Theorem grplcan 8025

Description: Left cancellation law for groups.

Hypothesis

Ref	Expression
grplcan.1	|- X = ran G

Assertion

Ref	Expression
grplcan	|- ((G e. Grp /\ (A e. X /\ B e. X /\ C e. X)) -> ((CGA) = (CGB) <-> A = B))

Proof of Theorem grplcan

Step	Hyp	Ref	Expression
1		opreq2 3960	. . . . . . 7 |- ((CGA) = (CGB) -> (((inv` G)` C)G(CGA)) = (((inv` G)` C)G(CGB)))
2	1	adantl 388	. . . . . 6 |- ((((G e. Grp /\ A e. X) /\ (B e. X /\ C e. X)) /\ (CGA) = (CGB)) -> (((inv` G)` C)G(CGA)) = (((inv` G)` C)G(CGB)))
3		grplcan.1	. . . . . . . . . . . 12 |- X = ran G
4		eqid 1473	. . . . . . . . . . . 12 |- (Id` G) = (Id` G)
5		eqid 1473	. . . . . . . . . . . 12 |- (inv` G) = (inv` G)
6	3, 4, 5	grplinv 8020	. . . . . . . . . . 11 |- ((G e. Grp /\ C e. X) -> (((inv` G)` C)GC) = (Id` G))
7	6	adantlr 393	. . . . . . . . . 10 |- (((G e. Grp /\ A e. X) /\ C e. X) -> (((inv` G)` C)GC) = (Id` G))
8	7	opreq1d 3966	. . . . . . . . 9 |- (((G e. Grp /\ A e. X) /\ C e. X) -> ((((inv` G)` C)GC)GA) = ((Id` G)GA))
9	3, 5	grpinvcl 8018	. . . . . . . . . . . . 13 |- ((G e. Grp /\ C e. X) -> ((inv` G)` C) e. X)
10	9	adantrl 394	. . . . . . . . . . . 12 |- ((G e. Grp /\ (A e. X /\ C e. X)) -> ((inv` G)` C) e. X)
11		simprr 415	. . . . . . . . . . . 12 |- ((G e. Grp /\ (A e. X /\ C e. X)) -> C e. X)
12		simprl 414	. . . . . . . . . . . 12 |- ((G e. Grp /\ (A e. X /\ C e. X)) -> A e. X)
13	10, 11, 12	3jca 818	. . . . . . . . . . 11 |- ((G e. Grp /\ (A e. X /\ C e. X)) -> (((inv` G)` C) e. X /\ C e. X /\ A e. X))
14	3	grpass 7997	. . . . . . . . . . 11 |- ((G e. Grp /\ (((inv` G)` C) e. X /\ C e. X /\ A e. X)) -> ((((inv` G)` C)GC)GA) = (((inv` G)` C)G(CGA)))
15	13, 14	syldan 467	. . . . . . . . . 10 |- ((G e. Grp /\ (A e. X /\ C e. X)) -> ((((inv` G)` C)GC)GA) = (((inv` G)` C)G(CGA)))
16	15	anassrs 441	. . . . . . . . 9 |- (((G e. Grp /\ A e. X) /\ C e. X) -> ((((inv` G)` C)GC)GA) = (((inv` G)` C)G(CGA)))
17	3, 4	grplid 8011	. . . . . . . . . 10 |- ((G e. Grp /\ A e. X) -> ((Id` G)GA) = A)
18	17	adantr 389	. . . . . . . . 9 |- (((G e. Grp /\ A e. X) /\ C e. X) -> ((Id` G)GA) = A)
19	8, 16, 18	3eqtr3d 1512	. . . . . . . 8 |- (((G e. Grp /\ A e. X) /\ C e. X) -> (((inv` G)` C)G(CGA)) = A)
20	19	adantrl 394	. . . . . . 7 |- (((G e. Grp /\ A e. X) /\ (B e. X /\ C e. X)) -> (((inv` G)` C)G(CGA)) = A)
21	20	adantr 389	. . . . . 6 |- ((((G e. Grp /\ A e. X) /\ (B e. X /\ C e. X)) /\ (CGA) = (CGB)) -> (((inv` G)` C)G(CGA)) = A)
22	6	adantrl 394	. . . . . . . . . 10 |- ((G e. Grp /\ (B e. X /\ C e. X)) -> (((inv` G)` C)GC) = (Id` G))
23	22	opreq1d 3966	. . . . . . . . 9 |- ((G e. Grp /\ (B e. X /\ C e. X)) -> ((((inv` G)` C)GC)GB) = ((Id` G)GB))
24	9	adantrl 394	. . . . . . . . . . 11 |- ((G e. Grp /\ (B e. X /\ C e. X)) -> ((inv` G)` C) e. X)
25		simprr 415	. . . . . . . . . . 11 |- ((G e. Grp /\ (B e. X /\ C e. X)) -> C e. X)
26		simprl 414	. . . . . . . . . . 11 |- ((G e. Grp /\ (B e. X /\ C e. X)) -> B e. X)
27	24, 25, 26	3jca 818	. . . . . . . . . 10 |- ((G e. Grp /\ (B e. X /\ C e. X)) -> (((inv` G)` C) e. X /\ C e. X /\ B e. X))
28	3	grpass 7997	. . . . . . . . . 10 |- ((G e. Grp /\ (((inv` G)` C) e. X /\ C e. X /\ B e. X)) -> ((((inv` G)` C)GC)GB) = (((inv` G)` C)G(CGB)))
29	27, 28	syldan 467	. . . . . . . . 9 |- ((G e. Grp /\ (B e. X /\ C e. X)) -> ((((inv` G)` C)GC)GB) = (((inv` G)` C)G(CGB)))
30	3, 4	grplid 8011	. . . . . . . . . 10 |- ((G e. Grp /\ B e. X) -> ((Id` G)GB) = B)
31	30	adantrr 395	. . . . . . . . 9 |- ((G e. Grp /\ (B e. X /\ C e. X)) -> ((Id` G)GB) = B)
32	23, 29, 31	3eqtr3d 1512	. . . . . . . 8 |- ((G e. Grp /\ (B e. X /\ C e. X)) -> (((inv` G)` C)G(CGB)) = B)
33	32	adantlr 393	. . . . . . 7 |- (((G e. Grp /\ A e. X) /\ (B e. X /\ C e. X)) -> (((inv` G)` C)G(CGB)) = B)
34	33	adantr 389	. . . . . 6 |- ((((G e. Grp /\ A e. X) /\ (B e. X /\ C e. X)) /\ (CGA) = (CGB)) -> (((inv` G)` C)G(CGB)) = B)
35	2, 21, 34	3eqtr3d 1512	. . . . 5 |- ((((G e. Grp /\ A e. X) /\ (B e. X /\ C e. X)) /\ (CGA) = (CGB)) -> A = B)
36	35	ex 373	. . . 4 |- (((G e. Grp /\ A e. X) /\ (B e. X /\ C e. X)) -> ((CGA) = (CGB) -> A = B))
37	36	exp43 384	. . 3 |- (G e. Grp -> (A e. X -> (B e. X -> (C e. X -> ((CGA) = (CGB) -> A = B)))))
38	37	3imp2 847	. 2 |- ((G e. Grp /\ (A e. X /\ B e. X /\ C e. X)) -> ((CGA) = (CGB) -> A = B))
39		opreq2 3960	. 2 |- (A = B -> (CGA) = (CGB))
40	38, 39	impbid1 516	1 |- ((G e. Grp /\ (A e. X /\ B e. X /\ C e. X)) -> ((CGA) = (CGB) <-> A = B))

Syntax hints:   -> wi 3   <-> wb 146   /\ 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:  grp2inv 8028  grplactf1o 8049  ringlcan 8110  vclcan 8136  nvlcan 8197

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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