Theorem grporcan 25088

Description: Right cancellation law for groups. (Contributed by NM, 26-Oct-2006.) (New usage is discouraged.)

Hypothesis

Ref	Expression
grprcan.1	|- X = ran G

Assertion

Ref	Expression
grporcan	|- ( ( G e. GrpOp /\ ( A e. X /\ B e. X /\ C e. X ) ) -> ( ( A G C ) = ( B G C ) <-> A = B ) )

Proof of Theorem grporcan

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		grprcan.1	. . . . . . . 8 |- X = ran G
2		eqid 2441	. . . . . . . 8 |- (GId ` G ) = (GId ` G )
3	1, 2	grpoidinv2 25085	. . . . . . 7 |- ( ( G e. GrpOp /\ C e. X ) -> ( ( ( (GId ` G ) G C ) = C /\ ( C G (GId ` G ) ) = C ) /\ E. y e. X ( ( y G C ) = (GId ` G ) /\ ( C G y ) = (GId ` G ) ) ) )
4		simpr 461	. . . . . . . . 9 |- ( ( ( y G C ) = (GId ` G ) /\ ( C G y ) = (GId ` G ) ) -> ( C G y ) = (GId ` G ) )
5	4	reximi 2909	. . . . . . . 8 |- ( E. y e. X ( ( y G C ) = (GId ` G ) /\ ( C G y ) = (GId ` G ) ) -> E. y e. X ( C G y ) = (GId ` G ) )
6	5	adantl 466	. . . . . . 7 |- ( ( ( ( (GId ` G ) G C ) = C /\ ( C G (GId ` G ) ) = C ) /\ E. y e. X ( ( y G C ) = (GId ` G ) /\ ( C G y ) = (GId ` G ) ) ) -> E. y e. X ( C G y ) = (GId ` G ) )
7	3, 6	syl 16	. . . . . 6 |- ( ( G e. GrpOp /\ C e. X ) -> E. y e. X ( C G y ) = (GId ` G ) )
8	7	ad2ant2rl 748	. . . . 5 |- ( ( ( G e. GrpOp /\ A e. X ) /\ ( B e. X /\ C e. X ) ) -> E. y e. X ( C G y ) = (GId ` G ) )
9		oveq1 6284	. . . . . . . . . . . 12 |- ( ( A G C ) = ( B G C ) -> ( ( A G C ) G y ) = ( ( B G C ) G y ) )
10	9	ad2antll 728	. . . . . . . . . . 11 |- ( ( ( ( G e. GrpOp /\ A e. X ) /\ ( B e. X /\ C e. X ) ) /\ ( y e. X /\ ( A G C ) = ( B G C ) ) ) -> ( ( A G C ) G y ) = ( ( B G C ) G y ) )
11	1	grpoass 25070	. . . . . . . . . . . . . 14 |- ( ( G e. GrpOp /\ ( A e. X /\ C e. X /\ y e. X ) ) -> ( ( A G C ) G y ) = ( A G ( C G y ) ) )
12	11	3anassrs 1217	. . . . . . . . . . . . 13 |- ( ( ( ( G e. GrpOp /\ A e. X ) /\ C e. X ) /\ y e. X ) -> ( ( A G C ) G y ) = ( A G ( C G y ) ) )
13	12	adantlrl 719	. . . . . . . . . . . 12 |- ( ( ( ( G e. GrpOp /\ A e. X ) /\ ( B e. X /\ C e. X ) ) /\ y e. X ) -> ( ( A G C ) G y ) = ( A G ( C G y ) ) )
14	13	adantrr 716	. . . . . . . . . . 11 |- ( ( ( ( G e. GrpOp /\ A e. X ) /\ ( B e. X /\ C e. X ) ) /\ ( y e. X /\ ( A G C ) = ( B G C ) ) ) -> ( ( A G C ) G y ) = ( A G ( C G y ) ) )
15	1	grpoass 25070	. . . . . . . . . . . . . . 15 |- ( ( G e. GrpOp /\ ( B e. X /\ C e. X /\ y e. X ) ) -> ( ( B G C ) G y ) = ( B G ( C G y ) ) )
16	15	3exp2 1213	. . . . . . . . . . . . . 14 |- ( G e. GrpOp -> ( B e. X -> ( C e. X -> ( y e. X -> ( ( B G C ) G y ) = ( B G ( C G y ) ) ) ) ) )
17	16	imp42 594	. . . . . . . . . . . . 13 |- ( ( ( G e. GrpOp /\ ( B e. X /\ C e. X ) ) /\ y e. X ) -> ( ( B G C ) G y ) = ( B G ( C G y ) ) )
18	17	adantllr 718	. . . . . . . . . . . 12 |- ( ( ( ( G e. GrpOp /\ A e. X ) /\ ( B e. X /\ C e. X ) ) /\ y e. X ) -> ( ( B G C ) G y ) = ( B G ( C G y ) ) )
19	18	adantrr 716	. . . . . . . . . . 11 |- ( ( ( ( G e. GrpOp /\ A e. X ) /\ ( B e. X /\ C e. X ) ) /\ ( y e. X /\ ( A G C ) = ( B G C ) ) ) -> ( ( B G C ) G y ) = ( B G ( C G y ) ) )
20	10, 14, 19	3eqtr3d 2490	. . . . . . . . . 10 |- ( ( ( ( G e. GrpOp /\ A e. X ) /\ ( B e. X /\ C e. X ) ) /\ ( y e. X /\ ( A G C ) = ( B G C ) ) ) -> ( A G ( C G y ) ) = ( B G ( C G y ) ) )
21	20	adantrrl 723	. . . . . . . . 9 |- ( ( ( ( G e. GrpOp /\ A e. X ) /\ ( B e. X /\ C e. X ) ) /\ ( y e. X /\ ( ( C G y ) = (GId ` G ) /\ ( A G C ) = ( B G C ) ) ) ) -> ( A G ( C G y ) ) = ( B G ( C G y ) ) )
22		oveq2 6285	. . . . . . . . . . 11 |- ( ( C G y ) = (GId ` G ) -> ( A G ( C G y ) ) = ( A G (GId ` G ) ) )
23	22	ad2antrl 727	. . . . . . . . . 10 |- ( ( y e. X /\ ( ( C G y ) = (GId ` G ) /\ ( A G C ) = ( B G C ) ) ) -> ( A G ( C G y ) ) = ( A G (GId ` G ) ) )
24	23	adantl 466	. . . . . . . . 9 |- ( ( ( ( G e. GrpOp /\ A e. X ) /\ ( B e. X /\ C e. X ) ) /\ ( y e. X /\ ( ( C G y ) = (GId ` G ) /\ ( A G C ) = ( B G C ) ) ) ) -> ( A G ( C G y ) ) = ( A G (GId ` G ) ) )
25		oveq2 6285	. . . . . . . . . . 11 |- ( ( C G y ) = (GId ` G ) -> ( B G ( C G y ) ) = ( B G (GId ` G ) ) )
26	25	ad2antrl 727	. . . . . . . . . 10 |- ( ( y e. X /\ ( ( C G y ) = (GId ` G ) /\ ( A G C ) = ( B G C ) ) ) -> ( B G ( C G y ) ) = ( B G (GId ` G ) ) )
27	26	adantl 466	. . . . . . . . 9 |- ( ( ( ( G e. GrpOp /\ A e. X ) /\ ( B e. X /\ C e. X ) ) /\ ( y e. X /\ ( ( C G y ) = (GId ` G ) /\ ( A G C ) = ( B G C ) ) ) ) -> ( B G ( C G y ) ) = ( B G (GId ` G ) ) )
28	21, 24, 27	3eqtr3d 2490	. . . . . . . 8 |- ( ( ( ( G e. GrpOp /\ A e. X ) /\ ( B e. X /\ C e. X ) ) /\ ( y e. X /\ ( ( C G y ) = (GId ` G ) /\ ( A G C ) = ( B G C ) ) ) ) -> ( A G (GId ` G ) ) = ( B G (GId ` G ) ) )
29	1, 2	grporid 25087	. . . . . . . . 9 |- ( ( G e. GrpOp /\ A e. X ) -> ( A G (GId ` G ) ) = A )
30	29	ad2antrr 725	. . . . . . . 8 |- ( ( ( ( G e. GrpOp /\ A e. X ) /\ ( B e. X /\ C e. X ) ) /\ ( y e. X /\ ( ( C G y ) = (GId ` G ) /\ ( A G C ) = ( B G C ) ) ) ) -> ( A G (GId ` G ) ) = A )
31	1, 2	grporid 25087	. . . . . . . . . 10 |- ( ( G e. GrpOp /\ B e. X ) -> ( B G (GId ` G ) ) = B )
32	31	ad2ant2r 746	. . . . . . . . 9 |- ( ( ( G e. GrpOp /\ A e. X ) /\ ( B e. X /\ C e. X ) ) -> ( B G (GId ` G ) ) = B )
33	32	adantr 465	. . . . . . . 8 |- ( ( ( ( G e. GrpOp /\ A e. X ) /\ ( B e. X /\ C e. X ) ) /\ ( y e. X /\ ( ( C G y ) = (GId ` G ) /\ ( A G C ) = ( B G C ) ) ) ) -> ( B G (GId ` G ) ) = B )
34	28, 30, 33	3eqtr3d 2490	. . . . . . 7 |- ( ( ( ( G e. GrpOp /\ A e. X ) /\ ( B e. X /\ C e. X ) ) /\ ( y e. X /\ ( ( C G y ) = (GId ` G ) /\ ( A G C ) = ( B G C ) ) ) ) -> A = B )
35	34	exp45 614	. . . . . 6 |- ( ( ( G e. GrpOp /\ A e. X ) /\ ( B e. X /\ C e. X ) ) -> ( y e. X -> ( ( C G y ) = (GId ` G ) -> ( ( A G C ) = ( B G C ) -> A = B ) ) ) )
36	35	rexlimdv 2931	. . . . 5 |- ( ( ( G e. GrpOp /\ A e. X ) /\ ( B e. X /\ C e. X ) ) -> ( E. y e. X ( C G y ) = (GId ` G ) -> ( ( A G C ) = ( B G C ) -> A = B ) ) )
37	8, 36	mpd 15	. . . 4 |- ( ( ( G e. GrpOp /\ A e. X ) /\ ( B e. X /\ C e. X ) ) -> ( ( A G C ) = ( B G C ) -> A = B ) )
38		oveq1 6284	. . . 4 |- ( A = B -> ( A G C ) = ( B G C ) )
39	37, 38	impbid1 203	. . 3 |- ( ( ( G e. GrpOp /\ A e. X ) /\ ( B e. X /\ C e. X ) ) -> ( ( A G C ) = ( B G C ) <-> A = B ) )
40	39	exp43 612	. 2 |- ( G e. GrpOp -> ( A e. X -> ( B e. X -> ( C e. X -> ( ( A G C ) = ( B G C ) <-> A = B ) ) ) ) )
41	40	3imp2 1210	1 |- ( ( G e. GrpOp /\ ( A e. X /\ B e. X /\ C e. X ) ) -> ( ( A G C ) = ( B G C ) <-> A = B ) )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   /\ w3a 972   = wceq 1381   e. wcel 1802   E.wrex 2792   ran crn 4986   ` cfv 5574  (class class class)co 6277   GrpOpcgr 25053  GIdcgi 25054

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-sep 4554  ax-nul 4562  ax-pr 4672  ax-un 6573

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-ral 2796  df-rex 2797  df-reu 2798  df-rab 2800  df-v 3095  df-sbc 3312  df-csb 3418  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-nul 3768  df-if 3923  df-sn 4011  df-pr 4013  df-op 4017  df-uni 4231  df-iun 4313  df-br 4434  df-opab 4492  df-mpt 4493  df-id 4781  df-xp 4991  df-rel 4992  df-cnv 4993  df-co 4994  df-dm 4995  df-rn 4996  df-iota 5537  df-fun 5576  df-fn 5577  df-f 5578  df-fo 5580  df-fv 5582  df-riota 6238  df-ov 6280  df-grpo 25058  df-gid 25059

This theorem is referenced by:  grpoinveu  25089  grpoid  25090  grpodiveq  25123  rngorcan  25263  rngorz  25269  vcrcan  25322  nvrcan  25383  ghomdiv  30314