Theorem lsssssubg

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

	Ref	Expression
Hypothesis	lsssubg.s	⊢ 𝑆 = ( LSubSp ‘ 𝑊 )
Assertion	lsssssubg	⊢ ( 𝑊 ∈ LMod → 𝑆 ⊆ ( SubGrp ‘ 𝑊 ) )

Proof

Step	Hyp	Ref	Expression
1		lsssubg.s	⊢ 𝑆 = ( LSubSp ‘ 𝑊 )
2	1	lsssubg	⊢ ( ( 𝑊 ∈ LMod ∧ 𝑥 ∈ 𝑆 ) → 𝑥 ∈ ( SubGrp ‘ 𝑊 ) )
3	2	ex	⊢ ( 𝑊 ∈ LMod → ( 𝑥 ∈ 𝑆 → 𝑥 ∈ ( SubGrp ‘ 𝑊 ) ) )
4	3	ssrdv	⊢ ( 𝑊 ∈ LMod → 𝑆 ⊆ ( SubGrp ‘ 𝑊 ) )