Theorem grporcan 27372

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 2622	. . . . . . . 8 |- (GId ` G ) = (GId ` G )
3	1, 2	grpoidinv2 27369	. . . . . . 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 477	. . . . . . . . 9 |- ( ( ( y G C ) = (GId ` G ) /\ ( C G y ) = (GId ` G ) ) -> ( C G y ) = (GId ` G ) )
5	4	reximi 3011	. . . . . . . 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 482	. . . . . . 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 17	. . . . . 6 |- ( ( G e. GrpOp /\ C e. X ) -> E. y e. X ( C G y ) = (GId ` G ) )
8	7	ad2ant2rl 785	. . . . 5 |- ( ( ( G e. GrpOp /\ A e. X ) /\ ( B e. X /\ C e. X ) ) -> E. y e. X ( C G y ) = (GId ` G ) )
9		oveq1 6657	. . . . . . . . . . . 12 |- ( ( A G C ) = ( B G C ) -> ( ( A G C ) G y ) = ( ( B G C ) G y ) )
10	9	ad2antll 765	. . . . . . . . . . 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 27357	. . . . . . . . . . . . . 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 1290	. . . . . . . . . . . . 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 756	. . . . . . . . . . . 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 753	. . . . . . . . . . 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 27357	. . . . . . . . . . . . . . 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 1285	. . . . . . . . . . . . . 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 620	. . . . . . . . . . . . 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 755	. . . . . . . . . . . 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 753	. . . . . . . . . . 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 2664	. . . . . . . . . 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 760	. . . . . . . . 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 6658	. . . . . . . . . . 11 |- ( ( C G y ) = (GId ` G ) -> ( A G ( C G y ) ) = ( A G (GId ` G ) ) )
23	22	ad2antrl 764	. . . . . . . . . 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 482	. . . . . . . . 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 6658	. . . . . . . . . . 11 |- ( ( C G y ) = (GId ` G ) -> ( B G ( C G y ) ) = ( B G (GId ` G ) ) )
26	25	ad2antrl 764	. . . . . . . . . 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 482	. . . . . . . . 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 2664	. . . . . . . 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 27371	. . . . . . . . 9 |- ( ( G e. GrpOp /\ A e. X ) -> ( A G (GId ` G ) ) = A )
30	29	ad2antrr 762	. . . . . . . 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 27371	. . . . . . . . . 10 |- ( ( G e. GrpOp /\ B e. X ) -> ( B G (GId ` G ) ) = B )
32	31	ad2ant2r 783	. . . . . . . . 9 |- ( ( ( G e. GrpOp /\ A e. X ) /\ ( B e. X /\ C e. X ) ) -> ( B G (GId ` G ) ) = B )
33	32	adantr 481	. . . . . . . 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 2664	. . . . . . 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 642	. . . . . 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 3030	. . . . 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 6657	. . . 4 |- ( A = B -> ( A G C ) = ( B G C ) )
39	37, 38	impbid1 215	. . 3 |- ( ( ( G e. GrpOp /\ A e. X ) /\ ( B e. X /\ C e. X ) ) -> ( ( A G C ) = ( B G C ) <-> A = B ) )
40	39	exp43 640	. 2 |- ( G e. GrpOp -> ( A e. X -> ( B e. X -> ( C e. X -> ( ( A G C ) = ( B G C ) <-> A = B ) ) ) ) )
41	40	3imp2 1282	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 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   E.wrex 2913   ran crn 5115   ` cfv 5888  (class class class)co 6650   GrpOpcgr 27343  GIdcgi 27344

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fo 5894  df-fv 5896  df-riota 6611  df-ov 6653  df-grpo 27347  df-gid 27348

This theorem is referenced by:  grpoinveu  27373  grpoid  27374  nvrcan  27479  ghomdiv  33691  rngorcan  33716  rngorz  33722