Theorem grpinvid2 9357

Description: The inverse of a group element expressed in terms of the identity element.

Hypotheses

Ref	Expression
grpinv.1	|- X = ran G
grpinv.2	|- U = (Id` G)
grpinv.3	|- N = (inv` G)

Assertion

Ref	Expression
grpinvid2	|- ((G e. Grp /\ A e. X /\ B e. X) -> ((N` A) = B <-> (BGA) = U))

Proof of Theorem grpinvid2

Step	Hyp	Ref	Expression
1		opreq1 4889	. . . 4 |- ((N` A) = B -> ((N` A)GA) = (BGA))
2	1	adantl 424	. . 3 |- (((G e. Grp /\ A e. X /\ B e. X) /\ (N` A) = B) -> ((N` A)GA) = (BGA))
3		grpinv.1	. . . . . 6 |- X = ran G
4		grpinv.2	. . . . . 6 |- U = (Id` G)
5		grpinv.3	. . . . . 6 |- N = (inv` G)
6	3, 4, 5	grplinv 9354	. . . . 5 |- ((G e. Grp /\ A e. X) -> ((N` A)GA) = U)
7	6	3adant3 896	. . . 4 |- ((G e. Grp /\ A e. X /\ B e. X) -> ((N` A)GA) = U)
8	7	adantr 425	. . 3 |- (((G e. Grp /\ A e. X /\ B e. X) /\ (N` A) = B) -> ((N` A)GA) = U)
9	2, 8	eqtr3d 1927	. 2 |- (((G e. Grp /\ A e. X /\ B e. X) /\ (N` A) = B) -> (BGA) = U)
10	3, 5	grpinvcl 9352	. . . . . . 7 |- ((G e. Grp /\ A e. X) -> (N` A) e. X)
11	3, 4	grplid 9345	. . . . . . 7 |- ((G e. Grp /\ (N` A) e. X) -> (UG(N` A)) = (N` A))
12	10, 11	syldan 516	. . . . . 6 |- ((G e. Grp /\ A e. X) -> (UG(N` A)) = (N` A))
13	12	3adant3 896	. . . . 5 |- ((G e. Grp /\ A e. X /\ B e. X) -> (UG(N` A)) = (N` A))
14	13	eqcomd 1889	. . . 4 |- ((G e. Grp /\ A e. X /\ B e. X) -> (N` A) = (UG(N` A)))
15	14	adantr 425	. . 3 |- (((G e. Grp /\ A e. X /\ B e. X) /\ (BGA) = U) -> (N` A) = (UG(N` A)))
16		opreq1 4889	. . . 4 |- ((BGA) = U -> ((BGA)G(N` A)) = (UG(N` A)))
17	16	adantl 424	. . 3 |- (((G e. Grp /\ A e. X /\ B e. X) /\ (BGA) = U) -> ((BGA)G(N` A)) = (UG(N` A)))
18		simprr 451	. . . . . . . 8 |- ((G e. Grp /\ (A e. X /\ B e. X)) -> B e. X)
19		simprl 450	. . . . . . . 8 |- ((G e. Grp /\ (A e. X /\ B e. X)) -> A e. X)
20	10	adantrr 431	. . . . . . . 8 |- ((G e. Grp /\ (A e. X /\ B e. X)) -> (N` A) e. X)
21	18, 19, 20	3jca 1050	. . . . . . 7 |- ((G e. Grp /\ (A e. X /\ B e. X)) -> (B e. X /\ A e. X /\ (N` A) e. X))
22	3	grpass 9327	. . . . . . 7 |- ((G e. Grp /\ (B e. X /\ A e. X /\ (N` A) e. X)) -> ((BGA)G(N` A)) = (BG(AG(N` A))))
23	21, 22	syldan 516	. . . . . 6 |- ((G e. Grp /\ (A e. X /\ B e. X)) -> ((BGA)G(N` A)) = (BG(AG(N` A))))
24	23	3impb 1063	. . . . 5 |- ((G e. Grp /\ A e. X /\ B e. X) -> ((BGA)G(N` A)) = (BG(AG(N` A))))
25	3, 4, 5	grprinv 9355	. . . . . . 7 |- ((G e. Grp /\ A e. X) -> (AG(N` A)) = U)
26	25	opreq2d 4898	. . . . . 6 |- ((G e. Grp /\ A e. X) -> (BG(AG(N` A))) = (BGU))
27	26	3adant3 896	. . . . 5 |- ((G e. Grp /\ A e. X /\ B e. X) -> (BG(AG(N` A))) = (BGU))
28	3, 4	grprid 9346	. . . . . 6 |- ((G e. Grp /\ B e. X) -> (BGU) = B)
29	28	3adant2 895	. . . . 5 |- ((G e. Grp /\ A e. X /\ B e. X) -> (BGU) = B)
30	24, 27, 29	3eqtrd 1929	. . . 4 |- ((G e. Grp /\ A e. X /\ B e. X) -> ((BGA)G(N` A)) = B)
31	30	adantr 425	. . 3 |- (((G e. Grp /\ A e. X /\ B e. X) /\ (BGA) = U) -> ((BGA)G(N` A)) = B)
32	15, 17, 31	3eqtr2d 1932	. 2 |- (((G e. Grp /\ A e. X /\ B e. X) /\ (BGA) = U) -> (N` A) = B)
33	9, 32	impbida 577	1 |- ((G e. Grp /\ A e. X /\ B e. X) -> ((N` A) = B <-> (BGA) = U))

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:  ghomf1olem 13637  ringnegmn1r 16103

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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