Theorem dmncan2 30077

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 30056	. . . 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 30001	. . . . . 6 |- ( ( R e. CRingOps /\ A e. X /\ C e. X ) -> ( A H C ) = ( C H A ) )
6	5	3adant3r2 1206	. . . . 5 |- ( ( R e. CRingOps /\ ( A e. X /\ B e. X /\ C e. X ) ) -> ( A H C ) = ( C H A ) )
7	2, 3, 4	crngocom 30001	. . . . . 6 |- ( ( R e. CRingOps /\ B e. X /\ C e. X ) -> ( B H C ) = ( C H B ) )
8	7	3adant3r1 1205	. . . . 5 |- ( ( R e. CRingOps /\ ( A e. X /\ B e. X /\ C e. X ) ) -> ( B H C ) = ( C H B ) )
9	6, 8	eqeq12d 2489	. . . 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 978	. . . 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 30076	. . 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 973   = wceq 1379   e. wcel 1767   =/= wne 2662   ran crn 5000   ` cfv 5586  (class class class)co 6282   1stc1st 6779   2ndc2nd 6780  GIdcgi 24865  CRingOpsccring 29995   Dmncdmn 30047

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 6574

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 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-1st 6781  df-2nd 6782  df-1o 7127  df-er 7308  df-en 7514  df-dom 7515  df-sdom 7516  df-fin 7517  df-grpo 24869  df-gid 24870  df-ginv 24871  df-gdiv 24872  df-ablo 24960  df-ass 24991  df-exid 24993  df-mgm 24997  df-sgr 25009  df-mndo 25016  df-rngo 25054  df-com2 25089  df-crngo 29996  df-idl 30010  df-pridl 30011  df-prrngo 30048  df-dmn 30049  df-igen 30060

This theorem is referenced by: (None)