Theorem lsatlssel

Description: An atom is a subspace. (Contributed by NM, 25-Aug-2014)

	Ref	Expression
Hypotheses	lsatlss.s	⊢ 𝑆 = ( LSubSp ‘ 𝑊 )
	lsatlss.a	⊢ 𝐴 = ( LSAtoms ‘ 𝑊 )
	lssatssel.w	⊢ ( 𝜑 → 𝑊 ∈ LMod )
	lssatssel.u	⊢ ( 𝜑 → 𝑈 ∈ 𝐴 )
Assertion	lsatlssel	⊢ ( 𝜑 → 𝑈 ∈ 𝑆 )

Proof

Step	Hyp	Ref	Expression
1		lsatlss.s	⊢ 𝑆 = ( LSubSp ‘ 𝑊 )
2		lsatlss.a	⊢ 𝐴 = ( LSAtoms ‘ 𝑊 )
3		lssatssel.w	⊢ ( 𝜑 → 𝑊 ∈ LMod )
4		lssatssel.u	⊢ ( 𝜑 → 𝑈 ∈ 𝐴 )
5	1 2	lsatlss	⊢ ( 𝑊 ∈ LMod → 𝐴 ⊆ 𝑆 )
6	3 5	syl	⊢ ( 𝜑 → 𝐴 ⊆ 𝑆 )
7	6 4	sseldd	⊢ ( 𝜑 → 𝑈 ∈ 𝑆 )