Theorem hhmet

Description: The induced metric 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	hhmet	$$⊢D\in \mathrm{Met}\left(\mathscr{H}\right)$$

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	4 2	imsmet	$$⊢U\in \mathrm{NrmCVec}\to D\in \mathrm{Met}\left(\mathscr{H}\right)$$
6	3 5	ax-mp	$$⊢D\in \mathrm{Met}\left(\mathscr{H}\right)$$