Theorem dmncan1 28876

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 28857	. . . . . 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 2443	. . . . . . 7 |- ( /g ` G ) = ( /g ` G )
6	2, 3, 4, 5	rngosubdi 28759	. . . . . 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 2451	. . 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 23877	. . . . . . . . . . . 12 |- ( R e. RingOps -> G e. GrpOp )
11	1, 10	syl 16	. . . . . . . . . . 11 |- ( R e. Dmn -> G e. GrpOp )
12	4, 5	grpodivcl 23734	. . . . . . . . . . . 12 |- ( ( G e. GrpOp /\ B e. X /\ C e. X ) -> ( B ( /g ` G ) C ) e. X )
13	12	3expb 1188	. . . . . . . . . . 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 28875	. . . . . . . . . . 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 1205	. . . . . . . . . 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 1202	. . . . . 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 2696	. . . . . 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 23869	. . . . . 6 |- ( ( R e. RingOps /\ A e. X /\ B e. X ) -> ( A H B ) e. X )
30	29	3adant3r3 1198	. . . . 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 23869	. . . . . 6 |- ( ( R e. RingOps /\ A e. X /\ C e. X ) -> ( A H C ) e. X )
33	32	3adant3r2 1197	. . . . 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 28746	. . . 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 1218	. . 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 28746	. . . . . 6 |- ( ( G e. GrpOp /\ B e. X /\ C e. X ) -> ( B = C <-> ( B ( /g ` G ) C ) = Z ) )
39	38	3expb 1188	. . . . 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 1147	. . 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 965   = wceq 1369   e. wcel 1756   =/= wne 2606   ran crn 4841   ` cfv 5418  (class class class)co 6091   1stc1st 6575   2ndc2nd 6576   GrpOpcgr 23673  GIdcgi 23674   /g cgs 23676   RingOpscrngo 23862   Dmncdmn 28847

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-rep 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  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 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-reu 2722  df-rmo 2723  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-uni 4092  df-int 4129  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-tr 4386  df-eprel 4632  df-id 4636  df-po 4641  df-so 4642  df-fr 4679  df-we 4681  df-ord 4722  df-on 4723  df-lim 4724  df-suc 4725  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-riota 6052  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-om 6477  df-1st 6577  df-2nd 6578  df-1o 6920  df-er 7101  df-en 7311  df-dom 7312  df-sdom 7313  df-fin 7314  df-grpo 23678  df-gid 23679  df-ginv 23680  df-gdiv 23681  df-ablo 23769  df-ass 23800  df-exid 23802  df-mgm 23806  df-sgr 23818  df-mndo 23825  df-rngo 23863  df-com2 23898  df-crngo 28796  df-idl 28810  df-pridl 28811  df-prrngo 28848  df-dmn 28849  df-igen 28860

This theorem is referenced by:  dmncan2  28877