Theorem grpdrcan 14738

Description: Right cancellation law for group "subtraction" (or "division").

Hypotheses

Ref	Expression
grpdrcan.1	|- X = ran G
grpdrcan.2	|- D = ( /g ` G)

Assertion

Ref	Expression
grpdrcan	|- ((G e. Grp /\ (A e. X /\ B e. X /\ C e. X)) -> ((ADC) = (BDC) <-> A = B))

Proof of Theorem grpdrcan

Step	Hyp	Ref	Expression
1		grpdrcan.1	. . . 4 |- X = ran G
2		eqid 1884	. . . 4 |- (inv` G) = (inv` G)
3		grpdrcan.2	. . . 4 |- D = ( /g ` G)
4	1, 2, 3	grpdivval 9367	. . 3 |- ((G e. Grp /\ A e. X /\ C e. X) -> (ADC) = (AG((inv` G)` C)))
5	4	3adant3r2 1078	. 2 |- ((G e. Grp /\ (A e. X /\ B e. X /\ C e. X)) -> (ADC) = (AG((inv` G)` C)))
6		eqeq1 1890	. . . 4 |- ((ADC) = (AG((inv` G)` C)) -> ((ADC) = (BDC) <-> (AG((inv` G)` C)) = (BDC)))
7	6	bibi1d 681	. . 3 |- ((ADC) = (AG((inv` G)` C)) -> (((ADC) = (BDC) <-> A = B) <-> ((AG((inv` G)` C)) = (BDC) <-> A = B)))
8	1, 2, 3	grpdivval 9367	. . . . 5 |- ((G e. Grp /\ B e. X /\ C e. X) -> (BDC) = (BG((inv` G)` C)))
9	8	3adant3r1 1077	. . . 4 |- ((G e. Grp /\ (A e. X /\ B e. X /\ C e. X)) -> (BDC) = (BG((inv` G)` C)))
10		eqeq2 1893	. . . . . 6 |- ((BDC) = (BG((inv` G)` C)) -> ((AG((inv` G)` C)) = (BDC) <-> (AG((inv` G)` C)) = (BG((inv` G)` C))))
11	10	bibi1d 681	. . . . 5 |- ((BDC) = (BG((inv` G)` C)) -> (((AG((inv` G)` C)) = (BDC) <-> A = B) <-> ((AG((inv` G)` C)) = (BG((inv` G)` C)) <-> A = B)))
12		simpr1 882	. . . . . . 7 |- ((G e. Grp /\ (A e. X /\ B e. X /\ C e. X)) -> A e. X)
13		simpr2 883	. . . . . . 7 |- ((G e. Grp /\ (A e. X /\ B e. X /\ C e. X)) -> B e. X)
14	1, 2	grpinvcl 9352	. . . . . . . 8 |- ((G e. Grp /\ C e. X) -> ((inv` G)` C) e. X)
15	14	3ad2antr3 1043	. . . . . . 7 |- ((G e. Grp /\ (A e. X /\ B e. X /\ C e. X)) -> ((inv` G)` C) e. X)
16	12, 13, 15	3jca 1050	. . . . . 6 |- ((G e. Grp /\ (A e. X /\ B e. X /\ C e. X)) -> (A e. X /\ B e. X /\ ((inv` G)` C) e. X))
17	1	grprcan 9347	. . . . . 6 |- ((G e. Grp /\ (A e. X /\ B e. X /\ ((inv` G)` C) e. X)) -> ((AG((inv` G)` C)) = (BG((inv` G)` C)) <-> A = B))
18	16, 17	syldan 516	. . . . 5 |- ((G e. Grp /\ (A e. X /\ B e. X /\ C e. X)) -> ((AG((inv` G)` C)) = (BG((inv` G)` C)) <-> A = B))
19	11, 18	syl5bir 227	. . . 4 |- ((BDC) = (BG((inv` G)` C)) -> ((G e. Grp /\ (A e. X /\ B e. X /\ C e. X)) -> ((AG((inv` G)` C)) = (BDC) <-> A = B)))
20	9, 19	mpcom 60	. . 3 |- ((G e. Grp /\ (A e. X /\ B e. X /\ C e. X)) -> ((AG((inv` G)` C)) = (BDC) <-> A = B))
21	7, 20	syl5bir 227	. 2 |- ((ADC) = (AG((inv` G)` C)) -> ((G e. Grp /\ (A e. X /\ B e. X /\ C e. X)) -> ((ADC) = (BDC) <-> A = B)))
22	5, 21	mpcom 60	1 |- ((G e. Grp /\ (A e. X /\ B e. X /\ C e. X)) -> ((ADC) = (BDC) <-> A = B))

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

This theorem is referenced by:  vecsrcan 14812  mvecrtol 14816  mvecrtol2 14820

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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