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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 30576 analog.) (Contributed by NM, 9-Dec-2013.) (Revised by Mario Carneiro, 19-Jun-2014.) |
| Ref | Expression |
|---|---|
| lspval.v | β’ π = (Baseβπ) |
| lspval.s | β’ π = (LSubSpβπ) |
| lspval.n | β’ π = (LSpanβπ) |
| Ref | Expression |
|---|---|
| lspcl | β’ ((π β LMod β§ π β π) β (πβπ) β π) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | lspval.v | . . 3 β’ π = (Baseβπ) | |
| 2 | lspval.s | . . 3 β’ π = (LSubSpβπ) | |
| 3 | lspval.n | . . 3 β’ π = (LSpanβπ) | |
| 4 | 1, 2, 3 | lspf 20577 | . 2 β’ (π β LMod β π:π« πβΆπ) |
| 5 | 1 | fvexi 6902 | . . . 4 β’ π β V |
| 6 | 5 | elpw2 5344 | . . 3 β’ (π β π« π β π β π) |
| 7 | 6 | biimpri 227 | . 2 β’ (π β π β π β π« π) |
| 8 | ffvelcdm 7080 | . 2 β’ ((π:π« πβΆπ β§ π β π« π) β (πβπ) β π) | |
| 9 | 4, 7, 8 | syl2an 596 | 1 β’ ((π β LMod β§ π β π) β (πβπ) β π) |
| Colors of variables: wff setvar class |
| Syntax hints: β wi 4 β§ wa 396 = wceq 1541 β wcel 2106 β wss 3947 π« cpw 4601 βΆwf 6536 βcfv 6540 Basecbs 17140 LModclmod 20463 LSubSpclss 20534 LSpanclspn 20574 |
| 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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 |
| This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7852 df-1st 7971 df-2nd 7972 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-er 8699 df-en 8936 df-dom 8937 df-sdom 8938 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-nn 12209 df-2 12271 df-sets 17093 df-slot 17111 df-ndx 17123 df-base 17141 df-plusg 17206 df-0g 17383 df-mgm 18557 df-sgrp 18606 df-mnd 18622 df-grp 18818 df-minusg 18819 df-sbg 18820 df-mgp 19982 df-ur 19999 df-ring 20051 df-lmod 20465 df-lss 20535 df-lsp 20575 |
| This theorem is referenced by: lspsncl 20580 lspprcl 20581 lsptpcl 20582 lspssv 20586 lspidm 20589 lspsnvsi 20607 lsp0 20612 lspun0 20614 lsslsp 20618 lmhmlsp 20652 lsmsp 20689 lsmsp2 20690 lspvadd 20699 lspsolvlem 20747 lspsolv 20748 lsppratlem2 20753 lsppratlem3 20754 islbs2 20759 islbs3 20760 lbsextlem2 20764 rspcl 20839 obselocv 21274 frlmsslsp 21342 islinds3 21380 0ellsp 32470 lsmidl 32499 lbslsat 32689 lsatdim 32690 drngdimgt0 32691 lindsunlem 32697 lbsdiflsp0 32699 dimkerim 32700 lindsadd 36469 lindsenlbs 36471 islshpsm 37838 lssats 37870 dvh4dimlem 40302 islssfgi 41799 lmhmfgsplit 41813 |
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