Theorem grpoinveu 27373

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 27369	. . . 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 473	. . . . . 6 |- ( ( ( y G A ) = U /\ ( A G y ) = U ) -> ( y G A ) = U )
5	4	reximi 3011	. . . . 5 |- ( E. y e. X ( ( y G A ) = U /\ ( A G y ) = U ) -> E. y e. X ( y G A ) = U )
6	5	adantl 482	. . . 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 2643	. . . . . . . . . . . 12 |- ( ( ( y G A ) = U /\ ( z G A ) = U ) -> ( y G A ) = ( z G A ) )
9	1	grporcan 27372	. . . . . . . . . . . 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 234	. . . . . . . . . . 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 1285	. . . . . . . . . 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 95	. . . . . . . . 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 619	. . . . . . . 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 846	. . . . . . 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 452	. . . . . 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 2969	. . . . 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 576	. . . 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 3017	. . 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 6657	. . . 4 |- ( y = z -> ( y G A ) = ( z G A ) )
21	20	eqeq1d 2624	. . 3 |- ( y = z -> ( ( y G A ) = U <-> ( z G A ) = U ) )
22	21	reu8 3402	. 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 224	1 |- ( ( G e. GrpOp /\ A e. X ) -> E! y e. X ( y G A ) = U )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   E.wrex 2913   E!wreu 2914   ran crn 5115   ` cfv 5888  (class class class)co 6650   GrpOpcgr 27343  GIdcgi 27344

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fo 5894  df-fv 5896  df-riota 6611  df-ov 6653  df-grpo 27347  df-gid 27348

This theorem is referenced by:  grpoinvcl  27378  grpoinv  27379