Theorem ge0divd

Description: Division of a nonnegative number by a positive number. (Contributed by Mario Carneiro, 28-May-2016)

	Ref	Expression
Hypotheses	rpgecld.1	⊢ ( 𝜑 → 𝐴 ∈ ℝ )
	rpgecld.2	⊢ ( 𝜑 → 𝐵 ∈ ℝ+ )
Assertion	ge0divd	⊢ ( 𝜑 → ( 0 ≤ 𝐴 ↔ 0 ≤ ( 𝐴 / 𝐵 ) ) )

Proof

Step	Hyp	Ref	Expression
1		rpgecld.1	⊢ ( 𝜑 → 𝐴 ∈ ℝ )
2		rpgecld.2	⊢ ( 𝜑 → 𝐵 ∈ ℝ+ )
3	2	rpred	⊢ ( 𝜑 → 𝐵 ∈ ℝ )
4	2	rpgt0d	⊢ ( 𝜑 → 0 < 𝐵 )
5		ge0div	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 0 < 𝐵 ) → ( 0 ≤ 𝐴 ↔ 0 ≤ ( 𝐴 / 𝐵 ) ) )
6	1 3 4 5	syl3anc	⊢ ( 𝜑 → ( 0 ≤ 𝐴 ↔ 0 ≤ ( 𝐴 / 𝐵 ) ) )