Theorem grpoinveu 25942

Description: The left inverse element of a group is unique. Lemma 2.2.1(b) of [Herstein] p. 55. (Contributed by NM, 27-Oct-2006.) (New usage is discouraged.)

Hypotheses

Ref	Expression
grpinveu.1	|- X = ran G
grpinveu.2	|- U = (GId ` G )

Assertion

Ref	Expression
grpoinveu	|- ( ( G e. GrpOp /\ A e. X ) -> E! y e. X ( y G A ) = U )

Distinct variable groups:   y, A   y, G   y, U   y, X

Proof of Theorem grpoinveu

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		grpinveu.1	. . . . 5 |- X = ran G
2		grpinveu.2	. . . . 5 |- U = (GId ` G )
3	1, 2	grpoidinv2 25938	. . . 4 |- ( ( G e. GrpOp /\ A e. X ) -> ( ( ( U G A ) = A /\ ( A G U ) = A ) /\ E. y e. X ( ( y G A ) = U /\ ( A G y ) = U ) ) )
4		simpl 459	. . . . . 6 |- ( ( ( y G A ) = U /\ ( A G y ) = U ) -> ( y G A ) = U )
5	4	reximi 2894	. . . . 5 |- ( E. y e. X ( ( y G A ) = U /\ ( A G y ) = U ) -> E. y e. X ( y G A ) = U )
6	5	adantl 468	. . . 4 |- ( ( ( ( U G A ) = A /\ ( A G U ) = A ) /\ E. y e. X ( ( y G A ) = U /\ ( A G y ) = U ) ) -> E. y e. X ( y G A ) = U )
7	3, 6	syl 17	. . 3 |- ( ( G e. GrpOp /\ A e. X ) -> E. y e. X ( y G A ) = U )
8		eqtr3 2451	. . . . . . . . . . . 12 |- ( ( ( y G A ) = U /\ ( z G A ) = U ) -> ( y G A ) = ( z G A ) )
9	1	grporcan 25941	. . . . . . . . . . . 12 |- ( ( G e. GrpOp /\ ( y e. X /\ z e. X /\ A e. X ) ) -> ( ( y G A ) = ( z G A ) <-> y = z ) )
10	8, 9	syl5ib 223	. . . . . . . . . . 11 |- ( ( G e. GrpOp /\ ( y e. X /\ z e. X /\ A e. X ) ) -> ( ( ( y G A ) = U /\ ( z G A ) = U ) -> y = z ) )
11	10	3exp2 1224	. . . . . . . . . 10 |- ( G e. GrpOp -> ( y e. X -> ( z e. X -> ( A e. X -> ( ( ( y G A ) = U /\ ( z G A ) = U ) -> y = z ) ) ) ) )
12	11	com24 91	. . . . . . . . 9 |- ( G e. GrpOp -> ( A e. X -> ( z e. X -> ( y e. X -> ( ( ( y G A ) = U /\ ( z G A ) = U ) -> y = z ) ) ) ) )
13	12	imp41 597	. . . . . . . 8 |- ( ( ( ( G e. GrpOp /\ A e. X ) /\ z e. X ) /\ y e. X ) -> ( ( ( y G A ) = U /\ ( z G A ) = U ) -> y = z ) )
14	13	an32s 812	. . . . . . 7 |- ( ( ( ( G e. GrpOp /\ A e. X ) /\ y e. X ) /\ z e. X ) -> ( ( ( y G A ) = U /\ ( z G A ) = U ) -> y = z ) )
15	14	expd 438	. . . . . 6 |- ( ( ( ( G e. GrpOp /\ A e. X ) /\ y e. X ) /\ z e. X ) -> ( ( y G A ) = U -> ( ( z G A ) = U -> y = z ) ) )
16	15	ralrimdva 2844	. . . . 5 |- ( ( ( G e. GrpOp /\ A e. X ) /\ y e. X ) -> ( ( y G A ) = U -> A. z e. X ( ( z G A ) = U -> y = z ) ) )
17	16	ancld 556	. . . 4 |- ( ( ( G e. GrpOp /\ A e. X ) /\ y e. X ) -> ( ( y G A ) = U -> ( ( y G A ) = U /\ A. z e. X ( ( z G A ) = U -> y = z ) ) ) )
18	17	reximdva 2901	. . 3 |- ( ( G e. GrpOp /\ A e. X ) -> ( E. y e. X ( y G A ) = U -> E. y e. X ( ( y G A ) = U /\ A. z e. X ( ( z G A ) = U -> y = z ) ) ) )
19	7, 18	mpd 15	. 2 |- ( ( G e. GrpOp /\ A e. X ) -> E. y e. X ( ( y G A ) = U /\ A. z e. X ( ( z G A ) = U -> y = z ) ) )
20		oveq1 6310	. . . 4 |- ( y = z -> ( y G A ) = ( z G A ) )
21	20	eqeq1d 2425	. . 3 |- ( y = z -> ( ( y G A ) = U <-> ( z G A ) = U ) )
22	21	reu8 3268	. 2 |- ( E! y e. X ( y G A ) = U <-> E. y e. X ( ( y G A ) = U /\ A. z e. X ( ( z G A ) = U -> y = z ) ) )
23	19, 22	sylibr 216	1 |- ( ( G e. GrpOp /\ A e. X ) -> E! y e. X ( y G A ) = U )

Syntax hints:   -> wi 4   /\ wa 371   /\ w3a 983   = wceq 1438   e. wcel 1869   A.wral 2776   E.wrex 2777   E!wreu 2778   ran crn 4852   ` cfv 5599  (class class class)co 6303   GrpOpcgr 25906  GIdcgi 25907

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1666  ax-4 1679  ax-5 1749  ax-6 1795  ax-7 1840  ax-8 1871  ax-9 1873  ax-10 1888  ax-11 1893  ax-12 1906  ax-13 2054  ax-ext 2401  ax-sep 4544  ax-nul 4553  ax-pr 4658  ax-un 6595

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 985  df-tru 1441  df-ex 1661  df-nf 1665  df-sb 1788  df-eu 2270  df-mo 2271  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2573  df-ne 2621  df-ral 2781  df-rex 2782  df-reu 2783  df-rab 2785  df-v 3084  df-sbc 3301  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3763  df-if 3911  df-sn 3998  df-pr 4000  df-op 4004  df-uni 4218  df-iun 4299  df-br 4422  df-opab 4481  df-mpt 4482  df-id 4766  df-xp 4857  df-rel 4858  df-cnv 4859  df-co 4860  df-dm 4861  df-rn 4862  df-iota 5563  df-fun 5601  df-fn 5602  df-f 5603  df-fo 5605  df-fv 5607  df-riota 6265  df-ov 6306  df-grpo 25911  df-gid 25912

This theorem is referenced by:  grpoinvcl  25946  grpoinv  25947