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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
StepHypRef 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
52, 3, 4crngocom 30001 . . . . . 6  |-  ( ( R  e. CRingOps  /\  A  e.  X  /\  C  e.  X )  ->  ( A H C )  =  ( C H A ) )
653adant3r2 1206 . . . . 5  |-  ( ( R  e. CRingOps  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( A H C )  =  ( C H A ) )
72, 3, 4crngocom 30001 . . . . . 6  |-  ( ( R  e. CRingOps  /\  B  e.  X  /\  C  e.  X )  ->  ( B H C )  =  ( C H B ) )
873adant3r1 1205 . . . . 5  |-  ( ( R  e. CRingOps  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( B H C )  =  ( C H B ) )
96, 8eqeq12d 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 ) ) )
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 978 . . . 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 30076 . . 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 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)
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