Theorem oc0

Description: The zero vector belongs to an orthogonal complement of a Hilbert subspace. (Contributed by NM, 11-Oct-1999) (New usage is discouraged.)

	Ref	Expression
Assertion	oc0	$$⊢H\in {\mathbf{S}}_{\mathscr{H}}\to 0_{ℎ}\in \perp \left(H\right)$$

Proof

Step	Hyp	Ref	Expression
1		shocsh	$$⊢H\in {\mathbf{S}}_{\mathscr{H}}\to \perp \left(H\right)\in {\mathbf{S}}_{\mathscr{H}}$$
2		sh0	$$⊢\perp \left(H\right)\in {\mathbf{S}}_{\mathscr{H}}\to 0_{ℎ}\in \perp \left(H\right)$$
3	1 2	syl	$$⊢H\in {\mathbf{S}}_{\mathscr{H}}\to 0_{ℎ}\in \perp \left(H\right)$$