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Metamath Proof Explorer

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 )