Theorem divge0

Description: The ratio of nonnegative and positive numbers is nonnegative. (Contributed by NM, 27-Sep-1999)

	Ref	Expression
Assertion	divge0	⊢ ( ( ( 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ) ∧ ( 𝐵 ∈ ℝ ∧ 0 < 𝐵 ) ) → 0 ≤ ( 𝐴 / 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		ge0div	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 0 < 𝐵 ) → ( 0 ≤ 𝐴 ↔ 0 ≤ ( 𝐴 / 𝐵 ) ) )
2	1	biimpd	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 0 < 𝐵 ) → ( 0 ≤ 𝐴 → 0 ≤ ( 𝐴 / 𝐵 ) ) )
3	2	3exp	⊢ ( 𝐴 ∈ ℝ → ( 𝐵 ∈ ℝ → ( 0 < 𝐵 → ( 0 ≤ 𝐴 → 0 ≤ ( 𝐴 / 𝐵 ) ) ) ) )
4	3	com34	⊢ ( 𝐴 ∈ ℝ → ( 𝐵 ∈ ℝ → ( 0 ≤ 𝐴 → ( 0 < 𝐵 → 0 ≤ ( 𝐴 / 𝐵 ) ) ) ) )
5	4	com23	⊢ ( 𝐴 ∈ ℝ → ( 0 ≤ 𝐴 → ( 𝐵 ∈ ℝ → ( 0 < 𝐵 → 0 ≤ ( 𝐴 / 𝐵 ) ) ) ) )
6	5	imp43	⊢ ( ( ( 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ) ∧ ( 𝐵 ∈ ℝ ∧ 0 < 𝐵 ) ) → 0 ≤ ( 𝐴 / 𝐵 ) )