ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  df-dvds GIF version

Definition df-dvds 10109
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 10108 . 2 class
2 vx . . . . . . 7 setvar 𝑥
32cv 1258 . . . . . 6 class 𝑥
4 cz 8302 . . . . . 6 class
53, 4wcel 1409 . . . . 5 wff 𝑥 ∈ ℤ
6 vy . . . . . . 7 setvar 𝑦
76cv 1258 . . . . . 6 class 𝑦
87, 4wcel 1409 . . . . 5 wff 𝑦 ∈ ℤ
95, 8wa 101 . . . 4 wff (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)
10 vn . . . . . . . 8 setvar 𝑛
1110cv 1258 . . . . . . 7 class 𝑛
12 cmul 6952 . . . . . . 7 class ·
1311, 3, 12co 5540 . . . . . 6 class (𝑛 · 𝑥)
1413, 7wceq 1259 . . . . 5 wff (𝑛 · 𝑥) = 𝑦
1514, 10, 4wrex 2324 . . . 4 wff 𝑛 ∈ ℤ (𝑛 · 𝑥) = 𝑦
169, 15wa 101 . . 3 wff ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ ∃𝑛 ∈ ℤ (𝑛 · 𝑥) = 𝑦)
1716, 2, 6copab 3845 . 2 class {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ ∃𝑛 ∈ ℤ (𝑛 · 𝑥) = 𝑦)}
181, 17wceq 1259 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  10110  dvdszrcl  10113
  Copyright terms: Public domain W3C validator