Theorem grprcan 8063

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 1475	. . . . . . . 8 |- (Id` G) = (Id` G)
3	1, 2	grpidinv2 8060	. . . . . . 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		pm3.27 323	. . . . . . . . 9 |- (((yGC) = (Id` G) /\ (CGy) = (Id` G)) -> (CGy) = (Id` G))
5	4	r19.22si 1734	. . . . . . . 8 |- (E.y e. X ((yGC) = (Id` G) /\ (CGy) = (Id` G)) -> E.y e. X (CGy) = (Id` G))
6	5	adantl 388	. . . . . . 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 10	. . . . . 6 |- ((G e. Grp /\ C e. X) -> E.y e. X (CGy) = (Id` G))
8	7	ad2ant2rl 411	. . . . 5 |- (((G e. Grp /\ A e. X) /\ (B e. X /\ C e. X)) -> E.y e. X (CGy) = (Id` G))
9		opreq1 3968	. . . . . . . . . . . 12 |- ((AGC) = (BGC) -> ((AGC)Gy) = ((BGC)Gy))
10	9	ad2antll 407	. . . . . . . . . . 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 8047	. . . . . . . . . . . . . . 15 |- ((G e. Grp /\ (A e. X /\ C e. X /\ y e. X)) -> ((AGC)Gy) = (AG(CGy)))
12	11	3exp2 851	. . . . . . . . . . . . . 14 |- (G e. Grp -> (A e. X -> (C e. X -> (y e. X -> ((AGC)Gy) = (AG(CGy))))))
13	12	imp41 368	. . . . . . . . . . . . 13 |- ((((G e. Grp /\ A e. X) /\ C e. X) /\ y e. X) -> ((AGC)Gy) = (AG(CGy)))
14	13	adantlrl 398	. . . . . . . . . . . 12 |- ((((G e. Grp /\ A e. X) /\ (B e. X /\ C e. X)) /\ y e. X) -> ((AGC)Gy) = (AG(CGy)))
15	14	adantrr 395	. . . . . . . . . . 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 8047	. . . . . . . . . . . . . . 15 |- ((G e. Grp /\ (B e. X /\ C e. X /\ y e. X)) -> ((BGC)Gy) = (BG(CGy)))
17	16	3exp2 851	. . . . . . . . . . . . . 14 |- (G e. Grp -> (B e. X -> (C e. X -> (y e. X -> ((BGC)Gy) = (BG(CGy))))))
18	17	imp42 369	. . . . . . . . . . . . 13 |- (((G e. Grp /\ (B e. X /\ C e. X)) /\ y e. X) -> ((BGC)Gy) = (BG(CGy)))
19	18	adantllr 397	. . . . . . . . . . . 12 |- ((((G e. Grp /\ A e. X) /\ (B e. X /\ C e. X)) /\ y e. X) -> ((BGC)Gy) = (BG(CGy)))
20	19	adantrr 395	. . . . . . . . . . 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 1515	. . . . . . . . . 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 402	. . . . . . . . 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 3969	. . . . . . . . . . 11 |- ((CGy) = (Id` G) -> (AG(CGy)) = (AG(Id` G)))
24	23	ad2antrl 406	. . . . . . . . . 10 |- ((y e. X /\ ((CGy) = (Id` G) /\ (AGC) = (BGC))) -> (AG(CGy)) = (AG(Id` G)))
25	24	adantl 388	. . . . . . . . 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 3969	. . . . . . . . . . 11 |- ((CGy) = (Id` G) -> (BG(CGy)) = (BG(Id` G)))
27	26	ad2antrl 406	. . . . . . . . . 10 |- ((y e. X /\ ((CGy) = (Id` G) /\ (AGC) = (BGC))) -> (BG(CGy)) = (BG(Id` G)))
28	27	adantl 388	. . . . . . . . 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 1515	. . . . . . . 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 8062	. . . . . . . . 9 |- ((G e. Grp /\ A e. X) -> (AG(Id` G)) = A)
31	30	ad2antrr 404	. . . . . . . 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 8062	. . . . . . . . . 10 |- ((G e. Grp /\ B e. X) -> (BG(Id` G)) = B)
33	32	ad2ant2r 409	. . . . . . . . 9 |- (((G e. Grp /\ A e. X) /\ (B e. X /\ C e. X)) -> (BG(Id` G)) = B)
34	33	adantr 389	. . . . . . . 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 1515	. . . . . . 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 386	. . . . . 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 1746	. . . . 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 26	. . . 4 |- (((G e. Grp /\ A e. X) /\ (B e. X /\ C e. X)) -> ((AGC) = (BGC) -> A = B))
39		opreq1 3968	. . . 4 |- (A = B -> (AGC) = (BGC))
40	38, 39	impbid1 517	. . 3 |- (((G e. Grp /\ A e. X) /\ (B e. X /\ C e. X)) -> ((AGC) = (BGC) <-> A = B))
41	40	exp43 384	. 2 |- (G e. Grp -> (A e. X -> (B e. X -> (C e. X -> ((AGC) = (BGC) <-> A = B)))))
42	41	3imp2 848	1 |- ((G e. Grp /\ (A e. X /\ B e. X /\ C e. X)) -> ((AGC) = (BGC) <-> A = B))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   /\ w3a 775   = wceq 956   e. wcel 958  E.wrex 1646  ran crn 3171  ` cfv 3182  (class class class)co 3963  Grpcgr 8033  Idcgi 8034

This theorem is referenced by:  grpinveu 8064  grpid 8065  ringrcan 8157  vcrcan 8183  nvrcan 8244

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-sep 2703  ax-pow 2742  ax-pr 2779  ax-un 2866

This theorem depends on definitions:  df-bi 147  df-or 224