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	$$⊢U=⟨⟨{+}_{ℎ},{\cdot }_{ℎ}⟩,{norm}_{ℎ}⟩$$
	hhims2.2	$$⊢D=\mathrm{IndMet}\left(U\right)$$
Assertion	hhmetdval	$$⊢\left(A\in \mathscr{H}\wedge B\in \mathscr{H}\right)\to ADB={norm}_{ℎ}\left(A{-}_{ℎ}B\right)$$

Proof

Step	Hyp	Ref	Expression
1		hhnv.1	$$⊢U=⟨⟨{+}_{ℎ},{\cdot }_{ℎ}⟩,{norm}_{ℎ}⟩$$
2		hhims2.2	$$⊢D=\mathrm{IndMet}\left(U\right)$$
3	1	hhnv	$$⊢U\in \mathrm{NrmCVec}$$
4	1	hhba	$$⊢\mathscr{H}=\mathrm{BaseSet}\left(U\right)$$
5	1 3 4 2	h2hmetdval	$$⊢\left(A\in \mathscr{H}\wedge B\in \mathscr{H}\right)\to ADB={norm}_{ℎ}\left(A{-}_{ℎ}B\right)$$