Theorem lspid

Description: The span of a subspace is itself. ( spanid analog.) (Contributed by NM, 15-Dec-2013) (Revised by Mario Carneiro, 19-Jun-2014)

	Ref	Expression
Hypotheses	lspid.s	|- S = ( LSubSp ` W )
	lspid.n	|- N = ( LSpan ` W )
Assertion	lspid	|- ( ( W e. LMod /\ U e. S ) -> ( N ` U ) = U )

Proof

Step	Hyp	Ref	Expression
1		lspid.s	|- S = ( LSubSp ` W )
2		lspid.n	|- N = ( LSpan ` W )
3		eqid	|- ( Base ` W ) = ( Base ` W )
4	3 1	lssss	|- ( U e. S -> U C_ ( Base ` W ) )
5	3 1 2	lspval	|- ( ( W e. LMod /\ U C_ ( Base ` W ) ) -> ( N ` U ) = |^| { t e. S | U C_ t } )
6	4 5	sylan2	|- ( ( W e. LMod /\ U e. S ) -> ( N ` U ) = |^| { t e. S | U C_ t } )
7		intmin	|- ( U e. S -> |^| { t e. S | U C_ t } = U )
8	7	adantl	|- ( ( W e. LMod /\ U e. S ) -> |^| { t e. S | U C_ t } = U )
9	6 8	eqtrd	|- ( ( W e. LMod /\ U e. S ) -> ( N ` U ) = U )