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Theorem grpideu 12410
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  |-  P  =  ( +g  `  G )
grpinvex.p  |-  Z  =  ( 0g `  G )
Assertion
Ref Expression
grpideu  |-  ( G  e.  Grp  ->  E! u  e.  B  A. x  e.  B  ( (
u P x )  =  x  /\  (
x P u )  =  x ) )
Distinct variable groups:    x, u, B    u, G, x    u, P, x    x, Z
Allowed substitution hint:    Z( u)

Proof of Theorem grpideu
StepHypRef Expression
1 grpmnd 12406 . 2  |-  ( G  e.  Grp  ->  G  e.  Mnd )
2 grpcl.b . . 3  |-  B  =  ( Base `  G )
3 grpcl.p . . 3  |-  P  =  ( +g  `  G )
42, 3mndideu 12210 . 2  |-  ( G  e.  Mnd  ->  E! u  e.  B  A. x  e.  B  ( (
u P x )  =  x  /\  (
x P u )  =  x ) )
51, 4syl 15 1  |-  ( G  e.  Grp  ->  E! u  e.  B  A. x  e.  B  ( (
u P x )  =  x  /\  (
x P u )  =  x ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 357    = wceq 1536    e. wcel 1538   A.wral 2291   E!wreu 2293   ` cfv 4287  (class class class)co 5354   Basecbs 11602   +g cplusg 11655   0gc0g 11828   Mndcmnd 12196   Grpcgrp 12197
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-5 1451  ax-6 1452  ax-7 1453  ax-gen 1454  ax-8 1540  ax-11 1541  ax-13 1542  ax-14 1543  ax-17 1545  ax-12o 1578  ax-10 1592  ax-9 1598  ax-4 1606  ax-16 1793  ax-ext 2064  ax-sep 3715  ax-nul 3723  ax-pr 3783  ax-un 4075
This theorem depends on definitions:  df-bi 175  df-or 358  df-an 359  df-3an 905  df-ex 1456  df-sb 1754  df-eu 1976  df-mo 1977  df-clab 2070  df-cleq 2075  df-clel 2078  df-ne 2201  df-ral 2295  df-rex 2296  df-reu 2297  df-rab 2298  df-v 2494  df-sbc 2668  df-dif 2813  df-un 2815  df-in 2817  df-ss 2821  df-nul 3089  df-if 3199  df-sn 3278  df-pr 3279  df-op 3281  df-uni 3439  df-br 3601  df-opab 3655  df-xp 4289  df-cnv 4291  df-dm 4293  df-rn 4294  df-res 4295  df-ima 4296  df-fv 4303  df-ov 5357  df-mnd 12202  df-grp 12401
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