Theorem shseli

Description: Membership in subspace sum. (Contributed by NM, 4-May-2000) (New usage is discouraged.)

	Ref	Expression
Hypotheses	shscl.1	|- A e. SH
	shscl.2	|- B e. SH
Assertion	shseli	|- ( C e. ( A +H B ) <-> E. x e. A E. y e. B C = ( x +h y ) )

Proof

Step	Hyp	Ref	Expression
1		shscl.1	|- A e. SH
2		shscl.2	|- B e. SH
3		shsel	|- ( ( A e. SH /\ B e. SH ) -> ( C e. ( A +H B ) <-> E. x e. A E. y e. B C = ( x +h y ) ) )
4	1 2 3	mp2an	|- ( C e. ( A +H B ) <-> E. x e. A E. y e. B C = ( x +h y ) )