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Metamath Proof Explorer

Theorem cjrebd

Description: A number is real iff it equals its complex conjugate. Proposition 10-3.4(f) of Gleason p. 133. (Contributed by Mario Carneiro, 29-May-2016)

	Ref	Expression
Hypotheses	recld.1	$⊢ φ \to A \in ℂ$
	cjrebd.2	$⊢ φ \to \overline{A} = A$
Assertion	cjrebd	$⊢ φ \to A \in ℝ$

Proof

Step	Hyp	Ref	Expression
1		recld.1	$⊢ φ \to A \in ℂ$
2		cjrebd.2	$⊢ φ \to \overline{A} = A$
3		cjreb	$⊢ A \in ℂ \to (A \in ℝ \leftrightarrow \overline{A} = A)$
4	1 3	syl	$⊢ φ \to (A \in ℝ \leftrightarrow \overline{A} = A)$
5	2 4	mpbird	$⊢ φ \to A \in ℝ$