Theorem gacan 9460

Description: Group inverses cancel in a group action. (Contributed by Jeff Hankins, 11-Aug-2009.)

Hypotheses

Ref	Expression
gacan.1	|- X = ran G
gacan.2	|- Y = ran M
gacan.3	|- N = (inv` G)

Assertion

Ref	Expression
gacan	|- ((M e. K /\ <.G, M>. e. GrpAct /\ (A e. X /\ B e. Y /\ C e. Y)) -> ((AMB) = C <-> ((N` A)MC) = B))

Proof of Theorem gacan

Step	Hyp	Ref	Expression
1		simpl1 879	. . . 4 |- (((M e. K /\ <.G, M>. e. GrpAct /\ (A e. X /\ B e. Y /\ C e. Y)) /\ (AMB) = C) -> M e. K)
2		simpl2 880	. . . 4 |- (((M e. K /\ <.G, M>. e. GrpAct /\ (A e. X /\ B e. Y /\ C e. Y)) /\ (AMB) = C) -> <.G, M>. e. GrpAct)
3		gacan.1	. . . . . . . . 9 |- X = ran G
4		gacan.3	. . . . . . . . 9 |- N = (inv` G)
5	3, 4	grpinvcl 9352	. . . . . . . 8 |- ((G e. Grp /\ A e. X) -> (N` A) e. X)
6		gagrp 9456	. . . . . . . 8 |- ((M e. K /\ <.G, M>. e. GrpAct) -> G e. Grp)
7	5, 6	sylan 497	. . . . . . 7 |- (((M e. K /\ <.G, M>. e. GrpAct) /\ A e. X) -> (N` A) e. X)
8	7	3impa 1062	. . . . . 6 |- ((M e. K /\ <.G, M>. e. GrpAct /\ A e. X) -> (N` A) e. X)
9		simp1 876	. . . . . 6 |- ((A e. X /\ B e. Y /\ C e. Y) -> A e. X)
10	8, 9	syl3an3 1132	. . . . 5 |- ((M e. K /\ <.G, M>. e. GrpAct /\ (A e. X /\ B e. Y /\ C e. Y)) -> (N` A) e. X)
11	10	adantr 425	. . . 4 |- (((M e. K /\ <.G, M>. e. GrpAct /\ (A e. X /\ B e. Y /\ C e. Y)) /\ (AMB) = C) -> (N` A) e. X)
12		simpl31 957	. . . 4 |- (((M e. K /\ <.G, M>. e. GrpAct /\ (A e. X /\ B e. Y /\ C e. Y)) /\ (AMB) = C) -> A e. X)
13		simpl32 958	. . . 4 |- (((M e. K /\ <.G, M>. e. GrpAct /\ (A e. X /\ B e. Y /\ C e. Y)) /\ (AMB) = C) -> B e. Y)
14		gacan.2	. . . . 5 |- Y = ran M
15	3, 14	gaass 9459	. . . 4 |- ((M e. K /\ <.G, M>. e. GrpAct /\ ((N` A) e. X /\ A e. X /\ B e. Y)) -> (((N` A)GA)MB) = ((N` A)M(AMB)))
16	1, 2, 11, 12, 13, 15	syl113anc 1112	. . 3 |- (((M e. K /\ <.G, M>. e. GrpAct /\ (A e. X /\ B e. Y /\ C e. Y)) /\ (AMB) = C) -> (((N` A)GA)MB) = ((N` A)M(AMB)))
17		eqid 1884	. . . . . . . . 9 |- (Id` G) = (Id` G)
18	3, 17, 4	grplinv 9354	. . . . . . . 8 |- ((G e. Grp /\ A e. X) -> ((N` A)GA) = (Id` G))
19	18, 6, 9	syl2an 503	. . . . . . 7 |- (((M e. K /\ <.G, M>. e. GrpAct) /\ (A e. X /\ B e. Y /\ C e. Y)) -> ((N` A)GA) = (Id` G))
20	19	3impa 1062	. . . . . 6 |- ((M e. K /\ <.G, M>. e. GrpAct /\ (A e. X /\ B e. Y /\ C e. Y)) -> ((N` A)GA) = (Id` G))
21	20	opreq1d 4897	. . . . 5 |- ((M e. K /\ <.G, M>. e. GrpAct /\ (A e. X /\ B e. Y /\ C e. Y)) -> (((N` A)GA)MB) = ((Id` G)MB))
22	14, 17	gagrpid 9458	. . . . . 6 |- ((M e. K /\ <.G, M>. e. GrpAct /\ B e. Y) -> ((Id` G)MB) = B)
23		simp2 877	. . . . . 6 |- ((A e. X /\ B e. Y /\ C e. Y) -> B e. Y)
24	22, 23	syl3an3 1132	. . . . 5 |- ((M e. K /\ <.G, M>. e. GrpAct /\ (A e. X /\ B e. Y /\ C e. Y)) -> ((Id` G)MB) = B)
25	21, 24	eqtrd 1925	. . . 4 |- ((M e. K /\ <.G, M>. e. GrpAct /\ (A e. X /\ B e. Y /\ C e. Y)) -> (((N` A)GA)MB) = B)
26	25	adantr 425	. . 3 |- (((M e. K /\ <.G, M>. e. GrpAct /\ (A e. X /\ B e. Y /\ C e. Y)) /\ (AMB) = C) -> (((N` A)GA)MB) = B)
27		opreq2 4890	. . . 4 |- ((AMB) = C -> ((N` A)M(AMB)) = ((N` A)MC))
28	27	adantl 424	. . 3 |- (((M e. K /\ <.G, M>. e. GrpAct /\ (A e. X /\ B e. Y /\ C e. Y)) /\ (AMB) = C) -> ((N` A)M(AMB)) = ((N` A)MC))
29	16, 26, 28	3eqtr3rd 1936	. 2 |- (((M e. K /\ <.G, M>. e. GrpAct /\ (A e. X /\ B e. Y /\ C e. Y)) /\ (AMB) = C) -> ((N` A)MC) = B)
30	3, 14	gaass 9459	. . . . 5 |- ((M e. K /\ <.G, M>. e. GrpAct /\ (A e. X /\ (N` A) e. X /\ C e. Y)) -> ((AG(N` A))MC) = (AM((N` A)MC)))
31	9	3ad2ant3 899	. . . . . 6 |- ((M e. K /\ <.G, M>. e. GrpAct /\ (A e. X /\ B e. Y /\ C e. Y)) -> A e. X)
32	5, 6, 9	syl2an 503	. . . . . . 7 |- (((M e. K /\ <.G, M>. e. GrpAct) /\ (A e. X /\ B e. Y /\ C e. Y)) -> (N` A) e. X)
33	32	3impa 1062	. . . . . 6 |- ((M e. K /\ <.G, M>. e. GrpAct /\ (A e. X /\ B e. Y /\ C e. Y)) -> (N` A) e. X)
34		simp3 878	. . . . . . 7 |- ((A e. X /\ B e. Y /\ C e. Y) -> C e. Y)
35	34	3ad2ant3 899	. . . . . 6 |- ((M e. K /\ <.G, M>. e. GrpAct /\ (A e. X /\ B e. Y /\ C e. Y)) -> C e. Y)
36	31, 33, 35	3jca 1050	. . . . 5 |- ((M e. K /\ <.G, M>. e. GrpAct /\ (A e. X /\ B e. Y /\ C e. Y)) -> (A e. X /\ (N` A) e. X /\ C e. Y))
37	30, 36	syld3an3 1142	. . . 4 |- ((M e. K /\ <.G, M>. e. GrpAct /\ (A e. X /\ B e. Y /\ C e. Y)) -> ((AG(N` A))MC) = (AM((N` A)MC)))
38	37	adantr 425	. . 3 |- (((M e. K /\ <.G, M>. e. GrpAct /\ (A e. X /\ B e. Y /\ C e. Y)) /\ ((N` A)MC) = B) -> ((AG(N` A))MC) = (AM((N` A)MC)))
39	3, 17, 4	grprinv 9355	. . . . . . . 8 |- ((G e. Grp /\ A e. X) -> (AG(N` A)) = (Id` G))
40	39, 6, 9	syl2an 503	. . . . . . 7 |- (((M e. K /\ <.G, M>. e. GrpAct) /\ (A e. X /\ B e. Y /\ C e. Y)) -> (AG(N` A)) = (Id` G))
41	40	3impa 1062	. . . . . 6 |- ((M e. K /\ <.G, M>. e. GrpAct /\ (A e. X /\ B e. Y /\ C e. Y)) -> (AG(N` A)) = (Id` G))
42	41	opreq1d 4897	. . . . 5 |- ((M e. K /\ <.G, M>. e. GrpAct /\ (A e. X /\ B e. Y /\ C e. Y)) -> ((AG(N` A))MC) = ((Id` G)MC))
43	14, 17	gagrpid 9458	. . . . . 6 |- ((M e. K /\ <.G, M>. e. GrpAct /\ C e. Y) -> ((Id` G)MC) = C)
44	43, 34	syl3an3 1132	. . . . 5 |- ((M e. K /\ <.G, M>. e. GrpAct /\ (A e. X /\ B e. Y /\ C e. Y)) -> ((Id` G)MC) = C)
45	42, 44	eqtrd 1925	. . . 4 |- ((M e. K /\ <.G, M>. e. GrpAct /\ (A e. X /\ B e. Y /\ C e. Y)) -> ((AG(N` A))MC) = C)
46	45	adantr 425	. . 3 |- (((M e. K /\ <.G, M>. e. GrpAct /\ (A e. X /\ B e. Y /\ C e. Y)) /\ ((N` A)MC) = B) -> ((AG(N` A))MC) = C)
47		opreq2 4890	. . . 4 |- (((N` A)MC) = B -> (AM((N` A)MC)) = (AMB))
48	47	adantl 424	. . 3 |- (((M e. K /\ <.G, M>. e. GrpAct /\ (A e. X /\ B e. Y /\ C e. Y)) /\ ((N` A)MC) = B) -> (AM((N` A)MC)) = (AMB))
49	38, 46, 48	3eqtr3rd 1936	. 2 |- (((M e. K /\ <.G, M>. e. GrpAct /\ (A e. X /\ B e. Y /\ C e. Y)) /\ ((N` A)MC) = B) -> (AMB) = C)
50	29, 49	impbida 577	1 |- ((M e. K /\ <.G, M>. e. GrpAct /\ (A e. X /\ B e. Y /\ C e. Y)) -> ((AMB) = C <-> ((N` A)MC) = B))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300  <.cop 3046  ran crn 3987  ` cfv 3998  (class class class)co 4884  Grpcgr 9311  Idcgi 9312  invcgn 9313  GrpActcga 9447

This theorem is referenced by:  gapmlem 9461  gapm 9462

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  df-ga 9448

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