Theorem lspcl

Description: The span of a set of vectors is a subspace. ( spancl analog.) (Contributed by NM, 9-Dec-2013) (Revised by Mario Carneiro, 19-Jun-2014)

	Ref	Expression
Hypotheses	lspval.v	|- V = ( Base ` W )
	lspval.s	|- S = ( LSubSp ` W )
	lspval.n	|- N = ( LSpan ` W )
Assertion	lspcl	|- ( ( W e. LMod /\ U C_ V ) -> ( N ` U ) 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	1 2 3	lspf	|- ( W e. LMod -> N : ~P V --> S )
5	1	fvexi	|- V e. _V
6	5	elpw2	|- ( U e. ~P V <-> U C_ V )
7	6	biimpri	|- ( U C_ V -> U e. ~P V )
8		ffvelrn	|- ( ( N : ~P V --> S /\ U e. ~P V ) -> ( N ` U ) e. S )
9	4 7 8	syl2an	|- ( ( W e. LMod /\ U C_ V ) -> ( N ` U ) e. S )