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Theorem grpideu 13262
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 13258 . 2  |-  ( G  e.  Grp  ->  G  e.  Mnd )
2 grpcl.b . . 3  |-  B  =  ( Base `  G
)
3 grpcl.p . . 3  |-  .+  =  ( +g  `  G )
42, 3mndideu 12981 . 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 1531    e. wcel 1533   A.wral 2311   E!wreu 2313   ` cfv 4312  (class class class)co 5404   Basecbs 12306   +g cplusg 12361   0gc0g 12539   Mndcmnd 12967   Grpcgrp 12968
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-5 1452  ax-6 1453  ax-7 1454  ax-gen 1455  ax-8 1535  ax-11 1536  ax-13 1537  ax-14 1538  ax-17 1540  ax-12o 1574  ax-10 1588  ax-9 1594  ax-4 1601  ax-16 1787  ax-ext 2082  ax-sep 3745  ax-nul 3753  ax-pr 3813  ax-un 4105
This theorem depends on definitions:  df-bi 175  df-or 357  df-an 358  df-3an 902  df-ex 1457  df-sb 1748  df-eu 1970  df-mo 1971  df-clab 2088  df-cleq 2093  df-clel 2096  df-ne 2220  df-ral 2315  df-rex 2316  df-reu 2317  df-rab 2318  df-v 2514  df-sbc 2688  df-dif 2833  df-un 2835  df-in 2837  df-ss 2841  df-nul 3111  df-if 3221  df-sn 3300  df-pr 3301  df-op 3303  df-uni 3469  df-br 3631  df-opab 3685  df-xp 4314  df-cnv 4316  df-dm 4318  df-rn 4319  df-res 4320  df-ima 4321  df-fv 4328  df-ov 5407  df-mnd 12973  df-grp 13253
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