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Metamath Proof Explorer

Theorem mulgt0i

Description: The product of two positive numbers is positive. (Contributed by NM, 16-May-1999)

Ref Expression
Hypotheses lt.1 ⊢ 𝐴 ∈ ℝ
lt.2 ⊢ 𝐵 ∈ ℝ
Assertion mulgt0i ( ( 0 < 𝐴 ∧ 0 < 𝐵 ) → 0 < ( 𝐴 · 𝐵 ) )

Proof

Step Hyp Ref Expression
1 lt.1 ⊢ 𝐴 ∈ ℝ
2 lt.2 ⊢ 𝐵 ∈ ℝ
3 axmulgt0 ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( ( 0 < 𝐴 ∧ 0 < 𝐵 ) → 0 < ( 𝐴 · 𝐵 ) ) )
4 1 2 3 mp2an ⊢ ( ( 0 < 𝐴 ∧ 0 < 𝐵 ) → 0 < ( 𝐴 · 𝐵 ) )