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Theorem grpinvop 8030
Description: The inverse of the group operation reverses the arguments. Lemma 2.2.1(d) of [Herstein] p. 55.
Hypotheses
Ref Expression
grpasscan1.1 |- X = ran G
grpasscan1.2 |- N = (inv` G)
Assertion
Ref Expression
grpinvop |- ((G e. Grp /\ A e. X /\ B e. X) -> (N` (AGB)) = ((N` B)G(N` A)))

Proof of Theorem grpinvop
StepHypRef Expression
1 grpasscan1.1 . . . . 5 |- X = ran G
21grpass 7997 . . . 4 |- ((G e. Grp /\ (A e. X /\ B e. X /\ ((N` B)G(N` A)) e. X)) -> ((AGB)G((N` B)G(N` A))) = (AG(BG((N` B)G(N` A)))))
3 3simp1 787 . . . 4 |- ((G e. Grp /\ A e. X /\ B e. X) -> G e. Grp)
4 3simp2 788 . . . . 5 |- ((G e. Grp /\ A e. X /\ B e. X) -> A e. X)
5 3simp3 789 . . . . 5 |- ((G e. Grp /\ A e. X /\ B e. X) -> B e. X)
61grpcl 7994 . . . . . 6 |- ((G e. Grp /\ (N` B) e. X /\ (N` A) e. X) -> ((N` B)G(N` A)) e. X)
7 grpasscan1.2 . . . . . . . 8 |- N = (inv` G)
81, 7grpinvcl 8018 . . . . . . 7 |- ((G e. Grp /\ B e. X) -> (N` B) e. X)
983adant2 797 . . . . . 6 |- ((G e. Grp /\ A e. X /\ B e. X) -> (N` B) e. X)
101, 7grpinvcl 8018 . . . . . . 7 |- ((G e. Grp /\ A e. X) -> (N` A) e. X)
11103adant3 798 . . . . . 6 |- ((G e. Grp /\ A e. X /\ B e. X) -> (N` A) e. X)
126, 3, 9, 11syl3anc 857 . . . . 5 |- ((G e. Grp /\ A e. X /\ B e. X) -> ((N` B)G(N` A)) e. X)
134, 5, 123jca 818 . . . 4 |- ((G e. Grp /\ A e. X /\ B e. X) -> (A e. X /\ B e. X /\ ((N` B)G(N` A)) e. X))
142, 3, 13sylanc 471 . . 3 |- ((G e. Grp /\ A e. X /\ B e. X) -> ((AGB)G((N` B)G(N` A))) = (AG(BG((N` B)G(N` A)))))
15 eqid 1473 . . . . . . . 8 |- (Id` G) = (Id` G)
161, 15, 7grprinv 8021 . . . . . . 7 |- ((G e. Grp /\ B e. X) -> (BG(N` B)) = (Id` G))
17163adant2 797 . . . . . 6 |- ((G e. Grp /\ A e. X /\ B e. X) -> (BG(N` B)) = (Id` G))
1817opreq1d 3966 . . . . 5 |- ((G e. Grp /\ A e. X /\ B e. X) -> ((BG(N` B))G(N` A)) = ((Id`
G)G(N` A)))
191grpass 7997 . . . . . 6 |- ((G e. Grp /\ (B e. X /\ (N` B) e. X /\ (N` A) e. X)) -> ((BG(N` B))G(N` A)) = (BG((N` B)G(N` A))))
205, 9, 113jca 818 . . . . . 6 |- ((G e. Grp /\ A e. X /\ B e. X) -> (B e. X /\ (N` B) e. X /\ (N` A) e. X))
2119, 3, 20sylanc 471 . . . . 5 |- ((G e. Grp /\ A e. X /\ B e. X) -> ((BG(N` B))G(N` A)) = (BG((N` B)G(N` A))))
221, 15grplid 8011 . . . . . . 7 |- ((G e. Grp /\ (N` A) e. X) -> ((Id` G)G(N` A)) = (N` A))
2310, 22syldan 467 . . . . . 6 |- ((G e. Grp /\ A e. X) -> ((Id` G)G(N` A)) = (N` A))
24233adant3 798 . . . . 5 |- ((G e. Grp /\ A e. X /\ B e. X) -> ((Id` G)G(N` A)) = (N` A))
2518, 21, 243eqtr3d 1512 . . . 4 |- ((G e. Grp /\ A e. X /\ B e. X) -> (BG((N` B)G(N` A))) = (N` A))
2625opreq2d 3967 . . 3 |- ((G e. Grp /\ A e. X /\ B e. X) -> (AG(BG((N` B)G(N` A)))) = (AG(N` A)))
271, 15, 7grprinv 8021 . . . 4 |- ((G e. Grp /\ A e. X) -> (AG(N` A)) = (Id` G))
28273adant3 798 . . 3 |- ((G e. Grp /\ A e. X /\ B e. X) -> (AG(N` A)) = (Id` G))
2914, 26, 283eqtrd 1508 . 2 |- ((G e. Grp /\ A e. X /\ B e. X) -> ((AGB)G((N` B)G(N` A))) = (Id` G))
301, 15, 7grpinvid1 8022 . . 3 |- ((G e. Grp /\ (AGB) e. X /\ ((N` B)G(N` A)) e. X) -> ((N` (AGB)) = ((N` B)G(N` A)) <-> ((AGB)G((N` B)G(N` A))) = (Id`
G)))
311grpcl 7994 . . 3 |- ((G e. Grp /\ A e. X /\ B e. X) -> (AGB) e. X)
3230, 3, 31, 12syl3anc 857 . 2 |- ((G e. Grp /\ A e. X /\ B e. X) -> ((N` (AGB)) = ((N` B)G(N` A)) <-> ((AGB)G((N` B)G(N` A))) = (Id` G)))
3329, 32mpbird 196 1 |- ((G e. Grp /\ A e. X /\ B e. X) -> (N` (AGB)) = ((N` B)G(N` A)))
Colors of variables: wff set class
Syntax hints:   -> wi 3   <-> wb 146   /\ w3a 774   = wceq 954   e. wcel 956  ran crn 3166  ` cfv 3177  (class class class)co 3954  Grpcgr 7983  Idcgi 7984  invcgn 7985
This theorem is referenced by:  grpinvdiv 8034  grppnpcan2 8042
This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-9 963  ax-10 964  ax-11 965  ax-12 966  ax-13 967  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-rep 2688  ax-sep 2698  ax-pow 2737  ax-pr 2774  ax-un 2861
This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 776  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-ral 1646  df-rex 1647  df-reu 1648  df-rab 1649  df-v 1808  df-sbc 1938  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-nul 2277  df-pw 2398  df-sn 2408  df-pr 2409  df-op 2412  df-uni 2499  df-br 2615  df-opab 2662  df-id 2830  df-xp 3179  df-rel 3180  df-cnv 3181  df-co 3182  df-dm 3183  df-rn 3184  df-res 3185  df-ima 3186  df-fun 3187  df-fn 3188  df-f 3189  df-fo 3191  df-fv 3193  df-opr 3956  df-grp 7987  df-gid 7988  df-ginv 7989
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