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Theorem rngmgp 15363
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 2296 . . 3  |-  ( Base `  R )  =  (
Base `  R )
2 rngmgp.g . . 3  |-  G  =  (mulGrp `  R )
3 eqid 2296 . . 3  |-  ( +g  `  R )  =  ( +g  `  R )
4 eqid 2296 . . 3  |-  ( .r
`  R )  =  ( .r `  R
)
51, 2, 3, 4isrng 15361 . 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 971 1  |-  ( R  e.  Ring  ->  G  e. 
Mnd )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696   A.wral 2556   ` cfv 5271  (class class class)co 5874   Basecbs 13164   +g cplusg 13224   .rcmulr 13225   Mndcmnd 14377   Grpcgrp 14378  mulGrpcmgp 15341   Ringcrg 15353
This theorem is referenced by:  mgpf  15368  rngcl  15370  iscrng2  15372  rngass  15373  rngideu  15374  rngidcl  15377  rngidmlem  15379  unitsubm  15468  dfrhm2  15514  isrhm2d  15522  pwsco1rhm  15526  pwsco2rhm  15527  subrgcrng  15565  subrgsubm  15574  issubrg3  15589  cntzsubr  15593  pwsdiagrhm  15594  psrcrng  16173  mplcoe3  16226  ply1tmcl  16364  coe1pwmul  16371  ply1coe  16384  cnfldexp  16423  expmhm  16465  nrgtrg  18216  evl1expd  19437  deg1pwle  19521  deg1pw  19522  plypf1  19610  amgm  20301  wilthlem2  20323  wilthlem3  20324  dchrelbas3  20493  lgsqrlem2  20597  lgsqrlem3  20598  lgsqrlem4  20599  iistmd  23301  hbtlem4  27433  rngvcl  27556  subrgacs  27611  idomrootle  27614  isdomn3  27626  mon1psubm  27628
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-nul 4165
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-iota 5235  df-fv 5279  df-ov 5877  df-rng 15356
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