 b )  <RR  ( ( F `  a
)  +  ( iota_ r  e.  RR  ( a  x.  r )  =  1 ) ) )  <-> 
( ( F `  <. [ <. ( <. { l  |  l  <Q  [ <. n ,  1o >. ]  ~Q  } ,  { u  |  [ <. n ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >. )  <RR  ( ( F `  b )  +  ( iota_ r  e.  RR  ( <. [ <. (
<. { l  |  l 
<Q  [ <. n ,  1o >. ]  ~Q  } ,  { u  |  [ <. n ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  x.  r )  =  1 ) )  /\  ( F `  b )  <RR  ( ( F `  <. [ <. (
<. { l  |  l 
<Q  [ <. n ,  1o >. ]  ~Q  } ,  { u  |  [ <. n ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >. )  +  (
iota_ r  e.  RR  ( <. [ <. ( <. { l  |  l 
<Q  [ <. n ,  1o >. ]  ~Q  } ,  { u  |  [ <. n ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  x.  r )  =  1 ) ) ) ) )
4232, 41imbi12d 232 . . . . . . . . . . . . 13  |-  ( a  =  <. [ <. ( <. { l  |  l 
<Q  [ <. n ,  1o >. ]  ~Q  } ,  { u  |  [ <. n ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  ->  ( (
a  <RR  b  ->  (
( F `  a
)  <RR  ( ( F `
 b )  +  ( iota_ r  e.  RR  ( a  x.  r
)  =  1 ) )  /\  ( F `
 b )  <RR  ( ( F `  a
)  +  ( iota_ r  e.  RR  ( a  x.  r )  =  1 ) ) ) )  <->  ( <. [ <. (
<. { l  |  l 
<Q  [ <. n ,  1o >. ]  ~Q  } ,  { u  |  [ <. n ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  <RR  b  ->  (
( F `  <. [
<. ( <. { l  |  l  <Q  [ <. n ,  1o >. ]  ~Q  } ,  { u  |  [ <. n ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >. )  <RR  ( ( F `  b )  +  ( iota_ r  e.  RR  ( <. [ <. (
<. { l  |  l 
<Q  [ <. n ,  1o >. ]  ~Q  } ,  { u  |  [ <. n ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  x.  r )  =  1 ) )  /\  ( F `  b )  <RR  ( ( F `  <. [ <. (
<. { l  |  l 
<Q  [ <. n ,  1o >. ]  ~Q  } ,  { u  |  [ <. n ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >. )  +  (
iota_ r  e.  RR  ( <. [ <. ( <. { l  |  l 
<Q  [ <. n ,  1o >. ]  ~Q  } ,  { u  |  [ <. n ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  x.  r )  =  1 ) ) ) ) ) )
4342ralbidv 2369 . . . . . . . . . . . 12  |-  ( a  =  <. [ <. ( <. { l  |  l 
<Q  [ <. n ,  1o >. ]  ~Q  } ,  { u  |  [ <. n ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  ->  ( A. b  e.  N  (
a  <RR  b  ->  (
( F `  a
)  <RR  ( ( F `
 b )  +  ( iota_ r  e.  RR  ( a  x.  r
)  =  1 ) )  /\  ( F `
 b )  <RR  ( ( F `  a
)  +  ( iota_ r  e.  RR  ( a  x.  r )  =  1 ) ) ) )  <->  A. b  e.  N  ( <. [ <. ( <. { l  |  l 
<Q  [ <. n ,  1o >. ]  ~Q  } ,  { u  |  [ <. n ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  <RR  b  ->  (
( F `  <. [
<. ( <. { l  |  l  <Q  [ <. n ,  1o >. ]  ~Q  } ,  { u  |  [ <. n ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >. )  <RR  ( ( F `  b )  +  ( iota_ r  e.  RR  ( <. [ <. (
<. { l  |  l 
<Q  [ <. n ,  1o >. ]  ~Q  } ,  { u  |  [ <. n ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  x.  r )  =  1 ) )  /\  ( F `  b )  <RR  ( ( F `  <. [ <. (
<. { l  |  l 
<Q  [ <. n ,  1o >. ]  ~Q  } ,  { u  |  [ <. n ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >. )  +  (
iota_ r  e.  RR  ( <. [ <. ( <. { l  |  l 
<Q  [ <. n ,  1o >. ]  ~Q  } ,  { u  |  [ <. n ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  x.  r )  =  1 ) ) ) ) ) )
4443rspcva 2700 . . . . . . . . . . 11  |-  ( (
<. [ <. ( <. { l  |  l  <Q  [ <. n ,  1o >. ]  ~Q  } ,  { u  |  [ <. n ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  e.  N  /\  A. a  e.  N  A. b  e.  N  (
a  <RR  b  ->  (
( F `  a
)  <RR  ( ( F `
 b )  +  ( iota_ r  e.  RR  ( a  x.  r
)  =  1 ) )  /\  ( F `
 b )  <RR  ( ( F `  a
)  +  ( iota_ r  e.  RR  ( a  x.  r )  =  1 ) ) ) ) )  ->  A. b  e.  N  ( <. [
<. ( <. { l  |  l  <Q  [ <. n ,  1o >. ]  ~Q  } ,  { u  |  [ <. n ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  <RR  b  ->  (
( F `  <. [
<. ( <. { l  |  l  <Q  [ <. n ,  1o >. ]  ~Q  } ,  { u  |  [ <. n ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >. )  <RR  ( ( F `  b )  +  ( iota_ r  e.  RR  ( <. [ <. (
<. { l  |  l 
<Q  [ <. n ,  1o >. ]  ~Q  } ,  { u  |  [ <. n ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  x.  r )  =  1 ) )  /\  ( F `  b )  <RR  ( ( F `  <. [ <. (
<. { l  |  l 
<Q  [ <. n ,  1o >. ]  ~Q  } ,  { u  |  [ <. n ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >. )  +  (
iota_ r  e.  RR  ( <. [ <. ( <. { l  |  l 
<Q  [ <. n ,  1o >. ]  ~Q  } ,  { u  |  [ <. n ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  x.  r )  =  1 ) ) ) ) )
459, 31, 44syl2anc 403 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  n  e.  N. )  /\  k  e.  N. )  /\  n  <N  k
)  ->  A. b  e.  N  ( <. [
<. ( <. { l  |  l  <Q  [ <. n ,  1o >. ]  ~Q  } ,  { u  |  [ <. n ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  <RR  b  ->  (
( F `  <. [
<. ( <. { l  |  l  <Q  [ <. n ,  1o >. ]  ~Q  } ,  { u  |  [ <. n ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >. )  <RR  ( ( F `  b )  +  ( iota_ r  e.  RR  ( <. [ <. (
<. { l  |  l 
<Q  [ <. n ,  1o >. ]  ~Q  } ,  { u  |  [ <. n ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  x.  r )  =  1 ) )  /\  ( F `  b )  <RR  ( ( F `  <. [ <. (
<. { l  |  l 
<Q  [ <. n ,  1o >. ]  ~Q  } ,  { u  |  [ <. n ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >. )  +  (
iota_ r  e.  RR  ( <. [ <. ( <. { l  |  l 
<Q  [ <. n ,  1o >. ]  ~Q  } ,  { u  |  [ <. n ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  x.  r )  =  1 ) ) ) ) )
46 breq2 3797 . . . . . . . . . . . 12  |-  ( b  =  <. [ <. ( <. { l  |  l 
<Q  [ <. k ,  1o >. ]  ~Q  } ,  { u  |  [ <. k ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  ->  ( <. [
<. ( <. { l  |  l  <Q  [ <. n ,  1o >. ]  ~Q  } ,  { u  |  [ <. n ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  <RR  b  <->  <. [ <. (
<. { l  |  l 
<Q  [ <. n ,  1o >. ]  ~Q  } ,  { u  |  [ <. n ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  <RR  <. [ <. ( <. { l  |  l 
<Q  [ <. k ,  1o >. ]  ~Q  } ,  { u  |  [ <. k ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >. ) )
47 fveq2 5209 . . . . . . . . . . . . . . 15  |-  ( b  =  <. [ <. ( <. { l  |  l 
<Q  [ <. k ,  1o >. ]  ~Q  } ,  { u  |  [ <. k ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  ->  ( F `  b )  =  ( F `  <. [ <. (
<. { l  |  l 
<Q  [ <. k ,  1o >. ]  ~Q  } ,  { u  |  [ <. k ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >. ) )
4847oveq1d 5558 . . . . . . . . . . . . . 14  |-  ( b  =  <. [ <. ( <. { l  |  l 
<Q  [ <. k ,  1o >. ]  ~Q  } ,  { u  |  [ <. k ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  ->  ( ( F `  b )  +  ( iota_ r  e.  RR  ( <. [ <. (
<. { l  |  l 
<Q  [ <. n ,  1o >. ]  ~Q  } ,  { u  |  [ <. n ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  x.  r )  =  1 ) )  =  ( ( F `
 <. [ <. ( <. { l  |  l 
<Q  [ <. k ,  1o >. ]  ~Q  } ,  { u  |  [ <. k ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >. )  +  (
iota_ r  e.  RR  ( <. [ <. ( <. { l  |  l 
<Q  [ <. n ,  1o >. ]  ~Q  } ,  { u  |  [ <. n ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  x.  r )  =  1 ) ) )
4948breq2d 3805 . . . . . . . . . . . . 13  |-  ( b  =  <. [ <. ( <. { l  |  l 
<Q  [ <. k ,  1o >. ]  ~Q  } ,  { u  |  [ <. k ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  ->  ( ( F `  <. [ <. (
<. { l  |  l 
<Q  [ <. n ,  1o >. ]  ~Q  } ,  { u  |  [ <. n ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >. )  <RR  ( ( F `  b )  +  ( iota_ r  e.  RR  ( <. [ <. (
<. { l  |  l 
<Q  [ <. n ,  1o >. ]  ~Q  } ,  { u  |  [ <. n ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  x.  r )  =  1 ) )  <-> 
( F `  <. [
<. ( <. { l  |  l  <Q  [ <. n ,  1o >. ]  ~Q  } ,  { u  |  [ <. n ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >. )  <RR  ( ( F `  <. [ <. (
<. { l  |  l 
<Q  [ <. k ,  1o >. ]  ~Q  } ,  { u  |  [ <. k ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >. )  +  (
iota_ r  e.  RR  ( <. [ <. ( <. { l  |  l 
<Q  [ <. n ,  1o >. ]  ~Q  } ,  { u  |  [ <. n ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  x.  r )  =  1 ) ) ) )
5047breq1d 3803 . . . . . . . . . . . . 13  |-  ( b  =  <. [ <. ( <. { l  |  l 
<Q  [ <. k ,  1o >. ]  ~Q  } ,  { u  |  [ <. k ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  ->  ( ( F `  b )  <RR  ( ( F `  <. [ <. ( <. { l  |  l  <Q  [ <. n ,  1o >. ]  ~Q  } ,  { u  |  [ <. n ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >. )  +  (
iota_ r  e.  RR  ( <. [ <. ( <. { l  |  l 
<Q  [ <. n ,  1o >. ]  ~Q  } ,  { u  |  [ <. n ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  x.  r )  =  1 ) )  <-> 
( F `  <. [
<. ( <. { l  |  l  <Q  [ <. k ,  1o >. ]  ~Q  } ,  { u  |  [ <. k ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >. )  <RR  ( ( F `  <. [ <. (
<. { l  |  l 
<Q  [ <. n ,  1o >. ]  ~Q  } ,  { u  |  [ <. n ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >. )  +  (
iota_ r  e.  RR  ( <. [ <. ( <. { l  |  l 
<Q  [ <. n ,  1o >. ]  ~Q  } ,  { u  |  [ <. n ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  x.  r )  =  1 ) ) ) )
5149, 50anbi12d 457 . . . . . . . . . . . 12  |-  ( b  =  <. [ <. ( <. { l  |  l 
<Q  [ <. k ,  1o >. ]  ~Q  } ,  { u  |  [ <. k ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  ->  ( (
( F `  <. [
<. ( <. { l  |  l  <Q  [ <. n ,  1o >. ]  ~Q  } ,  { u  |  [ <. n ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >. )  <RR  ( ( F `  b )  +  ( iota_ r  e.  RR  ( <. [ <. (
<. { l  |  l 
<Q  [ <. n ,  1o >. ]  ~Q  } ,  { u  |  [ <. n ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  x.  r )  =  1 ) )  /\  ( F `  b )  <RR  ( ( F `  <. [ <. (
<. { l  |  l 
<Q  [ <. n ,  1o >. ]  ~Q  } ,  { u  |  [ <. n ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >. )  +  (
iota_ r  e.  RR  ( <. [ <. ( <. { l  |  l 
<Q  [ <. n ,  1o >. ]  ~Q  } ,  { u  |  [ <. n ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  x.  r )  =  1 ) ) )  <->  ( ( F `
 <. [ <. ( <. { l  |  l 
<Q  [ <. n ,  1o >. ]  ~Q  } ,  { u  |  [ <. n ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >. )  <RR  ( ( F `  <. [ <. (
<. { l  |  l 
<Q  [ <. k ,  1o >. ]  ~Q  } ,  { u  |  [ <. k ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >. )  +  (
iota_ r  e.  RR  ( <. [ <. ( <. { l  |  l 
<Q  [ <. n ,  1o >. ]  ~Q  } ,  { u  |  [ <. n ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  x.  r )  =  1 ) )  /\  ( F `  <. [ <. ( <. { l  |  l  <Q  [ <. k ,  1o >. ]  ~Q  } ,  { u  |  [ <. k ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >. )  <RR  ( ( F `  <. [ <. (
<. { l  |  l 
<Q  [ <. n ,  1o >. ]  ~Q  } ,  { u  |  [ <. n ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >. )  +  (
iota_ r  e.  RR  ( <. [ <. ( <. { l  |  l 
<Q  [ <. n ,  1o >. ]  ~Q  } ,  { u  |  [ <. n ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  x.  r )  =  1 ) ) ) ) )
5246, 51imbi12d 232 . . . . . . . . . . 11  |-  ( b  =  <. [ <. ( <. { l  |  l 
<Q  [ <. k ,  1o >. ]  ~Q  } ,  { u  |  [ <. k ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  ->  ( ( <. [ <. ( <. { l  |  l  <Q  [ <. n ,  1o >. ]  ~Q  } ,  { u  |  [ <. n ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  <RR  b  ->  (
( F `  <. [
<. ( <. { l  |  l  <Q  [ <. n ,  1o >. ]  ~Q  } ,  { u  |  [ <. n ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >. )  <RR  ( ( F `  b )  +  ( iota_ r  e.  RR  ( <. [ <. (
<. { l  |  l 
<Q  [ <. n ,  1o >. ]  ~Q  } ,  { u  |  [ <. n ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  x.  r )  =  1 ) )  /\  ( F `  b )  <RR  ( ( F `  <. [ <. (
<. { l  |  l 
<Q  [ <. n ,  1o >. ]  ~Q  } ,  { u  |  [ <. n ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >. )  +  (
iota_ r  e.  RR  ( <. [ <. ( <. { l  |  l 
<Q  [ <. n ,  1o >. ]  ~Q  } ,  { u  |  [ <. n ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  x.  r )  =  1 ) ) ) )  <->  ( <. [
<. ( <. { l  |  l  <Q  [ <. n ,  1o >. ]  ~Q  } ,  { u  |  [ <. n ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  <RR  <. [ <. ( <. { l  |  l 
<Q  [ <. k ,  1o >. ]  ~Q  } ,  { u  |  [ <. k ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  ->  ( ( F `  <. [ <. (
<. { l  |  l 
<Q  [ <. n ,  1o >. ]  ~Q  } ,  { u  |  [ <. n ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >. )  <RR  ( ( F `  <. [ <. (
<. { l  |  l 
<Q  [ <. k ,  1o >. ]  ~Q  } ,  { u  |  [ <. k ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >. )  +  (
iota_ r  e.  RR  ( <. [ <. ( <. { l  |  l 
<Q  [ <. n ,  1o >. ]  ~Q  } ,  { u  |  [ <. n ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  x.  r )  =  1 ) )  /\  ( F `  <. [ <. ( <. { l  |  l  <Q  [ <. k ,  1o >. ]  ~Q  } ,  { u  |  [ <. k ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >. )  <RR  ( ( F `  <. [ <. (
<. { l  |  l 
<Q  [ <. n ,  1o >. ]  ~Q  } ,  { u  |  [ <. n ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >. )  +  (
iota_ r  e.  RR  ( <. [ <. ( <. { l  |  l 
<Q  [ <. n ,  1o >. ]  ~Q  } ,  { u  |  [ <. n ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  x.  r )  =  1 ) ) ) ) ) )
5352rspcva 2700 . . . . . . . . . 10  |-  ( (
<. [ <. ( <. { l  |  l  <Q  [ <. k ,  1o >. ]  ~Q  } ,  { u  |  [ <. k ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  e.  N  /\  A. b  e.  N  (
<. [ <. ( <. { l  |  l  <Q  [ <. n ,  1o >. ]  ~Q  } ,  { u  |  [ <. n ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  <RR  b  ->  (
( F `  <. [
<. ( <. { l  |  l  <Q  [ <. n ,  1o >. ]  ~Q  } ,  { u  |  [ <. n ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >. )  <RR  ( ( F `  b )  +  ( iota_ r  e.  RR  ( <. [ <. (
<. { l  |  l 
<Q  [ <. n ,  1o >. ]  ~Q  } ,  { u  |  [ <. n ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  x.  r )  =  1 ) )  /\  ( F `  b )  <RR  ( ( F `  <. [ <. (
<. { l  |  l 
<Q  [ <. n ,  1o >. ]  ~Q  } ,  { u  |  [ <. n ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >. )  +  (
iota_ r  e.  RR  ( <. [ <. ( <. { l  |  l 
<Q  [ <. n ,  1o >. ]  ~Q  } ,  { u  |  [ <. n ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  x.  r )  =  1 ) ) ) ) )  -> 
( <. [ <. ( <. { l  |  l 
<Q  [ <. n ,  1o >. ]  ~Q  } ,  { u  |  [ <. n ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  <RR  <. [ <. ( <. { l  |  l 
<Q  [ <. k ,  1o >. ]  ~Q  } ,  { u  |  [ <. k ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  ->  ( ( F `  <. [ <. (
<. { l  |  l 
<Q  [ <. n ,  1o >. ]  ~Q  } ,  { u  |  [ <. n ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >. )  <RR  ( ( F `  <. [ <. (
<. { l  |  l 
<Q  [ <. k ,  1o >. ]  ~Q  } ,  { u  |  [ <. k ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >. )  +  (
iota_ r  e.  RR  ( <. [ <. ( <. { l  |  l 
<Q  [ <. n ,  1o >. ]  ~Q  } ,  { u  |  [ <. n ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  x.  r )  =  1 ) )  /\  ( F `  <. [ <. ( <. { l  |  l  <Q  [ <. k ,  1o >. ]  ~Q  } ,  { u  |  [ <. k ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >. )  <RR  ( ( F `  <. [ <. (
<. { l  |  l 
<Q  [ <. n ,  1o >. ]  ~Q  } ,  { u  |  [ <. n ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >. )  +  (
iota_ r  e.  RR  ( <. [ <. ( <. { l  |  l 
<Q  [ <. n ,  1o >. ]  ~Q  } ,  { u  |  [ <. n ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  x.  r )  =  1 ) ) ) ) )
546, 45, 53syl2anc 403 . . . . . . . . 9  |-  ( ( ( ( ph  /\  n  e.  N. )  /\  k  e.  N. )  /\  n  <N  k
)  ->  ( <. [
<. ( <. { l  |  l  <Q  [ <. n ,  1o >. ]  ~Q  } ,  { u  |  [ <. n ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  <RR  <. [ <. ( <. { l  |  l 
<Q  [ <. k ,  1o >. ]  ~Q  } ,  { u  |  [ <. k ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  ->  ( ( F `  <. [ <. (
<. { l  |  l 
<Q  [ <. n ,  1o >. ]  ~Q  } ,  { u  |  [ <. n ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >. )  <RR  ( ( F `  <. [ <. (
<. { l  |  l 
<Q  [ <. k ,  1o >. ]  ~Q  } ,  { u  |  [ <. k ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >. )  +  (
iota_ r  e.  RR  ( <. [ <. ( <. { l  |  l 
<Q  [ <. n ,  1o >. ]  ~Q  } ,  { u  |  [ <. n ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  x.  r )  =  1 ) )  /\  ( F `  <. [ <. ( <. { l  |  l  <Q  [ <. k ,  1o >. ]  ~Q  } ,  { u  |  [ <. k ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >. )  <RR  ( ( F `  <. [ <. (
<. { l  |  l 
<Q  [ <. n ,  1o >. ]  ~Q  } ,  { u  |  [ <. n ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >. )  +  (
iota_ r  e.  RR  ( <. [ <. ( <. { l  |  l 
<Q  [ <. n ,  1o >. ]  ~Q  } ,  { u  |  [ <. n ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  x.  r )  =  1 ) ) ) ) )
552, 54mpd 13 . . . . . . . 8  |-  ( ( ( ( ph  /\  n  e.  N. )  /\  k  e.  N. )  /\  n  <N  k
)  ->  ( ( F `  <. [ <. (
<. { l  |  l 
<Q  [ <. n ,  1o >. ]  ~Q  } ,  { u  |  [ <. n ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >. )  <RR  ( ( F `  <. [ <. (
<. { l  |  l 
<Q  [ <. k ,  1o >. ]  ~Q  } ,  { u  |  [ <. k ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >. )  +  (
iota_ r  e.  RR  ( <. [ <. ( <. { l  |  l 
<Q  [ <. n ,  1o >. ]  ~Q  } ,  { u  |  [ <. n ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  x.  r )  =  1 ) )  /\  ( F `  <. [ <. ( <. { l  |  l  <Q  [ <. k ,  1o >. ]  ~Q  } ,  { u  |  [ <. k ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >. )  <RR  ( ( F `  <. [ <. (
<. { l  |  l 
<Q  [ <. n ,  1o >. ]  ~Q  } ,  { u  |  [ <. n ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >. )  +  (
iota_ r  e.  RR  ( <. [ <. ( <. { l  |  l 
<Q  [ <. n ,  1o >. ]  ~Q  } ,  { u  |  [ <. n ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  x.  r )  =  1 ) ) ) )
5655simpld 110 . . . . . . 7  |-  ( ( ( ( ph  /\  n  e.  N. )  /\  k  e.  N. )  /\  n  <N  k
)  ->  ( F `  <. [ <. ( <. { l  |  l 
<Q  [ <. n ,  1o >. ]  ~Q  } ,  { u  |  [ <. n ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >. )  <RR  ( ( F `  <. [ <. (
<. { l  |  l 
<Q  [ <. k ,  1o >. ]  ~Q  } ,  { u  |  [ <. k ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >. )  +  (
iota_ r  e.  RR  ( <. [ <. ( <. { l  |  l 
<Q  [ <. n ,  1o >. ]  ~Q  } ,  { u  |  [ <. n ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  x.  r )  =  1 ) ) )
57 axcaucvg.f . . . . . . . . 9  |-  ( ph  ->  F : N --> RR )
58 axcaucvg.g . . . . . . . . 9  |-  G  =  ( j  e.  N.  |->  ( iota_ z  e.  R.  ( F `  <. [ <. (
<. { l  |  l 
<Q  [ <. j ,  1o >. ]  ~Q  } ,  { u  |  [ <. j ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >. )  =  <. z ,  0R >. )
)
594, 57, 10, 58axcaucvglemval 7125 . . . . . . . 8  |-  ( (
ph  /\  n  e.  N. )  ->  ( F `
 <. [ <. ( <. { l  |  l 
<Q  [ <. n ,  1o >. ]  ~Q  } ,  { u  |  [ <. n ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >. )  =  <. ( G `  n ) ,  0R >. )
6059ad2antrr 472 . . . . . . 7  |-  ( ( ( ( ph  /\  n  e.  N. )  /\  k  e.  N. )  /\  n  <N  k
)  ->  ( F `  <. [ <. ( <. { l  |  l 
<Q  [ <. n ,  1o >. ]  ~Q  } ,  { u  |  [ <. n ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >. )  =  <. ( G `  n ) ,  0R >. )
614, 57, 10, 58axcaucvglemval 7125 . . . . . . . . . . 11  |-  ( (
ph  /\  k  e.  N. )  ->  ( F `
 <. [ <. ( <. { l  |  l 
<Q  [ <. k ,  1o >. ]  ~Q  } ,  { u  |  [ <. k ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >. )  =  <. ( G `  k ) ,  0R >. )
6261adantlr 461 . . . . . . . . . 10  |-  ( ( ( ph  /\  n  e.  N. )  /\  k  e.  N. )  ->  ( F `  <. [ <. (
<. { l  |  l 
<Q  [ <. k ,  1o >. ]  ~Q  } ,  { u  |  [ <. k ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >. )  =  <. ( G `  k ) ,  0R >. )
6362adantr 270 . . . . . . . . 9  |-  ( ( ( ( ph  /\  n  e.  N. )  /\  k  e.  N. )  /\  n  <N  k
)  ->  ( F `  <. [ <. ( <. { l  |  l 
<Q  [ <. k ,  1o >. ]  ~Q  } ,  { u  |  [ <. k ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >. )  =  <. ( G `  k ) ,  0R >. )
64 recriota 7118 . . . . . . . . . 10  |-  ( n  e.  N.  ->  ( iota_ r  e.  RR  ( <. [ <. ( <. { l  |  l  <Q  [ <. n ,  1o >. ]  ~Q  } ,  { u  |  [ <. n ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  x.  r )  =  1 )  = 
<. [ <. ( <. { l  |  l  <Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >. )
6564ad3antlr 477 . . . . . . . . 9  |-  ( ( ( ( ph  /\  n  e.  N. )  /\  k  e.  N. )  /\  n  <N  k
)  ->  ( iota_ r  e.  RR  ( <. [ <. ( <. { l  |  l  <Q  [ <. n ,  1o >. ]  ~Q  } ,  { u  |  [ <. n ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  x.  r )  =  1 )  = 
<. [ <. ( <. { l  |  l  <Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >. )
6663, 65oveq12d 5561 . . . . . . . 8  |-  ( ( ( ( ph  /\  n  e.  N. )  /\  k  e.  N. )  /\  n  <N  k
)  ->  ( ( F `  <. [ <. (
<. { l  |  l 
<Q  [ <. k ,  1o >. ]  ~Q  } ,  { u  |  [ <. k ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >. )  +  (
iota_ r  e.  RR  ( <. [ <. ( <. { l  |  l 
<Q  [ <. n ,  1o >. ]  ~Q  } ,  { u  |  [ <. n ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  x.  r )  =  1 ) )  =  ( <. ( G `  k ) ,  0R >.  +  <. [ <. (
<. { l  |  l 
<Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >. ) )
674, 57, 10, 58axcaucvglemf 7124 . . . . . . . . . . 11  |-  ( ph  ->  G : N. --> R. )
6867ad3antrrr 476 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  n  e.  N. )  /\  k  e.  N. )  /\  n  <N  k
)  ->  G : N.
--> R. )
69 simplr 497 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  n  e.  N. )  /\  k  e.  N. )  /\  n  <N  k
)  ->  k  e.  N. )
7068, 69ffvelrnd 5335 . . . . . . . . 9  |-  ( ( ( ( ph  /\  n  e.  N. )  /\  k  e.  N. )  /\  n  <N  k
)  ->  ( G `  k )  e.  R. )
71 recnnpr 6800 . . . . . . . . . . 11  |-  ( n  e.  N.  ->  <. { l  |  l  <Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >.  e.  P. )
72 prsrcl 7022 . . . . . . . . . . 11  |-  ( <. { 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. )
7371, 72syl 14 . . . . . . . . . 10  |-  ( n  e.  N.  ->  [ <. (
<. { l  |  l 
<Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  e.  R. )
7473ad3antlr 477 . . . . . . . . 9  |-  ( ( ( ( ph  /\  n  e.  N. )  /\  k  e.  N. )  /\  n  <N  k
)  ->  [ <. ( <. { l  |  l 
<Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  e.  R. )
75 addresr 7067 . . . . . . . . 9  |-  ( ( ( 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
) ,  0R >.  + 
<. [ <. ( <. { l  |  l  <Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >. )  =  <. ( ( G `  k
)  +R  [ <. (
<. { l  |  l 
<Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ) ,  0R >. )
7670, 74, 75syl2anc 403 . . . . . . . 8  |-  ( ( ( ( ph  /\  n  e.  N. )  /\  k  e.  N. )  /\  n  <N  k
)  ->  ( <. ( G `  k ) ,  0R >.  +  <. [
<. ( <. { l  |  l  <Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >. )  =  <. ( ( G `  k
)  +R  [ <. (
<. { l  |  l 
<Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ) ,  0R >. )
7766, 76eqtrd 2114 . . . . . . 7  |-  ( ( ( ( ph  /\  n  e.  N. )  /\  k  e.  N. )  /\  n  <N  k
)  ->  ( ( F `  <. [ <. (
<. { l  |  l 
<Q  [ <. k ,  1o >. ]  ~Q  } ,  { u  |  [ <. k ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >. )  +  (
iota_ r  e.  RR  ( <. [ <. ( <. { l  |  l 
<Q  [ <. n ,  1o >. ]  ~Q  } ,  { u  |  [ <. n ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  x.  r )  =  1 ) )  =  <. ( ( G `
 k )  +R 
[ <. ( <. { l  |  l  <Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ) ,  0R >. )
7856, 60, 773brtr3d 3822 . . . . . 6  |-  ( ( ( ( ph  /\  n  e.  N. )  /\  k  e.  N. )  /\  n  <N  k
)  ->  <. ( G `
 n ) ,  0R >.  <RR  <. (
( G `  k
)  +R  [ <. (
<. { l  |  l 
<Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ) ,  0R >. )
79 ltresr 7069 . . . . . 6  |-  ( <.
( G `  n
) ,  0R >.  <RR  <. ( ( G `  k )  +R  [ <. ( <. { l  |  l  <Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ) ,  0R >.  <->  ( G `  n )  <R  (
( G `  k
)  +R  [ <. (
<. { l  |  l 
<Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ) )
8078, 79sylib 120 . . . . 5  |-  ( ( ( ( ph  /\  n  e.  N. )  /\  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  ) )
8155simprd 112 . . . . . . 7  |-  ( ( ( ( ph  /\  n  e.  N. )  /\  k  e.  N. )  /\  n  <N  k
)  ->  ( F `  <. [ <. ( <. { l  |  l 
<Q  [ <. k ,  1o >. ]  ~Q  } ,  { u  |  [ <. k ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >. )  <RR  ( ( F `  <. [ <. (
<. { l  |  l 
<Q  [ <. n ,  1o >. ]  ~Q  } ,  { u  |  [ <. n ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >. )  +  (
iota_ r  e.  RR  ( <. [ <. ( <. { l  |  l 
<Q  [ <. n ,  1o >. ]  ~Q  } ,  { u  |  [ <. n ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  x.  r )  =  1 ) ) )
8260, 65oveq12d 5561 . . . . . . . 8  |-  ( ( ( ( ph  /\  n  e.  N. )  /\  k  e.  N. )  /\  n  <N  k
)  ->  ( ( F `  <. [ <. (
<. { l  |  l 
<Q  [ <. n ,  1o >. ]  ~Q  } ,  { u  |  [ <. n ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >. )  +  (
iota_ r  e.  RR  ( <. [ <. ( <. { l  |  l 
<Q  [ <. n ,  1o >. ]  ~Q  } ,  { u  |  [ <. n ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  x.  r )  =  1 ) )  =  ( <. ( G `  n ) ,  0R >.  +  <. [ <. (
<. { l  |  l 
<Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >. ) )
83 simpllr 501 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  n  e.  N. )  /\  k  e.  N. )  /\  n  <N  k
)  ->  n  e.  N. )
8468, 83ffvelrnd 5335 . . . . . . . . 9  |-  ( ( ( ( ph  /\  n  e.  N. )  /\  k  e.  N. )  /\  n  <N  k
)  ->  ( G `  n )  e.  R. )
85 addresr 7067 . . . . . . . . 9  |-  ( ( ( 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
) ,  0R >.  + 
<. [ <. ( <. { l  |  l  <Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >. )  =  <. ( ( G `  n
)  +R  [ <. (
<. { l  |  l 
<Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ) ,  0R >. )
8684, 74, 85syl2anc 403 . . . . . . . 8  |-  ( ( ( ( ph  /\  n  e.  N. )  /\  k  e.  N. )  /\  n  <N  k
)  ->  ( <. ( G `  n ) ,  0R >.  +  <. [
<. ( <. { l  |  l  <Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >. )  =  <. ( ( G `  n
)  +R  [ <. (
<. { l  |  l 
<Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ) ,  0R >. )
8782, 86eqtrd 2114 . . . . . . 7  |-  ( ( ( ( ph  /\  n  e.  N. )  /\  k  e.  N. )  /\  n  <N  k
)  ->  ( ( F `  <. [ <. (
<. { l  |  l 
<Q  [ <. n ,  1o >. ]  ~Q  } ,  { u  |  [ <. n ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >. )  +  (
iota_ r  e.  RR  ( <. [ <. ( <. { l  |  l 
<Q  [ <. n ,  1o >. ]  ~Q  } ,  { u  |  [ <. n ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  x.  r )  =  1 ) )  =  <. ( ( G `
 n )  +R 
[ <. ( <. { l  |  l  <Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ) ,  0R >. )
8881, 63, 873brtr3d 3822 . . . . . 6  |-  ( ( ( ( ph  /\  n  e.  N. )  /\  k  e.  N. )  /\  n  <N  k
)  ->  <. ( G `
 k ) ,  0R >.  <RR  <. (
( G `  n
)  +R  [ <. (
<. { l  |  l 
<Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ) ,  0R >. )
89 ltresr 7069 . . . . . 6  |-  ( <.
( G `  k
) ,  0R >.  <RR  <. ( ( G `  n )  +R  [ <. ( <. { l  |  l  <Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ) ,  0R >.  <->  ( G `  k )  <R  (
( G `  n
)  +R  [ <. (
<. { l  |  l 
<Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ) )
9088, 89sylib 120 . . . . 5  |-  ( ( ( ( ph  /\  n  e.  N. )  /\  k  e.  N. )  /\  n  <N  k
)  ->  ( G `  k )  <R  (
( G `  n
)  +R  [ <. (
<. { l  |  l 
<Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ) )
9180, 90jca 300 . . . 4  |-  ( ( ( ( ph  /\  n  e.  N. )  /\  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  ) ) )
9291ex 113 . . 3  |-  ( ( ( ph  /\  n  e.  N. )  /\  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  ) ) ) )
9392ralrimiva 2435 . 2  |-  ( (
ph  /\  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  ) ) ) )
9493ralrimiva 2435 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 102    = wceq 1285    e. wcel 1434   {cab 2068   A.wral 2349   <.cop 3409   |^|cint 3644   class class class wbr 3793    |-> cmpt 3847   -->wf 4928   ` cfv 4932   iota_crio 5498  (class class class)co 5543   1oc1o 6058   [cec 6170   N.cnpi 6524    <N clti 6527    ~Q ceq 6531   *Qcrq 6536    <Q cltq 6537   P.cnp 6543   1Pc1p 6544    +P. cpp 6545    ~R cer 6548   R.cnr 6549   0Rc0r 6550    +R cplr 6553    <R cltr 6555   RRcr 7042   1c1 7044    + caddc 7046    <RR cltrr 7047    x. cmul 7048
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 2064  ax-coll 3901  ax-sep 3904  ax-nul 3912  ax-pow 3956  ax-pr 3972  ax-un 4196  ax-setind 4288  ax-iinf 4337
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 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ne 2247  df-ral 2354  df-rex 2355  df-reu 2356  df-rmo 2357  df-rab 2358  df-v 2604  df-sbc 2817  df-csb 2910  df-dif 2976  df-un 2978  df-in 2980  df-ss 2987  df-nul 3259  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-uni 3610  df-int 3645  df-iun 3688  df-br 3794  df-opab 3848  df-mpt 3849  df-tr 3884  df-eprel 4052  df-id 4056  df-po 4059  df-iso 4060  df-iord 4129  df-on 4131  df-suc 4134  df-iom 4340  df-xp 4377  df-rel 4378  df-cnv 4379  df-co 4380  df-dm 4381  df-rn 4382  df-res 4383  df-ima 4384  df-iota 4897  df-fun 4934  df-fn 4935  df-f 4936  df-f1 4937  df-fo 4938  df-f1o 4939  df-fv 4940  df-riota 5499  df-ov 5546  df-oprab 5547  df-mpt2 5548  df-1st 5798  df-2nd 5799  df-recs 5954  df-irdg 6019  df-1o 6065  df-2o 6066  df-oadd 6069  df-omul 6070  df-er 6172  df-ec 6174  df-qs 6178  df-ni 6556  df-pli 6557  df-mi 6558  df-lti 6559  df-plpq 6596  df-mpq 6597  df-enq 6599  df-nqqs 6600  df-plqqs 6601  df-mqqs 6602  df-1nqqs 6603  df-rq 6604  df-ltnqqs 6605  df-enq0 6676  df-nq0 6677  df-0nq0 6678  df-plq0 6679  df-mq0 6680  df-inp 6718  df-i1p 6719  df-iplp 6720  df-imp 6721  df-iltp 6722  df-enr 6965  df-nr 6966  df-plr 6967  df-mr 6968  df-ltr 6969  df-0r 6970  df-1r 6971  df-m1r 6972  df-c 7049  df-0 7050  df-1 7051  df-r 7053  df-add 7054  df-mul 7055  df-lt 7056
This theorem is referenced by:  axcaucvglemres  7127
  Copyright terms: Public domain W3C validator