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Metamath Proof Explorer

Theorem hilxmet

Description: The Hilbert space norm determines a metric space. (Contributed by Mario Carneiro, 10-Sep-2015) (New usage is discouraged.)

		Ref	Expression
	Hypothesis	hilmet.1	$⊢ D = {norm}_{ℎ} \circ -_{ℎ}$
	Assertion	hilxmet	$⊢ D \in ∞Met (ℋ)$

Step	Hyp	Ref	Expression
1		hilmet.1	$⊢ D = {norm}_{ℎ} \circ -_{ℎ}$
2	1	hilmet	$⊢ D \in Met (ℋ)$
3		metxmet	$⊢ D \in Met (ℋ) \to D \in ∞Met (ℋ)$
4	2 3	ax-mp	$⊢ D \in ∞Met (ℋ)$