Theorem lspval

Description: The span of a set of vectors (in a left module). ( spanval analog.) (Contributed by NM, 8-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	lspval	⊢ ( ( 𝑊 ∈ LMod ∧ 𝑈 ⊆ 𝑉 ) → ( 𝑁 ‘ 𝑈 ) = ∩ { 𝑡 ∈ 𝑆 ∣ 𝑈 ⊆ 𝑡 } )

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	lspfval	⊢ ( 𝑊 ∈ LMod → 𝑁 = ( 𝑠 ∈ 𝒫 𝑉 ↦ ∩ { 𝑡 ∈ 𝑆 ∣ 𝑠 ⊆ 𝑡 } ) )
5	4	fveq1d	⊢ ( 𝑊 ∈ LMod → ( 𝑁 ‘ 𝑈 ) = ( ( 𝑠 ∈ 𝒫 𝑉 ↦ ∩ { 𝑡 ∈ 𝑆 ∣ 𝑠 ⊆ 𝑡 } ) ‘ 𝑈 ) )
6	5	adantr	⊢ ( ( 𝑊 ∈ LMod ∧ 𝑈 ⊆ 𝑉 ) → ( 𝑁 ‘ 𝑈 ) = ( ( 𝑠 ∈ 𝒫 𝑉 ↦ ∩ { 𝑡 ∈ 𝑆 ∣ 𝑠 ⊆ 𝑡 } ) ‘ 𝑈 ) )
7		eqid	⊢ ( 𝑠 ∈ 𝒫 𝑉 ↦ ∩ { 𝑡 ∈ 𝑆 ∣ 𝑠 ⊆ 𝑡 } ) = ( 𝑠 ∈ 𝒫 𝑉 ↦ ∩ { 𝑡 ∈ 𝑆 ∣ 𝑠 ⊆ 𝑡 } )
8		sseq1	⊢ ( 𝑠 = 𝑈 → ( 𝑠 ⊆ 𝑡 ↔ 𝑈 ⊆ 𝑡 ) )
9	8	rabbidv	⊢ ( 𝑠 = 𝑈 → { 𝑡 ∈ 𝑆 ∣ 𝑠 ⊆ 𝑡 } = { 𝑡 ∈ 𝑆 ∣ 𝑈 ⊆ 𝑡 } )
10	9	inteqd	⊢ ( 𝑠 = 𝑈 → ∩ { 𝑡 ∈ 𝑆 ∣ 𝑠 ⊆ 𝑡 } = ∩ { 𝑡 ∈ 𝑆 ∣ 𝑈 ⊆ 𝑡 } )
11		simpr	⊢ ( ( 𝑊 ∈ LMod ∧ 𝑈 ⊆ 𝑉 ) → 𝑈 ⊆ 𝑉 )
12	1	fvexi	⊢ 𝑉 ∈ V
13	12	elpw2	⊢ ( 𝑈 ∈ 𝒫 𝑉 ↔ 𝑈 ⊆ 𝑉 )
14	11 13	sylibr	⊢ ( ( 𝑊 ∈ LMod ∧ 𝑈 ⊆ 𝑉 ) → 𝑈 ∈ 𝒫 𝑉 )
15	1 2	lss1	⊢ ( 𝑊 ∈ LMod → 𝑉 ∈ 𝑆 )
16		sseq2	⊢ ( 𝑡 = 𝑉 → ( 𝑈 ⊆ 𝑡 ↔ 𝑈 ⊆ 𝑉 ) )
17	16	rspcev	⊢ ( ( 𝑉 ∈ 𝑆 ∧ 𝑈 ⊆ 𝑉 ) → ∃ 𝑡 ∈ 𝑆 𝑈 ⊆ 𝑡 )
18	15 17	sylan	⊢ ( ( 𝑊 ∈ LMod ∧ 𝑈 ⊆ 𝑉 ) → ∃ 𝑡 ∈ 𝑆 𝑈 ⊆ 𝑡 )
19		intexrab	⊢ ( ∃ 𝑡 ∈ 𝑆 𝑈 ⊆ 𝑡 ↔ ∩ { 𝑡 ∈ 𝑆 ∣ 𝑈 ⊆ 𝑡 } ∈ V )
20	18 19	sylib	⊢ ( ( 𝑊 ∈ LMod ∧ 𝑈 ⊆ 𝑉 ) → ∩ { 𝑡 ∈ 𝑆 ∣ 𝑈 ⊆ 𝑡 } ∈ V )
21	7 10 14 20	fvmptd3	⊢ ( ( 𝑊 ∈ LMod ∧ 𝑈 ⊆ 𝑉 ) → ( ( 𝑠 ∈ 𝒫 𝑉 ↦ ∩ { 𝑡 ∈ 𝑆 ∣ 𝑠 ⊆ 𝑡 } ) ‘ 𝑈 ) = ∩ { 𝑡 ∈ 𝑆 ∣ 𝑈 ⊆ 𝑡 } )
22	6 21	eqtrd	⊢ ( ( 𝑊 ∈ LMod ∧ 𝑈 ⊆ 𝑉 ) → ( 𝑁 ‘ 𝑈 ) = ∩ { 𝑡 ∈ 𝑆 ∣ 𝑈 ⊆ 𝑡 } )