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

Theorem isgrp 14493
Description: The predicate "is a group." (This theorem demonstrates the use of symbols as variable names, first proposed by FL in 2010.) (Contributed by NM, 17-Oct-2012.) (Revised by Mario Carneiro, 6-Jan-2015.)
Hypotheses
Ref Expression
isgrp.b  |-  B  =  ( Base `  G
)
isgrp.p  |-  .+  =  ( +g  `  G )
isgrp.z  |-  .0.  =  ( 0g `  G )
Assertion
Ref Expression
isgrp  |-  ( G  e.  Grp  <->  ( G  e.  Mnd  /\  A. a  e.  B  E. m  e.  B  ( m  .+  a )  =  .0.  ) )
Distinct variable groups:    m, a, B    G, a, m
Allowed substitution hints:    .+ ( m, a)    .0. ( m, a)

Proof of Theorem isgrp
Dummy variable  g is distinct from all other variables.
StepHypRef Expression
1 fveq2 5525 . . . 4  |-  ( g  =  G  ->  ( Base `  g )  =  ( Base `  G
) )
2 isgrp.b . . . 4  |-  B  =  ( Base `  G
)
31, 2syl6eqr 2333 . . 3  |-  ( g  =  G  ->  ( Base `  g )  =  B )
4 fveq2 5525 . . . . . . 7  |-  ( g  =  G  ->  ( +g  `  g )  =  ( +g  `  G
) )
5 isgrp.p . . . . . . 7  |-  .+  =  ( +g  `  G )
64, 5syl6eqr 2333 . . . . . 6  |-  ( g  =  G  ->  ( +g  `  g )  = 
.+  )
76oveqd 5875 . . . . 5  |-  ( g  =  G  ->  (
m ( +g  `  g
) a )  =  ( m  .+  a
) )
8 fveq2 5525 . . . . . 6  |-  ( g  =  G  ->  ( 0g `  g )  =  ( 0g `  G
) )
9 isgrp.z . . . . . 6  |-  .0.  =  ( 0g `  G )
108, 9syl6eqr 2333 . . . . 5  |-  ( g  =  G  ->  ( 0g `  g )  =  .0.  )
117, 10eqeq12d 2297 . . . 4  |-  ( g  =  G  ->  (
( m ( +g  `  g ) a )  =  ( 0g `  g )  <->  ( m  .+  a )  =  .0.  ) )
123, 11rexeqbidv 2749 . . 3  |-  ( g  =  G  ->  ( E. m  e.  ( Base `  g ) ( m ( +g  `  g
) a )  =  ( 0g `  g
)  <->  E. m  e.  B  ( m  .+  a )  =  .0.  ) )
133, 12raleqbidv 2748 . 2  |-  ( g  =  G  ->  ( A. a  e.  ( Base `  g ) E. m  e.  ( Base `  g ) ( m ( +g  `  g
) a )  =  ( 0g `  g
)  <->  A. a  e.  B  E. m  e.  B  ( m  .+  a )  =  .0.  ) )
14 df-grp 14489 . 2  |-  Grp  =  { g  e.  Mnd  | 
A. a  e.  (
Base `  g ) E. m  e.  ( Base `  g ) ( m ( +g  `  g
) a )  =  ( 0g `  g
) }
1513, 14elrab2 2925 1  |-  ( G  e.  Grp  <->  ( G  e.  Mnd  /\  A. a  e.  B  E. m  e.  B  ( m  .+  a )  =  .0.  ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543   E.wrex 2544   ` cfv 5255  (class class class)co 5858   Basecbs 13148   +g cplusg 13208   0gc0g 13400   Mndcmnd 14361   Grpcgrp 14362
This theorem is referenced by:  grpmnd  14494  grpinvex  14497  grppropd  14500  isgrpd2e  14504
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-iota 5219  df-fv 5263  df-ov 5861  df-grp 14489
  Copyright terms: Public domain W3C validator