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Metamath Proof Explorer

Theorem mulne0

Description: The product of two nonzero numbers is nonzero. (Contributed by NM, 30-Dec-2007)

		Ref	Expression
	Assertion	mulne0	$⊢ ((A \in ℂ \land A \neq 0) \land (B \in ℂ \land B \neq 0)) \to A B \neq 0$

Step	Hyp	Ref	Expression
1		mulne0b	$⊢ (A \in ℂ \land B \in ℂ) \to ((A \neq 0 \land B \neq 0) \leftrightarrow A B \neq 0)$
2	1	biimpa	$⊢ ((A \in ℂ \land B \in ℂ) \land (A \neq 0 \land B \neq 0)) \to A B \neq 0$
3	2	an4s	$⊢ ((A \in ℂ \land A \neq 0) \land (B \in ℂ \land B \neq 0)) \to A B \neq 0$