Theorem negreb

Description: The negative of a real is real. (Contributed by NM, 11-Aug-1999) (Revised by Mario Carneiro, 14-Jul-2014)

	Ref	Expression
Assertion	negreb	$$⊢A\in ℂ\to \left(-A\in ℝ\leftrightarrow A\in ℝ\right)$$

Proof

Step	Hyp	Ref	Expression
1		renegcl	$$⊢-A\in ℝ\to -\left(-A\right)\in ℝ$$
2		negneg	$$⊢A\in ℂ\to -\left(-A\right)=A$$
3	2	eleq1d	$$⊢A\in ℂ\to \left(-\left(-A\right)\in ℝ\leftrightarrow A\in ℝ\right)$$
4	1 3	syl5ib	$$⊢A\in ℂ\to \left(-A\in ℝ\to A\in ℝ\right)$$
5		renegcl	$$⊢A\in ℝ\to -A\in ℝ$$
6	4 5	impbid1	$$⊢A\in ℂ\to \left(-A\in ℝ\leftrightarrow A\in ℝ\right)$$