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Theorem cnpart 15187

Description: The specification of restriction to the right half-plane partitions the complex plane without 0 into two disjoint pieces, which are related by a reflection about the origin (under the map 𝑥 ↦ -𝑥). (Contributed by Mario Carneiro, 8-Jul-2013.)

**Assertion**
Ref	Expression
cnpart	⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → ((0 ≤ (ℜ‘𝐴) ∧ (i · 𝐴) ∉ ℝ⁺) ↔ ¬ (0 ≤ (ℜ‘-𝐴) ∧ (i · -𝐴) ∉ ℝ⁺)))

**Proof of Theorem cnpart**
Step	Hyp	Ref	Expression
1		df-nel 3048	. . . . . 6 ⊢ (-(i · 𝐴) ∉ ℝ⁺ ↔ ¬ -(i · 𝐴) ∈ ℝ⁺)
2		simpr 486	. . . . . . . 8 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (ℜ‘𝐴) = 0) → (ℜ‘𝐴) = 0)
3		0le0 12313	. . . . . . . 8 ⊢ 0 ≤ 0
4	2, 3	eqbrtrdi 5188	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (ℜ‘𝐴) = 0) → (ℜ‘𝐴) ≤ 0)
5	4	biantrurd 534	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (ℜ‘𝐴) = 0) → (-(i · 𝐴) ∉ ℝ⁺ ↔ ((ℜ‘𝐴) ≤ 0 ∧ -(i · 𝐴) ∉ ℝ⁺)))
6	1, 5	bitr3id 285	. . . . 5 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (ℜ‘𝐴) = 0) → (¬ -(i · 𝐴) ∈ ℝ⁺ ↔ ((ℜ‘𝐴) ≤ 0 ∧ -(i · 𝐴) ∉ ℝ⁺)))
7	6	con1bid 356	. . . 4 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (ℜ‘𝐴) = 0) → (¬ ((ℜ‘𝐴) ≤ 0 ∧ -(i · 𝐴) ∉ ℝ⁺) ↔ -(i · 𝐴) ∈ ℝ⁺))
8		ax-icn 11169	. . . . . . . . . . . 12 ⊢ i ∈ ℂ
9		mulcl 11194	. . . . . . . . . . . 12 ⊢ ((i ∈ ℂ ∧ 𝐴 ∈ ℂ) → (i · 𝐴) ∈ ℂ)
10	8, 9	mpan 689	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℂ → (i · 𝐴) ∈ ℂ)
11		reim0b 15066	. . . . . . . . . . 11 ⊢ ((i · 𝐴) ∈ ℂ → ((i · 𝐴) ∈ ℝ ↔ (ℑ‘(i · 𝐴)) = 0))
12	10, 11	syl 17	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℂ → ((i · 𝐴) ∈ ℝ ↔ (ℑ‘(i · 𝐴)) = 0))
13		imre 15055	. . . . . . . . . . . . 13 ⊢ ((i · 𝐴) ∈ ℂ → (ℑ‘(i · 𝐴)) = (ℜ‘(-i · (i · 𝐴))))
14	10, 13	syl 17	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ℂ → (ℑ‘(i · 𝐴)) = (ℜ‘(-i · (i · 𝐴))))
15		ine0 11649	. . . . . . . . . . . . . . . . 17 ⊢ i ≠ 0
16		divrec2 11889	. . . . . . . . . . . . . . . . 17 ⊢ (((i · 𝐴) ∈ ℂ ∧ i ∈ ℂ ∧ i ≠ 0) → ((i · 𝐴) / i) = ((1 / i) · (i · 𝐴)))
17	8, 15, 16	mp3an23 1454	. . . . . . . . . . . . . . . 16 ⊢ ((i · 𝐴) ∈ ℂ → ((i · 𝐴) / i) = ((1 / i) · (i · 𝐴)))
18	10, 17	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ ℂ → ((i · 𝐴) / i) = ((1 / i) · (i · 𝐴)))
19		irec 14165	. . . . . . . . . . . . . . . 16 ⊢ (1 / i) = -i
20	19	oveq1i 7419	. . . . . . . . . . . . . . 15 ⊢ ((1 / i) · (i · 𝐴)) = (-i · (i · 𝐴))
21	18, 20	eqtrdi 2789	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ ℂ → ((i · 𝐴) / i) = (-i · (i · 𝐴)))
22		divcan3 11898	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ ℂ ∧ i ∈ ℂ ∧ i ≠ 0) → ((i · 𝐴) / i) = 𝐴)
23	8, 15, 22	mp3an23 1454	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ ℂ → ((i · 𝐴) / i) = 𝐴)
24	21, 23	eqtr3d 2775	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ ℂ → (-i · (i · 𝐴)) = 𝐴)
25	24	fveq2d 6896	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ℂ → (ℜ‘(-i · (i · 𝐴))) = (ℜ‘𝐴))
26	14, 25	eqtrd 2773	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℂ → (ℑ‘(i · 𝐴)) = (ℜ‘𝐴))
27	26	eqeq1d 2735	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℂ → ((ℑ‘(i · 𝐴)) = 0 ↔ (ℜ‘𝐴) = 0))
28	12, 27	bitrd 279	. . . . . . . . 9 ⊢ (𝐴 ∈ ℂ → ((i · 𝐴) ∈ ℝ ↔ (ℜ‘𝐴) = 0))
29	28	biimpar 479	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) = 0) → (i · 𝐴) ∈ ℝ)
30	29	adantlr 714	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (ℜ‘𝐴) = 0) → (i · 𝐴) ∈ ℝ)
31		mulne0 11856	. . . . . . . . 9 ⊢ (((i ∈ ℂ ∧ i ≠ 0) ∧ (𝐴 ∈ ℂ ∧ 𝐴 ≠ 0)) → (i · 𝐴) ≠ 0)
32	8, 15, 31	mpanl12 701	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (i · 𝐴) ≠ 0)
33	32	adantr 482	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (ℜ‘𝐴) = 0) → (i · 𝐴) ≠ 0)
34		rpneg 13006	. . . . . . 7 ⊢ (((i · 𝐴) ∈ ℝ ∧ (i · 𝐴) ≠ 0) → ((i · 𝐴) ∈ ℝ⁺ ↔ ¬ -(i · 𝐴) ∈ ℝ⁺))
35	30, 33, 34	syl2anc 585	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (ℜ‘𝐴) = 0) → ((i · 𝐴) ∈ ℝ⁺ ↔ ¬ -(i · 𝐴) ∈ ℝ⁺))
36	35	con2bid 355	. . . . 5 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (ℜ‘𝐴) = 0) → (-(i · 𝐴) ∈ ℝ⁺ ↔ ¬ (i · 𝐴) ∈ ℝ⁺))
37		df-nel 3048	. . . . 5 ⊢ ((i · 𝐴) ∉ ℝ⁺ ↔ ¬ (i · 𝐴) ∈ ℝ⁺)
38	36, 37	bitr4di 289	. . . 4 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (ℜ‘𝐴) = 0) → (-(i · 𝐴) ∈ ℝ⁺ ↔ (i · 𝐴) ∉ ℝ⁺))
39	3, 2	breqtrrid 5187	. . . . 5 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (ℜ‘𝐴) = 0) → 0 ≤ (ℜ‘𝐴))
40	39	biantrurd 534	. . . 4 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (ℜ‘𝐴) = 0) → ((i · 𝐴) ∉ ℝ⁺ ↔ (0 ≤ (ℜ‘𝐴) ∧ (i · 𝐴) ∉ ℝ⁺)))
41	7, 38, 40	3bitrrd 306	. . 3 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (ℜ‘𝐴) = 0) → ((0 ≤ (ℜ‘𝐴) ∧ (i · 𝐴) ∉ ℝ⁺) ↔ ¬ ((ℜ‘𝐴) ≤ 0 ∧ -(i · 𝐴) ∉ ℝ⁺)))
42	28	adantr 482	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → ((i · 𝐴) ∈ ℝ ↔ (ℜ‘𝐴) = 0))
43	42	necon3bbid 2979	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (¬ (i · 𝐴) ∈ ℝ ↔ (ℜ‘𝐴) ≠ 0))
44	43	biimpar 479	. . . . . . . 8 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (ℜ‘𝐴) ≠ 0) → ¬ (i · 𝐴) ∈ ℝ)
45		rpre 12982	. . . . . . . 8 ⊢ ((i · 𝐴) ∈ ℝ⁺ → (i · 𝐴) ∈ ℝ)
46	44, 45	nsyl 140	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (ℜ‘𝐴) ≠ 0) → ¬ (i · 𝐴) ∈ ℝ⁺)
47	46, 37	sylibr 233	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (ℜ‘𝐴) ≠ 0) → (i · 𝐴) ∉ ℝ⁺)
48	47	biantrud 533	. . . . 5 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (ℜ‘𝐴) ≠ 0) → (0 ≤ (ℜ‘𝐴) ↔ (0 ≤ (ℜ‘𝐴) ∧ (i · 𝐴) ∉ ℝ⁺)))
49		simpr 486	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (ℜ‘𝐴) ≠ 0) → (ℜ‘𝐴) ≠ 0)
50	49	biantrud 533	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (ℜ‘𝐴) ≠ 0) → (0 ≤ (ℜ‘𝐴) ↔ (0 ≤ (ℜ‘𝐴) ∧ (ℜ‘𝐴) ≠ 0)))
51		0re 11216	. . . . . . . 8 ⊢ 0 ∈ ℝ
52		recl 15057	. . . . . . . 8 ⊢ (𝐴 ∈ ℂ → (ℜ‘𝐴) ∈ ℝ)
53		ltlen 11315	. . . . . . . . 9 ⊢ ((0 ∈ ℝ ∧ (ℜ‘𝐴) ∈ ℝ) → (0 < (ℜ‘𝐴) ↔ (0 ≤ (ℜ‘𝐴) ∧ (ℜ‘𝐴) ≠ 0)))
54		ltnle 11293	. . . . . . . . 9 ⊢ ((0 ∈ ℝ ∧ (ℜ‘𝐴) ∈ ℝ) → (0 < (ℜ‘𝐴) ↔ ¬ (ℜ‘𝐴) ≤ 0))
55	53, 54	bitr3d 281	. . . . . . . 8 ⊢ ((0 ∈ ℝ ∧ (ℜ‘𝐴) ∈ ℝ) → ((0 ≤ (ℜ‘𝐴) ∧ (ℜ‘𝐴) ≠ 0) ↔ ¬ (ℜ‘𝐴) ≤ 0))
56	51, 52, 55	sylancr 588	. . . . . . 7 ⊢ (𝐴 ∈ ℂ → ((0 ≤ (ℜ‘𝐴) ∧ (ℜ‘𝐴) ≠ 0) ↔ ¬ (ℜ‘𝐴) ≤ 0))
57	56	ad2antrr 725	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (ℜ‘𝐴) ≠ 0) → ((0 ≤ (ℜ‘𝐴) ∧ (ℜ‘𝐴) ≠ 0) ↔ ¬ (ℜ‘𝐴) ≤ 0))
58	50, 57	bitrd 279	. . . . 5 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (ℜ‘𝐴) ≠ 0) → (0 ≤ (ℜ‘𝐴) ↔ ¬ (ℜ‘𝐴) ≤ 0))
59	48, 58	bitr3d 281	. . . 4 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (ℜ‘𝐴) ≠ 0) → ((0 ≤ (ℜ‘𝐴) ∧ (i · 𝐴) ∉ ℝ⁺) ↔ ¬ (ℜ‘𝐴) ≤ 0))
60		renegcl 11523	. . . . . . . . . 10 ⊢ (-(i · 𝐴) ∈ ℝ → --(i · 𝐴) ∈ ℝ)
61	10	negnegd 11562	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ℂ → --(i · 𝐴) = (i · 𝐴))
62	61	eleq1d 2819	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℂ → (--(i · 𝐴) ∈ ℝ ↔ (i · 𝐴) ∈ ℝ))
63	62	ad2antrr 725	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (ℜ‘𝐴) ≠ 0) → (--(i · 𝐴) ∈ ℝ ↔ (i · 𝐴) ∈ ℝ))
64	60, 63	imbitrid 243	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (ℜ‘𝐴) ≠ 0) → (-(i · 𝐴) ∈ ℝ → (i · 𝐴) ∈ ℝ))
65	44, 64	mtod 197	. . . . . . . 8 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (ℜ‘𝐴) ≠ 0) → ¬ -(i · 𝐴) ∈ ℝ)
66		rpre 12982	. . . . . . . 8 ⊢ (-(i · 𝐴) ∈ ℝ⁺ → -(i · 𝐴) ∈ ℝ)
67	65, 66	nsyl 140	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (ℜ‘𝐴) ≠ 0) → ¬ -(i · 𝐴) ∈ ℝ⁺)
68	67, 1	sylibr 233	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (ℜ‘𝐴) ≠ 0) → -(i · 𝐴) ∉ ℝ⁺)
69	68	biantrud 533	. . . . 5 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (ℜ‘𝐴) ≠ 0) → ((ℜ‘𝐴) ≤ 0 ↔ ((ℜ‘𝐴) ≤ 0 ∧ -(i · 𝐴) ∉ ℝ⁺)))
70	69	notbid 318	. . . 4 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (ℜ‘𝐴) ≠ 0) → (¬ (ℜ‘𝐴) ≤ 0 ↔ ¬ ((ℜ‘𝐴) ≤ 0 ∧ -(i · 𝐴) ∉ ℝ⁺)))
71	59, 70	bitrd 279	. . 3 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (ℜ‘𝐴) ≠ 0) → ((0 ≤ (ℜ‘𝐴) ∧ (i · 𝐴) ∉ ℝ⁺) ↔ ¬ ((ℜ‘𝐴) ≤ 0 ∧ -(i · 𝐴) ∉ ℝ⁺)))
72	41, 71	pm2.61dane 3030	. 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → ((0 ≤ (ℜ‘𝐴) ∧ (i · 𝐴) ∉ ℝ⁺) ↔ ¬ ((ℜ‘𝐴) ≤ 0 ∧ -(i · 𝐴) ∉ ℝ⁺)))
73		reneg 15072	. . . . . . 7 ⊢ (𝐴 ∈ ℂ → (ℜ‘-𝐴) = -(ℜ‘𝐴))
74	73	breq2d 5161	. . . . . 6 ⊢ (𝐴 ∈ ℂ → (0 ≤ (ℜ‘-𝐴) ↔ 0 ≤ -(ℜ‘𝐴)))
75	52	le0neg1d 11785	. . . . . 6 ⊢ (𝐴 ∈ ℂ → ((ℜ‘𝐴) ≤ 0 ↔ 0 ≤ -(ℜ‘𝐴)))
76	74, 75	bitr4d 282	. . . . 5 ⊢ (𝐴 ∈ ℂ → (0 ≤ (ℜ‘-𝐴) ↔ (ℜ‘𝐴) ≤ 0))
77		mulneg2 11651	. . . . . . 7 ⊢ ((i ∈ ℂ ∧ 𝐴 ∈ ℂ) → (i · -𝐴) = -(i · 𝐴))
78	8, 77	mpan 689	. . . . . 6 ⊢ (𝐴 ∈ ℂ → (i · -𝐴) = -(i · 𝐴))
79		neleq1 3053	. . . . . 6 ⊢ ((i · -𝐴) = -(i · 𝐴) → ((i · -𝐴) ∉ ℝ⁺ ↔ -(i · 𝐴) ∉ ℝ⁺))
80	78, 79	syl 17	. . . . 5 ⊢ (𝐴 ∈ ℂ → ((i · -𝐴) ∉ ℝ⁺ ↔ -(i · 𝐴) ∉ ℝ⁺))
81	76, 80	anbi12d 632	. . . 4 ⊢ (𝐴 ∈ ℂ → ((0 ≤ (ℜ‘-𝐴) ∧ (i · -𝐴) ∉ ℝ⁺) ↔ ((ℜ‘𝐴) ≤ 0 ∧ -(i · 𝐴) ∉ ℝ⁺)))
82	81	notbid 318	. . 3 ⊢ (𝐴 ∈ ℂ → (¬ (0 ≤ (ℜ‘-𝐴) ∧ (i · -𝐴) ∉ ℝ⁺) ↔ ¬ ((ℜ‘𝐴) ≤ 0 ∧ -(i · 𝐴) ∉ ℝ⁺)))
83	82	adantr 482	. 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (¬ (0 ≤ (ℜ‘-𝐴) ∧ (i · -𝐴) ∉ ℝ⁺) ↔ ¬ ((ℜ‘𝐴) ≤ 0 ∧ -(i · 𝐴) ∉ ℝ⁺)))
84	72, 83	bitr4d 282	1 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → ((0 ≤ (ℜ‘𝐴) ∧ (i · 𝐴) ∉ ℝ⁺) ↔ ¬ (0 ≤ (ℜ‘-𝐴) ∧ (i · -𝐴) ∉ ℝ⁺)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ≠ wne 2941 ∉ wnel 3047 class class class wbr 5149 ‘cfv 6544 (class class class)co 7409 ℂcc 11108 ℝcr 11109 0cc0 11110 1c1 11111 ici 11112 · cmul 11115 < clt 11248 ≤ cle 11249 -cneg 11445 / cdiv 11871 ℝ⁺crp 12974 ℜcre 15044 ℑcim 15045

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-resscn 11167 ax-1cn 11168 ax-icn 11169 ax-addcl 11170 ax-addrcl 11171 ax-mulcl 11172 ax-mulrcl 11173 ax-mulcom 11174 ax-addass 11175 ax-mulass 11176 ax-distr 11177 ax-i2m1 11178 ax-1ne0 11179 ax-1rid 11180 ax-rnegex 11181 ax-rrecex 11182 ax-cnre 11183 ax-pre-lttri 11184 ax-pre-lttrn 11185 ax-pre-ltadd 11186 ax-pre-mulgt0 11187

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5575 df-po 5589 df-so 5590 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-er 8703 df-en 8940 df-dom 8941 df-sdom 8942 df-pnf 11250 df-mnf 11251 df-xr 11252 df-ltxr 11253 df-le 11254 df-sub 11446 df-neg 11447 df-div 11872 df-2 12275 df-rp 12975 df-cj 15046 df-re 15047 df-im 15048

This theorem is referenced by: sqrmo 15198

W3C validator