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	$$⊢\phi \to A\in ℝ$$
	crred.2	$$⊢\phi \to B\in ℝ$$
Assertion	crred	$$⊢\phi \to \Re \left(A+\mathrm{i}B\right)=A$$

Proof

Step	Hyp	Ref	Expression
1		crred.1	$$⊢\phi \to A\in ℝ$$
2		crred.2	$$⊢\phi \to B\in ℝ$$
3		crre	$$⊢\left(A\in ℝ\wedge B\in ℝ\right)\to \Re \left(A+\mathrm{i}B\right)=A$$
4	1 2 3	syl2anc	$$⊢\phi \to \Re \left(A+\mathrm{i}B\right)=A$$