Theorem grpinveu 13370

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 13342	. . 3 |- ( ( G e. Grp /\ X e. B ) -> E. y e. B ( y .+ X ) = .0. )
5		eqtr3 2225	. . . . . . . . . . . 12 |- ( ( ( y .+ X ) = .0. /\ ( z .+ X ) = .0. ) -> ( y .+ X ) = ( z .+ X ) )
6	1, 2	grprcan 13369	. . . . . . . . . . . 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 2586	. . . . 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 2608	. . 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 5951	. . . 4 |- ( y = z -> ( y .+ X ) = ( z .+ X ) )
18	17	eqeq1d 2214	. . 3 |- ( y = z -> ( ( y .+ X ) = .0. <-> ( z .+ X ) = .0. ) )
19	18	reu8 2969	. 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 2176   A.wral 2484   E.wrex 2485   E!wreu 2486   ` cfv 5271  (class class class)co 5944   Basecbs 12832   +g cplusg 12909   0gc0g 13088   Grpcgrp 13332

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 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-sep 4162  ax-pow 4218  ax-pr 4253  ax-un 4480  ax-cnex 8016  ax-resscn 8017  ax-1re 8019  ax-addrcl 8022

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-reu 2491  df-rmo 2492  df-rab 2493  df-v 2774  df-sbc 2999  df-csb 3094  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-int 3886  df-br 4045  df-opab 4106  df-mpt 4107  df-id 4340  df-xp 4681  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-rn 4686  df-res 4687  df-iota 5232  df-fun 5273  df-fn 5274  df-fv 5279  df-riota 5899  df-ov 5947  df-inn 9037  df-2 9095  df-ndx 12835  df-slot 12836  df-base 12838  df-plusg 12922  df-0g 13090  df-mgm 13188  df-sgrp 13234  df-mnd 13249  df-grp 13335

This theorem is referenced by:  grpinvf  13379  grplinv  13382  isgrpinv  13386