dya2icoseg2 - Mathbox for Thierry Arnoux

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Theorem dya2icoseg2 34280

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 2737	. . . . . 6 ⊢ (⌊‘(1 − (2 log_b 𝑑))) = (⌊‘(1 − (2 log_b 𝑑)))
4	1, 2, 3	dya2icoseg 34279	. . . . 5 ⊢ ((𝑋 ∈ ℝ ∧ 𝑑 ∈ ℝ⁺) → ∃𝑏 ∈ ran 𝐼(𝑋 ∈ 𝑏 ∧ 𝑏 ⊆ ((𝑋 − 𝑑)(,)(𝑋 + 𝑑))))
5	4	ralrimiva 3146	. . . 4 ⊢ (𝑋 ∈ ℝ → ∀𝑑 ∈ ℝ⁺ ∃𝑏 ∈ ran 𝐼(𝑋 ∈ 𝑏 ∧ 𝑏 ⊆ ((𝑋 − 𝑑)(,)(𝑋 + 𝑑))))
6	5	3ad2ant1 1134	. . 3 ⊢ ((𝑋 ∈ ℝ ∧ 𝐸 ∈ ran (,) ∧ 𝑋 ∈ 𝐸) → ∀𝑑 ∈ ℝ⁺ ∃𝑏 ∈ ran 𝐼(𝑋 ∈ 𝑏 ∧ 𝑏 ⊆ ((𝑋 − 𝑑)(,)(𝑋 + 𝑑))))
7		simp3 1139	. . . . 5 ⊢ ((𝑋 ∈ ℝ ∧ 𝐸 ∈ ran (,) ∧ 𝑋 ∈ 𝐸) → 𝑋 ∈ 𝐸)
8		iooex 13410	. . . . . . . . . 10 ⊢ (,) ∈ V
9	8	rnex 7932	. . . . . . . . 9 ⊢ ran (,) ∈ V
10		bastg 22973	. . . . . . . . 9 ⊢ (ran (,) ∈ V → ran (,) ⊆ (topGen‘ran (,)))
11	9, 10	ax-mp 5	. . . . . . . 8 ⊢ ran (,) ⊆ (topGen‘ran (,))
12		simp2 1138	. . . . . . . 8 ⊢ ((𝑋 ∈ ℝ ∧ 𝐸 ∈ ran (,) ∧ 𝑋 ∈ 𝐸) → 𝐸 ∈ ran (,))
13	11, 12	sselid 3981	. . . . . . 7 ⊢ ((𝑋 ∈ ℝ ∧ 𝐸 ∈ ran (,) ∧ 𝑋 ∈ 𝐸) → 𝐸 ∈ (topGen‘ran (,)))
14	13, 1	eleqtrrdi 2852	. . . . . 6 ⊢ ((𝑋 ∈ ℝ ∧ 𝐸 ∈ ran (,) ∧ 𝑋 ∈ 𝐸) → 𝐸 ∈ 𝐽)
15		eqid 2737	. . . . . . . . 9 ⊢ ((abs ∘ − ) ↾ (ℝ × ℝ)) = ((abs ∘ − ) ↾ (ℝ × ℝ))
16	15	rexmet 24812	. . . . . . . 8 ⊢ ((abs ∘ − ) ↾ (ℝ × ℝ)) ∈ (∞Met‘ℝ)
17		recms 25414	. . . . . . . . . . 11 ⊢ ℝ_fld ∈ CMetSp
18		cmsms 25382	. . . . . . . . . . 11 ⊢ (ℝ_fld ∈ CMetSp → ℝ_fld ∈ MetSp)
19		msxms 24464	. . . . . . . . . . 11 ⊢ (ℝ_fld ∈ MetSp → ℝ_fld ∈ ∞MetSp)
20	17, 18, 19	mp2b 10	. . . . . . . . . 10 ⊢ ℝ_fld ∈ ∞MetSp
21		retopn 25413	. . . . . . . . . . . 12 ⊢ (topGen‘ran (,)) = (TopOpen‘ℝ_fld)
22	1, 21	eqtri 2765	. . . . . . . . . . 11 ⊢ 𝐽 = (TopOpen‘ℝ_fld)
23		rebase 21624	. . . . . . . . . . 11 ⊢ ℝ = (Base‘ℝ_fld)
24		reds 21634	. . . . . . . . . . . 12 ⊢ (abs ∘ − ) = (dist‘ℝ_fld)
25	24	reseq1i 5993	. . . . . . . . . . 11 ⊢ ((abs ∘ − ) ↾ (ℝ × ℝ)) = ((dist‘ℝ_fld) ↾ (ℝ × ℝ))
26	22, 23, 25	xmstopn 24461	. . . . . . . . . 10 ⊢ (ℝ_fld ∈ ∞MetSp → 𝐽 = (MetOpen‘((abs ∘ − ) ↾ (ℝ × ℝ))))
27	20, 26	ax-mp 5	. . . . . . . . 9 ⊢ 𝐽 = (MetOpen‘((abs ∘ − ) ↾ (ℝ × ℝ)))
28	27	elmopn2 24455	. . . . . . . 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 7438	. . . . . . . 8 ⊢ (𝑥 = 𝑋 → (𝑥(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝑑) = (𝑋(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝑑))
33	32	sseq1d 4015	. . . . . . 7 ⊢ (𝑥 = 𝑋 → ((𝑥(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝑑) ⊆ 𝐸 ↔ (𝑋(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝑑) ⊆ 𝐸))
34	33	rexbidv 3179	. . . . . 6 ⊢ (𝑥 = 𝑋 → (∃𝑑 ∈ ℝ⁺ (𝑥(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝑑) ⊆ 𝐸 ↔ ∃𝑑 ∈ ℝ⁺ (𝑋(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝑑) ⊆ 𝐸))
35	34	rspcva 3620	. . . . 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 584	. . . 4 ⊢ ((𝑋 ∈ ℝ ∧ 𝐸 ∈ ran (,) ∧ 𝑋 ∈ 𝐸) → ∃𝑑 ∈ ℝ⁺ (𝑋(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝑑) ⊆ 𝐸)
37		rpre 13043	. . . . . . 7 ⊢ (𝑑 ∈ ℝ⁺ → 𝑑 ∈ ℝ)
38	15	bl2ioo 24813	. . . . . . . 8 ⊢ ((𝑋 ∈ ℝ ∧ 𝑑 ∈ ℝ) → (𝑋(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝑑) = ((𝑋 − 𝑑)(,)(𝑋 + 𝑑)))
39	38	sseq1d 4015	. . . . . . 7 ⊢ ((𝑋 ∈ ℝ ∧ 𝑑 ∈ ℝ) → ((𝑋(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝑑) ⊆ 𝐸 ↔ ((𝑋 − 𝑑)(,)(𝑋 + 𝑑)) ⊆ 𝐸))
40	37, 39	sylan2 593	. . . . . 6 ⊢ ((𝑋 ∈ ℝ ∧ 𝑑 ∈ ℝ⁺) → ((𝑋(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝑑) ⊆ 𝐸 ↔ ((𝑋 − 𝑑)(,)(𝑋 + 𝑑)) ⊆ 𝐸))
41	40	rexbidva 3177	. . . . 5 ⊢ (𝑋 ∈ ℝ → (∃𝑑 ∈ ℝ⁺ (𝑋(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝑑) ⊆ 𝐸 ↔ ∃𝑑 ∈ ℝ⁺ ((𝑋 − 𝑑)(,)(𝑋 + 𝑑)) ⊆ 𝐸))
42	41	3ad2ant1 1134	. . . 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 232	. . 3 ⊢ ((𝑋 ∈ ℝ ∧ 𝐸 ∈ ran (,) ∧ 𝑋 ∈ 𝐸) → ∃𝑑 ∈ ℝ⁺ ((𝑋 − 𝑑)(,)(𝑋 + 𝑑)) ⊆ 𝐸)
44		r19.29 3114	. . 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 584	. 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 3189	. . . 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 3992	. . . . . . 7 ⊢ ((𝑏 ⊆ ((𝑋 − 𝑑)(,)(𝑋 + 𝑑)) ∧ ((𝑋 − 𝑑)(,)(𝑋 + 𝑑)) ⊆ 𝐸) → 𝑏 ⊆ 𝐸)
48	47	anim2i 617	. . . . . 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 3084	. . . 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 235	. . 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 3151	. 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 206 ∧ wa 395 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 ∀wral 3061 ∃wrex 3070 Vcvv 3480 ⊆ wss 3951 × cxp 5683 ran crn 5686 ↾ cres 5687 ∘ ccom 5689 ‘cfv 6561 (class class class)co 7431 ∈ cmpo 7433 ℝcr 11154 1c1 11156 + caddc 11158 − cmin 11492 / cdiv 11920 2c2 12321 ℤcz 12613 ℝ⁺crp 13034 (,)cioo 13387 [,)cico 13389 ⌊cfl 13830 ↑cexp 14102 abscabs 15273 distcds 17306 TopOpenctopn 17466 topGenctg 17482 ∞Metcxmet 21349 ballcbl 21351 MetOpencmopn 21354 ℝ_fldcrefld 21622 ∞MetSpcxms 24327 MetSpcms 24328 CMetSpccms 25366 log_b clogb 26807

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-inf2 9681 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233 ax-addf 11234

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-iin 4994 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-se 5638 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-isom 6570 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-om 7888 df-1st 8014 df-2nd 8015 df-supp 8186 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-2o 8507 df-er 8745 df-map 8868 df-pm 8869 df-ixp 8938 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-fsupp 9402 df-fi 9451 df-sup 9482 df-inf 9483 df-oi 9550 df-card 9979 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-z 12614 df-dec 12734 df-uz 12879 df-q 12991 df-rp 13035 df-xneg 13154 df-xadd 13155 df-xmul 13156 df-ioo 13391 df-ioc 13392 df-ico 13393 df-icc 13394 df-fz 13548 df-fzo 13695 df-fl 13832 df-mod 13910 df-seq 14043 df-exp 14103 df-fac 14313 df-bc 14342 df-hash 14370 df-shft 15106 df-cj 15138 df-re 15139 df-im 15140 df-sqrt 15274 df-abs 15275 df-limsup 15507 df-clim 15524 df-rlim 15525 df-sum 15723 df-ef 16103 df-sin 16105 df-cos 16106 df-pi 16108 df-struct 17184 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-ress 17275 df-plusg 17310 df-mulr 17311 df-starv 17312 df-sca 17313 df-vsca 17314 df-ip 17315 df-tset 17316 df-ple 17317 df-ds 17319 df-unif 17320 df-hom 17321 df-cco 17322 df-rest 17467 df-topn 17468 df-0g 17486 df-gsum 17487 df-topgen 17488 df-pt 17489 df-prds 17492 df-xrs 17547 df-qtop 17552 df-imas 17553 df-xps 17555 df-mre 17629 df-mrc 17630 df-acs 17632 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-submnd 18797 df-mulg 19086 df-cntz 19335 df-cmn 19800 df-psmet 21356 df-xmet 21357 df-met 21358 df-bl 21359 df-mopn 21360 df-fbas 21361 df-fg 21362 df-cnfld 21365 df-refld 21623 df-top 22900 df-topon 22917 df-topsp 22939 df-bases 22953 df-cld 23027 df-ntr 23028 df-cls 23029 df-nei 23106 df-lp 23144 df-perf 23145 df-cn 23235 df-cnp 23236 df-haus 23323 df-cmp 23395 df-tx 23570 df-hmeo 23763 df-fil 23854 df-fm 23946 df-flim 23947 df-flf 23948 df-fcls 23949 df-xms 24330 df-ms 24331 df-tms 24332 df-cncf 24904 df-cfil 25289 df-cmet 25291 df-cms 25369 df-limc 25901 df-dv 25902 df-log 26598 df-cxp 26599 df-logb 26808

This theorem is referenced by: dya2iocnrect 34283

W3C validator