Theorem divge0d

Description: The ratio of nonnegative and positive numbers is nonnegative. (Contributed by Mario Carneiro, 28-May-2016)

	Ref	Expression
Hypotheses	rpgecld.1	⊢ ( 𝜑 → 𝐴 ∈ ℝ )
	rpgecld.2	⊢ ( 𝜑 → 𝐵 ∈ ℝ+ )
	divge0d.3	⊢ ( 𝜑 → 0 ≤ 𝐴 )
Assertion	divge0d	⊢ ( 𝜑 → 0 ≤ ( 𝐴 / 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		rpgecld.1	⊢ ( 𝜑 → 𝐴 ∈ ℝ )
2		rpgecld.2	⊢ ( 𝜑 → 𝐵 ∈ ℝ+ )
3		divge0d.3	⊢ ( 𝜑 → 0 ≤ 𝐴 )
4	2	rpregt0d	⊢ ( 𝜑 → ( 𝐵 ∈ ℝ ∧ 0 < 𝐵 ) )
5		divge0	⊢ ( ( ( 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ) ∧ ( 𝐵 ∈ ℝ ∧ 0 < 𝐵 ) ) → 0 ≤ ( 𝐴 / 𝐵 ) )
6	1 3 4 5	syl21anc	⊢ ( 𝜑 → 0 ≤ ( 𝐴 / 𝐵 ) )