Theorem dmncan1 33875

Description: Cancellation law for domains. (Contributed by Jeff Madsen, 6-Jan-2011.)

Hypotheses

Ref	Expression
dmncan.1	|- G = ( 1st ` R )
dmncan.2	|- H = ( 2nd ` R )
dmncan.3	|- X = ran G
dmncan.4	|- Z = (GId ` G )

Assertion

Ref	Expression
dmncan1	|- ( ( ( R e. Dmn /\ ( A e. X /\ B e. X /\ C e. X ) ) /\ A =/= Z ) -> ( ( A H B ) = ( A H C ) -> B = C ) )

Proof of Theorem dmncan1

Step	Hyp	Ref	Expression
1		dmnrngo 33856	. . . . . 6 |- ( R e. Dmn -> R e. RingOps )
2		dmncan.1	. . . . . . 7 |- G = ( 1st ` R )
3		dmncan.2	. . . . . . 7 |- H = ( 2nd ` R )
4		dmncan.3	. . . . . . 7 |- X = ran G
5		eqid 2622	. . . . . . 7 |- ( /g ` G ) = ( /g ` G )
6	2, 3, 4, 5	rngosubdi 33744	. . . . . 6 |- ( ( R e. RingOps /\ ( A e. X /\ B e. X /\ C e. X ) ) -> ( A H ( B ( /g ` G ) C ) ) = ( ( A H B ) ( /g ` G ) ( A H C ) ) )
7	1, 6	sylan 488	. . . . 5 |- ( ( R e. Dmn /\ ( A e. X /\ B e. X /\ C e. X ) ) -> ( A H ( B ( /g ` G ) C ) ) = ( ( A H B ) ( /g ` G ) ( A H C ) ) )
8	7	adantr 481	. . . 4 |- ( ( ( R e. Dmn /\ ( A e. X /\ B e. X /\ C e. X ) ) /\ A =/= Z ) -> ( A H ( B ( /g ` G ) C ) ) = ( ( A H B ) ( /g ` G ) ( A H C ) ) )
9	8	eqeq1d 2624	. . 3 |- ( ( ( R e. Dmn /\ ( A e. X /\ B e. X /\ C e. X ) ) /\ A =/= Z ) -> ( ( A H ( B ( /g ` G ) C ) ) = Z <-> ( ( A H B ) ( /g ` G ) ( A H C ) ) = Z ) )
10	2	rngogrpo 33709	. . . . . . . . . . . 12 |- ( R e. RingOps -> G e. GrpOp )
11	1, 10	syl 17	. . . . . . . . . . 11 |- ( R e. Dmn -> G e. GrpOp )
12	4, 5	grpodivcl 27393	. . . . . . . . . . . 12 |- ( ( G e. GrpOp /\ B e. X /\ C e. X ) -> ( B ( /g ` G ) C ) e. X )
13	12	3expb 1266	. . . . . . . . . . 11 |- ( ( G e. GrpOp /\ ( B e. X /\ C e. X ) ) -> ( B ( /g ` G ) C ) e. X )
14	11, 13	sylan 488	. . . . . . . . . 10 |- ( ( R e. Dmn /\ ( B e. X /\ C e. X ) ) -> ( B ( /g ` G ) C ) e. X )
15	14	adantlr 751	. . . . . . . . 9 |- ( ( ( R e. Dmn /\ A e. X ) /\ ( B e. X /\ C e. X ) ) -> ( B ( /g ` G ) C ) e. X )
16		dmncan.4	. . . . . . . . . . . 12 |- Z = (GId ` G )
17	2, 3, 4, 16	dmnnzd 33874	. . . . . . . . . . 11 |- ( ( R e. Dmn /\ ( A e. X /\ ( B ( /g ` G ) C ) e. X /\ ( A H ( B ( /g ` G ) C ) ) = Z ) ) -> ( A = Z \/ ( B ( /g ` G ) C ) = Z ) )
18	17	3exp2 1285	. . . . . . . . . 10 |- ( R e. Dmn -> ( A e. X -> ( ( B ( /g ` G ) C ) e. X -> ( ( A H ( B ( /g ` G ) C ) ) = Z -> ( A = Z \/ ( B ( /g ` G ) C ) = Z ) ) ) ) )
19	18	imp31 448	. . . . . . . . 9 |- ( ( ( R e. Dmn /\ A e. X ) /\ ( B ( /g ` G ) C ) e. X ) -> ( ( A H ( B ( /g ` G ) C ) ) = Z -> ( A = Z \/ ( B ( /g ` G ) C ) = Z ) ) )
20	15, 19	syldan 487	. . . . . . . 8 |- ( ( ( R e. Dmn /\ A e. X ) /\ ( B e. X /\ C e. X ) ) -> ( ( A H ( B ( /g ` G ) C ) ) = Z -> ( A = Z \/ ( B ( /g ` G ) C ) = Z ) ) )
21	20	exp43 640	. . . . . . 7 |- ( R e. Dmn -> ( A e. X -> ( B e. X -> ( C e. X -> ( ( A H ( B ( /g ` G ) C ) ) = Z -> ( A = Z \/ ( B ( /g ` G ) C ) = Z ) ) ) ) ) )
22	21	3imp2 1282	. . . . . 6 |- ( ( R e. Dmn /\ ( A e. X /\ B e. X /\ C e. X ) ) -> ( ( A H ( B ( /g ` G ) C ) ) = Z -> ( A = Z \/ ( B ( /g ` G ) C ) = Z ) ) )
23		neor 2885	. . . . . 6 |- ( ( A = Z \/ ( B ( /g ` G ) C ) = Z ) <-> ( A =/= Z -> ( B ( /g ` G ) C ) = Z ) )
24	22, 23	syl6ib 241	. . . . 5 |- ( ( R e. Dmn /\ ( A e. X /\ B e. X /\ C e. X ) ) -> ( ( A H ( B ( /g ` G ) C ) ) = Z -> ( A =/= Z -> ( B ( /g ` G ) C ) = Z ) ) )
25	24	com23 86	. . . 4 |- ( ( R e. Dmn /\ ( A e. X /\ B e. X /\ C e. X ) ) -> ( A =/= Z -> ( ( A H ( B ( /g ` G ) C ) ) = Z -> ( B ( /g ` G ) C ) = Z ) ) )
26	25	imp 445	. . 3 |- ( ( ( R e. Dmn /\ ( A e. X /\ B e. X /\ C e. X ) ) /\ A =/= Z ) -> ( ( A H ( B ( /g ` G ) C ) ) = Z -> ( B ( /g ` G ) C ) = Z ) )
27	9, 26	sylbird 250	. 2 |- ( ( ( R e. Dmn /\ ( A e. X /\ B e. X /\ C e. X ) ) /\ A =/= Z ) -> ( ( ( A H B ) ( /g ` G ) ( A H C ) ) = Z -> ( B ( /g ` G ) C ) = Z ) )
28	11	adantr 481	. . . 4 |- ( ( R e. Dmn /\ ( A e. X /\ B e. X /\ C e. X ) ) -> G e. GrpOp )
29	2, 3, 4	rngocl 33700	. . . . . 6 |- ( ( R e. RingOps /\ A e. X /\ B e. X ) -> ( A H B ) e. X )
30	29	3adant3r3 1276	. . . . 5 |- ( ( R e. RingOps /\ ( A e. X /\ B e. X /\ C e. X ) ) -> ( A H B ) e. X )
31	1, 30	sylan 488	. . . 4 |- ( ( R e. Dmn /\ ( A e. X /\ B e. X /\ C e. X ) ) -> ( A H B ) e. X )
32	2, 3, 4	rngocl 33700	. . . . . 6 |- ( ( R e. RingOps /\ A e. X /\ C e. X ) -> ( A H C ) e. X )
33	32	3adant3r2 1275	. . . . 5 |- ( ( R e. RingOps /\ ( A e. X /\ B e. X /\ C e. X ) ) -> ( A H C ) e. X )
34	1, 33	sylan 488	. . . 4 |- ( ( R e. Dmn /\ ( A e. X /\ B e. X /\ C e. X ) ) -> ( A H C ) e. X )
35	4, 16, 5	grpoeqdivid 33680	. . . 4 |- ( ( G e. GrpOp /\ ( A H B ) e. X /\ ( A H C ) e. X ) -> ( ( A H B ) = ( A H C ) <-> ( ( A H B ) ( /g ` G ) ( A H C ) ) = Z ) )
36	28, 31, 34, 35	syl3anc 1326	. . 3 |- ( ( R e. Dmn /\ ( A e. X /\ B e. X /\ C e. X ) ) -> ( ( A H B ) = ( A H C ) <-> ( ( A H B ) ( /g ` G ) ( A H C ) ) = Z ) )
37	36	adantr 481	. 2 |- ( ( ( R e. Dmn /\ ( A e. X /\ B e. X /\ C e. X ) ) /\ A =/= Z ) -> ( ( A H B ) = ( A H C ) <-> ( ( A H B ) ( /g ` G ) ( A H C ) ) = Z ) )
38	4, 16, 5	grpoeqdivid 33680	. . . . . 6 |- ( ( G e. GrpOp /\ B e. X /\ C e. X ) -> ( B = C <-> ( B ( /g ` G ) C ) = Z ) )
39	38	3expb 1266	. . . . 5 |- ( ( G e. GrpOp /\ ( B e. X /\ C e. X ) ) -> ( B = C <-> ( B ( /g ` G ) C ) = Z ) )
40	11, 39	sylan 488	. . . 4 |- ( ( R e. Dmn /\ ( B e. X /\ C e. X ) ) -> ( B = C <-> ( B ( /g ` G ) C ) = Z ) )
41	40	3adantr1 1220	. . 3 |- ( ( R e. Dmn /\ ( A e. X /\ B e. X /\ C e. X ) ) -> ( B = C <-> ( B ( /g ` G ) C ) = Z ) )
42	41	adantr 481	. 2 |- ( ( ( R e. Dmn /\ ( A e. X /\ B e. X /\ C e. X ) ) /\ A =/= Z ) -> ( B = C <-> ( B ( /g ` G ) C ) = Z ) )
43	27, 37, 42	3imtr4d 283	1 |- ( ( ( R e. Dmn /\ ( A e. X /\ B e. X /\ C e. X ) ) /\ A =/= Z ) -> ( ( A H B ) = ( A H C ) -> B = C ) )

Syntax hints:   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   ran crn 5115   ` cfv 5888  (class class class)co 6650   1stc1st 7166   2ndc2nd 7167   GrpOpcgr 27343  GIdcgi 27344   /g cgs 27346   RingOpscrngo 33693   Dmncdmn 33846

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-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  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-rmo 2920  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-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  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-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  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-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-1o 7560  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-grpo 27347  df-gid 27348  df-ginv 27349  df-gdiv 27350  df-ablo 27399  df-ass 33642  df-exid 33644  df-mgmOLD 33648  df-sgrOLD 33660  df-mndo 33666  df-rngo 33694  df-com2 33789  df-crngo 33793  df-idl 33809  df-pridl 33810  df-prrngo 33847  df-dmn 33848  df-igen 33859

This theorem is referenced by:  dmncan2  33876