Theorem grpinvid2 8073

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 3968	. . . 4 |- ((N` A) = B -> ((N` A)GA) = (BGA))
2	1	adantl 388	. . 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 8070	. . . . 5 |- ((G e. Grp /\ A e. X) -> ((N` A)GA) = U)
7	6	3adant3 799	. . . 4 |- ((G e. Grp /\ A e. X /\ B e. X) -> ((N` A)GA) = U)
8	7	adantr 389	. . 3 |- (((G e. Grp /\ A e. X /\ B e. X) /\ (N` A) = B) -> ((N` A)GA) = U)
9	2, 8	eqtr3d 1509	. 2 |- (((G e. Grp /\ A e. X /\ B e. X) /\ (N` A) = B) -> (BGA) = U)
10	3, 5	grpinvcl 8068	. . . . . . 7 |- ((G e. Grp /\ A e. X) -> (N` A) e. X)
11	3, 4	grplid 8061	. . . . . . 7 |- ((G e. Grp /\ (N` A) e. X) -> (UG(N` A)) = (N` A))
12	10, 11	syldan 467	. . . . . 6 |- ((G e. Grp /\ A e. X) -> (UG(N` A)) = (N` A))
13	12	3adant3 799	. . . . 5 |- ((G e. Grp /\ A e. X /\ B e. X) -> (UG(N` A)) = (N` A))
14	13	eqcomd 1480	. . . 4 |- ((G e. Grp /\ A e. X /\ B e. X) -> (N` A) = (UG(N` A)))
15	14	adantr 389	. . 3 |- (((G e. Grp /\ A e. X /\ B e. X) /\ (BGA) = U) -> (N` A) = (UG(N` A)))
16		opreq1 3968	. . . 4 |- ((BGA) = U -> ((BGA)G(N` A)) = (UG(N` A)))
17	16	adantl 388	. . 3 |- (((G e. Grp /\ A e. X /\ B e. X) /\ (BGA) = U) -> ((BGA)G(N` A)) = (UG(N` A)))
18		simprr 415	. . . . . . . 8 |- ((G e. Grp /\ (A e. X /\ B e. X)) -> B e. X)
19		simprl 414	. . . . . . . 8 |- ((G e. Grp /\ (A e. X /\ B e. X)) -> A e. X)
20	10	adantrr 395	. . . . . . . 8 |- ((G e. Grp /\ (A e. X /\ B e. X)) -> (N` A) e. X)
21	18, 19, 20	3jca 819	. . . . . . 7 |- ((G e. Grp /\ (A e. X /\ B e. X)) -> (B e. X /\ A e. X /\ (N` A) e. X))
22	3	grpass 8047	. . . . . . 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 467	. . . . . 6 |- ((G e. Grp /\ (A e. X /\ B e. X)) -> ((BGA)G(N` A)) = (BG(AG(N` A))))
24	23	3impb 829	. . . . 5 |- ((G e. Grp /\ A e. X /\ B e. X) -> ((BGA)G(N` A)) = (BG(AG(N` A))))
25	3, 4, 5	grprinv 8071	. . . . . . 7 |- ((G e. Grp /\ A e. X) -> (AG(N` A)) = U)
26	25	opreq2d 3976	. . . . . 6 |- ((G e. Grp /\ A e. X) -> (BG(AG(N` A))) = (BGU))
27	26	3adant3 799	. . . . 5 |- ((G e. Grp /\ A e. X /\ B e. X) -> (BG(AG(N` A))) = (BGU))
28	3, 4	grprid 8062	. . . . . 6 |- ((G e. Grp /\ B e. X) -> (BGU) = B)
29	28	3adant2 798	. . . . 5 |- ((G e. Grp /\ A e. X /\ B e. X) -> (BGU) = B)
30	24, 27, 29	3eqtrd 1511	. . . 4 |- ((G e. Grp /\ A e. X /\ B e. X) -> ((BGA)G(N` A)) = B)
31	30	adantr 389	. . 3 |- (((G e. Grp /\ A e. X /\ B e. X) /\ (BGA) = U) -> ((BGA)G(N` A)) = B)
32	15, 17, 31	3eqtr2d 1513	. 2 |- (((G e. Grp /\ A e. X /\ B e. X) /\ (BGA) = U) -> (N` A) = B)
33	9, 32	impbida 519	1 |- ((G e. Grp /\ A e. X /\ B e. X) -> ((N` A) = B <-> (BGA) = U))

Syntax hints:   -> wi 3   <-> wb 146   /\ 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

This theorem is referenced by:  ghomf1olem 10396

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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