Theorem cjre

Description: A real number equals its complex conjugate. Proposition 10-3.4(f) of Gleason p. 133. (Contributed by NM, 8-Oct-1999)

	Ref	Expression
Assertion	cjre	$$⊢A\in ℝ\to \bar{A}=A$$

Proof

Step	Hyp	Ref	Expression
1		recn	$$⊢A\in ℝ\to A\in ℂ$$
2		cjreb	$$⊢A\in ℂ\to \left(A\in ℝ\leftrightarrow \bar{A}=A\right)$$
3	2	biimpd	$$⊢A\in ℂ\to \left(A\in ℝ\to \bar{A}=A\right)$$
4	1 3	mpcom	$$⊢A\in ℝ\to \bar{A}=A$$