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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 28782 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 2778 | . . . 4 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
2 | lspid.s | . . . 4 ⊢ 𝑆 = (LSubSp‘𝑊) | |
3 | 1, 2 | lssss 19333 | . . 3 ⊢ (𝑈 ∈ 𝑆 → 𝑈 ⊆ (Base‘𝑊)) |
4 | lspid.n | . . . 4 ⊢ 𝑁 = (LSpan‘𝑊) | |
5 | 1, 2, 4 | lspval 19374 | . . 3 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ⊆ (Base‘𝑊)) → (𝑁‘𝑈) = ∩ {𝑡 ∈ 𝑆 ∣ 𝑈 ⊆ 𝑡}) |
6 | 3, 5 | sylan2 586 | . 2 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆) → (𝑁‘𝑈) = ∩ {𝑡 ∈ 𝑆 ∣ 𝑈 ⊆ 𝑡}) |
7 | intmin 4732 | . . 3 ⊢ (𝑈 ∈ 𝑆 → ∩ {𝑡 ∈ 𝑆 ∣ 𝑈 ⊆ 𝑡} = 𝑈) | |
8 | 7 | adantl 475 | . 2 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆) → ∩ {𝑡 ∈ 𝑆 ∣ 𝑈 ⊆ 𝑡} = 𝑈) |
9 | 6, 8 | eqtrd 2814 | 1 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆) → (𝑁‘𝑈) = 𝑈) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 = wceq 1601 ∈ wcel 2107 {crab 3094 ⊆ wss 3792 ∩ cint 4712 ‘cfv 6137 Basecbs 16259 LModclmod 19259 LSubSpclss 19328 LSpanclspn 19370 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-rep 5008 ax-sep 5019 ax-nul 5027 ax-pow 5079 ax-pr 5140 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-ral 3095 df-rex 3096 df-reu 3097 df-rmo 3098 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-op 4405 df-uni 4674 df-int 4713 df-iun 4757 df-br 4889 df-opab 4951 df-mpt 4968 df-id 5263 df-xp 5363 df-rel 5364 df-cnv 5365 df-co 5366 df-dm 5367 df-rn 5368 df-res 5369 df-ima 5370 df-iota 6101 df-fun 6139 df-fn 6140 df-f 6141 df-f1 6142 df-fo 6143 df-f1o 6144 df-fv 6145 df-riota 6885 df-ov 6927 df-0g 16492 df-mgm 17632 df-sgrp 17674 df-mnd 17685 df-grp 17816 df-lmod 19261 df-lss 19329 df-lsp 19371 |
This theorem is referenced by: lspidm 19385 lspssp 19387 lspsn0 19407 lspsolvlem 19542 lbsextlem3 19561 islshpsm 35139 lshpnel2N 35144 lssats 35171 lkrlsp3 35263 dochspocN 37539 dochsatshp 37610 filnm 38629 |
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