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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 18934 | . 2 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Abel) | |
2 | ablcmn 18552 | . 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 2166 CMndccmn 18546 Abelcabl 18547 Ringcrg 18901 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2803 ax-rep 4994 ax-sep 5005 ax-nul 5013 ax-pow 5065 ax-pr 5127 ax-un 7209 ax-cnex 10308 ax-resscn 10309 ax-1cn 10310 ax-icn 10311 ax-addcl 10312 ax-addrcl 10313 ax-mulcl 10314 ax-mulrcl 10315 ax-mulcom 10316 ax-addass 10317 ax-mulass 10318 ax-distr 10319 ax-i2m1 10320 ax-1ne0 10321 ax-1rid 10322 ax-rnegex 10323 ax-rrecex 10324 ax-cnre 10325 ax-pre-lttri 10326 ax-pre-lttrn 10327 ax-pre-ltadd 10328 ax-pre-mulgt0 10329 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3or 1114 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-reu 3124 df-rmo 3125 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4145 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-tp 4402 df-op 4404 df-uni 4659 df-iun 4742 df-br 4874 df-opab 4936 df-mpt 4953 df-tr 4976 df-id 5250 df-eprel 5255 df-po 5263 df-so 5264 df-fr 5301 df-we 5303 df-xp 5348 df-rel 5349 df-cnv 5350 df-co 5351 df-dm 5352 df-rn 5353 df-res 5354 df-ima 5355 df-pred 5920 df-ord 5966 df-on 5967 df-lim 5968 df-suc 5969 df-iota 6086 df-fun 6125 df-fn 6126 df-f 6127 df-f1 6128 df-fo 6129 df-f1o 6130 df-fv 6131 df-riota 6866 df-ov 6908 df-oprab 6909 df-mpt2 6910 df-om 7327 df-wrecs 7672 df-recs 7734 df-rdg 7772 df-er 8009 df-en 8223 df-dom 8224 df-sdom 8225 df-pnf 10393 df-mnf 10394 df-xr 10395 df-ltxr 10396 df-le 10397 df-sub 10587 df-neg 10588 df-nn 11351 df-2 11414 df-ndx 16225 df-slot 16226 df-base 16228 df-sets 16229 df-plusg 16318 df-0g 16455 df-mgm 17595 df-sgrp 17637 df-mnd 17648 df-grp 17779 df-minusg 17780 df-cmn 18548 df-abl 18549 df-mgp 18844 df-ur 18856 df-ring 18903 |
This theorem is referenced by: ringsrg 18943 gsummulc1 18960 gsummulc2 18961 gsumdixp 18963 psrmulcllem 19748 psrlidm 19764 psrridm 19765 psrass1 19766 psrdi 19767 psrdir 19768 psrcom 19770 mplmonmul 19825 mplcoe1 19826 evlslem2 19872 evlslem1 19875 psropprmul 19968 coe1mul2 19999 coe1fzgsumdlem 20031 gsumsmonply1 20033 gsummoncoe1 20034 lply1binom 20036 evls1gsumadd 20049 evl1gsumdlem 20080 gsumfsum 20173 nn0srg 20176 rge0srg 20177 regsumsupp 20329 ip2di 20348 frlmphl 20487 mamucl 20574 mamuass 20575 mamudi 20576 mamudir 20577 mat1dimmul 20650 dmatmul 20671 mavmulcl 20721 mavmulass 20723 mdetleib2 20762 mdetf 20769 mdetrlin 20776 mdetralt 20782 m2detleib 20805 madugsum 20817 smadiadetlem3lem2 20842 smadiadet 20845 mat2pmatmul 20906 m2pmfzgsumcl 20923 decpmatmul 20947 pmatcollpw1 20951 pmatcollpwfi 20957 pmatcollpw3fi1lem1 20961 pm2mpcl 20972 mply1topmatcl 20980 mp2pm2mplem2 20982 mp2pm2mplem4 20984 mp2pm2mp 20986 pm2mpghm 20991 pm2mpmhmlem2 20994 pm2mp 21000 chfacfscmulgsum 21035 chfacfpmmulgsum 21039 cpmadugsumlemF 21051 cpmadugsumfi 21052 cayhamlem4 21063 tdeglem1 24217 tdeglem3 24218 tdeglem4 24219 plypf1 24367 taylfvallem 24511 taylf 24514 tayl0 24515 taylpfval 24518 jensenlem1 25126 jensenlem2 25127 jensen 25128 amgm 25130 ofldchr 30359 mdetpmtr1 30434 matunitlindflem1 33949 lfladdcl 35146 ply1mulgsum 43025 amgmwlem 43444 |
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