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	$$⊢\phi \to A\in ℂ$$
Assertion	msq0d	$$⊢\phi \to \left(AA=0\leftrightarrow A=0\right)$$

Proof

Step	Hyp	Ref	Expression
1		msq0d.1	$$⊢\phi \to A\in ℂ$$
2		mul0or	$$⊢\left(A\in ℂ\wedge A\in ℂ\right)\to \left(AA=0\leftrightarrow \left(A=0\vee A=0\right)\right)$$
3	1 1 2	syl2anc	$$⊢\phi \to \left(AA=0\leftrightarrow \left(A=0\vee A=0\right)\right)$$
4		oridm	$$⊢\left(A=0\vee A=0\right)\leftrightarrow A=0$$
5	3 4	bitrdi	$$⊢\phi \to \left(AA=0\leftrightarrow A=0\right)$$