Theorem hhmetdval

Description: Value of the distance function of the metric space 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	hhmetdval	⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ) → ( 𝐴 𝐷 𝐵 ) = ( normℎ ‘ ( 𝐴 −ℎ 𝐵 ) ) )

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	1 3 4 2	h2hmetdval	⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ) → ( 𝐴 𝐷 𝐵 ) = ( normℎ ‘ ( 𝐴 −ℎ 𝐵 ) ) )