Metamath Proof Explorer


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 )