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Theorem caucvgsr 6867
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 6791 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 6866).

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 6862).

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 6791 to get a limit (see caucvgsrlemgt1 6860).

4. Map the resulting limit from positive reals back to signed reals (see caucvgsrlemgt1 6860).

5. Offset that limit so that we get the limit of the original sequence rather than the limit of the offsetted sequence (see caucvgsrlemoffres 6865). (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 1pi 6394 . . . . . . . . . . 11  |-  1o  e.  N.
4 breq1 3764 . . . . . . . . . . . . . 14  |-  ( n  =  1o  ->  (
n  <N  k  <->  1o  <N  k ) )
5 fveq2 5165 . . . . . . . . . . . . . . . 16  |-  ( n  =  1o  ->  ( F `  n )  =  ( F `  1o ) )
6 opeq1 3546 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( n  =  1o  ->  <. n ,  1o >.  =  <. 1o ,  1o >. )
76eceq1d 6129 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( n  =  1o  ->  [ <. n ,  1o >. ]  ~Q  =  [ <. 1o ,  1o >. ]  ~Q  )
87fveq2d 5169 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( n  =  1o  ->  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  =  ( *Q `  [ <. 1o ,  1o >. ]  ~Q  ) )
98breq2d 3773 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( n  =  1o  ->  (
l  <Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <->  l  <Q  ( *Q `  [ <. 1o ,  1o >. ]  ~Q  ) ) )
109abbidv 2155 . . . . . . . . . . . . . . . . . . . . 21  |-  ( n  =  1o  ->  { l  |  l  <Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) }  =  { l  |  l  <Q  ( *Q
`  [ <. 1o ,  1o >. ]  ~Q  ) } )
118breq1d 3771 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( n  =  1o  ->  (
( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u  <->  ( *Q `  [ <. 1o ,  1o >. ]  ~Q  )  <Q  u ) )
1211abbidv 2155 . . . . . . . . . . . . . . . . . . . . 21  |-  ( n  =  1o  ->  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u }  =  {
u  |  ( *Q
`  [ <. 1o ,  1o >. ]  ~Q  )  <Q  u } )
1310, 12opeq12d 3554 . . . . . . . . . . . . . . . . . . . 20  |-  ( 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 } >. )
1413oveq1d 5514 . . . . . . . . . . . . . . . . . . 19  |-  ( 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 ) )
1514opeq1d 3552 . . . . . . . . . . . . . . . . . 18  |-  ( 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 >. )
1615eceq1d 6129 . . . . . . . . . . . . . . . . 17  |-  ( 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  )
1716oveq2d 5515 . . . . . . . . . . . . . . . 16  |-  ( 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  ) )
185, 17breq12d 3774 . . . . . . . . . . . . . . 15  |-  ( 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  ) ) )
195, 16oveq12d 5517 . . . . . . . . . . . . . . . 16  |-  ( 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  ) )
2019breq2d 3773 . . . . . . . . . . . . . . 15  |-  ( 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  ) ) )
2118, 20anbi12d 442 . . . . . . . . . . . . . 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 `  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  ) ) ) )
224, 21imbi12d 223 . . . . . . . . . . . . 13  |-  ( 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  ) ) ) ) )
2322ralbidv 2323 . . . . . . . . . . . 12  |-  ( 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  ) ) ) ) )
2423rspcv 2649 . . . . . . . . . . 11  |-  ( 1o  e.  N.  ->  ( A. n  e.  N.  A. k  e.  N.  (
n  <N  k  ->  (
( F `  n
)  <R  ( ( F `
 k )  +R 
[ <. ( <. { l  |  l  <Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  )  /\  ( F `  k )  <R  (
( F `  n
)  +R  [ <. (
<. { l  |  l 
<Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ) ) )  ->  A. 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  ) ) ) ) )
253, 2, 24mpsyl 59 . . . . . . . . . 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 102 . . . . . . . . . . . 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 2381 . . . . . . . . . 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 3765 . . . . . . . . . . 11  |-  ( k  =  m  ->  ( 1o  <N  k  <->  1o  <N  m ) )
31 fveq2 5165 . . . . . . . . . . . . 13  |-  ( k  =  m  ->  ( F `  k )  =  ( F `  m ) )
3231oveq1d 5514 . . . . . . . . . . . 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 3773 . . . . . . . . . . 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 223 . . . . . . . . . 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 2649 . . . . . . . . 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 265 . . . . . . . 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 6430 . . . . . . . . . . . . . . . . . . . 20  |-  1Q  =  [ <. 1o ,  1o >. ]  ~Q
3837fveq2i 5168 . . . . . . . . . . . . . . . . . . 19  |-  ( *Q
`  1Q )  =  ( *Q `  [ <. 1o ,  1o >. ]  ~Q  )
39 rec1nq 6474 . . . . . . . . . . . . . . . . . . 19  |-  ( *Q
`  1Q )  =  1Q
4038, 39eqtr3i 2062 . . . . . . . . . . . . . . . . . 18  |-  ( *Q
`  [ <. 1o ,  1o >. ]  ~Q  )  =  1Q
4140breq2i 3769 . . . . . . . . . . . . . . . . 17  |-  ( l 
<Q  ( *Q `  [ <. 1o ,  1o >. ]  ~Q  )  <->  l  <Q  1Q )
4241abbii 2153 . . . . . . . . . . . . . . . 16  |-  { l  |  l  <Q  ( *Q `  [ <. 1o ,  1o >. ]  ~Q  ) }  =  { l  |  l  <Q  1Q }
4340breq1i 3768 . . . . . . . . . . . . . . . . 17  |-  ( ( *Q `  [ <. 1o ,  1o >. ]  ~Q  )  <Q  u  <->  1Q  <Q  u )
4443abbii 2153 . . . . . . . . . . . . . . . 16  |-  { u  |  ( *Q `  [ <. 1o ,  1o >. ]  ~Q  )  <Q  u }  =  {
u  |  1Q  <Q  u }
4542, 44opeq12i 3551 . . . . . . . . . . . . . . 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 6546 . . . . . . . . . . . . . . 15  |-  1P  =  <. { l  |  l 
<Q  1Q } ,  {
u  |  1Q  <Q  u } >.
4745, 46eqtr4i 2063 . . . . . . . . . . . . . 14  |-  <. { l  |  l  <Q  ( *Q `  [ <. 1o ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. 1o ,  1o >. ]  ~Q  )  <Q  u } >.  =  1P
4847oveq1i 5509 . . . . . . . . . . . . 13  |-  ( <. { l  |  l 
<Q  ( *Q `  [ <. 1o ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. 1o ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P )  =  ( 1P  +P.  1P )
4948opeq1i 3549 . . . . . . . . . . . 12  |-  <. ( <. { l  |  l 
<Q  ( *Q `  [ <. 1o ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. 1o ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P ) ,  1P >.  =  <. ( 1P  +P.  1P ) ,  1P >.
50 eceq1 6128 . . . . . . . . . . . 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 6798 . . . . . . . . . . 11  |-  1R  =  [ <. ( 1P  +P.  1P ) ,  1P >. ]  ~R
5351, 52eqtr4i 2063 . . . . . . . . . 10  |-  [ <. (
<. { l  |  l 
<Q  ( *Q `  [ <. 1o ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. 1o ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  =  1R
5453oveq2i 5510 . . . . . . . . 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 3769 . . . . . . . 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 150 . . . . . . 7  |-  ( (
ph  /\  m  e.  N. )  ->  ( 1o 
<N  m  ->  ( F `
 1o )  <R 
( ( F `  m )  +R  1R ) ) )
5756imp 115 . . . . . 6  |-  ( ( ( ph  /\  m  e.  N. )  /\  1o  <N  m )  ->  ( F `  1o )  <R  ( ( F `  m )  +R  1R ) )
581adantr 261 . . . . . . . . . 10  |-  ( (
ph  /\  m  e.  N. )  ->  F : N.
--> R. )
593a1i 9 . . . . . . . . . 10  |-  ( (
ph  /\  m  e.  N. )  ->  1o  e.  N. )
6058, 59ffvelrnd 5290 . . . . . . . . 9  |-  ( (
ph  /\  m  e.  N. )  ->  ( F `
 1o )  e. 
R. )
61 ltadd1sr 6842 . . . . . . . . 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 261 . . . . . . 7  |-  ( ( ( ph  /\  m  e.  N. )  /\  1o  =  m )  ->  ( F `  1o )  <R  ( ( F `  1o )  +R  1R )
)
64 fveq2 5165 . . . . . . . . 9  |-  ( 1o  =  m  ->  ( F `  1o )  =  ( F `  m ) )
6564oveq1d 5514 . . . . . . . 8  |-  ( 1o  =  m  ->  (
( F `  1o )  +R  1R )  =  ( ( F `  m )  +R  1R ) )
6665adantl 262 . . . . . . 7  |-  ( ( ( ph  /\  m  e.  N. )  /\  1o  =  m )  ->  (
( F `  1o )  +R  1R )  =  ( ( F `  m )  +R  1R ) )
6763, 66breqtrd 3785 . . . . . 6  |-  ( ( ( ph  /\  m  e.  N. )  /\  1o  =  m )  ->  ( F `  1o )  <R  ( ( F `  m )  +R  1R ) )
68 nlt1pig 6420 . . . . . . . . 9  |-  ( m  e.  N.  ->  -.  m  <N  1o )
6968adantl 262 . . . . . . . 8  |-  ( (
ph  /\  m  e.  N. )  ->  -.  m  <N  1o )
7069pm2.21d 549 . . . . . . 7  |-  ( (
ph  /\  m  e.  N. )  ->  ( m 
<N  1o  ->  ( F `  1o )  <R  (
( F `  m
)  +R  1R )
) )
7170imp 115 . . . . . 6  |-  ( ( ( ph  /\  m  e.  N. )  /\  m  <N  1o )  ->  ( F `  1o )  <R  ( ( F `  m )  +R  1R ) )
72 pitri3or 6401 . . . . . . . 8  |-  ( ( 1o  e.  N.  /\  m  e.  N. )  ->  ( 1o  <N  m  \/  1o  =  m  \/  m  <N  1o )
)
733, 72mpan 400 . . . . . . 7  |-  ( m  e.  N.  ->  ( 1o  <N  m  \/  1o  =  m  \/  m  <N  1o ) )
7473adantl 262 . . . . . 6  |-  ( (
ph  /\  m  e.  N. )  ->  ( 1o 
<N  m  \/  1o  =  m  \/  m  <N  1o ) )
7557, 67, 71, 74mpjao3dan 1202 . . . . 5  |-  ( (
ph  /\  m  e.  N. )  ->  ( F `
 1o )  <R 
( ( F `  m )  +R  1R ) )
76 ltasrg 6836 . . . . . . 7  |-  ( ( f  e.  R.  /\  g  e.  R.  /\  h  e.  R. )  ->  (
f  <R  g  <->  ( h  +R  f )  <R  (
h  +R  g ) ) )
7776adantl 262 . . . . . 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 5289 . . . . . . 7  |-  ( (
ph  /\  m  e.  N. )  ->  ( F `
 m )  e. 
R. )
79 1sr 6817 . . . . . . 7  |-  1R  e.  R.
80 addclsr 6819 . . . . . . 7  |-  ( ( ( F `  m
)  e.  R.  /\  1R  e.  R. )  -> 
( ( F `  m )  +R  1R )  e.  R. )
8178, 79, 80sylancl 392 . . . . . 6  |-  ( (
ph  /\  m  e.  N. )  ->  ( ( F `  m )  +R  1R )  e. 
R. )
82 m1r 6818 . . . . . . 7  |-  -1R  e.  R.
8382a1i 9 . . . . . 6  |-  ( (
ph  /\  m  e.  N. )  ->  -1R  e.  R. )
84 addcomsrg 6821 . . . . . . 7  |-  ( ( f  e.  R.  /\  g  e.  R. )  ->  ( f  +R  g
)  =  ( g  +R  f ) )
8584adantl 262 . . . . . 6  |-  ( ( ( ph  /\  m  e.  N. )  /\  (
f  e.  R.  /\  g  e.  R. )
)  ->  ( f  +R  g )  =  ( g  +R  f ) )
8677, 60, 81, 83, 85caovord2d 5657 . . . . 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 135 . . . 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 6822 . . . . . 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 1135 . . . . 5  |-  ( (
ph  /\  m  e.  N. )  ->  ( ( ( F `  m
)  +R  1R )  +R  -1R )  =  ( ( F `  m
)  +R  ( 1R 
+R  -1R ) ) )
91 addcomsrg 6821 . . . . . . . . 9  |-  ( ( 1R  e.  R.  /\  -1R  e.  R. )  -> 
( 1R  +R  -1R )  =  ( -1R  +R 
1R ) )
9279, 82, 91mp2an 402 . . . . . . . 8  |-  ( 1R 
+R  -1R )  =  ( -1R  +R  1R )
93 m1p1sr 6826 . . . . . . . 8  |-  ( -1R 
+R  1R )  =  0R
9492, 93eqtri 2060 . . . . . . 7  |-  ( 1R 
+R  -1R )  =  0R
9594oveq2i 5510 . . . . . 6  |-  ( ( F `  m )  +R  ( 1R  +R  -1R ) )  =  ( ( F `  m
)  +R  0R )
96 0idsr 6833 . . . . . . 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 2084 . . . . 5  |-  ( (
ph  /\  m  e.  N. )  ->  ( ( F `  m )  +R  ( 1R  +R  -1R ) )  =  ( F `  m ) )
9990, 98eqtrd 2072 . . . 4  |-  ( (
ph  /\  m  e.  N. )  ->  ( ( ( F `  m
)  +R  1R )  +R  -1R )  =  ( F `  m ) )
10087, 99breqtrd 3785 . . 3  |-  ( (
ph  /\  m  e.  N. )  ->  ( ( F `  1o )  +R  -1R )  <R 
( F `  m
) )
101100ralrimiva 2389 . 2  |-  ( ph  ->  A. m  e.  N.  ( ( F `  1o )  +R  -1R )  <R  ( F `  m
) )
1021, 2, 101caucvgsrlembnd 6866 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 97    <-> wb 98    \/ w3o 884    /\ w3a 885    = wceq 1243    e. wcel 1393   {cab 2026   A.wral 2303   E.wrex 2304   <.cop 3375   class class class wbr 3761   -->wf 4885   ` cfv 4889  (class class class)co 5499   1oc1o 5981   [cec 6091   N.cnpi 6351    <N clti 6354    ~Q ceq 6358   1Qc1q 6360   *Qcrq 6363    <Q cltq 6364   1Pc1p 6371    +P. cpp 6372    ~R cer 6375   R.cnr 6376   0Rc0r 6377   1Rc1r 6378   -1Rcm1r 6379    +R cplr 6380    <R cltr 6382
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-coll 3869  ax-sep 3872  ax-nul 3880  ax-pow 3924  ax-pr 3941  ax-un 4166  ax-setind 4256  ax-iinf 4298
This theorem depends on definitions:  df-bi 110  df-dc 743  df-3or 886  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2308  df-rex 2309  df-reu 2310  df-rmo 2311  df-rab 2312  df-v 2556  df-sbc 2762  df-csb 2850  df-dif 2917  df-un 2919  df-in 2921  df-ss 2928  df-nul 3222  df-pw 3358  df-sn 3378  df-pr 3379  df-op 3381  df-uni 3578  df-int 3613  df-iun 3656  df-br 3762  df-opab 3816  df-mpt 3817  df-tr 3852  df-eprel 4023  df-id 4027  df-po 4030  df-iso 4031  df-iord 4099  df-on 4101  df-suc 4104  df-iom 4301  df-xp 4338  df-rel 4339  df-cnv 4340  df-co 4341  df-dm 4342  df-rn 4343  df-res 4344  df-ima 4345  df-iota 4854  df-fun 4891  df-fn 4892  df-f 4893  df-f1 4894  df-fo 4895  df-f1o 4896  df-fv 4897  df-riota 5455  df-ov 5502  df-oprab 5503  df-mpt2 5504  df-1st 5754  df-2nd 5755  df-recs 5907  df-irdg 5944  df-1o 5988  df-2o 5989  df-oadd 5992  df-omul 5993  df-er 6093  df-ec 6095  df-qs 6099  df-ni 6383  df-pli 6384  df-mi 6385  df-lti 6386  df-plpq 6423  df-mpq 6424  df-enq 6426  df-nqqs 6427  df-plqqs 6428  df-mqqs 6429  df-1nqqs 6430  df-rq 6431  df-ltnqqs 6432  df-enq0 6503  df-nq0 6504  df-0nq0 6505  df-plq0 6506  df-mq0 6507  df-inp 6545  df-i1p 6546  df-iplp 6547  df-imp 6548  df-iltp 6549  df-enr 6792  df-nr 6793  df-plr 6794  df-mr 6795  df-ltr 6796  df-0r 6797  df-1r 6798  df-m1r 6799
This theorem is referenced by:  axcaucvglemres  6954
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