Theorem chslej

Description: Subspace sum is smaller than subspace join. Remark in Kalmbach p. 65. (Contributed by NM, 12-Jul-2004) (New usage is discouraged.)

	Ref	Expression
Assertion	chslej	$$⊢\left(A\in {\mathbf{C}}_{\mathscr{H}}\wedge B\in {\mathbf{C}}_{\mathscr{H}}\right)\to A{+}_{\mathscr{H}}B\subseteq A{\vee }_{\mathscr{H}}B$$

Proof

Step	Hyp	Ref	Expression
1		chsh	$$⊢A\in {\mathbf{C}}_{\mathscr{H}}\to A\in {\mathbf{S}}_{\mathscr{H}}$$
2		chsh	$$⊢B\in {\mathbf{C}}_{\mathscr{H}}\to B\in {\mathbf{S}}_{\mathscr{H}}$$
3		shslej	$$⊢\left(A\in {\mathbf{S}}_{\mathscr{H}}\wedge B\in {\mathbf{S}}_{\mathscr{H}}\right)\to A{+}_{\mathscr{H}}B\subseteq A{\vee }_{\mathscr{H}}B$$
4	1 2 3	syl2an	$$⊢\left(A\in {\mathbf{C}}_{\mathscr{H}}\wedge B\in {\mathbf{C}}_{\mathscr{H}}\right)\to A{+}_{\mathscr{H}}B\subseteq A{\vee }_{\mathscr{H}}B$$