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Metamath Proof Explorer

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 ) ) )