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Theorem axcaucvg 7898
Description: Real number completeness axiom. A Cauchy sequence with a modulus of convergence converges. 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).

Because we are stating this axiom before we have introduced notations for  NN or division, we use  N for the natural numbers and express a reciprocal in terms of  iota_.

This construction-dependent theorem should not be referenced directly; instead, use ax-caucvg 7930. (Contributed by Jim Kingdon, 8-Jul-2021.) (New usage is discouraged.)

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 ) ) ) ) )
Assertion
Ref Expression
axcaucvg  |-  ( ph  ->  E. y  e.  RR  A. x  e.  RR  (
0  <RR  x  ->  E. j  e.  N  A. k  e.  N  ( j  <RR  k  ->  ( ( F `  k )  <RR  ( y  +  x
)  /\  y  <RR  ( ( F `  k
)  +  x ) ) ) ) )
Distinct variable groups:    j, F, k, n    x, F, y, j, k    j, N, k, n    x, N, y    ph, j, k, n   
k, r, n    ph, x
Allowed substitution hints:    ph( y, r)    F( r)    N( r)

Proof of Theorem axcaucvg
Dummy variables  a  l  u  z  b  c are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 axcaucvg.n . 2  |-  N  = 
|^| { x  |  ( 1  e.  x  /\  A. y  e.  x  ( y  +  1 )  e.  x ) }
2 axcaucvg.f . 2  |-  ( ph  ->  F : N --> RR )
3 axcaucvg.cau . 2  |-  ( 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 ) ) ) ) )
4 breq1 4006 . . . . . . . . . . . . 13  |-  ( b  =  l  ->  (
b  <Q  [ <. j ,  1o >. ]  ~Q  <->  l  <Q  [
<. j ,  1o >. ]  ~Q  ) )
54cbvabv 2302 . . . . . . . . . . . 12  |-  { b  |  b  <Q  [ <. j ,  1o >. ]  ~Q  }  =  { l  |  l  <Q  [ <. j ,  1o >. ]  ~Q  }
6 breq2 4007 . . . . . . . . . . . . 13  |-  ( c  =  u  ->  ( [ <. j ,  1o >. ]  ~Q  <Q  c  <->  [
<. j ,  1o >. ]  ~Q  <Q  u )
)
76cbvabv 2302 . . . . . . . . . . . 12  |-  { c  |  [ <. j ,  1o >. ]  ~Q  <Q  c }  =  { u  |  [ <. j ,  1o >. ]  ~Q  <Q  u }
85, 7opeq12i 3783 . . . . . . . . . . 11  |-  <. { b  |  b  <Q  [ <. j ,  1o >. ]  ~Q  } ,  { c  |  [ <. j ,  1o >. ]  ~Q  <Q  c } >.  =  <. { l  |  l  <Q  [ <. j ,  1o >. ]  ~Q  } ,  { u  |  [ <. j ,  1o >. ]  ~Q  <Q  u } >.
98oveq1i 5884 . . . . . . . . . 10  |-  ( <. { b  |  b 
<Q  [ <. j ,  1o >. ]  ~Q  } ,  { c  |  [ <. j ,  1o >. ]  ~Q  <Q  c } >.  +P.  1P )  =  ( <. { l  |  l  <Q  [ <. j ,  1o >. ]  ~Q  } ,  { u  |  [ <. j ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P )
109opeq1i 3781 . . . . . . . . 9  |-  <. ( <. { b  |  b 
<Q  [ <. j ,  1o >. ]  ~Q  } ,  { c  |  [ <. j ,  1o >. ]  ~Q  <Q  c } >.  +P.  1P ) ,  1P >.  =  <. (
<. { l  |  l 
<Q  [ <. j ,  1o >. ]  ~Q  } ,  { u  |  [ <. j ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >.
11 eceq1 6569 . . . . . . . . 9  |-  ( <.
( <. { b  |  b  <Q  [ <. j ,  1o >. ]  ~Q  } ,  { c  |  [ <. j ,  1o >. ]  ~Q  <Q  c } >.  +P.  1P ) ,  1P >.  =  <. (
<. { l  |  l 
<Q  [ <. j ,  1o >. ]  ~Q  } ,  { u  |  [ <. j ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >.  ->  [ <. (
<. { b  |  b 
<Q  [ <. j ,  1o >. ]  ~Q  } ,  { c  |  [ <. j ,  1o >. ]  ~Q  <Q  c } >.  +P.  1P ) ,  1P >. ]  ~R  =  [ <. ( <. { l  |  l  <Q  [ <. j ,  1o >. ]  ~Q  } ,  { u  |  [ <. j ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  )
1210, 11ax-mp 5 . . . . . . . 8  |-  [ <. (
<. { b  |  b 
<Q  [ <. j ,  1o >. ]  ~Q  } ,  { c  |  [ <. j ,  1o >. ]  ~Q  <Q  c } >.  +P.  1P ) ,  1P >. ]  ~R  =  [ <. ( <. { l  |  l  <Q  [ <. j ,  1o >. ]  ~Q  } ,  { u  |  [ <. j ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R
1312opeq1i 3781 . . . . . . 7  |-  <. [ <. (
<. { b  |  b 
<Q  [ <. j ,  1o >. ]  ~Q  } ,  { c  |  [ <. j ,  1o >. ]  ~Q  <Q  c } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  =  <. [ <. (
<. { l  |  l 
<Q  [ <. j ,  1o >. ]  ~Q  } ,  { u  |  [ <. j ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.
1413fveq2i 5518 . . . . . 6  |-  ( F `
 <. [ <. ( <. { b  |  b 
<Q  [ <. j ,  1o >. ]  ~Q  } ,  { c  |  [ <. j ,  1o >. ]  ~Q  <Q  c } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >. )  =  ( F `  <. [ <. (
<. { l  |  l 
<Q  [ <. j ,  1o >. ]  ~Q  } ,  { u  |  [ <. j ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >. )
1514a1i 9 . . . . 5  |-  ( a  =  z  ->  ( F `  <. [ <. (
<. { b  |  b 
<Q  [ <. j ,  1o >. ]  ~Q  } ,  { c  |  [ <. j ,  1o >. ]  ~Q  <Q  c } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >. )  =  ( F `  <. [ <. (
<. { l  |  l 
<Q  [ <. j ,  1o >. ]  ~Q  } ,  { u  |  [ <. j ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >. ) )
16 opeq1 3778 . . . . 5  |-  ( a  =  z  ->  <. a ,  0R >.  =  <. z ,  0R >. )
1715, 16eqeq12d 2192 . . . 4  |-  ( a  =  z  ->  (
( F `  <. [
<. ( <. { b  |  b  <Q  [ <. j ,  1o >. ]  ~Q  } ,  { c  |  [ <. j ,  1o >. ]  ~Q  <Q  c } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >. )  =  <. a ,  0R >.  <->  ( F `  <. [ <. ( <. { l  |  l 
<Q  [ <. j ,  1o >. ]  ~Q  } ,  { u  |  [ <. j ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >. )  =  <. z ,  0R >. )
)
1817cbvriotav 5841 . . 3  |-  ( iota_ a  e.  R.  ( F `
 <. [ <. ( <. { b  |  b 
<Q  [ <. j ,  1o >. ]  ~Q  } ,  { c  |  [ <. j ,  1o >. ]  ~Q  <Q  c } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >. )  =  <. a ,  0R >. )  =  ( iota_ z  e. 
R.  ( F `  <. [ <. ( <. { l  |  l  <Q  [ <. j ,  1o >. ]  ~Q  } ,  { u  |  [ <. j ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >. )  =  <. z ,  0R >. )
1918mpteq2i 4090 . 2  |-  ( j  e.  N.  |->  ( iota_ a  e.  R.  ( F `
 <. [ <. ( <. { b  |  b 
<Q  [ <. j ,  1o >. ]  ~Q  } ,  { c  |  [ <. j ,  1o >. ]  ~Q  <Q  c } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >. )  =  <. a ,  0R >. )
)  =  ( 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 >. )
)
201, 2, 3, 19axcaucvglemres 7897 1  |-  ( ph  ->  E. y  e.  RR  A. x  e.  RR  (
0  <RR  x  ->  E. j  e.  N  A. k  e.  N  ( j  <RR  k  ->  ( ( F `  k )  <RR  ( y  +  x
)  /\  y  <RR  ( ( F `  k
)  +  x ) ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    = wceq 1353    e. wcel 2148   {cab 2163   A.wral 2455   E.wrex 2456   <.cop 3595   |^|cint 3844   class class class wbr 4003    |-> cmpt 4064   -->wf 5212   ` cfv 5216   iota_crio 5829  (class class class)co 5874   1oc1o 6409   [cec 6532   N.cnpi 7270    ~Q ceq 7277    <Q cltq 7283   1Pc1p 7290    +P. cpp 7291    ~R cer 7294   R.cnr 7295   0Rc0r 7296   RRcr 7809   0cc0 7810   1c1 7811    + caddc 7813    <RR cltrr 7814    x. cmul 7815
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4118  ax-sep 4121  ax-nul 4129  ax-pow 4174  ax-pr 4209  ax-un 4433  ax-setind 4536  ax-iinf 4587
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-iun 3888  df-br 4004  df-opab 4065  df-mpt 4066  df-tr 4102  df-eprel 4289  df-id 4293  df-po 4296  df-iso 4297  df-iord 4366  df-on 4368  df-suc 4371  df-iom 4590  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-res 4638  df-ima 4639  df-iota 5178  df-fun 5218  df-fn 5219  df-f 5220  df-f1 5221  df-fo 5222  df-f1o 5223  df-fv 5224  df-riota 5830  df-ov 5877  df-oprab 5878  df-mpo 5879  df-1st 6140  df-2nd 6141  df-recs 6305  df-irdg 6370  df-1o 6416  df-2o 6417  df-oadd 6420  df-omul 6421  df-er 6534  df-ec 6536  df-qs 6540  df-ni 7302  df-pli 7303  df-mi 7304  df-lti 7305  df-plpq 7342  df-mpq 7343  df-enq 7345  df-nqqs 7346  df-plqqs 7347  df-mqqs 7348  df-1nqqs 7349  df-rq 7350  df-ltnqqs 7351  df-enq0 7422  df-nq0 7423  df-0nq0 7424  df-plq0 7425  df-mq0 7426  df-inp 7464  df-i1p 7465  df-iplp 7466  df-imp 7467  df-iltp 7468  df-enr 7724  df-nr 7725  df-plr 7726  df-mr 7727  df-ltr 7728  df-0r 7729  df-1r 7730  df-m1r 7731  df-c 7816  df-0 7817  df-1 7818  df-r 7820  df-add 7821  df-mul 7822  df-lt 7823
This theorem is referenced by: (None)
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