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Mirrors > Home > MPE Home > Th. List > ringass | Structured version Visualization version GIF version |
Description: Associative law for the multiplication operation of 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 2821 | . . 3 ⊢ (mulGrp‘𝑅) = (mulGrp‘𝑅) | |
2 | 1 | ringmgp 19297 | . 2 ⊢ (𝑅 ∈ Ring → (mulGrp‘𝑅) ∈ Mnd) |
3 | ringcl.b | . . . 4 ⊢ 𝐵 = (Base‘𝑅) | |
4 | 1, 3 | mgpbas 19239 | . . 3 ⊢ 𝐵 = (Base‘(mulGrp‘𝑅)) |
5 | ringcl.t | . . . 4 ⊢ · = (.r‘𝑅) | |
6 | 1, 5 | mgpplusg 19237 | . . 3 ⊢ · = (+g‘(mulGrp‘𝑅)) |
7 | 4, 6 | mndass 17914 | . 2 ⊢ (((mulGrp‘𝑅) ∈ Mnd ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((𝑋 · 𝑌) · 𝑍) = (𝑋 · (𝑌 · 𝑍))) |
8 | 2, 7 | sylan 582 | 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 398 ∧ w3a 1083 = wceq 1533 ∈ wcel 2110 ‘cfv 6350 (class class class)co 7150 Basecbs 16477 .rcmulr 16560 Mndcmnd 17905 mulGrpcmgp 19233 Ringcrg 19291 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4833 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5455 df-eprel 5460 df-po 5469 df-so 5470 df-fr 5509 df-we 5511 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-pred 6143 df-ord 6189 df-on 6190 df-lim 6191 df-suc 6192 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-nn 11633 df-2 11694 df-ndx 16480 df-slot 16481 df-base 16483 df-sets 16484 df-plusg 16572 df-sgrp 17895 df-mnd 17906 df-mgp 19234 df-ring 19293 |
This theorem is referenced by: ringinvnzdiv 19337 ringmneg1 19340 ringmneg2 19341 imasring 19363 opprring 19375 dvdsrtr 19396 dvdsrmul1 19397 unitgrp 19411 dvrass 19434 dvrcan1 19435 drngmul0or 19517 isdrngd 19521 subrginv 19545 issubrg2 19549 sralmod 19953 unitrrg 20060 sraassa 20093 psrlmod 20175 psrass1 20179 psrass23l 20182 psrass23 20184 frlmphl 20919 mamuass 21005 mamuvs1 21008 mavmulass 21152 mdetrsca 21206 chfacfpmmulgsum2 21467 nrginvrcnlem 23294 ply1divex 24724 rdivmuldivd 30857 dvrcan5 30859 ornglmullt 30875 mxidlprm 30972 fedgmullem1 31020 fedgmullem2 31021 mdetpmtr1 31083 mdetpmtr12 31085 mdetlap 31092 matunitlindflem1 34882 lflvscl 36207 lflvsass 36211 eqlkr3 36231 lkrlsp 36232 lcfl7lem 38629 lclkrlem2m 38649 lcfrlem1 38672 hgmapvvlem1 39053 |
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