Theorem grpasscan1 8027

Description: An associative cancellation law for groups. (Contributed by Paul Chapman, 25-Feb-2008.)

Hypotheses

Ref	Expression
grpasscan1.1	|- X = ran G
grpasscan1.2	|- N = (inv` G)

Assertion

Ref	Expression
grpasscan1	|- ((G e. Grp /\ A e. X /\ B e. X) -> (AG((N` A)GB)) = B)

Proof of Theorem grpasscan1

Step	Hyp	Ref	Expression
1		grpasscan1.1	. . . . 5 |- X = ran G
2		eqid 1473	. . . . 5 |- (Id` G) = (Id` G)
3		grpasscan1.2	. . . . 5 |- N = (inv` G)
4	1, 2, 3	grprinv 8021	. . . 4 |- ((G e. Grp /\ A e. X) -> (AG(N` A)) = (Id` G))
5	4	3adant3 798	. . 3 |- ((G e. Grp /\ A e. X /\ B e. X) -> (AG(N` A)) = (Id` G))
6	5	opreq1d 3966	. 2 |- ((G e. Grp /\ A e. X /\ B e. X) -> ((AG(N` A))GB) = ((Id` G)GB))
7	1, 3	grpinvcl 8018	. . . 4 |- ((G e. Grp /\ A e. X) -> (N` A) e. X)
8	1	grpass 7997	. . . . . 6 |- ((G e. Grp /\ (A e. X /\ (N` A) e. X /\ B e. X)) -> ((AG(N` A))GB) = (AG((N` A)GB)))
9	8	3exp2 850	. . . . 5 |- (G e. Grp -> (A e. X -> ((N` A) e. X -> (B e. X -> ((AG(N` A))GB) = (AG((N` A)GB))))))
10	9	imp 350	. . . 4 |- ((G e. Grp /\ A e. X) -> ((N` A) e. X -> (B e. X -> ((AG(N` A))GB) = (AG((N` A)GB)))))
11	7, 10	mpd 26	. . 3 |- ((G e. Grp /\ A e. X) -> (B e. X -> ((AG(N` A))GB) = (AG((N` A)GB))))
12	11	3impia 829	. 2 |- ((G e. Grp /\ A e. X /\ B e. X) -> ((AG(N` A))GB) = (AG((N` A)GB)))
13	1, 2	grplid 8011	. . 3 |- ((G e. Grp /\ B e. X) -> ((Id` G)GB) = B)
14	13	3adant2 797	. 2 |- ((G e. Grp /\ A e. X /\ B e. X) -> ((Id` G)GB) = B)
15	6, 12, 14	3eqtr3d 1512	1 |- ((G e. Grp /\ A e. X /\ B e. X) -> (AG((N` A)GB)) = B)

Syntax hints:   -> wi 3   /\ wa 223   /\ 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:  grplactf1o 8049  ghgrpilem3 8087

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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