Theorem mndideu 17304

Description: The two-sided identity element of a monoid is unique. Lemma 2.2.1(a) of [Herstein] p. 55. (Contributed by Mario Carneiro, 8-Dec-2014.)

Hypotheses

Ref	Expression
mndcl.b	|- B = ( Base ` G )
mndcl.p	|- .+ = ( +g ` G )

Assertion

Ref	Expression
mndideu	|- ( G e. Mnd -> E! u e. B A. x e. B ( ( u .+ x ) = x /\ ( x .+ u ) = x ) )

Distinct variable groups:   x, B   x, G   x, .+ , u   u, B   u, G   u, .+

Proof of Theorem mndideu

Step	Hyp	Ref	Expression
1		mndcl.b	. . 3 |- B = ( Base ` G )
2		mndcl.p	. . 3 |- .+ = ( +g ` G )
3	1, 2	mndid 17303	. 2 |- ( G e. Mnd -> E. u e. B A. x e. B ( ( u .+ x ) = x /\ ( x .+ u ) = x ) )
4		mgmidmo 17259	. . 3 |- E* u e. B A. x e. B ( ( u .+ x ) = x /\ ( x .+ u ) = x )
5	4	a1i 11	. 2 |- ( G e. Mnd -> E* u e. B A. x e. B ( ( u .+ x ) = x /\ ( x .+ u ) = x ) )
6		reu5 3159	. 2 |- ( E! u e. B A. x e. B ( ( u .+ x ) = x /\ ( x .+ u ) = x ) <-> ( E. u e. B A. x e. B ( ( u .+ x ) = x /\ ( x .+ u ) = x ) /\ E* u e. B A. x e. B ( ( u .+ x ) = x /\ ( x .+ u ) = x ) ) )
7	3, 5, 6	sylanbrc 698	1 |- ( G e. Mnd -> 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.wrex 2913   E!wreu 2914   E*wrmo 2915   ` cfv 5888  (class class class)co 6650   Basecbs 15857   +g cplusg 15941   Mndcmnd 17294

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

This theorem is referenced by:  grpideu  17433  srgideu  18514  ringideu  18565