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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 2728 | . . 3 โข (mulGrpโ๐ ) = (mulGrpโ๐ ) | |
2 | 1 | ringmgp 20178 | . 2 โข (๐ โ Ring โ (mulGrpโ๐ ) โ Mnd) |
3 | ringcl.b | . . . 4 โข ๐ต = (Baseโ๐ ) | |
4 | 1, 3 | mgpbas 20079 | . . 3 โข ๐ต = (Baseโ(mulGrpโ๐ )) |
5 | ringcl.t | . . . 4 โข ยท = (.rโ๐ ) | |
6 | 1, 5 | mgpplusg 20077 | . . 3 โข ยท = (+gโ(mulGrpโ๐ )) |
7 | 4, 6 | mndass 18702 | . 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 1085 = wceq 1534 โ wcel 2099 โcfv 6548 (class class class)co 7420 Basecbs 17179 .rcmulr 17233 Mndcmnd 18693 mulGrpcmgp 20073 Ringcrg 20172 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-cnex 11194 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 ax-pre-mulgt0 11215 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7871 df-2nd 7994 df-frecs 8286 df-wrecs 8317 df-recs 8391 df-rdg 8430 df-er 8724 df-en 8964 df-dom 8965 df-sdom 8966 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-sub 11476 df-neg 11477 df-nn 12243 df-2 12305 df-sets 17132 df-slot 17150 df-ndx 17162 df-base 17180 df-plusg 17245 df-sgrp 18678 df-mnd 18694 df-mgp 20074 df-ring 20174 |
This theorem is referenced by: ringassd 20196 ringinvnzdiv 20236 ringmneg1 20239 ringmneg2 20240 imasring 20265 dvdsrtr 20306 dvdsrmul1 20307 unitgrp 20321 dvrass 20346 dvrcan1 20347 rdivmuldivd 20351 subrginv 20526 issubrg2 20530 drngmul0or 20652 isdrngd 20656 isdrngdOLD 20658 sralmod 21079 unitrrg 21239 frlmphl 21714 sraassaOLD 21802 psrlmod 21902 psrass1 21906 psrass23l 21909 psrass23 21911 mamuass 22301 mamuvs1 22304 mavmulass 22450 mdetrsca 22504 chfacfpmmulgsum2 22766 nrginvrcnlem 24607 ply1divex 26071 dvrcan5 32944 ornglmullt 33022 mxidlprm 33183 fedgmullem1 33323 fedgmullem2 33324 mdetpmtr1 33424 mdetpmtr12 33426 mdetlap 33433 matunitlindflem1 37089 lflvscl 38549 lflvsass 38553 eqlkr3 38573 lkrlsp 38574 lcfl7lem 40972 lclkrlem2m 40992 lcfrlem1 41015 hgmapvvlem1 41396 |
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