dya2icoseg2 - Mathbox for Thierry Arnoux

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Theorem dya2icoseg2 33888

Description: For any point and any open interval of ℝ containing that point, there is a closed-below open-above dyadic rational interval which contains that point and is included in the original interval. (Contributed by Thierry Arnoux, 12-Oct-2017.)

**Hypotheses**
Ref	Expression
sxbrsiga.0	⊢ 𝐽 = (topGen‘ran (,))
dya2ioc.1	⊢ 𝐼 = (𝑥 ∈ ℤ, 𝑛 ∈ ℤ ↦ ((𝑥 / (2↑𝑛))[,)((𝑥 + 1) / (2↑𝑛))))

**Assertion**
Ref	Expression
dya2icoseg2	⊢ ((𝑋 ∈ ℝ ∧ 𝐸 ∈ ran (,) ∧ 𝑋 ∈ 𝐸) → ∃𝑏 ∈ ran 𝐼(𝑋 ∈ 𝑏 ∧ 𝑏 ⊆ 𝐸))

Distinct variable groups: 𝑥,𝑛 𝑥,𝐼 𝑛,𝑏,𝑥 𝐸,𝑏,𝑥 𝐼,𝑏 𝑋,𝑏,𝑥 Allowed substitution hints: 𝐸(𝑛) 𝐼(𝑛) 𝐽(𝑥,𝑛,𝑏) 𝑋(𝑛)

Proof of Theorem dya2icoseg2 Dummy variable 𝑑 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		sxbrsiga.0	. . . . . 6 ⊢ 𝐽 = (topGen‘ran (,))
2		dya2ioc.1	. . . . . 6 ⊢ 𝐼 = (𝑥 ∈ ℤ, 𝑛 ∈ ℤ ↦ ((𝑥 / (2↑𝑛))[,)((𝑥 + 1) / (2↑𝑛))))
3		eqid 2727	. . . . . 6 ⊢ (⌊‘(1 − (2 log_b 𝑑))) = (⌊‘(1 − (2 log_b 𝑑)))
4	1, 2, 3	dya2icoseg 33887	. . . . 5 ⊢ ((𝑋 ∈ ℝ ∧ 𝑑 ∈ ℝ⁺) → ∃𝑏 ∈ ran 𝐼(𝑋 ∈ 𝑏 ∧ 𝑏 ⊆ ((𝑋 − 𝑑)(,)(𝑋 + 𝑑))))
5	4	ralrimiva 3141	. . . 4 ⊢ (𝑋 ∈ ℝ → ∀𝑑 ∈ ℝ⁺ ∃𝑏 ∈ ran 𝐼(𝑋 ∈ 𝑏 ∧ 𝑏 ⊆ ((𝑋 − 𝑑)(,)(𝑋 + 𝑑))))
6	5	3ad2ant1 1131	. . 3 ⊢ ((𝑋 ∈ ℝ ∧ 𝐸 ∈ ran (,) ∧ 𝑋 ∈ 𝐸) → ∀𝑑 ∈ ℝ⁺ ∃𝑏 ∈ ran 𝐼(𝑋 ∈ 𝑏 ∧ 𝑏 ⊆ ((𝑋 − 𝑑)(,)(𝑋 + 𝑑))))
7		simp3 1136	. . . . 5 ⊢ ((𝑋 ∈ ℝ ∧ 𝐸 ∈ ran (,) ∧ 𝑋 ∈ 𝐸) → 𝑋 ∈ 𝐸)
8		iooex 13373	. . . . . . . . . 10 ⊢ (,) ∈ V
9	8	rnex 7912	. . . . . . . . 9 ⊢ ran (,) ∈ V
10		bastg 22862	. . . . . . . . 9 ⊢ (ran (,) ∈ V → ran (,) ⊆ (topGen‘ran (,)))
11	9, 10	ax-mp 5	. . . . . . . 8 ⊢ ran (,) ⊆ (topGen‘ran (,))
12		simp2 1135	. . . . . . . 8 ⊢ ((𝑋 ∈ ℝ ∧ 𝐸 ∈ ran (,) ∧ 𝑋 ∈ 𝐸) → 𝐸 ∈ ran (,))
13	11, 12	sselid 3976	. . . . . . 7 ⊢ ((𝑋 ∈ ℝ ∧ 𝐸 ∈ ran (,) ∧ 𝑋 ∈ 𝐸) → 𝐸 ∈ (topGen‘ran (,)))
14	13, 1	eleqtrrdi 2839	. . . . . 6 ⊢ ((𝑋 ∈ ℝ ∧ 𝐸 ∈ ran (,) ∧ 𝑋 ∈ 𝐸) → 𝐸 ∈ 𝐽)
15		eqid 2727	. . . . . . . . 9 ⊢ ((abs ∘ − ) ↾ (ℝ × ℝ)) = ((abs ∘ − ) ↾ (ℝ × ℝ))
16	15	rexmet 24700	. . . . . . . 8 ⊢ ((abs ∘ − ) ↾ (ℝ × ℝ)) ∈ (∞Met‘ℝ)
17		recms 25301	. . . . . . . . . . 11 ⊢ ℝ_fld ∈ CMetSp
18		cmsms 25269	. . . . . . . . . . 11 ⊢ (ℝ_fld ∈ CMetSp → ℝ_fld ∈ MetSp)
19		msxms 24353	. . . . . . . . . . 11 ⊢ (ℝ_fld ∈ MetSp → ℝ_fld ∈ ∞MetSp)
20	17, 18, 19	mp2b 10	. . . . . . . . . 10 ⊢ ℝ_fld ∈ ∞MetSp
21		retopn 25300	. . . . . . . . . . . 12 ⊢ (topGen‘ran (,)) = (TopOpen‘ℝ_fld)
22	1, 21	eqtri 2755	. . . . . . . . . . 11 ⊢ 𝐽 = (TopOpen‘ℝ_fld)
23		rebase 21531	. . . . . . . . . . 11 ⊢ ℝ = (Base‘ℝ_fld)
24		reds 21541	. . . . . . . . . . . 12 ⊢ (abs ∘ − ) = (dist‘ℝ_fld)
25	24	reseq1i 5975	. . . . . . . . . . 11 ⊢ ((abs ∘ − ) ↾ (ℝ × ℝ)) = ((dist‘ℝ_fld) ↾ (ℝ × ℝ))
26	22, 23, 25	xmstopn 24350	. . . . . . . . . 10 ⊢ (ℝ_fld ∈ ∞MetSp → 𝐽 = (MetOpen‘((abs ∘ − ) ↾ (ℝ × ℝ))))
27	20, 26	ax-mp 5	. . . . . . . . 9 ⊢ 𝐽 = (MetOpen‘((abs ∘ − ) ↾ (ℝ × ℝ)))
28	27	elmopn2 24344	. . . . . . . 8 ⊢ (((abs ∘ − ) ↾ (ℝ × ℝ)) ∈ (∞Met‘ℝ) → (𝐸 ∈ 𝐽 ↔ (𝐸 ⊆ ℝ ∧ ∀𝑥 ∈ 𝐸 ∃𝑑 ∈ ℝ⁺ (𝑥(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝑑) ⊆ 𝐸)))
29	16, 28	ax-mp 5	. . . . . . 7 ⊢ (𝐸 ∈ 𝐽 ↔ (𝐸 ⊆ ℝ ∧ ∀𝑥 ∈ 𝐸 ∃𝑑 ∈ ℝ⁺ (𝑥(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝑑) ⊆ 𝐸))
30	29	simprbi 496	. . . . . 6 ⊢ (𝐸 ∈ 𝐽 → ∀𝑥 ∈ 𝐸 ∃𝑑 ∈ ℝ⁺ (𝑥(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝑑) ⊆ 𝐸)
31	14, 30	syl 17	. . . . 5 ⊢ ((𝑋 ∈ ℝ ∧ 𝐸 ∈ ran (,) ∧ 𝑋 ∈ 𝐸) → ∀𝑥 ∈ 𝐸 ∃𝑑 ∈ ℝ⁺ (𝑥(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝑑) ⊆ 𝐸)
32		oveq1 7421	. . . . . . . 8 ⊢ (𝑥 = 𝑋 → (𝑥(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝑑) = (𝑋(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝑑))
33	32	sseq1d 4009	. . . . . . 7 ⊢ (𝑥 = 𝑋 → ((𝑥(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝑑) ⊆ 𝐸 ↔ (𝑋(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝑑) ⊆ 𝐸))
34	33	rexbidv 3173	. . . . . 6 ⊢ (𝑥 = 𝑋 → (∃𝑑 ∈ ℝ⁺ (𝑥(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝑑) ⊆ 𝐸 ↔ ∃𝑑 ∈ ℝ⁺ (𝑋(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝑑) ⊆ 𝐸))
35	34	rspcva 3605	. . . . 5 ⊢ ((𝑋 ∈ 𝐸 ∧ ∀𝑥 ∈ 𝐸 ∃𝑑 ∈ ℝ⁺ (𝑥(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝑑) ⊆ 𝐸) → ∃𝑑 ∈ ℝ⁺ (𝑋(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝑑) ⊆ 𝐸)
36	7, 31, 35	syl2anc 583	. . . 4 ⊢ ((𝑋 ∈ ℝ ∧ 𝐸 ∈ ran (,) ∧ 𝑋 ∈ 𝐸) → ∃𝑑 ∈ ℝ⁺ (𝑋(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝑑) ⊆ 𝐸)
37		rpre 13008	. . . . . . 7 ⊢ (𝑑 ∈ ℝ⁺ → 𝑑 ∈ ℝ)
38	15	bl2ioo 24701	. . . . . . . 8 ⊢ ((𝑋 ∈ ℝ ∧ 𝑑 ∈ ℝ) → (𝑋(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝑑) = ((𝑋 − 𝑑)(,)(𝑋 + 𝑑)))
39	38	sseq1d 4009	. . . . . . 7 ⊢ ((𝑋 ∈ ℝ ∧ 𝑑 ∈ ℝ) → ((𝑋(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝑑) ⊆ 𝐸 ↔ ((𝑋 − 𝑑)(,)(𝑋 + 𝑑)) ⊆ 𝐸))
40	37, 39	sylan2 592	. . . . . 6 ⊢ ((𝑋 ∈ ℝ ∧ 𝑑 ∈ ℝ⁺) → ((𝑋(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝑑) ⊆ 𝐸 ↔ ((𝑋 − 𝑑)(,)(𝑋 + 𝑑)) ⊆ 𝐸))
41	40	rexbidva 3171	. . . . 5 ⊢ (𝑋 ∈ ℝ → (∃𝑑 ∈ ℝ⁺ (𝑋(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝑑) ⊆ 𝐸 ↔ ∃𝑑 ∈ ℝ⁺ ((𝑋 − 𝑑)(,)(𝑋 + 𝑑)) ⊆ 𝐸))
42	41	3ad2ant1 1131	. . . 4 ⊢ ((𝑋 ∈ ℝ ∧ 𝐸 ∈ ran (,) ∧ 𝑋 ∈ 𝐸) → (∃𝑑 ∈ ℝ⁺ (𝑋(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝑑) ⊆ 𝐸 ↔ ∃𝑑 ∈ ℝ⁺ ((𝑋 − 𝑑)(,)(𝑋 + 𝑑)) ⊆ 𝐸))
43	36, 42	mpbid 231	. . 3 ⊢ ((𝑋 ∈ ℝ ∧ 𝐸 ∈ ran (,) ∧ 𝑋 ∈ 𝐸) → ∃𝑑 ∈ ℝ⁺ ((𝑋 − 𝑑)(,)(𝑋 + 𝑑)) ⊆ 𝐸)
44		r19.29 3109	. . 3 ⊢ ((∀𝑑 ∈ ℝ⁺ ∃𝑏 ∈ ran 𝐼(𝑋 ∈ 𝑏 ∧ 𝑏 ⊆ ((𝑋 − 𝑑)(,)(𝑋 + 𝑑))) ∧ ∃𝑑 ∈ ℝ⁺ ((𝑋 − 𝑑)(,)(𝑋 + 𝑑)) ⊆ 𝐸) → ∃𝑑 ∈ ℝ⁺ (∃𝑏 ∈ ran 𝐼(𝑋 ∈ 𝑏 ∧ 𝑏 ⊆ ((𝑋 − 𝑑)(,)(𝑋 + 𝑑))) ∧ ((𝑋 − 𝑑)(,)(𝑋 + 𝑑)) ⊆ 𝐸))
45	6, 43, 44	syl2anc 583	. 2 ⊢ ((𝑋 ∈ ℝ ∧ 𝐸 ∈ ran (,) ∧ 𝑋 ∈ 𝐸) → ∃𝑑 ∈ ℝ⁺ (∃𝑏 ∈ ran 𝐼(𝑋 ∈ 𝑏 ∧ 𝑏 ⊆ ((𝑋 − 𝑑)(,)(𝑋 + 𝑑))) ∧ ((𝑋 − 𝑑)(,)(𝑋 + 𝑑)) ⊆ 𝐸))
46		r19.41v 3183	. . . 4 ⊢ (∃𝑏 ∈ ran 𝐼((𝑋 ∈ 𝑏 ∧ 𝑏 ⊆ ((𝑋 − 𝑑)(,)(𝑋 + 𝑑))) ∧ ((𝑋 − 𝑑)(,)(𝑋 + 𝑑)) ⊆ 𝐸) ↔ (∃𝑏 ∈ ran 𝐼(𝑋 ∈ 𝑏 ∧ 𝑏 ⊆ ((𝑋 − 𝑑)(,)(𝑋 + 𝑑))) ∧ ((𝑋 − 𝑑)(,)(𝑋 + 𝑑)) ⊆ 𝐸))
47		sstr 3986	. . . . . . 7 ⊢ ((𝑏 ⊆ ((𝑋 − 𝑑)(,)(𝑋 + 𝑑)) ∧ ((𝑋 − 𝑑)(,)(𝑋 + 𝑑)) ⊆ 𝐸) → 𝑏 ⊆ 𝐸)
48	47	anim2i 616	. . . . . 6 ⊢ ((𝑋 ∈ 𝑏 ∧ (𝑏 ⊆ ((𝑋 − 𝑑)(,)(𝑋 + 𝑑)) ∧ ((𝑋 − 𝑑)(,)(𝑋 + 𝑑)) ⊆ 𝐸)) → (𝑋 ∈ 𝑏 ∧ 𝑏 ⊆ 𝐸))
49	48	anassrs 467	. . . . 5 ⊢ (((𝑋 ∈ 𝑏 ∧ 𝑏 ⊆ ((𝑋 − 𝑑)(,)(𝑋 + 𝑑))) ∧ ((𝑋 − 𝑑)(,)(𝑋 + 𝑑)) ⊆ 𝐸) → (𝑋 ∈ 𝑏 ∧ 𝑏 ⊆ 𝐸))
50	49	reximi 3079	. . . 4 ⊢ (∃𝑏 ∈ ran 𝐼((𝑋 ∈ 𝑏 ∧ 𝑏 ⊆ ((𝑋 − 𝑑)(,)(𝑋 + 𝑑))) ∧ ((𝑋 − 𝑑)(,)(𝑋 + 𝑑)) ⊆ 𝐸) → ∃𝑏 ∈ ran 𝐼(𝑋 ∈ 𝑏 ∧ 𝑏 ⊆ 𝐸))
51	46, 50	sylbir 234	. . 3 ⊢ ((∃𝑏 ∈ ran 𝐼(𝑋 ∈ 𝑏 ∧ 𝑏 ⊆ ((𝑋 − 𝑑)(,)(𝑋 + 𝑑))) ∧ ((𝑋 − 𝑑)(,)(𝑋 + 𝑑)) ⊆ 𝐸) → ∃𝑏 ∈ ran 𝐼(𝑋 ∈ 𝑏 ∧ 𝑏 ⊆ 𝐸))
52	51	rexlimivw 3146	. 2 ⊢ (∃𝑑 ∈ ℝ⁺ (∃𝑏 ∈ ran 𝐼(𝑋 ∈ 𝑏 ∧ 𝑏 ⊆ ((𝑋 − 𝑑)(,)(𝑋 + 𝑑))) ∧ ((𝑋 − 𝑑)(,)(𝑋 + 𝑑)) ⊆ 𝐸) → ∃𝑏 ∈ ran 𝐼(𝑋 ∈ 𝑏 ∧ 𝑏 ⊆ 𝐸))
53	45, 52	syl 17	1 ⊢ ((𝑋 ∈ ℝ ∧ 𝐸 ∈ ran (,) ∧ 𝑋 ∈ 𝐸) → ∃𝑏 ∈ ran 𝐼(𝑋 ∈ 𝑏 ∧ 𝑏 ⊆ 𝐸))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 ∧ w3a 1085 = wceq 1534 ∈ wcel 2099 ∀wral 3056 ∃wrex 3065 Vcvv 3469 ⊆ wss 3944 × cxp 5670 ran crn 5673 ↾ cres 5674 ∘ ccom 5676 ‘cfv 6542 (class class class)co 7414 ∈ cmpo 7416 ℝcr 11131 1c1 11133 + caddc 11135 − cmin 11468 / cdiv 11895 2c2 12291 ℤcz 12582 ℝ⁺crp 13000 (,)cioo 13350 [,)cico 13352 ⌊cfl 13781 ↑cexp 14052 abscabs 15207 distcds 17235 TopOpenctopn 17396 topGenctg 17412 ∞Metcxmet 21257 ballcbl 21259 MetOpencmopn 21262 ℝ_fldcrefld 21529 ∞MetSpcxms 24216 MetSpcms 24217 CMetSpccms 25253 log_b clogb 26689

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-inf2 9658 ax-cnex 11188 ax-resscn 11189 ax-1cn 11190 ax-icn 11191 ax-addcl 11192 ax-addrcl 11193 ax-mulcl 11194 ax-mulrcl 11195 ax-mulcom 11196 ax-addass 11197 ax-mulass 11198 ax-distr 11199 ax-i2m1 11200 ax-1ne0 11201 ax-1rid 11202 ax-rnegex 11203 ax-rrecex 11204 ax-cnre 11205 ax-pre-lttri 11206 ax-pre-lttrn 11207 ax-pre-ltadd 11208 ax-pre-mulgt0 11209 ax-pre-sup 11210 ax-addf 11211

This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-rmo 3371 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-iin 4994 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-se 5628 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-isom 6551 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-of 7679 df-om 7865 df-1st 7987 df-2nd 7988 df-supp 8160 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-2o 8481 df-er 8718 df-map 8840 df-pm 8841 df-ixp 8910 df-en 8958 df-dom 8959 df-sdom 8960 df-fin 8961 df-fsupp 9380 df-fi 9428 df-sup 9459 df-inf 9460 df-oi 9527 df-card 9956 df-pnf 11274 df-mnf 11275 df-xr 11276 df-ltxr 11277 df-le 11278 df-sub 11470 df-neg 11471 df-div 11896 df-nn 12237 df-2 12299 df-3 12300 df-4 12301 df-5 12302 df-6 12303 df-7 12304 df-8 12305 df-9 12306 df-n0 12497 df-z 12583 df-dec 12702 df-uz 12847 df-q 12957 df-rp 13001 df-xneg 13118 df-xadd 13119 df-xmul 13120 df-ioo 13354 df-ioc 13355 df-ico 13356 df-icc 13357 df-fz 13511 df-fzo 13654 df-fl 13783 df-mod 13861 df-seq 13993 df-exp 14053 df-fac 14259 df-bc 14288 df-hash 14316 df-shft 15040 df-cj 15072 df-re 15073 df-im 15074 df-sqrt 15208 df-abs 15209 df-limsup 15441 df-clim 15458 df-rlim 15459 df-sum 15659 df-ef 16037 df-sin 16039 df-cos 16040 df-pi 16042 df-struct 17109 df-sets 17126 df-slot 17144 df-ndx 17156 df-base 17174 df-ress 17203 df-plusg 17239 df-mulr 17240 df-starv 17241 df-sca 17242 df-vsca 17243 df-ip 17244 df-tset 17245 df-ple 17246 df-ds 17248 df-unif 17249 df-hom 17250 df-cco 17251 df-rest 17397 df-topn 17398 df-0g 17416 df-gsum 17417 df-topgen 17418 df-pt 17419 df-prds 17422 df-xrs 17477 df-qtop 17482 df-imas 17483 df-xps 17485 df-mre 17559 df-mrc 17560 df-acs 17562 df-mgm 18593 df-sgrp 18672 df-mnd 18688 df-submnd 18734 df-mulg 19017 df-cntz 19261 df-cmn 19730 df-psmet 21264 df-xmet 21265 df-met 21266 df-bl 21267 df-mopn 21268 df-fbas 21269 df-fg 21270 df-cnfld 21273 df-refld 21530 df-top 22789 df-topon 22806 df-topsp 22828 df-bases 22842 df-cld 22916 df-ntr 22917 df-cls 22918 df-nei 22995 df-lp 23033 df-perf 23034 df-cn 23124 df-cnp 23125 df-haus 23212 df-cmp 23284 df-tx 23459 df-hmeo 23652 df-fil 23743 df-fm 23835 df-flim 23836 df-flf 23837 df-fcls 23838 df-xms 24219 df-ms 24220 df-tms 24221 df-cncf 24791 df-cfil 25176 df-cmet 25178 df-cms 25256 df-limc 25788 df-dv 25789 df-log 26483 df-cxp 26484 df-logb 26690

This theorem is referenced by: dya2iocnrect 33891

W3C validator