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Metamath Proof Explorer

Theorem absge0

Description: Absolute value is nonnegative. (Contributed by NM, 20-Nov-2004) (Revised by Mario Carneiro, 29-May-2016)

		Ref	Expression
	Assertion	absge0	$⊢ A \in ℂ \to 0 \leq \|A\|$

Step	Hyp	Ref	Expression
1		cjmulrcl	$⊢ A \in ℂ \to A \overline{A} \in ℝ$
2		cjmulge0	$⊢ A \in ℂ \to 0 \leq A \overline{A}$
3		sqrtge0	$⊢ (A \overline{A} \in ℝ \land 0 \leq A \overline{A}) \to 0 \leq \sqrt{A \overline{A}}$
4	1 2 3	syl2anc	$⊢ A \in ℂ \to 0 \leq \sqrt{A \overline{A}}$
5		absval	$⊢ A \in ℂ \to \|A\| = \sqrt{A \overline{A}}$
6	4 5	breqtrrd	$⊢ A \in ℂ \to 0 \leq \|A\|$