This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem reim0bi

Description: A number is real iff its imaginary part is 0. (Contributed by NM, 29-May-1999)

		Ref	Expression
	Hypothesis	recl.1	$⊢ A \in ℂ$
	Assertion	reim0bi	$⊢ A \in ℝ \leftrightarrow ℑ (A) = 0$

Step	Hyp	Ref	Expression
1		recl.1	$⊢ A \in ℂ$
2		reim0b	$⊢ A \in ℂ \to (A \in ℝ \leftrightarrow ℑ (A) = 0)$
3	1 2	ax-mp	$⊢ A \in ℝ \leftrightarrow ℑ (A) = 0$