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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 31145 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 20857 | . 2 β’ (π β LMod β π:π« πβΆπ) |
| 5 | 1 | fvexi 6911 | . . . 4 β’ π β V |
| 6 | 5 | elpw2 5347 | . . 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 7091 | . 2 β’ ((π:π« πβΆπ β§ π β π« π) β (πβπ) β π) | |
| 9 | 4, 7, 8 | syl2an 595 | 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 395 = wceq 1534 β wcel 2099 β wss 3947 π« cpw 4603 βΆwf 6544 βcfv 6548 Basecbs 17179 LModclmod 20742 LSubSpclss 20814 LSpanclspn 20854 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-cnex 11194 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 ax-pre-mulgt0 11215 |
| This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3373 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-int 4950 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7871 df-1st 7993 df-2nd 7994 df-frecs 8286 df-wrecs 8317 df-recs 8391 df-rdg 8430 df-er 8724 df-en 8964 df-dom 8965 df-sdom 8966 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-sub 11476 df-neg 11477 df-nn 12243 df-2 12305 df-sets 17132 df-slot 17150 df-ndx 17162 df-base 17180 df-plusg 17245 df-0g 17422 df-mgm 18599 df-sgrp 18678 df-mnd 18694 df-grp 18892 df-minusg 18893 df-sbg 18894 df-mgp 20074 df-ur 20121 df-ring 20174 df-lmod 20744 df-lss 20815 df-lsp 20855 |
| This theorem is referenced by: lspsncl 20860 lspprcl 20861 lsptpcl 20862 lspssv 20866 lspidm 20869 lspsnvsi 20887 lsp0 20892 lspun0 20894 lsslsp 20898 lsslspOLD 20899 lmhmlsp 20933 lsmsp 20970 lsmsp2 20971 lspvadd 20980 lspsolvlem 21029 lspsolv 21030 lsppratlem2 21035 lsppratlem3 21036 islbs2 21041 islbs3 21042 lbsextlem2 21046 rspcl 21130 obselocv 21661 frlmsslsp 21729 islinds3 21767 0ellsp 33081 lsmidl 33110 lbslsat 33310 lsatdim 33311 drngdimgt0 33312 lindsunlem 33318 lbsdiflsp0 33320 dimkerim 33321 lindsadd 37086 lindsenlbs 37088 islshpsm 38452 lssats 38484 dvh4dimlem 40916 islssfgi 42496 lmhmfgsplit 42510 |
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