Theorem shsub2

Description: Subspace sum is an upper bound of its arguments. (Contributed by NM, 17-Dec-2004) (New usage is discouraged.)

	Ref	Expression
Assertion	shsub2	$$⊢\left(A\in {\mathbf{S}}_{\mathscr{H}}\wedge B\in {\mathbf{S}}_{\mathscr{H}}\right)\to A\subseteq B{+}_{\mathscr{H}}A$$

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		shscom	$$⊢\left(A\in {\mathbf{S}}_{\mathscr{H}}\wedge B\in {\mathbf{S}}_{\mathscr{H}}\right)\to A{+}_{\mathscr{H}}B=B{+}_{\mathscr{H}}A$$
3	1 2	sseqtrd	$$⊢\left(A\in {\mathbf{S}}_{\mathscr{H}}\wedge B\in {\mathbf{S}}_{\mathscr{H}}\right)\to A\subseteq B{+}_{\mathscr{H}}A$$