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Theorem rngmgp 15662
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 2435 . . 3  |-  ( Base `  R )  =  (
Base `  R )
2 rngmgp.g . . 3  |-  G  =  (mulGrp `  R )
3 eqid 2435 . . 3  |-  ( +g  `  R )  =  ( +g  `  R )
4 eqid 2435 . . 3  |-  ( .r
`  R )  =  ( .r `  R
)
51, 2, 3, 4isrng 15660 . 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 1652    e. wcel 1725   A.wral 2697   ` cfv 5446  (class class class)co 6073   Basecbs 13461   +g cplusg 13521   .rcmulr 13522   Mndcmnd 14676   Grpcgrp 14677  mulGrpcmgp 15640   Ringcrg 15652
This theorem is referenced by:  mgpf  15667  rngcl  15669  iscrng2  15671  rngass  15672  rngideu  15673  rngidcl  15676  rngidmlem  15678  unitsubm  15767  dfrhm2  15813  isrhm2d  15821  pwsco1rhm  15825  pwsco2rhm  15826  subrgcrng  15864  subrgsubm  15873  issubrg3  15888  cntzsubr  15892  pwsdiagrhm  15893  psrcrng  16468  mplcoe3  16521  ply1tmcl  16656  coe1pwmul  16663  ply1coe  16676  cnfldexp  16726  expmhm  16768  nrgtrg  18717  evl1expd  19950  deg1pwle  20034  deg1pw  20035  plypf1  20123  amgm  20821  wilthlem2  20844  wilthlem3  20845  dchrelbas3  21014  lgsqrlem2  21118  lgsqrlem3  21119  lgsqrlem4  21120  iistmd  24292  hbtlem4  27298  rngvcl  27421  subrgacs  27476  idomrootle  27479  isdomn3  27491  mon1psubm  27493
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-nul 4330
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-iota 5410  df-fv 5454  df-ov 6076  df-rng 15655
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