Theorem shsel1

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	shsel1	|- ( ( A e. SH /\ B e. SH ) -> ( C e. A -> C e. ( A +H B ) ) )

Proof

Step	Hyp	Ref	Expression
1		shel	|- ( ( A e. SH /\ C e. A ) -> C e. ~H )
2		ax-hvaddid	|- ( C e. ~H -> ( C +h 0h ) = C )
3	1 2	syl	|- ( ( A e. SH /\ C e. A ) -> ( C +h 0h ) = C )
4	3	adantlr	|- ( ( ( A e. SH /\ B e. SH ) /\ C e. A ) -> ( C +h 0h ) = C )
5		sh0	|- ( B e. SH -> 0h e. B )
6	5	adantl	|- ( ( A e. SH /\ B e. SH ) -> 0h e. B )
7		shsva	|- ( ( A e. SH /\ B e. SH ) -> ( ( C e. A /\ 0h e. B ) -> ( C +h 0h ) e. ( A +H B ) ) )
8	6 7	mpan2d	|- ( ( A e. SH /\ B e. SH ) -> ( C e. A -> ( C +h 0h ) e. ( A +H B ) ) )
9	8	imp	|- ( ( ( A e. SH /\ B e. SH ) /\ C e. A ) -> ( C +h 0h ) e. ( A +H B ) )
10	4 9	eqeltrrd	|- ( ( ( A e. SH /\ B e. SH ) /\ C e. A ) -> C e. ( A +H B ) )
11	10	ex	|- ( ( A e. SH /\ B e. SH ) -> ( C e. A -> C e. ( A +H B ) ) )