Theorem grpinveu 13701

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 13673	. . 3 |- ( ( G e. Grp /\ X e. B ) -> E. y e. B ( y .+ X ) = .0. )
5		eqtr3 2251	. . . . . . . . . . . 12 |- ( ( ( y .+ X ) = .0. /\ ( z .+ X ) = .0. ) -> ( y .+ X ) = ( z .+ X ) )
6	1, 2	grprcan 13700	. . . . . . . . . . . 12 |- ( ( G e. Grp /\ ( y e. B /\ z e. B /\ X e. B ) ) -> ( ( y .+ X ) = ( z .+ X ) <-> y = z ) )
7	5, 6	imbitrid 154	. . . . . . . . . . 11 |- ( ( G e. Grp /\ ( y e. B /\ z e. B /\ X e. B ) ) -> ( ( ( y .+ X ) = .0. /\ ( z .+ X ) = .0. ) -> y = z ) )
8	7	3exp2 1252	. . . . . . . . . 10 |- ( G e. Grp -> ( y e. B -> ( z e. B -> ( X e. B -> ( ( ( y .+ X ) = .0. /\ ( z .+ X ) = .0. ) -> y = z ) ) ) ) )
9	8	com24 87	. . . . . . . . 9 |- ( G e. Grp -> ( X e. B -> ( z e. B -> ( y e. B -> ( ( ( y .+ X ) = .0. /\ ( z .+ X ) = .0. ) -> y = z ) ) ) ) )
10	9	imp41 353	. . . . . . . 8 |- ( ( ( ( G e. Grp /\ X e. B ) /\ z e. B ) /\ y e. B ) -> ( ( ( y .+ X ) = .0. /\ ( z .+ X ) = .0. ) -> y = z ) )
11	10	an32s 570	. . . . . . 7 |- ( ( ( ( G e. Grp /\ X e. B ) /\ y e. B ) /\ z e. B ) -> ( ( ( y .+ X ) = .0. /\ ( z .+ X ) = .0. ) -> y = z ) )
12	11	expd 258	. . . . . 6 |- ( ( ( ( G e. Grp /\ X e. B ) /\ y e. B ) /\ z e. B ) -> ( ( y .+ X ) = .0. -> ( ( z .+ X ) = .0. -> y = z ) ) )
13	12	ralrimdva 2613	. . . . 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 325	. . . 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 2635	. . 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 13	. 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 6035	. . . 4 |- ( y = z -> ( y .+ X ) = ( z .+ X ) )
18	17	eqeq1d 2240	. . 3 |- ( y = z -> ( ( y .+ X ) = .0. <-> ( z .+ X ) = .0. ) )
19	18	reu8 3003	. 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 134	1 |- ( ( G e. Grp /\ X e. B ) -> E! y e. B ( y .+ X ) = .0. )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 1005   = wceq 1398   e. wcel 2202   A.wral 2511   E.wrex 2512   E!wreu 2513   ` cfv 5333  (class class class)co 6028   Basecbs 13162   +g cplusg 13240   0gc0g 13419   Grpcgrp 13663

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-cnex 8183  ax-resscn 8184  ax-1re 8186  ax-addrcl 8189

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-reu 2518  df-rmo 2519  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-br 4094  df-opab 4156  df-mpt 4157  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-iota 5293  df-fun 5335  df-fn 5336  df-fv 5341  df-riota 5981  df-ov 6031  df-inn 9203  df-2 9261  df-ndx 13165  df-slot 13166  df-base 13168  df-plusg 13253  df-0g 13421  df-mgm 13519  df-sgrp 13565  df-mnd 13580  df-grp 13666

This theorem is referenced by:  grpinvf  13710  grplinv  13713  isgrpinv  13717