Theorem shsub1i

Description: Subspace sum is an upper bound of its arguments. (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	shsub1i	$$⊢A\subseteq A{+}_{\mathscr{H}}B$$

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	shsel1i	$$⊢x\in A\to x\in \left(A{+}_{\mathscr{H}}B\right)$$
4	3	ssriv	$$⊢A\subseteq A{+}_{\mathscr{H}}B$$