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

Metamath Proof Explorer

Theorem abscjd

Description: The absolute value of a number and its conjugate are the same. Proposition 10-3.7(b) of Gleason p. 133. (Contributed by Mario Carneiro, 29-May-2016)

		Ref	Expression
	Hypothesis	abscld.1	$⊢ φ \to A \in ℂ$
	Assertion	abscjd	$⊢ φ \to \|\overline{A}\| = \|A\|$

Proof

Step	Hyp	Ref	Expression
1		abscld.1	$⊢ φ \to A \in ℂ$
2		abscj	$⊢ A \in ℂ \to \|\overline{A}\| = \|A\|$
3	1 2	syl	$⊢ φ \to \|\overline{A}\| = \|A\|$