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 ) ) ) |
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 ) ) ) |