Theorem lsptpcl

Description: The span of an unordered triple is a subspace (frequently used special case of lspcl ). (Contributed by NM, 22-May-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 )
	lsptpcl.z	|- ( ph -> Z e. V )
Assertion	lsptpcl	|- ( ph -> ( N ` { X , Y , Z } ) 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		lsptpcl.z	|- ( ph -> Z e. V )
8		df-tp	|- { X , Y , Z } = ( { X , Y } u. { Z } )
9	5 6	prssd	|- ( ph -> { X , Y } C_ V )
10	7	snssd	|- ( ph -> { Z } C_ V )
11	9 10	unssd	|- ( ph -> ( { X , Y } u. { Z } ) C_ V )
12	8 11	eqsstrid	|- ( ph -> { X , Y , Z } C_ V )
13	1 2 3	lspcl	|- ( ( W e. LMod /\ { X , Y , Z } C_ V ) -> ( N ` { X , Y , Z } ) e. S )
14	4 12 13	syl2anc	|- ( ph -> ( N ` { X , Y , Z } ) e. S )