Theorem axcaucvg 7841

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 7873. (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 3985	. . . . . . . . . . . . 13 |- ( b = l -> ( b <Q [ <. j , 1o >. ] ~Q <-> l <Q [ <. j , 1o >. ] ~Q ) )
5	4	cbvabv 2291	. . . . . . . . . . . 12 |- { b | b <Q [ <. j , 1o >. ] ~Q } = { l | l <Q [ <. j , 1o >. ] ~Q }
6		breq2 3986	. . . . . . . . . . . . 13 |- ( c = u -> ( [ <. j , 1o >. ] ~Q <Q c <-> [ <. j , 1o >. ] ~Q <Q u ) )
7	6	cbvabv 2291	. . . . . . . . . . . 12 |- { c | [ <. j , 1o >. ] ~Q <Q c } = { u | [ <. j , 1o >. ] ~Q <Q u }
8	5, 7	opeq12i 3763	. . . . . . . . . . 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 5852	. . . . . . . . . 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 3761	. . . . . . . . 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 6536	. . . . . . . . 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 3761	. . . . . . 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 5489	. . . . . 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 3758	. . . . 5 |- ( a = z -> <. a , 0R >. = <. z , 0R >. )
17	15, 16	eqeq12d 2180	. . . 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 5809	. . 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 4069	. 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 7840	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 103   = wceq 1343   e. wcel 2136   {cab 2151   A.wral 2444   E.wrex 2445   <.cop 3579   |^|cint 3824   class class class wbr 3982   |-> cmpt 4043   -->wf 5184   ` cfv 5188   iota_crio 5797  (class class class)co 5842   1oc1o 6377   [cec 6499   N.cnpi 7213   ~Q ceq 7220   <Q cltq 7226   1Pc1p 7233   +P. cpp 7234   ~R cer 7237   R.cnr 7238   0Rc0r 7239   RRcr 7752   0cc0 7753   1c1 7754   + caddc 7756   <RR cltrr 7757   x. cmul 7758

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-coll 4097  ax-sep 4100  ax-nul 4108  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-iinf 4565

This theorem depends on definitions:  df-bi 116  df-dc 825  df-3or 969  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-ral 2449  df-rex 2450  df-reu 2451  df-rmo 2452  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-int 3825  df-iun 3868  df-br 3983  df-opab 4044  df-mpt 4045  df-tr 4081  df-eprel 4267  df-id 4271  df-po 4274  df-iso 4275  df-iord 4344  df-on 4346  df-suc 4349  df-iom 4568  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-riota 5798  df-ov 5845  df-oprab 5846  df-mpo 5847  df-1st 6108  df-2nd 6109  df-recs 6273  df-irdg 6338  df-1o 6384  df-2o 6385  df-oadd 6388  df-omul 6389  df-er 6501  df-ec 6503  df-qs 6507  df-ni 7245  df-pli 7246  df-mi 7247  df-lti 7248  df-plpq 7285  df-mpq 7286  df-enq 7288  df-nqqs 7289  df-plqqs 7290  df-mqqs 7291  df-1nqqs 7292  df-rq 7293  df-ltnqqs 7294  df-enq0 7365  df-nq0 7366  df-0nq0 7367  df-plq0 7368  df-mq0 7369  df-inp 7407  df-i1p 7408  df-iplp 7409  df-imp 7410  df-iltp 7411  df-enr 7667  df-nr 7668  df-plr 7669  df-mr 7670  df-ltr 7671  df-0r 7672  df-1r 7673  df-m1r 7674  df-c 7759  df-0 7760  df-1 7761  df-r 7763  df-add 7764  df-mul 7765  df-lt 7766

This theorem is referenced by: (None)