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Theorem grpideu 14295
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 14291 . 2  |-  ( G  e.  Grp  ->  G  e.  Mnd )
2 grpcl.b . . 3  |-  B  =  ( Base `  G
)
3 grpcl.p . . 3  |-  .+  =  ( +g  `  G )
42, 3mndideu 14172 . 2  |-  ( G  e.  Mnd  ->  E! u  e.  B  A. x  e.  B  (
( u  .+  x
)  =  x  /\  ( x  .+  u )  =  x ) )
51, 4syl 17 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 6    /\ wa 360    = wceq 1619    e. wcel 1621   A.wral 2507   E!wreu 2509   ` cfv 4589  (class class class)co 5707   Basecbs 12984   +g cplusg 13044   0gc0g 13236   Mndcmnd 14158   Grpcgrp 14159
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234  ax-sep 4035  ax-nul 4043  ax-pr 4105  ax-un 4400
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-ral 2511  df-rex 2512  df-reu 2513  df-rab 2514  df-v 2727  df-sbc 2920  df-dif 3078  df-un 3080  df-in 3082  df-ss 3086  df-nul 3360  df-if 3468  df-sn 3547  df-pr 3548  df-op 3550  df-uni 3725  df-br 3918  df-opab 3972  df-xp 4591  df-cnv 4593  df-dm 4595  df-rn 4596  df-res 4597  df-ima 4598  df-fv 4605  df-ov 5710  df-mnd 14164  df-grp 14286
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