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| Mirrors > Home > MPE Home > Th. List > lspid | Structured version Visualization version GIF version | ||
| Description: The span of a subspace is itself. (spanid 31366 analog.) (Contributed by NM, 15-Dec-2013.) (Revised by Mario Carneiro, 19-Jun-2014.) |
| Ref | Expression |
|---|---|
| lspid.s | ⊢ 𝑆 = (LSubSp‘𝑊) |
| lspid.n | ⊢ 𝑁 = (LSpan‘𝑊) |
| Ref | Expression |
|---|---|
| lspid | ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆) → (𝑁‘𝑈) = 𝑈) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | eqid 2737 | . . . 4 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
| 2 | lspid.s | . . . 4 ⊢ 𝑆 = (LSubSp‘𝑊) | |
| 3 | 1, 2 | lssss 20934 | . . 3 ⊢ (𝑈 ∈ 𝑆 → 𝑈 ⊆ (Base‘𝑊)) |
| 4 | lspid.n | . . . 4 ⊢ 𝑁 = (LSpan‘𝑊) | |
| 5 | 1, 2, 4 | lspval 20973 | . . 3 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ⊆ (Base‘𝑊)) → (𝑁‘𝑈) = ∩ {𝑡 ∈ 𝑆 ∣ 𝑈 ⊆ 𝑡}) |
| 6 | 3, 5 | sylan2 593 | . 2 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆) → (𝑁‘𝑈) = ∩ {𝑡 ∈ 𝑆 ∣ 𝑈 ⊆ 𝑡}) |
| 7 | intmin 4968 | . . 3 ⊢ (𝑈 ∈ 𝑆 → ∩ {𝑡 ∈ 𝑆 ∣ 𝑈 ⊆ 𝑡} = 𝑈) | |
| 8 | 7 | adantl 481 | . 2 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆) → ∩ {𝑡 ∈ 𝑆 ∣ 𝑈 ⊆ 𝑡} = 𝑈) |
| 9 | 6, 8 | eqtrd 2777 | 1 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆) → (𝑁‘𝑈) = 𝑈) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 {crab 3436 ⊆ wss 3951 ∩ cint 4946 ‘cfv 6561 Basecbs 17247 LModclmod 20858 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-rmo 3380 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-riota 7388 df-ov 7434 df-0g 17486 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-grp 18954 df-lmod 20860 df-lss 20930 df-lsp 20970 |
| This theorem is referenced by: lspidm 20984 lspssp 20986 lspsn0 21006 lspsolvlem 21144 lbsextlem3 21162 islshpsm 38981 lshpnel2N 38986 lssats 39013 lkrlsp3 39105 dochspocN 41382 dochsatshp 41453 filnm 43102 |
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