MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  rngmgp Unicode version

Theorem rngmgp 15598
Description: A ring is a monoid under multiplication. (Contributed by Mario Carneiro, 6-Jan-2015.)
Hypothesis
Ref Expression
rngmgp.g  |-  G  =  (mulGrp `  R )
Assertion
Ref Expression
rngmgp  |-  ( R  e.  Ring  ->  G  e. 
Mnd )

Proof of Theorem rngmgp
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2388 . . 3  |-  ( Base `  R )  =  (
Base `  R )
2 rngmgp.g . . 3  |-  G  =  (mulGrp `  R )
3 eqid 2388 . . 3  |-  ( +g  `  R )  =  ( +g  `  R )
4 eqid 2388 . . 3  |-  ( .r
`  R )  =  ( .r `  R
)
51, 2, 3, 4isrng 15596 . 2  |-  ( R  e.  Ring  <->  ( R  e. 
Grp  /\  G  e.  Mnd  /\  A. x  e.  ( Base `  R
) A. y  e.  ( Base `  R
) A. z  e.  ( Base `  R
) ( ( x ( .r `  R
) ( y ( +g  `  R ) z ) )  =  ( ( x ( .r `  R ) y ) ( +g  `  R ) ( x ( .r `  R
) z ) )  /\  ( ( x ( +g  `  R
) y ) ( .r `  R ) z )  =  ( ( x ( .r
`  R ) z ) ( +g  `  R
) ( y ( .r `  R ) z ) ) ) ) )
65simp2bi 973 1  |-  ( R  e.  Ring  ->  G  e. 
Mnd )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1717   A.wral 2650   ` cfv 5395  (class class class)co 6021   Basecbs 13397   +g cplusg 13457   .rcmulr 13458   Mndcmnd 14612   Grpcgrp 14613  mulGrpcmgp 15576   Ringcrg 15588
This theorem is referenced by:  mgpf  15603  rngcl  15605  iscrng2  15607  rngass  15608  rngideu  15609  rngidcl  15612  rngidmlem  15614  unitsubm  15703  dfrhm2  15749  isrhm2d  15757  pwsco1rhm  15761  pwsco2rhm  15762  subrgcrng  15800  subrgsubm  15809  issubrg3  15824  cntzsubr  15828  pwsdiagrhm  15829  psrcrng  16404  mplcoe3  16457  ply1tmcl  16592  coe1pwmul  16599  ply1coe  16612  cnfldexp  16658  expmhm  16700  nrgtrg  18597  evl1expd  19826  deg1pwle  19910  deg1pw  19911  plypf1  19999  amgm  20697  wilthlem2  20720  wilthlem3  20721  dchrelbas3  20890  lgsqrlem2  20994  lgsqrlem3  20995  lgsqrlem4  20996  iistmd  24105  hbtlem4  27000  rngvcl  27123  subrgacs  27178  idomrootle  27181  isdomn3  27193  mon1psubm  27195
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-nul 4280
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-ral 2655  df-rex 2656  df-rab 2659  df-v 2902  df-sbc 3106  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-nul 3573  df-if 3684  df-sn 3764  df-pr 3765  df-op 3767  df-uni 3959  df-br 4155  df-iota 5359  df-fv 5403  df-ov 6024  df-rng 15591
  Copyright terms: Public domain W3C validator