Metamath Proof Explorer


Theorem halfpos2

Description: A number is positive iff its half is positive. (Contributed by NM, 10-Apr-2005)

Ref Expression
Assertion halfpos2 ( 𝐴 ∈ ℝ → ( 0 < 𝐴 ↔ 0 < ( 𝐴 / 2 ) ) )

Proof

Step Hyp Ref Expression
1 2re 2 ∈ ℝ
2 2pos 0 < 2
3 gt0div ( ( 𝐴 ∈ ℝ ∧ 2 ∈ ℝ ∧ 0 < 2 ) → ( 0 < 𝐴 ↔ 0 < ( 𝐴 / 2 ) ) )
4 1 2 3 mp3an23 ( 𝐴 ∈ ℝ → ( 0 < 𝐴 ↔ 0 < ( 𝐴 / 2 ) ) )