Theorem grplcan 9359

Description: Left cancellation law for groups.

Hypothesis

Ref	Expression
grplcan.1	|- X = ran G

Assertion

Ref	Expression
grplcan	|- ((G e. Grp /\ (A e. X /\ B e. X /\ C e. X)) -> ((CGA) = (CGB) <-> A = B))

Proof of Theorem grplcan

Step	Hyp	Ref	Expression
1		opreq2 4890	. . . . . 6 |- ((CGA) = (CGB) -> (((inv` G)` C)G(CGA)) = (((inv` G)` C)G(CGB)))
2	1	adantl 424	. . . . 5 |- ((((G e. Grp /\ A e. X) /\ (B e. X /\ C e. X)) /\ (CGA) = (CGB)) -> (((inv` G)` C)G(CGA)) = (((inv` G)` C)G(CGB)))
3		grplcan.1	. . . . . . . . . . 11 |- X = ran G
4		eqid 1884	. . . . . . . . . . 11 |- (Id` G) = (Id` G)
5		eqid 1884	. . . . . . . . . . 11 |- (inv` G) = (inv` G)
6	3, 4, 5	grplinv 9354	. . . . . . . . . 10 |- ((G e. Grp /\ C e. X) -> (((inv` G)` C)GC) = (Id` G))
7	6	adantlr 429	. . . . . . . . 9 |- (((G e. Grp /\ A e. X) /\ C e. X) -> (((inv` G)` C)GC) = (Id` G))
8	7	opreq1d 4897	. . . . . . . 8 |- (((G e. Grp /\ A e. X) /\ C e. X) -> ((((inv` G)` C)GC)GA) = ((Id` G)GA))
9	3, 5	grpinvcl 9352	. . . . . . . . . . . 12 |- ((G e. Grp /\ C e. X) -> ((inv` G)` C) e. X)
10	9	adantrl 430	. . . . . . . . . . 11 |- ((G e. Grp /\ (A e. X /\ C e. X)) -> ((inv` G)` C) e. X)
11		simprr 451	. . . . . . . . . . 11 |- ((G e. Grp /\ (A e. X /\ C e. X)) -> C e. X)
12		simprl 450	. . . . . . . . . . 11 |- ((G e. Grp /\ (A e. X /\ C e. X)) -> A e. X)
13	10, 11, 12	3jca 1050	. . . . . . . . . 10 |- ((G e. Grp /\ (A e. X /\ C e. X)) -> (((inv` G)` C) e. X /\ C e. X /\ A e. X))
14	3	grpass 9327	. . . . . . . . . 10 |- ((G e. Grp /\ (((inv` G)` C) e. X /\ C e. X /\ A e. X)) -> ((((inv` G)` C)GC)GA) = (((inv` G)` C)G(CGA)))
15	13, 14	syldan 516	. . . . . . . . 9 |- ((G e. Grp /\ (A e. X /\ C e. X)) -> ((((inv` G)` C)GC)GA) = (((inv` G)` C)G(CGA)))
16	15	anassrs 489	. . . . . . . 8 |- (((G e. Grp /\ A e. X) /\ C e. X) -> ((((inv` G)` C)GC)GA) = (((inv` G)` C)G(CGA)))
17	3, 4	grplid 9345	. . . . . . . . 9 |- ((G e. Grp /\ A e. X) -> ((Id` G)GA) = A)
18	17	adantr 425	. . . . . . . 8 |- (((G e. Grp /\ A e. X) /\ C e. X) -> ((Id` G)GA) = A)
19	8, 16, 18	3eqtr3d 1934	. . . . . . 7 |- (((G e. Grp /\ A e. X) /\ C e. X) -> (((inv` G)` C)G(CGA)) = A)
20	19	adantrl 430	. . . . . 6 |- (((G e. Grp /\ A e. X) /\ (B e. X /\ C e. X)) -> (((inv` G)` C)G(CGA)) = A)
21	20	adantr 425	. . . . 5 |- ((((G e. Grp /\ A e. X) /\ (B e. X /\ C e. X)) /\ (CGA) = (CGB)) -> (((inv` G)` C)G(CGA)) = A)
22	6	adantrl 430	. . . . . . . . 9 |- ((G e. Grp /\ (B e. X /\ C e. X)) -> (((inv` G)` C)GC) = (Id` G))
23	22	opreq1d 4897	. . . . . . . 8 |- ((G e. Grp /\ (B e. X /\ C e. X)) -> ((((inv` G)` C)GC)GB) = ((Id` G)GB))
24	9	adantrl 430	. . . . . . . . . 10 |- ((G e. Grp /\ (B e. X /\ C e. X)) -> ((inv` G)` C) e. X)
25		simprr 451	. . . . . . . . . 10 |- ((G e. Grp /\ (B e. X /\ C e. X)) -> C e. X)
26		simprl 450	. . . . . . . . . 10 |- ((G e. Grp /\ (B e. X /\ C e. X)) -> B e. X)
27	24, 25, 26	3jca 1050	. . . . . . . . 9 |- ((G e. Grp /\ (B e. X /\ C e. X)) -> (((inv` G)` C) e. X /\ C e. X /\ B e. X))
28	3	grpass 9327	. . . . . . . . 9 |- ((G e. Grp /\ (((inv` G)` C) e. X /\ C e. X /\ B e. X)) -> ((((inv` G)` C)GC)GB) = (((inv` G)` C)G(CGB)))
29	27, 28	syldan 516	. . . . . . . 8 |- ((G e. Grp /\ (B e. X /\ C e. X)) -> ((((inv` G)` C)GC)GB) = (((inv` G)` C)G(CGB)))
30	3, 4	grplid 9345	. . . . . . . . 9 |- ((G e. Grp /\ B e. X) -> ((Id` G)GB) = B)
31	30	adantrr 431	. . . . . . . 8 |- ((G e. Grp /\ (B e. X /\ C e. X)) -> ((Id` G)GB) = B)
32	23, 29, 31	3eqtr3d 1934	. . . . . . 7 |- ((G e. Grp /\ (B e. X /\ C e. X)) -> (((inv` G)` C)G(CGB)) = B)
33	32	adantlr 429	. . . . . 6 |- (((G e. Grp /\ A e. X) /\ (B e. X /\ C e. X)) -> (((inv` G)` C)G(CGB)) = B)
34	33	adantr 425	. . . . 5 |- ((((G e. Grp /\ A e. X) /\ (B e. X /\ C e. X)) /\ (CGA) = (CGB)) -> (((inv` G)` C)G(CGB)) = B)
35	2, 21, 34	3eqtr3d 1934	. . . 4 |- ((((G e. Grp /\ A e. X) /\ (B e. X /\ C e. X)) /\ (CGA) = (CGB)) -> A = B)
36	35	exp53 419	. . 3 |- (G e. Grp -> (A e. X -> (B e. X -> (C e. X -> ((CGA) = (CGB) -> A = B)))))
37	36	3imp2 1083	. 2 |- ((G e. Grp /\ (A e. X /\ B e. X /\ C e. X)) -> ((CGA) = (CGB) -> A = B))
38		opreq2 4890	. 2 |- (A = B -> (CGA) = (CGB))
39	37, 38	impbid1 575	1 |- ((G e. Grp /\ (A e. X /\ B e. X /\ C e. X)) -> ((CGA) = (CGB) <-> 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  Idcgi 9312  invcgn 9313

This theorem is referenced by:  grp2inv 9363  grplactf1o 9406  ringlcan 9483  ringlz 9487  vclcan 9516  nvlcan 9577  grpdlcan 14739  veclcan 14819  mulinvsca 14823

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  df-ginv 9318

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