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Definition df-dvds 10722
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
StepHypRef Expression
1 cdvds 10721 . 2 class
2 vx . . . . . . 7 setvar 𝑥
32cv 1286 . . . . . 6 class 𝑥
4 cz 8686 . . . . . 6 class
53, 4wcel 1436 . . . . 5 wff 𝑥 ∈ ℤ
6 vy . . . . . . 7 setvar 𝑦
76cv 1286 . . . . . 6 class 𝑦
87, 4wcel 1436 . . . . 5 wff 𝑦 ∈ ℤ
95, 8wa 102 . . . 4 wff (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)
10 vn . . . . . . . 8 setvar 𝑛
1110cv 1286 . . . . . . 7 class 𝑛
12 cmul 7302 . . . . . . 7 class ·
1311, 3, 12co 5615 . . . . . 6 class (𝑛 · 𝑥)
1413, 7wceq 1287 . . . . 5 wff (𝑛 · 𝑥) = 𝑦
1514, 10, 4wrex 2356 . . . 4 wff 𝑛 ∈ ℤ (𝑛 · 𝑥) = 𝑦
169, 15wa 102 . . 3 wff ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ ∃𝑛 ∈ ℤ (𝑛 · 𝑥) = 𝑦)
1716, 2, 6copab 3875 . 2 class {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ ∃𝑛 ∈ ℤ (𝑛 · 𝑥) = 𝑦)}
181, 17wceq 1287 1 wff ∥ = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ ∃𝑛 ∈ ℤ (𝑛 · 𝑥) = 𝑦)}
Colors of variables: wff set class
This definition is referenced by:  divides  10723  dvdszrcl  10726
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