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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 2610 | . . 3 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
2 | eqid 2610 | . . 3 ⊢ (+g‘𝑊) = (+g‘𝑊) | |
3 | eqid 2610 | . . 3 ⊢ ( ·𝑠 ‘𝑊) = ( ·𝑠 ‘𝑊) | |
4 | eqid 2610 | . . 3 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊) | |
5 | eqid 2610 | . . 3 ⊢ (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊)) | |
6 | eqid 2610 | . . 3 ⊢ (+g‘(Scalar‘𝑊)) = (+g‘(Scalar‘𝑊)) | |
7 | eqid 2610 | . . 3 ⊢ (.r‘(Scalar‘𝑊)) = (.r‘(Scalar‘𝑊)) | |
8 | eqid 2610 | . . 3 ⊢ (1r‘(Scalar‘𝑊)) = (1r‘(Scalar‘𝑊)) | |
9 | 1, 2, 3, 4, 5, 6, 7, 8 | islmod 18690 | . 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 1069 | 1 ⊢ (𝑊 ∈ LMod → 𝑊 ∈ Grp) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ∀wral 2896 ‘cfv 5804 (class class class)co 6549 Basecbs 15695 +gcplusg 15768 .rcmulr 15769 Scalarcsca 15771 ·𝑠 cvsca 15772 Grpcgrp 17245 1rcur 18324 Ringcrg 18370 LModclmod 18686 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-nul 4717 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-iota 5768 df-fv 5812 df-ov 6552 df-lmod 18688 |
This theorem is referenced by: lmodbn0 18696 lmodvacl 18700 lmodass 18701 lmodlcan 18702 lmod0vcl 18715 lmod0vlid 18716 lmod0vrid 18717 lmod0vid 18718 lmodvsmmulgdi 18721 lmodfopne 18724 lmodvnegcl 18727 lmodvnegid 18728 lmodvsubcl 18731 lmodcom 18732 lmodabl 18733 lmodvpncan 18739 lmodvnpcan 18740 lmodsubeq0 18745 lmodsubid 18746 lmodvsghm 18747 lmodprop2d 18748 lsssubg 18778 islss3 18780 lssacs 18788 prdslmodd 18790 lspsnneg 18827 lspsnsub 18828 lmodindp1 18835 lmodvsinv2 18858 islmhm2 18859 0lmhm 18861 idlmhm 18862 pwsdiaglmhm 18878 pwssplit3 18882 lspexch 18950 lspsolvlem 18963 mplind 19323 ip0l 19800 ipsubdir 19806 ipsubdi 19807 ip2eq 19817 lsmcss 19855 dsmmlss 19907 frlm0 19917 frlmsubgval 19927 frlmup1 19956 islindf4 19996 matgrp 20055 tlmtgp 21809 clmgrp 22676 ncvspi 22764 cphtchnm 22837 ipcau2 22841 tchcphlem1 22842 tchcph 22844 rrxnm 22987 rrxds 22989 pjthlem2 23017 lclkrlem2m 35826 mapdpglem14 35992 baerlem3lem1 36014 baerlem5amN 36023 baerlem5bmN 36024 baerlem5abmN 36025 mapdh6bN 36044 mapdh6cN 36045 hdmap1l6b 36119 hdmap1l6c 36120 hdmap1neglem1N 36135 hdmap11 36158 kercvrlsm 36671 pwssplit4 36677 pwslnmlem2 36681 mendring 36781 zlmodzxzsub 41931 lmodvsmdi 41957 lincvalsng 41999 lincvalsc0 42004 linc0scn0 42006 linc1 42008 lcoel0 42011 lindslinindimp2lem4 42044 snlindsntor 42054 lincresunit3 42064 |
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