Theorem mul0ord

Description: If a product is zero, one of its factors must be zero. Theorem I.11 of Apostol p. 18. (Contributed by Mario Carneiro, 27-May-2016)

	Ref	Expression
Hypotheses	mul0ord.1	$$⊢\phi \to A\in ℂ$$
	mul0ord.2	$$⊢\phi \to B\in ℂ$$
Assertion	mul0ord	$$⊢\phi \to \left(AB=0\leftrightarrow \left(A=0\vee B=0\right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		mul0ord.1	$$⊢\phi \to A\in ℂ$$
2		mul0ord.2	$$⊢\phi \to B\in ℂ$$
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	syl2anc	$$⊢\phi \to \left(AB=0\leftrightarrow \left(A=0\vee B=0\right)\right)$$