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Mirrors > Home > MPE Home > Th. List > lspcl | Structured version Visualization version GIF version |
Description: The span of a set of vectors is a subspace. (spancl 31364 analog.) (Contributed by NM, 9-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 |
---|---|
lspcl | ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ⊆ 𝑉) → (𝑁‘𝑈) ∈ 𝑆) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lspval.v | . . 3 ⊢ 𝑉 = (Base‘𝑊) | |
2 | lspval.s | . . 3 ⊢ 𝑆 = (LSubSp‘𝑊) | |
3 | lspval.n | . . 3 ⊢ 𝑁 = (LSpan‘𝑊) | |
4 | 1, 2, 3 | lspf 20989 | . 2 ⊢ (𝑊 ∈ LMod → 𝑁:𝒫 𝑉⟶𝑆) |
5 | 1 | fvexi 6920 | . . . 4 ⊢ 𝑉 ∈ V |
6 | 5 | elpw2 5339 | . . 3 ⊢ (𝑈 ∈ 𝒫 𝑉 ↔ 𝑈 ⊆ 𝑉) |
7 | 6 | biimpri 228 | . 2 ⊢ (𝑈 ⊆ 𝑉 → 𝑈 ∈ 𝒫 𝑉) |
8 | ffvelcdm 7100 | . 2 ⊢ ((𝑁:𝒫 𝑉⟶𝑆 ∧ 𝑈 ∈ 𝒫 𝑉) → (𝑁‘𝑈) ∈ 𝑆) | |
9 | 4, 7, 8 | syl2an 596 | 1 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ⊆ 𝑉) → (𝑁‘𝑈) ∈ 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1536 ∈ wcel 2105 ⊆ wss 3962 𝒫 cpw 4604 ⟶wf 6558 ‘cfv 6562 Basecbs 17244 LModclmod 20874 LSubSpclss 20946 LSpanclspn 20986 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-rep 5284 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-cnex 11208 ax-resscn 11209 ax-1cn 11210 ax-icn 11211 ax-addcl 11212 ax-addrcl 11213 ax-mulcl 11214 ax-mulrcl 11215 ax-mulcom 11216 ax-addass 11217 ax-mulass 11218 ax-distr 11219 ax-i2m1 11220 ax-1ne0 11221 ax-1rid 11222 ax-rnegex 11223 ax-rrecex 11224 ax-cnre 11225 ax-pre-lttri 11226 ax-pre-lttrn 11227 ax-pre-ltadd 11228 ax-pre-mulgt0 11229 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3377 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-int 4951 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-om 7887 df-1st 8012 df-2nd 8013 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-er 8743 df-en 8984 df-dom 8985 df-sdom 8986 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-sub 11491 df-neg 11492 df-nn 12264 df-2 12326 df-sets 17197 df-slot 17215 df-ndx 17227 df-base 17245 df-plusg 17310 df-0g 17487 df-mgm 18665 df-sgrp 18744 df-mnd 18760 df-grp 18966 df-minusg 18967 df-sbg 18968 df-mgp 20152 df-ur 20199 df-ring 20252 df-lmod 20876 df-lss 20947 df-lsp 20987 |
This theorem is referenced by: lspsncl 20992 lspprcl 20993 lsptpcl 20994 lspssv 20998 lspidm 21001 lspsnvsi 21019 lsp0 21024 lspun0 21026 lsslsp 21030 lsslspOLD 21031 lmhmlsp 21065 lsmsp 21102 lsmsp2 21103 lspvadd 21112 lspsolvlem 21161 lspsolv 21162 lsppratlem2 21167 lsppratlem3 21168 islbs2 21173 islbs3 21174 lbsextlem2 21178 rspcl 21262 obselocv 21765 frlmsslsp 21833 islinds3 21871 0ellsp 33376 lsmidl 33408 lbslsat 33643 lsatdim 33644 drngdimgt0 33645 lindsunlem 33651 lbsdiflsp0 33653 dimkerim 33654 lindsadd 37599 lindsenlbs 37601 islshpsm 38961 lssats 38993 dvh4dimlem 41425 islssfgi 43060 lmhmfgsplit 43074 |
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