Metamath Proof Explorer


Theorem gt0divd

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

Ref Expression
Hypotheses rpgecld.1 ( 𝜑𝐴 ∈ ℝ )
rpgecld.2 ( 𝜑𝐵 ∈ ℝ+ )
Assertion gt0divd ( 𝜑 → ( 0 < 𝐴 ↔ 0 < ( 𝐴 / 𝐵 ) ) )

Proof

Step Hyp Ref Expression
1 rpgecld.1 ( 𝜑𝐴 ∈ ℝ )
2 rpgecld.2 ( 𝜑𝐵 ∈ ℝ+ )
3 2 rpred ( 𝜑𝐵 ∈ ℝ )
4 2 rpgt0d ( 𝜑 → 0 < 𝐵 )
5 gt0div ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 0 < 𝐵 ) → ( 0 < 𝐴 ↔ 0 < ( 𝐴 / 𝐵 ) ) )
6 1 3 4 5 syl3anc ( 𝜑 → ( 0 < 𝐴 ↔ 0 < ( 𝐴 / 𝐵 ) ) )