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Definition df-gz 15418
Description: Define the set of gaussian integers, which are complex numbers whose real and imaginary parts are integers. (Note that the [i] is actually part of the symbol token and has no independent meaning.) (Contributed by Mario Carneiro, 14-Jul-2014.)
Assertion
Ref Expression
df-gz ℤ[i] = {𝑥 ∈ ℂ ∣ ((ℜ‘𝑥) ∈ ℤ ∧ (ℑ‘𝑥) ∈ ℤ)}

Detailed syntax breakdown of Definition df-gz
StepHypRef Expression
1 cgz 15417 . 2 class ℤ[i]
2 vx . . . . . . 7 setvar 𝑥
32cv 1473 . . . . . 6 class 𝑥
4 cre 13631 . . . . . 6 class
53, 4cfv 5790 . . . . 5 class (ℜ‘𝑥)
6 cz 11210 . . . . 5 class
75, 6wcel 1976 . . . 4 wff (ℜ‘𝑥) ∈ ℤ
8 cim 13632 . . . . . 6 class
93, 8cfv 5790 . . . . 5 class (ℑ‘𝑥)
109, 6wcel 1976 . . . 4 wff (ℑ‘𝑥) ∈ ℤ
117, 10wa 382 . . 3 wff ((ℜ‘𝑥) ∈ ℤ ∧ (ℑ‘𝑥) ∈ ℤ)
12 cc 9790 . . 3 class
1311, 2, 12crab 2899 . 2 class {𝑥 ∈ ℂ ∣ ((ℜ‘𝑥) ∈ ℤ ∧ (ℑ‘𝑥) ∈ ℤ)}
141, 13wceq 1474 1 wff ℤ[i] = {𝑥 ∈ ℂ ∣ ((ℜ‘𝑥) ∈ ℤ ∧ (ℑ‘𝑥) ∈ ℤ)}
Colors of variables: wff setvar class
This definition is referenced by:  elgz  15419
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