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Mirrors > Home > MPE Home > Th. List > lspval | Structured version Visualization version GIF version |
Description: The span of a set of vectors (in a left module). (spanval 29109 analog.) (Contributed by NM, 8-Dec-2013.) (Revised by Mario Carneiro, 19-Jun-2014.) |
Ref | Expression |
---|---|
lspval.v | ⊢ 𝑉 = (Base‘𝑊) |
lspval.s | ⊢ 𝑆 = (LSubSp‘𝑊) |
lspval.n | ⊢ 𝑁 = (LSpan‘𝑊) |
Ref | Expression |
---|---|
lspval | ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ⊆ 𝑉) → (𝑁‘𝑈) = ∩ {𝑡 ∈ 𝑆 ∣ 𝑈 ⊆ 𝑡}) |
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 19744 | . . . 4 ⊢ (𝑊 ∈ LMod → 𝑁 = (𝑠 ∈ 𝒫 𝑉 ↦ ∩ {𝑡 ∈ 𝑆 ∣ 𝑠 ⊆ 𝑡})) |
5 | 4 | fveq1d 6671 | . . 3 ⊢ (𝑊 ∈ LMod → (𝑁‘𝑈) = ((𝑠 ∈ 𝒫 𝑉 ↦ ∩ {𝑡 ∈ 𝑆 ∣ 𝑠 ⊆ 𝑡})‘𝑈)) |
6 | 5 | adantr 483 | . 2 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ⊆ 𝑉) → (𝑁‘𝑈) = ((𝑠 ∈ 𝒫 𝑉 ↦ ∩ {𝑡 ∈ 𝑆 ∣ 𝑠 ⊆ 𝑡})‘𝑈)) |
7 | eqid 2821 | . . 3 ⊢ (𝑠 ∈ 𝒫 𝑉 ↦ ∩ {𝑡 ∈ 𝑆 ∣ 𝑠 ⊆ 𝑡}) = (𝑠 ∈ 𝒫 𝑉 ↦ ∩ {𝑡 ∈ 𝑆 ∣ 𝑠 ⊆ 𝑡}) | |
8 | sseq1 3991 | . . . . 5 ⊢ (𝑠 = 𝑈 → (𝑠 ⊆ 𝑡 ↔ 𝑈 ⊆ 𝑡)) | |
9 | 8 | rabbidv 3480 | . . . 4 ⊢ (𝑠 = 𝑈 → {𝑡 ∈ 𝑆 ∣ 𝑠 ⊆ 𝑡} = {𝑡 ∈ 𝑆 ∣ 𝑈 ⊆ 𝑡}) |
10 | 9 | inteqd 4880 | . . 3 ⊢ (𝑠 = 𝑈 → ∩ {𝑡 ∈ 𝑆 ∣ 𝑠 ⊆ 𝑡} = ∩ {𝑡 ∈ 𝑆 ∣ 𝑈 ⊆ 𝑡}) |
11 | simpr 487 | . . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ⊆ 𝑉) → 𝑈 ⊆ 𝑉) | |
12 | 1 | fvexi 6683 | . . . . 5 ⊢ 𝑉 ∈ V |
13 | 12 | elpw2 5247 | . . . 4 ⊢ (𝑈 ∈ 𝒫 𝑉 ↔ 𝑈 ⊆ 𝑉) |
14 | 11, 13 | sylibr 236 | . . 3 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ⊆ 𝑉) → 𝑈 ∈ 𝒫 𝑉) |
15 | 1, 2 | lss1 19709 | . . . . 5 ⊢ (𝑊 ∈ LMod → 𝑉 ∈ 𝑆) |
16 | sseq2 3992 | . . . . . 6 ⊢ (𝑡 = 𝑉 → (𝑈 ⊆ 𝑡 ↔ 𝑈 ⊆ 𝑉)) | |
17 | 16 | rspcev 3622 | . . . . 5 ⊢ ((𝑉 ∈ 𝑆 ∧ 𝑈 ⊆ 𝑉) → ∃𝑡 ∈ 𝑆 𝑈 ⊆ 𝑡) |
18 | 15, 17 | sylan 582 | . . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ⊆ 𝑉) → ∃𝑡 ∈ 𝑆 𝑈 ⊆ 𝑡) |
19 | intexrab 5242 | . . . 4 ⊢ (∃𝑡 ∈ 𝑆 𝑈 ⊆ 𝑡 ↔ ∩ {𝑡 ∈ 𝑆 ∣ 𝑈 ⊆ 𝑡} ∈ V) | |
20 | 18, 19 | sylib 220 | . . 3 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ⊆ 𝑉) → ∩ {𝑡 ∈ 𝑆 ∣ 𝑈 ⊆ 𝑡} ∈ V) |
21 | 7, 10, 14, 20 | fvmptd3 6790 | . 2 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ⊆ 𝑉) → ((𝑠 ∈ 𝒫 𝑉 ↦ ∩ {𝑡 ∈ 𝑆 ∣ 𝑠 ⊆ 𝑡})‘𝑈) = ∩ {𝑡 ∈ 𝑆 ∣ 𝑈 ⊆ 𝑡}) |
22 | 6, 21 | eqtrd 2856 | 1 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ⊆ 𝑉) → (𝑁‘𝑈) = ∩ {𝑡 ∈ 𝑆 ∣ 𝑈 ⊆ 𝑡}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ∃wrex 3139 {crab 3142 Vcvv 3494 ⊆ wss 3935 𝒫 cpw 4538 ∩ cint 4875 ↦ cmpt 5145 ‘cfv 6354 Basecbs 16482 LModclmod 19633 LSubSpclss 19702 LSpanclspn 19742 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5189 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-op 4573 df-uni 4838 df-int 4876 df-iun 4920 df-br 5066 df-opab 5128 df-mpt 5146 df-id 5459 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-riota 7113 df-ov 7158 df-0g 16714 df-mgm 17851 df-sgrp 17900 df-mnd 17911 df-grp 18105 df-lmod 19635 df-lss 19703 df-lsp 19743 |
This theorem is referenced by: lspid 19753 lspss 19755 lspssid 19756 dochspss 38513 lcosslsp 44492 |
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