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Theorem isgrp 16260
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 5848 . . . 4  |-  ( g  =  G  ->  ( Base `  g )  =  ( Base `  G
) )
2 isgrp.b . . . 4  |-  B  =  ( Base `  G
)
31, 2syl6eqr 2513 . . 3  |-  ( g  =  G  ->  ( Base `  g )  =  B )
4 fveq2 5848 . . . . . . 7  |-  ( g  =  G  ->  ( +g  `  g )  =  ( +g  `  G
) )
5 isgrp.p . . . . . . 7  |-  .+  =  ( +g  `  G )
64, 5syl6eqr 2513 . . . . . 6  |-  ( g  =  G  ->  ( +g  `  g )  = 
.+  )
76oveqd 6287 . . . . 5  |-  ( g  =  G  ->  (
m ( +g  `  g
) a )  =  ( m  .+  a
) )
8 fveq2 5848 . . . . . 6  |-  ( g  =  G  ->  ( 0g `  g )  =  ( 0g `  G
) )
9 isgrp.z . . . . . 6  |-  .0.  =  ( 0g `  G )
108, 9syl6eqr 2513 . . . . 5  |-  ( g  =  G  ->  ( 0g `  g )  =  .0.  )
117, 10eqeq12d 2476 . . . 4  |-  ( g  =  G  ->  (
( m ( +g  `  g ) a )  =  ( 0g `  g )  <->  ( m  .+  a )  =  .0.  ) )
123, 11rexeqbidv 3066 . . 3  |-  ( g  =  G  ->  ( E. m  e.  ( Base `  g ) ( m ( +g  `  g
) a )  =  ( 0g `  g
)  <->  E. m  e.  B  ( m  .+  a )  =  .0.  ) )
133, 12raleqbidv 3065 . 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 16256 . 2  |-  Grp  =  { g  e.  Mnd  | 
A. a  e.  (
Base `  g ) E. m  e.  ( Base `  g ) ( m ( +g  `  g
) a )  =  ( 0g `  g
) }
1513, 14elrab2 3256 1  |-  ( G  e.  Grp  <->  ( G  e.  Mnd  /\  A. a  e.  B  E. m  e.  B  ( m  .+  a )  =  .0.  ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 367    = wceq 1398    e. wcel 1823   A.wral 2804   E.wrex 2805   ` cfv 5570  (class class class)co 6270   Basecbs 14716   +g cplusg 14784   0gc0g 14929   Mndcmnd 16118   Grpcgrp 16252
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-iota 5534  df-fv 5578  df-ov 6273  df-grp 16256
This theorem is referenced by:  grpmnd  16261  grpinvex  16264  grppropd  16267  isgrpd2e  16271  grp1  16341  ghmgrp  16393  2zrngagrp  33003
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