Theorem rerebd

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

	Ref	Expression
Hypotheses	recld.1	|- ( ph -> A e. CC )
	rerebd.2	|- ( ph -> ( Re ` A ) = A )
Assertion	rerebd	|- ( ph -> A e. RR )

Proof

Step	Hyp	Ref	Expression
1		recld.1	|- ( ph -> A e. CC )
2		rerebd.2	|- ( ph -> ( Re ` A ) = A )
3		rereb	|- ( A e. CC -> ( A e. RR <-> ( Re ` A ) = A ) )
4	1 3	syl	|- ( ph -> ( A e. RR <-> ( Re ` A ) = A ) )
5	2 4	mpbird	|- ( ph -> A e. RR )