Theorem spansna

Description: The span of the singleton of a vector is an atom. (Contributed by NM, 18-Dec-2004) (New usage is discouraged.)

	Ref	Expression
Assertion	spansna	|- ( ( A e. ~H /\ A =/= 0h ) -> ( span ` { A } ) e. HAtoms )

Proof

Step	Hyp	Ref	Expression
1		spansn	|- ( A e. ~H -> ( span ` { A } ) = ( _|_ ` ( _|_ ` { A } ) ) )
2	1	adantr	|- ( ( A e. ~H /\ A =/= 0h ) -> ( span ` { A } ) = ( _|_ ` ( _|_ ` { A } ) ) )
3		h1da	|- ( ( A e. ~H /\ A =/= 0h ) -> ( _|_ ` ( _|_ ` { A } ) ) e. HAtoms )
4	2 3	eqeltrd	|- ( ( A e. ~H /\ A =/= 0h ) -> ( span ` { A } ) e. HAtoms )