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

Metamath Proof Explorer

Theorem nmval

Description: The value of the norm as the distance to zero. Problem 1 of Kreyszig p. 63. (Contributed by NM, 4-Dec-2006) (Revised by Mario Carneiro, 2-Oct-2015)

	Ref	Expression
Hypotheses	nmfval.n	$⊢ N = norm (W)$
	nmfval.x	$⊢ X = {Base}_{W}$
	nmfval.z	$⊢ \underset{˙}{0} = 0_{W}$
	nmfval.d	$⊢ D = dist (W)$
Assertion	nmval	$⊢ A \in X \to N (A) = A D \underset{˙}{0}$

Proof

Step	Hyp	Ref	Expression
1		nmfval.n	$⊢ N = norm (W)$
2		nmfval.x	$⊢ X = {Base}_{W}$
3		nmfval.z	$⊢ \underset{˙}{0} = 0_{W}$
4		nmfval.d	$⊢ D = dist (W)$
5		oveq1	$⊢ x = A \to x D \underset{˙}{0} = A D \underset{˙}{0}$
6	1 2 3 4	nmfval	$⊢ N = (x \in X ⟼ x D \underset{˙}{0})$
7		ovex	$⊢ A D \underset{˙}{0} \in V$
8	5 6 7	fvmpt	$⊢ A \in X \to N (A) = A D \underset{˙}{0}$