Theorem crred

Description: The real part of a complex number representation. Definition 10-3.1 of Gleason p. 132. (Contributed by Mario Carneiro, 29-May-2016)

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

Proof

Step	Hyp	Ref	Expression
1		crred.1	|- ( ph -> A e. RR )
2		crred.2	|- ( ph -> B e. RR )
3		crre	|- ( ( A e. RR /\ B e. RR ) -> ( Re ` ( A + ( _i x. B ) ) ) = A )
4	1 2 3	syl2anc	|- ( ph -> ( Re ` ( A + ( _i x. B ) ) ) = A )