Metamath Proof Explorer

Theorem mulgt0

Description: The product of two positive numbers is positive. (Contributed by NM, 10-Mar-2008)

		Ref	Expression
	Assertion	mulgt0	$⊢ ((A \in ℝ \land 0 < A) \land (B \in ℝ \land 0 < B)) \to 0 < A B$

Step	Hyp	Ref	Expression
1		axmulgt0	$⊢ (A \in ℝ \land B \in ℝ) \to ((0 < A \land 0 < B) \to 0 < A B)$
2	1	imp	$⊢ ((A \in ℝ \land B \in ℝ) \land (0 < A \land 0 < B)) \to 0 < A B$
3	2	an4s	$⊢ ((A \in ℝ \land 0 < A) \land (B \in ℝ \land 0 < B)) \to 0 < A B$