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 )