Theorem lspprcl

Description: The span of a pair is a subspace (frequently used special case of lspcl ). (Contributed by NM, 11-Apr-2015)

	Ref	Expression
Hypotheses	lspval.v	|- V = ( Base ` W )
	lspval.s	|- S = ( LSubSp ` W )
	lspval.n	|- N = ( LSpan ` W )
	lspprcl.w	|- ( ph -> W e. LMod )
	lspprcl.x	|- ( ph -> X e. V )
	lspprcl.y	|- ( ph -> Y e. V )
Assertion	lspprcl	|- ( ph -> ( N ` { X , Y } ) 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		lspprcl.w	|- ( ph -> W e. LMod )
5		lspprcl.x	|- ( ph -> X e. V )
6		lspprcl.y	|- ( ph -> Y e. V )
7	5 6	prssd	|- ( ph -> { X , Y } C_ V )
8	1 2 3	lspcl	|- ( ( W e. LMod /\ { X , Y } C_ V ) -> ( N ` { X , Y } ) e. S )
9	4 7 8	syl2anc	|- ( ph -> ( N ` { X , Y } ) e. S )