Theorem shsub1

Description: Subspace sum is an upper bound of its arguments. (Contributed by NM, 14-Dec-2004) (New usage is discouraged.)

	Ref	Expression
Assertion	shsub1	$$⊢\left(A\in {\mathbf{S}}_{\mathscr{H}}\wedge B\in {\mathbf{S}}_{\mathscr{H}}\right)\to A\subseteq A{+}_{\mathscr{H}}B$$

Proof

Step	Hyp	Ref	Expression
1		shsel1	$$⊢\left(A\in {\mathbf{S}}_{\mathscr{H}}\wedge B\in {\mathbf{S}}_{\mathscr{H}}\right)\to \left(x\in A\to x\in \left(A{+}_{\mathscr{H}}B\right)\right)$$
2	1	ssrdv	$$⊢\left(A\in {\mathbf{S}}_{\mathscr{H}}\wedge B\in {\mathbf{S}}_{\mathscr{H}}\right)\to A\subseteq A{+}_{\mathscr{H}}B$$