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 e. SH
	shincl.2	|- B e. SH
	shless.1	|- C e. SH
Assertion	shlessi	|- ( A C_ B -> ( A +H C ) C_ ( B +H C ) )

Proof

Step	Hyp	Ref	Expression
1		shincl.1	|- A e. SH
2		shincl.2	|- B e. SH
3		shless.1	|- C e. SH
4		shless	|- ( ( ( A e. SH /\ B e. SH /\ C e. SH ) /\ A C_ B ) -> ( A +H C ) C_ ( B +H C ) )
5	4	ex	|- ( ( A e. SH /\ B e. SH /\ C e. SH ) -> ( A C_ B -> ( A +H C ) C_ ( B +H C ) ) )
6	1 2 3 5	mp3an	|- ( A C_ B -> ( A +H C ) C_ ( B +H C ) )