Theorem shub1i

Description: Hilbert lattice join is an upper bound of two subspaces. (Contributed by NM, 19-Oct-1999) (New usage is discouraged.)

	Ref	Expression
Hypotheses	shincl.1	$$⊢A\in {\mathbf{S}}_{\mathscr{H}}$$
	shincl.2	$$⊢B\in {\mathbf{S}}_{\mathscr{H}}$$
Assertion	shub1i	$$⊢A\subseteq A{\vee }_{\mathscr{H}}B$$

Proof

Step	Hyp	Ref	Expression
1		shincl.1	$$⊢A\in {\mathbf{S}}_{\mathscr{H}}$$
2		shincl.2	$$⊢B\in {\mathbf{S}}_{\mathscr{H}}$$
3	1 2	shsub1i	$$⊢A\subseteq A{+}_{\mathscr{H}}B$$
4	1 2	shsleji	$$⊢A{+}_{\mathscr{H}}B\subseteq A{\vee }_{\mathscr{H}}B$$
5	3 4	sstri	$$⊢A\subseteq A{\vee }_{\mathscr{H}}B$$