Theorem grpdivzer 14740

Description: Condition for a "subtraction" (or "division") value to be equal to the identity element.

Hypotheses

Ref	Expression
grpdivzer.1	|- X = ran G
grpdivzer.2	|- U = (Id` G)
grpdivzer.3	|- D = ( /g ` G)

Assertion

Ref	Expression
grpdivzer	|- ((G e. Grp /\ A e. X /\ B e. X) -> ((ADB) = U <-> A = B))

Proof of Theorem grpdivzer

Step	Hyp	Ref	Expression
1		grpdivzer.1	. . . 4 |- X = ran G
2		eqid 1884	. . . 4 |- (inv` G) = (inv` G)
3		grpdivzer.3	. . . 4 |- D = ( /g ` G)
4	1, 2, 3	grpdivval 9367	. . 3 |- ((G e. Grp /\ A e. X /\ B e. X) -> (ADB) = (AG((inv` G)` B)))
5	4	eqeq1d 1892	. 2 |- ((G e. Grp /\ A e. X /\ B e. X) -> ((ADB) = U <-> (AG((inv` G)` B)) = U))
6		grpdivzer.2	. . . 4 |- U = (Id` G)
7	1, 6, 2	grpinvid1 9356	. . 3 |- ((G e. Grp /\ A e. X /\ ((inv` G)` B) e. X) -> (((inv` G)` A) = ((inv` G)` B) <-> (AG((inv` G)` B)) = U))
8	1, 2	grpinvcl 9352	. . . 4 |- ((G e. Grp /\ B e. X) -> ((inv` G)` B) e. X)
9	8	3adant2 895	. . 3 |- ((G e. Grp /\ A e. X /\ B e. X) -> ((inv` G)` B) e. X)
10	7, 9	syld3an3 1142	. 2 |- ((G e. Grp /\ A e. X /\ B e. X) -> (((inv` G)` A) = ((inv` G)` B) <-> (AG((inv` G)` B)) = U))
11	1, 2	grpinvf 9364	. . . . 5 |- (G e. Grp -> (inv` G):X-1-1-onto->X)
12		f1of1 4634	. . . . 5 |- ((inv` G):X-1-1-onto->X -> (inv` G):X-1-1->X)
13	11, 12	syl 12	. . . 4 |- (G e. Grp -> (inv` G):X-1-1->X)
14	13	3ad2ant1 897	. . 3 |- ((G e. Grp /\ A e. X /\ B e. X) -> (inv` G):X-1-1->X)
15		3simpc 874	. . 3 |- ((G e. Grp /\ A e. X /\ B e. X) -> (A e. X /\ B e. X))
16		f1fveq 4852	. . 3 |- (((inv` G):X-1-1->X /\ (A e. X /\ B e. X)) -> (((inv` G)` A) = ((inv` G)` B) <-> A = B))
17	14, 15, 16	syl11anc 524	. 2 |- ((G e. Grp /\ A e. X /\ B e. X) -> (((inv` G)` A) = ((inv` G)` B) <-> A = B))
18	5, 10, 17	3bitr2d 605	1 |- ((G e. Grp /\ A e. X /\ B e. X) -> ((ADB) = U <-> A = B))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300  ran crn 3987  -1-1->wf1 3995  -1-1-onto->wf1o 3997  ` cfv 3998  (class class class)co 4884  Grpcgr 9311  Idcgi 9312  invcgn 9313   /g cgs 9314

This theorem is referenced by:  svli2 14826

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-oprab 4887  df-grp 9316  df-gid 9317  df-ginv 9318  df-gdiv 9319

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