Theorem shub1

Description: Hilbert lattice join is an upper bound of two subspaces. (Contributed by NM, 22-Jun-2004) (New usage is discouraged.)

	Ref	Expression
Assertion	shub1	$$⊢\left(A\in {\mathbf{S}}_{\mathscr{H}}\wedge B\in {\mathbf{S}}_{\mathscr{H}}\right)\to A\subseteq A{\vee }_{\mathscr{H}}B$$

Proof

Step	Hyp	Ref	Expression
1		shsub1	$$⊢\left(A\in {\mathbf{S}}_{\mathscr{H}}\wedge B\in {\mathbf{S}}_{\mathscr{H}}\right)\to A\subseteq A{+}_{\mathscr{H}}B$$
2		shslej	$$⊢\left(A\in {\mathbf{S}}_{\mathscr{H}}\wedge B\in {\mathbf{S}}_{\mathscr{H}}\right)\to A{+}_{\mathscr{H}}B\subseteq A{\vee }_{\mathscr{H}}B$$
3	1 2	sstrd	$$⊢\left(A\in {\mathbf{S}}_{\mathscr{H}}\wedge B\in {\mathbf{S}}_{\mathscr{H}}\right)\to A\subseteq A{\vee }_{\mathscr{H}}B$$