Theorem grpinvop 8080

Description: The inverse of the group operation reverses the arguments. Lemma 2.2.1(d) of [Herstein] p. 55.

Hypotheses

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

Assertion

Ref	Expression
grpinvop	|- ((G e. Grp /\ A e. X /\ B e. X) -> (N` (AGB)) = ((N` B)G(N` A)))

Proof of Theorem grpinvop

Step	Hyp	Ref	Expression
1		grpasscan1.1	. . . . 5 |- X = ran G
2	1	grpass 8047	. . . 4 |- ((G e. Grp /\ (A e. X /\ B e. X /\ ((N` B)G(N` A)) e. X)) -> ((AGB)G((N` B)G(N` A))) = (AG(BG((N` B)G(N` A)))))
3		3simp1 788	. . . 4 |- ((G e. Grp /\ A e. X /\ B e. X) -> G e. Grp)
4		3simp2 789	. . . . 5 |- ((G e. Grp /\ A e. X /\ B e. X) -> A e. X)
5		3simp3 790	. . . . 5 |- ((G e. Grp /\ A e. X /\ B e. X) -> B e. X)
6	1	grpcl 8044	. . . . . 6 |- ((G e. Grp /\ (N` B) e. X /\ (N` A) e. X) -> ((N` B)G(N` A)) e. X)
7		grpasscan1.2	. . . . . . . 8 |- N = (inv` G)
8	1, 7	grpinvcl 8068	. . . . . . 7 |- ((G e. Grp /\ B e. X) -> (N` B) e. X)
9	8	3adant2 798	. . . . . 6 |- ((G e. Grp /\ A e. X /\ B e. X) -> (N` B) e. X)
10	1, 7	grpinvcl 8068	. . . . . . 7 |- ((G e. Grp /\ A e. X) -> (N` A) e. X)
11	10	3adant3 799	. . . . . 6 |- ((G e. Grp /\ A e. X /\ B e. X) -> (N` A) e. X)
12	6, 3, 9, 11	syl3anc 858	. . . . 5 |- ((G e. Grp /\ A e. X /\ B e. X) -> ((N` B)G(N` A)) e. X)
13	4, 5, 12	3jca 819	. . . 4 |- ((G e. Grp /\ A e. X /\ B e. X) -> (A e. X /\ B e. X /\ ((N` B)G(N` A)) e. X))
14	2, 3, 13	sylanc 471	. . 3 |- ((G e. Grp /\ A e. X /\ B e. X) -> ((AGB)G((N` B)G(N` A))) = (AG(BG((N` B)G(N` A)))))
15		eqid 1475	. . . . . . . 8 |- (Id` G) = (Id` G)
16	1, 15, 7	grprinv 8071	. . . . . . 7 |- ((G e. Grp /\ B e. X) -> (BG(N` B)) = (Id` G))
17	16	3adant2 798	. . . . . 6 |- ((G e. Grp /\ A e. X /\ B e. X) -> (BG(N` B)) = (Id` G))
18	17	opreq1d 3975	. . . . 5 |- ((G e. Grp /\ A e. X /\ B e. X) -> ((BG(N` B))G(N` A)) = ((Id` G)G(N` A)))
19	1	grpass 8047	. . . . . 6 |- ((G e. Grp /\ (B e. X /\ (N` B) e. X /\ (N` A) e. X)) -> ((BG(N` B))G(N` A)) = (BG((N` B)G(N` A))))
20	5, 9, 11	3jca 819	. . . . . 6 |- ((G e. Grp /\ A e. X /\ B e. X) -> (B e. X /\ (N` B) e. X /\ (N` A) e. X))
21	19, 3, 20	sylanc 471	. . . . 5 |- ((G e. Grp /\ A e. X /\ B e. X) -> ((BG(N` B))G(N` A)) = (BG((N` B)G(N` A))))
22	1, 15	grplid 8061	. . . . . . 7 |- ((G e. Grp /\ (N` A) e. X) -> ((Id` G)G(N` A)) = (N` A))
23	10, 22	syldan 467	. . . . . 6 |- ((G e. Grp /\ A e. X) -> ((Id` G)G(N` A)) = (N` A))
24	23	3adant3 799	. . . . 5 |- ((G e. Grp /\ A e. X /\ B e. X) -> ((Id` G)G(N` A)) = (N` A))
25	18, 21, 24	3eqtr3d 1515	. . . 4 |- ((G e. Grp /\ A e. X /\ B e. X) -> (BG((N` B)G(N` A))) = (N` A))
26	25	opreq2d 3976	. . 3 |- ((G e. Grp /\ A e. X /\ B e. X) -> (AG(BG((N` B)G(N` A)))) = (AG(N` A)))
27	1, 15, 7	grprinv 8071	. . . 4 |- ((G e. Grp /\ A e. X) -> (AG(N` A)) = (Id` G))
28	27	3adant3 799	. . 3 |- ((G e. Grp /\ A e. X /\ B e. X) -> (AG(N` A)) = (Id` G))
29	14, 26, 28	3eqtrd 1511	. 2 |- ((G e. Grp /\ A e. X /\ B e. X) -> ((AGB)G((N` B)G(N` A))) = (Id` G))
30	1, 15, 7	grpinvid1 8072	. . 3 |- ((G e. Grp /\ (AGB) e. X /\ ((N` B)G(N` A)) e. X) -> ((N` (AGB)) = ((N` B)G(N` A)) <-> ((AGB)G((N` B)G(N` A))) = (Id` G)))
31	1	grpcl 8044	. . 3 |- ((G e. Grp /\ A e. X /\ B e. X) -> (AGB) e. X)
32	30, 3, 31, 12	syl3anc 858	. 2 |- ((G e. Grp /\ A e. X /\ B e. X) -> ((N` (AGB)) = ((N` B)G(N` A)) <-> ((AGB)G((N` B)G(N` A))) = (Id` G)))
33	29, 32	mpbird 196	1 |- ((G e. Grp /\ A e. X /\ B e. X) -> (N` (AGB)) = ((N` B)G(N` A)))

Syntax hints:   -> wi 3   <-> wb 146   /\ w3a 775   = wceq 956   e. wcel 958  ran crn 3171  ` cfv 3182  (class class class)co 3963  Grpcgr 8033  Idcgi 8034  invcgn 8035

This theorem is referenced by:  grpinvdiv 8084  grppnpcan2 8092

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-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-grp 8037  df-gid 8038  df-ginv 8039

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