Theorem hhxmet

Description: The induced metric of Hilbert space. (Contributed by Mario Carneiro, 10-Sep-2015) (New usage is discouraged.)

	Ref	Expression
Hypotheses	hhnv.1	$$⊢U=⟨⟨{+}_{ℎ},{\cdot }_{ℎ}⟩,{norm}_{ℎ}⟩$$
	hhims2.2	$$⊢D=\mathrm{IndMet}\left(U\right)$$
Assertion	hhxmet	$$⊢D\in \mathrm{\infty Met}\left(\mathscr{H}\right)$$

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 2	hhmet	$$⊢D\in \mathrm{Met}\left(\mathscr{H}\right)$$
4		metxmet	$$⊢D\in \mathrm{Met}\left(\mathscr{H}\right)\to D\in \mathrm{\infty Met}\left(\mathscr{H}\right)$$
5	3 4	ax-mp	$$⊢D\in \mathrm{\infty Met}\left(\mathscr{H}\right)$$