Theorem axcaucvg 6974

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 7004. (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.

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 3767	. . . . . . . . . . . . 13 |- ( b = l -> ( b <Q [ <. j , 1o >. ] ~Q <-> l <Q [ <. j , 1o >. ] ~Q ) )
5	4	cbvabv 2161	. . . . . . . . . . . 12 |- { b | b <Q [ <. j , 1o >. ] ~Q } = { l | l <Q [ <. j , 1o >. ] ~Q }
6		breq2 3768	. . . . . . . . . . . . 13 |- ( c = u -> ( [ <. j , 1o >. ] ~Q <Q c <-> [ <. j , 1o >. ] ~Q <Q u ) )
7	6	cbvabv 2161	. . . . . . . . . . . 12 |- { c | [ <. j , 1o >. ] ~Q <Q c } = { u | [ <. j , 1o >. ] ~Q <Q u }
8	5, 7	opeq12i 3554	. . . . . . . . . . 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 5522	. . . . . . . . . 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 3552	. . . . . . . . 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 6141	. . . . . . . . 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 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
13	12	opeq1i 3552	. . . . . . 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 5181	. . . . . 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 3549	. . . . 5 |- ( a = z -> <. a , 0R >. = <. z , 0R >. )
17	15, 16	eqeq12d 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 >. ) )
18	17	cbvriotav 5479	. . 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 3844	. 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 6973	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 97   = wceq 1243   e. wcel 1393   {cab 2026   A.wral 2306   E.wrex 2307   <.cop 3378   |^|cint 3615   class class class wbr 3764   |-> cmpt 3818   -->wf 4898   ` cfv 4902   iota_crio 5467  (class class class)co 5512   1oc1o 5994   [cec 6104   N.cnpi 6370   ~Q ceq 6377   <Q cltq 6383   1Pc1p 6390   +P. cpp 6391   ~R cer 6394   R.cnr 6395   0Rc0r 6396   RRcr 6888   0cc0 6889   1c1 6890   + caddc 6892   <RR cltrr 6893   x. cmul 6894

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 3872  ax-sep 3875  ax-nul 3883  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-setind 4262  ax-iinf 4311

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 2311  df-rex 2312  df-reu 2313  df-rmo 2314  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-int 3616  df-iun 3659  df-br 3765  df-opab 3819  df-mpt 3820  df-tr 3855  df-eprel 4026  df-id 4030  df-po 4033  df-iso 4034  df-iord 4103  df-on 4105  df-suc 4108  df-iom 4314  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909  df-fv 4910  df-riota 5468  df-ov 5515  df-oprab 5516  df-mpt2 5517  df-1st 5767  df-2nd 5768  df-recs 5920  df-irdg 5957  df-1o 6001  df-2o 6002  df-oadd 6005  df-omul 6006  df-er 6106  df-ec 6108  df-qs 6112  df-ni 6402  df-pli 6403  df-mi 6404  df-lti 6405  df-plpq 6442  df-mpq 6443  df-enq 6445  df-nqqs 6446  df-plqqs 6447  df-mqqs 6448  df-1nqqs 6449  df-rq 6450  df-ltnqqs 6451  df-enq0 6522  df-nq0 6523  df-0nq0 6524  df-plq0 6525  df-mq0 6526  df-inp 6564  df-i1p 6565  df-iplp 6566  df-imp 6567  df-iltp 6568  df-enr 6811  df-nr 6812  df-plr 6813  df-mr 6814  df-ltr 6815  df-0r 6816  df-1r 6817  df-m1r 6818  df-c 6895  df-0 6896  df-1 6897  df-r 6899  df-add 6900  df-mul 6901  df-lt 6902

This theorem is referenced by: (None)