Theorem hl0cl

Description: The Hilbert space zero vector. (Contributed by NM, 7-Sep-2007) (New usage is discouraged.)

	Ref	Expression
Hypotheses	hl0cl.1	$$⊢X=\mathrm{BaseSet}\left(U\right)$$
	hl0cl.5	$$⊢Z=0_{\mathrm{vec}}\left(U\right)$$
Assertion	hl0cl	$$⊢U\in {CHil}_{\mathrm{OLD}}\to Z\in X$$

Proof

Step	Hyp	Ref	Expression
1		hl0cl.1	$$⊢X=\mathrm{BaseSet}\left(U\right)$$
2		hl0cl.5	$$⊢Z=0_{\mathrm{vec}}\left(U\right)$$
3		hlnv	$$⊢U\in {CHil}_{\mathrm{OLD}}\to U\in \mathrm{NrmCVec}$$
4	1 2	nvzcl	$$⊢U\in \mathrm{NrmCVec}\to Z\in X$$
5	3 4	syl	$$⊢U\in {CHil}_{\mathrm{OLD}}\to Z\in X$$