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Mirrors > Home > MPE Home > Th. List > lmodgrp | Structured version Visualization version GIF version |
Description: A left module is a group. (Contributed by NM, 8-Dec-2013.) (Revised by Mario Carneiro, 25-Jun-2014.) |
Ref | Expression |
---|---|
lmodgrp | β’ (π β LMod β π β Grp) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2733 | . . 3 β’ (Baseβπ) = (Baseβπ) | |
2 | eqid 2733 | . . 3 β’ (+gβπ) = (+gβπ) | |
3 | eqid 2733 | . . 3 β’ ( Β·π βπ) = ( Β·π βπ) | |
4 | eqid 2733 | . . 3 β’ (Scalarβπ) = (Scalarβπ) | |
5 | eqid 2733 | . . 3 β’ (Baseβ(Scalarβπ)) = (Baseβ(Scalarβπ)) | |
6 | eqid 2733 | . . 3 β’ (+gβ(Scalarβπ)) = (+gβ(Scalarβπ)) | |
7 | eqid 2733 | . . 3 β’ (.rβ(Scalarβπ)) = (.rβ(Scalarβπ)) | |
8 | eqid 2733 | . . 3 β’ (1rβ(Scalarβπ)) = (1rβ(Scalarβπ)) | |
9 | 1, 2, 3, 4, 5, 6, 7, 8 | islmod 20340 | . 2 β’ (π β LMod β (π β Grp β§ (Scalarβπ) β Ring β§ βπ β (Baseβ(Scalarβπ))βπ β (Baseβ(Scalarβπ))βπ₯ β (Baseβπ)βπ€ β (Baseβπ)(((π( Β·π βπ)π€) β (Baseβπ) β§ (π( Β·π βπ)(π€(+gβπ)π₯)) = ((π( Β·π βπ)π€)(+gβπ)(π( Β·π βπ)π₯)) β§ ((π(+gβ(Scalarβπ))π)( Β·π βπ)π€) = ((π( Β·π βπ)π€)(+gβπ)(π( Β·π βπ)π€))) β§ (((π(.rβ(Scalarβπ))π)( Β·π βπ)π€) = (π( Β·π βπ)(π( Β·π βπ)π€)) β§ ((1rβ(Scalarβπ))( Β·π βπ)π€) = π€)))) |
10 | 9 | simp1bi 1146 | 1 β’ (π β LMod β π β Grp) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 397 β§ w3a 1088 = wceq 1542 β wcel 2107 βwral 3061 βcfv 6497 (class class class)co 7358 Basecbs 17088 +gcplusg 17138 .rcmulr 17139 Scalarcsca 17141 Β·π cvsca 17142 Grpcgrp 18753 1rcur 19918 Ringcrg 19969 LModclmod 20336 |
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-ext 2704 ax-nul 5264 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-ne 2941 df-ral 3062 df-rab 3407 df-v 3446 df-sbc 3741 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-br 5107 df-iota 6449 df-fv 6505 df-ov 7361 df-lmod 20338 |
This theorem is referenced by: lmodgrpd 20346 lmodbn0 20347 lmodvacl 20351 lmodass 20352 lmodlcan 20353 lmod0vcl 20366 lmod0vlid 20367 lmod0vrid 20368 lmod0vid 20369 lmodvsmmulgdi 20372 lmodfopne 20375 lmodvnegcl 20378 lmodvnegid 20379 lmodvsubcl 20382 lmodcom 20383 lmodabl 20384 lmodvpncan 20390 lmodvnpcan 20391 lmodsubeq0 20396 lmodsubid 20397 lmodvsghm 20398 lmodprop2d 20399 lsssubg 20433 islss3 20435 lssacs 20443 prdslmodd 20445 lspsnneg 20482 lspsnsub 20483 lmodindp1 20490 lmodvsinv2 20513 islmhm2 20514 0lmhm 20516 idlmhm 20517 pwsdiaglmhm 20533 pwssplit3 20537 lspexch 20606 lspsolvlem 20619 ip0l 21056 ipsubdir 21062 ipsubdi 21063 ip2eq 21073 lsmcss 21112 dsmmlss 21166 frlm0 21176 frlmsubgval 21187 frlmplusgvalb 21191 frlmup1 21220 islindf4 21260 mplind 21494 matgrp 21795 tlmtgp 23563 clmgrp 24447 ncvspi 24536 cphtcphnm 24610 ipcau2 24614 tcphcphlem1 24615 tcphcph 24617 rrxnm 24771 rrxds 24773 pjthlem2 24818 lmodvslmhm 31941 eqgvscpbl 32189 imaslmod 32192 quslmod 32193 linds2eq 32216 lbslsat 32368 lindsunlem 32376 lbsdiflsp0 32378 dimkerim 32379 lclkrlem2m 40028 mapdpglem14 40194 baerlem3lem1 40216 baerlem5amN 40225 baerlem5bmN 40226 baerlem5abmN 40227 mapdh6bN 40246 mapdh6cN 40247 hdmap1l6b 40320 hdmap1l6c 40321 hdmap11 40357 frlmsnic 40771 kercvrlsm 41453 pwssplit4 41459 pwslnmlem2 41463 mendring 41562 zlmodzxzsub 46522 lmodvsmdi 46544 lincvalsng 46583 lincvalsc0 46588 linc0scn0 46590 linc1 46592 lcoel0 46595 lindslinindimp2lem4 46628 snlindsntor 46638 lincresunit3 46648 |
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