Theorem lspid

Description: The span of a subspace is itself. ( spanid analog.) (Contributed by NM, 15-Dec-2013) (Revised by Mario Carneiro, 19-Jun-2014)

	Ref	Expression
Hypotheses	lspid.s	⊢ 𝑆 = ( LSubSp ‘ 𝑊 )
	lspid.n	⊢ 𝑁 = ( LSpan ‘ 𝑊 )
Assertion	lspid	⊢ ( ( 𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆 ) → ( 𝑁 ‘ 𝑈 ) = 𝑈 )

Proof

Step	Hyp	Ref	Expression
1		lspid.s	⊢ 𝑆 = ( LSubSp ‘ 𝑊 )
2		lspid.n	⊢ 𝑁 = ( LSpan ‘ 𝑊 )
3		eqid	⊢ ( Base ‘ 𝑊 ) = ( Base ‘ 𝑊 )
4	3 1	lssss	⊢ ( 𝑈 ∈ 𝑆 → 𝑈 ⊆ ( Base ‘ 𝑊 ) )
5	3 1 2	lspval	⊢ ( ( 𝑊 ∈ LMod ∧ 𝑈 ⊆ ( Base ‘ 𝑊 ) ) → ( 𝑁 ‘ 𝑈 ) = ∩ { 𝑡 ∈ 𝑆 ∣ 𝑈 ⊆ 𝑡 } )
6	4 5	sylan2	⊢ ( ( 𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆 ) → ( 𝑁 ‘ 𝑈 ) = ∩ { 𝑡 ∈ 𝑆 ∣ 𝑈 ⊆ 𝑡 } )
7		intmin	⊢ ( 𝑈 ∈ 𝑆 → ∩ { 𝑡 ∈ 𝑆 ∣ 𝑈 ⊆ 𝑡 } = 𝑈 )
8	7	adantl	⊢ ( ( 𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆 ) → ∩ { 𝑡 ∈ 𝑆 ∣ 𝑈 ⊆ 𝑡 } = 𝑈 )
9	6 8	eqtrd	⊢ ( ( 𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆 ) → ( 𝑁 ‘ 𝑈 ) = 𝑈 )