Metamath Proof Explorer


Theorem rered

Description: A real number equals its real part. One direction of Proposition 10-3.4(f) of Gleason p. 133. (Contributed by Mario Carneiro, 29-May-2016)

Ref Expression
Hypothesis crred.1
|- ( ph -> A e. RR )
Assertion rered
|- ( ph -> ( Re ` A ) = A )

Proof

Step Hyp Ref Expression
1 crred.1
 |-  ( ph -> A e. RR )
2 rere
 |-  ( A e. RR -> ( Re ` A ) = A )
3 1 2 syl
 |-  ( ph -> ( Re ` A ) = A )