Theorem rere

Description: A real number equals its real part. One direction of Proposition 10-3.4(f) of Gleason p. 133. (Contributed by Paul Chapman, 7-Sep-2007)

	Ref	Expression
Assertion	rere	$$⊢A\in ℝ\to \Re \left(A\right)=A$$

Proof

Step	Hyp	Ref	Expression
1		recn	$$⊢A\in ℝ\to A\in ℂ$$
2		rereb	$$⊢A\in ℂ\to \left(A\in ℝ\leftrightarrow \Re \left(A\right)=A\right)$$
3	1 2	syl	$$⊢A\in ℝ\to \left(A\in ℝ\leftrightarrow \Re \left(A\right)=A\right)$$
4	3	ibi	$$⊢A\in ℝ\to \Re \left(A\right)=A$$