Theorem hlcmet

Description: The induced metric on a complex Hilbert space is complete. (Contributed by NM, 8-Sep-2007) (New usage is discouraged.)

	Ref	Expression
Hypotheses	hlcmet.x	$$⊢X=\mathrm{BaseSet}\left(U\right)$$
	hlcmet.8	$$⊢D=\mathrm{IndMet}\left(U\right)$$
Assertion	hlcmet	$$⊢U\in {CHil}_{\mathrm{OLD}}\to D\in \mathrm{CMet}\left(X\right)$$

Proof

Step	Hyp	Ref	Expression
1		hlcmet.x	$$⊢X=\mathrm{BaseSet}\left(U\right)$$
2		hlcmet.8	$$⊢D=\mathrm{IndMet}\left(U\right)$$
3		hlobn	$$⊢U\in {CHil}_{\mathrm{OLD}}\to U\in \mathrm{CBan}$$
4	1 2	cbncms	$$⊢U\in \mathrm{CBan}\to D\in \mathrm{CMet}\left(X\right)$$
5	3 4	syl	$$⊢U\in {CHil}_{\mathrm{OLD}}\to D\in \mathrm{CMet}\left(X\right)$$