Theorem lspval 21230

Description: The span of a set of vectors (in a left module). (spanval 31917 analog.) (Contributed by NM, 8-Dec-2013.) (Revised by Mario Carneiro, 19-Jun-2014.)

Hypotheses

Ref	Expression
lspval.v	⊢ 𝑉 = (Base‘𝑊)
lspval.s	⊢ 𝑆 = (LSubSp‘𝑊)
lspval.n	⊢ 𝑁 = (LSpan‘𝑊)

Assertion

Ref	Expression
lspval	⊢ ((𝑊 ∈ LMod ∧ 𝑈 ⊆ 𝑉) → (𝑁‘𝑈) = ∩ {𝑡 ∈ 𝑆 ∣ 𝑈 ⊆ 𝑡})

Distinct variable groups:   𝑡,𝑆   𝑡,𝑈   𝑡,𝑉

Allowed substitution hints:   𝑁(𝑡)   𝑊(𝑡)

Proof of Theorem lspval

Dummy variable 𝑠 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		lspval.v	. . . . 5 ⊢ 𝑉 = (Base‘𝑊)
2		lspval.s	. . . . 5 ⊢ 𝑆 = (LSubSp‘𝑊)
3		lspval.n	. . . . 5 ⊢ 𝑁 = (LSpan‘𝑊)
4	1, 2, 3	lspfval 21228	. . . 4 ⊢ (𝑊 ∈ LMod → 𝑁 = (𝑠 ∈ 𝒫 𝑉 ↦ ∩ {𝑡 ∈ 𝑆 ∣ 𝑠 ⊆ 𝑡}))
5	4	fveq1d 6879	. . 3 ⊢ (𝑊 ∈ LMod → (𝑁‘𝑈) = ((𝑠 ∈ 𝒫 𝑉 ↦ ∩ {𝑡 ∈ 𝑆 ∣ 𝑠 ⊆ 𝑡})‘𝑈))
6	5	adantr 486	. 2 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ⊆ 𝑉) → (𝑁‘𝑈) = ((𝑠 ∈ 𝒫 𝑉 ↦ ∩ {𝑡 ∈ 𝑆 ∣ 𝑠 ⊆ 𝑡})‘𝑈))
7		eqid 2761	. . 3 ⊢ (𝑠 ∈ 𝒫 𝑉 ↦ ∩ {𝑡 ∈ 𝑆 ∣ 𝑠 ⊆ 𝑡}) = (𝑠 ∈ 𝒫 𝑉 ↦ ∩ {𝑡 ∈ 𝑆 ∣ 𝑠 ⊆ 𝑡})
8		sseq1 3956	. . . . 5 ⊢ (𝑠 = 𝑈 → (𝑠 ⊆ 𝑡 ↔ 𝑈 ⊆ 𝑡))
9	8	rabbidv 3420	. . . 4 ⊢ (𝑠 = 𝑈 → {𝑡 ∈ 𝑆 ∣ 𝑠 ⊆ 𝑡} = {𝑡 ∈ 𝑆 ∣ 𝑈 ⊆ 𝑡})
10	9	inteqd 4912	. . 3 ⊢ (𝑠 = 𝑈 → ∩ {𝑡 ∈ 𝑆 ∣ 𝑠 ⊆ 𝑡} = ∩ {𝑡 ∈ 𝑆 ∣ 𝑈 ⊆ 𝑡})
11	1	fvexi 6891	. . . . 5 ⊢ 𝑉 ∈ V
12	11	elpw2 5296	. . . 4 ⊢ (𝑈 ∈ 𝒫 𝑉 ↔ 𝑈 ⊆ 𝑉)
13	12	bilanri 512	. . 3 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ⊆ 𝑉) → 𝑈 ∈ 𝒫 𝑉)
14	1, 2	lss1 21193	. . . . 5 ⊢ (𝑊 ∈ LMod → 𝑉 ∈ 𝑆)
15		sseq2 3957	. . . . . 6 ⊢ (𝑡 = 𝑉 → (𝑈 ⊆ 𝑡 ↔ 𝑈 ⊆ 𝑉))
16	15	rspcev 3577	. . . . 5 ⊢ ((𝑉 ∈ 𝑆 ∧ 𝑈 ⊆ 𝑉) → ∃𝑡 ∈ 𝑆 𝑈 ⊆ 𝑡)
17	14, 16	sylan 592	. . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ⊆ 𝑉) → ∃𝑡 ∈ 𝑆 𝑈 ⊆ 𝑡)
18		intexrab 5308	. . . 4 ⊢ (∃𝑡 ∈ 𝑆 𝑈 ⊆ 𝑡 ↔ ∩ {𝑡 ∈ 𝑆 ∣ 𝑈 ⊆ 𝑡} ∈ V)
19	17, 18	sylib 221	. . 3 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ⊆ 𝑉) → ∩ {𝑡 ∈ 𝑆 ∣ 𝑈 ⊆ 𝑡} ∈ V)
20	7, 10, 13, 19	fvmptd3 7009	. 2 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ⊆ 𝑉) → ((𝑠 ∈ 𝒫 𝑉 ↦ ∩ {𝑡 ∈ 𝑆 ∣ 𝑠 ⊆ 𝑡})‘𝑈) = ∩ {𝑡 ∈ 𝑆 ∣ 𝑈 ⊆ 𝑡})
21	6, 20	eqtrd 2796	1 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ⊆ 𝑉) → (𝑁‘𝑈) = ∩ {𝑡 ∈ 𝑆 ∣ 𝑈 ⊆ 𝑡})

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∃wrex 3087  {crab 3413  Vcvv 3451   ⊆ wss 3899  𝒫 cpw 4557  ∩ cint 4907   ↦ cmpt 5186  ‘cfv 6531  Basecbs 17367  LModclmod 21115  LSubSpclss 21186  LSpanclspn 21226

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6487  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-riota 7369  df-ov 7415  df-0g 17592  df-mgm 18796  df-sgrp 18888  df-mnd 18904  df-grp 19127  df-lmod 21117  df-lss 21187  df-lsp 21227

This theorem is used by:  lspid  21237  lspss  21239  lspssid  21240  rspvalint  21503  dochspss  42403  lcosslsp  49494