Metamath Proof Explorer


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 = ⟨ ⟨ + , · ⟩ , norm
hhims2.2 𝐷 = ( IndMet ‘ 𝑈 )
Assertion hhmetdval ( ( 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ) → ( 𝐴 𝐷 𝐵 ) = ( norm ‘ ( 𝐴 𝐵 ) ) )

Proof

Step Hyp Ref Expression
1 hhnv.1 𝑈 = ⟨ ⟨ + , · ⟩ , norm
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 ) = ( norm ‘ ( 𝐴 𝐵 ) ) )