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Theorem grpideu 12673
Description: The two-sided identity element of a group is unique. Lemma 2.2.1(a) of [Herstein] p. 55. (Contributed by NM, 16-Aug-2011.) (Revised by Mario Carneiro, 8-Dec-2014.)
Hypotheses
Ref Expression
grpcl.b  |-  B  =  ( Base `  G
)
grpcl.p  |-  .+  =  ( +g  `  G )
grpinvex.p  |-  .0.  =  ( 0g `  G )
Assertion
Ref Expression
grpideu  |-  ( G  e.  Grp  ->  E! u  e.  B  A. x  e.  B  (
( u  .+  x
)  =  x  /\  ( x  .+  u )  =  x ) )
Distinct variable groups:    x, u, B    u, G, x    u,  .+ , x    x,  .0.
Allowed substitution hint:    .0. ( u)

Proof of Theorem grpideu
StepHypRef Expression
1 grpmnd 12669 . 2  |-  ( G  e.  Grp  ->  G  e.  Mnd )
2 grpcl.b . . 3  |-  B  =  ( Base `  G
)
3 grpcl.p . . 3  |-  .+  =  ( +g  `  G )
42, 3mndideu 12392 . 2  |-  ( G  e.  Mnd  ->  E! u  e.  B  A. x  e.  B  (
( u  .+  x
)  =  x  /\  ( x  .+  u )  =  x ) )
51, 4syl 15 1  |-  ( G  e.  Grp  ->  E! u  e.  B  A. x  e.  B  (
( u  .+  x
)  =  x  /\  ( x  .+  u )  =  x ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 356    = wceq 1518    e. wcel 1520   A.wral 2272   E!wreu 2274   ` cfv 4266  (class class class)co 5356   Basecbs 11718   +g cplusg 11772   0gc0g 11950   Mndcmnd 12378   Grpcgrp 12379
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-5 1440  ax-6 1441  ax-7 1442  ax-gen 1443  ax-8 1522  ax-11 1523  ax-13 1524  ax-14 1525  ax-17 1527  ax-12o 1560  ax-10 1574  ax-9 1580  ax-4 1587  ax-16 1773  ax-ext 2044  ax-sep 3699  ax-nul 3707  ax-pr 3767  ax-un 4059
This theorem depends on definitions:  df-bi 175  df-or 357  df-an 358  df-3an 896  df-ex 1445  df-sb 1734  df-eu 1956  df-mo 1957  df-clab 2050  df-cleq 2055  df-clel 2058  df-ne 2182  df-ral 2276  df-rex 2277  df-reu 2278  df-rab 2279  df-v 2475  df-sbc 2649  df-dif 2794  df-un 2796  df-in 2798  df-ss 2802  df-nul 3071  df-if 3180  df-sn 3259  df-pr 3260  df-op 3262  df-uni 3423  df-br 3585  df-opab 3639  df-xp 4268  df-cnv 4270  df-dm 4272  df-rn 4273  df-res 4274  df-ima 4275  df-fv 4282  df-ov 5359  df-mnd 12384  df-grp 12664
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