Theorem halfnneg2

Description: A number is nonnegative iff its half is nonnegative. (Contributed by NM, 9-Dec-2005)

	Ref	Expression
Assertion	halfnneg2	⊢ ( 𝐴 ∈ ℝ → ( 0 ≤ 𝐴 ↔ 0 ≤ ( 𝐴 / 2 ) ) )

Proof

Step	Hyp	Ref	Expression
1		2re	⊢ 2 ∈ ℝ
2		2pos	⊢ 0 < 2
3		ge0div	⊢ ( ( 𝐴 ∈ ℝ ∧ 2 ∈ ℝ ∧ 0 < 2 ) → ( 0 ≤ 𝐴 ↔ 0 ≤ ( 𝐴 / 2 ) ) )
4	1 2 3	mp3an23	⊢ ( 𝐴 ∈ ℝ → ( 0 ≤ 𝐴 ↔ 0 ≤ ( 𝐴 / 2 ) ) )