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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
StepHypRef 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
52, 3, 4crngocom 31693 . . . . . 6  |-  ( ( R  e. CRingOps  /\  A  e.  X  /\  C  e.  X )  ->  ( A H C )  =  ( C H A ) )
653adant3r2 1209 . . . . 5  |-  ( ( R  e. CRingOps  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( A H C )  =  ( C H A ) )
72, 3, 4crngocom 31693 . . . . . 6  |-  ( ( R  e. CRingOps  /\  B  e.  X  /\  C  e.  X )  ->  ( B H C )  =  ( C H B ) )
873adant3r1 1208 . . . . 5  |-  ( ( R  e. CRingOps  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( B H C )  =  ( C H B ) )
96, 8eqeq12d 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 ) ) )
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 981 . . . 4  |-  ( ( C  e.  X  /\  A  e.  X  /\  B  e.  X )  <->  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)
1312biimpri 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 )
152, 3, 4, 14dmncan1 31768 . . 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 217 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 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)
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