MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  grpideu Unicode version

Theorem grpideu 13228
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 13224 . 2  |-  ( G  e.  Grp  ->  G  e.  Mnd )
2 grpcl.b . . 3  |-  B  =  ( Base `  G
)
3 grpcl.p . . 3  |-  .+  =  ( +g  `  G )
42, 3mndideu 12947 . 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 1524    e. wcel 1526   A.wral 2279   E!wreu 2281   ` cfv 4278  (class class class)co 5370   Basecbs 12272   +g cplusg 12327   0gc0g 12505   Mndcmnd 12933   Grpcgrp 12934
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-5 1446  ax-6 1447  ax-7 1448  ax-gen 1449  ax-8 1528  ax-11 1529  ax-13 1530  ax-14 1531  ax-17 1533  ax-12o 1567  ax-10 1581  ax-9 1587  ax-4 1594  ax-16 1780  ax-ext 2051  ax-sep 3711  ax-nul 3719  ax-pr 3779  ax-un 4071
This theorem depends on definitions:  df-bi 175  df-or 357  df-an 358  df-3an 898  df-ex 1451  df-sb 1741  df-eu 1963  df-mo 1964  df-clab 2057  df-cleq 2062  df-clel 2065  df-ne 2189  df-ral 2283  df-rex 2284  df-reu 2285  df-rab 2286  df-v 2482  df-sbc 2656  df-dif 2801  df-un 2803  df-in 2805  df-ss 2809  df-nul 3078  df-if 3187  df-sn 3266  df-pr 3267  df-op 3269  df-uni 3435  df-br 3597  df-opab 3651  df-xp 4280  df-cnv 4282  df-dm 4284  df-rn 4285  df-res 4286  df-ima 4287  df-fv 4294  df-ov 5373  df-mnd 12939  df-grp 13219
  Copyright terms: Public domain W3C validator