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 e. RR* /\ -e A e. RR ) -> A e. RR ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xnegneg | |- ( A e. RR* -> -e -e A = A ) |
|
2 | 1 | adantr | |- ( ( A e. RR* /\ -e A e. RR ) -> -e -e A = A ) |
3 | xnegrecl | |- ( -e A e. RR -> -e -e A e. RR ) |
|
4 | 3 | adantl | |- ( ( A e. RR* /\ -e A e. RR ) -> -e -e A e. RR ) |
5 | 2 4 | eqeltrrd | |- ( ( A e. RR* /\ -e A e. RR ) -> A e. RR ) |