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Theorem ringmgp 20062
Description: A ring is a monoid under multiplication. (Contributed by Mario Carneiro, 6-Jan-2015.)
Hypothesis
Ref Expression
ringmgp.g 𝐺 = (mulGrp‘𝑅)
Assertion
Ref Expression
ringmgp (𝑅 ∈ Ring → 𝐺 ∈ Mnd)

Proof of Theorem ringmgp
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2733 . . 3 (Base‘𝑅) = (Base‘𝑅)
2 ringmgp.g . . 3 𝐺 = (mulGrp‘𝑅)
3 eqid 2733 . . 3 (+g𝑅) = (+g𝑅)
4 eqid 2733 . . 3 (.r𝑅) = (.r𝑅)
51, 2, 3, 4isring 20060 . 2 (𝑅 ∈ Ring ↔ (𝑅 ∈ Grp ∧ 𝐺 ∈ Mnd ∧ ∀𝑥 ∈ (Base‘𝑅)∀𝑦 ∈ (Base‘𝑅)∀𝑧 ∈ (Base‘𝑅)((𝑥(.r𝑅)(𝑦(+g𝑅)𝑧)) = ((𝑥(.r𝑅)𝑦)(+g𝑅)(𝑥(.r𝑅)𝑧)) ∧ ((𝑥(+g𝑅)𝑦)(.r𝑅)𝑧) = ((𝑥(.r𝑅)𝑧)(+g𝑅)(𝑦(.r𝑅)𝑧)))))
65simp2bi 1147 1 (𝑅 ∈ Ring → 𝐺 ∈ Mnd)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  wcel 2107  wral 3062  cfv 6544  (class class class)co 7409  Basecbs 17144  +gcplusg 17197  .rcmulr 17198  Mndcmnd 18625  Grpcgrp 18819  mulGrpcmgp 19987  Ringcrg 20056
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-nul 5307
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2942  df-ral 3063  df-rab 3434  df-v 3477  df-sbc 3779  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-iota 6496  df-fv 6552  df-ov 7412  df-ring 20058
This theorem is referenced by:  mgpf  20071  ringcl  20073  iscrng2  20075  ringass  20076  ringideu  20077  ringidcl  20083  ringidmlem  20085  ringsrg  20109  pwspjmhmmgpd  20141  pwsexpg  20142  unitsubm  20200  dfrhm2  20253  isrhm2d  20265  idrhm  20268  pwsco1rhm  20277  pwsco2rhm  20278  subrgcrng  20323  subrgsubm  20332  issubrg3  20347  cntzsubr  20353  pwsdiagrhm  20354  subrgacs  20416  cnfldexp  20978  expmhm  21014  nn0srg  21015  rge0srg  21016  assamulgscmlem2  21454  psrcrng  21533  mplcoe3  21593  mplcoe5lem  21594  mplcoe5  21595  evlsgsummul  21655  mhppwdeg  21693  ply1moncl  21793  coe1pwmul  21801  ply1coefsupp  21819  ply1coe  21820  gsummoncoe1  21828  lply1binomsc  21831  evls1gsummul  21844  evl1expd  21864  evl1gsummul  21879  evl1scvarpw  21882  evl1scvarpwval  21883  evl1gsummon  21884  ringvcl  21900  mat1mhm  21986  scmatmhm  22036  m1detdiag  22099  mdetdiaglem  22100  m2detleiblem2  22130  mat2pmatmhm  22235  pmatcollpwscmatlem1  22291  mply1topmatcllem  22305  mply1topmatcl  22307  pm2mpghm  22318  pm2mpmhm  22322  monmat2matmon  22326  pm2mp  22327  chpscmatgsumbin  22346  chpscmatgsummon  22347  chfacfscmulcl  22359  chfacfscmul0  22360  chfacfpmmulcl  22363  chfacfpmmul0  22364  chfacfpmmulgsum2  22367  cayhamlem1  22368  cpmadugsumlemB  22376  cpmadugsumlemC  22377  cpmadugsumlemF  22378  cayhamlem2  22386  cayhamlem4  22390  nrgtrg  24207  deg1pw  25638  plypf1  25726  efsubm  26060  amgm  26495  wilthlem2  26573  wilthlem3  26574  dchrelbas3  26741  lgsqrlem2  26850  lgsqrlem3  26851  lgsqrlem4  26852  cntrcrng  32214  psgnid  32256  cnmsgn0g  32305  altgnsg  32308  freshmansdream  32381  frobrhm  32382  fermltlchr  32478  znfermltl  32479  ringlsmss  32505  evls1fpws  32646  gsummoncoe1fzo  32668  ply1degltdimlem  32707  ply1degltdim  32708  iistmd  32882  evlsvvvallem  41133  evlsvvval  41135  evlsexpval  41139  selvvvval  41157  evlselv  41159  mhphf  41169  hbtlem4  41868  idomrootle  41937  isdomn3  41946  mon1psubm  41948  amgm2d  42950  amgm3d  42951  amgm4d  42952  c0rhm  46711  c0rnghm  46712  invginvrid  47043  ply1mulgsumlem4  47070  ply1mulgsum  47071  amgmw2d  47851
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