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

Definition df-dvds 15028
 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 15027 . 2 class
2 vx . . . . . . 7 setvar 𝑥
32cv 1522 . . . . . 6 class 𝑥
4 cz 11415 . . . . . 6 class
53, 4wcel 2030 . . . . 5 wff 𝑥 ∈ ℤ
6 vy . . . . . . 7 setvar 𝑦
76cv 1522 . . . . . 6 class 𝑦
87, 4wcel 2030 . . . . 5 wff 𝑦 ∈ ℤ
95, 8wa 383 . . . 4 wff (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)
10 vn . . . . . . . 8 setvar 𝑛
1110cv 1522 . . . . . . 7 class 𝑛
12 cmul 9979 . . . . . . 7 class ·
1311, 3, 12co 6690 . . . . . 6 class (𝑛 · 𝑥)
1413, 7wceq 1523 . . . . 5 wff (𝑛 · 𝑥) = 𝑦
1514, 10, 4wrex 2942 . . . 4 wff 𝑛 ∈ ℤ (𝑛 · 𝑥) = 𝑦
169, 15wa 383 . . 3 wff ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ ∃𝑛 ∈ ℤ (𝑛 · 𝑥) = 𝑦)
1716, 2, 6copab 4745 . 2 class {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ ∃𝑛 ∈ ℤ (𝑛 · 𝑥) = 𝑦)}
181, 17wceq 1523 1 wff ∥ = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ ∃𝑛 ∈ ℤ (𝑛 · 𝑥) = 𝑦)}
 Colors of variables: wff setvar class This definition is referenced by:  divides  15029  dvdszrcl  15032  dvdsrzring  19879  reldvds  38831
 Copyright terms: Public domain W3C validator