Theorem dmncan1 30092

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 30073	. . . . . 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 2467	. . . . . . 7 |- ( /g ` G ) = ( /g ` G )
6	2, 3, 4, 5	rngosubdi 29975	. . . . . 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 471	. . . . 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 465	. . . 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 2469	. . 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 25084	. . . . . . . . . . . 12 |- ( R e. RingOps -> G e. GrpOp )
11	1, 10	syl 16	. . . . . . . . . . 11 |- ( R e. Dmn -> G e. GrpOp )
12	4, 5	grpodivcl 24941	. . . . . . . . . . . 12 |- ( ( G e. GrpOp /\ B e. X /\ C e. X ) -> ( B ( /g ` G ) C ) e. X )
13	12	3expb 1197	. . . . . . . . . . 11 |- ( ( G e. GrpOp /\ ( B e. X /\ C e. X ) ) -> ( B ( /g ` G ) C ) e. X )
14	11, 13	sylan 471	. . . . . . . . . 10 |- ( ( R e. Dmn /\ ( B e. X /\ C e. X ) ) -> ( B ( /g ` G ) C ) e. X )
15	14	adantlr 714	. . . . . . . . 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 30091	. . . . . . . . . . 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 1214	. . . . . . . . . 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 432	. . . . . . . . 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 470	. . . . . . . 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 612	. . . . . . 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 1211	. . . . . 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 2791	. . . . . 6 |- ( ( A = Z \/ ( B ( /g ` G ) C ) = Z ) <-> ( A =/= Z -> ( B ( /g ` G ) C ) = Z ) )
24	22, 23	syl6ib 226	. . . . 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 78	. . . 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 429	. . 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 235	. 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 465	. . . 4 |- ( ( R e. Dmn /\ ( A e. X /\ B e. X /\ C e. X ) ) -> G e. GrpOp )
29	2, 3, 4	rngocl 25076	. . . . . 6 |- ( ( R e. RingOps /\ A e. X /\ B e. X ) -> ( A H B ) e. X )
30	29	3adant3r3 1207	. . . . 5 |- ( ( R e. RingOps /\ ( A e. X /\ B e. X /\ C e. X ) ) -> ( A H B ) e. X )
31	1, 30	sylan 471	. . . 4 |- ( ( R e. Dmn /\ ( A e. X /\ B e. X /\ C e. X ) ) -> ( A H B ) e. X )
32	2, 3, 4	rngocl 25076	. . . . . 6 |- ( ( R e. RingOps /\ A e. X /\ C e. X ) -> ( A H C ) e. X )
33	32	3adant3r2 1206	. . . . 5 |- ( ( R e. RingOps /\ ( A e. X /\ B e. X /\ C e. X ) ) -> ( A H C ) e. X )
34	1, 33	sylan 471	. . . 4 |- ( ( R e. Dmn /\ ( A e. X /\ B e. X /\ C e. X ) ) -> ( A H C ) e. X )
35	4, 16, 5	grpoeqdivid 29962	. . . 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 1228	. . 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 465	. 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 29962	. . . . . 6 |- ( ( G e. GrpOp /\ B e. X /\ C e. X ) -> ( B = C <-> ( B ( /g ` G ) C ) = Z ) )
39	38	3expb 1197	. . . . 5 |- ( ( G e. GrpOp /\ ( B e. X /\ C e. X ) ) -> ( B = C <-> ( B ( /g ` G ) C ) = Z ) )
40	11, 39	sylan 471	. . . 4 |- ( ( R e. Dmn /\ ( B e. X /\ C e. X ) ) -> ( B = C <-> ( B ( /g ` G ) C ) = Z ) )
41	40	3adantr1 1155	. . 3 |- ( ( R e. Dmn /\ ( A e. X /\ B e. X /\ C e. X ) ) -> ( B = C <-> ( B ( /g ` G ) C ) = Z ) )
42	41	adantr 465	. 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 268	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 184   \/ wo 368   /\ wa 369   /\ w3a 973   = wceq 1379   e. wcel 1767   =/= wne 2662   ran crn 5000   ` cfv 5587  (class class class)co 6283   1stc1st 6782   2ndc2nd 6783   GrpOpcgr 24880  GIdcgi 24881   /g cgs 24883   RingOpscrngo 25069   Dmncdmn 30063

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6575

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5550  df-fun 5589  df-fn 5590  df-f 5591  df-f1 5592  df-fo 5593  df-f1o 5594  df-fv 5595  df-riota 6244  df-ov 6286  df-oprab 6287  df-mpt2 6288  df-om 6680  df-1st 6784  df-2nd 6785  df-1o 7130  df-er 7311  df-en 7517  df-dom 7518  df-sdom 7519  df-fin 7520  df-grpo 24885  df-gid 24886  df-ginv 24887  df-gdiv 24888  df-ablo 24976  df-ass 25007  df-exid 25009  df-mgm 25013  df-sgr 25025  df-mndo 25032  df-rngo 25070  df-com2 25105  df-crngo 30012  df-idl 30026  df-pridl 30027  df-prrngo 30064  df-dmn 30065  df-igen 30076

This theorem is referenced by:  dmncan2  30093