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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 | ⊢ 𝐵 = (Base‘𝑅) |
ringcl.t | ⊢ · = (.r‘𝑅) |
Ref | Expression |
---|---|
ringass | ⊢ ((𝑅 ∈ Ring ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((𝑋 · 𝑌) · 𝑍) = (𝑋 · (𝑌 · 𝑍))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2731 | . . 3 ⊢ (mulGrp‘𝑅) = (mulGrp‘𝑅) | |
2 | 1 | ringmgp 20134 | . 2 ⊢ (𝑅 ∈ Ring → (mulGrp‘𝑅) ∈ Mnd) |
3 | ringcl.b | . . . 4 ⊢ 𝐵 = (Base‘𝑅) | |
4 | 1, 3 | mgpbas 20035 | . . 3 ⊢ 𝐵 = (Base‘(mulGrp‘𝑅)) |
5 | ringcl.t | . . . 4 ⊢ · = (.r‘𝑅) | |
6 | 1, 5 | mgpplusg 20033 | . . 3 ⊢ · = (+g‘(mulGrp‘𝑅)) |
7 | 4, 6 | mndass 18669 | . 2 ⊢ (((mulGrp‘𝑅) ∈ Mnd ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((𝑋 · 𝑌) · 𝑍) = (𝑋 · (𝑌 · 𝑍))) |
8 | 2, 7 | sylan 579 | 1 ⊢ ((𝑅 ∈ Ring ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((𝑋 · 𝑌) · 𝑍) = (𝑋 · (𝑌 · 𝑍))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1086 = wceq 1540 ∈ wcel 2105 ‘cfv 6543 (class class class)co 7412 Basecbs 17149 .rcmulr 17203 Mndcmnd 18660 mulGrpcmgp 20029 Ringcrg 20128 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-cnex 11170 ax-resscn 11171 ax-1cn 11172 ax-icn 11173 ax-addcl 11174 ax-addrcl 11175 ax-mulcl 11176 ax-mulrcl 11177 ax-mulcom 11178 ax-addass 11179 ax-mulass 11180 ax-distr 11181 ax-i2m1 11182 ax-1ne0 11183 ax-1rid 11184 ax-rnegex 11185 ax-rrecex 11186 ax-cnre 11187 ax-pre-lttri 11188 ax-pre-lttrn 11189 ax-pre-ltadd 11190 ax-pre-mulgt0 11191 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-reu 3376 df-rab 3432 df-v 3475 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 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7860 df-2nd 7980 df-frecs 8270 df-wrecs 8301 df-recs 8375 df-rdg 8414 df-er 8707 df-en 8944 df-dom 8945 df-sdom 8946 df-pnf 11255 df-mnf 11256 df-xr 11257 df-ltxr 11258 df-le 11259 df-sub 11451 df-neg 11452 df-nn 12218 df-2 12280 df-sets 17102 df-slot 17120 df-ndx 17132 df-base 17150 df-plusg 17215 df-sgrp 18645 df-mnd 18661 df-mgp 20030 df-ring 20130 |
This theorem is referenced by: ringassd 20151 ringinvnzdiv 20190 ringmneg1 20193 ringmneg2 20194 imasring 20219 dvdsrtr 20260 dvdsrmul1 20261 unitgrp 20275 dvrass 20300 dvrcan1 20301 rdivmuldivd 20305 subrginv 20479 issubrg2 20483 drngmul0or 20530 isdrngd 20534 isdrngdOLD 20536 sralmod 20955 unitrrg 21110 frlmphl 21556 sraassaOLD 21644 psrlmod 21741 psrass1 21745 psrass23l 21748 psrass23 21750 mamuass 22123 mamuvs1 22126 mavmulass 22272 mdetrsca 22326 chfacfpmmulgsum2 22588 nrginvrcnlem 24429 ply1divex 25890 dvrcan5 32656 ornglmullt 32696 mxidlprm 32861 fedgmullem1 33003 fedgmullem2 33004 mdetpmtr1 33102 mdetpmtr12 33104 mdetlap 33111 matunitlindflem1 36788 lflvscl 38251 lflvsass 38255 eqlkr3 38275 lkrlsp 38276 lcfl7lem 40674 lclkrlem2m 40694 lcfrlem1 40717 hgmapvvlem1 41098 |
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