Theorem grplcan 8071

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 3975	. . . . . . 7 |- ((CGA) = (CGB) -> (((inv` G)` C)G(CGA)) = (((inv` G)` C)G(CGB)))
2	1	adantl 390	. . . . . 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 1478	. . . . . . . . . . . 12 |- (Id` G) = (Id` G)
5		eqid 1478	. . . . . . . . . . . 12 |- (inv` G) = (inv` G)
6	3, 4, 5	grplinv 8066	. . . . . . . . . . 11 |- ((G e. Grp /\ C e. X) -> (((inv` G)` C)GC) = (Id` G))
7	6	adantlr 395	. . . . . . . . . 10 |- (((G e. Grp /\ A e. X) /\ C e. X) -> (((inv` G)` C)GC) = (Id` G))
8	7	opreq1d 3981	. . . . . . . . 9 |- (((G e. Grp /\ A e. X) /\ C e. X) -> ((((inv` G)` C)GC)GA) = ((Id` G)GA))
9	3, 5	grpinvcl 8064	. . . . . . . . . . . . 13 |- ((G e. Grp /\ C e. X) -> ((inv` G)` C) e. X)
10	9	adantrl 396	. . . . . . . . . . . 12 |- ((G e. Grp /\ (A e. X /\ C e. X)) -> ((inv` G)` C) e. X)
11		simprr 417	. . . . . . . . . . . 12 |- ((G e. Grp /\ (A e. X /\ C e. X)) -> C e. X)
12		simprl 416	. . . . . . . . . . . 12 |- ((G e. Grp /\ (A e. X /\ C e. X)) -> A e. X)
13	10, 11, 12	3jca 821	. . . . . . . . . . 11 |- ((G e. Grp /\ (A e. X /\ C e. X)) -> (((inv` G)` C) e. X /\ C e. X /\ A e. X))
14	3	grpass 8044	. . . . . . . . . . 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 469	. . . . . . . . . 10 |- ((G e. Grp /\ (A e. X /\ C e. X)) -> ((((inv` G)` C)GC)GA) = (((inv` G)` C)G(CGA)))
16	15	anassrs 443	. . . . . . . . 9 |- (((G e. Grp /\ A e. X) /\ C e. X) -> ((((inv` G)` C)GC)GA) = (((inv` G)` C)G(CGA)))
17	3, 4	grplid 8057	. . . . . . . . . 10 |- ((G e. Grp /\ A e. X) -> ((Id` G)GA) = A)
18	17	adantr 391	. . . . . . . . 9 |- (((G e. Grp /\ A e. X) /\ C e. X) -> ((Id` G)GA) = A)
19	8, 16, 18	3eqtr3d 1518	. . . . . . . 8 |- (((G e. Grp /\ A e. X) /\ C e. X) -> (((inv` G)` C)G(CGA)) = A)
20	19	adantrl 396	. . . . . . 7 |- (((G e. Grp /\ A e. X) /\ (B e. X /\ C e. X)) -> (((inv` G)` C)G(CGA)) = A)
21	20	adantr 391	. . . . . 6 |- ((((G e. Grp /\ A e. X) /\ (B e. X /\ C e. X)) /\ (CGA) = (CGB)) -> (((inv` G)` C)G(CGA)) = A)
22	6	adantrl 396	. . . . . . . . . 10 |- ((G e. Grp /\ (B e. X /\ C e. X)) -> (((inv` G)` C)GC) = (Id` G))
23	22	opreq1d 3981	. . . . . . . . 9 |- ((G e. Grp /\ (B e. X /\ C e. X)) -> ((((inv` G)` C)GC)GB) = ((Id` G)GB))
24	9	adantrl 396	. . . . . . . . . . 11 |- ((G e. Grp /\ (B e. X /\ C e. X)) -> ((inv` G)` C) e. X)
25		simprr 417	. . . . . . . . . . 11 |- ((G e. Grp /\ (B e. X /\ C e. X)) -> C e. X)
26		simprl 416	. . . . . . . . . . 11 |- ((G e. Grp /\ (B e. X /\ C e. X)) -> B e. X)
27	24, 25, 26	3jca 821	. . . . . . . . . 10 |- ((G e. Grp /\ (B e. X /\ C e. X)) -> (((inv` G)` C) e. X /\ C e. X /\ B e. X))
28	3	grpass 8044	. . . . . . . . . 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 469	. . . . . . . . 9 |- ((G e. Grp /\ (B e. X /\ C e. X)) -> ((((inv` G)` C)GC)GB) = (((inv` G)` C)G(CGB)))
30	3, 4	grplid 8057	. . . . . . . . . 10 |- ((G e. Grp /\ B e. X) -> ((Id` G)GB) = B)
31	30	adantrr 397	. . . . . . . . 9 |- ((G e. Grp /\ (B e. X /\ C e. X)) -> ((Id` G)GB) = B)
32	23, 29, 31	3eqtr3d 1518	. . . . . . . 8 |- ((G e. Grp /\ (B e. X /\ C e. X)) -> (((inv` G)` C)G(CGB)) = B)
33	32	adantlr 395	. . . . . . 7 |- (((G e. Grp /\ A e. X) /\ (B e. X /\ C e. X)) -> (((inv` G)` C)G(CGB)) = B)
34	33	adantr 391	. . . . . 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 1518	. . . . 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 386	. . 3 |- (G e. Grp -> (A e. X -> (B e. X -> (C e. X -> ((CGA) = (CGB) -> A = B)))))
38	37	3imp2 850	. 2 |- ((G e. Grp /\ (A e. X /\ B e. X /\ C e. X)) -> ((CGA) = (CGB) -> A = B))
39		opreq2 3975	. 2 |- (A = B -> (CGA) = (CGB))
40	38, 39	impbid1 519	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 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:  grp2inv 8074  grplactf1o 8094  ringlcan 8154  vclcan 8180  nvlcan 8241

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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