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Metamath Proof Explorer

Theorem halfnneg2

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

		Ref	Expression
	Assertion	halfnneg2	$⊢ A \in ℝ \to (0 \leq A \leftrightarrow 0 \leq \frac{A}{2})$

Step	Hyp	Ref	Expression
1		2re	$⊢ 2 \in ℝ$
2		2pos	$⊢ 0 < 2$
3		ge0div	$⊢ (A \in ℝ \land 2 \in ℝ \land 0 < 2) \to (0 \leq A \leftrightarrow 0 \leq \frac{A}{2})$
4	1 2 3	mp3an23	$⊢ A \in ℝ \to (0 \leq A \leftrightarrow 0 \leq \frac{A}{2})$