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Mirrors > Home > MPE Home > Th. List > lmodabl | Structured version Visualization version GIF version |
Description: A left module is an abelian group (of vectors, under addition). (Contributed by NM, 8-Dec-2013.) (Revised by Mario Carneiro, 25-Jun-2014.) |
Ref | Expression |
---|---|
lmodabl | ⊢ (𝑊 ∈ LMod → 𝑊 ∈ Abel) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqidd 2623 | . 2 ⊢ (𝑊 ∈ LMod → (Base‘𝑊) = (Base‘𝑊)) | |
2 | eqidd 2623 | . 2 ⊢ (𝑊 ∈ LMod → (+g‘𝑊) = (+g‘𝑊)) | |
3 | lmodgrp 18870 | . 2 ⊢ (𝑊 ∈ LMod → 𝑊 ∈ Grp) | |
4 | eqid 2622 | . . 3 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
5 | eqid 2622 | . . 3 ⊢ (+g‘𝑊) = (+g‘𝑊) | |
6 | 4, 5 | lmodcom 18909 | . 2 ⊢ ((𝑊 ∈ LMod ∧ 𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊)) → (𝑥(+g‘𝑊)𝑦) = (𝑦(+g‘𝑊)𝑥)) |
7 | 1, 2, 3, 6 | isabld 18206 | 1 ⊢ (𝑊 ∈ LMod → 𝑊 ∈ Abel) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 1990 ‘cfv 5888 Basecbs 15857 +gcplusg 15941 Abelcabl 18194 LModclmod 18863 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-2 11079 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-plusg 15954 df-0g 16102 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-grp 17425 df-minusg 17426 df-cmn 18195 df-abl 18196 df-mgp 18490 df-ur 18502 df-ring 18549 df-lmod 18865 |
This theorem is referenced by: lmodcmn 18911 lmodnegadd 18912 lmodvsubadd 18914 lmodvaddsub4 18915 lssvancl1 18945 invlmhm 19042 lmhmplusg 19044 lsmcl 19083 lspprabs 19095 pj1lmhm 19100 pj1lmhm2 19101 lvecindp 19138 lvecindp2 19139 lsmcv 19141 zlmlmod 19871 pjdm2 20055 pjf2 20058 pjfo 20059 ocvpj 20061 frlmsslsp 20135 nlmtlm 22498 ngpocelbl 22508 nmhmplusg 22561 clmabl 22869 cvsi 22930 minveclem2 23197 pjthlem2 23209 ttgcontlem1 25765 bj-modssabl 33142 lcvexchlem3 34323 lcvexchlem4 34324 lcvexchlem5 34325 lsatcvatlem 34336 lsatcvat 34337 lsatcvat3 34339 l1cvat 34342 lshpsmreu 34396 lshpkrlem5 34401 dia2dimlem5 36357 dihjatc3 36602 dihmeetlem9N 36604 dihjatcclem1 36707 dihjat 36712 lclkrlem2b 36797 baerlem3lem1 36996 baerlem5alem1 36997 baerlem5blem1 36998 baerlem3lem2 36999 baerlem5alem2 37000 baerlem5blem2 37001 hdmap1neglem1N 37117 hdmaprnlem7N 37147 isnumbasgrplem3 37675 gsumlsscl 42164 |
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