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Theorem dvdsmulgcd 16603

Description: A divisibility equivalent for odmulg 19598. (Contributed by Stefan O'Rear, 6-Sep-2015.)

**Assertion**
Ref	Expression
dvdsmulgcd	⊢ ((𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) → (𝐴 ∥ (𝐵 · 𝐶) ↔ 𝐴 ∥ (𝐵 · (𝐶 gcd 𝐴))))

Proof of Theorem dvdsmulgcd Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simplr 768	. . . 4 ⊢ (((𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) ∧ 𝐴 ∥ (𝐵 · 𝐶)) → 𝐶 ∈ ℤ)
2		dvdszrcl 16307	. . . . . 6 ⊢ (𝐴 ∥ (𝐵 · 𝐶) → (𝐴 ∈ ℤ ∧ (𝐵 · 𝐶) ∈ ℤ))
3	2	adantl 481	. . . . 5 ⊢ (((𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) ∧ 𝐴 ∥ (𝐵 · 𝐶)) → (𝐴 ∈ ℤ ∧ (𝐵 · 𝐶) ∈ ℤ))
4	3	simpld 494	. . . 4 ⊢ (((𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) ∧ 𝐴 ∥ (𝐵 · 𝐶)) → 𝐴 ∈ ℤ)
5		bezout 16590	. . . 4 ⊢ ((𝐶 ∈ ℤ ∧ 𝐴 ∈ ℤ) → ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℤ (𝐶 gcd 𝐴) = ((𝐶 · 𝑥) + (𝐴 · 𝑦)))
6	1, 4, 5	syl2anc 583	. . 3 ⊢ (((𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) ∧ 𝐴 ∥ (𝐵 · 𝐶)) → ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℤ (𝐶 gcd 𝐴) = ((𝐶 · 𝑥) + (𝐴 · 𝑦)))
7	4	adantr 480	. . . . . . 7 ⊢ ((((𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) ∧ 𝐴 ∥ (𝐵 · 𝐶)) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) → 𝐴 ∈ ℤ)
8		simplll 774	. . . . . . . 8 ⊢ ((((𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) ∧ 𝐴 ∥ (𝐵 · 𝐶)) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) → 𝐵 ∈ ℤ)
9		simpllr 775	. . . . . . . . 9 ⊢ ((((𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) ∧ 𝐴 ∥ (𝐵 · 𝐶)) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) → 𝐶 ∈ ℤ)
10		simprl 770	. . . . . . . . 9 ⊢ ((((𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) ∧ 𝐴 ∥ (𝐵 · 𝐶)) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) → 𝑥 ∈ ℤ)
11	9, 10	zmulcld 12753	. . . . . . . 8 ⊢ ((((𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) ∧ 𝐴 ∥ (𝐵 · 𝐶)) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) → (𝐶 · 𝑥) ∈ ℤ)
12	8, 11	zmulcld 12753	. . . . . . 7 ⊢ ((((𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) ∧ 𝐴 ∥ (𝐵 · 𝐶)) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) → (𝐵 · (𝐶 · 𝑥)) ∈ ℤ)
13		simprr 772	. . . . . . . . 9 ⊢ ((((𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) ∧ 𝐴 ∥ (𝐵 · 𝐶)) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) → 𝑦 ∈ ℤ)
14	7, 13	zmulcld 12753	. . . . . . . 8 ⊢ ((((𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) ∧ 𝐴 ∥ (𝐵 · 𝐶)) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) → (𝐴 · 𝑦) ∈ ℤ)
15	8, 14	zmulcld 12753	. . . . . . 7 ⊢ ((((𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) ∧ 𝐴 ∥ (𝐵 · 𝐶)) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) → (𝐵 · (𝐴 · 𝑦)) ∈ ℤ)
16	8, 9	zmulcld 12753	. . . . . . . . 9 ⊢ ((((𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) ∧ 𝐴 ∥ (𝐵 · 𝐶)) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) → (𝐵 · 𝐶) ∈ ℤ)
17		simplr 768	. . . . . . . . 9 ⊢ ((((𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) ∧ 𝐴 ∥ (𝐵 · 𝐶)) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) → 𝐴 ∥ (𝐵 · 𝐶))
18	7, 16, 10, 17	dvdsmultr1d 16345	. . . . . . . 8 ⊢ ((((𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) ∧ 𝐴 ∥ (𝐵 · 𝐶)) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) → 𝐴 ∥ ((𝐵 · 𝐶) · 𝑥))
19	8	zcnd 12748	. . . . . . . . 9 ⊢ ((((𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) ∧ 𝐴 ∥ (𝐵 · 𝐶)) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) → 𝐵 ∈ ℂ)
20	9	zcnd 12748	. . . . . . . . 9 ⊢ ((((𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) ∧ 𝐴 ∥ (𝐵 · 𝐶)) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) → 𝐶 ∈ ℂ)
21	10	zcnd 12748	. . . . . . . . 9 ⊢ ((((𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) ∧ 𝐴 ∥ (𝐵 · 𝐶)) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) → 𝑥 ∈ ℂ)
22	19, 20, 21	mulassd 11313	. . . . . . . 8 ⊢ ((((𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) ∧ 𝐴 ∥ (𝐵 · 𝐶)) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) → ((𝐵 · 𝐶) · 𝑥) = (𝐵 · (𝐶 · 𝑥)))
23	18, 22	breqtrd 5192	. . . . . . 7 ⊢ ((((𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) ∧ 𝐴 ∥ (𝐵 · 𝐶)) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) → 𝐴 ∥ (𝐵 · (𝐶 · 𝑥)))
24	8, 13	zmulcld 12753	. . . . . . . . 9 ⊢ ((((𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) ∧ 𝐴 ∥ (𝐵 · 𝐶)) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) → (𝐵 · 𝑦) ∈ ℤ)
25		dvdsmul1 16326	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℤ ∧ (𝐵 · 𝑦) ∈ ℤ) → 𝐴 ∥ (𝐴 · (𝐵 · 𝑦)))
26	7, 24, 25	syl2anc 583	. . . . . . . 8 ⊢ ((((𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) ∧ 𝐴 ∥ (𝐵 · 𝐶)) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) → 𝐴 ∥ (𝐴 · (𝐵 · 𝑦)))
27	7	zcnd 12748	. . . . . . . . 9 ⊢ ((((𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) ∧ 𝐴 ∥ (𝐵 · 𝐶)) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) → 𝐴 ∈ ℂ)
28	13	zcnd 12748	. . . . . . . . 9 ⊢ ((((𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) ∧ 𝐴 ∥ (𝐵 · 𝐶)) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) → 𝑦 ∈ ℂ)
29	19, 27, 28	mul12d 11499	. . . . . . . 8 ⊢ ((((𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) ∧ 𝐴 ∥ (𝐵 · 𝐶)) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) → (𝐵 · (𝐴 · 𝑦)) = (𝐴 · (𝐵 · 𝑦)))
30	26, 29	breqtrrd 5194	. . . . . . 7 ⊢ ((((𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) ∧ 𝐴 ∥ (𝐵 · 𝐶)) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) → 𝐴 ∥ (𝐵 · (𝐴 · 𝑦)))
31	7, 12, 15, 23, 30	dvds2addd 16340	. . . . . 6 ⊢ ((((𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) ∧ 𝐴 ∥ (𝐵 · 𝐶)) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) → 𝐴 ∥ ((𝐵 · (𝐶 · 𝑥)) + (𝐵 · (𝐴 · 𝑦))))
32	11	zcnd 12748	. . . . . . 7 ⊢ ((((𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) ∧ 𝐴 ∥ (𝐵 · 𝐶)) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) → (𝐶 · 𝑥) ∈ ℂ)
33	14	zcnd 12748	. . . . . . 7 ⊢ ((((𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) ∧ 𝐴 ∥ (𝐵 · 𝐶)) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) → (𝐴 · 𝑦) ∈ ℂ)
34	19, 32, 33	adddid 11314	. . . . . 6 ⊢ ((((𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) ∧ 𝐴 ∥ (𝐵 · 𝐶)) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) → (𝐵 · ((𝐶 · 𝑥) + (𝐴 · 𝑦))) = ((𝐵 · (𝐶 · 𝑥)) + (𝐵 · (𝐴 · 𝑦))))
35	31, 34	breqtrrd 5194	. . . . 5 ⊢ ((((𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) ∧ 𝐴 ∥ (𝐵 · 𝐶)) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) → 𝐴 ∥ (𝐵 · ((𝐶 · 𝑥) + (𝐴 · 𝑦))))
36		oveq2 7456	. . . . . 6 ⊢ ((𝐶 gcd 𝐴) = ((𝐶 · 𝑥) + (𝐴 · 𝑦)) → (𝐵 · (𝐶 gcd 𝐴)) = (𝐵 · ((𝐶 · 𝑥) + (𝐴 · 𝑦))))
37	36	breq2d 5178	. . . . 5 ⊢ ((𝐶 gcd 𝐴) = ((𝐶 · 𝑥) + (𝐴 · 𝑦)) → (𝐴 ∥ (𝐵 · (𝐶 gcd 𝐴)) ↔ 𝐴 ∥ (𝐵 · ((𝐶 · 𝑥) + (𝐴 · 𝑦)))))
38	35, 37	syl5ibrcom 247	. . . 4 ⊢ ((((𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) ∧ 𝐴 ∥ (𝐵 · 𝐶)) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) → ((𝐶 gcd 𝐴) = ((𝐶 · 𝑥) + (𝐴 · 𝑦)) → 𝐴 ∥ (𝐵 · (𝐶 gcd 𝐴))))
39	38	rexlimdvva 3219	. . 3 ⊢ (((𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) ∧ 𝐴 ∥ (𝐵 · 𝐶)) → (∃𝑥 ∈ ℤ ∃𝑦 ∈ ℤ (𝐶 gcd 𝐴) = ((𝐶 · 𝑥) + (𝐴 · 𝑦)) → 𝐴 ∥ (𝐵 · (𝐶 gcd 𝐴))))
40	6, 39	mpd 15	. 2 ⊢ (((𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) ∧ 𝐴 ∥ (𝐵 · 𝐶)) → 𝐴 ∥ (𝐵 · (𝐶 gcd 𝐴)))
41		dvdszrcl 16307	. . . . 5 ⊢ (𝐴 ∥ (𝐵 · (𝐶 gcd 𝐴)) → (𝐴 ∈ ℤ ∧ (𝐵 · (𝐶 gcd 𝐴)) ∈ ℤ))
42	41	adantl 481	. . . 4 ⊢ (((𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) ∧ 𝐴 ∥ (𝐵 · (𝐶 gcd 𝐴))) → (𝐴 ∈ ℤ ∧ (𝐵 · (𝐶 gcd 𝐴)) ∈ ℤ))
43	42	simpld 494	. . 3 ⊢ (((𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) ∧ 𝐴 ∥ (𝐵 · (𝐶 gcd 𝐴))) → 𝐴 ∈ ℤ)
44	42	simprd 495	. . 3 ⊢ (((𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) ∧ 𝐴 ∥ (𝐵 · (𝐶 gcd 𝐴))) → (𝐵 · (𝐶 gcd 𝐴)) ∈ ℤ)
45		zmulcl 12692	. . . 4 ⊢ ((𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) → (𝐵 · 𝐶) ∈ ℤ)
46	45	adantr 480	. . 3 ⊢ (((𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) ∧ 𝐴 ∥ (𝐵 · (𝐶 gcd 𝐴))) → (𝐵 · 𝐶) ∈ ℤ)
47		simpr 484	. . 3 ⊢ (((𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) ∧ 𝐴 ∥ (𝐵 · (𝐶 gcd 𝐴))) → 𝐴 ∥ (𝐵 · (𝐶 gcd 𝐴)))
48		simplr 768	. . . . . 6 ⊢ (((𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) ∧ 𝐴 ∥ (𝐵 · (𝐶 gcd 𝐴))) → 𝐶 ∈ ℤ)
49		gcddvds 16549	. . . . . 6 ⊢ ((𝐶 ∈ ℤ ∧ 𝐴 ∈ ℤ) → ((𝐶 gcd 𝐴) ∥ 𝐶 ∧ (𝐶 gcd 𝐴) ∥ 𝐴))
50	48, 43, 49	syl2anc 583	. . . . 5 ⊢ (((𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) ∧ 𝐴 ∥ (𝐵 · (𝐶 gcd 𝐴))) → ((𝐶 gcd 𝐴) ∥ 𝐶 ∧ (𝐶 gcd 𝐴) ∥ 𝐴))
51	50	simpld 494	. . . 4 ⊢ (((𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) ∧ 𝐴 ∥ (𝐵 · (𝐶 gcd 𝐴))) → (𝐶 gcd 𝐴) ∥ 𝐶)
52	48, 43	gcdcld 16554	. . . . . 6 ⊢ (((𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) ∧ 𝐴 ∥ (𝐵 · (𝐶 gcd 𝐴))) → (𝐶 gcd 𝐴) ∈ ℕ₀)
53	52	nn0zd 12665	. . . . 5 ⊢ (((𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) ∧ 𝐴 ∥ (𝐵 · (𝐶 gcd 𝐴))) → (𝐶 gcd 𝐴) ∈ ℤ)
54		simpll 766	. . . . 5 ⊢ (((𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) ∧ 𝐴 ∥ (𝐵 · (𝐶 gcd 𝐴))) → 𝐵 ∈ ℤ)
55		dvdscmul 16331	. . . . 5 ⊢ (((𝐶 gcd 𝐴) ∈ ℤ ∧ 𝐶 ∈ ℤ ∧ 𝐵 ∈ ℤ) → ((𝐶 gcd 𝐴) ∥ 𝐶 → (𝐵 · (𝐶 gcd 𝐴)) ∥ (𝐵 · 𝐶)))
56	53, 48, 54, 55	syl3anc 1371	. . . 4 ⊢ (((𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) ∧ 𝐴 ∥ (𝐵 · (𝐶 gcd 𝐴))) → ((𝐶 gcd 𝐴) ∥ 𝐶 → (𝐵 · (𝐶 gcd 𝐴)) ∥ (𝐵 · 𝐶)))
57	51, 56	mpd 15	. . 3 ⊢ (((𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) ∧ 𝐴 ∥ (𝐵 · (𝐶 gcd 𝐴))) → (𝐵 · (𝐶 gcd 𝐴)) ∥ (𝐵 · 𝐶))
58	43, 44, 46, 47, 57	dvdstrd 16343	. 2 ⊢ (((𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) ∧ 𝐴 ∥ (𝐵 · (𝐶 gcd 𝐴))) → 𝐴 ∥ (𝐵 · 𝐶))
59	40, 58	impbida 800	1 ⊢ ((𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) → (𝐴 ∥ (𝐵 · 𝐶) ↔ 𝐴 ∥ (𝐵 · (𝐶 gcd 𝐴))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1537 ∈ wcel 2108 ∃wrex 3076 class class class wbr 5166 (class class class)co 7448 + caddc 11187 · cmul 11189 ℤcz 12639 ∥ cdvds 16302 gcd cgcd 16540

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770 ax-cnex 11240 ax-resscn 11241 ax-1cn 11242 ax-icn 11243 ax-addcl 11244 ax-addrcl 11245 ax-mulcl 11246 ax-mulrcl 11247 ax-mulcom 11248 ax-addass 11249 ax-mulass 11250 ax-distr 11251 ax-i2m1 11252 ax-1ne0 11253 ax-1rid 11254 ax-rnegex 11255 ax-rrecex 11256 ax-cnre 11257 ax-pre-lttri 11258 ax-pre-lttrn 11259 ax-pre-ltadd 11260 ax-pre-mulgt0 11261 ax-pre-sup 11262

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-rmo 3388 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-iun 5017 df-br 5167 df-opab 5229 df-mpt 5250 df-tr 5284 df-id 5593 df-eprel 5599 df-po 5607 df-so 5608 df-fr 5652 df-we 5654 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-pred 6332 df-ord 6398 df-on 6399 df-lim 6400 df-suc 6401 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-f1 6578 df-fo 6579 df-f1o 6580 df-fv 6581 df-riota 7404 df-ov 7451 df-oprab 7452 df-mpo 7453 df-om 7904 df-2nd 8031 df-frecs 8322 df-wrecs 8353 df-recs 8427 df-rdg 8466 df-er 8763 df-en 9004 df-dom 9005 df-sdom 9006 df-sup 9511 df-inf 9512 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330 df-sub 11522 df-neg 11523 df-div 11948 df-nn 12294 df-2 12356 df-3 12357 df-n0 12554 df-z 12640 df-uz 12904 df-rp 13058 df-fl 13843 df-mod 13921 df-seq 14053 df-exp 14113 df-cj 15148 df-re 15149 df-im 15150 df-sqrt 15284 df-abs 15285 df-dvds 16303 df-gcd 16541

This theorem is referenced by: coprmdvds 16700 odmulg 19598 fpprwpprb 47614

W3C validator