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Theorem axcaucvg 6955
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 6985. (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, j, k, y    j, N, k, n    y, N, x    ph, j, k, n   
k, r, n    ph, x    x, y
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 3764 . . . . . . . . . . . . 13  |-  ( b  =  l  ->  (
b  <Q  [ <. j ,  1o >. ]  ~Q  <->  l  <Q  [
<. j ,  1o >. ]  ~Q  ) )
54cbvabv 2161 . . . . . . . . . . . 12  |-  { b  |  b  <Q  [ <. j ,  1o >. ]  ~Q  }  =  { l  |  l  <Q  [ <. j ,  1o >. ]  ~Q  }
6 breq2 3765 . . . . . . . . . . . . 13  |-  ( c  =  u  ->  ( [ <. j ,  1o >. ]  ~Q  <Q  c  <->  [
<. j ,  1o >. ]  ~Q  <Q  u )
)
76cbvabv 2161 . . . . . . . . . . . 12  |-  { c  |  [ <. j ,  1o >. ]  ~Q  <Q  c }  =  { u  |  [ <. j ,  1o >. ]  ~Q  <Q  u }
85, 7opeq12i 3551 . . . . . . . . . . 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 5509 . . . . . . . . . 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 3549 . . . . . . . . 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 6128 . . . . . . . . 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 3549 . . . . . . 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 5168 . . . . . 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 3546 . . . . 5  |-  ( a  =  z  ->  <. a ,  0R >.  =  <. z ,  0R >. )
1715, 16eqeq12d 2054 . . . 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 5466 . . 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 3841 . 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 6954 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 97    = wceq 1243    e. wcel 1393   {cab 2026   A.wral 2303   E.wrex 2304   <.cop 3375   |^|cint 3612   class class class wbr 3761    |-> cmpt 3815   -->wf 4885   ` cfv 4889   iota_crio 5454  (class class class)co 5499   1oc1o 5981   [cec 6091   N.cnpi 6351    ~Q ceq 6358    <Q cltq 6364   1Pc1p 6371    +P. cpp 6372    ~R cer 6375   R.cnr 6376   0Rc0r 6377   RRcr 6869   0cc0 6870   1c1 6871    + caddc 6873    <RR cltrr 6874    x. cmul 6875
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  df-c 6876  df-0 6877  df-1 6878  df-r 6880  df-add 6881  df-mul 6882  df-lt 6883
This theorem is referenced by: (None)
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