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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 2825 | . 2 ⊢ (𝑊 ∈ LMod → (Base‘𝑊) = (Base‘𝑊)) | |
2 | eqidd 2825 | . 2 ⊢ (𝑊 ∈ LMod → (+g‘𝑊) = (+g‘𝑊)) | |
3 | lmodgrp 19644 | . 2 ⊢ (𝑊 ∈ LMod → 𝑊 ∈ Grp) | |
4 | eqid 2824 | . . 3 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
5 | eqid 2824 | . . 3 ⊢ (+g‘𝑊) = (+g‘𝑊) | |
6 | 4, 5 | lmodcom 19683 | . 2 ⊢ ((𝑊 ∈ LMod ∧ 𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊)) → (𝑥(+g‘𝑊)𝑦) = (𝑦(+g‘𝑊)𝑥)) |
7 | 1, 2, 3, 6 | isabld 18923 | 1 ⊢ (𝑊 ∈ LMod → 𝑊 ∈ Abel) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2113 ‘cfv 6358 Basecbs 16486 +gcplusg 16568 Abelcabl 18910 LModclmod 19637 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 ax-cnex 10596 ax-resscn 10597 ax-1cn 10598 ax-icn 10599 ax-addcl 10600 ax-addrcl 10601 ax-mulcl 10602 ax-mulrcl 10603 ax-mulcom 10604 ax-addass 10605 ax-mulass 10606 ax-distr 10607 ax-i2m1 10608 ax-1ne0 10609 ax-1rid 10610 ax-rnegex 10611 ax-rrecex 10612 ax-cnre 10613 ax-pre-lttri 10614 ax-pre-lttrn 10615 ax-pre-ltadd 10616 ax-pre-mulgt0 10617 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-nel 3127 df-ral 3146 df-rex 3147 df-reu 3148 df-rmo 3149 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-pss 3957 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-tp 4575 df-op 4577 df-uni 4842 df-iun 4924 df-br 5070 df-opab 5132 df-mpt 5150 df-tr 5176 df-id 5463 df-eprel 5468 df-po 5477 df-so 5478 df-fr 5517 df-we 5519 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-pred 6151 df-ord 6197 df-on 6198 df-lim 6199 df-suc 6200 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7117 df-ov 7162 df-oprab 7163 df-mpo 7164 df-om 7584 df-wrecs 7950 df-recs 8011 df-rdg 8049 df-er 8292 df-en 8513 df-dom 8514 df-sdom 8515 df-pnf 10680 df-mnf 10681 df-xr 10682 df-ltxr 10683 df-le 10684 df-sub 10875 df-neg 10876 df-nn 11642 df-2 11703 df-ndx 16489 df-slot 16490 df-base 16492 df-sets 16493 df-plusg 16581 df-0g 16718 df-mgm 17855 df-sgrp 17904 df-mnd 17915 df-grp 18109 df-minusg 18110 df-cmn 18911 df-abl 18912 df-mgp 19243 df-ur 19255 df-ring 19302 df-lmod 19639 |
This theorem is referenced by: lmodcmn 19685 lmodnegadd 19686 lmodvsubadd 19688 lmodvaddsub4 19689 lssvancl1 19719 invlmhm 19817 lmhmplusg 19819 lsmcl 19858 lspprabs 19870 pj1lmhm 19875 pj1lmhm2 19876 lvecindp 19913 lvecindp2 19914 lsmcv 19916 zlmlmod 20673 pjdm2 20858 pjf2 20861 pjfo 20862 ocvpj 20864 frlmsslsp 20943 nlmtlm 23306 ngpocelbl 23316 nmhmplusg 23369 clmabl 23676 cvsi 23737 minveclem2 24032 pjthlem2 24044 ttgcontlem1 26674 quslmod 30927 quslmhm 30928 lindsunlem 31024 qusdimsum 31028 fedgmullem2 31030 bj-modssabl 34566 lcvexchlem3 36176 lcvexchlem4 36177 lcvexchlem5 36178 lsatcvatlem 36189 lsatcvat 36190 lsatcvat3 36192 l1cvat 36195 lshpsmreu 36249 lshpkrlem5 36254 dia2dimlem5 38208 dihjatc3 38453 dihmeetlem9N 38455 dihjatcclem1 38558 dihjat 38563 lclkrlem2b 38648 baerlem3lem1 38847 baerlem5alem1 38848 baerlem5blem1 38849 baerlem3lem2 38850 baerlem5alem2 38851 baerlem5blem2 38852 hdmaprnlem7N 38995 isnumbasgrplem3 39711 gsumlsscl 44438 |
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