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

Proof

Step	Hyp	Ref	Expression
1		crre.1	$$⊢A\in ℝ$$
2		crre.2	$$⊢B\in ℝ$$
3		crre	$$⊢\left(A\in ℝ\wedge B\in ℝ\right)\to \Re \left(A+\mathrm{i}B\right)=A$$
4	1 2 3	mp2an	$$⊢\Re \left(A+\mathrm{i}B\right)=A$$