Theorem lspsnss2

Description: Comparable spans of singletons must have proportional vectors. See lspsneq for equal span version. (Contributed by NM, 7-Jun-2015)

	Ref	Expression
Hypotheses	lspsnss2.v	|- V = ( Base ` W )
	lspsnss2.s	|- S = ( Scalar ` W )
	lspsnss2.k	|- K = ( Base ` S )
	lspsnss2.t	|- .x. = ( .s ` W )
	lspsnss2.n	|- N = ( LSpan ` W )
	lspsnss2.w	|- ( ph -> W e. LMod )
	lspsnss2.x	|- ( ph -> X e. V )
	lspsnss2.y	|- ( ph -> Y e. V )
Assertion	lspsnss2	|- ( ph -> ( ( N ` { X } ) C_ ( N ` { Y } ) <-> E. k e. K X = ( k .x. Y ) ) )

Proof

Step	Hyp	Ref	Expression
1		lspsnss2.v	|- V = ( Base ` W )
2		lspsnss2.s	|- S = ( Scalar ` W )
3		lspsnss2.k	|- K = ( Base ` S )
4		lspsnss2.t	|- .x. = ( .s ` W )
5		lspsnss2.n	|- N = ( LSpan ` W )
6		lspsnss2.w	|- ( ph -> W e. LMod )
7		lspsnss2.x	|- ( ph -> X e. V )
8		lspsnss2.y	|- ( ph -> Y e. V )
9		eqid	|- ( LSubSp ` W ) = ( LSubSp ` W )
10	1 9 5	lspsncl	|- ( ( W e. LMod /\ Y e. V ) -> ( N ` { Y } ) e. ( LSubSp ` W ) )
11	6 8 10	syl2anc	|- ( ph -> ( N ` { Y } ) e. ( LSubSp ` W ) )
12	1 9 5 6 11 7	ellspsn5b	|- ( ph -> ( X e. ( N ` { Y } ) <-> ( N ` { X } ) C_ ( N ` { Y } ) ) )
13	2 3 1 4 5	ellspsn	|- ( ( W e. LMod /\ Y e. V ) -> ( X e. ( N ` { Y } ) <-> E. k e. K X = ( k .x. Y ) ) )
14	6 8 13	syl2anc	|- ( ph -> ( X e. ( N ` { Y } ) <-> E. k e. K X = ( k .x. Y ) ) )
15	12 14	bitr3d	|- ( ph -> ( ( N ` { X } ) C_ ( N ` { Y } ) <-> E. k e. K X = ( k .x. Y ) ) )