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Metamath Proof Explorer

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	$⊢ (A \in ℝ^{*} \land - A \in ℝ) \to A \in ℝ$

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