Theorem shlessi

Description: Subset implies subset of subspace sum. (Contributed by NM, 18-Nov-2000) (New usage is discouraged.)

	Ref	Expression
Hypotheses	shincl.1	$$⊢A\in {\mathbf{S}}_{\mathscr{H}}$$
	shincl.2	$$⊢B\in {\mathbf{S}}_{\mathscr{H}}$$
	shless.1	$$⊢C\in {\mathbf{S}}_{\mathscr{H}}$$
Assertion	shlessi	$$⊢A\subseteq B\to A{+}_{\mathscr{H}}C\subseteq B{+}_{\mathscr{H}}C$$

Proof

Step	Hyp	Ref	Expression
1		shincl.1	$$⊢A\in {\mathbf{S}}_{\mathscr{H}}$$
2		shincl.2	$$⊢B\in {\mathbf{S}}_{\mathscr{H}}$$
3		shless.1	$$⊢C\in {\mathbf{S}}_{\mathscr{H}}$$
4		shless	$$⊢\left(\left(A\in {\mathbf{S}}_{\mathscr{H}}\wedge B\in {\mathbf{S}}_{\mathscr{H}}\wedge C\in {\mathbf{S}}_{\mathscr{H}}\right)\wedge A\subseteq B\right)\to A{+}_{\mathscr{H}}C\subseteq B{+}_{\mathscr{H}}C$$
5	4	ex	$$⊢\left(A\in {\mathbf{S}}_{\mathscr{H}}\wedge B\in {\mathbf{S}}_{\mathscr{H}}\wedge C\in {\mathbf{S}}_{\mathscr{H}}\right)\to \left(A\subseteq B\to A{+}_{\mathscr{H}}C\subseteq B{+}_{\mathscr{H}}C\right)$$
6	1 2 3 5	mp3an	$$⊢A\subseteq B\to A{+}_{\mathscr{H}}C\subseteq B{+}_{\mathscr{H}}C$$