Theorem lspval 20926

Description: The span of a set of vectors (in a left module). (spanval 31408 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 20924	. . . 4 ⊢ (𝑊 ∈ LMod → 𝑁 = (𝑠 ∈ 𝒫 𝑉 ↦ ∩ {𝑡 ∈ 𝑆 ∣ 𝑠 ⊆ 𝑡}))
5	4	fveq1d 6836	. . 3 ⊢ (𝑊 ∈ LMod → (𝑁‘𝑈) = ((𝑠 ∈ 𝒫 𝑉 ↦ ∩ {𝑡 ∈ 𝑆 ∣ 𝑠 ⊆ 𝑡})‘𝑈))
6	5	adantr 480	. 2 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ⊆ 𝑉) → (𝑁‘𝑈) = ((𝑠 ∈ 𝒫 𝑉 ↦ ∩ {𝑡 ∈ 𝑆 ∣ 𝑠 ⊆ 𝑡})‘𝑈))
7		eqid 2736	. . 3 ⊢ (𝑠 ∈ 𝒫 𝑉 ↦ ∩ {𝑡 ∈ 𝑆 ∣ 𝑠 ⊆ 𝑡}) = (𝑠 ∈ 𝒫 𝑉 ↦ ∩ {𝑡 ∈ 𝑆 ∣ 𝑠 ⊆ 𝑡})
8		sseq1 3959	. . . . 5 ⊢ (𝑠 = 𝑈 → (𝑠 ⊆ 𝑡 ↔ 𝑈 ⊆ 𝑡))
9	8	rabbidv 3406	. . . 4 ⊢ (𝑠 = 𝑈 → {𝑡 ∈ 𝑆 ∣ 𝑠 ⊆ 𝑡} = {𝑡 ∈ 𝑆 ∣ 𝑈 ⊆ 𝑡})
10	9	inteqd 4907	. . 3 ⊢ (𝑠 = 𝑈 → ∩ {𝑡 ∈ 𝑆 ∣ 𝑠 ⊆ 𝑡} = ∩ {𝑡 ∈ 𝑆 ∣ 𝑈 ⊆ 𝑡})
11		simpr 484	. . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ⊆ 𝑉) → 𝑈 ⊆ 𝑉)
12	1	fvexi 6848	. . . . 5 ⊢ 𝑉 ∈ V
13	12	elpw2 5279	. . . 4 ⊢ (𝑈 ∈ 𝒫 𝑉 ↔ 𝑈 ⊆ 𝑉)
14	11, 13	sylibr 234	. . 3 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ⊆ 𝑉) → 𝑈 ∈ 𝒫 𝑉)
15	1, 2	lss1 20889	. . . . 5 ⊢ (𝑊 ∈ LMod → 𝑉 ∈ 𝑆)
16		sseq2 3960	. . . . . 6 ⊢ (𝑡 = 𝑉 → (𝑈 ⊆ 𝑡 ↔ 𝑈 ⊆ 𝑉))
17	16	rspcev 3576	. . . . 5 ⊢ ((𝑉 ∈ 𝑆 ∧ 𝑈 ⊆ 𝑉) → ∃𝑡 ∈ 𝑆 𝑈 ⊆ 𝑡)
18	15, 17	sylan 580	. . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ⊆ 𝑉) → ∃𝑡 ∈ 𝑆 𝑈 ⊆ 𝑡)
19		intexrab 5292	. . . 4 ⊢ (∃𝑡 ∈ 𝑆 𝑈 ⊆ 𝑡 ↔ ∩ {𝑡 ∈ 𝑆 ∣ 𝑈 ⊆ 𝑡} ∈ V)
20	18, 19	sylib 218	. . 3 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ⊆ 𝑉) → ∩ {𝑡 ∈ 𝑆 ∣ 𝑈 ⊆ 𝑡} ∈ V)
21	7, 10, 14, 20	fvmptd3 6964	. 2 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ⊆ 𝑉) → ((𝑠 ∈ 𝒫 𝑉 ↦ ∩ {𝑡 ∈ 𝑆 ∣ 𝑠 ⊆ 𝑡})‘𝑈) = ∩ {𝑡 ∈ 𝑆 ∣ 𝑈 ⊆ 𝑡})
22	6, 21	eqtrd 2771	1 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ⊆ 𝑉) → (𝑁‘𝑈) = ∩ {𝑡 ∈ 𝑆 ∣ 𝑈 ⊆ 𝑡})

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2113  ∃wrex 3060  {crab 3399  Vcvv 3440   ⊆ wss 3901  𝒫 cpw 4554  ∩ cint 4902   ↦ cmpt 5179  ‘cfv 6492  Basecbs 17136  LModclmod 20811  LSubSpclss 20882  LSpanclspn 20922

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-rep 5224  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rmo 3350  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-int 4903  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7315  df-ov 7361  df-0g 17361  df-mgm 18565  df-sgrp 18644  df-mnd 18660  df-grp 18866  df-lmod 20813  df-lss 20883  df-lsp 20923

This theorem is referenced by:  lspid  20933  lspss  20935  lspssid  20936  dochspss  41634  lcosslsp  48680