Metamath Proof Explorer


Theorem xnegred

Description: An extended real is real if and only if its extended negative is real. (Contributed by Glauco Siliprandi, 2-Jan-2022)

Ref Expression
Hypothesis xnegred.1
|- ( ph -> A e. RR* )
Assertion xnegred
|- ( ph -> ( A e. RR <-> -e A e. RR ) )

Proof

Step Hyp Ref Expression
1 xnegred.1
 |-  ( ph -> A e. RR* )
2 xnegre
 |-  ( A e. RR* -> ( A e. RR <-> -e A e. RR ) )
3 1 2 syl
 |-  ( ph -> ( A e. RR <-> -e A e. RR ) )