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 ∈ ℝ
	prodgt0.2	⊢ 𝐵 ∈ ℝ
Assertion	prodgt0i	⊢ ( ( 0 ≤ 𝐴 ∧ 0 < ( 𝐴 · 𝐵 ) ) → 0 < 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		ltplus1.1	⊢ 𝐴 ∈ ℝ
2		prodgt0.2	⊢ 𝐵 ∈ ℝ
3		prodgt0	⊢ ( ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) ∧ ( 0 ≤ 𝐴 ∧ 0 < ( 𝐴 · 𝐵 ) ) ) → 0 < 𝐵 )
4	1 2 3	mpanl12	⊢ ( ( 0 ≤ 𝐴 ∧ 0 < ( 𝐴 · 𝐵 ) ) → 0 < 𝐵 )