Theorem chjoi

Description: The join of a closed subspace and its orthocomplement. (Contributed by NM, 24-Oct-1999) (New usage is discouraged.)

	Ref	Expression
Hypothesis	ch0le.1	$$⊢A\in {\mathbf{C}}_{\mathscr{H}}$$
Assertion	chjoi	$$⊢A{\vee }_{\mathscr{H}}\perp \left(A\right)=\mathscr{H}$$

Proof

Step	Hyp	Ref	Expression
1		ch0le.1	$$⊢A\in {\mathbf{C}}_{\mathscr{H}}$$
2	1	chssii	$$⊢A\subseteq \mathscr{H}$$
3		ssjo	$$⊢A\subseteq \mathscr{H}\to A{\vee }_{\mathscr{H}}\perp \left(A\right)=\mathscr{H}$$
4	2 3	ax-mp	$$⊢A{\vee }_{\mathscr{H}}\perp \left(A\right)=\mathscr{H}$$