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Theorem grpmnd 12774
Description: A group is a monoid. (Contributed by Mario Carneiro, 6-Jan-2015.)
Assertion
Ref Expression
grpmnd  |-  ( G  e.  Grp  ->  G  e.  Mnd )

Proof of Theorem grpmnd
Dummy variables  m  a are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2177 . . 3  |-  ( Base `  G )  =  (
Base `  G )
2 eqid 2177 . . 3  |-  ( +g  `  G )  =  ( +g  `  G )
3 eqid 2177 . . 3  |-  ( 0g
`  G )  =  ( 0g `  G
)
41, 2, 3isgrp 12773 . 2  |-  ( G  e.  Grp  <->  ( G  e.  Mnd  /\  A. a  e.  ( Base `  G
) E. m  e.  ( Base `  G
) ( m ( +g  `  G ) a )  =  ( 0g `  G ) ) )
54simplbi 274 1  |-  ( G  e.  Grp  ->  G  e.  Mnd )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1353    e. wcel 2148   A.wral 2455   E.wrex 2456   ` cfv 5212  (class class class)co 5869   Basecbs 12445   +g cplusg 12518   0gc0g 12653   Mndcmnd 12709   Grpcgrp 12767
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2739  df-un 3133  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-br 4001  df-iota 5174  df-fv 5220  df-ov 5872  df-grp 12770
This theorem is referenced by:  grpcl  12775  grpass  12776  grpideu  12778  grpmndd  12779  grpplusf  12781  grpplusfo  12782  grpsgrp  12791  dfgrp2  12792  grpidcl  12794  grplid  12796  grprid  12797  dfgrp3m  12858  mulgaddcom  12895  mulginvcom  12896  mulgz  12899  mulgneg2  12905  mulgass  12908  isabl2  12924
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