Metamath Proof Explorer


Theorem cjrebi

Description: A number is real iff it equals its complex conjugate. Proposition 10-3.4(f) of Gleason p. 133. (Contributed by NM, 11-Oct-1999)

Ref Expression
Hypothesis recl.1
|- A e. CC
Assertion cjrebi
|- ( A e. RR <-> ( * ` A ) = A )

Proof

Step Hyp Ref Expression
1 recl.1
 |-  A e. CC
2 cjreb
 |-  ( A e. CC -> ( A e. RR <-> ( * ` A ) = A ) )
3 1 2 ax-mp
 |-  ( A e. RR <-> ( * ` A ) = A )