Theorem lsatssv

Description: An atom is a set of vectors. (Contributed by NM, 27-Feb-2015)

	Ref	Expression
Hypotheses	lsatssv.v	$$⊢V={\mathrm{Base}}_W$$
	lsatssv.a	$$⊢A=\mathrm{LSAtoms}\left(W\right)$$
	lsatssv.w	$$⊢\phi \to W\in \mathrm{LMod}$$
	lsatssv.g	$$⊢\phi \to Q\in A$$
Assertion	lsatssv	$$⊢\phi \to Q\subseteq V$$

Proof

Step	Hyp	Ref	Expression
1		lsatssv.v	$$⊢V={\mathrm{Base}}_W$$
2		lsatssv.a	$$⊢A=\mathrm{LSAtoms}\left(W\right)$$
3		lsatssv.w	$$⊢\phi \to W\in \mathrm{LMod}$$
4		lsatssv.g	$$⊢\phi \to Q\in A$$
5		eqid	$$⊢\mathrm{LSubSp}\left(W\right)=\mathrm{LSubSp}\left(W\right)$$
6	5 2 3 4	lsatlssel	$$⊢\phi \to Q\in \mathrm{LSubSp}\left(W\right)$$
7	1 5	lssss	$$⊢Q\in \mathrm{LSubSp}\left(W\right)\to Q\subseteq V$$
8	6 7	syl	$$⊢\phi \to Q\subseteq V$$