Theorem lspidm

Description: The span of a set of vectors is idempotent. (Contributed by NM, 22-Feb-2014) (Revised by Mario Carneiro, 19-Jun-2014)

	Ref	Expression
Hypotheses	lspss.v	|- V = ( Base ` W )
	lspss.n	|- N = ( LSpan ` W )
Assertion	lspidm	|- ( ( W e. LMod /\ U C_ V ) -> ( N ` ( N ` U ) ) = ( N ` U ) )

Proof

Step	Hyp	Ref	Expression
1		lspss.v	|- V = ( Base ` W )
2		lspss.n	|- N = ( LSpan ` W )
3		eqid	|- ( LSubSp ` W ) = ( LSubSp ` W )
4	1 3 2	lspcl	|- ( ( W e. LMod /\ U C_ V ) -> ( N ` U ) e. ( LSubSp ` W ) )
5	3 2	lspid	|- ( ( W e. LMod /\ ( N ` U ) e. ( LSubSp ` W ) ) -> ( N ` ( N ` U ) ) = ( N ` U ) )
6	4 5	syldan	|- ( ( W e. LMod /\ U C_ V ) -> ( N ` ( N ` U ) ) = ( N ` U ) )