Theorem mul0ori

Description: If a product is zero, one of its factors must be zero. Theorem I.11 of Apostol p. 18. (Contributed by NM, 7-Oct-1999)

	Ref	Expression
Hypotheses	mul0or.1	$$⊢A\in ℂ$$
	mul0or.2	$$⊢B\in ℂ$$
Assertion	mul0ori	$$⊢AB=0\leftrightarrow \left(A=0\vee B=0\right)$$

Proof

Step	Hyp	Ref	Expression
1		mul0or.1	$$⊢A\in ℂ$$
2		mul0or.2	$$⊢B\in ℂ$$
3		mul0or	$$⊢\left(A\in ℂ\wedge B\in ℂ\right)\to \left(AB=0\leftrightarrow \left(A=0\vee B=0\right)\right)$$
4	1 2 3	mp2an	$$⊢AB=0\leftrightarrow \left(A=0\vee B=0\right)$$