Theorem lsssssubg

Description: All subspaces are subgroups. (Contributed by Mario Carneiro, 19-Apr-2016)

	Ref	Expression
Hypothesis	lsssubg.s	$$⊢S=\mathrm{LSubSp}\left(W\right)$$
Assertion	lsssssubg	$$⊢W\in \mathrm{LMod}\to S\subseteq \mathrm{SubGrp}\left(W\right)$$

Proof

Step	Hyp	Ref	Expression
1		lsssubg.s	$$⊢S=\mathrm{LSubSp}\left(W\right)$$
2	1	lsssubg	$$⊢\left(W\in \mathrm{LMod}\wedge x\in S\right)\to x\in \mathrm{SubGrp}\left(W\right)$$
3	2	ex	$$⊢W\in \mathrm{LMod}\to \left(x\in S\to x\in \mathrm{SubGrp}\left(W\right)\right)$$
4	3	ssrdv	$$⊢W\in \mathrm{LMod}\to S\subseteq \mathrm{SubGrp}\left(W\right)$$