Theorem shocel

Description: Membership in orthogonal complement of H subspace. (Contributed by NM, 9-Oct-1999) (New usage is discouraged.)

	Ref	Expression
Assertion	shocel	|- ( H e. SH -> ( A e. ( _|_ ` H ) <-> ( A e. ~H /\ A. x e. H ( A .ih x ) = 0 ) ) )

Proof

Step	Hyp	Ref	Expression
1		shss	|- ( H e. SH -> H C_ ~H )
2		ocel	|- ( H C_ ~H -> ( A e. ( _|_ ` H ) <-> ( A e. ~H /\ A. x e. H ( A .ih x ) = 0 ) ) )
3	1 2	syl	|- ( H e. SH -> ( A e. ( _|_ ` H ) <-> ( A e. ~H /\ A. x e. H ( A .ih x ) = 0 ) ) )