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 = 〈 〈 +ℎ , ·ℎ 〉 , normℎ 〉 | |
h2h.2 | ⊢ 𝑈 ∈ NrmCVec | ||
h2hm.4 | ⊢ ℋ = ( BaseSet ‘ 𝑈 ) | ||
h2hm.5 | ⊢ 𝐷 = ( IndMet ‘ 𝑈 ) | ||
Assertion | h2hmetdval | ⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ) → ( 𝐴 𝐷 𝐵 ) = ( normℎ ‘ ( 𝐴 −ℎ 𝐵 ) ) ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | h2h.1 | ⊢ 𝑈 = 〈 〈 +ℎ , ·ℎ 〉 , normℎ 〉 | |
2 | h2h.2 | ⊢ 𝑈 ∈ NrmCVec | |
3 | h2hm.4 | ⊢ ℋ = ( BaseSet ‘ 𝑈 ) | |
4 | h2hm.5 | ⊢ 𝐷 = ( IndMet ‘ 𝑈 ) | |
5 | 1 2 3 | h2hvs | ⊢ −ℎ = ( −𝑣 ‘ 𝑈 ) |
6 | 1 2 | h2hnm | ⊢ normℎ = ( normCV ‘ 𝑈 ) |
7 | 3 5 6 4 | imsdval | ⊢ ( ( 𝑈 ∈ NrmCVec ∧ 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ) → ( 𝐴 𝐷 𝐵 ) = ( normℎ ‘ ( 𝐴 −ℎ 𝐵 ) ) ) |
8 | 2 7 | mp3an1 | ⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ) → ( 𝐴 𝐷 𝐵 ) = ( normℎ ‘ ( 𝐴 −ℎ 𝐵 ) ) ) |