Theorem hhmet

Description: The induced metric of Hilbert space. (Contributed by NM, 10-Apr-2008) (New usage is discouraged.)

	Ref	Expression
Hypotheses	hhnv.1	⊢ 𝑈 = ⟨ ⟨ +ℎ , ·ℎ ⟩ , normℎ ⟩
	hhims2.2	⊢ 𝐷 = ( IndMet ‘ 𝑈 )
Assertion	hhmet	⊢ 𝐷 ∈ ( Met ‘ ℋ )

Proof

Step	Hyp	Ref	Expression
1		hhnv.1	⊢ 𝑈 = ⟨ ⟨ +ℎ , ·ℎ ⟩ , normℎ ⟩
2		hhims2.2	⊢ 𝐷 = ( IndMet ‘ 𝑈 )
3	1	hhnv	⊢ 𝑈 ∈ NrmCVec
4	1	hhba	⊢ ℋ = ( BaseSet ‘ 𝑈 )
5	4 2	imsmet	⊢ ( 𝑈 ∈ NrmCVec → 𝐷 ∈ ( Met ‘ ℋ ) )
6	3 5	ax-mp	⊢ 𝐷 ∈ ( Met ‘ ℋ )