Theorem shss

Description: A subspace is a subset of Hilbert space. (Contributed by NM, 9-Oct-1999) (Revised by Mario Carneiro, 23-Dec-2013) (New usage is discouraged.)

	Ref	Expression
Assertion	shss	$$⊢H\in {\mathbf{S}}_{\mathscr{H}}\to H\subseteq \mathscr{H}$$

Proof

Step	Hyp	Ref	Expression
1		issh	$$⊢H\in {\mathbf{S}}_{\mathscr{H}}\leftrightarrow \left(\left(H\subseteq \mathscr{H}\wedge 0_{ℎ}\in H\right)\wedge \left({+}_{ℎ}\left[H\times H\right]\subseteq H\wedge {\cdot }_{ℎ}\left[ℂ\times H\right]\subseteq H\right)\right)$$
2	1	simplbi	$$⊢H\in {\mathbf{S}}_{\mathscr{H}}\to \left(H\subseteq \mathscr{H}\wedge 0_{ℎ}\in H\right)$$
3	2	simpld	$$⊢H\in {\mathbf{S}}_{\mathscr{H}}\to H\subseteq \mathscr{H}$$