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Theorem isgrp 14818
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 5730 . . . 4  |-  ( g  =  G  ->  ( Base `  g )  =  ( Base `  G
) )
2 isgrp.b . . . 4  |-  B  =  ( Base `  G
)
31, 2syl6eqr 2488 . . 3  |-  ( g  =  G  ->  ( Base `  g )  =  B )
4 fveq2 5730 . . . . . . 7  |-  ( g  =  G  ->  ( +g  `  g )  =  ( +g  `  G
) )
5 isgrp.p . . . . . . 7  |-  .+  =  ( +g  `  G )
64, 5syl6eqr 2488 . . . . . 6  |-  ( g  =  G  ->  ( +g  `  g )  = 
.+  )
76oveqd 6100 . . . . 5  |-  ( g  =  G  ->  (
m ( +g  `  g
) a )  =  ( m  .+  a
) )
8 fveq2 5730 . . . . . 6  |-  ( g  =  G  ->  ( 0g `  g )  =  ( 0g `  G
) )
9 isgrp.z . . . . . 6  |-  .0.  =  ( 0g `  G )
108, 9syl6eqr 2488 . . . . 5  |-  ( g  =  G  ->  ( 0g `  g )  =  .0.  )
117, 10eqeq12d 2452 . . . 4  |-  ( g  =  G  ->  (
( m ( +g  `  g ) a )  =  ( 0g `  g )  <->  ( m  .+  a )  =  .0.  ) )
123, 11rexeqbidv 2919 . . 3  |-  ( g  =  G  ->  ( E. m  e.  ( Base `  g ) ( m ( +g  `  g
) a )  =  ( 0g `  g
)  <->  E. m  e.  B  ( m  .+  a )  =  .0.  ) )
133, 12raleqbidv 2918 . 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 14814 . 2  |-  Grp  =  { g  e.  Mnd  | 
A. a  e.  (
Base `  g ) E. m  e.  ( Base `  g ) ( m ( +g  `  g
) a )  =  ( 0g `  g
) }
1513, 14elrab2 3096 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 178    /\ wa 360    = wceq 1653    e. wcel 1726   A.wral 2707   E.wrex 2708   ` cfv 5456  (class class class)co 6083   Basecbs 13471   +g cplusg 13531   0gc0g 13725   Mndcmnd 14686   Grpcgrp 14687
This theorem is referenced by:  grpmnd  14819  grpinvex  14822  grppropd  14825  isgrpd2e  14829
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4215  df-iota 5420  df-fv 5464  df-ov 6086  df-grp 14814
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