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Theorem axcaucvg 7701
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 7733. (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 3927 . . . . . . . . . . . . 13  |-  ( b  =  l  ->  (
b  <Q  [ <. j ,  1o >. ]  ~Q  <->  l  <Q  [
<. j ,  1o >. ]  ~Q  ) )
54cbvabv 2262 . . . . . . . . . . . 12  |-  { b  |  b  <Q  [ <. j ,  1o >. ]  ~Q  }  =  { l  |  l  <Q  [ <. j ,  1o >. ]  ~Q  }
6 breq2 3928 . . . . . . . . . . . . 13  |-  ( c  =  u  ->  ( [ <. j ,  1o >. ]  ~Q  <Q  c  <->  [
<. j ,  1o >. ]  ~Q  <Q  u )
)
76cbvabv 2262 . . . . . . . . . . . 12  |-  { c  |  [ <. j ,  1o >. ]  ~Q  <Q  c }  =  { u  |  [ <. j ,  1o >. ]  ~Q  <Q  u }
85, 7opeq12i 3705 . . . . . . . . . . 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 5777 . . . . . . . . . 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 3703 . . . . . . . . 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 6457 . . . . . . . . 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 3703 . . . . . . 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 5417 . . . . . 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 3700 . . . . 5  |-  ( a  =  z  ->  <. a ,  0R >.  =  <. z ,  0R >. )
1715, 16eqeq12d 2152 . . . 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 5734 . . 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 4010 . 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 7700 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 103    = wceq 1331    e. wcel 1480   {cab 2123   A.wral 2414   E.wrex 2415   <.cop 3525   |^|cint 3766   class class class wbr 3924    |-> cmpt 3984   -->wf 5114   ` cfv 5118   iota_crio 5722  (class class class)co 5767   1oc1o 6299   [cec 6420   N.cnpi 7073    ~Q ceq 7080    <Q cltq 7086   1Pc1p 7093    +P. cpp 7094    ~R cer 7097   R.cnr 7098   0Rc0r 7099   RRcr 7612   0cc0 7613   1c1 7614    + caddc 7616    <RR cltrr 7617    x. cmul 7618
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-coll 4038  ax-sep 4041  ax-nul 4049  ax-pow 4093  ax-pr 4126  ax-un 4350  ax-setind 4447  ax-iinf 4497
This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ne 2307  df-ral 2419  df-rex 2420  df-reu 2421  df-rmo 2422  df-rab 2423  df-v 2683  df-sbc 2905  df-csb 2999  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-nul 3359  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-int 3767  df-iun 3810  df-br 3925  df-opab 3985  df-mpt 3986  df-tr 4022  df-eprel 4206  df-id 4210  df-po 4213  df-iso 4214  df-iord 4283  df-on 4285  df-suc 4288  df-iom 4500  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-res 4546  df-ima 4547  df-iota 5083  df-fun 5120  df-fn 5121  df-f 5122  df-f1 5123  df-fo 5124  df-f1o 5125  df-fv 5126  df-riota 5723  df-ov 5770  df-oprab 5771  df-mpo 5772  df-1st 6031  df-2nd 6032  df-recs 6195  df-irdg 6260  df-1o 6306  df-2o 6307  df-oadd 6310  df-omul 6311  df-er 6422  df-ec 6424  df-qs 6428  df-ni 7105  df-pli 7106  df-mi 7107  df-lti 7108  df-plpq 7145  df-mpq 7146  df-enq 7148  df-nqqs 7149  df-plqqs 7150  df-mqqs 7151  df-1nqqs 7152  df-rq 7153  df-ltnqqs 7154  df-enq0 7225  df-nq0 7226  df-0nq0 7227  df-plq0 7228  df-mq0 7229  df-inp 7267  df-i1p 7268  df-iplp 7269  df-imp 7270  df-iltp 7271  df-enr 7527  df-nr 7528  df-plr 7529  df-mr 7530  df-ltr 7531  df-0r 7532  df-1r 7533  df-m1r 7534  df-c 7619  df-0 7620  df-1 7621  df-r 7623  df-add 7624  df-mul 7625  df-lt 7626
This theorem is referenced by: (None)
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