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Theorem axcaucvg 7414
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 7444. (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 3840 . . . . . . . . . . . . 13  |-  ( b  =  l  ->  (
b  <Q  [ <. j ,  1o >. ]  ~Q  <->  l  <Q  [
<. j ,  1o >. ]  ~Q  ) )
54cbvabv 2211 . . . . . . . . . . . 12  |-  { b  |  b  <Q  [ <. j ,  1o >. ]  ~Q  }  =  { l  |  l  <Q  [ <. j ,  1o >. ]  ~Q  }
6 breq2 3841 . . . . . . . . . . . . 13  |-  ( c  =  u  ->  ( [ <. j ,  1o >. ]  ~Q  <Q  c  <->  [
<. j ,  1o >. ]  ~Q  <Q  u )
)
76cbvabv 2211 . . . . . . . . . . . 12  |-  { c  |  [ <. j ,  1o >. ]  ~Q  <Q  c }  =  { u  |  [ <. j ,  1o >. ]  ~Q  <Q  u }
85, 7opeq12i 3622 . . . . . . . . . . 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 5644 . . . . . . . . . 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 3620 . . . . . . . . 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 6307 . . . . . . . . 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 7 . . . . . . . 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 3620 . . . . . . 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 5292 . . . . . 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 3617 . . . . 5  |-  ( a  =  z  ->  <. a ,  0R >.  =  <. z ,  0R >. )
1715, 16eqeq12d 2102 . . . 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 5601 . . 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 3917 . 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 7413 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 102    = wceq 1289    e. wcel 1438   {cab 2074   A.wral 2359   E.wrex 2360   <.cop 3444   |^|cint 3683   class class class wbr 3837    |-> cmpt 3891   -->wf 4998   ` cfv 5002   iota_crio 5589  (class class class)co 5634   1oc1o 6156   [cec 6270   N.cnpi 6810    ~Q ceq 6817    <Q cltq 6823   1Pc1p 6830    +P. cpp 6831    ~R cer 6834   R.cnr 6835   0Rc0r 6836   RRcr 7328   0cc0 7329   1c1 7330    + caddc 7332    <RR cltrr 7333    x. cmul 7334
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  df-c 7335  df-0 7336  df-1 7337  df-r 7339  df-add 7340  df-mul 7341  df-lt 7342
This theorem is referenced by: (None)
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