Theorem dmncan2 29018

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 28997	. . . 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 28942	. . . . . 6 |- ( ( R e. CRingOps /\ A e. X /\ C e. X ) -> ( A H C ) = ( C H A ) )
6	5	3adant3r2 1198	. . . . 5 |- ( ( R e. CRingOps /\ ( A e. X /\ B e. X /\ C e. X ) ) -> ( A H C ) = ( C H A ) )
7	2, 3, 4	crngocom 28942	. . . . . 6 |- ( ( R e. CRingOps /\ B e. X /\ C e. X ) -> ( B H C ) = ( C H B ) )
8	7	3adant3r1 1197	. . . . 5 |- ( ( R e. CRingOps /\ ( A e. X /\ B e. X /\ C e. X ) ) -> ( B H C ) = ( C H B ) )
9	6, 8	eqeq12d 2473	. . . 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 29017	. . 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 1370   e. wcel 1758   =/= wne 2644   ran crn 4942   ` cfv 5519  (class class class)co 6193   1stc1st 6678   2ndc2nd 6679  GIdcgi 23819  CRingOpsccring 28936   Dmncdmn 28988

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4504  ax-sep 4514  ax-nul 4522  ax-pow 4571  ax-pr 4632  ax-un 6475

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3073  df-sbc 3288  df-csb 3390  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-pss 3445  df-nul 3739  df-if 3893  df-pw 3963  df-sn 3979  df-pr 3981  df-tp 3983  df-op 3985  df-uni 4193  df-int 4230  df-iun 4274  df-br 4394  df-opab 4452  df-mpt 4453  df-tr 4487  df-eprel 4733  df-id 4737  df-po 4742  df-so 4743  df-fr 4780  df-we 4782  df-ord 4823  df-on 4824  df-lim 4825  df-suc 4826  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-rn 4952  df-res 4953  df-ima 4954  df-iota 5482  df-fun 5521  df-fn 5522  df-f 5523  df-f1 5524  df-fo 5525  df-f1o 5526  df-fv 5527  df-riota 6154  df-ov 6196  df-oprab 6197  df-mpt2 6198  df-om 6580  df-1st 6680  df-2nd 6681  df-1o 7023  df-er 7204  df-en 7414  df-dom 7415  df-sdom 7416  df-fin 7417  df-grpo 23823  df-gid 23824  df-ginv 23825  df-gdiv 23826  df-ablo 23914  df-ass 23945  df-exid 23947  df-mgm 23951  df-sgr 23963  df-mndo 23970  df-rngo 24008  df-com2 24043  df-crngo 28937  df-idl 28951  df-pridl 28952  df-prrngo 28989  df-dmn 28990  df-igen 29001

This theorem is referenced by: (None)