Definition df-dvds 16064

Description: Define the divides relation, see definition in [ApostolNT] p. 14. (Contributed by Paul Chapman, 21-Mar-2011.)

Assertion

Ref	Expression
df-dvds	⊢ ∥ = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ ∃𝑛 ∈ ℤ (𝑛 · 𝑥) = 𝑦)}

Distinct variable group:   𝑥,𝑛,𝑦

Detailed syntax breakdown of Definition df-dvds

Step	Hyp	Ref	Expression
1		cdvds 16063	. 2 class ∥
2		vx	. . . . . . 7 setvar 𝑥
3	2	cv 1539	. . . . . 6 class 𝑥
4		cz 12421	. . . . . 6 class ℤ
5	3, 4	wcel 2105	. . . . 5 wff 𝑥 ∈ ℤ
6		vy	. . . . . . 7 setvar 𝑦
7	6	cv 1539	. . . . . 6 class 𝑦
8	7, 4	wcel 2105	. . . . 5 wff 𝑦 ∈ ℤ
9	5, 8	wa 396	. . . 4 wff (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)
10		vn	. . . . . . . 8 setvar 𝑛
11	10	cv 1539	. . . . . . 7 class 𝑛
12		cmul 10978	. . . . . . 7 class ·
13	11, 3, 12	co 7338	. . . . . 6 class (𝑛 · 𝑥)
14	13, 7	wceq 1540	. . . . 5 wff (𝑛 · 𝑥) = 𝑦
15	14, 10, 4	wrex 3070	. . . 4 wff ∃𝑛 ∈ ℤ (𝑛 · 𝑥) = 𝑦
16	9, 15	wa 396	. . 3 wff ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ ∃𝑛 ∈ ℤ (𝑛 · 𝑥) = 𝑦)
17	16, 2, 6	copab 5155	. 2 class {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ ∃𝑛 ∈ ℤ (𝑛 · 𝑥) = 𝑦)}
18	1, 17	wceq 1540	1 wff ∥ = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ ∃𝑛 ∈ ℤ (𝑛 · 𝑥) = 𝑦)}

This definition is referenced by:  divides  16065  dvdszrcl  16068  dvdsrzring  20790  reldvds  42306