Theorem lspcl

Description: The span of a set of vectors is a subspace. ( spancl analog.) (Contributed by NM, 9-Dec-2013) (Revised by Mario Carneiro, 19-Jun-2014)

	Ref	Expression
Hypotheses	lspval.v	⊢ 𝑉 = ( Base ‘ 𝑊 )
	lspval.s	⊢ 𝑆 = ( LSubSp ‘ 𝑊 )
	lspval.n	⊢ 𝑁 = ( LSpan ‘ 𝑊 )
Assertion	lspcl	⊢ ( ( 𝑊 ∈ LMod ∧ 𝑈 ⊆ 𝑉 ) → ( 𝑁 ‘ 𝑈 ) ∈ 𝑆 )

Proof

Step	Hyp	Ref	Expression
1		lspval.v	⊢ 𝑉 = ( Base ‘ 𝑊 )
2		lspval.s	⊢ 𝑆 = ( LSubSp ‘ 𝑊 )
3		lspval.n	⊢ 𝑁 = ( LSpan ‘ 𝑊 )
4	1 2 3	lspf	⊢ ( 𝑊 ∈ LMod → 𝑁 : 𝒫 𝑉 ⟶ 𝑆 )
5	1	fvexi	⊢ 𝑉 ∈ V
6	5	elpw2	⊢ ( 𝑈 ∈ 𝒫 𝑉 ↔ 𝑈 ⊆ 𝑉 )
7	6	biimpri	⊢ ( 𝑈 ⊆ 𝑉 → 𝑈 ∈ 𝒫 𝑉 )
8		ffvelcdm	⊢ ( ( 𝑁 : 𝒫 𝑉 ⟶ 𝑆 ∧ 𝑈 ∈ 𝒫 𝑉 ) → ( 𝑁 ‘ 𝑈 ) ∈ 𝑆 )
9	4 7 8	syl2an	⊢ ( ( 𝑊 ∈ LMod ∧ 𝑈 ⊆ 𝑉 ) → ( 𝑁 ‘ 𝑈 ) ∈ 𝑆 )