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Theorem grpideu 14179
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 14175 . 2  |-  ( G  e.  Grp  ->  G  e.  Mnd )
2 grpcl.b . . 3  |-  B  =  ( Base `  G
)
3 grpcl.p . . 3  |-  .+  =  ( +g  `  G )
42, 3mndideu 14056 . 2  |-  ( G  e.  Mnd  ->  E! u  e.  B  A. x  e.  B  (
( u  .+  x
)  =  x  /\  ( x  .+  u )  =  x ) )
51, 4syl 16 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 5    /\ wa 357    = wceq 1608    e. wcel 1610   A.wral 2495   E!wreu 2497   ` cfv 4571  (class class class)co 5685   Basecbs 12956   +g cplusg 13016   0gc0g 13208   Mndcmnd 14042   Grpcgrp 14043
This theorem was proved from axioms:  ax-1 6  ax-2 7  ax-3 8  ax-mp 9  ax-5 1522  ax-6 1523  ax-7 1524  ax-gen 1525  ax-8 1612  ax-11 1613  ax-13 1614  ax-14 1615  ax-17 1617  ax-12o 1653  ax-10 1667  ax-9 1673  ax-4 1681  ax-16 1915  ax-ext 2222  ax-sep 4017  ax-nul 4025  ax-pr 4087  ax-un 4382
This theorem depends on definitions:  df-bi 176  df-or 358  df-an 359  df-3an 935  df-tru 1309  df-ex 1527  df-nf 1529  df-sb 1872  df-eu 2106  df-mo 2107  df-clab 2228  df-cleq 2234  df-clel 2237  df-nfc 2362  df-ne 2402  df-ral 2499  df-rex 2500  df-reu 2501  df-rab 2502  df-v 2714  df-sbc 2907  df-dif 3061  df-un 3063  df-in 3065  df-ss 3069  df-nul 3343  df-if 3451  df-sn 3530  df-pr 3531  df-op 3533  df-uni 3708  df-br 3901  df-opab 3955  df-xp 4573  df-cnv 4575  df-dm 4577  df-rn 4578  df-res 4579  df-ima 4580  df-fv 4587  df-ov 5688  df-mnd 14048  df-grp 14170
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