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Mirrors > Home > MPE Home > Th. List > ringabl | Structured version Visualization version GIF version |
Description: A ring is an Abelian group. (Contributed by NM, 26-Aug-2011.) |
Ref | Expression |
---|---|
ringabl | ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Abel) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqidd 2741 | . 2 ⊢ (𝑅 ∈ Ring → (Base‘𝑅) = (Base‘𝑅)) | |
2 | eqidd 2741 | . 2 ⊢ (𝑅 ∈ Ring → (+g‘𝑅) = (+g‘𝑅)) | |
3 | ringgrp 19784 | . 2 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Grp) | |
4 | eqid 2740 | . . 3 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
5 | eqid 2740 | . . 3 ⊢ (+g‘𝑅) = (+g‘𝑅) | |
6 | 4, 5 | ringcom 19814 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → (𝑥(+g‘𝑅)𝑦) = (𝑦(+g‘𝑅)𝑥)) |
7 | 1, 2, 3, 6 | isabld 19396 | 1 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Abel) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2110 ‘cfv 6431 Basecbs 16908 +gcplusg 16958 Abelcabl 19383 Ringcrg 19779 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-sep 5227 ax-nul 5234 ax-pow 5292 ax-pr 5356 ax-un 7580 ax-cnex 10926 ax-resscn 10927 ax-1cn 10928 ax-icn 10929 ax-addcl 10930 ax-addrcl 10931 ax-mulcl 10932 ax-mulrcl 10933 ax-mulcom 10934 ax-addass 10935 ax-mulass 10936 ax-distr 10937 ax-i2m1 10938 ax-1ne0 10939 ax-1rid 10940 ax-rnegex 10941 ax-rrecex 10942 ax-cnre 10943 ax-pre-lttri 10944 ax-pre-lttrn 10945 ax-pre-ltadd 10946 ax-pre-mulgt0 10947 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ne 2946 df-nel 3052 df-ral 3071 df-rex 3072 df-reu 3073 df-rmo 3074 df-rab 3075 df-v 3433 df-sbc 3721 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-pss 3911 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4846 df-iun 4932 df-br 5080 df-opab 5142 df-mpt 5163 df-tr 5197 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6200 df-ord 6267 df-on 6268 df-lim 6269 df-suc 6270 df-iota 6389 df-fun 6433 df-fn 6434 df-f 6435 df-f1 6436 df-fo 6437 df-f1o 6438 df-fv 6439 df-riota 7226 df-ov 7272 df-oprab 7273 df-mpo 7274 df-om 7705 df-2nd 7823 df-frecs 8086 df-wrecs 8117 df-recs 8191 df-rdg 8230 df-er 8479 df-en 8715 df-dom 8716 df-sdom 8717 df-pnf 11010 df-mnf 11011 df-xr 11012 df-ltxr 11013 df-le 11014 df-sub 11205 df-neg 11206 df-nn 11972 df-2 12034 df-sets 16861 df-slot 16879 df-ndx 16891 df-base 16909 df-plusg 16971 df-0g 17148 df-mgm 18322 df-sgrp 18371 df-mnd 18382 df-grp 18576 df-minusg 18577 df-cmn 19384 df-abl 19385 df-mgp 19717 df-ur 19734 df-ring 19781 |
This theorem is referenced by: ringcmn 19816 2idlcpbl 20501 qus1 20502 qusrhm 20504 zringabl 20670 ip2subdi 20845 zlmassa 21102 mplbas2 21239 mdetralt 21753 mdetuni0 21766 cayhamlem1 22011 cpmadugsumlemF 22021 nrgtgp 23832 ply1divmo 25296 r1pid 25320 efabl 25702 jensenlem2 26133 amgmlem 26135 lidlnsg 31615 idlsrgcmnd 31654 cnzh 31914 rezh 31915 matunitlindflem1 35767 lflsub 37075 lfladdcom 37080 lflnegcl 37083 baerlem3lem1 39715 ringabld 40239 isnumbasgrplem3 40925 ringrng 45404 lidlabl 45449 cznabel 45479 |
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