Metamath Proof Explorer


Theorem h2hmetdval

Description: Value of the distance function of the metric space of Hilbert space. (Contributed by NM, 6-Jun-2008) (New usage is discouraged.)

Ref Expression
Hypotheses h2h.1 𝑈 = ⟨ ⟨ + , · ⟩ , norm
h2h.2 𝑈 ∈ NrmCVec
h2hm.4 ℋ = ( BaseSet ‘ 𝑈 )
h2hm.5 𝐷 = ( IndMet ‘ 𝑈 )
Assertion h2hmetdval ( ( 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ) → ( 𝐴 𝐷 𝐵 ) = ( norm ‘ ( 𝐴 𝐵 ) ) )

Proof

Step Hyp Ref Expression
1 h2h.1 𝑈 = ⟨ ⟨ + , · ⟩ , norm
2 h2h.2 𝑈 ∈ NrmCVec
3 h2hm.4 ℋ = ( BaseSet ‘ 𝑈 )
4 h2hm.5 𝐷 = ( IndMet ‘ 𝑈 )
5 1 2 3 h2hvs = ( −𝑣𝑈 )
6 1 2 h2hnm norm = ( normCV𝑈 )
7 3 5 6 4 imsdval ( ( 𝑈 ∈ NrmCVec ∧ 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ) → ( 𝐴 𝐷 𝐵 ) = ( norm ‘ ( 𝐴 𝐵 ) ) )
8 2 7 mp3an1 ( ( 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ) → ( 𝐴 𝐷 𝐵 ) = ( norm ‘ ( 𝐴 𝐵 ) ) )