Theorem axcaucvg 7962

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 7994. (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 4033	. . . . . . . . . . . . 13 |- ( b = l -> ( b <Q [ <. j , 1o >. ] ~Q <-> l <Q [ <. j , 1o >. ] ~Q ) )
5	4	cbvabv 2318	. . . . . . . . . . . 12 |- { b | b <Q [ <. j , 1o >. ] ~Q } = { l | l <Q [ <. j , 1o >. ] ~Q }
6		breq2 4034	. . . . . . . . . . . . 13 |- ( c = u -> ( [ <. j , 1o >. ] ~Q <Q c <-> [ <. j , 1o >. ] ~Q <Q u ) )
7	6	cbvabv 2318	. . . . . . . . . . . 12 |- { c | [ <. j , 1o >. ] ~Q <Q c } = { u | [ <. j , 1o >. ] ~Q <Q u }
8	5, 7	opeq12i 3810	. . . . . . . . . . 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 5929	. . . . . . . . . 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 3808	. . . . . . . . 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 6624	. . . . . . . . 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 3808	. . . . . . 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 5558	. . . . . 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 3805	. . . . 5 |- ( a = z -> <. a , 0R >. = <. z , 0R >. )
17	15, 16	eqeq12d 2208	. . . 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 5886	. . 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 4117	. 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 7961	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 ) ) ) ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1364   e. wcel 2164   {cab 2179   A.wral 2472   E.wrex 2473   <.cop 3622   |^|cint 3871   class class class wbr 4030   |-> cmpt 4091   -->wf 5251   ` cfv 5255   iota_crio 5873  (class class class)co 5919   1oc1o 6464   [cec 6587   N.cnpi 7334   ~Q ceq 7341   <Q cltq 7347   1Pc1p 7354   +P. cpp 7355   ~R cer 7358   R.cnr 7359   0Rc0r 7360   RRcr 7873   0cc0 7874   1c1 7875   + caddc 7877   <RR cltrr 7878   x. cmul 7879

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4145  ax-sep 4148  ax-nul 4156  ax-pow 4204  ax-pr 4239  ax-un 4465  ax-setind 4570  ax-iinf 4621

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-rex 2478  df-reu 2479  df-rmo 2480  df-rab 2481  df-v 2762  df-sbc 2987  df-csb 3082  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3448  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-int 3872  df-iun 3915  df-br 4031  df-opab 4092  df-mpt 4093  df-tr 4129  df-eprel 4321  df-id 4325  df-po 4328  df-iso 4329  df-iord 4398  df-on 4400  df-suc 4403  df-iom 4624  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-ima 4673  df-iota 5216  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-riota 5874  df-ov 5922  df-oprab 5923  df-mpo 5924  df-1st 6195  df-2nd 6196  df-recs 6360  df-irdg 6425  df-1o 6471  df-2o 6472  df-oadd 6475  df-omul 6476  df-er 6589  df-ec 6591  df-qs 6595  df-ni 7366  df-pli 7367  df-mi 7368  df-lti 7369  df-plpq 7406  df-mpq 7407  df-enq 7409  df-nqqs 7410  df-plqqs 7411  df-mqqs 7412  df-1nqqs 7413  df-rq 7414  df-ltnqqs 7415  df-enq0 7486  df-nq0 7487  df-0nq0 7488  df-plq0 7489  df-mq0 7490  df-inp 7528  df-i1p 7529  df-iplp 7530  df-imp 7531  df-iltp 7532  df-enr 7788  df-nr 7789  df-plr 7790  df-mr 7791  df-ltr 7792  df-0r 7793  df-1r 7794  df-m1r 7795  df-c 7880  df-0 7881  df-1 7882  df-r 7884  df-add 7885  df-mul 7886  df-lt 7887

This theorem is referenced by: (None)