Metamath Proof Explorer


Theorem negrebd

Description: The negative of a real is real. (Contributed by Mario Carneiro, 28-May-2016)

Ref Expression
Hypotheses negidd.1
|- ( ph -> A e. CC )
negrebd.2
|- ( ph -> -u A e. RR )
Assertion negrebd
|- ( ph -> A e. RR )

Proof

Step Hyp Ref Expression
1 negidd.1
 |-  ( ph -> A e. CC )
2 negrebd.2
 |-  ( ph -> -u A e. RR )
3 negreb
 |-  ( A e. CC -> ( -u A e. RR <-> A e. RR ) )
4 1 3 syl
 |-  ( ph -> ( -u A e. RR <-> A e. RR ) )
5 2 4 mpbid
 |-  ( ph -> A e. RR )