Metamath Proof Explorer


Theorem sqeq0

Description: A number is zero iff its square is zero. (Contributed by NM, 11-Mar-2006)

Ref Expression
Assertion sqeq0 ( 𝐴 ∈ ℂ → ( ( 𝐴 ↑ 2 ) = 0 ↔ 𝐴 = 0 ) )

Proof

Step Hyp Ref Expression
1 2nn 2 ∈ ℕ
2 expeq0 ( ( 𝐴 ∈ ℂ ∧ 2 ∈ ℕ ) → ( ( 𝐴 ↑ 2 ) = 0 ↔ 𝐴 = 0 ) )
3 1 2 mpan2 ( 𝐴 ∈ ℂ → ( ( 𝐴 ↑ 2 ) = 0 ↔ 𝐴 = 0 ) )