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Mirrors > Home > MPE Home > Th. List > crngcom | Structured version Visualization version GIF version |
Description: A commutative ring's multiplication operation is commutative. (Contributed by Mario Carneiro, 7-Jan-2015.) |
Ref | Expression |
---|---|
ringcl.b | โข ๐ต = (Baseโ๐ ) |
ringcl.t | โข ยท = (.rโ๐ ) |
Ref | Expression |
---|---|
crngcom | โข ((๐ โ CRing โง ๐ โ ๐ต โง ๐ โ ๐ต) โ (๐ ยท ๐) = (๐ ยท ๐)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2730 | . . 3 โข (mulGrpโ๐ ) = (mulGrpโ๐ ) | |
2 | 1 | crngmgp 20135 | . 2 โข (๐ โ CRing โ (mulGrpโ๐ ) โ CMnd) |
3 | ringcl.b | . . . 4 โข ๐ต = (Baseโ๐ ) | |
4 | 1, 3 | mgpbas 20034 | . . 3 โข ๐ต = (Baseโ(mulGrpโ๐ )) |
5 | ringcl.t | . . . 4 โข ยท = (.rโ๐ ) | |
6 | 1, 5 | mgpplusg 20032 | . . 3 โข ยท = (+gโ(mulGrpโ๐ )) |
7 | 4, 6 | cmncom 19707 | . 2 โข (((mulGrpโ๐ ) โ CMnd โง ๐ โ ๐ต โง ๐ โ ๐ต) โ (๐ ยท ๐) = (๐ ยท ๐)) |
8 | 2, 7 | syl3an1 1161 | 1 โข ((๐ โ CRing โง ๐ โ ๐ต โง ๐ โ ๐ต) โ (๐ ยท ๐) = (๐ ยท ๐)) |
Colors of variables: wff setvar class |
Syntax hints: โ wi 4 โง w3a 1085 = wceq 1539 โ wcel 2104 โcfv 6542 (class class class)co 7411 Basecbs 17148 .rcmulr 17202 CMndccmn 19689 mulGrpcmgp 20028 CRingccrg 20128 |
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 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7727 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-reu 3375 df-rab 3431 df-v 3474 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-om 7858 df-2nd 7978 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 df-er 8705 df-en 8942 df-dom 8943 df-sdom 8944 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-nn 12217 df-2 12279 df-sets 17101 df-slot 17119 df-ndx 17131 df-base 17149 df-plusg 17214 df-cmn 19691 df-mgp 20029 df-cring 20130 |
This theorem is referenced by: crngoppr 20229 unitmulclb 20272 rdivmuldivd 20304 idsrngd 20613 rmodislmodlem 20683 lmhmvsca 20800 quscrng 21029 frlmphl 21555 psrcom 21748 psrass23 21749 mplcoe2 21815 mamutpos 22180 mat1dimcrng 22199 dmatcrng 22224 scmatcrng 22243 mdetrsca 22325 mdetmul 22345 cramerimp 22408 fta1glem1 25918 idomrcan 32647 ringlsmss1 32780 mxidlprm 32860 r1pid2 32954 extdg1id 33030 minplyirred 33059 mdetlap 33110 crngcomd 41391 imacrhmcl 41393 |
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