Theorem mulne0b

Description: The product of two nonzero numbers is nonzero. (Contributed by NM, 1-Aug-2004) (Proof shortened by Andrew Salmon, 19-Nov-2011)

	Ref	Expression
Assertion	mulne0b	$$⊢\left(A\in ℂ\wedge B\in ℂ\right)\to \left(\left(A\ne 0\wedge B\ne 0\right)\leftrightarrow AB\ne 0\right)$$

Proof

Step	Hyp	Ref	Expression
1		neanior	$$⊢\left(A\ne 0\wedge B\ne 0\right)\leftrightarrow \neg \left(A=0\vee B=0\right)$$
2		mul0or	$$⊢\left(A\in ℂ\wedge B\in ℂ\right)\to \left(AB=0\leftrightarrow \left(A=0\vee B=0\right)\right)$$
3	2	necon3abid	$$⊢\left(A\in ℂ\wedge B\in ℂ\right)\to \left(AB\ne 0\leftrightarrow \neg \left(A=0\vee B=0\right)\right)$$
4	1 3	bitr4id	$$⊢\left(A\in ℂ\wedge B\in ℂ\right)\to \left(\left(A\ne 0\wedge B\ne 0\right)\leftrightarrow AB\ne 0\right)$$