Theorem grporcan 25421

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 2454	. . . . . . . 8 |- (GId ` G ) = (GId ` G )
3	1, 2	grpoidinv2 25418	. . . . . . 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 459	. . . . . . . . 9 |- ( ( ( y G C ) = (GId ` G ) /\ ( C G y ) = (GId ` G ) ) -> ( C G y ) = (GId ` G ) )
5	4	reximi 2922	. . . . . . . 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 464	. . . . . . 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 746	. . . . 5 |- ( ( ( G e. GrpOp /\ A e. X ) /\ ( B e. X /\ C e. X ) ) -> E. y e. X ( C G y ) = (GId ` G ) )
9		oveq1 6277	. . . . . . . . . . . 12 |- ( ( A G C ) = ( B G C ) -> ( ( A G C ) G y ) = ( ( B G C ) G y ) )
10	9	ad2antll 726	. . . . . . . . . . 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 25403	. . . . . . . . . . . . . 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 1216	. . . . . . . . . . . . 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 717	. . . . . . . . . . . 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 714	. . . . . . . . . . 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 25403	. . . . . . . . . . . . . . 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 1212	. . . . . . . . . . . . . 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 592	. . . . . . . . . . . . 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 716	. . . . . . . . . . . 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 714	. . . . . . . . . . 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 2503	. . . . . . . . . 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 721	. . . . . . . . 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 6278	. . . . . . . . . . 11 |- ( ( C G y ) = (GId ` G ) -> ( A G ( C G y ) ) = ( A G (GId ` G ) ) )
23	22	ad2antrl 725	. . . . . . . . . 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 464	. . . . . . . . 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 6278	. . . . . . . . . . 11 |- ( ( C G y ) = (GId ` G ) -> ( B G ( C G y ) ) = ( B G (GId ` G ) ) )
26	25	ad2antrl 725	. . . . . . . . . 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 464	. . . . . . . . 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 2503	. . . . . . . 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 25420	. . . . . . . . 9 |- ( ( G e. GrpOp /\ A e. X ) -> ( A G (GId ` G ) ) = A )
30	29	ad2antrr 723	. . . . . . . 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 25420	. . . . . . . . . 10 |- ( ( G e. GrpOp /\ B e. X ) -> ( B G (GId ` G ) ) = B )
32	31	ad2ant2r 744	. . . . . . . . 9 |- ( ( ( G e. GrpOp /\ A e. X ) /\ ( B e. X /\ C e. X ) ) -> ( B G (GId ` G ) ) = B )
33	32	adantr 463	. . . . . . . 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 2503	. . . . . . 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 612	. . . . . 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 2944	. . . . 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 6277	. . . 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 610	. 2 |- ( G e. GrpOp -> ( A e. X -> ( B e. X -> ( C e. X -> ( ( A G C ) = ( B G C ) <-> A = B ) ) ) ) )
41	40	3imp2 1209	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 367   /\ w3a 971   = wceq 1398   e. wcel 1823   E.wrex 2805   ran crn 4989   ` cfv 5570  (class class class)co 6270   GrpOpcgr 25386  GIdcgi 25387

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pr 4676  ax-un 6565

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-fo 5576  df-fv 5578  df-riota 6232  df-ov 6273  df-grpo 25391  df-gid 25392

This theorem is referenced by:  grpoinveu  25422  grpoid  25423  grpodiveq  25456  rngorcan  25596  rngorz  25602  vcrcan  25655  nvrcan  25716  ghomdiv  30586