Theorem grpinveu 14794

Description: The left inverse element of a group is unique. Lemma 2.2.1(b) of [Herstein] p. 55. (Contributed by NM, 24-Aug-2011.)

Hypotheses

Ref	Expression
grpinveu.b	|- B = ( Base ` G )
grpinveu.p	|- .+ = ( +g ` G )
grpinveu.o	|- .0. = ( 0g ` G )

Assertion

Ref	Expression
grpinveu	|- ( ( G e. Grp /\ X e. B ) -> E! y e. B ( y .+ X ) = .0. )

Distinct variable groups:   y, B   y, G   y, .+   y, .0.   y, X

Proof of Theorem grpinveu

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		grpinveu.b	. . . 4 |- B = ( Base ` G )
2		grpinveu.p	. . . 4 |- .+ = ( +g ` G )
3		grpinveu.o	. . . 4 |- .0. = ( 0g ` G )
4	1, 2, 3	grpinvex 14775	. . 3 |- ( ( G e. Grp /\ X e. B ) -> E. y e. B ( y .+ X ) = .0. )
5		eqtr3 2423	. . . . . . . . . . . 12 |- ( ( ( y .+ X ) = .0. /\ ( z .+ X ) = .0. ) -> ( y .+ X ) = ( z .+ X ) )
6	1, 2	grprcan 14793	. . . . . . . . . . . 12 |- ( ( G e. Grp /\ ( y e. B /\ z e. B /\ X e. B ) ) -> ( ( y .+ X ) = ( z .+ X ) <-> y = z ) )
7	5, 6	syl5ib 211	. . . . . . . . . . 11 |- ( ( G e. Grp /\ ( y e. B /\ z e. B /\ X e. B ) ) -> ( ( ( y .+ X ) = .0. /\ ( z .+ X ) = .0. ) -> y = z ) )
8	7	3exp2 1171	. . . . . . . . . 10 |- ( G e. Grp -> ( y e. B -> ( z e. B -> ( X e. B -> ( ( ( y .+ X ) = .0. /\ ( z .+ X ) = .0. ) -> y = z ) ) ) ) )
9	8	com24 83	. . . . . . . . 9 |- ( G e. Grp -> ( X e. B -> ( z e. B -> ( y e. B -> ( ( ( y .+ X ) = .0. /\ ( z .+ X ) = .0. ) -> y = z ) ) ) ) )
10	9	imp41 577	. . . . . . . 8 |- ( ( ( ( G e. Grp /\ X e. B ) /\ z e. B ) /\ y e. B ) -> ( ( ( y .+ X ) = .0. /\ ( z .+ X ) = .0. ) -> y = z ) )
11	10	an32s 780	. . . . . . 7 |- ( ( ( ( G e. Grp /\ X e. B ) /\ y e. B ) /\ z e. B ) -> ( ( ( y .+ X ) = .0. /\ ( z .+ X ) = .0. ) -> y = z ) )
12	11	exp3a 426	. . . . . 6 |- ( ( ( ( G e. Grp /\ X e. B ) /\ y e. B ) /\ z e. B ) -> ( ( y .+ X ) = .0. -> ( ( z .+ X ) = .0. -> y = z ) ) )
13	12	ralrimdva 2756	. . . . 5 |- ( ( ( G e. Grp /\ X e. B ) /\ y e. B ) -> ( ( y .+ X ) = .0. -> A. z e. B ( ( z .+ X ) = .0. -> y = z ) ) )
14	13	ancld 537	. . . 4 |- ( ( ( G e. Grp /\ X e. B ) /\ y e. B ) -> ( ( y .+ X ) = .0. -> ( ( y .+ X ) = .0. /\ A. z e. B ( ( z .+ X ) = .0. -> y = z ) ) ) )
15	14	reximdva 2778	. . 3 |- ( ( G e. Grp /\ X e. B ) -> ( E. y e. B ( y .+ X ) = .0. -> E. y e. B ( ( y .+ X ) = .0. /\ A. z e. B ( ( z .+ X ) = .0. -> y = z ) ) ) )
16	4, 15	mpd 15	. 2 |- ( ( G e. Grp /\ X e. B ) -> E. y e. B ( ( y .+ X ) = .0. /\ A. z e. B ( ( z .+ X ) = .0. -> y = z ) ) )
17		oveq1 6047	. . . 4 |- ( y = z -> ( y .+ X ) = ( z .+ X ) )
18	17	eqeq1d 2412	. . 3 |- ( y = z -> ( ( y .+ X ) = .0. <-> ( z .+ X ) = .0. ) )
19	18	reu8 3090	. 2 |- ( E! y e. B ( y .+ X ) = .0. <-> E. y e. B ( ( y .+ X ) = .0. /\ A. z e. B ( ( z .+ X ) = .0. -> y = z ) ) )
20	16, 19	sylibr 204	1 |- ( ( G e. Grp /\ X e. B ) -> E! y e. B ( y .+ X ) = .0. )

Syntax hints:   -> wi 4   /\ wa 359   /\ w3a 936   = wceq 1649   e. wcel 1721   A.wral 2666   E.wrex 2667   E!wreu 2668   ` cfv 5413  (class class class)co 6040   Basecbs 13424   +g cplusg 13484   0gc0g 13678   Grpcgrp 14640

This theorem is referenced by:  grpinvf  14804  grplinv  14806  isgrpinv  14810

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363

This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-iota 5377  df-fun 5415  df-fv 5421  df-ov 6043  df-riota 6508  df-0g 13682  df-mnd 14645  df-grp 14767