Theorem msq0d

Description: A number is zero iff its square is zero (where square is represented using multiplication). (Contributed by Mario Carneiro, 27-May-2016)

	Ref	Expression
Hypothesis	msq0d.1	⊢ ( 𝜑 → 𝐴 ∈ ℂ )
Assertion	msq0d	⊢ ( 𝜑 → ( ( 𝐴 · 𝐴 ) = 0 ↔ 𝐴 = 0 ) )

Proof

Step	Hyp	Ref	Expression
1		msq0d.1	⊢ ( 𝜑 → 𝐴 ∈ ℂ )
2		mul0or	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ∈ ℂ ) → ( ( 𝐴 · 𝐴 ) = 0 ↔ ( 𝐴 = 0 ∨ 𝐴 = 0 ) ) )
3	1 1 2	syl2anc	⊢ ( 𝜑 → ( ( 𝐴 · 𝐴 ) = 0 ↔ ( 𝐴 = 0 ∨ 𝐴 = 0 ) ) )
4		oridm	⊢ ( ( 𝐴 = 0 ∨ 𝐴 = 0 ) ↔ 𝐴 = 0 )
5	3 4	bitrdi	⊢ ( 𝜑 → ( ( 𝐴 · 𝐴 ) = 0 ↔ 𝐴 = 0 ) )