Theorem shsel2

Description: A subspace sum contains a member of one of its subspaces. (Contributed by NM, 15-Dec-2004) (New usage is discouraged.)

	Ref	Expression
Assertion	shsel2	|- ( ( A e. SH /\ B e. SH ) -> ( C e. B -> C e. ( A +H B ) ) )

Proof

Step	Hyp	Ref	Expression
1		shsel1	|- ( ( B e. SH /\ A e. SH ) -> ( C e. B -> C e. ( B +H A ) ) )
2	1	ancoms	|- ( ( A e. SH /\ B e. SH ) -> ( C e. B -> C e. ( B +H A ) ) )
3		shscom	|- ( ( A e. SH /\ B e. SH ) -> ( A +H B ) = ( B +H A ) )
4	3	eleq2d	|- ( ( A e. SH /\ B e. SH ) -> ( C e. ( A +H B ) <-> C e. ( B +H A ) ) )
5	2 4	sylibrd	|- ( ( A e. SH /\ B e. SH ) -> ( C e. B -> C e. ( A +H B ) ) )