Theorem grprcan 9347

Description: Right cancellation law for groups.

Hypothesis

Ref	Expression
grprcan.1	|- X = ran G

Assertion

Ref	Expression
grprcan	|- ((G e. Grp /\ (A e. X /\ B e. X /\ C e. X)) -> ((AGC) = (BGC) <-> A = B))

Proof of Theorem grprcan

Step	Hyp	Ref	Expression
1		grprcan.1	. . . . . . . 8 |- X = ran G
2		eqid 1884	. . . . . . . 8 |- (Id` G) = (Id` G)
3	1, 2	grpidinv2 9344	. . . . . . 7 |- ((G e. Grp /\ C e. X) -> ((((Id` G)GC) = C /\ (CG(Id` G)) = C) /\ E.y e. X ((yGC) = (Id` G) /\ (CGy) = (Id` G))))
4		simpr 350	. . . . . . . . 9 |- (((yGC) = (Id` G) /\ (CGy) = (Id` G)) -> (CGy) = (Id` G))
5	4	reximi 2198	. . . . . . . 8 |- (E.y e. X ((yGC) = (Id` G) /\ (CGy) = (Id` G)) -> E.y e. X (CGy) = (Id` G))
6	5	adantl 424	. . . . . . 7 |- (((((Id` G)GC) = C /\ (CG(Id` G)) = C) /\ E.y e. X ((yGC) = (Id` G) /\ (CGy) = (Id` G))) -> E.y e. X (CGy) = (Id` G))
7	3, 6	syl 12	. . . . . 6 |- ((G e. Grp /\ C e. X) -> E.y e. X (CGy) = (Id` G))
8	7	ad2ant2rl 447	. . . . 5 |- (((G e. Grp /\ A e. X) /\ (B e. X /\ C e. X)) -> E.y e. X (CGy) = (Id` G))
9		opreq1 4889	. . . . . . . . . . . 12 |- ((AGC) = (BGC) -> ((AGC)Gy) = ((BGC)Gy))
10	9	ad2antll 443	. . . . . . . . . . 11 |- ((((G e. Grp /\ A e. X) /\ (B e. X /\ C e. X)) /\ (y e. X /\ (AGC) = (BGC))) -> ((AGC)Gy) = ((BGC)Gy))
11	1	grpass 9327	. . . . . . . . . . . . . . 15 |- ((G e. Grp /\ (A e. X /\ C e. X /\ y e. X)) -> ((AGC)Gy) = (AG(CGy)))
12	11	3exp2 1086	. . . . . . . . . . . . . 14 |- (G e. Grp -> (A e. X -> (C e. X -> (y e. X -> ((AGC)Gy) = (AG(CGy))))))
13	12	imp41 395	. . . . . . . . . . . . 13 |- ((((G e. Grp /\ A e. X) /\ C e. X) /\ y e. X) -> ((AGC)Gy) = (AG(CGy)))
14	13	adantlrl 434	. . . . . . . . . . . 12 |- ((((G e. Grp /\ A e. X) /\ (B e. X /\ C e. X)) /\ y e. X) -> ((AGC)Gy) = (AG(CGy)))
15	14	adantrr 431	. . . . . . . . . . 11 |- ((((G e. Grp /\ A e. X) /\ (B e. X /\ C e. X)) /\ (y e. X /\ (AGC) = (BGC))) -> ((AGC)Gy) = (AG(CGy)))
16	1	grpass 9327	. . . . . . . . . . . . . . 15 |- ((G e. Grp /\ (B e. X /\ C e. X /\ y e. X)) -> ((BGC)Gy) = (BG(CGy)))
17	16	3exp2 1086	. . . . . . . . . . . . . 14 |- (G e. Grp -> (B e. X -> (C e. X -> (y e. X -> ((BGC)Gy) = (BG(CGy))))))
18	17	imp42 396	. . . . . . . . . . . . 13 |- (((G e. Grp /\ (B e. X /\ C e. X)) /\ y e. X) -> ((BGC)Gy) = (BG(CGy)))
19	18	adantllr 433	. . . . . . . . . . . 12 |- ((((G e. Grp /\ A e. X) /\ (B e. X /\ C e. X)) /\ y e. X) -> ((BGC)Gy) = (BG(CGy)))
20	19	adantrr 431	. . . . . . . . . . 11 |- ((((G e. Grp /\ A e. X) /\ (B e. X /\ C e. X)) /\ (y e. X /\ (AGC) = (BGC))) -> ((BGC)Gy) = (BG(CGy)))
21	10, 15, 20	3eqtr3d 1934	. . . . . . . . . 10 |- ((((G e. Grp /\ A e. X) /\ (B e. X /\ C e. X)) /\ (y e. X /\ (AGC) = (BGC))) -> (AG(CGy)) = (BG(CGy)))
22	21	adantrrl 438	. . . . . . . . 9 |- ((((G e. Grp /\ A e. X) /\ (B e. X /\ C e. X)) /\ (y e. X /\ ((CGy) = (Id` G) /\ (AGC) = (BGC)))) -> (AG(CGy)) = (BG(CGy)))
23		opreq2 4890	. . . . . . . . . . 11 |- ((CGy) = (Id` G) -> (AG(CGy)) = (AG(Id` G)))
24	23	ad2antrl 442	. . . . . . . . . 10 |- ((y e. X /\ ((CGy) = (Id` G) /\ (AGC) = (BGC))) -> (AG(CGy)) = (AG(Id` G)))
25	24	adantl 424	. . . . . . . . 9 |- ((((G e. Grp /\ A e. X) /\ (B e. X /\ C e. X)) /\ (y e. X /\ ((CGy) = (Id` G) /\ (AGC) = (BGC)))) -> (AG(CGy)) = (AG(Id` G)))
26		opreq2 4890	. . . . . . . . . . 11 |- ((CGy) = (Id` G) -> (BG(CGy)) = (BG(Id` G)))
27	26	ad2antrl 442	. . . . . . . . . 10 |- ((y e. X /\ ((CGy) = (Id` G) /\ (AGC) = (BGC))) -> (BG(CGy)) = (BG(Id` G)))
28	27	adantl 424	. . . . . . . . 9 |- ((((G e. Grp /\ A e. X) /\ (B e. X /\ C e. X)) /\ (y e. X /\ ((CGy) = (Id` G) /\ (AGC) = (BGC)))) -> (BG(CGy)) = (BG(Id` G)))
29	22, 25, 28	3eqtr3d 1934	. . . . . . . 8 |- ((((G e. Grp /\ A e. X) /\ (B e. X /\ C e. X)) /\ (y e. X /\ ((CGy) = (Id` G) /\ (AGC) = (BGC)))) -> (AG(Id` G)) = (BG(Id` G)))
30	1, 2	grprid 9346	. . . . . . . . 9 |- ((G e. Grp /\ A e. X) -> (AG(Id` G)) = A)
31	30	ad2antrr 440	. . . . . . . 8 |- ((((G e. Grp /\ A e. X) /\ (B e. X /\ C e. X)) /\ (y e. X /\ ((CGy) = (Id` G) /\ (AGC) = (BGC)))) -> (AG(Id` G)) = A)
32	1, 2	grprid 9346	. . . . . . . . . 10 |- ((G e. Grp /\ B e. X) -> (BG(Id` G)) = B)
33	32	ad2ant2r 445	. . . . . . . . 9 |- (((G e. Grp /\ A e. X) /\ (B e. X /\ C e. X)) -> (BG(Id` G)) = B)
34	33	adantr 425	. . . . . . . 8 |- ((((G e. Grp /\ A e. X) /\ (B e. X /\ C e. X)) /\ (y e. X /\ ((CGy) = (Id` G) /\ (AGC) = (BGC)))) -> (BG(Id` G)) = B)
35	29, 31, 34	3eqtr3d 1934	. . . . . . 7 |- ((((G e. Grp /\ A e. X) /\ (B e. X /\ C e. X)) /\ (y e. X /\ ((CGy) = (Id` G) /\ (AGC) = (BGC)))) -> A = B)
36	35	exp45 417	. . . . . 6 |- (((G e. Grp /\ A e. X) /\ (B e. X /\ C e. X)) -> (y e. X -> ((CGy) = (Id` G) -> ((AGC) = (BGC) -> A = B))))
37	36	r19.23adv 2215	. . . . 5 |- (((G e. Grp /\ A e. X) /\ (B e. X /\ C e. X)) -> (E.y e. X (CGy) = (Id` G) -> ((AGC) = (BGC) -> A = B)))
38	8, 37	mpd 29	. . . 4 |- (((G e. Grp /\ A e. X) /\ (B e. X /\ C e. X)) -> ((AGC) = (BGC) -> A = B))
39		opreq1 4889	. . . 4 |- (A = B -> (AGC) = (BGC))
40	38, 39	impbid1 575	. . 3 |- (((G e. Grp /\ A e. X) /\ (B e. X /\ C e. X)) -> ((AGC) = (BGC) <-> A = B))
41	40	exp43 415	. 2 |- (G e. Grp -> (A e. X -> (B e. X -> (C e. X -> ((AGC) = (BGC) <-> A = B)))))
42	41	3imp2 1083	1 |- ((G e. Grp /\ (A e. X /\ B e. X /\ C e. X)) -> ((AGC) = (BGC) <-> A = B))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300  E.wrex 2106  ran crn 3987  ` cfv 3998  (class class class)co 4884  Grpcgr 9311  Idcgi 9312

This theorem is referenced by:  grpinveu 9348  grpid 9349  ringrcan 9482  ringrz 9488  vcrcan 9515  nvrcan 9576  grpdrcan 14738  trooo 14758  vecrcan 14818  ghomdiv 16041

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-reu 2111  df-rab 2112  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-if 2983  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-id 3586  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-f1 4011  df-fo 4012  df-f1o 4013  df-fv 4014  df-opr 4886  df-grp 9316  df-gid 9317

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