Theorem prodgt0i

Description: Infer that a multiplicand is positive from a nonnegative multiplier and positive product. (Contributed by NM, 15-May-1999)

	Ref	Expression
Hypotheses	ltplus1.1	$$⊢A\in ℝ$$
	prodgt0.2	$$⊢B\in ℝ$$
Assertion	prodgt0i	$$⊢\left(0\le A\wedge 0<AB\right)\to 0<B$$

Proof

Step	Hyp	Ref	Expression
1		ltplus1.1	$$⊢A\in ℝ$$
2		prodgt0.2	$$⊢B\in ℝ$$
3		prodgt0	$$⊢\left(\left(A\in ℝ\wedge B\in ℝ\right)\wedge \left(0\le A\wedge 0<AB\right)\right)\to 0<B$$
4	1 2 3	mpanl12	$$⊢\left(0\le A\wedge 0<AB\right)\to 0<B$$