Theorem grpoinveu 23854

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 23850	. . . 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 2922	. . . . 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 2479	. . . . . . . . . . . 12 |- ( ( ( y G A ) = U /\ ( z G A ) = U ) -> ( y G A ) = ( z G A ) )
9	1	grporcan 23853	. . . . . . . . . . . 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 1206	. . . . . . . . . 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 2905	. . . . 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 2927	. . 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 6200	. . . 4 |- ( y = z -> ( y G A ) = ( z G A ) )
21	20	eqeq1d 2453	. . 3 |- ( y = z -> ( ( y G A ) = U <-> ( z G A ) = U ) )
22	21	reu8 3255	. 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 965   = wceq 1370   e. wcel 1758   A.wral 2795   E.wrex 2796   E!wreu 2797   ran crn 4942   ` cfv 5519  (class class class)co 6193   GrpOpcgr 23818  GIdcgi 23819

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4514  ax-nul 4522  ax-pr 4632  ax-un 6475

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-reu 2802  df-rab 2804  df-v 3073  df-sbc 3288  df-csb 3390  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-nul 3739  df-if 3893  df-sn 3979  df-pr 3981  df-op 3985  df-uni 4193  df-iun 4274  df-br 4394  df-opab 4452  df-mpt 4453  df-id 4737  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-rn 4952  df-iota 5482  df-fun 5521  df-fn 5522  df-f 5523  df-fo 5525  df-fv 5527  df-riota 6154  df-ov 6196  df-grpo 23823  df-gid 23824

This theorem is referenced by:  grpoinvcl  23858  grpoinv  23859