Theorem lspidm

Description: The span of a set of vectors is idempotent. (Contributed by NM, 22-Feb-2014) (Revised by Mario Carneiro, 19-Jun-2014)

	Ref	Expression
Hypotheses	lspss.v	⊢ 𝑉 = ( Base ‘ 𝑊 )
	lspss.n	⊢ 𝑁 = ( LSpan ‘ 𝑊 )
Assertion	lspidm	⊢ ( ( 𝑊 ∈ LMod ∧ 𝑈 ⊆ 𝑉 ) → ( 𝑁 ‘ ( 𝑁 ‘ 𝑈 ) ) = ( 𝑁 ‘ 𝑈 ) )

Proof

Step	Hyp	Ref	Expression
1		lspss.v	⊢ 𝑉 = ( Base ‘ 𝑊 )
2		lspss.n	⊢ 𝑁 = ( LSpan ‘ 𝑊 )
3		eqid	⊢ ( LSubSp ‘ 𝑊 ) = ( LSubSp ‘ 𝑊 )
4	1 3 2	lspcl	⊢ ( ( 𝑊 ∈ LMod ∧ 𝑈 ⊆ 𝑉 ) → ( 𝑁 ‘ 𝑈 ) ∈ ( LSubSp ‘ 𝑊 ) )
5	3 2	lspid	⊢ ( ( 𝑊 ∈ LMod ∧ ( 𝑁 ‘ 𝑈 ) ∈ ( LSubSp ‘ 𝑊 ) ) → ( 𝑁 ‘ ( 𝑁 ‘ 𝑈 ) ) = ( 𝑁 ‘ 𝑈 ) )
6	4 5	syldan	⊢ ( ( 𝑊 ∈ LMod ∧ 𝑈 ⊆ 𝑉 ) → ( 𝑁 ‘ ( 𝑁 ‘ 𝑈 ) ) = ( 𝑁 ‘ 𝑈 ) )