Theorem xnegrecl2

Description: If the extended real negative is real, then the number itself is real. (Contributed by Glauco Siliprandi, 2-Jan-2022)

	Ref	Expression
Assertion	xnegrecl2	$$⊢\left(A\in {ℝ}^{*}\wedge -A\in ℝ\right)\to A\in ℝ$$

Proof

Step	Hyp	Ref	Expression
1		xnegneg	$$⊢A\in {ℝ}^{*}\to -\left(-A\right)=A$$
2	1	adantr	$$⊢\left(A\in {ℝ}^{*}\wedge -A\in ℝ\right)\to -\left(-A\right)=A$$
3		xnegrecl	$$⊢-A\in ℝ\to -\left(-A\right)\in ℝ$$
4	3	adantl	$$⊢\left(A\in {ℝ}^{*}\wedge -A\in ℝ\right)\to -\left(-A\right)\in ℝ$$
5	2 4	eqeltrrd	$$⊢\left(A\in {ℝ}^{*}\wedge -A\in ℝ\right)\to A\in ℝ$$