Theorem dmncan2 28830

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
dmncan2	|- ( ( ( R e. Dmn /\ ( A e. X /\ B e. X /\ C e. X ) ) /\ C =/= Z ) -> ( ( A H C ) = ( B H C ) -> A = B ) )

Proof of Theorem dmncan2

Step	Hyp	Ref	Expression
1		dmncrng 28809	. . . 4 |- ( R e. Dmn -> R e. CRingOps )
2		dmncan.1	. . . . . . 7 |- G = ( 1st ` R )
3		dmncan.2	. . . . . . 7 |- H = ( 2nd ` R )
4		dmncan.3	. . . . . . 7 |- X = ran G
5	2, 3, 4	crngocom 28754	. . . . . 6 |- ( ( R e. CRingOps /\ A e. X /\ C e. X ) -> ( A H C ) = ( C H A ) )
6	5	3adant3r2 1197	. . . . 5 |- ( ( R e. CRingOps /\ ( A e. X /\ B e. X /\ C e. X ) ) -> ( A H C ) = ( C H A ) )
7	2, 3, 4	crngocom 28754	. . . . . 6 |- ( ( R e. CRingOps /\ B e. X /\ C e. X ) -> ( B H C ) = ( C H B ) )
8	7	3adant3r1 1196	. . . . 5 |- ( ( R e. CRingOps /\ ( A e. X /\ B e. X /\ C e. X ) ) -> ( B H C ) = ( C H B ) )
9	6, 8	eqeq12d 2452	. . . 4 |- ( ( R e. CRingOps /\ ( A e. X /\ B e. X /\ C e. X ) ) -> ( ( A H C ) = ( B H C ) <-> ( C H A ) = ( C H B ) ) )
10	1, 9	sylan 471	. . 3 |- ( ( R e. Dmn /\ ( A e. X /\ B e. X /\ C e. X ) ) -> ( ( A H C ) = ( B H C ) <-> ( C H A ) = ( C H B ) ) )
11	10	adantr 465	. 2 |- ( ( ( R e. Dmn /\ ( A e. X /\ B e. X /\ C e. X ) ) /\ C =/= Z ) -> ( ( A H C ) = ( B H C ) <-> ( C H A ) = ( C H B ) ) )
12		3anrot 970	. . . 4 |- ( ( C e. X /\ A e. X /\ B e. X ) <-> ( A e. X /\ B e. X /\ C e. X ) )
13	12	biimpri 206	. . 3 |- ( ( A e. X /\ B e. X /\ C e. X ) -> ( C e. X /\ A e. X /\ B e. X ) )
14		dmncan.4	. . . 4 |- Z = (GId ` G )
15	2, 3, 4, 14	dmncan1 28829	. . 3 |- ( ( ( R e. Dmn /\ ( C e. X /\ A e. X /\ B e. X ) ) /\ C =/= Z ) -> ( ( C H A ) = ( C H B ) -> A = B ) )
16	13, 15	sylanl2 651	. 2 |- ( ( ( R e. Dmn /\ ( A e. X /\ B e. X /\ C e. X ) ) /\ C =/= Z ) -> ( ( C H A ) = ( C H B ) -> A = B ) )
17	11, 16	sylbid 215	1 |- ( ( ( R e. Dmn /\ ( A e. X /\ B e. X /\ C e. X ) ) /\ C =/= Z ) -> ( ( A H C ) = ( B H C ) -> A = B ) )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   /\ w3a 965   = wceq 1369   e. wcel 1756   =/= wne 2601   ran crn 4836   ` cfv 5413  (class class class)co 6086   1stc1st 6570   2ndc2nd 6571  GIdcgi 23625  CRingOpsccring 28748   Dmncdmn 28800

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 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367

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 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2715  df-rex 2716  df-reu 2717  df-rmo 2718  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-uni 4087  df-int 4124  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-tr 4381  df-eprel 4627  df-id 4631  df-po 4636  df-so 4637  df-fr 4674  df-we 4676  df-ord 4717  df-on 4718  df-lim 4719  df-suc 4720  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-riota 6047  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-om 6472  df-1st 6572  df-2nd 6573  df-1o 6912  df-er 7093  df-en 7303  df-dom 7304  df-sdom 7305  df-fin 7306  df-grpo 23629  df-gid 23630  df-ginv 23631  df-gdiv 23632  df-ablo 23720  df-ass 23751  df-exid 23753  df-mgm 23757  df-sgr 23769  df-mndo 23776  df-rngo 23814  df-com2 23849  df-crngo 28749  df-idl 28763  df-pridl 28764  df-prrngo 28801  df-dmn 28802  df-igen 28813

This theorem is referenced by: (None)