Theorem axcaucvg 8087

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 8119. (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 4086	. . . . . . . . . . . . 13 |- ( b = l -> ( b <Q [ <. j , 1o >. ] ~Q <-> l <Q [ <. j , 1o >. ] ~Q ) )
5	4	cbvabv 2354	. . . . . . . . . . . 12 |- { b | b <Q [ <. j , 1o >. ] ~Q } = { l | l <Q [ <. j , 1o >. ] ~Q }
6		breq2 4087	. . . . . . . . . . . . 13 |- ( c = u -> ( [ <. j , 1o >. ] ~Q <Q c <-> [ <. j , 1o >. ] ~Q <Q u ) )
7	6	cbvabv 2354	. . . . . . . . . . . 12 |- { c | [ <. j , 1o >. ] ~Q <Q c } = { u | [ <. j , 1o >. ] ~Q <Q u }
8	5, 7	opeq12i 3862	. . . . . . . . . . 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 6011	. . . . . . . . . 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 3860	. . . . . . . . 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 6715	. . . . . . . . 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 3860	. . . . . . 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 5630	. . . . . 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 3857	. . . . 5 |- ( a = z -> <. a , 0R >. = <. z , 0R >. )
17	15, 16	eqeq12d 2244	. . . 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 5967	. . 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 4171	. 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 8086	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 1395   e. wcel 2200   {cab 2215   A.wral 2508   E.wrex 2509   <.cop 3669   |^|cint 3923   class class class wbr 4083   |-> cmpt 4145   -->wf 5314   ` cfv 5318   iota_crio 5953  (class class class)co 6001   1oc1o 6555   [cec 6678   N.cnpi 7459   ~Q ceq 7466   <Q cltq 7472   1Pc1p 7479   +P. cpp 7480   ~R cer 7483   R.cnr 7484   0Rc0r 7485   RRcr 7998   0cc0 7999   1c1 8000   + caddc 8002   <RR cltrr 8003   x. cmul 8004

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-nul 4210  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-iinf 4680

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-tr 4183  df-eprel 4380  df-id 4384  df-po 4387  df-iso 4388  df-iord 4457  df-on 4459  df-suc 4462  df-iom 4683  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-riota 5954  df-ov 6004  df-oprab 6005  df-mpo 6006  df-1st 6286  df-2nd 6287  df-recs 6451  df-irdg 6516  df-1o 6562  df-2o 6563  df-oadd 6566  df-omul 6567  df-er 6680  df-ec 6682  df-qs 6686  df-ni 7491  df-pli 7492  df-mi 7493  df-lti 7494  df-plpq 7531  df-mpq 7532  df-enq 7534  df-nqqs 7535  df-plqqs 7536  df-mqqs 7537  df-1nqqs 7538  df-rq 7539  df-ltnqqs 7540  df-enq0 7611  df-nq0 7612  df-0nq0 7613  df-plq0 7614  df-mq0 7615  df-inp 7653  df-i1p 7654  df-iplp 7655  df-imp 7656  df-iltp 7657  df-enr 7913  df-nr 7914  df-plr 7915  df-mr 7916  df-ltr 7917  df-0r 7918  df-1r 7919  df-m1r 7920  df-c 8005  df-0 8006  df-1 8007  df-r 8009  df-add 8010  df-mul 8011  df-lt 8012

This theorem is referenced by: (None)