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	$$⊢U=⟨⟨{+}_{ℎ},{\cdot }_{ℎ}⟩,{norm}_{ℎ}⟩$$
	h2h.2	$$⊢U\in \mathrm{NrmCVec}$$
	h2hm.4	$$⊢\mathscr{H}=\mathrm{BaseSet}\left(U\right)$$
	h2hm.5	$$⊢D=\mathrm{IndMet}\left(U\right)$$
Assertion	h2hmetdval	$$⊢\left(A\in \mathscr{H}\wedge B\in \mathscr{H}\right)\to ADB={norm}_{ℎ}\left(A{-}_{ℎ}B\right)$$

Proof

Step	Hyp	Ref	Expression
1		h2h.1	$$⊢U=⟨⟨{+}_{ℎ},{\cdot }_{ℎ}⟩,{norm}_{ℎ}⟩$$
2		h2h.2	$$⊢U\in \mathrm{NrmCVec}$$
3		h2hm.4	$$⊢\mathscr{H}=\mathrm{BaseSet}\left(U\right)$$
4		h2hm.5	$$⊢D=\mathrm{IndMet}\left(U\right)$$
5	1 2 3	h2hvs	$$⊢{-}_{ℎ}={-}_v\left(U\right)$$
6	1 2	h2hnm	$$⊢{norm}_{ℎ}={norm}_{CV}\left(U\right)$$
7	3 5 6 4	imsdval	$$⊢\left(U\in \mathrm{NrmCVec}\wedge A\in \mathscr{H}\wedge B\in \mathscr{H}\right)\to ADB={norm}_{ℎ}\left(A{-}_{ℎ}B\right)$$
8	2 7	mp3an1	$$⊢\left(A\in \mathscr{H}\wedge B\in \mathscr{H}\right)\to ADB={norm}_{ℎ}\left(A{-}_{ℎ}B\right)$$