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Metamath Proof Explorer

Theorem mulgt0ii

Description: The product of two positive numbers is positive. (Contributed by NM, 18-May-1999)

Ref Expression
Hypotheses lt.1 ⊢ 𝐴 ∈ ℝ
lt.2 ⊢ 𝐵 ∈ ℝ
mulgt0i.3 ⊢ 0 < 𝐴
mulgt0i.4 ⊢ 0 < 𝐵
Assertion mulgt0ii 0 < ( 𝐴 · 𝐵 )

Proof

Step Hyp Ref Expression
1 lt.1 ⊢ 𝐴 ∈ ℝ
2 lt.2 ⊢ 𝐵 ∈ ℝ
3 mulgt0i.3 ⊢ 0 < 𝐴
4 mulgt0i.4 ⊢ 0 < 𝐵
5 1 2 mulgt0i ⊢ ( ( 0 < 𝐴 ∧ 0 < 𝐵 ) → 0 < ( 𝐴 · 𝐵 ) )
6 3 4 5 mp2an ⊢ 0 < ( 𝐴 · 𝐵 )