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Mirrors > Home > MPE Home > Th. List > lspssid | Structured version Visualization version GIF version |
Description: A set of vectors is a subset of its span. (spanss2 29128 analog.) (Contributed by NM, 6-Feb-2014.) (Revised by Mario Carneiro, 19-Jun-2014.) |
Ref | Expression |
---|---|
lspss.v | ⊢ 𝑉 = (Base‘𝑊) |
lspss.n | ⊢ 𝑁 = (LSpan‘𝑊) |
Ref | Expression |
---|---|
lspssid | ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ⊆ 𝑉) → 𝑈 ⊆ (𝑁‘𝑈)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssintub 4856 | . 2 ⊢ 𝑈 ⊆ ∩ {𝑡 ∈ (LSubSp‘𝑊) ∣ 𝑈 ⊆ 𝑡} | |
2 | lspss.v | . . 3 ⊢ 𝑉 = (Base‘𝑊) | |
3 | eqid 2798 | . . 3 ⊢ (LSubSp‘𝑊) = (LSubSp‘𝑊) | |
4 | lspss.n | . . 3 ⊢ 𝑁 = (LSpan‘𝑊) | |
5 | 2, 3, 4 | lspval 19740 | . 2 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ⊆ 𝑉) → (𝑁‘𝑈) = ∩ {𝑡 ∈ (LSubSp‘𝑊) ∣ 𝑈 ⊆ 𝑡}) |
6 | 1, 5 | sseqtrrid 3968 | 1 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ⊆ 𝑉) → 𝑈 ⊆ (𝑁‘𝑈)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 {crab 3110 ⊆ wss 3881 ∩ cint 4838 ‘cfv 6324 Basecbs 16475 LModclmod 19627 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-rmo 3114 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-riota 7093 df-ov 7138 df-0g 16707 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-grp 18098 df-lmod 19629 df-lss 19697 df-lsp 19737 |
This theorem is referenced by: lspun 19752 lspsnid 19758 lsslsp 19780 lmhmlsp 19814 lsmsp 19851 lsmssspx 19853 lspvadd 19861 lspsolvlem 19907 lspsolv 19908 lsppratlem3 19914 lsppratlem4 19915 islbs3 19920 lbsextlem2 19924 lbsextlem4 19926 rspssid 19989 ocvlsp 20365 obselocv 20417 frlmsslsp 20485 lindff1 20509 islinds3 20523 mxidlprm 31048 lbslsat 31102 lindsunlem 31108 dimkerim 31111 lindsenlbs 35052 dochocsp 38675 djhunssN 38705 islssfg2 40015 |
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