Theorem ch0

Description: The zero vector belongs to any closed subspace of a Hilbert space. (Contributed by NM, 24-Aug-1999) (New usage is discouraged.)

	Ref	Expression
Assertion	ch0	$$⊢H\in {\mathbf{C}}_{\mathscr{H}}\to 0_{ℎ}\in H$$

Proof

Step	Hyp	Ref	Expression
1		chsh	$$⊢H\in {\mathbf{C}}_{\mathscr{H}}\to H\in {\mathbf{S}}_{\mathscr{H}}$$
2		sh0	$$⊢H\in {\mathbf{S}}_{\mathscr{H}}\to 0_{ℎ}\in H$$
3	1 2	syl	$$⊢H\in {\mathbf{C}}_{\mathscr{H}}\to 0_{ℎ}\in H$$