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 )