Theorem axcaucvg 8267

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 8299. (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.

Step	Hyp	Ref	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 4133	. . . . . . . . . . . . 13 |- ( b = l -> ( b <Q [ <. j , 1o >. ] ~Q <-> l <Q [ <. j , 1o >. ] ~Q ) )
5	4	cbvabv 2365	. . . . . . . . . . . 12 |- { b | b <Q [ <. j , 1o >. ] ~Q } = { l | l <Q [ <. j , 1o >. ] ~Q }
6		breq2 4134	. . . . . . . . . . . . 13 |- ( c = u -> ( [ <. j , 1o >. ] ~Q <Q c <-> [ <. j , 1o >. ] ~Q <Q u ) )
7	6	cbvabv 2365	. . . . . . . . . . . 12 |- { c | [ <. j , 1o >. ] ~Q <Q c } = { u | [ <. j , 1o >. ] ~Q <Q u }
8	5, 7	opeq12i 3909	. . . . . . . . . . 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 } >.
9	8	oveq1i 6095	. . . . . . . . . 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 )
10	9	opeq1i 3907	. . . . . . . . 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 6842	. . . . . . . . 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 )
12	10, 11	ax-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
13	12	opeq1i 3907	. . . . . . 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 >.
14	13	fveq2i 5698	. . . . . 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 >. )
15	14	a1i 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 3904	. . . . 5 |- ( a = z -> <. a , 0R >. = <. z , 0R >. )
17	15, 16	eqeq12d 2253	. . . 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 >. ) )
18	17	cbvriotav 6051	. . 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 >. )
19	18	mpteq2i 4218	. 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 >. ) )
20	1, 2, 3, 19	axcaucvglemres 8266	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 ) ) ) ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   = wceq 1402   e. wcel 2209   {cab 2224   A.wral 2528   E.wrex 2529   <.cop 3712   |^|cint 3970   class class class wbr 4130   |-> cmpt 4192   -->wf 5373   ` cfv 5377   iota_crio 6037  (class class class)co 6085   1oc1o 6680   [cec 6805   N.cnpi 7639   ~Q ceq 7646   <Q cltq 7652   1Pc1p 7659   +P. cpp 7660   ~R cer 7663   R.cnr 7664   0Rc0r 7665   RRcr 8178   0cc0 8179   1c1 8180   + caddc 8182   <RR cltrr 8183   x. cmul 8184

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4246  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-iinf 4735

This proof depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-tr 4230  df-eprel 4434  df-id 4438  df-po 4441  df-iso 4442  df-iord 4511  df-on 4513  df-suc 4516  df-iom 4738  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-recs 6576  df-irdg 6641  df-1o 6687  df-2o 6688  df-oadd 6691  df-omul 6692  df-er 6807  df-ec 6809  df-qs 6813  df-ni 7671  df-pli 7672  df-mi 7673  df-lti 7674  df-plpq 7711  df-mpq 7712  df-enq 7714  df-nqqs 7715  df-plqqs 7716  df-mqqs 7717  df-1nqqs 7718  df-rq 7719  df-ltnqqs 7720  df-enq0 7791  df-nq0 7792  df-0nq0 7793  df-plq0 7794  df-mq0 7795  df-inp 7833  df-i1p 7834  df-iplp 7835  df-imp 7836  df-iltp 7837  df-enr 8093  df-nr 8094  df-plr 8095  df-mr 8096  df-ltr 8097  df-0r 8098  df-1r 8099  df-m1r 8100  df-c 8185  df-0 8186  df-1 8187  df-r 8189  df-add 8190  df-mul 8191  df-lt 8192

This theorem is used by: (None)