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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 28755 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 4717 | . 2 ⊢ 𝑈 ⊆ ∩ {𝑡 ∈ (LSubSp‘𝑊) ∣ 𝑈 ⊆ 𝑡} | |
2 | lspss.v | . . 3 ⊢ 𝑉 = (Base‘𝑊) | |
3 | eqid 2825 | . . 3 ⊢ (LSubSp‘𝑊) = (LSubSp‘𝑊) | |
4 | lspss.n | . . 3 ⊢ 𝑁 = (LSpan‘𝑊) | |
5 | 2, 3, 4 | lspval 19341 | . 2 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ⊆ 𝑉) → (𝑁‘𝑈) = ∩ {𝑡 ∈ (LSubSp‘𝑊) ∣ 𝑈 ⊆ 𝑡}) |
6 | 1, 5 | syl5sseqr 3879 | 1 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ⊆ 𝑉) → 𝑈 ⊆ (𝑁‘𝑈)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 = wceq 1656 ∈ wcel 2164 {crab 3121 ⊆ wss 3798 ∩ cint 4699 ‘cfv 6127 Basecbs 16229 LModclmod 19226 LSubSpclss 19295 LSpanclspn 19337 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-rep 4996 ax-sep 5007 ax-nul 5015 ax-pow 5067 ax-pr 5129 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-ral 3122 df-rex 3123 df-reu 3124 df-rmo 3125 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4147 df-if 4309 df-pw 4382 df-sn 4400 df-pr 4402 df-op 4406 df-uni 4661 df-int 4700 df-iun 4744 df-br 4876 df-opab 4938 df-mpt 4955 df-id 5252 df-xp 5352 df-rel 5353 df-cnv 5354 df-co 5355 df-dm 5356 df-rn 5357 df-res 5358 df-ima 5359 df-iota 6090 df-fun 6129 df-fn 6130 df-f 6131 df-f1 6132 df-fo 6133 df-f1o 6134 df-fv 6135 df-riota 6871 df-ov 6913 df-0g 16462 df-mgm 17602 df-sgrp 17644 df-mnd 17655 df-grp 17786 df-lmod 19228 df-lss 19296 df-lsp 19338 |
This theorem is referenced by: lspun 19353 lspsnid 19359 lsslsp 19381 lmhmlsp 19415 lsmsp 19452 lsmssspx 19454 lspvadd 19462 lspsolvlem 19509 lspsolv 19510 lsppratlem3 19517 lsppratlem4 19518 islbs3 19523 lbsextlem2 19527 lbsextlem4 19529 rspssid 19591 ocvlsp 20390 obselocv 20442 frlmsslsp 20509 lindff1 20533 islinds3 20547 lindsenlbs 33943 dochocsp 37449 djhunssN 37479 islssfg2 38479 |
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