Theorem shub2

Description: A subspace is a subset of its Hilbert lattice join with another. (Contributed by NM, 22-Jun-2004) (New usage is discouraged.)

	Ref	Expression
Assertion	shub2	$$⊢\left(A\in {\mathbf{S}}_{\mathscr{H}}\wedge B\in {\mathbf{S}}_{\mathscr{H}}\right)\to A\subseteq B{\vee }_{\mathscr{H}}A$$

Proof

Step	Hyp	Ref	Expression
1		shub1	$$⊢\left(A\in {\mathbf{S}}_{\mathscr{H}}\wedge B\in {\mathbf{S}}_{\mathscr{H}}\right)\to A\subseteq A{\vee }_{\mathscr{H}}B$$
2		shjcom	$$⊢\left(A\in {\mathbf{S}}_{\mathscr{H}}\wedge B\in {\mathbf{S}}_{\mathscr{H}}\right)\to A{\vee }_{\mathscr{H}}B=B{\vee }_{\mathscr{H}}A$$
3	1 2	sseqtrd	$$⊢\left(A\in {\mathbf{S}}_{\mathscr{H}}\wedge B\in {\mathbf{S}}_{\mathscr{H}}\right)\to A\subseteq B{\vee }_{\mathscr{H}}A$$