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

Metamath Proof Explorer

Theorem normge0

Description: The norm of a vector is nonnegative. (Contributed by NM, 29-May-1999) (New usage is discouraged.)

		Ref	Expression
	Assertion	normge0	$⊢ A \in ℋ \to 0 \leq {norm}_{ℎ} (A)$

Step	Hyp	Ref	Expression
1		hiidrcl	$⊢ A \in ℋ \to A \cdot_{ih} A \in ℝ$
2		hiidge0	$⊢ A \in ℋ \to 0 \leq A \cdot_{ih} A$
3	1 2	sqrtge0d	$⊢ A \in ℋ \to 0 \leq \sqrt{A \cdot_{ih} A}$
4		normval	$⊢ A \in ℋ \to {norm}_{ℎ} (A) = \sqrt{A \cdot_{ih} A}$
5	3 4	breqtrrd	$⊢ A \in ℋ \to 0 \leq {norm}_{ℎ} (A)$