HomeHome Metamath Proof Explorer < Previous   Next >
Related theorems
Unicode version

Theorem grpideu 13207
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 13203 . 2  |-  ( G  e.  Grp  ->  G  e.  Mnd )
2 grpcl.b . . 3  |-  B  =  ( Base `  G
)
3 grpcl.p . . 3  |-  .+  =  ( +g  `  G )
42, 3mndideu 12926 . 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 1520    e. wcel 1522   A.wral 2274   E!wreu 2276   ` cfv 4268  (class class class)co 5360   Basecbs 12252   +g cplusg 12306   0gc0g 12484   Mndcmnd 12912   Grpcgrp 12913
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-5 1442  ax-6 1443  ax-7 1444  ax-gen 1445  ax-8 1524  ax-11 1525  ax-13 1526  ax-14 1527  ax-17 1529  ax-12o 1562  ax-10 1576  ax-9 1582  ax-4 1589  ax-16 1775  ax-ext 2046  ax-sep 3701  ax-nul 3709  ax-pr 3769  ax-un 4061
This theorem depends on definitions:  df-bi 175  df-or 357  df-an 358  df-3an 898  df-ex 1447  df-sb 1736  df-eu 1958  df-mo 1959  df-clab 2052  df-cleq 2057  df-clel 2060  df-ne 2184  df-ral 2278  df-rex 2279  df-reu 2280  df-rab 2281  df-v 2477  df-sbc 2651  df-dif 2796  df-un 2798  df-in 2800  df-ss 2804  df-nul 3073  df-if 3182  df-sn 3261  df-pr 3262  df-op 3264  df-uni 3425  df-br 3587  df-opab 3641  df-xp 4270  df-cnv 4272  df-dm 4274  df-rn 4275  df-res 4276  df-ima 4277  df-fv 4284  df-ov 5363  df-mnd 12918  df-grp 13198
Copyright terms: Public domain