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Metamath Proof Explorer

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 ) )