Theorem grpinvid1 8034

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
grpinvid1	|- ((G e. Grp /\ A e. X /\ B e. X) -> ((N` A) = B <-> (AGB) = U))

Proof of Theorem grpinvid1

Step	Hyp	Ref	Expression
1		opreq2 3964	. . . 4 |- ((N` A) = B -> (AG(N` A)) = (AGB))
2	1	adantl 388	. . 3 |- (((G e. Grp /\ A e. X /\ B e. X) /\ (N` A) = B) -> (AG(N` A)) = (AGB))
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	grprinv 8033	. . . . 5 |- ((G e. Grp /\ A e. X) -> (AG(N` A)) = U)
7	6	3adant3 798	. . . 4 |- ((G e. Grp /\ A e. X /\ B e. X) -> (AG(N` A)) = U)
8	7	adantr 389	. . 3 |- (((G e. Grp /\ A e. X /\ B e. X) /\ (N` A) = B) -> (AG(N` A)) = U)
9	2, 8	eqtr3d 1507	. 2 |- (((G e. Grp /\ A e. X /\ B e. X) /\ (N` A) = B) -> (AGB) = U)
10		opreq2 3964	. . . 4 |- ((AGB) = U -> ((N` A)G(AGB)) = ((N` A)GU))
11	10	adantl 388	. . 3 |- (((G e. Grp /\ A e. X /\ B e. X) /\ (AGB) = U) -> ((N` A)G(AGB)) = ((N` A)GU))
12	3, 4, 5	grplinv 8032	. . . . . . 7 |- ((G e. Grp /\ A e. X) -> ((N` A)GA) = U)
13	12	opreq1d 3970	. . . . . 6 |- ((G e. Grp /\ A e. X) -> (((N` A)GA)GB) = (UGB))
14	13	3adant3 798	. . . . 5 |- ((G e. Grp /\ A e. X /\ B e. X) -> (((N` A)GA)GB) = (UGB))
15	3, 5	grpinvcl 8030	. . . . . . . . 9 |- ((G e. Grp /\ A e. X) -> (N` A) e. X)
16	15	adantrr 395	. . . . . . . 8 |- ((G e. Grp /\ (A e. X /\ B e. X)) -> (N` A) e. X)
17		simprl 414	. . . . . . . 8 |- ((G e. Grp /\ (A e. X /\ B e. X)) -> A e. X)
18		simprr 415	. . . . . . . 8 |- ((G e. Grp /\ (A e. X /\ B e. X)) -> B e. X)
19	16, 17, 18	3jca 818	. . . . . . 7 |- ((G e. Grp /\ (A e. X /\ B e. X)) -> ((N` A) e. X /\ A e. X /\ B e. X))
20	3	grpass 8009	. . . . . . 7 |- ((G e. Grp /\ ((N` A) e. X /\ A e. X /\ B e. X)) -> (((N` A)GA)GB) = ((N` A)G(AGB)))
21	19, 20	syldan 467	. . . . . 6 |- ((G e. Grp /\ (A e. X /\ B e. X)) -> (((N` A)GA)GB) = ((N` A)G(AGB)))
22	21	3impb 828	. . . . 5 |- ((G e. Grp /\ A e. X /\ B e. X) -> (((N` A)GA)GB) = ((N` A)G(AGB)))
23	3, 4	grplid 8023	. . . . . 6 |- ((G e. Grp /\ B e. X) -> (UGB) = B)
24	23	3adant2 797	. . . . 5 |- ((G e. Grp /\ A e. X /\ B e. X) -> (UGB) = B)
25	14, 22, 24	3eqtr3d 1513	. . . 4 |- ((G e. Grp /\ A e. X /\ B e. X) -> ((N` A)G(AGB)) = B)
26	25	adantr 389	. . 3 |- (((G e. Grp /\ A e. X /\ B e. X) /\ (AGB) = U) -> ((N` A)G(AGB)) = B)
27	3, 4	grprid 8024	. . . . . 6 |- ((G e. Grp /\ (N` A) e. X) -> ((N` A)GU) = (N` A))
28	15, 27	syldan 467	. . . . 5 |- ((G e. Grp /\ A e. X) -> ((N` A)GU) = (N` A))
29	28	3adant3 798	. . . 4 |- ((G e. Grp /\ A e. X /\ B e. X) -> ((N` A)GU) = (N` A))
30	29	adantr 389	. . 3 |- (((G e. Grp /\ A e. X /\ B e. X) /\ (AGB) = U) -> ((N` A)GU) = (N` A))
31	11, 26, 30	3eqtr3rd 1514	. 2 |- (((G e. Grp /\ A e. X /\ B e. X) /\ (AGB) = U) -> (N` A) = B)
32	9, 31	impbida 518	1 |- ((G e. Grp /\ A e. X /\ B e. X) -> ((N` A) = B <-> (AGB) = U))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   /\ w3a 774   = wceq 955   e. wcel 957  ran crn 3167  ` cfv 3178  (class class class)co 3958  Grpcgr 7995  Idcgi 7996  invcgn 7997

This theorem is referenced by:  grpinvid 8036  grpinvop 8042  ghomgrpilem2 10342

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 1209  ax-11o 1217  ax-ext 1458  ax-rep 2689  ax-sep 2699  ax-pow 2738  ax-pr 2775  ax-un 2862

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 776  df-ex 980  df-sb 1171  df-eu 1381  df-mo 1382  df-clab 1463  df-cleq 1468  df-clel 1471  df-ne 1585  df-ral 1647  df-rex 1648  df-reu 1649  df-rab 1650  df-v 1809  df-sbc 1939  df-dif 2046  df-un 2047  df-in 2048  df-ss 2050  df-nul 2278  df-pw 2399  df-sn 2409  df-pr 2410  df-op 2413  df-uni 2500  df-br 2616  df-opab 2663  df-id 2831  df-xp 3180  df-rel 3181  df-cnv 3182  df-co 3183  df-dm 3184  df-rn 3185  df-res 3186  df-ima 3187  df-fun 3188  df-fn 3189  df-f 3190  df-fo 3192  df-fv 3194  df-opr 3960  df-grp 7999  df-gid 8000  df-ginv 8001

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