Theorem grpolcan 27384

Description: Left cancellation law for groups. (Contributed by NM, 27-Oct-2006.) (New usage is discouraged.)

Hypothesis

Ref	Expression
grplcan.1	|- X = ran G

Assertion

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

Proof of Theorem grpolcan

Step	Hyp	Ref	Expression
1		oveq2 6658	. . . . . 6 |- ( ( C G A ) = ( C G B ) -> ( ( ( inv ` G ) ` C ) G ( C G A ) ) = ( ( ( inv ` G ) ` C ) G ( C G B ) ) )
2	1	adantl 482	. . . . 5 |- ( ( ( ( G e. GrpOp /\ A e. X ) /\ ( B e. X /\ C e. X ) ) /\ ( C G A ) = ( C G B ) ) -> ( ( ( inv ` G ) ` C ) G ( C G A ) ) = ( ( ( inv ` G ) ` C ) G ( C G B ) ) )
3		grplcan.1	. . . . . . . . . . 11 |- X = ran G
4		eqid 2622	. . . . . . . . . . 11 |- (GId ` G ) = (GId ` G )
5		eqid 2622	. . . . . . . . . . 11 |- ( inv ` G ) = ( inv ` G )
6	3, 4, 5	grpolinv 27380	. . . . . . . . . 10 |- ( ( G e. GrpOp /\ C e. X ) -> ( ( ( inv ` G ) ` C ) G C ) = (GId ` G ) )
7	6	adantlr 751	. . . . . . . . 9 |- ( ( ( G e. GrpOp /\ A e. X ) /\ C e. X ) -> ( ( ( inv ` G ) ` C ) G C ) = (GId ` G ) )
8	7	oveq1d 6665	. . . . . . . 8 |- ( ( ( G e. GrpOp /\ A e. X ) /\ C e. X ) -> ( ( ( ( inv ` G ) ` C ) G C ) G A ) = ( (GId ` G ) G A ) )
9	3, 5	grpoinvcl 27378	. . . . . . . . . . . 12 |- ( ( G e. GrpOp /\ C e. X ) -> ( ( inv ` G ) ` C ) e. X )
10	9	adantrl 752	. . . . . . . . . . 11 |- ( ( G e. GrpOp /\ ( A e. X /\ C e. X ) ) -> ( ( inv ` G ) ` C ) e. X )
11		simprr 796	. . . . . . . . . . 11 |- ( ( G e. GrpOp /\ ( A e. X /\ C e. X ) ) -> C e. X )
12		simprl 794	. . . . . . . . . . 11 |- ( ( G e. GrpOp /\ ( A e. X /\ C e. X ) ) -> A e. X )
13	10, 11, 12	3jca 1242	. . . . . . . . . 10 |- ( ( G e. GrpOp /\ ( A e. X /\ C e. X ) ) -> ( ( ( inv ` G ) ` C ) e. X /\ C e. X /\ A e. X ) )
14	3	grpoass 27357	. . . . . . . . . 10 |- ( ( G e. GrpOp /\ ( ( ( inv ` G ) ` C ) e. X /\ C e. X /\ A e. X ) ) -> ( ( ( ( inv ` G ) ` C ) G C ) G A ) = ( ( ( inv ` G ) ` C ) G ( C G A ) ) )
15	13, 14	syldan 487	. . . . . . . . 9 |- ( ( G e. GrpOp /\ ( A e. X /\ C e. X ) ) -> ( ( ( ( inv ` G ) ` C ) G C ) G A ) = ( ( ( inv ` G ) ` C ) G ( C G A ) ) )
16	15	anassrs 680	. . . . . . . 8 |- ( ( ( G e. GrpOp /\ A e. X ) /\ C e. X ) -> ( ( ( ( inv ` G ) ` C ) G C ) G A ) = ( ( ( inv ` G ) ` C ) G ( C G A ) ) )
17	3, 4	grpolid 27370	. . . . . . . . 9 |- ( ( G e. GrpOp /\ A e. X ) -> ( (GId ` G ) G A ) = A )
18	17	adantr 481	. . . . . . . 8 |- ( ( ( G e. GrpOp /\ A e. X ) /\ C e. X ) -> ( (GId ` G ) G A ) = A )
19	8, 16, 18	3eqtr3d 2664	. . . . . . 7 |- ( ( ( G e. GrpOp /\ A e. X ) /\ C e. X ) -> ( ( ( inv ` G ) ` C ) G ( C G A ) ) = A )
20	19	adantrl 752	. . . . . 6 |- ( ( ( G e. GrpOp /\ A e. X ) /\ ( B e. X /\ C e. X ) ) -> ( ( ( inv ` G ) ` C ) G ( C G A ) ) = A )
21	20	adantr 481	. . . . 5 |- ( ( ( ( G e. GrpOp /\ A e. X ) /\ ( B e. X /\ C e. X ) ) /\ ( C G A ) = ( C G B ) ) -> ( ( ( inv ` G ) ` C ) G ( C G A ) ) = A )
22	6	adantrl 752	. . . . . . . . 9 |- ( ( G e. GrpOp /\ ( B e. X /\ C e. X ) ) -> ( ( ( inv ` G ) ` C ) G C ) = (GId ` G ) )
23	22	oveq1d 6665	. . . . . . . 8 |- ( ( G e. GrpOp /\ ( B e. X /\ C e. X ) ) -> ( ( ( ( inv ` G ) ` C ) G C ) G B ) = ( (GId ` G ) G B ) )
24	9	adantrl 752	. . . . . . . . . 10 |- ( ( G e. GrpOp /\ ( B e. X /\ C e. X ) ) -> ( ( inv ` G ) ` C ) e. X )
25		simprr 796	. . . . . . . . . 10 |- ( ( G e. GrpOp /\ ( B e. X /\ C e. X ) ) -> C e. X )
26		simprl 794	. . . . . . . . . 10 |- ( ( G e. GrpOp /\ ( B e. X /\ C e. X ) ) -> B e. X )
27	24, 25, 26	3jca 1242	. . . . . . . . 9 |- ( ( G e. GrpOp /\ ( B e. X /\ C e. X ) ) -> ( ( ( inv ` G ) ` C ) e. X /\ C e. X /\ B e. X ) )
28	3	grpoass 27357	. . . . . . . . 9 |- ( ( G e. GrpOp /\ ( ( ( inv ` G ) ` C ) e. X /\ C e. X /\ B e. X ) ) -> ( ( ( ( inv ` G ) ` C ) G C ) G B ) = ( ( ( inv ` G ) ` C ) G ( C G B ) ) )
29	27, 28	syldan 487	. . . . . . . 8 |- ( ( G e. GrpOp /\ ( B e. X /\ C e. X ) ) -> ( ( ( ( inv ` G ) ` C ) G C ) G B ) = ( ( ( inv ` G ) ` C ) G ( C G B ) ) )
30	3, 4	grpolid 27370	. . . . . . . . 9 |- ( ( G e. GrpOp /\ B e. X ) -> ( (GId ` G ) G B ) = B )
31	30	adantrr 753	. . . . . . . 8 |- ( ( G e. GrpOp /\ ( B e. X /\ C e. X ) ) -> ( (GId ` G ) G B ) = B )
32	23, 29, 31	3eqtr3d 2664	. . . . . . 7 |- ( ( G e. GrpOp /\ ( B e. X /\ C e. X ) ) -> ( ( ( inv ` G ) ` C ) G ( C G B ) ) = B )
33	32	adantlr 751	. . . . . 6 |- ( ( ( G e. GrpOp /\ A e. X ) /\ ( B e. X /\ C e. X ) ) -> ( ( ( inv ` G ) ` C ) G ( C G B ) ) = B )
34	33	adantr 481	. . . . 5 |- ( ( ( ( G e. GrpOp /\ A e. X ) /\ ( B e. X /\ C e. X ) ) /\ ( C G A ) = ( C G B ) ) -> ( ( ( inv ` G ) ` C ) G ( C G B ) ) = B )
35	2, 21, 34	3eqtr3d 2664	. . . 4 |- ( ( ( ( G e. GrpOp /\ A e. X ) /\ ( B e. X /\ C e. X ) ) /\ ( C G A ) = ( C G B ) ) -> A = B )
36	35	exp53 647	. . 3 |- ( G e. GrpOp -> ( A e. X -> ( B e. X -> ( C e. X -> ( ( C G A ) = ( C G B ) -> A = B ) ) ) ) )
37	36	3imp2 1282	. 2 |- ( ( G e. GrpOp /\ ( A e. X /\ B e. X /\ C e. X ) ) -> ( ( C G A ) = ( C G B ) -> A = B ) )
38		oveq2 6658	. 2 |- ( A = B -> ( C G A ) = ( C G B ) )
39	37, 38	impbid1 215	1 |- ( ( G e. GrpOp /\ ( A e. X /\ B e. X /\ C e. X ) ) -> ( ( C G A ) = ( C G B ) <-> A = B ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   ran crn 5115   ` cfv 5888  (class class class)co 6650   GrpOpcgr 27343  GIdcgi 27344   invcgn 27345

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-rep 4771  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-ne 2795  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-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-grpo 27347  df-gid 27348  df-ginv 27349

This theorem is referenced by:  grpo2inv  27385  vclcan  27426  rngolcan  33717  rngolz  33721