Metamath Proof Explorer

Theorem cjre

Description: A real number equals its complex conjugate. Proposition 10-3.4(f) of Gleason p. 133. (Contributed by NM, 8-Oct-1999)

Ref Expression
Assertion cjre 
|- ( A e. RR -> ( * ` A ) = A )

Proof

Step Hyp Ref Expression
1 recn 
 |-  ( A e. RR -> A e. CC )
2 cjreb 
 |-  ( A e. CC -> ( A e. RR <-> ( * ` A ) = A ) )
3 2 biimpd 
 |-  ( A e. CC -> ( A e. RR -> ( * ` A ) = A ) )
4 1 3 mpcom 
 |-  ( A e. RR -> ( * ` A ) = A )