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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 30320 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 20450 | . 2 β’ (π β LMod β π:π« πβΆπ) |
5 | 1 | fvexi 6857 | . . . 4 β’ π β V |
6 | 5 | elpw2 5303 | . . 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 7033 | . 2 β’ ((π:π« πβΆπ β§ π β π« π) β (πβπ) β π) | |
9 | 4, 7, 8 | syl2an 597 | 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 397 = wceq 1542 β wcel 2107 β wss 3911 π« cpw 4561 βΆwf 6493 βcfv 6497 Basecbs 17088 LModclmod 20336 LSubSpclss 20407 LSpanclspn 20447 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5243 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-cnex 11112 ax-resscn 11113 ax-1cn 11114 ax-icn 11115 ax-addcl 11116 ax-addrcl 11117 ax-mulcl 11118 ax-mulrcl 11119 ax-mulcom 11120 ax-addass 11121 ax-mulass 11122 ax-distr 11123 ax-i2m1 11124 ax-1ne0 11125 ax-1rid 11126 ax-rnegex 11127 ax-rrecex 11128 ax-cnre 11129 ax-pre-lttri 11130 ax-pre-lttrn 11131 ax-pre-ltadd 11132 ax-pre-mulgt0 11133 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-int 4909 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7804 df-1st 7922 df-2nd 7923 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-er 8651 df-en 8887 df-dom 8888 df-sdom 8889 df-pnf 11196 df-mnf 11197 df-xr 11198 df-ltxr 11199 df-le 11200 df-sub 11392 df-neg 11393 df-nn 12159 df-2 12221 df-sets 17041 df-slot 17059 df-ndx 17071 df-base 17089 df-plusg 17151 df-0g 17328 df-mgm 18502 df-sgrp 18551 df-mnd 18562 df-grp 18756 df-minusg 18757 df-sbg 18758 df-mgp 19902 df-ur 19919 df-ring 19971 df-lmod 20338 df-lss 20408 df-lsp 20448 |
This theorem is referenced by: lspsncl 20453 lspprcl 20454 lsptpcl 20455 lspssv 20459 lspidm 20462 lspsnvsi 20480 lsp0 20485 lspun0 20487 lsslsp 20491 lmhmlsp 20525 lsmsp 20562 lsmsp2 20563 lspvadd 20572 lspsolvlem 20619 lspsolv 20620 lsppratlem2 20625 lsppratlem3 20626 islbs2 20631 islbs3 20632 lbsextlem2 20636 rspcl 20708 obselocv 21150 frlmsslsp 21218 islinds3 21256 0ellsp 32205 lsmidl 32230 lbslsat 32368 lsatdim 32369 drngdimgt0 32370 lindsunlem 32376 lbsdiflsp0 32378 dimkerim 32379 lindsadd 36117 lindsenlbs 36119 islshpsm 37488 lssats 37520 dvh4dimlem 39952 islssfgi 41442 lmhmfgsplit 41456 |
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