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Mirrors > Home > MPE Home > Th. List > lspfval | Structured version Visualization version GIF version |
Description: The span function for a left vector space (or a left module). (df-span 29013 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 |
---|---|
lspfval | ⊢ (𝑊 ∈ 𝑋 → 𝑁 = (𝑠 ∈ 𝒫 𝑉 ↦ ∩ {𝑡 ∈ 𝑆 ∣ 𝑠 ⊆ 𝑡})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lspval.n | . 2 ⊢ 𝑁 = (LSpan‘𝑊) | |
2 | elex 3510 | . . 3 ⊢ (𝑊 ∈ 𝑋 → 𝑊 ∈ V) | |
3 | fveq2 6663 | . . . . . . 7 ⊢ (𝑤 = 𝑊 → (Base‘𝑤) = (Base‘𝑊)) | |
4 | lspval.v | . . . . . . 7 ⊢ 𝑉 = (Base‘𝑊) | |
5 | 3, 4 | syl6eqr 2871 | . . . . . 6 ⊢ (𝑤 = 𝑊 → (Base‘𝑤) = 𝑉) |
6 | 5 | pweqd 4540 | . . . . 5 ⊢ (𝑤 = 𝑊 → 𝒫 (Base‘𝑤) = 𝒫 𝑉) |
7 | fveq2 6663 | . . . . . . . 8 ⊢ (𝑤 = 𝑊 → (LSubSp‘𝑤) = (LSubSp‘𝑊)) | |
8 | lspval.s | . . . . . . . 8 ⊢ 𝑆 = (LSubSp‘𝑊) | |
9 | 7, 8 | syl6eqr 2871 | . . . . . . 7 ⊢ (𝑤 = 𝑊 → (LSubSp‘𝑤) = 𝑆) |
10 | 9 | rabeqdv 3482 | . . . . . 6 ⊢ (𝑤 = 𝑊 → {𝑡 ∈ (LSubSp‘𝑤) ∣ 𝑠 ⊆ 𝑡} = {𝑡 ∈ 𝑆 ∣ 𝑠 ⊆ 𝑡}) |
11 | 10 | inteqd 4872 | . . . . 5 ⊢ (𝑤 = 𝑊 → ∩ {𝑡 ∈ (LSubSp‘𝑤) ∣ 𝑠 ⊆ 𝑡} = ∩ {𝑡 ∈ 𝑆 ∣ 𝑠 ⊆ 𝑡}) |
12 | 6, 11 | mpteq12dv 5142 | . . . 4 ⊢ (𝑤 = 𝑊 → (𝑠 ∈ 𝒫 (Base‘𝑤) ↦ ∩ {𝑡 ∈ (LSubSp‘𝑤) ∣ 𝑠 ⊆ 𝑡}) = (𝑠 ∈ 𝒫 𝑉 ↦ ∩ {𝑡 ∈ 𝑆 ∣ 𝑠 ⊆ 𝑡})) |
13 | df-lsp 19673 | . . . 4 ⊢ LSpan = (𝑤 ∈ V ↦ (𝑠 ∈ 𝒫 (Base‘𝑤) ↦ ∩ {𝑡 ∈ (LSubSp‘𝑤) ∣ 𝑠 ⊆ 𝑡})) | |
14 | 4 | fvexi 6677 | . . . . . 6 ⊢ 𝑉 ∈ V |
15 | 14 | pwex 5272 | . . . . 5 ⊢ 𝒫 𝑉 ∈ V |
16 | 15 | mptex 6977 | . . . 4 ⊢ (𝑠 ∈ 𝒫 𝑉 ↦ ∩ {𝑡 ∈ 𝑆 ∣ 𝑠 ⊆ 𝑡}) ∈ V |
17 | 12, 13, 16 | fvmpt 6761 | . . 3 ⊢ (𝑊 ∈ V → (LSpan‘𝑊) = (𝑠 ∈ 𝒫 𝑉 ↦ ∩ {𝑡 ∈ 𝑆 ∣ 𝑠 ⊆ 𝑡})) |
18 | 2, 17 | syl 17 | . 2 ⊢ (𝑊 ∈ 𝑋 → (LSpan‘𝑊) = (𝑠 ∈ 𝒫 𝑉 ↦ ∩ {𝑡 ∈ 𝑆 ∣ 𝑠 ⊆ 𝑡})) |
19 | 1, 18 | syl5eq 2865 | 1 ⊢ (𝑊 ∈ 𝑋 → 𝑁 = (𝑠 ∈ 𝒫 𝑉 ↦ ∩ {𝑡 ∈ 𝑆 ∣ 𝑠 ⊆ 𝑡})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1528 ∈ wcel 2105 {crab 3139 Vcvv 3492 ⊆ wss 3933 𝒫 cpw 4535 ∩ cint 4867 ↦ cmpt 5137 ‘cfv 6348 Basecbs 16471 LSubSpclss 19632 LSpanclspn 19672 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-lsp 19673 |
This theorem is referenced by: lspf 19675 lspval 19676 00lsp 19682 mrclsp 19690 lsppropd 19719 |
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