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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 31083 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 20817 | . 2 β’ (π β LMod β π:π« πβΆπ) |
| 5 | 1 | fvexi 6896 | . . . 4 β’ π β V |
| 6 | 5 | elpw2 5336 | . . 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 7074 | . 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 1533 β wcel 2098 β wss 3941 π« cpw 4595 βΆwf 6530 βcfv 6534 Basecbs 17149 LModclmod 20702 LSubSpclss 20774 LSpanclspn 20814 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-rep 5276 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 ax-cnex 11163 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 ax-pre-mulgt0 11184 |
| This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3960 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-int 4942 df-iun 4990 df-br 5140 df-opab 5202 df-mpt 5223 df-tr 5257 df-id 5565 df-eprel 5571 df-po 5579 df-so 5580 df-fr 5622 df-we 5624 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-pred 6291 df-ord 6358 df-on 6359 df-lim 6360 df-suc 6361 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-riota 7358 df-ov 7405 df-oprab 7406 df-mpo 7407 df-om 7850 df-1st 7969 df-2nd 7970 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-er 8700 df-en 8937 df-dom 8938 df-sdom 8939 df-pnf 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-le 11253 df-sub 11445 df-neg 11446 df-nn 12212 df-2 12274 df-sets 17102 df-slot 17120 df-ndx 17132 df-base 17150 df-plusg 17215 df-0g 17392 df-mgm 18569 df-sgrp 18648 df-mnd 18664 df-grp 18862 df-minusg 18863 df-sbg 18864 df-mgp 20036 df-ur 20083 df-ring 20136 df-lmod 20704 df-lss 20775 df-lsp 20815 |
| This theorem is referenced by: lspsncl 20820 lspprcl 20821 lsptpcl 20822 lspssv 20826 lspidm 20829 lspsnvsi 20847 lsp0 20852 lspun0 20854 lsslsp 20858 lsslspOLD 20859 lmhmlsp 20893 lsmsp 20930 lsmsp2 20931 lspvadd 20940 lspsolvlem 20989 lspsolv 20990 lsppratlem2 20995 lsppratlem3 20996 islbs2 21001 islbs3 21002 lbsextlem2 21006 rspcl 21090 obselocv 21612 frlmsslsp 21680 islinds3 21718 0ellsp 32977 lsmidl 33006 lbslsat 33208 lsatdim 33209 drngdimgt0 33210 lindsunlem 33216 lbsdiflsp0 33218 dimkerim 33219 lindsadd 36984 lindsenlbs 36986 islshpsm 38353 lssats 38385 dvh4dimlem 40817 islssfgi 42364 lmhmfgsplit 42378 |
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