Theorem hlmet

Description: The induced metric on a complex Hilbert space. (Contributed by NM, 7-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	hlmet	$$⊢U\in {CHil}_{\mathrm{OLD}}\to D\in \mathrm{Met}\left(X\right)$$

Proof

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