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 ) )