Theorem grpinvid1 8068

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 3975	. . . 4 |- ((N` A) = B -> (AG(N` A)) = (AGB))
2	1	adantl 390	. . 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 8067	. . . . 5 |- ((G e. Grp /\ A e. X) -> (AG(N` A)) = U)
7	6	3adant3 801	. . . 4 |- ((G e. Grp /\ A e. X /\ B e. X) -> (AG(N` A)) = U)
8	7	adantr 391	. . 3 |- (((G e. Grp /\ A e. X /\ B e. X) /\ (N` A) = B) -> (AG(N` A)) = U)
9	2, 8	eqtr3d 1512	. 2 |- (((G e. Grp /\ A e. X /\ B e. X) /\ (N` A) = B) -> (AGB) = U)
10		opreq2 3975	. . . 4 |- ((AGB) = U -> ((N` A)G(AGB)) = ((N` A)GU))
11	10	adantl 390	. . 3 |- (((G e. Grp /\ A e. X /\ B e. X) /\ (AGB) = U) -> ((N` A)G(AGB)) = ((N` A)GU))
12	3, 4, 5	grplinv 8066	. . . . . . 7 |- ((G e. Grp /\ A e. X) -> ((N` A)GA) = U)
13	12	opreq1d 3981	. . . . . 6 |- ((G e. Grp /\ A e. X) -> (((N` A)GA)GB) = (UGB))
14	13	3adant3 801	. . . . 5 |- ((G e. Grp /\ A e. X /\ B e. X) -> (((N` A)GA)GB) = (UGB))
15	3, 5	grpinvcl 8064	. . . . . . . . 9 |- ((G e. Grp /\ A e. X) -> (N` A) e. X)
16	15	adantrr 397	. . . . . . . 8 |- ((G e. Grp /\ (A e. X /\ B e. X)) -> (N` A) e. X)
17		simprl 416	. . . . . . . 8 |- ((G e. Grp /\ (A e. X /\ B e. X)) -> A e. X)
18		simprr 417	. . . . . . . 8 |- ((G e. Grp /\ (A e. X /\ B e. X)) -> B e. X)
19	16, 17, 18	3jca 821	. . . . . . 7 |- ((G e. Grp /\ (A e. X /\ B e. X)) -> ((N` A) e. X /\ A e. X /\ B e. X))
20	3	grpass 8044	. . . . . . 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 469	. . . . . 6 |- ((G e. Grp /\ (A e. X /\ B e. X)) -> (((N` A)GA)GB) = ((N` A)G(AGB)))
22	21	3impb 831	. . . . 5 |- ((G e. Grp /\ A e. X /\ B e. X) -> (((N` A)GA)GB) = ((N` A)G(AGB)))
23	3, 4	grplid 8057	. . . . . 6 |- ((G e. Grp /\ B e. X) -> (UGB) = B)
24	23	3adant2 800	. . . . 5 |- ((G e. Grp /\ A e. X /\ B e. X) -> (UGB) = B)
25	14, 22, 24	3eqtr3d 1518	. . . 4 |- ((G e. Grp /\ A e. X /\ B e. X) -> ((N` A)G(AGB)) = B)
26	25	adantr 391	. . 3 |- (((G e. Grp /\ A e. X /\ B e. X) /\ (AGB) = U) -> ((N` A)G(AGB)) = B)
27	3, 4	grprid 8058	. . . . . 6 |- ((G e. Grp /\ (N` A) e. X) -> ((N` A)GU) = (N` A))
28	15, 27	syldan 469	. . . . 5 |- ((G e. Grp /\ A e. X) -> ((N` A)GU) = (N` A))
29	28	3adant3 801	. . . 4 |- ((G e. Grp /\ A e. X /\ B e. X) -> ((N` A)GU) = (N` A))
30	29	adantr 391	. . 3 |- (((G e. Grp /\ A e. X /\ B e. X) /\ (AGB) = U) -> ((N` A)GU) = (N` A))
31	11, 26, 30	3eqtr3rd 1519	. 2 |- (((G e. Grp /\ A e. X /\ B e. X) /\ (AGB) = U) -> (N` A) = B)
32	9, 31	impbida 521	1 |- ((G e. Grp /\ A e. X /\ B e. X) -> ((N` A) = B <-> (AGB) = U))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   /\ w3a 777   = wceq 958   e. wcel 960  ran crn 3177  ` cfv 3188  (class class class)co 3969  Grpcgr 8030  Idcgi 8031  invcgn 8032

This theorem is referenced by:  grpinvid 8070  grpinvop 8076  ghomgrpilem2 10381

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-9 967  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-rep 2698  ax-sep 2708  ax-pow 2748  ax-pr 2785  ax-un 2872

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 779  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-ral 1652  df-rex 1653  df-reu 1654  df-rab 1655  df-v 1815  df-sbc 1945  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-op 2420  df-uni 2508  df-br 2625  df-opab 2672  df-id 2841  df-xp 3190  df-rel 3191  df-cnv 3192  df-co 3193  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-fun 3198  df-fn 3199  df-f 3200  df-fo 3202  df-fv 3204  df-opr 3971  df-grp 8034  df-gid 8035  df-ginv 8036

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