Theorem lsatssv

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

	Ref	Expression
Hypotheses	lsatssv.v	⊢ 𝑉 = ( Base ‘ 𝑊 )
	lsatssv.a	⊢ 𝐴 = ( LSAtoms ‘ 𝑊 )
	lsatssv.w	⊢ ( 𝜑 → 𝑊 ∈ LMod )
	lsatssv.g	⊢ ( 𝜑 → 𝑄 ∈ 𝐴 )
Assertion	lsatssv	⊢ ( 𝜑 → 𝑄 ⊆ 𝑉 )

Proof

Step	Hyp	Ref	Expression
1		lsatssv.v	⊢ 𝑉 = ( Base ‘ 𝑊 )
2		lsatssv.a	⊢ 𝐴 = ( LSAtoms ‘ 𝑊 )
3		lsatssv.w	⊢ ( 𝜑 → 𝑊 ∈ LMod )
4		lsatssv.g	⊢ ( 𝜑 → 𝑄 ∈ 𝐴 )
5		eqid	⊢ ( LSubSp ‘ 𝑊 ) = ( LSubSp ‘ 𝑊 )
6	5 2 3 4	lsatlssel	⊢ ( 𝜑 → 𝑄 ∈ ( LSubSp ‘ 𝑊 ) )
7	1 5	lssss	⊢ ( 𝑄 ∈ ( LSubSp ‘ 𝑊 ) → 𝑄 ⊆ 𝑉 )
8	6 7	syl	⊢ ( 𝜑 → 𝑄 ⊆ 𝑉 )