Theorem lspsncl

Description: The span of a singleton is a subspace (frequently used special case of lspcl ). (Contributed by NM, 17-Jul-2014)

	Ref	Expression
Hypotheses	lspval.v	|- V = ( Base ` W )
	lspval.s	|- S = ( LSubSp ` W )
	lspval.n	|- N = ( LSpan ` W )
Assertion	lspsncl	|- ( ( W e. LMod /\ X e. V ) -> ( N ` { X } ) e. S )

Proof

Step	Hyp	Ref	Expression
1		lspval.v	|- V = ( Base ` W )
2		lspval.s	|- S = ( LSubSp ` W )
3		lspval.n	|- N = ( LSpan ` W )
4		snssi	|- ( X e. V -> { X } C_ V )
5	1 2 3	lspcl	|- ( ( W e. LMod /\ { X } C_ V ) -> ( N ` { X } ) e. S )
6	4 5	sylan2	|- ( ( W e. LMod /\ X e. V ) -> ( N ` { X } ) e. S )