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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 2738 | . 2 ⊢ (𝑊 ∈ LMod → (Base‘𝑊) = (Base‘𝑊)) | |
| 2 | eqidd 2738 | . 2 ⊢ (𝑊 ∈ LMod → (+g‘𝑊) = (+g‘𝑊)) | |
| 3 | lmodgrp 20865 | . 2 ⊢ (𝑊 ∈ LMod → 𝑊 ∈ Grp) | |
| 4 | eqid 2737 | . . 3 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
| 5 | eqid 2737 | . . 3 ⊢ (+g‘𝑊) = (+g‘𝑊) | |
| 6 | 4, 5 | lmodcom 20906 | . 2 ⊢ ((𝑊 ∈ LMod ∧ 𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊)) → (𝑥(+g‘𝑊)𝑦) = (𝑦(+g‘𝑊)𝑥)) |
| 7 | 1, 2, 3, 6 | isabld 19813 | 1 ⊢ (𝑊 ∈ LMod → 𝑊 ∈ Abel) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∈ wcel 2108 ‘cfv 6561 Basecbs 17247 +gcplusg 17297 Abelcabl 19799 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-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-2 12329 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-plusg 17310 df-0g 17486 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-grp 18954 df-minusg 18955 df-cmn 19800 df-abl 19801 df-mgp 20138 df-ur 20179 df-ring 20232 df-lmod 20860 |
| This theorem is referenced by: lmodcmn 20908 lmodnegadd 20909 lmodvsubadd 20911 lmodvaddsub4 20912 lssvancl1 20943 invlmhm 21041 lmhmplusg 21043 lsmcl 21082 lspprabs 21094 pj1lmhm 21099 pj1lmhm2 21100 lvecindp 21140 lvecindp2 21141 lsmcv 21143 zlmlmod 21537 pjdm2 21731 pjf2 21734 pjfo 21735 ocvpj 21737 frlmsslsp 21816 nlmtlm 24715 ngpocelbl 24725 nmhmplusg 24778 clmabl 25102 cvsi 25163 minveclem2 25460 pjthlem2 25472 ttgcontlem1 28899 quslmod 33386 quslmhm 33387 lindsunlem 33675 qusdimsum 33679 fedgmullem2 33681 bj-modssabl 37281 lcvexchlem3 39037 lcvexchlem4 39038 lcvexchlem5 39039 lsatcvatlem 39050 lsatcvat 39051 lsatcvat3 39053 l1cvat 39056 lshpsmreu 39110 lshpkrlem5 39115 dia2dimlem5 41070 dihjatc3 41315 dihmeetlem9N 41317 dihjatcclem1 41420 dihjat 41425 lclkrlem2b 41510 baerlem3lem1 41709 baerlem5alem1 41710 baerlem5blem1 41711 baerlem3lem2 41712 baerlem5alem2 41713 baerlem5blem2 41714 hdmaprnlem7N 41857 isnumbasgrplem3 43117 gsumlsscl 48296 |
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