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Theorem axcaucvg 7901
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 7933. (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 4008 . . . . . . . . . . . . 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 4009 . . . . . . . . . . . . 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 3785 . . . . . . . . . . 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 5887 . . . . . . . . . 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 3783 . . . . . . . . 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 6572 . . . . . . . . 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 3783 . . . . . . 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 5520 . . . . . 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 3780 . . . . 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 5844 . . 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 4092 . 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 7900 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 3597   |^|cint 3846   class class class wbr 4005    |-> cmpt 4066   -->wf 5214   ` cfv 5218   iota_crio 5832  (class class class)co 5877   1oc1o 6412   [cec 6535   N.cnpi 7273    ~Q ceq 7280    <Q cltq 7286   1Pc1p 7293    +P. cpp 7294    ~R cer 7297   R.cnr 7298   0Rc0r 7299   RRcr 7812   0cc0 7813   1c1 7814    + caddc 7816    <RR cltrr 7817    x. cmul 7818
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 4120  ax-sep 4123  ax-nul 4131  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-iinf 4589
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 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-tr 4104  df-eprel 4291  df-id 4295  df-po 4298  df-iso 4299  df-iord 4368  df-on 4370  df-suc 4373  df-iom 4592  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-riota 5833  df-ov 5880  df-oprab 5881  df-mpo 5882  df-1st 6143  df-2nd 6144  df-recs 6308  df-irdg 6373  df-1o 6419  df-2o 6420  df-oadd 6423  df-omul 6424  df-er 6537  df-ec 6539  df-qs 6543  df-ni 7305  df-pli 7306  df-mi 7307  df-lti 7308  df-plpq 7345  df-mpq 7346  df-enq 7348  df-nqqs 7349  df-plqqs 7350  df-mqqs 7351  df-1nqqs 7352  df-rq 7353  df-ltnqqs 7354  df-enq0 7425  df-nq0 7426  df-0nq0 7427  df-plq0 7428  df-mq0 7429  df-inp 7467  df-i1p 7468  df-iplp 7469  df-imp 7470  df-iltp 7471  df-enr 7727  df-nr 7728  df-plr 7729  df-mr 7730  df-ltr 7731  df-0r 7732  df-1r 7733  df-m1r 7734  df-c 7819  df-0 7820  df-1 7821  df-r 7823  df-add 7824  df-mul 7825  df-lt 7826
This theorem is referenced by: (None)
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