Theorem grplcan 8075

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 3969	. . . . . . 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 1475	. . . . . . . . . . . 12 |- (Id` G) = (Id` G)
5		eqid 1475	. . . . . . . . . . . 12 |- (inv` G) = (inv` G)
6	3, 4, 5	grplinv 8070	. . . . . . . . . . 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 3975	. . . . . . . . 9 |- (((G e. Grp /\ A e. X) /\ C e. X) -> ((((inv` G)` C)GC)GA) = ((Id` G)GA))
9	3, 5	grpinvcl 8068	. . . . . . . . . . . . 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 819	. . . . . . . . . . 11 |- ((G e. Grp /\ (A e. X /\ C e. X)) -> (((inv` G)` C) e. X /\ C e. X /\ A e. X))
14	3	grpass 8047	. . . . . . . . . . 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 8061	. . . . . . . . . 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 1515	. . . . . . . 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 3975	. . . . . . . . 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 819	. . . . . . . . . 10 |- ((G e. Grp /\ (B e. X /\ C e. X)) -> (((inv` G)` C) e. X /\ C e. X /\ B e. X))
28	3	grpass 8047	. . . . . . . . . 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 8061	. . . . . . . . . 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 1515	. . . . . . . 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 1515	. . . . 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 848	. 2 |- ((G e. Grp /\ (A e. X /\ B e. X /\ C e. X)) -> ((CGA) = (CGB) -> A = B))
39		opreq2 3969	. 2 |- (A = B -> (CGA) = (CGB))
40	38, 39	impbid1 517	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 775   = wceq 956   e. wcel 958  ran crn 3171  ` cfv 3182  (class class class)co 3963  Grpcgr 8033  Idcgi 8034  invcgn 8035

This theorem is referenced by:  grp2inv 8078  grplactf1o 8098  ringlcan 8158  vclcan 8184  nvlcan 8245

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-9 965  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-rep 2693  ax-sep 2703  ax-pow 2742  ax-pr 2779  ax-un 2866

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-rex 1650  df-reu 1651  df-rab 1652  df-v 1812  df-sbc 1942  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-uni 2504  df-br 2620  df-opab 2667  df-id 2835  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fun 3192  df-fn 3193  df-f 3194  df-fo 3196  df-fv 3198  df-opr 3965  df-grp 8037  df-gid 8038  df-ginv 8039

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