dya2icoseg2 - Mathbox for Thierry Arnoux

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Theorem dya2icoseg2 34957

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 2760	. . . . . 6 ⊢ (⌊‘(1 − (2 log_b 𝑑))) = (⌊‘(1 − (2 log_b 𝑑)))
4	1, 2, 3	dya2icoseg 34956	. . . . 5 ⊢ ((𝑋 ∈ ℝ ∧ 𝑑 ∈ ℝ⁺) → ∃𝑏 ∈ ran 𝐼(𝑋 ∈ 𝑏 ∧ 𝑏 ⊆ ((𝑋 − 𝑑)(,)(𝑋 + 𝑑))))
5	4	ralrimiva 3154	. . . 4 ⊢ (𝑋 ∈ ℝ → ∀𝑑 ∈ ℝ⁺ ∃𝑏 ∈ ran 𝐼(𝑋 ∈ 𝑏 ∧ 𝑏 ⊆ ((𝑋 − 𝑑)(,)(𝑋 + 𝑑))))
6	5	3ad2ant1 1151	. . 3 ⊢ ((𝑋 ∈ ℝ ∧ 𝐸 ∈ ran (,) ∧ 𝑋 ∈ 𝐸) → ∀𝑑 ∈ ℝ⁺ ∃𝑏 ∈ ran 𝐼(𝑋 ∈ 𝑏 ∧ 𝑏 ⊆ ((𝑋 − 𝑑)(,)(𝑋 + 𝑑))))
7		simp3 1156	. . . . 5 ⊢ ((𝑋 ∈ ℝ ∧ 𝐸 ∈ ran (,) ∧ 𝑋 ∈ 𝐸) → 𝑋 ∈ 𝐸)
8		iooex 13503	. . . . . . . . . 10 ⊢ (,) ∈ V
9	8	rnex 7923	. . . . . . . . 9 ⊢ ran (,) ∈ V
10		bastg 23290	. . . . . . . . 9 ⊢ (ran (,) ∈ V → ran (,) ⊆ (topGen‘ran (,)))
11	9, 10	ax-mp 5	. . . . . . . 8 ⊢ ran (,) ⊆ (topGen‘ran (,))
12		simp2 1155	. . . . . . . 8 ⊢ ((𝑋 ∈ ℝ ∧ 𝐸 ∈ ran (,) ∧ 𝑋 ∈ 𝐸) → 𝐸 ∈ ran (,))
13	11, 12	sselid 3906	. . . . . . 7 ⊢ ((𝑋 ∈ ℝ ∧ 𝐸 ∈ ran (,) ∧ 𝑋 ∈ 𝐸) → 𝐸 ∈ (topGen‘ran (,)))
14	13, 1	eleqtrrdi 2871	. . . . . 6 ⊢ ((𝑋 ∈ ℝ ∧ 𝐸 ∈ ran (,) ∧ 𝑋 ∈ 𝐸) → 𝐸 ∈ 𝐽)
15		eqid 2760	. . . . . . . . 9 ⊢ ((abs ∘ − ) ↾ (ℝ × ℝ)) = ((abs ∘ − ) ↾ (ℝ × ℝ))
16	15	rexmet 25116	. . . . . . . 8 ⊢ ((abs ∘ − ) ↾ (ℝ × ℝ)) ∈ (∞Met‘ℝ)
17		recms 25707	. . . . . . . . . . 11 ⊢ ℝ_fld ∈ CMetSp
18		cmsms 25675	. . . . . . . . . . 11 ⊢ (ℝ_fld ∈ CMetSp → ℝ_fld ∈ MetSp)
19		msxms 24779	. . . . . . . . . . 11 ⊢ (ℝ_fld ∈ MetSp → ℝ_fld ∈ ∞MetSp)
20	17, 18, 19	mp2b 10	. . . . . . . . . 10 ⊢ ℝ_fld ∈ ∞MetSp
21		retopn 25706	. . . . . . . . . . . 12 ⊢ (topGen‘ran (,)) = (TopOpen‘ℝ_fld)
22	1, 21	eqtri 2783	. . . . . . . . . . 11 ⊢ 𝐽 = (TopOpen‘ℝ_fld)
23		rebase 21918	. . . . . . . . . . 11 ⊢ ℝ = (Base‘ℝ_fld)
24		reds 21928	. . . . . . . . . . . 12 ⊢ (abs ∘ − ) = (dist‘ℝ_fld)
25	24	reseq1i 5965	. . . . . . . . . . 11 ⊢ ((abs ∘ − ) ↾ (ℝ × ℝ)) = ((dist‘ℝ_fld) ↾ (ℝ × ℝ))
26	22, 23, 25	xmstopn 24776	. . . . . . . . . 10 ⊢ (ℝ_fld ∈ ∞MetSp → 𝐽 = (MetOpen‘((abs ∘ − ) ↾ (ℝ × ℝ))))
27	20, 26	ax-mp 5	. . . . . . . . 9 ⊢ 𝐽 = (MetOpen‘((abs ∘ − ) ↾ (ℝ × ℝ)))
28	27	elmopn2 24770	. . . . . . . 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 503	. . . . . 6 ⊢ (𝐸 ∈ 𝐽 → ∀𝑥 ∈ 𝐸 ∃𝑑 ∈ ℝ⁺ (𝑥(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝑑) ⊆ 𝐸)
31	14, 30	syl 18	. . . . 5 ⊢ ((𝑋 ∈ ℝ ∧ 𝐸 ∈ ran (,) ∧ 𝑋 ∈ 𝐸) → ∀𝑥 ∈ 𝐸 ∃𝑑 ∈ ℝ⁺ (𝑥(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝑑) ⊆ 𝐸)
32		oveq1 7428	. . . . . . . 8 ⊢ (𝑥 = 𝑋 → (𝑥(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝑑) = (𝑋(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝑑))
33	32	sseq1d 3939	. . . . . . 7 ⊢ (𝑥 = 𝑋 → ((𝑥(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝑑) ⊆ 𝐸 ↔ (𝑋(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝑑) ⊆ 𝐸))
34	33	rexbidv 3186	. . . . . 6 ⊢ (𝑥 = 𝑋 → (∃𝑑 ∈ ℝ⁺ (𝑥(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝑑) ⊆ 𝐸 ↔ ∃𝑑 ∈ ℝ⁺ (𝑋(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝑑) ⊆ 𝐸))
35	34	rspcva 3574	. . . . 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 596	. . . 4 ⊢ ((𝑋 ∈ ℝ ∧ 𝐸 ∈ ran (,) ∧ 𝑋 ∈ 𝐸) → ∃𝑑 ∈ ℝ⁺ (𝑋(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝑑) ⊆ 𝐸)
37		rpre 13133	. . . . . . 7 ⊢ (𝑑 ∈ ℝ⁺ → 𝑑 ∈ ℝ)
38	15	bl2ioo 25117	. . . . . . . 8 ⊢ ((𝑋 ∈ ℝ ∧ 𝑑 ∈ ℝ) → (𝑋(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝑑) = ((𝑋 − 𝑑)(,)(𝑋 + 𝑑)))
39	38	sseq1d 3939	. . . . . . 7 ⊢ ((𝑋 ∈ ℝ ∧ 𝑑 ∈ ℝ) → ((𝑋(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝑑) ⊆ 𝐸 ↔ ((𝑋 − 𝑑)(,)(𝑋 + 𝑑)) ⊆ 𝐸))
40	37, 39	sylan2 605	. . . . . 6 ⊢ ((𝑋 ∈ ℝ ∧ 𝑑 ∈ ℝ⁺) → ((𝑋(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝑑) ⊆ 𝐸 ↔ ((𝑋 − 𝑑)(,)(𝑋 + 𝑑)) ⊆ 𝐸))
41	40	rexbidva 3184	. . . . 5 ⊢ (𝑋 ∈ ℝ → (∃𝑑 ∈ ℝ⁺ (𝑋(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝑑) ⊆ 𝐸 ↔ ∃𝑑 ∈ ℝ⁺ ((𝑋 − 𝑑)(,)(𝑋 + 𝑑)) ⊆ 𝐸))
42	41	3ad2ant1 1151	. . . 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 235	. . 3 ⊢ ((𝑋 ∈ ℝ ∧ 𝐸 ∈ ran (,) ∧ 𝑋 ∈ 𝐸) → ∃𝑑 ∈ ℝ⁺ ((𝑋 − 𝑑)(,)(𝑋 + 𝑑)) ⊆ 𝐸)
44		r19.29 3125	. . 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 596	. 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 3192	. . . 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 3916	. . . . . . 7 ⊢ ((𝑏 ⊆ ((𝑋 − 𝑑)(,)(𝑋 + 𝑑)) ∧ ((𝑋 − 𝑑)(,)(𝑋 + 𝑑)) ⊆ 𝐸) → 𝑏 ⊆ 𝐸)
48	47	anim2i 629	. . . . . 6 ⊢ ((𝑋 ∈ 𝑏 ∧ (𝑏 ⊆ ((𝑋 − 𝑑)(,)(𝑋 + 𝑑)) ∧ ((𝑋 − 𝑑)(,)(𝑋 + 𝑑)) ⊆ 𝐸)) → (𝑋 ∈ 𝑏 ∧ 𝑏 ⊆ 𝐸))
49	48	anassrs 473	. . . . 5 ⊢ (((𝑋 ∈ 𝑏 ∧ 𝑏 ⊆ ((𝑋 − 𝑑)(,)(𝑋 + 𝑑))) ∧ ((𝑋 − 𝑑)(,)(𝑋 + 𝑑)) ⊆ 𝐸) → (𝑋 ∈ 𝑏 ∧ 𝑏 ⊆ 𝐸))
50	49	reximi 3100	. . . 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 238	. . 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 3159	. 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 18	1 ⊢ ((𝑋 ∈ ℝ ∧ 𝐸 ∈ ran (,) ∧ 𝑋 ∈ 𝐸) → ∃𝑏 ∈ ran 𝐼(𝑋 ∈ 𝑏 ∧ 𝑏 ⊆ 𝐸))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 ∧ w3a 1103 = wceq 1569 ∈ wcel 2144 ∀wral 3076 ∃wrex 3086 Vcvv 3450 ⊆ wss 3895 × cxp 5648 ran crn 5651 ↾ cres 5652 ∘ ccom 5654 ‘cfv 6539 (class class class)co 7421 ∈ cmpo 7423 ℝcr 11203 1c1 11205 + caddc 11207 − cmin 11545 / cdiv 11977 2c2 12401 ℤcz 12697 ℝ⁺crp 13124 (,)cioo 13480 [,)cico 13482 ⌊cfl 13934 ↑cexp 14208 abscabs 15407 distcds 17443 TopOpenctopn 17598 topGenctg 17614 ∞Metcxmet 21669 ballcbl 21671 MetOpencmopn 21674 ℝ_fldcrefld 21916 ∞MetSpcxms 24642 MetSpcms 24643 CMetSpccms 25659 log_b clogb 27098

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1942 ax-6 1999 ax-7 2040 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2212 ax-ext 2732 ax-rep 5231 ax-sep 5248 ax-nul 5259 ax-pow 5326 ax-pr 5390 ax-un 7752 ax-inf2 9646 ax-cnex 11260 ax-resscn 11261 ax-1cn 11262 ax-icn 11263 ax-addcl 11264 ax-addrcl 11265 ax-mulcl 11266 ax-mulrcl 11267 ax-mulcom 11268 ax-addass 11269 ax-mulass 11270 ax-distr 11271 ax-i2m1 11272 ax-1ne0 11273 ax-1rid 11274 ax-rnegex 11275 ax-rrecex 11276 ax-cnre 11277 ax-pre-lttri 11278 ax-pre-lttrn 11279 ax-pre-ltadd 11280 ax-pre-mulgt0 11281 ax-pre-sup 11282 ax-addf 11283

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2099 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3739 df-csb 3847 df-ss 3896 df-pss 4014 df-dif 4047 df-in 4091 df-un 4151 df-nul 4279 df-if 4482 df-pw 4558 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4835 df-uni 4867 df-int 4907 df-iun 4952 df-iin 4953 df-br 5103 df-opab 5167 df-mpt 5186 df-tr 5212 df-id 5545 df-eprel 5550 df-po 5558 df-so 5559 df-fr 5603 df-se 5604 df-we 5605 df-xp 5656 df-rel 5657 df-cnv 5658 df-co 5659 df-dm 5660 df-rn 5661 df-res 5662 df-ima 5663 df-pred 6304 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6494 df-fun 6541 df-fn 6542 df-f 6543 df-f1 6544 df-fo 6545 df-f1o 6546 df-fv 6547 df-isom 6548 df-riota 7378 df-ov 7424 df-oprab 7425 df-mpo 7426 df-of 7694 df-om 7879 df-1st 8002 df-2nd 8003 df-supp 8181 df-frecs 8303 df-wrecs 8334 df-recs 8383 df-rdg 8422 df-1o 8480 df-2o 8481 df-er 8721 df-map 8853 df-pm 8854 df-ixp 8930 df-en 8978 df-dom 8979 df-sdom 8980 df-fin 8981 df-fsupp 9358 df-fi 9407 df-sup 9438 df-inf 9439 df-oi 9508 df-card 10024 df-pnf 11349 df-mnf 11350 df-xr 11351 df-ltxr 11352 df-le 11353 df-sub 11547 df-neg 11548 df-div 11978 df-nn 12340 df-2 12409 df-3 12410 df-4 12411 df-5 12412 df-6 12413 df-7 12414 df-8 12415 df-9 12416 df-n0 12611 df-z 12698 df-dec 12819 df-uz 12970 df-q 13080 df-rp 13125 df-xneg 13245 df-xadd 13246 df-xmul 13247 df-ioo 13484 df-ioc 13485 df-ico 13486 df-icc 13487 df-fz 13644 df-fzo 13793 df-fl 13936 df-mod 14014 df-seq 14149 df-exp 14209 df-fac 14422 df-bc 14451 df-hash 14479 df-shft 15226 df-cj 15272 df-re 15273 df-im 15274 df-sqrt 15408 df-abs 15409 df-limsup 15644 df-clim 15661 df-rlim 15662 df-sum 15860 df-ef 16239 df-sin 16241 df-cos 16242 df-pi 16244 df-struct 17331 df-sets 17348 df-slot 17366 df-ndx 17378 df-base 17394 df-ress 17415 df-plusg 17447 df-mulr 17448 df-starv 17449 df-sca 17450 df-vsca 17451 df-ip 17452 df-tset 17453 df-ple 17454 df-ds 17456 df-unif 17457 df-hom 17458 df-cco 17459 df-rest 17599 df-topn 17600 df-0g 17618 df-gsum 17619 df-topgen 17620 df-pt 17621 df-prds 17624 df-xrs 17680 df-qtop 17685 df-imas 17686 df-xps 17688 df-mre 17762 df-mrc 17763 df-acs 17765 df-mgm 18822 df-sgrp 18914 df-mnd 18930 df-submnd 18985 df-mulg 19284 df-cntz 19537 df-cmn 20002 df-psmet 21676 df-xmet 21677 df-met 21678 df-bl 21679 df-mopn 21680 df-fbas 21681 df-fg 21682 df-cnfld 21685 df-refld 21917 df-top 23218 df-topon 23235 df-topsp 23257 df-bases 23270 df-cld 23343 df-ntr 23344 df-cls 23345 df-nei 23422 df-lp 23460 df-perf 23461 df-cn 23551 df-cnp 23552 df-haus 23639 df-cmp 23711 df-tx 23887 df-hmeo 24080 df-fil 24171 df-fm 24263 df-flim 24264 df-flf 24265 df-fcls 24266 df-xms 24645 df-ms 24646 df-tms 24647 df-cncf 25205 df-cfil 25582 df-cmet 25584 df-cms 25662 df-limc 26192 df-dv 26193 df-log 26890 df-cxp 26891 df-logb 27099

This theorem is used by: dya2iocnrect 34960

W3C validator