Theorem sqeq0

Description: A complex 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 ) )