Theorem h1did

Description: A generating vector belongs to the 1-dimensional subspace it generates. (Contributed by NM, 22-Jul-2001) (New usage is discouraged.)

	Ref	Expression
Assertion	h1did	|- ( A e. ~H -> A e. ( _|_ ` ( _|_ ` { A } ) ) )

Proof

Step	Hyp	Ref	Expression
1		snssi	|- ( A e. ~H -> { A } C_ ~H )
2		ococss	|- ( { A } C_ ~H -> { A } C_ ( _|_ ` ( _|_ ` { A } ) ) )
3	1 2	syl	|- ( A e. ~H -> { A } C_ ( _|_ ` ( _|_ ` { A } ) ) )
4		snssg	|- ( A e. ~H -> ( A e. ( _|_ ` ( _|_ ` { A } ) ) <-> { A } C_ ( _|_ ` ( _|_ ` { A } ) ) ) )
5	3 4	mpbird	|- ( A e. ~H -> A e. ( _|_ ` ( _|_ ` { A } ) ) )