Metamath Proof Explorer

Theorem crrei

Description: The real part of a complex number representation. Definition 10-3.1 of Gleason p. 132. (Contributed by NM, 10-May-1999)

Ref Expression
Hypotheses crre.1 
|- A e. RR
crre.2 
|- B e. RR
Assertion crrei 
|- ( Re ` ( A + ( _i x. B ) ) ) = A

Proof

Step Hyp Ref Expression
1 crre.1 
 |-  A e. RR
2 crre.2 
 |-  B e. RR
3 crre 
 |-  ( ( A e. RR /\ B e. RR ) -> ( Re ` ( A + ( _i x. B ) ) ) = A )
4 1 2 3 mp2an 
 |-  ( Re ` ( A + ( _i x. B ) ) ) = A