Theorem grpideu 17433

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

Step	Hyp	Ref	Expression
1		grpmnd 17429	. 2 |- ( G e. Grp -> G e. Mnd )
2		grpcl.b	. . 3 |- B = ( Base ` G )
3		grpcl.p	. . 3 |- .+ = ( +g ` G )
4	2, 3	mndideu 17304	. 2 |- ( G e. Mnd -> E! u e. B A. x e. B ( ( u .+ x ) = x /\ ( x .+ u ) = x ) )
5	1, 4	syl 17	1 |- ( G e. Grp -> E! u e. B A. x e. B ( ( u .+ x ) = x /\ ( x .+ u ) = x ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   E!wreu 2914   ` cfv 5888  (class class class)co 6650   Basecbs 15857   +g cplusg 15941   0gc0g 16100   Mndcmnd 17294   Grpcgrp 17422

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-nul 4789  ax-pow 4843

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-iota 5851  df-fv 5896  df-ov 6653  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-grp 17425

This theorem is referenced by: (None)