Theorem dmncan2 31769

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 31748	. . . 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 31693	. . . . . 6 |- ( ( R e. CRingOps /\ A e. X /\ C e. X ) -> ( A H C ) = ( C H A ) )
6	5	3adant3r2 1209	. . . . 5 |- ( ( R e. CRingOps /\ ( A e. X /\ B e. X /\ C e. X ) ) -> ( A H C ) = ( C H A ) )
7	2, 3, 4	crngocom 31693	. . . . . 6 |- ( ( R e. CRingOps /\ B e. X /\ C e. X ) -> ( B H C ) = ( C H B ) )
8	7	3adant3r1 1208	. . . . 5 |- ( ( R e. CRingOps /\ ( A e. X /\ B e. X /\ C e. X ) ) -> ( B H C ) = ( C H B ) )
9	6, 8	eqeq12d 2426	. . . 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 981	. . . 4 |- ( ( C e. X /\ A e. X /\ B e. X ) <-> ( A e. X /\ B e. X /\ C e. X ) )
13	12	biimpri 208	. . 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 31768	. . 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 217	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 186   /\ wa 369   /\ w3a 976   = wceq 1407   e. wcel 1844   =/= wne 2600   ran crn 4826   ` cfv 5571  (class class class)co 6280   1stc1st 6784   2ndc2nd 6785  GIdcgi 25616  CRingOpsccring 31687   Dmncdmn 31739

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-8 1846  ax-9 1848  ax-10 1863  ax-11 1868  ax-12 1880  ax-13 2028  ax-ext 2382  ax-rep 4509  ax-sep 4519  ax-nul 4527  ax-pow 4574  ax-pr 4632  ax-un 6576

This theorem depends on definitions:  df-bi 187  df-or 370  df-an 371  df-3or 977  df-3an 978  df-tru 1410  df-ex 1636  df-nf 1640  df-sb 1766  df-eu 2244  df-mo 2245  df-clab 2390  df-cleq 2396  df-clel 2399  df-nfc 2554  df-ne 2602  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3063  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-pss 3432  df-nul 3741  df-if 3888  df-pw 3959  df-sn 3975  df-pr 3977  df-tp 3979  df-op 3981  df-uni 4194  df-int 4230  df-iun 4275  df-br 4398  df-opab 4456  df-mpt 4457  df-tr 4492  df-eprel 4736  df-id 4740  df-po 4746  df-so 4747  df-fr 4784  df-we 4786  df-xp 4831  df-rel 4832  df-cnv 4833  df-co 4834  df-dm 4835  df-rn 4836  df-res 4837  df-ima 4838  df-ord 5415  df-on 5416  df-lim 5417  df-suc 5418  df-iota 5535  df-fun 5573  df-fn 5574  df-f 5575  df-f1 5576  df-fo 5577  df-f1o 5578  df-fv 5579  df-riota 6242  df-ov 6283  df-oprab 6284  df-mpt2 6285  df-om 6686  df-1st 6786  df-2nd 6787  df-1o 7169  df-er 7350  df-en 7557  df-dom 7558  df-sdom 7559  df-fin 7560  df-grpo 25620  df-gid 25621  df-ginv 25622  df-gdiv 25623  df-ablo 25711  df-ass 25742  df-exid 25744  df-mgmOLD 25748  df-sgrOLD 25760  df-mndo 25767  df-rngo 25805  df-com2 25840  df-crngo 31688  df-idl 31702  df-pridl 31703  df-prrngo 31740  df-dmn 31741  df-igen 31752

This theorem is referenced by: (None)