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Mirrors > Home > MPE Home > Th. List > ringass | Structured version Visualization version GIF version |
Description: Associative law for multiplication in a ring. (Contributed by NM, 27-Aug-2011.) (Revised by Mario Carneiro, 6-Jan-2015.) |
Ref | Expression |
---|---|
ringcl.b | โข ๐ต = (Baseโ๐ ) |
ringcl.t | โข ยท = (.rโ๐ ) |
Ref | Expression |
---|---|
ringass | โข ((๐ โ Ring โง (๐ โ ๐ต โง ๐ โ ๐ต โง ๐ โ ๐ต)) โ ((๐ ยท ๐) ยท ๐) = (๐ ยท (๐ ยท ๐))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2724 | . . 3 โข (mulGrpโ๐ ) = (mulGrpโ๐ ) | |
2 | 1 | ringmgp 20140 | . 2 โข (๐ โ Ring โ (mulGrpโ๐ ) โ Mnd) |
3 | ringcl.b | . . . 4 โข ๐ต = (Baseโ๐ ) | |
4 | 1, 3 | mgpbas 20041 | . . 3 โข ๐ต = (Baseโ(mulGrpโ๐ )) |
5 | ringcl.t | . . . 4 โข ยท = (.rโ๐ ) | |
6 | 1, 5 | mgpplusg 20039 | . . 3 โข ยท = (+gโ(mulGrpโ๐ )) |
7 | 4, 6 | mndass 18672 | . 2 โข (((mulGrpโ๐ ) โ Mnd โง (๐ โ ๐ต โง ๐ โ ๐ต โง ๐ โ ๐ต)) โ ((๐ ยท ๐) ยท ๐) = (๐ ยท (๐ ยท ๐))) |
8 | 2, 7 | sylan 579 | 1 โข ((๐ โ Ring โง (๐ โ ๐ต โง ๐ โ ๐ต โง ๐ โ ๐ต)) โ ((๐ ยท ๐) ยท ๐) = (๐ ยท (๐ ยท ๐))) |
Colors of variables: wff setvar class |
Syntax hints: โ wi 4 โง wa 395 โง w3a 1084 = wceq 1533 โ wcel 2098 โcfv 6534 (class class class)co 7402 Basecbs 17149 .rcmulr 17203 Mndcmnd 18663 mulGrpcmgp 20035 Ringcrg 20134 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 ax-cnex 11163 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 ax-pre-mulgt0 11184 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3960 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-iun 4990 df-br 5140 df-opab 5202 df-mpt 5223 df-tr 5257 df-id 5565 df-eprel 5571 df-po 5579 df-so 5580 df-fr 5622 df-we 5624 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-pred 6291 df-ord 6358 df-on 6359 df-lim 6360 df-suc 6361 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-riota 7358 df-ov 7405 df-oprab 7406 df-mpo 7407 df-om 7850 df-2nd 7970 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-er 8700 df-en 8937 df-dom 8938 df-sdom 8939 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 17102 df-slot 17120 df-ndx 17132 df-base 17150 df-plusg 17215 df-sgrp 18648 df-mnd 18664 df-mgp 20036 df-ring 20136 |
This theorem is referenced by: ringassd 20157 ringinvnzdiv 20196 ringmneg1 20199 ringmneg2 20200 imasring 20225 dvdsrtr 20266 dvdsrmul1 20267 unitgrp 20281 dvrass 20306 dvrcan1 20307 rdivmuldivd 20311 subrginv 20486 issubrg2 20490 drngmul0or 20612 isdrngd 20616 isdrngdOLD 20618 sralmod 21039 unitrrg 21199 frlmphl 21665 sraassaOLD 21753 psrlmod 21852 psrass1 21856 psrass23l 21859 psrass23 21861 mamuass 22246 mamuvs1 22249 mavmulass 22395 mdetrsca 22449 chfacfpmmulgsum2 22711 nrginvrcnlem 24552 ply1divex 26016 dvrcan5 32876 ornglmullt 32917 mxidlprm 33081 fedgmullem1 33221 fedgmullem2 33222 mdetpmtr1 33322 mdetpmtr12 33324 mdetlap 33331 matunitlindflem1 36987 lflvscl 38450 lflvsass 38454 eqlkr3 38474 lkrlsp 38475 lcfl7lem 40873 lclkrlem2m 40893 lcfrlem1 40916 hgmapvvlem1 41297 |
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