Theorem grpasscan1 9361

Description: An associative cancellation law for groups. (Contributed by Paul Chapman, 25-Feb-2008.)

Hypotheses

Ref	Expression
grpasscan1.1	|- X = ran G
grpasscan1.2	|- N = (inv` G)

Assertion

Ref	Expression
grpasscan1	|- ((G e. Grp /\ A e. X /\ B e. X) -> (AG((N` A)GB)) = B)

Proof of Theorem grpasscan1

Step	Hyp	Ref	Expression
1		grpasscan1.1	. . . . 5 |- X = ran G
2		eqid 1884	. . . . 5 |- (Id` G) = (Id` G)
3		grpasscan1.2	. . . . 5 |- N = (inv` G)
4	1, 2, 3	grprinv 9355	. . . 4 |- ((G e. Grp /\ A e. X) -> (AG(N` A)) = (Id` G))
5	4	3adant3 896	. . 3 |- ((G e. Grp /\ A e. X /\ B e. X) -> (AG(N` A)) = (Id` G))
6	5	opreq1d 4897	. 2 |- ((G e. Grp /\ A e. X /\ B e. X) -> ((AG(N` A))GB) = ((Id` G)GB))
7	1, 3	grpinvcl 9352	. . . 4 |- ((G e. Grp /\ A e. X) -> (N` A) e. X)
8	1	grpass 9327	. . . . . 6 |- ((G e. Grp /\ (A e. X /\ (N` A) e. X /\ B e. X)) -> ((AG(N` A))GB) = (AG((N` A)GB)))
9	8	3exp2 1086	. . . . 5 |- (G e. Grp -> (A e. X -> ((N` A) e. X -> (B e. X -> ((AG(N` A))GB) = (AG((N` A)GB))))))
10	9	imp 377	. . . 4 |- ((G e. Grp /\ A e. X) -> ((N` A) e. X -> (B e. X -> ((AG(N` A))GB) = (AG((N` A)GB)))))
11	7, 10	mpd 29	. . 3 |- ((G e. Grp /\ A e. X) -> (B e. X -> ((AG(N` A))GB) = (AG((N` A)GB))))
12	11	3impia 1064	. 2 |- ((G e. Grp /\ A e. X /\ B e. X) -> ((AG(N` A))GB) = (AG((N` A)GB)))
13	1, 2	grplid 9345	. . 3 |- ((G e. Grp /\ B e. X) -> ((Id` G)GB) = B)
14	13	3adant2 895	. 2 |- ((G e. Grp /\ A e. X /\ B e. X) -> ((Id` G)GB) = B)
15	6, 12, 14	3eqtr3d 1934	1 |- ((G e. Grp /\ A e. X /\ B e. X) -> (AG((N` A)GB)) = B)

Syntax hints:   -> wi 3   /\ 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:  gxcom 9392  gxsuc 9395  grplactf1o 9406  ghgrpilem3 9443

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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