Theorem divgt0i

Description: The ratio of two positive numbers is positive. (Contributed by NM, 16-May-1999)

	Ref	Expression
Hypotheses	ltplus1.1	⊢ 𝐴 ∈ ℝ
	prodgt0.2	⊢ 𝐵 ∈ ℝ
Assertion	divgt0i	⊢ ( ( 0 < 𝐴 ∧ 0 < 𝐵 ) → 0 < ( 𝐴 / 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		ltplus1.1	⊢ 𝐴 ∈ ℝ
2		prodgt0.2	⊢ 𝐵 ∈ ℝ
3		divgt0	⊢ ( ( ( 𝐴 ∈ ℝ ∧ 0 < 𝐴 ) ∧ ( 𝐵 ∈ ℝ ∧ 0 < 𝐵 ) ) → 0 < ( 𝐴 / 𝐵 ) )
4	2 3	mpanr1	⊢ ( ( ( 𝐴 ∈ ℝ ∧ 0 < 𝐴 ) ∧ 0 < 𝐵 ) → 0 < ( 𝐴 / 𝐵 ) )
5	1 4	mpanl1	⊢ ( ( 0 < 𝐴 ∧ 0 < 𝐵 ) → 0 < ( 𝐴 / 𝐵 ) )