dvdsmulgcd - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > dvdsmulgcd		Structured version Visualization version GIF version

Theorem dvdsmulgcd 16514

Description: A divisibility equivalent for odmulg 19520. (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 769	. . . 4 ⊢ (((𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) ∧ 𝐴 ∥ (𝐵 · 𝐶)) → 𝐶 ∈ ℤ)
2		dvdszrcl 16215	. . . . . 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 16501	. . . 4 ⊢ ((𝐶 ∈ ℤ ∧ 𝐴 ∈ ℤ) → ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℤ (𝐶 gcd 𝐴) = ((𝐶 · 𝑥) + (𝐴 · 𝑦)))
6	1, 4, 5	syl2anc 585	. . 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 775	. . . . . . . 8 ⊢ ((((𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) ∧ 𝐴 ∥ (𝐵 · 𝐶)) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) → 𝐵 ∈ ℤ)
9		simpllr 776	. . . . . . . . 9 ⊢ ((((𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) ∧ 𝐴 ∥ (𝐵 · 𝐶)) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) → 𝐶 ∈ ℤ)
10		simprl 771	. . . . . . . . 9 ⊢ ((((𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) ∧ 𝐴 ∥ (𝐵 · 𝐶)) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) → 𝑥 ∈ ℤ)
11	9, 10	zmulcld 12628	. . . . . . . 8 ⊢ ((((𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) ∧ 𝐴 ∥ (𝐵 · 𝐶)) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) → (𝐶 · 𝑥) ∈ ℤ)
12	8, 11	zmulcld 12628	. . . . . . 7 ⊢ ((((𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) ∧ 𝐴 ∥ (𝐵 · 𝐶)) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) → (𝐵 · (𝐶 · 𝑥)) ∈ ℤ)
13		simprr 773	. . . . . . . . 9 ⊢ ((((𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) ∧ 𝐴 ∥ (𝐵 · 𝐶)) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) → 𝑦 ∈ ℤ)
14	7, 13	zmulcld 12628	. . . . . . . 8 ⊢ ((((𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) ∧ 𝐴 ∥ (𝐵 · 𝐶)) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) → (𝐴 · 𝑦) ∈ ℤ)
15	8, 14	zmulcld 12628	. . . . . . 7 ⊢ ((((𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) ∧ 𝐴 ∥ (𝐵 · 𝐶)) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) → (𝐵 · (𝐴 · 𝑦)) ∈ ℤ)
16	8, 9	zmulcld 12628	. . . . . . . . 9 ⊢ ((((𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) ∧ 𝐴 ∥ (𝐵 · 𝐶)) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) → (𝐵 · 𝐶) ∈ ℤ)
17		simplr 769	. . . . . . . . 9 ⊢ ((((𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) ∧ 𝐴 ∥ (𝐵 · 𝐶)) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) → 𝐴 ∥ (𝐵 · 𝐶))
18	7, 16, 10, 17	dvdsmultr1d 16255	. . . . . . . 8 ⊢ ((((𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) ∧ 𝐴 ∥ (𝐵 · 𝐶)) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) → 𝐴 ∥ ((𝐵 · 𝐶) · 𝑥))
19	8	zcnd 12623	. . . . . . . . 9 ⊢ ((((𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) ∧ 𝐴 ∥ (𝐵 · 𝐶)) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) → 𝐵 ∈ ℂ)
20	9	zcnd 12623	. . . . . . . . 9 ⊢ ((((𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) ∧ 𝐴 ∥ (𝐵 · 𝐶)) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) → 𝐶 ∈ ℂ)
21	10	zcnd 12623	. . . . . . . . 9 ⊢ ((((𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) ∧ 𝐴 ∥ (𝐵 · 𝐶)) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) → 𝑥 ∈ ℂ)
22	19, 20, 21	mulassd 11157	. . . . . . . 8 ⊢ ((((𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) ∧ 𝐴 ∥ (𝐵 · 𝐶)) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) → ((𝐵 · 𝐶) · 𝑥) = (𝐵 · (𝐶 · 𝑥)))
23	18, 22	breqtrd 5100	. . . . . . 7 ⊢ ((((𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) ∧ 𝐴 ∥ (𝐵 · 𝐶)) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) → 𝐴 ∥ (𝐵 · (𝐶 · 𝑥)))
24	8, 13	zmulcld 12628	. . . . . . . . 9 ⊢ ((((𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) ∧ 𝐴 ∥ (𝐵 · 𝐶)) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) → (𝐵 · 𝑦) ∈ ℤ)
25		dvdsmul1 16235	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℤ ∧ (𝐵 · 𝑦) ∈ ℤ) → 𝐴 ∥ (𝐴 · (𝐵 · 𝑦)))
26	7, 24, 25	syl2anc 585	. . . . . . . 8 ⊢ ((((𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) ∧ 𝐴 ∥ (𝐵 · 𝐶)) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) → 𝐴 ∥ (𝐴 · (𝐵 · 𝑦)))
27	7	zcnd 12623	. . . . . . . . 9 ⊢ ((((𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) ∧ 𝐴 ∥ (𝐵 · 𝐶)) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) → 𝐴 ∈ ℂ)
28	13	zcnd 12623	. . . . . . . . 9 ⊢ ((((𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) ∧ 𝐴 ∥ (𝐵 · 𝐶)) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) → 𝑦 ∈ ℂ)
29	19, 27, 28	mul12d 11344	. . . . . . . 8 ⊢ ((((𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) ∧ 𝐴 ∥ (𝐵 · 𝐶)) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) → (𝐵 · (𝐴 · 𝑦)) = (𝐴 · (𝐵 · 𝑦)))
30	26, 29	breqtrrd 5102	. . . . . . 7 ⊢ ((((𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) ∧ 𝐴 ∥ (𝐵 · 𝐶)) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) → 𝐴 ∥ (𝐵 · (𝐴 · 𝑦)))
31	7, 12, 15, 23, 30	dvds2addd 16250	. . . . . 6 ⊢ ((((𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) ∧ 𝐴 ∥ (𝐵 · 𝐶)) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) → 𝐴 ∥ ((𝐵 · (𝐶 · 𝑥)) + (𝐵 · (𝐴 · 𝑦))))
32	11	zcnd 12623	. . . . . . 7 ⊢ ((((𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) ∧ 𝐴 ∥ (𝐵 · 𝐶)) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) → (𝐶 · 𝑥) ∈ ℂ)
33	14	zcnd 12623	. . . . . . 7 ⊢ ((((𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) ∧ 𝐴 ∥ (𝐵 · 𝐶)) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) → (𝐴 · 𝑦) ∈ ℂ)
34	19, 32, 33	adddid 11158	. . . . . 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 5102	. . . . 5 ⊢ ((((𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) ∧ 𝐴 ∥ (𝐵 · 𝐶)) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) → 𝐴 ∥ (𝐵 · ((𝐶 · 𝑥) + (𝐴 · 𝑦))))
36		oveq2 7364	. . . . . 6 ⊢ ((𝐶 gcd 𝐴) = ((𝐶 · 𝑥) + (𝐴 · 𝑦)) → (𝐵 · (𝐶 gcd 𝐴)) = (𝐵 · ((𝐶 · 𝑥) + (𝐴 · 𝑦))))
37	36	breq2d 5086	. . . . 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 3192	. . 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 16215	. . . . 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 12565	. . . 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 769	. . . . . 6 ⊢ (((𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) ∧ 𝐴 ∥ (𝐵 · (𝐶 gcd 𝐴))) → 𝐶 ∈ ℤ)
49		gcddvds 16461	. . . . . 6 ⊢ ((𝐶 ∈ ℤ ∧ 𝐴 ∈ ℤ) → ((𝐶 gcd 𝐴) ∥ 𝐶 ∧ (𝐶 gcd 𝐴) ∥ 𝐴))
50	48, 43, 49	syl2anc 585	. . . . 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 16466	. . . . . 6 ⊢ (((𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) ∧ 𝐴 ∥ (𝐵 · (𝐶 gcd 𝐴))) → (𝐶 gcd 𝐴) ∈ ℕ₀)
53	52	nn0zd 12538	. . . . 5 ⊢ (((𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) ∧ 𝐴 ∥ (𝐵 · (𝐶 gcd 𝐴))) → (𝐶 gcd 𝐴) ∈ ℤ)
54		simpll 767	. . . . 5 ⊢ (((𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) ∧ 𝐴 ∥ (𝐵 · (𝐶 gcd 𝐴))) → 𝐵 ∈ ℤ)
55		dvdscmul 16240	. . . . 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 1374	. . . 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 16253	. 2 ⊢ (((𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) ∧ 𝐴 ∥ (𝐵 · (𝐶 gcd 𝐴))) → 𝐴 ∥ (𝐵 · 𝐶))
59	40, 58	impbida 801	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 1542 ∈ wcel 2114 ∃wrex 3059 class class class wbr 5074 (class class class)co 7356 + caddc 11030 · cmul 11032 ℤcz 12513 ∥ cdvds 16210 gcd cgcd 16452

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2184 ax-ext 2707 ax-sep 5220 ax-nul 5230 ax-pow 5296 ax-pr 5364 ax-un 7678 ax-cnex 11083 ax-resscn 11084 ax-1cn 11085 ax-icn 11086 ax-addcl 11087 ax-addrcl 11088 ax-mulcl 11089 ax-mulrcl 11090 ax-mulcom 11091 ax-addass 11092 ax-mulass 11093 ax-distr 11094 ax-i2m1 11095 ax-1ne0 11096 ax-1rid 11097 ax-rnegex 11098 ax-rrecex 11099 ax-cnre 11100 ax-pre-lttri 11101 ax-pre-lttrn 11102 ax-pre-ltadd 11103 ax-pre-mulgt0 11104 ax-pre-sup 11105

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2538 df-eu 2568 df-clab 2714 df-cleq 2727 df-clel 2810 df-nfc 2884 df-ne 2931 df-nel 3035 df-ral 3050 df-rex 3060 df-rmo 3340 df-reu 3341 df-rab 3388 df-v 3429 df-sbc 3726 df-csb 3834 df-dif 3888 df-un 3890 df-in 3892 df-ss 3902 df-pss 3905 df-nul 4264 df-if 4457 df-pw 4533 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4841 df-iun 4925 df-br 5075 df-opab 5137 df-mpt 5156 df-tr 5182 df-id 5515 df-eprel 5520 df-po 5528 df-so 5529 df-fr 5573 df-we 5575 df-xp 5626 df-rel 5627 df-cnv 5628 df-co 5629 df-dm 5630 df-rn 5631 df-res 5632 df-ima 5633 df-pred 6254 df-ord 6315 df-on 6316 df-lim 6317 df-suc 6318 df-iota 6443 df-fun 6489 df-fn 6490 df-f 6491 df-f1 6492 df-fo 6493 df-f1o 6494 df-fv 6495 df-riota 7313 df-ov 7359 df-oprab 7360 df-mpo 7361 df-om 7807 df-2nd 7932 df-frecs 8220 df-wrecs 8251 df-recs 8300 df-rdg 8338 df-er 8632 df-en 8883 df-dom 8884 df-sdom 8885 df-sup 9344 df-inf 9345 df-pnf 11170 df-mnf 11171 df-xr 11172 df-ltxr 11173 df-le 11174 df-sub 11368 df-neg 11369 df-div 11797 df-nn 12164 df-2 12233 df-3 12234 df-n0 12427 df-z 12514 df-uz 12778 df-rp 12932 df-fl 13740 df-mod 13818 df-seq 13953 df-exp 14013 df-cj 15050 df-re 15051 df-im 15052 df-sqrt 15186 df-abs 15187 df-dvds 16211 df-gcd 16453

This theorem is referenced by: coprmdvds 16611 odmulg 19520 fpprwpprb 48204

W3C validator