Theorem lspval 20910

Description: The span of a set of vectors (in a left module). (spanval 31315 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 20908	. . . 4 ⊢ (𝑊 ∈ LMod → 𝑁 = (𝑠 ∈ 𝒫 𝑉 ↦ ∩ {𝑡 ∈ 𝑆 ∣ 𝑠 ⊆ 𝑡}))
5	4	fveq1d 6830	. . 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 2733	. . 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 3403	. . . 4 ⊢ (𝑠 = 𝑈 → {𝑡 ∈ 𝑆 ∣ 𝑠 ⊆ 𝑡} = {𝑡 ∈ 𝑆 ∣ 𝑈 ⊆ 𝑡})
10	9	inteqd 4902	. . 3 ⊢ (𝑠 = 𝑈 → ∩ {𝑡 ∈ 𝑆 ∣ 𝑠 ⊆ 𝑡} = ∩ {𝑡 ∈ 𝑆 ∣ 𝑈 ⊆ 𝑡})
11		simpr 484	. . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ⊆ 𝑉) → 𝑈 ⊆ 𝑉)
12	1	fvexi 6842	. . . . 5 ⊢ 𝑉 ∈ V
13	12	elpw2 5274	. . . 4 ⊢ (𝑈 ∈ 𝒫 𝑉 ↔ 𝑈 ⊆ 𝑉)
14	11, 13	sylibr 234	. . 3 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ⊆ 𝑉) → 𝑈 ∈ 𝒫 𝑉)
15	1, 2	lss1 20873	. . . . 5 ⊢ (𝑊 ∈ LMod → 𝑉 ∈ 𝑆)
16		sseq2 3957	. . . . . 6 ⊢ (𝑡 = 𝑉 → (𝑈 ⊆ 𝑡 ↔ 𝑈 ⊆ 𝑉))
17	16	rspcev 3573	. . . . 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 5287	. . . 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 6958	. 2 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ⊆ 𝑉) → ((𝑠 ∈ 𝒫 𝑉 ↦ ∩ {𝑡 ∈ 𝑆 ∣ 𝑠 ⊆ 𝑡})‘𝑈) = ∩ {𝑡 ∈ 𝑆 ∣ 𝑈 ⊆ 𝑡})
22	6, 21	eqtrd 2768	1 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ⊆ 𝑉) → (𝑁‘𝑈) = ∩ {𝑡 ∈ 𝑆 ∣ 𝑈 ⊆ 𝑡})

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2113  ∃wrex 3057  {crab 3396  Vcvv 3437   ⊆ wss 3898  𝒫 cpw 4549  ∩ cint 4897   ↦ cmpt 5174  ‘cfv 6486  Basecbs 17122  LModclmod 20795  LSubSpclss 20866  LSpanclspn 20906

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 2182  ax-ext 2705  ax-rep 5219  ax-sep 5236  ax-nul 5246  ax-pow 5305  ax-pr 5372

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 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-ral 3049  df-rex 3058  df-rmo 3347  df-reu 3348  df-rab 3397  df-v 3439  df-sbc 3738  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-if 4475  df-pw 4551  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-int 4898  df-iun 4943  df-br 5094  df-opab 5156  df-mpt 5175  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-riota 7309  df-ov 7355  df-0g 17347  df-mgm 18550  df-sgrp 18629  df-mnd 18645  df-grp 18851  df-lmod 20797  df-lss 20867  df-lsp 20907

This theorem is referenced by:  lspid  20917  lspss  20919  lspssid  20920  dochspss  41497  lcosslsp  48563