Theorem shsvai

Description: Vector sum belongs to subspace sum. (Contributed by NM, 17-Oct-1999) (New usage is discouraged.)

	Ref	Expression
Hypotheses	shincl.1	|- A e. SH
	shincl.2	|- B e. SH
Assertion	shsvai	|- ( ( C e. A /\ D e. B ) -> ( C +h D ) e. ( A +H B ) )

Proof

Step	Hyp	Ref	Expression
1		shincl.1	|- A e. SH
2		shincl.2	|- B e. SH
3		shsva	|- ( ( A e. SH /\ B e. SH ) -> ( ( C e. A /\ D e. B ) -> ( C +h D ) e. ( A +H B ) ) )
4	1 2 3	mp2an	|- ( ( C e. A /\ D e. B ) -> ( C +h D ) e. ( A +H B ) )