Theorem chsh

Description: A closed subspace is a subspace. (Contributed by NM, 19-Oct-1999) (Revised by Mario Carneiro, 23-Dec-2013) (New usage is discouraged.)

	Ref	Expression
Assertion	chsh	$$⊢H\in {\mathbf{C}}_{\mathscr{H}}\to H\in {\mathbf{S}}_{\mathscr{H}}$$

Proof

Step	Hyp	Ref	Expression
1		isch	$$⊢H\in {\mathbf{C}}_{\mathscr{H}}\leftrightarrow \left(H\in {\mathbf{S}}_{\mathscr{H}}\wedge \underset{v}{⇝}\left[H^{\mathbb{N}}\right]\subseteq H\right)$$
2	1	simplbi	$$⊢H\in {\mathbf{C}}_{\mathscr{H}}\to H\in {\mathbf{S}}_{\mathscr{H}}$$