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Theorem caucvgsr 7326
Description: A Cauchy sequence of signed reals with a modulus of convergence converges to a signed real. This is basically Corollary 11.2.13 of [HoTT], p. (varies). The HoTT book theorem has a modulus of convergence (that is, a rate of convergence) specified by (11.2.9) in HoTT whereas this theorem fixes the rate of convergence to say that all terms after the nth term must be within  1  /  n of the nth term (it should later be able to prove versions of this theorem with a different fixed rate or a modulus of convergence supplied as a hypothesis).

This is similar to caucvgprpr 7250 but is for signed reals rather than positive reals.

Here is an outline of how we prove it:

1. Choose a lower bound for the sequence (see caucvgsrlembnd 7325).

2. Offset each element of the sequence so that each element of the resulting sequence is greater than one (greater than zero would not suffice, because the limit as well as the elements of the sequence need to be positive) (see caucvgsrlemofff 7321).

3. Since a signed real (element of  R.) which is greater than zero can be mapped to a positive real (element of  P.), perform that mapping on each element of the sequence and invoke caucvgprpr 7250 to get a limit (see caucvgsrlemgt1 7319).

4. Map the resulting limit from positive reals back to signed reals (see caucvgsrlemgt1 7319).

5. Offset that limit so that we get the limit of the original sequence rather than the limit of the offsetted sequence (see caucvgsrlemoffres 7324). (Contributed by Jim Kingdon, 20-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  ) ) ) )
Assertion
Ref Expression
caucvgsr  |-  ( ph  ->  E. y  e.  R.  A. x  e.  R.  ( 0R  <R  x  ->  E. j  e.  N.  A. k  e. 
N.  ( j  <N 
k  ->  ( ( F `  k )  <R  ( y  +R  x
)  /\  y  <R  ( ( F `  k
)  +R  x ) ) ) ) )
Distinct variable groups:    j, F, k, l, u    n, F, k, l, u    x, F, y, j, k    ph, j,
k, x    ph, n
Allowed substitution hints:    ph( y, u, l)

Proof of Theorem caucvgsr
Dummy variables  f  g  h  m are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 caucvgsr.f . 2  |-  ( ph  ->  F : N. --> R. )
2 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  ) ) ) )
3 breq1 3840 . . . . . . . . . . . . 13  |-  ( n  =  1o  ->  (
n  <N  k  <->  1o  <N  k ) )
4 fveq2 5289 . . . . . . . . . . . . . . 15  |-  ( n  =  1o  ->  ( F `  n )  =  ( F `  1o ) )
5 opeq1 3617 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( n  =  1o  ->  <. n ,  1o >.  =  <. 1o ,  1o >. )
65eceq1d 6308 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( n  =  1o  ->  [ <. n ,  1o >. ]  ~Q  =  [ <. 1o ,  1o >. ]  ~Q  )
76fveq2d 5293 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( n  =  1o  ->  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  =  ( *Q `  [ <. 1o ,  1o >. ]  ~Q  ) )
87breq2d 3849 . . . . . . . . . . . . . . . . . . . . 21  |-  ( n  =  1o  ->  (
l  <Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <->  l  <Q  ( *Q `  [ <. 1o ,  1o >. ]  ~Q  ) ) )
98abbidv 2205 . . . . . . . . . . . . . . . . . . . 20  |-  ( n  =  1o  ->  { l  |  l  <Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) }  =  { l  |  l  <Q  ( *Q
`  [ <. 1o ,  1o >. ]  ~Q  ) } )
107breq1d 3847 . . . . . . . . . . . . . . . . . . . . 21  |-  ( n  =  1o  ->  (
( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u  <->  ( *Q `  [ <. 1o ,  1o >. ]  ~Q  )  <Q  u ) )
1110abbidv 2205 . . . . . . . . . . . . . . . . . . . 20  |-  ( n  =  1o  ->  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u }  =  {
u  |  ( *Q
`  [ <. 1o ,  1o >. ]  ~Q  )  <Q  u } )
129, 11opeq12d 3625 . . . . . . . . . . . . . . . . . . 19  |-  ( n  =  1o  ->  <. { l  |  l  <Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >.  =  <. { l  |  l  <Q 
( *Q `  [ <. 1o ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. 1o ,  1o >. ]  ~Q  )  <Q  u } >. )
1312oveq1d 5649 . . . . . . . . . . . . . . . . . 18  |-  ( n  =  1o  ->  ( <. { l  |  l 
<Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P )  =  ( <. { l  |  l  <Q 
( *Q `  [ <. 1o ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. 1o ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P ) )
1413opeq1d 3623 . . . . . . . . . . . . . . . . 17  |-  ( n  =  1o  ->  <. ( <. { l  |  l 
<Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P ) ,  1P >.  =  <. (
<. { l  |  l 
<Q  ( *Q `  [ <. 1o ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. 1o ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P ) ,  1P >. )
1514eceq1d 6308 . . . . . . . . . . . . . . . 16  |-  ( n  =  1o  ->  [ <. (
<. { l  |  l 
<Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  =  [ <. ( <. { l  |  l  <Q  ( *Q `  [ <. 1o ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. 1o ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  )
1615oveq2d 5650 . . . . . . . . . . . . . . 15  |-  ( n  =  1o  ->  (
( F `  k
)  +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 `  [ <. 1o ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. 1o ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ) )
174, 16breq12d 3850 . . . . . . . . . . . . . 14  |-  ( n  =  1o  ->  (
( 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 `  1o )  <R  ( ( F `
 k )  +R 
[ <. ( <. { l  |  l  <Q  ( *Q `  [ <. 1o ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. 1o ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ) ) )
184, 15oveq12d 5652 . . . . . . . . . . . . . . 15  |-  ( n  =  1o  ->  (
( F `  n
)  +R  [ <. (
<. { l  |  l 
<Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  )  =  ( ( F `  1o )  +R  [ <. ( <. { l  |  l  <Q  ( *Q `  [ <. 1o ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. 1o ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ) )
1918breq2d 3849 . . . . . . . . . . . . . 14  |-  ( n  =  1o  ->  (
( 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  ( ( F `
 1o )  +R 
[ <. ( <. { l  |  l  <Q  ( *Q `  [ <. 1o ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. 1o ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ) ) )
2017, 19anbi12d 457 . . . . . . . . . . . . 13  |-  ( n  =  1o  ->  (
( ( 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  ) )  <->  ( ( F `  1o )  <R  ( ( F `  k )  +R  [ <. ( <. { l  |  l  <Q  ( *Q `  [ <. 1o ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. 1o ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  )  /\  ( F `  k
)  <R  ( ( F `
 1o )  +R 
[ <. ( <. { l  |  l  <Q  ( *Q `  [ <. 1o ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. 1o ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ) ) ) )
213, 20imbi12d 232 . . . . . . . . . . . 12  |-  ( n  =  1o  ->  (
( 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  ) ) )  <->  ( 1o  <N  k  ->  ( ( F `  1o )  <R  ( ( F `  k )  +R  [ <. ( <. { l  |  l  <Q  ( *Q `  [ <. 1o ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. 1o ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  )  /\  ( F `  k
)  <R  ( ( F `
 1o )  +R 
[ <. ( <. { l  |  l  <Q  ( *Q `  [ <. 1o ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. 1o ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ) ) ) ) )
2221ralbidv 2380 . . . . . . . . . . 11  |-  ( n  =  1o  ->  ( 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.  ( 1o  <N  k  ->  ( ( F `
 1o )  <R 
( ( F `  k )  +R  [ <. ( <. { l  |  l  <Q  ( *Q `  [ <. 1o ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. 1o ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  )  /\  ( F `  k
)  <R  ( ( F `
 1o )  +R 
[ <. ( <. { l  |  l  <Q  ( *Q `  [ <. 1o ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. 1o ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ) ) ) ) )
23 1pi 6853 . . . . . . . . . . . 12  |-  1o  e.  N.
2423a1i 9 . . . . . . . . . . 11  |-  ( ph  ->  1o  e.  N. )
2522, 2, 24rspcdva 2727 . . . . . . . . . 10  |-  ( ph  ->  A. k  e.  N.  ( 1o  <N  k  -> 
( ( F `  1o )  <R  ( ( F `  k )  +R  [ <. ( <. { l  |  l 
<Q  ( *Q `  [ <. 1o ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. 1o ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  )  /\  ( F `  k )  <R  (
( F `  1o )  +R  [ <. ( <. { l  |  l 
<Q  ( *Q `  [ <. 1o ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. 1o ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ) ) ) )
26 simpl 107 . . . . . . . . . . . 12  |-  ( ( ( F `  1o )  <R  ( ( F `
 k )  +R 
[ <. ( <. { l  |  l  <Q  ( *Q `  [ <. 1o ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. 1o ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  )  /\  ( F `  k )  <R  (
( F `  1o )  +R  [ <. ( <. { l  |  l 
<Q  ( *Q `  [ <. 1o ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. 1o ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ) )  ->  ( F `  1o )  <R  ( ( F `  k )  +R  [ <. ( <. { l  |  l  <Q  ( *Q `  [ <. 1o ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. 1o ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  )
)
2726imim2i 12 . . . . . . . . . . 11  |-  ( ( 1o  <N  k  ->  ( ( F `  1o )  <R  ( ( F `
 k )  +R 
[ <. ( <. { l  |  l  <Q  ( *Q `  [ <. 1o ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. 1o ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  )  /\  ( F `  k )  <R  (
( F `  1o )  +R  [ <. ( <. { l  |  l 
<Q  ( *Q `  [ <. 1o ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. 1o ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ) ) )  -> 
( 1o  <N  k  ->  ( F `  1o )  <R  ( ( F `
 k )  +R 
[ <. ( <. { l  |  l  <Q  ( *Q `  [ <. 1o ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. 1o ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ) ) )
2827ralimi 2438 . . . . . . . . . 10  |-  ( A. k  e.  N.  ( 1o  <N  k  ->  (
( F `  1o )  <R  ( ( F `
 k )  +R 
[ <. ( <. { l  |  l  <Q  ( *Q `  [ <. 1o ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. 1o ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  )  /\  ( F `  k )  <R  (
( F `  1o )  +R  [ <. ( <. { l  |  l 
<Q  ( *Q `  [ <. 1o ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. 1o ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ) ) )  ->  A. k  e.  N.  ( 1o  <N  k  -> 
( F `  1o )  <R  ( ( F `
 k )  +R 
[ <. ( <. { l  |  l  <Q  ( *Q `  [ <. 1o ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. 1o ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ) ) )
2925, 28syl 14 . . . . . . . . 9  |-  ( ph  ->  A. k  e.  N.  ( 1o  <N  k  -> 
( F `  1o )  <R  ( ( F `
 k )  +R 
[ <. ( <. { l  |  l  <Q  ( *Q `  [ <. 1o ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. 1o ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ) ) )
30 breq2 3841 . . . . . . . . . . 11  |-  ( k  =  m  ->  ( 1o  <N  k  <->  1o  <N  m ) )
31 fveq2 5289 . . . . . . . . . . . . 13  |-  ( k  =  m  ->  ( F `  k )  =  ( F `  m ) )
3231oveq1d 5649 . . . . . . . . . . . 12  |-  ( k  =  m  ->  (
( F `  k
)  +R  [ <. (
<. { l  |  l 
<Q  ( *Q `  [ <. 1o ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. 1o ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  )  =  ( ( F `  m )  +R  [ <. ( <. { l  |  l  <Q  ( *Q `  [ <. 1o ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. 1o ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ) )
3332breq2d 3849 . . . . . . . . . . 11  |-  ( k  =  m  ->  (
( F `  1o )  <R  ( ( F `
 k )  +R 
[ <. ( <. { l  |  l  <Q  ( *Q `  [ <. 1o ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. 1o ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ) 
<->  ( F `  1o )  <R  ( ( F `
 m )  +R 
[ <. ( <. { l  |  l  <Q  ( *Q `  [ <. 1o ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. 1o ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ) ) )
3430, 33imbi12d 232 . . . . . . . . . 10  |-  ( k  =  m  ->  (
( 1o  <N  k  ->  ( F `  1o )  <R  ( ( F `
 k )  +R 
[ <. ( <. { l  |  l  <Q  ( *Q `  [ <. 1o ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. 1o ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ) )  <->  ( 1o  <N  m  ->  ( F `  1o )  <R  (
( F `  m
)  +R  [ <. (
<. { l  |  l 
<Q  ( *Q `  [ <. 1o ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. 1o ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ) ) ) )
3534rspcv 2718 . . . . . . . . 9  |-  ( m  e.  N.  ->  ( A. k  e.  N.  ( 1o  <N  k  -> 
( F `  1o )  <R  ( ( F `
 k )  +R 
[ <. ( <. { l  |  l  <Q  ( *Q `  [ <. 1o ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. 1o ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ) )  ->  ( 1o  <N  m  ->  ( F `  1o )  <R  ( ( F `  m )  +R  [ <. ( <. { l  |  l  <Q  ( *Q `  [ <. 1o ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. 1o ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  )
) ) )
3629, 35mpan9 275 . . . . . . . 8  |-  ( (
ph  /\  m  e.  N. )  ->  ( 1o 
<N  m  ->  ( F `
 1o )  <R 
( ( F `  m )  +R  [ <. ( <. { l  |  l  <Q  ( *Q `  [ <. 1o ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. 1o ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  )
) )
37 df-1nqqs 6889 . . . . . . . . . . . . . . . . . . . 20  |-  1Q  =  [ <. 1o ,  1o >. ]  ~Q
3837fveq2i 5292 . . . . . . . . . . . . . . . . . . 19  |-  ( *Q
`  1Q )  =  ( *Q `  [ <. 1o ,  1o >. ]  ~Q  )
39 rec1nq 6933 . . . . . . . . . . . . . . . . . . 19  |-  ( *Q
`  1Q )  =  1Q
4038, 39eqtr3i 2110 . . . . . . . . . . . . . . . . . 18  |-  ( *Q
`  [ <. 1o ,  1o >. ]  ~Q  )  =  1Q
4140breq2i 3845 . . . . . . . . . . . . . . . . 17  |-  ( l 
<Q  ( *Q `  [ <. 1o ,  1o >. ]  ~Q  )  <->  l  <Q  1Q )
4241abbii 2203 . . . . . . . . . . . . . . . 16  |-  { l  |  l  <Q  ( *Q `  [ <. 1o ,  1o >. ]  ~Q  ) }  =  { l  |  l  <Q  1Q }
4340breq1i 3844 . . . . . . . . . . . . . . . . 17  |-  ( ( *Q `  [ <. 1o ,  1o >. ]  ~Q  )  <Q  u  <->  1Q  <Q  u )
4443abbii 2203 . . . . . . . . . . . . . . . 16  |-  { u  |  ( *Q `  [ <. 1o ,  1o >. ]  ~Q  )  <Q  u }  =  {
u  |  1Q  <Q  u }
4542, 44opeq12i 3622 . . . . . . . . . . . . . . 15  |-  <. { l  |  l  <Q  ( *Q `  [ <. 1o ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. 1o ,  1o >. ]  ~Q  )  <Q  u } >.  =  <. { l  |  l  <Q  1Q } ,  { u  |  1Q  <Q  u } >.
46 df-i1p 7005 . . . . . . . . . . . . . . 15  |-  1P  =  <. { l  |  l 
<Q  1Q } ,  {
u  |  1Q  <Q  u } >.
4745, 46eqtr4i 2111 . . . . . . . . . . . . . 14  |-  <. { l  |  l  <Q  ( *Q `  [ <. 1o ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. 1o ,  1o >. ]  ~Q  )  <Q  u } >.  =  1P
4847oveq1i 5644 . . . . . . . . . . . . 13  |-  ( <. { l  |  l 
<Q  ( *Q `  [ <. 1o ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. 1o ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P )  =  ( 1P  +P.  1P )
4948opeq1i 3620 . . . . . . . . . . . 12  |-  <. ( <. { l  |  l 
<Q  ( *Q `  [ <. 1o ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. 1o ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P ) ,  1P >.  =  <. ( 1P  +P.  1P ) ,  1P >.
50 eceq1 6307 . . . . . . . . . . . 12  |-  ( <.
( <. { l  |  l  <Q  ( *Q `  [ <. 1o ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. 1o ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P ) ,  1P >.  =  <. ( 1P  +P.  1P ) ,  1P >.  ->  [ <. (
<. { l  |  l 
<Q  ( *Q `  [ <. 1o ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. 1o ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  =  [ <. ( 1P  +P.  1P ) ,  1P >. ]  ~R  )
5149, 50ax-mp 7 . . . . . . . . . . 11  |-  [ <. (
<. { l  |  l 
<Q  ( *Q `  [ <. 1o ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. 1o ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  =  [ <. ( 1P  +P.  1P ) ,  1P >. ]  ~R
52 df-1r 7257 . . . . . . . . . . 11  |-  1R  =  [ <. ( 1P  +P.  1P ) ,  1P >. ]  ~R
5351, 52eqtr4i 2111 . . . . . . . . . 10  |-  [ <. (
<. { l  |  l 
<Q  ( *Q `  [ <. 1o ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. 1o ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  =  1R
5453oveq2i 5645 . . . . . . . . 9  |-  ( ( F `  m )  +R  [ <. ( <. { l  |  l 
<Q  ( *Q `  [ <. 1o ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. 1o ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  )  =  ( ( F `  m )  +R  1R )
5554breq2i 3845 . . . . . . . 8  |-  ( ( F `  1o ) 
<R  ( ( F `  m )  +R  [ <. ( <. { l  |  l  <Q  ( *Q `  [ <. 1o ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. 1o ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  )  <->  ( F `  1o ) 
<R  ( ( F `  m )  +R  1R ) )
5636, 55syl6ib 159 . . . . . . 7  |-  ( (
ph  /\  m  e.  N. )  ->  ( 1o 
<N  m  ->  ( F `
 1o )  <R 
( ( F `  m )  +R  1R ) ) )
5756imp 122 . . . . . 6  |-  ( ( ( ph  /\  m  e.  N. )  /\  1o  <N  m )  ->  ( F `  1o )  <R  ( ( F `  m )  +R  1R ) )
581adantr 270 . . . . . . . . . 10  |-  ( (
ph  /\  m  e.  N. )  ->  F : N.
--> R. )
5923a1i 9 . . . . . . . . . 10  |-  ( (
ph  /\  m  e.  N. )  ->  1o  e.  N. )
6058, 59ffvelrnd 5419 . . . . . . . . 9  |-  ( (
ph  /\  m  e.  N. )  ->  ( F `
 1o )  e. 
R. )
61 ltadd1sr 7301 . . . . . . . . 9  |-  ( ( F `  1o )  e.  R.  ->  ( F `  1o )  <R  ( ( F `  1o )  +R  1R )
)
6260, 61syl 14 . . . . . . . 8  |-  ( (
ph  /\  m  e.  N. )  ->  ( F `
 1o )  <R 
( ( F `  1o )  +R  1R )
)
6362adantr 270 . . . . . . 7  |-  ( ( ( ph  /\  m  e.  N. )  /\  1o  =  m )  ->  ( F `  1o )  <R  ( ( F `  1o )  +R  1R )
)
64 fveq2 5289 . . . . . . . . 9  |-  ( 1o  =  m  ->  ( F `  1o )  =  ( F `  m ) )
6564oveq1d 5649 . . . . . . . 8  |-  ( 1o  =  m  ->  (
( F `  1o )  +R  1R )  =  ( ( F `  m )  +R  1R ) )
6665adantl 271 . . . . . . 7  |-  ( ( ( ph  /\  m  e.  N. )  /\  1o  =  m )  ->  (
( F `  1o )  +R  1R )  =  ( ( F `  m )  +R  1R ) )
6763, 66breqtrd 3861 . . . . . 6  |-  ( ( ( ph  /\  m  e.  N. )  /\  1o  =  m )  ->  ( F `  1o )  <R  ( ( F `  m )  +R  1R ) )
68 nlt1pig 6879 . . . . . . . . 9  |-  ( m  e.  N.  ->  -.  m  <N  1o )
6968adantl 271 . . . . . . . 8  |-  ( (
ph  /\  m  e.  N. )  ->  -.  m  <N  1o )
7069pm2.21d 584 . . . . . . 7  |-  ( (
ph  /\  m  e.  N. )  ->  ( m 
<N  1o  ->  ( F `  1o )  <R  (
( F `  m
)  +R  1R )
) )
7170imp 122 . . . . . 6  |-  ( ( ( ph  /\  m  e.  N. )  /\  m  <N  1o )  ->  ( F `  1o )  <R  ( ( F `  m )  +R  1R ) )
72 pitri3or 6860 . . . . . . . 8  |-  ( ( 1o  e.  N.  /\  m  e.  N. )  ->  ( 1o  <N  m  \/  1o  =  m  \/  m  <N  1o )
)
7323, 72mpan 415 . . . . . . 7  |-  ( m  e.  N.  ->  ( 1o  <N  m  \/  1o  =  m  \/  m  <N  1o ) )
7473adantl 271 . . . . . 6  |-  ( (
ph  /\  m  e.  N. )  ->  ( 1o 
<N  m  \/  1o  =  m  \/  m  <N  1o ) )
7557, 67, 71, 74mpjao3dan 1243 . . . . 5  |-  ( (
ph  /\  m  e.  N. )  ->  ( F `
 1o )  <R 
( ( F `  m )  +R  1R ) )
76 ltasrg 7295 . . . . . . 7  |-  ( ( f  e.  R.  /\  g  e.  R.  /\  h  e.  R. )  ->  (
f  <R  g  <->  ( h  +R  f )  <R  (
h  +R  g ) ) )
7776adantl 271 . . . . . 6  |-  ( ( ( ph  /\  m  e.  N. )  /\  (
f  e.  R.  /\  g  e.  R.  /\  h  e.  R. ) )  -> 
( f  <R  g  <->  ( h  +R  f ) 
<R  ( h  +R  g
) ) )
781ffvelrnda 5418 . . . . . . 7  |-  ( (
ph  /\  m  e.  N. )  ->  ( F `
 m )  e. 
R. )
79 1sr 7276 . . . . . . 7  |-  1R  e.  R.
80 addclsr 7278 . . . . . . 7  |-  ( ( ( F `  m
)  e.  R.  /\  1R  e.  R. )  -> 
( ( F `  m )  +R  1R )  e.  R. )
8178, 79, 80sylancl 404 . . . . . 6  |-  ( (
ph  /\  m  e.  N. )  ->  ( ( F `  m )  +R  1R )  e. 
R. )
82 m1r 7277 . . . . . . 7  |-  -1R  e.  R.
8382a1i 9 . . . . . 6  |-  ( (
ph  /\  m  e.  N. )  ->  -1R  e.  R. )
84 addcomsrg 7280 . . . . . . 7  |-  ( ( f  e.  R.  /\  g  e.  R. )  ->  ( f  +R  g
)  =  ( g  +R  f ) )
8584adantl 271 . . . . . 6  |-  ( ( ( ph  /\  m  e.  N. )  /\  (
f  e.  R.  /\  g  e.  R. )
)  ->  ( f  +R  g )  =  ( g  +R  f ) )
8677, 60, 81, 83, 85caovord2d 5796 . . . . 5  |-  ( (
ph  /\  m  e.  N. )  ->  ( ( F `  1o ) 
<R  ( ( F `  m )  +R  1R ) 
<->  ( ( F `  1o )  +R  -1R )  <R  ( ( ( F `
 m )  +R 
1R )  +R  -1R ) ) )
8775, 86mpbid 145 . . . 4  |-  ( (
ph  /\  m  e.  N. )  ->  ( ( F `  1o )  +R  -1R )  <R 
( ( ( F `
 m )  +R 
1R )  +R  -1R ) )
8879a1i 9 . . . . . 6  |-  ( (
ph  /\  m  e.  N. )  ->  1R  e.  R. )
89 addasssrg 7281 . . . . . 6  |-  ( ( ( F `  m
)  e.  R.  /\  1R  e.  R.  /\  -1R  e.  R. )  ->  (
( ( F `  m )  +R  1R )  +R  -1R )  =  ( ( F `  m )  +R  ( 1R  +R  -1R ) ) )
9078, 88, 83, 89syl3anc 1174 . . . . 5  |-  ( (
ph  /\  m  e.  N. )  ->  ( ( ( F `  m
)  +R  1R )  +R  -1R )  =  ( ( F `  m
)  +R  ( 1R 
+R  -1R ) ) )
91 addcomsrg 7280 . . . . . . . . 9  |-  ( ( 1R  e.  R.  /\  -1R  e.  R. )  -> 
( 1R  +R  -1R )  =  ( -1R  +R 
1R ) )
9279, 82, 91mp2an 417 . . . . . . . 8  |-  ( 1R 
+R  -1R )  =  ( -1R  +R  1R )
93 m1p1sr 7285 . . . . . . . 8  |-  ( -1R 
+R  1R )  =  0R
9492, 93eqtri 2108 . . . . . . 7  |-  ( 1R 
+R  -1R )  =  0R
9594oveq2i 5645 . . . . . 6  |-  ( ( F `  m )  +R  ( 1R  +R  -1R ) )  =  ( ( F `  m
)  +R  0R )
96 0idsr 7292 . . . . . . 7  |-  ( ( F `  m )  e.  R.  ->  (
( F `  m
)  +R  0R )  =  ( F `  m ) )
9778, 96syl 14 . . . . . 6  |-  ( (
ph  /\  m  e.  N. )  ->  ( ( F `  m )  +R  0R )  =  ( F `  m
) )
9895, 97syl5eq 2132 . . . . 5  |-  ( (
ph  /\  m  e.  N. )  ->  ( ( F `  m )  +R  ( 1R  +R  -1R ) )  =  ( F `  m ) )
9990, 98eqtrd 2120 . . . 4  |-  ( (
ph  /\  m  e.  N. )  ->  ( ( ( F `  m
)  +R  1R )  +R  -1R )  =  ( F `  m ) )
10087, 99breqtrd 3861 . . 3  |-  ( (
ph  /\  m  e.  N. )  ->  ( ( F `  1o )  +R  -1R )  <R 
( F `  m
) )
101100ralrimiva 2446 . 2  |-  ( ph  ->  A. m  e.  N.  ( ( F `  1o )  +R  -1R )  <R  ( F `  m
) )
1021, 2, 101caucvgsrlembnd 7325 1  |-  ( ph  ->  E. y  e.  R.  A. x  e.  R.  ( 0R  <R  x  ->  E. j  e.  N.  A. k  e. 
N.  ( j  <N 
k  ->  ( ( F `  k )  <R  ( y  +R  x
)  /\  y  <R  ( ( F `  k
)  +R  x ) ) ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 102    <-> wb 103    \/ w3o 923    /\ w3a 924    = wceq 1289    e. wcel 1438   {cab 2074   A.wral 2359   E.wrex 2360   <.cop 3444   class class class wbr 3837   -->wf 4998   ` cfv 5002  (class class class)co 5634   1oc1o 6156   [cec 6270   N.cnpi 6810    <N clti 6813    ~Q ceq 6817   1Qc1q 6819   *Qcrq 6822    <Q cltq 6823   1Pc1p 6830    +P. cpp 6831    ~R cer 6834   R.cnr 6835   0Rc0r 6836   1Rc1r 6837   -1Rcm1r 6838    +R cplr 6839    <R cltr 6841
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 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-coll 3946  ax-sep 3949  ax-nul 3957  ax-pow 4001  ax-pr 4027  ax-un 4251  ax-setind 4343  ax-iinf 4393
This theorem depends on definitions:  df-bi 115  df-dc 781  df-3or 925  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-ral 2364  df-rex 2365  df-reu 2366  df-rmo 2367  df-rab 2368  df-v 2621  df-sbc 2839  df-csb 2932  df-dif 2999  df-un 3001  df-in 3003  df-ss 3010  df-nul 3285  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-int 3684  df-iun 3727  df-br 3838  df-opab 3892  df-mpt 3893  df-tr 3929  df-eprel 4107  df-id 4111  df-po 4114  df-iso 4115  df-iord 4184  df-on 4186  df-suc 4189  df-iom 4396  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-rn 4439  df-res 4440  df-ima 4441  df-iota 4967  df-fun 5004  df-fn 5005  df-f 5006  df-f1 5007  df-fo 5008  df-f1o 5009  df-fv 5010  df-riota 5590  df-ov 5637  df-oprab 5638  df-mpt2 5639  df-1st 5893  df-2nd 5894  df-recs 6052  df-irdg 6117  df-1o 6163  df-2o 6164  df-oadd 6167  df-omul 6168  df-er 6272  df-ec 6274  df-qs 6278  df-ni 6842  df-pli 6843  df-mi 6844  df-lti 6845  df-plpq 6882  df-mpq 6883  df-enq 6885  df-nqqs 6886  df-plqqs 6887  df-mqqs 6888  df-1nqqs 6889  df-rq 6890  df-ltnqqs 6891  df-enq0 6962  df-nq0 6963  df-0nq0 6964  df-plq0 6965  df-mq0 6966  df-inp 7004  df-i1p 7005  df-iplp 7006  df-imp 7007  df-iltp 7008  df-enr 7251  df-nr 7252  df-plr 7253  df-mr 7254  df-ltr 7255  df-0r 7256  df-1r 7257  df-m1r 7258
This theorem is referenced by:  axcaucvglemres  7413
  Copyright terms: Public domain W3C validator