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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 2732 | . . 3 โข (mulGrpโ๐ ) = (mulGrpโ๐ ) | |
2 | 1 | crngmgp 20063 | . 2 โข (๐ โ CRing โ (mulGrpโ๐ ) โ CMnd) |
3 | ringcl.b | . . . 4 โข ๐ต = (Baseโ๐ ) | |
4 | 1, 3 | mgpbas 19992 | . . 3 โข ๐ต = (Baseโ(mulGrpโ๐ )) |
5 | ringcl.t | . . . 4 โข ยท = (.rโ๐ ) | |
6 | 1, 5 | mgpplusg 19990 | . . 3 โข ยท = (+gโ(mulGrpโ๐ )) |
7 | 4, 6 | cmncom 19665 | . 2 โข (((mulGrpโ๐ ) โ CMnd โง ๐ โ ๐ต โง ๐ โ ๐ต) โ (๐ ยท ๐) = (๐ ยท ๐)) |
8 | 2, 7 | syl3an1 1163 | 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 1087 = wceq 1541 โ wcel 2106 โcfv 6543 (class class class)co 7408 Basecbs 17143 .rcmulr 17197 CMndccmn 19647 mulGrpcmgp 19986 CRingccrg 20056 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7724 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7364 df-ov 7411 df-oprab 7412 df-mpo 7413 df-om 7855 df-2nd 7975 df-frecs 8265 df-wrecs 8296 df-recs 8370 df-rdg 8409 df-er 8702 df-en 8939 df-dom 8940 df-sdom 8941 df-pnf 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-le 11253 df-sub 11445 df-neg 11446 df-nn 12212 df-2 12274 df-sets 17096 df-slot 17114 df-ndx 17126 df-base 17144 df-plusg 17209 df-cmn 19649 df-mgp 19987 df-cring 20058 |
This theorem is referenced by: crngoppr 20153 unitmulclb 20194 rdivmuldivd 20226 idsrngd 20469 rmodislmodlem 20538 lmhmvsca 20655 quscrng 20877 frlmphl 21335 psrcom 21528 psrass23 21529 mplcoe2 21595 mamutpos 21959 mat1dimcrng 21978 dmatcrng 22003 scmatcrng 22022 mdetrsca 22104 mdetmul 22124 cramerimp 22187 fta1glem1 25682 idomrcan 32372 ringlsmss1 32501 mxidlprm 32581 extdg1id 32737 minplyirred 32765 mdetlap 32807 crngcomd 41089 imacrhmcl 41091 |
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