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Mirrors > Home > MPE Home > Th. List > ringcmn | Structured version Visualization version GIF version |
Description: A ring is a commutative monoid. (Contributed by Mario Carneiro, 7-Jan-2015.) |
Ref | Expression |
---|---|
ringcmn | ⊢ (𝑅 ∈ Ring → 𝑅 ∈ CMnd) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ringabl 18626 | . 2 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Abel) | |
2 | ablcmn 18245 | . 2 ⊢ (𝑅 ∈ Abel → 𝑅 ∈ CMnd) | |
3 | 1, 2 | syl 17 | 1 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ CMnd) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2030 CMndccmn 18239 Abelcabl 18240 Ringcrg 18593 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-rep 4804 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 ax-cnex 10030 ax-resscn 10031 ax-1cn 10032 ax-icn 10033 ax-addcl 10034 ax-addrcl 10035 ax-mulcl 10036 ax-mulrcl 10037 ax-mulcom 10038 ax-addass 10039 ax-mulass 10040 ax-distr 10041 ax-i2m1 10042 ax-1ne0 10043 ax-1rid 10044 ax-rnegex 10045 ax-rrecex 10046 ax-cnre 10047 ax-pre-lttri 10048 ax-pre-lttrn 10049 ax-pre-ltadd 10050 ax-pre-mulgt0 10051 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1055 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-nel 2927 df-ral 2946 df-rex 2947 df-reu 2948 df-rmo 2949 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-pss 3623 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-tp 4215 df-op 4217 df-uni 4469 df-iun 4554 df-br 4686 df-opab 4746 df-mpt 4763 df-tr 4786 df-id 5053 df-eprel 5058 df-po 5064 df-so 5065 df-fr 5102 df-we 5104 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-pred 5718 df-ord 5764 df-on 5765 df-lim 5766 df-suc 5767 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 df-riota 6651 df-ov 6693 df-oprab 6694 df-mpt2 6695 df-om 7108 df-wrecs 7452 df-recs 7513 df-rdg 7551 df-er 7787 df-en 7998 df-dom 7999 df-sdom 8000 df-pnf 10114 df-mnf 10115 df-xr 10116 df-ltxr 10117 df-le 10118 df-sub 10306 df-neg 10307 df-nn 11059 df-2 11117 df-ndx 15907 df-slot 15908 df-base 15910 df-sets 15911 df-plusg 16001 df-0g 16149 df-mgm 17289 df-sgrp 17331 df-mnd 17342 df-grp 17472 df-minusg 17473 df-cmn 18241 df-abl 18242 df-mgp 18536 df-ur 18548 df-ring 18595 |
This theorem is referenced by: ringsrg 18635 gsummulc1 18652 gsummulc2 18653 gsumdixp 18655 psrmulcllem 19435 psrlidm 19451 psrridm 19452 psrass1 19453 psrdi 19454 psrdir 19455 psrcom 19457 mplmonmul 19512 mplcoe1 19513 evlslem2 19560 evlslem1 19563 psropprmul 19656 coe1mul2 19687 coe1fzgsumdlem 19719 gsumsmonply1 19721 gsummoncoe1 19722 lply1binom 19724 evls1gsumadd 19737 evl1gsumdlem 19768 gsumfsum 19861 nn0srg 19864 rge0srg 19865 regsumsupp 20016 ip2di 20034 frlmphl 20168 mamucl 20255 mamuass 20256 mamudi 20257 mamudir 20258 mat1dimmul 20330 dmatmul 20351 mavmulcl 20401 mavmulass 20403 mdetleib2 20442 mdetf 20449 mdetrlin 20456 mdetralt 20462 m2detleib 20485 madugsum 20497 smadiadetlem3lem2 20521 smadiadet 20524 mat2pmatmul 20584 m2pmfzgsumcl 20601 decpmatmul 20625 pmatcollpw1 20629 pmatcollpwfi 20635 pmatcollpw3fi1lem1 20639 pm2mpcl 20650 mply1topmatcl 20658 mp2pm2mplem2 20660 mp2pm2mplem4 20662 mp2pm2mp 20664 pm2mpghm 20669 pm2mpmhmlem2 20672 pm2mp 20678 chfacfscmulgsum 20713 chfacfpmmulgsum 20717 cpmadugsumlemF 20729 cpmadugsumfi 20730 cayhamlem4 20741 tdeglem1 23863 tdeglem3 23864 tdeglem4 23865 plypf1 24013 taylfvallem 24157 taylf 24160 tayl0 24161 taylpfval 24164 jensenlem1 24758 jensenlem2 24759 jensen 24760 amgm 24762 ofldchr 29942 mdetpmtr1 30017 matunitlindflem1 33535 lfladdcl 34676 ply1mulgsum 42503 amgmwlem 42876 |
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