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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 | ⊢ (𝑊 ∈ LMod → 𝑊 ∈ Grp) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | eqid 2737 | . . 3 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
| 2 | eqid 2737 | . . 3 ⊢ (+g‘𝑊) = (+g‘𝑊) | |
| 3 | eqid 2737 | . . 3 ⊢ ( ·𝑠 ‘𝑊) = ( ·𝑠 ‘𝑊) | |
| 4 | eqid 2737 | . . 3 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊) | |
| 5 | eqid 2737 | . . 3 ⊢ (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊)) | |
| 6 | eqid 2737 | . . 3 ⊢ (+g‘(Scalar‘𝑊)) = (+g‘(Scalar‘𝑊)) | |
| 7 | eqid 2737 | . . 3 ⊢ (.r‘(Scalar‘𝑊)) = (.r‘(Scalar‘𝑊)) | |
| 8 | eqid 2737 | . . 3 ⊢ (1r‘(Scalar‘𝑊)) = (1r‘(Scalar‘𝑊)) | |
| 9 | 1, 2, 3, 4, 5, 6, 7, 8 | islmod 20862 | . 2 ⊢ (𝑊 ∈ LMod ↔ (𝑊 ∈ Grp ∧ (Scalar‘𝑊) ∈ Ring ∧ ∀𝑞 ∈ (Base‘(Scalar‘𝑊))∀𝑟 ∈ (Base‘(Scalar‘𝑊))∀𝑥 ∈ (Base‘𝑊)∀𝑤 ∈ (Base‘𝑊)(((𝑟( ·𝑠 ‘𝑊)𝑤) ∈ (Base‘𝑊) ∧ (𝑟( ·𝑠 ‘𝑊)(𝑤(+g‘𝑊)𝑥)) = ((𝑟( ·𝑠 ‘𝑊)𝑤)(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑥)) ∧ ((𝑞(+g‘(Scalar‘𝑊))𝑟)( ·𝑠 ‘𝑊)𝑤) = ((𝑞( ·𝑠 ‘𝑊)𝑤)(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑤))) ∧ (((𝑞(.r‘(Scalar‘𝑊))𝑟)( ·𝑠 ‘𝑊)𝑤) = (𝑞( ·𝑠 ‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑤)) ∧ ((1r‘(Scalar‘𝑊))( ·𝑠 ‘𝑊)𝑤) = 𝑤)))) |
| 10 | 9 | simp1bi 1146 | 1 ⊢ (𝑊 ∈ LMod → 𝑊 ∈ Grp) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 ∀wral 3061 ‘cfv 6561 (class class class)co 7431 Basecbs 17247 +gcplusg 17297 .rcmulr 17298 Scalarcsca 17300 ·𝑠 cvsca 17301 Grpcgrp 18951 1rcur 20178 Ringcrg 20230 LModclmod 20858 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-nul 5306 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-ral 3062 df-rab 3437 df-v 3482 df-sbc 3789 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-iota 6514 df-fv 6569 df-ov 7434 df-lmod 20860 |
| This theorem is referenced by: lmodgrpd 20868 lmodbn0 20869 lmodvacl 20873 lmodass 20874 lmodlcan 20875 lmod0vcl 20889 lmod0vlid 20890 lmod0vrid 20891 lmod0vid 20892 lmodvsmmulgdi 20895 lmodfopne 20898 lmodvnegcl 20901 lmodvnegid 20902 lmodvsubcl 20905 lmodcom 20906 lmodabl 20907 lmodvpncan 20913 lmodvnpcan 20914 lmodsubeq0 20919 lmodsubid 20920 lmodvsghm 20921 lmodprop2d 20922 lsssubg 20955 islss3 20957 lssacs 20965 prdslmodd 20967 lspsnneg 21004 lspsnsub 21005 lmodindp1 21012 lmodvsinv2 21036 islmhm2 21037 0lmhm 21039 idlmhm 21040 pwsdiaglmhm 21056 pwssplit3 21060 lspexch 21131 lspsolvlem 21144 ip0l 21654 ipsubdir 21660 ipsubdi 21661 ip2eq 21671 lsmcss 21710 dsmmlss 21764 frlm0 21774 frlmsubgval 21785 frlmplusgvalb 21789 frlmup1 21818 islindf4 21858 mplind 22094 matgrp 22436 tlmtgp 24204 clmgrp 25101 ncvspi 25190 cphtcphnm 25264 ipcau2 25268 tcphcphlem1 25269 tcphcph 25271 rrxnm 25425 rrxds 25427 pjthlem2 25472 lmodvslmhm 33053 eqgvscpbl 33378 imaslmod 33381 quslmod 33386 linds2eq 33409 lbslsat 33667 lindsunlem 33675 lbsdiflsp0 33677 dimkerim 33678 lclkrlem2m 41521 mapdpglem14 41687 baerlem3lem1 41709 baerlem5amN 41718 baerlem5bmN 41719 baerlem5abmN 41720 mapdh6bN 41739 mapdh6cN 41740 hdmap1l6b 41813 hdmap1l6c 41814 hdmap11 41850 frlmsnic 42550 kercvrlsm 43095 pwssplit4 43101 pwslnmlem2 43105 mendring 43200 zlmodzxzsub 48276 lmodvsmdi 48295 lincvalsng 48333 lincvalsc0 48338 linc0scn0 48340 linc1 48342 lcoel0 48345 lindslinindimp2lem4 48378 snlindsntor 48388 lincresunit3 48398 |
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