Theorem divgt0ii

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

	Ref	Expression
Hypotheses	ltplus1.1	⊢ 𝐴 ∈ ℝ
	prodgt0.2	⊢ 𝐵 ∈ ℝ
	ltreci.3	⊢ 0 < 𝐴
	ltreci.4	⊢ 0 < 𝐵
Assertion	divgt0ii	⊢ 0 < ( 𝐴 / 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		ltplus1.1	⊢ 𝐴 ∈ ℝ
2		prodgt0.2	⊢ 𝐵 ∈ ℝ
3		ltreci.3	⊢ 0 < 𝐴
4		ltreci.4	⊢ 0 < 𝐵
5	1 2 4	divgt0i2i	⊢ ( 0 < 𝐴 → 0 < ( 𝐴 / 𝐵 ) )
6	3 5	ax-mp	⊢ 0 < ( 𝐴 / 𝐵 )