Theorem xnegrecl2d

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

	Ref	Expression
Hypotheses	xnegrecl2d.1	|- ( ph -> A e. RR* )
	xnegrecl2d.2	|- ( ph -> -e A e. RR )
Assertion	xnegrecl2d	|- ( ph -> A e. RR )

Proof

Step	Hyp	Ref	Expression
1		xnegrecl2d.1	|- ( ph -> A e. RR* )
2		xnegrecl2d.2	|- ( ph -> -e A e. RR )
3		xnegrecl2	|- ( ( A e. RR* /\ -e A e. RR ) -> A e. RR )
4	1 2 3	syl2anc	|- ( ph -> A e. RR )