Theorem msq0i

Description: A number is zero iff its square is zero (where square is represented using multiplication). (Contributed by NM, 28-Jul-1999)

	Ref	Expression
Hypothesis	mul0or.1	$$⊢A\in ℂ$$
Assertion	msq0i	$$⊢AA=0\leftrightarrow A=0$$

Proof

Step	Hyp	Ref	Expression
1		mul0or.1	$$⊢A\in ℂ$$
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	mp2an	$$⊢AA=0\leftrightarrow \left(A=0\vee A=0\right)$$
4		oridm	$$⊢\left(A=0\vee A=0\right)\leftrightarrow A=0$$
5	3 4	bitri	$$⊢AA=0\leftrightarrow A=0$$