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Metamath Proof Explorer

Theorem remimd

Description: Value of the conjugate of a complex number. The value is the real part minus _i times the imaginary part. Definition 10-3.2 of Gleason p. 132. (Contributed by Mario Carneiro, 29-May-2016)

		Ref	Expression
	Hypothesis	recld.1	$⊢ φ \to A \in ℂ$
	Assertion	remimd	$⊢ φ \to \overline{A} = ℜ (A) - i ℑ (A)$

Proof

Step	Hyp	Ref	Expression
1		recld.1	$⊢ φ \to A \in ℂ$
2		remim	$⊢ A \in ℂ \to \overline{A} = ℜ (A) - i ℑ (A)$
3	1 2	syl	$⊢ φ \to \overline{A} = ℜ (A) - i ℑ (A)$