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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 29092 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 3459 | . . 3 ⊢ (𝑊 ∈ 𝑋 → 𝑊 ∈ V) | |
3 | fveq2 6645 | . . . . . . 7 ⊢ (𝑤 = 𝑊 → (Base‘𝑤) = (Base‘𝑊)) | |
4 | lspval.v | . . . . . . 7 ⊢ 𝑉 = (Base‘𝑊) | |
5 | 3, 4 | eqtr4di 2851 | . . . . . 6 ⊢ (𝑤 = 𝑊 → (Base‘𝑤) = 𝑉) |
6 | 5 | pweqd 4516 | . . . . 5 ⊢ (𝑤 = 𝑊 → 𝒫 (Base‘𝑤) = 𝒫 𝑉) |
7 | fveq2 6645 | . . . . . . . 8 ⊢ (𝑤 = 𝑊 → (LSubSp‘𝑤) = (LSubSp‘𝑊)) | |
8 | lspval.s | . . . . . . . 8 ⊢ 𝑆 = (LSubSp‘𝑊) | |
9 | 7, 8 | eqtr4di 2851 | . . . . . . 7 ⊢ (𝑤 = 𝑊 → (LSubSp‘𝑤) = 𝑆) |
10 | 9 | rabeqdv 3432 | . . . . . 6 ⊢ (𝑤 = 𝑊 → {𝑡 ∈ (LSubSp‘𝑤) ∣ 𝑠 ⊆ 𝑡} = {𝑡 ∈ 𝑆 ∣ 𝑠 ⊆ 𝑡}) |
11 | 10 | inteqd 4843 | . . . . 5 ⊢ (𝑤 = 𝑊 → ∩ {𝑡 ∈ (LSubSp‘𝑤) ∣ 𝑠 ⊆ 𝑡} = ∩ {𝑡 ∈ 𝑆 ∣ 𝑠 ⊆ 𝑡}) |
12 | 6, 11 | mpteq12dv 5115 | . . . 4 ⊢ (𝑤 = 𝑊 → (𝑠 ∈ 𝒫 (Base‘𝑤) ↦ ∩ {𝑡 ∈ (LSubSp‘𝑤) ∣ 𝑠 ⊆ 𝑡}) = (𝑠 ∈ 𝒫 𝑉 ↦ ∩ {𝑡 ∈ 𝑆 ∣ 𝑠 ⊆ 𝑡})) |
13 | df-lsp 19737 | . . . 4 ⊢ LSpan = (𝑤 ∈ V ↦ (𝑠 ∈ 𝒫 (Base‘𝑤) ↦ ∩ {𝑡 ∈ (LSubSp‘𝑤) ∣ 𝑠 ⊆ 𝑡})) | |
14 | 4 | fvexi 6659 | . . . . . 6 ⊢ 𝑉 ∈ V |
15 | 14 | pwex 5246 | . . . . 5 ⊢ 𝒫 𝑉 ∈ V |
16 | 15 | mptex 6963 | . . . 4 ⊢ (𝑠 ∈ 𝒫 𝑉 ↦ ∩ {𝑡 ∈ 𝑆 ∣ 𝑠 ⊆ 𝑡}) ∈ V |
17 | 12, 13, 16 | fvmpt 6745 | . . 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 2845 | 1 ⊢ (𝑊 ∈ 𝑋 → 𝑁 = (𝑠 ∈ 𝒫 𝑉 ↦ ∩ {𝑡 ∈ 𝑆 ∣ 𝑠 ⊆ 𝑡})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1538 ∈ wcel 2111 {crab 3110 Vcvv 3441 ⊆ wss 3881 𝒫 cpw 4497 ∩ cint 4838 ↦ cmpt 5110 ‘cfv 6324 Basecbs 16475 LSubSpclss 19696 LSpanclspn 19736 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-lsp 19737 |
This theorem is referenced by: lspf 19739 lspval 19740 00lsp 19746 mrclsp 19754 lsppropd 19783 |
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