Theorem dmncan1 32042

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 32023	. . . . . 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 2420	. . . . . . 7 |- ( /g ` G ) = ( /g ` G )
6	2, 3, 4, 5	rngosubdi 31925	. . . . . 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 473	. . . . 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 466	. . . 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 2422	. . 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 25989	. . . . . . . . . . . 12 |- ( R e. RingOps -> G e. GrpOp )
11	1, 10	syl 17	. . . . . . . . . . 11 |- ( R e. Dmn -> G e. GrpOp )
12	4, 5	grpodivcl 25846	. . . . . . . . . . . 12 |- ( ( G e. GrpOp /\ B e. X /\ C e. X ) -> ( B ( /g ` G ) C ) e. X )
13	12	3expb 1206	. . . . . . . . . . 11 |- ( ( G e. GrpOp /\ ( B e. X /\ C e. X ) ) -> ( B ( /g ` G ) C ) e. X )
14	11, 13	sylan 473	. . . . . . . . . 10 |- ( ( R e. Dmn /\ ( B e. X /\ C e. X ) ) -> ( B ( /g ` G ) C ) e. X )
15	14	adantlr 719	. . . . . . . . 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 32041	. . . . . . . . . . 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 1223	. . . . . . . . . 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 433	. . . . . . . . 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 472	. . . . . . . 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 615	. . . . . . 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 1220	. . . . . 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 2746	. . . . . 6 |- ( ( A = Z \/ ( B ( /g ` G ) C ) = Z ) <-> ( A =/= Z -> ( B ( /g ` G ) C ) = Z ) )
24	22, 23	syl6ib 229	. . . . 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 81	. . . 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 430	. . 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 238	. 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 466	. . . 4 |- ( ( R e. Dmn /\ ( A e. X /\ B e. X /\ C e. X ) ) -> G e. GrpOp )
29	2, 3, 4	rngocl 25981	. . . . . 6 |- ( ( R e. RingOps /\ A e. X /\ B e. X ) -> ( A H B ) e. X )
30	29	3adant3r3 1216	. . . . 5 |- ( ( R e. RingOps /\ ( A e. X /\ B e. X /\ C e. X ) ) -> ( A H B ) e. X )
31	1, 30	sylan 473	. . . 4 |- ( ( R e. Dmn /\ ( A e. X /\ B e. X /\ C e. X ) ) -> ( A H B ) e. X )
32	2, 3, 4	rngocl 25981	. . . . . 6 |- ( ( R e. RingOps /\ A e. X /\ C e. X ) -> ( A H C ) e. X )
33	32	3adant3r2 1215	. . . . 5 |- ( ( R e. RingOps /\ ( A e. X /\ B e. X /\ C e. X ) ) -> ( A H C ) e. X )
34	1, 33	sylan 473	. . . 4 |- ( ( R e. Dmn /\ ( A e. X /\ B e. X /\ C e. X ) ) -> ( A H C ) e. X )
35	4, 16, 5	grpoeqdivid 31912	. . . 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 1264	. . 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 466	. 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 31912	. . . . . 6 |- ( ( G e. GrpOp /\ B e. X /\ C e. X ) -> ( B = C <-> ( B ( /g ` G ) C ) = Z ) )
39	38	3expb 1206	. . . . 5 |- ( ( G e. GrpOp /\ ( B e. X /\ C e. X ) ) -> ( B = C <-> ( B ( /g ` G ) C ) = Z ) )
40	11, 39	sylan 473	. . . 4 |- ( ( R e. Dmn /\ ( B e. X /\ C e. X ) ) -> ( B = C <-> ( B ( /g ` G ) C ) = Z ) )
41	40	3adantr1 1164	. . 3 |- ( ( R e. Dmn /\ ( A e. X /\ B e. X /\ C e. X ) ) -> ( B = C <-> ( B ( /g ` G ) C ) = Z ) )
42	41	adantr 466	. 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 271	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 187   \/ wo 369   /\ wa 370   /\ w3a 982   = wceq 1437   e. wcel 1867   =/= wne 2616   ran crn 4846   ` cfv 5592  (class class class)co 6296   1stc1st 6796   2ndc2nd 6797   GrpOpcgr 25785  GIdcgi 25786   /g cgs 25788   RingOpscrngo 25974   Dmncdmn 32013

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-8 1869  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-rep 4529  ax-sep 4539  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6588

This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-ral 2778  df-rex 2779  df-reu 2780  df-rmo 2781  df-rab 2782  df-v 3080  df-sbc 3297  df-csb 3393  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-pss 3449  df-nul 3759  df-if 3907  df-pw 3978  df-sn 3994  df-pr 3996  df-tp 3998  df-op 4000  df-uni 4214  df-int 4250  df-iun 4295  df-br 4418  df-opab 4476  df-mpt 4477  df-tr 4512  df-eprel 4756  df-id 4760  df-po 4766  df-so 4767  df-fr 4804  df-we 4806  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-ord 5436  df-on 5437  df-lim 5438  df-suc 5439  df-iota 5556  df-fun 5594  df-fn 5595  df-f 5596  df-f1 5597  df-fo 5598  df-f1o 5599  df-fv 5600  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6698  df-1st 6798  df-2nd 6799  df-1o 7181  df-er 7362  df-en 7569  df-dom 7570  df-sdom 7571  df-fin 7572  df-grpo 25790  df-gid 25791  df-ginv 25792  df-gdiv 25793  df-ablo 25881  df-ass 25912  df-exid 25914  df-mgmOLD 25918  df-sgrOLD 25930  df-mndo 25937  df-rngo 25975  df-com2 26010  df-crngo 31962  df-idl 31976  df-pridl 31977  df-prrngo 32014  df-dmn 32015  df-igen 32026

This theorem is referenced by:  dmncan2  32043