Theorem divge0i

Description: The ratio of nonnegative and positive numbers is nonnegative. (Contributed by NM, 12-Aug-1999)

	Ref	Expression
Hypotheses	ltplus1.1	⊢ 𝐴 ∈ ℝ
	prodgt0.2	⊢ 𝐵 ∈ ℝ
Assertion	divge0i	⊢ ( ( 0 ≤ 𝐴 ∧ 0 < 𝐵 ) → 0 ≤ ( 𝐴 / 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		ltplus1.1	⊢ 𝐴 ∈ ℝ
2		prodgt0.2	⊢ 𝐵 ∈ ℝ
3		divge0	⊢ ( ( ( 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 ) )