Theorem grpnpcan 8091

Description: Group theory analog of npcant 5399.

Hypotheses

Ref	Expression
grpdivf.1	|- X = ran G
grpdivf.3	|- D = ( /g ` G)

Assertion

Ref	Expression
grpnpcan	|- ((G e. Grp /\ A e. X /\ B e. X) -> ((ADB)GB) = A)

Proof of Theorem grpnpcan

Step	Hyp	Ref	Expression
1		grpdivf.1	. . . 4 |- X = ran G
2		eqid 1475	. . . 4 |- (inv` G) = (inv` G)
3		grpdivf.3	. . . 4 |- D = ( /g ` G)
4	1, 2, 3	grpdivval 8082	. . 3 |- ((G e. Grp /\ A e. X /\ B e. X) -> (ADB) = (AG((inv` G)` B)))
5	4	opreq1d 3975	. 2 |- ((G e. Grp /\ A e. X /\ B e. X) -> ((ADB)GB) = ((AG((inv` G)` B))GB))
6	1	grpass 8047	. . 3 |- ((G e. Grp /\ (A e. X /\ ((inv` G)` B) e. X /\ B e. X)) -> ((AG((inv` G)` B))GB) = (AG(((inv` G)` B)GB)))
7		3simp1 788	. . 3 |- ((G e. Grp /\ A e. X /\ B e. X) -> G e. Grp)
8		3simp2 789	. . . 4 |- ((G e. Grp /\ A e. X /\ B e. X) -> A e. X)
9	1, 2	grpinvcl 8068	. . . . 5 |- ((G e. Grp /\ B e. X) -> ((inv` G)` B) e. X)
10	9	3adant2 798	. . . 4 |- ((G e. Grp /\ A e. X /\ B e. X) -> ((inv` G)` B) e. X)
11		3simp3 790	. . . 4 |- ((G e. Grp /\ A e. X /\ B e. X) -> B e. X)
12	8, 10, 11	3jca 819	. . 3 |- ((G e. Grp /\ A e. X /\ B e. X) -> (A e. X /\ ((inv` G)` B) e. X /\ B e. X))
13	6, 7, 12	sylanc 471	. 2 |- ((G e. Grp /\ A e. X /\ B e. X) -> ((AG((inv` G)` B))GB) = (AG(((inv` G)` B)GB)))
14		eqid 1475	. . . . . 6 |- (Id` G) = (Id` G)
15	1, 14, 2	grplinv 8070	. . . . 5 |- ((G e. Grp /\ B e. X) -> (((inv` G)` B)GB) = (Id` G))
16	15	opreq2d 3976	. . . 4 |- ((G e. Grp /\ B e. X) -> (AG(((inv` G)` B)GB)) = (AG(Id` G)))
17	16	3adant2 798	. . 3 |- ((G e. Grp /\ A e. X /\ B e. X) -> (AG(((inv` G)` B)GB)) = (AG(Id` G)))
18	1, 14	grprid 8062	. . . 4 |- ((G e. Grp /\ A e. X) -> (AG(Id` G)) = A)
19	18	3adant3 799	. . 3 |- ((G e. Grp /\ A e. X /\ B e. X) -> (AG(Id` G)) = A)
20	17, 19	eqtrd 1507	. 2 |- ((G e. Grp /\ A e. X /\ B e. X) -> (AG(((inv` G)` B)GB)) = A)
21	5, 13, 20	3eqtrd 1511	1 |- ((G e. Grp /\ A e. X /\ B e. X) -> ((ADB)GB) = A)

Syntax hints:   -> wi 3   /\ wa 223   /\ w3a 775   = wceq 956   e. wcel 958  ran crn 3171  ` cfv 3182  (class class class)co 3963  Grpcgr 8033  Idcgi 8034  invcgn 8035   /g cgs 8036

This theorem is referenced by:  grpnpncan 8094  ablnnncan 8111  ghgrpilem3 8135

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-9 965  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-rep 2693  ax-sep 2703  ax-pow 2742  ax-pr 2779  ax-un 2866

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-rex 1650  df-reu 1651  df-rab 1652  df-v 1812  df-sbc 1942  df-csb 2002  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-uni 2504  df-br 2620  df-opab 2667  df-id 2835  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fun 3192  df-fn 3193  df-f 3194  df-fo 3196  df-fv 3198  df-opr 3965  df-oprab 3966  df-grp 8037  df-gid 8038  df-ginv 8039  df-gdiv 8040

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