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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 31328 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 3501 | . . 3 ⊢ (𝑊 ∈ 𝑋 → 𝑊 ∈ V) | |
| 3 | fveq2 6906 | . . . . . . 7 ⊢ (𝑤 = 𝑊 → (Base‘𝑤) = (Base‘𝑊)) | |
| 4 | lspval.v | . . . . . . 7 ⊢ 𝑉 = (Base‘𝑊) | |
| 5 | 3, 4 | eqtr4di 2795 | . . . . . 6 ⊢ (𝑤 = 𝑊 → (Base‘𝑤) = 𝑉) |
| 6 | 5 | pweqd 4617 | . . . . 5 ⊢ (𝑤 = 𝑊 → 𝒫 (Base‘𝑤) = 𝒫 𝑉) |
| 7 | fveq2 6906 | . . . . . . . 8 ⊢ (𝑤 = 𝑊 → (LSubSp‘𝑤) = (LSubSp‘𝑊)) | |
| 8 | lspval.s | . . . . . . . 8 ⊢ 𝑆 = (LSubSp‘𝑊) | |
| 9 | 7, 8 | eqtr4di 2795 | . . . . . . 7 ⊢ (𝑤 = 𝑊 → (LSubSp‘𝑤) = 𝑆) |
| 10 | 9 | rabeqdv 3452 | . . . . . 6 ⊢ (𝑤 = 𝑊 → {𝑡 ∈ (LSubSp‘𝑤) ∣ 𝑠 ⊆ 𝑡} = {𝑡 ∈ 𝑆 ∣ 𝑠 ⊆ 𝑡}) |
| 11 | 10 | inteqd 4951 | . . . . 5 ⊢ (𝑤 = 𝑊 → ∩ {𝑡 ∈ (LSubSp‘𝑤) ∣ 𝑠 ⊆ 𝑡} = ∩ {𝑡 ∈ 𝑆 ∣ 𝑠 ⊆ 𝑡}) |
| 12 | 6, 11 | mpteq12dv 5233 | . . . 4 ⊢ (𝑤 = 𝑊 → (𝑠 ∈ 𝒫 (Base‘𝑤) ↦ ∩ {𝑡 ∈ (LSubSp‘𝑤) ∣ 𝑠 ⊆ 𝑡}) = (𝑠 ∈ 𝒫 𝑉 ↦ ∩ {𝑡 ∈ 𝑆 ∣ 𝑠 ⊆ 𝑡})) |
| 13 | df-lsp 20970 | . . . 4 ⊢ LSpan = (𝑤 ∈ V ↦ (𝑠 ∈ 𝒫 (Base‘𝑤) ↦ ∩ {𝑡 ∈ (LSubSp‘𝑤) ∣ 𝑠 ⊆ 𝑡})) | |
| 14 | 4 | fvexi 6920 | . . . . . 6 ⊢ 𝑉 ∈ V |
| 15 | 14 | pwex 5380 | . . . . 5 ⊢ 𝒫 𝑉 ∈ V |
| 16 | 15 | mptex 7243 | . . . 4 ⊢ (𝑠 ∈ 𝒫 𝑉 ↦ ∩ {𝑡 ∈ 𝑆 ∣ 𝑠 ⊆ 𝑡}) ∈ V |
| 17 | 12, 13, 16 | fvmpt 7016 | . . 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 | eqtrid 2789 | 1 ⊢ (𝑊 ∈ 𝑋 → 𝑁 = (𝑠 ∈ 𝒫 𝑉 ↦ ∩ {𝑡 ∈ 𝑆 ∣ 𝑠 ⊆ 𝑡})) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 {crab 3436 Vcvv 3480 ⊆ wss 3951 𝒫 cpw 4600 ∩ cint 4946 ↦ cmpt 5225 ‘cfv 6561 Basecbs 17247 LSubSpclss 20929 LSpanclspn 20969 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-lsp 20970 |
| This theorem is referenced by: lspf 20972 lspval 20973 00lsp 20979 mrclsp 20987 lsppropd 21017 |
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