Theorem grpinveu 9348

Description: The left inverse element of a group is unique. Lemma 2.2.1(b) of [Herstein] p. 55.

Hypotheses

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

Assertion

Ref	Expression
grpinveu	|- ((G e. Grp /\ A e. X) -> E!y e. X (yGA) = U)

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

Proof of Theorem grpinveu

Step	Hyp	Ref	Expression
1		grpinveu.1	. . . . 5 |- X = ran G
2		grpinveu.2	. . . . 5 |- U = (Id` G)
3	1, 2	grpidinv2 9344	. . . 4 |- ((G e. Grp /\ A e. X) -> (((UGA) = A /\ (AGU) = A) /\ E.y e. X ((yGA) = U /\ (AGy) = U)))
4		simpl 346	. . . . . 6 |- (((yGA) = U /\ (AGy) = U) -> (yGA) = U)
5	4	reximi 2198	. . . . 5 |- (E.y e. X ((yGA) = U /\ (AGy) = U) -> E.y e. X (yGA) = U)
6	5	adantl 424	. . . 4 |- ((((UGA) = A /\ (AGU) = A) /\ E.y e. X ((yGA) = U /\ (AGy) = U)) -> E.y e. X (yGA) = U)
7	3, 6	syl 12	. . 3 |- ((G e. Grp /\ A e. X) -> E.y e. X (yGA) = U)
8	1	grprcan 9347	. . . . . . . . . . . 12 |- ((G e. Grp /\ (y e. X /\ z e. X /\ A e. X)) -> ((yGA) = (zGA) <-> y = z))
9		eqtr3 1907	. . . . . . . . . . . 12 |- (((yGA) = U /\ (zGA) = U) -> (yGA) = (zGA))
10	8, 9	syl5bi 225	. . . . . . . . . . 11 |- ((G e. Grp /\ (y e. X /\ z e. X /\ A e. X)) -> (((yGA) = U /\ (zGA) = U) -> y = z))
11	10	3exp2 1086	. . . . . . . . . 10 |- (G e. Grp -> (y e. X -> (z e. X -> (A e. X -> (((yGA) = U /\ (zGA) = U) -> y = z)))))
12	11	com24 41	. . . . . . . . 9 |- (G e. Grp -> (A e. X -> (z e. X -> (y e. X -> (((yGA) = U /\ (zGA) = U) -> y = z)))))
13	12	imp41 395	. . . . . . . 8 |- ((((G e. Grp /\ A e. X) /\ z e. X) /\ y e. X) -> (((yGA) = U /\ (zGA) = U) -> y = z))
14	13	an1rs 547	. . . . . . 7 |- ((((G e. Grp /\ A e. X) /\ y e. X) /\ z e. X) -> (((yGA) = U /\ (zGA) = U) -> y = z))
15	14	exp3a 405	. . . . . 6 |- ((((G e. Grp /\ A e. X) /\ y e. X) /\ z e. X) -> ((yGA) = U -> ((zGA) = U -> y = z)))
16	15	r19.21adva 2182	. . . . 5 |- (((G e. Grp /\ A e. X) /\ y e. X) -> ((yGA) = U -> A.z e. X ((zGA) = U -> y = z)))
17	16	ancld 322	. . . 4 |- (((G e. Grp /\ A e. X) /\ y e. X) -> ((yGA) = U -> ((yGA) = U /\ A.z e. X ((zGA) = U -> y = z))))
18	17	reximdva 2203	. . 3 |- ((G e. Grp /\ A e. X) -> (E.y e. X (yGA) = U -> E.y e. X ((yGA) = U /\ A.z e. X ((zGA) = U -> y = z))))
19	7, 18	mpd 29	. 2 |- ((G e. Grp /\ A e. X) -> E.y e. X ((yGA) = U /\ A.z e. X ((zGA) = U -> y = z)))
20		opreq1 4889	. . . 4 |- (y = z -> (yGA) = (zGA))
21	20	eqeq1d 1892	. . 3 |- (y = z -> ((yGA) = U <-> (zGA) = U))
22	21	reu8 2448	. 2 |- (E!y e. X (yGA) = U <-> E.y e. X ((yGA) = U /\ A.z e. X ((zGA) = U -> y = z)))
23	19, 22	sylibr 217	1 |- ((G e. Grp /\ A e. X) -> E!y e. X (yGA) = U)

Syntax hints:   -> wi 3   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300  A.wral 2105  E.wrex 2106  E!wreu 2107  ran crn 3987  ` cfv 3998  (class class class)co 4884  Grpcgr 9311  Idcgi 9312

This theorem is referenced by:  grpinvcl 9352  grpinv 9353

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-reu 2111  df-rab 2112  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-if 2983  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-id 3586  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-f1 4011  df-fo 4012  df-f1o 4013  df-fv 4014  df-opr 4886  df-grp 9316  df-gid 9317

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