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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
StepHypRef 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
52, 3, 4crngocom 28754 . . . . . 6  |-  ( ( R  e. CRingOps  /\  A  e.  X  /\  C  e.  X )  ->  ( A H C )  =  ( C H A ) )
653adant3r2 1197 . . . . 5  |-  ( ( R  e. CRingOps  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( A H C )  =  ( C H A ) )
72, 3, 4crngocom 28754 . . . . . 6  |-  ( ( R  e. CRingOps  /\  B  e.  X  /\  C  e.  X )  ->  ( B H C )  =  ( C H B ) )
873adant3r1 1196 . . . . 5  |-  ( ( R  e. CRingOps  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( B H C )  =  ( C H B ) )
96, 8eqeq12d 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 ) ) )
101, 9sylan 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 ) ) )
1110adantr 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 )
)
1312biimpri 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 )
152, 3, 4, 14dmncan1 28829 . . 3  |-  ( ( ( R  e.  Dmn  /\  ( C  e.  X  /\  A  e.  X  /\  B  e.  X
) )  /\  C  =/=  Z )  ->  (
( C H A )  =  ( C H B )  ->  A  =  B )
)
1613, 15sylanl2 651 . 2  |-  ( ( ( R  e.  Dmn  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X
) )  /\  C  =/=  Z )  ->  (
( C H A )  =  ( C H B )  ->  A  =  B )
)
1711, 16sylbid 215 1  |-  ( ( ( R  e.  Dmn  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X
) )  /\  C  =/=  Z )  ->  (
( A H C )  =  ( B H C )  ->  A  =  B )
)
Colors of variables: wff setvar class
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)
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