Theorem grpoinveu 25638

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 25634	. . . 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 455	. . . . . 6 |- ( ( ( y G A ) = U /\ ( A G y ) = U ) -> ( y G A ) = U )
5	4	reximi 2872	. . . . 5 |- ( E. y e. X ( ( y G A ) = U /\ ( A G y ) = U ) -> E. y e. X ( y G A ) = U )
6	5	adantl 464	. . . 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 2430	. . . . . . . . . . . 12 |- ( ( ( y G A ) = U /\ ( z G A ) = U ) -> ( y G A ) = ( z G A ) )
9	1	grporcan 25637	. . . . . . . . . . . 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 1215	. . . . . . . . . 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 591	. . . . . . . 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 805	. . . . . . 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 434	. . . . . 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 2822	. . . . 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 551	. . . 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 2879	. . 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 6285	. . . 4 |- ( y = z -> ( y G A ) = ( z G A ) )
21	20	eqeq1d 2404	. . 3 |- ( y = z -> ( ( y G A ) = U <-> ( z G A ) = U ) )
22	21	reu8 3245	. 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 367   /\ w3a 974   = wceq 1405   e. wcel 1842   A.wral 2754   E.wrex 2755   E!wreu 2756   ran crn 4824   ` cfv 5569  (class class class)co 6278   GrpOpcgr 25602  GIdcgi 25603

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4517  ax-nul 4525  ax-pr 4630  ax-un 6574

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2759  df-rex 2760  df-reu 2761  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4192  df-iun 4273  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4738  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-fo 5575  df-fv 5577  df-riota 6240  df-ov 6281  df-grpo 25607  df-gid 25608

This theorem is referenced by:  grpoinvcl  25642  grpoinv  25643