Theorem grpoinveu 24886

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 24882	. . . 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 457	. . . . . 6 |- ( ( ( y G A ) = U /\ ( A G y ) = U ) -> ( y G A ) = U )
5	4	reximi 2925	. . . . 5 |- ( E. y e. X ( ( y G A ) = U /\ ( A G y ) = U ) -> E. y e. X ( y G A ) = U )
6	5	adantl 466	. . . 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 16	. . 3 |- ( ( G e. GrpOp /\ A e. X ) -> E. y e. X ( y G A ) = U )
8		eqtr3 2488	. . . . . . . . . . . 12 |- ( ( ( y G A ) = U /\ ( z G A ) = U ) -> ( y G A ) = ( z G A ) )
9	1	grporcan 24885	. . . . . . . . . . . 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 219	. . . . . . . . . . 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 1209	. . . . . . . . . 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 87	. . . . . . . . 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 593	. . . . . . . 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 802	. . . . . . 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 436	. . . . . 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 2875	. . . . 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 553	. . . 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 2931	. . 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 6282	. . . 4 |- ( y = z -> ( y G A ) = ( z G A ) )
21	20	eqeq1d 2462	. . 3 |- ( y = z -> ( ( y G A ) = U <-> ( z G A ) = U ) )
22	21	reu8 3292	. 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 212	1 |- ( ( G e. GrpOp /\ A e. X ) -> E! y e. X ( y G A ) = U )

Syntax hints:   -> wi 4   /\ wa 369   /\ w3a 968   = wceq 1374   e. wcel 1762   A.wral 2807   E.wrex 2808   E!wreu 2809   ran crn 4993   ` cfv 5579  (class class class)co 6275   GrpOpcgr 24850  GIdcgi 24851

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pr 4679  ax-un 6567

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-id 4788  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-fo 5585  df-fv 5587  df-riota 6236  df-ov 6278  df-grpo 24855  df-gid 24856

This theorem is referenced by:  grpoinvcl  24890  grpoinv  24891