Theorem grpinveu 13485

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 13457	. . 3 |- ( ( G e. Grp /\ X e. B ) -> E. y e. B ( y .+ X ) = .0. )
5		eqtr3 2227	. . . . . . . . . . . 12 |- ( ( ( y .+ X ) = .0. /\ ( z .+ X ) = .0. ) -> ( y .+ X ) = ( z .+ X ) )
6	1, 2	grprcan 13484	. . . . . . . . . . . 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 1228	. . . . . . . . . 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 568	. . . . . . 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 2588	. . . . 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 2610	. . 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 5974	. . . 4 |- ( y = z -> ( y .+ X ) = ( z .+ X ) )
18	17	eqeq1d 2216	. . 3 |- ( y = z -> ( ( y .+ X ) = .0. <-> ( z .+ X ) = .0. ) )
19	18	reu8 2976	. 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 981   = wceq 1373   e. wcel 2178   A.wral 2486   E.wrex 2487   E!wreu 2488   ` cfv 5290  (class class class)co 5967   Basecbs 12947   +g cplusg 13024   0gc0g 13203   Grpcgrp 13447

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-pow 4234  ax-pr 4269  ax-un 4498  ax-cnex 8051  ax-resscn 8052  ax-1re 8054  ax-addrcl 8057

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-rex 2492  df-reu 2493  df-rmo 2494  df-rab 2495  df-v 2778  df-sbc 3006  df-csb 3102  df-un 3178  df-in 3180  df-ss 3187  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-int 3900  df-br 4060  df-opab 4122  df-mpt 4123  df-id 4358  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-res 4705  df-iota 5251  df-fun 5292  df-fn 5293  df-fv 5298  df-riota 5922  df-ov 5970  df-inn 9072  df-2 9130  df-ndx 12950  df-slot 12951  df-base 12953  df-plusg 13037  df-0g 13205  df-mgm 13303  df-sgrp 13349  df-mnd 13364  df-grp 13450

This theorem is referenced by:  grpinvf  13494  grplinv  13497  isgrpinv  13501