Theorem grpinvid2 8056

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 3965	. . . 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 8053	. . . . 5 |- ((G e. Grp /\ A e. X) -> ((N` A)GA) = U)
7	6	3adant3 798	. . . 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 1508	. 2 |- (((G e. Grp /\ A e. X /\ B e. X) /\ (N` A) = B) -> (BGA) = U)
10	3, 5	grpinvcl 8051	. . . . . . 7 |- ((G e. Grp /\ A e. X) -> (N` A) e. X)
11	3, 4	grplid 8044	. . . . . . 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 798	. . . . 5 |- ((G e. Grp /\ A e. X /\ B e. X) -> (UG(N` A)) = (N` A))
14	13	eqcomd 1479	. . . 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 3965	. . . 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 818	. . . . . . 7 |- ((G e. Grp /\ (A e. X /\ B e. X)) -> (B e. X /\ A e. X /\ (N` A) e. X))
22	3	grpass 8030	. . . . . . 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 828	. . . . 5 |- ((G e. Grp /\ A e. X /\ B e. X) -> ((BGA)G(N` A)) = (BG(AG(N` A))))
25	3, 4, 5	grprinv 8054	. . . . . . 7 |- ((G e. Grp /\ A e. X) -> (AG(N` A)) = U)
26	25	opreq2d 3973	. . . . . 6 |- ((G e. Grp /\ A e. X) -> (BG(AG(N` A))) = (BGU))
27	26	3adant3 798	. . . . 5 |- ((G e. Grp /\ A e. X /\ B e. X) -> (BG(AG(N` A))) = (BGU))
28	3, 4	grprid 8045	. . . . . 6 |- ((G e. Grp /\ B e. X) -> (BGU) = B)
29	28	3adant2 797	. . . . 5 |- ((G e. Grp /\ A e. X /\ B e. X) -> (BGU) = B)
30	24, 27, 29	3eqtrd 1510	. . . 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 1512	. 2 |- (((G e. Grp /\ A e. X /\ B e. X) /\ (BGA) = U) -> (N` A) = B)
33	9, 32	impbida 518	1 |- ((G e. Grp /\ A e. X /\ B e. X) -> ((N` A) = B <-> (BGA) = U))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   /\ w3a 774   = wceq 955   e. wcel 957  ran crn 3168  ` cfv 3179  (class class class)co 3960  Grpcgr 8016  Idcgi 8017  invcgn 8018

This theorem is referenced by:  ghomf1olem 10387

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-9 964  ax-10 965  ax-11 966  ax-12 967  ax-13 968  ax-14 969  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-rep 2690  ax-sep 2700  ax-pow 2739  ax-pr 2776  ax-un 2863

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 776  df-ex 980  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1586  df-ral 1648  df-rex 1649  df-reu 1650  df-rab 1651  df-v 1810  df-sbc 1940  df-dif 2047  df-un 2048  df-in 2049  df-ss 2051  df-nul 2279  df-pw 2400  df-sn 2410  df-pr 2411  df-op 2414  df-uni 2501  df-br 2617  df-opab 2664  df-id 2832  df-xp 3181  df-rel 3182  df-cnv 3183  df-co 3184  df-dm 3185  df-rn 3186  df-res 3187  df-ima 3188  df-fun 3189  df-fn 3190  df-f 3191  df-fo 3193  df-fv 3195  df-opr 3962  df-grp 8020  df-gid 8021  df-ginv 8022

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