Theorem grporcan 23720

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 2443	. . . . . . . 8 |- (GId ` G ) = (GId ` G )
3	1, 2	grpoidinv2 23717	. . . . . . 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 2835	. . . . . . . 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 6110	. . . . . . . . . . . 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 23702	. . . . . . . . . . . . . 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 1209	. . . . . . . . . . . . 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 23702	. . . . . . . . . . . . . . 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 1205	. . . . . . . . . . . . . 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 2483	. . . . . . . . . 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 6111	. . . . . . . . . . 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 6111	. . . . . . . . . . 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 2483	. . . . . . . 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 23719	. . . . . . . . 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 23719	. . . . . . . . . 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 2483	. . . . . . 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 2852	. . . . 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 6110	. . . 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 1202	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 965   = wceq 1369   e. wcel 1756   E.wrex 2728   ran crn 4853   ` cfv 5430  (class class class)co 6103   GrpOpcgr 23685  GIdcgi 23686

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4425  ax-nul 4433  ax-pr 4543  ax-un 6384

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2620  df-ral 2732  df-rex 2733  df-reu 2734  df-rab 2736  df-v 2986  df-sbc 3199  df-csb 3301  df-dif 3343  df-un 3345  df-in 3347  df-ss 3354  df-nul 3650  df-if 3804  df-sn 3890  df-pr 3892  df-op 3896  df-uni 4104  df-iun 4185  df-br 4305  df-opab 4363  df-mpt 4364  df-id 4648  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-iota 5393  df-fun 5432  df-fn 5433  df-f 5434  df-fo 5436  df-fv 5438  df-riota 6064  df-ov 6106  df-grpo 23690  df-gid 23691

This theorem is referenced by:  grpoinveu  23721  grpoid  23722  grpodiveq  23755  rngorcan  23895  rngorz  23901  vcrcan  23954  nvrcan  24015  ghomdiv  28761