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Theorem dmncan2 26702
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 26681 . . . 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 26626 . . . . . 6  |-  ( ( R  e. CRingOps  /\  A  e.  X  /\  C  e.  X )  ->  ( A H C )  =  ( C H A ) )
653adant3r2 1161 . . . . 5  |-  ( ( R  e. CRingOps  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( A H C )  =  ( C H A ) )
72, 3, 4crngocom 26626 . . . . . 6  |-  ( ( R  e. CRingOps  /\  B  e.  X  /\  C  e.  X )  ->  ( B H C )  =  ( C H B ) )
873adant3r1 1160 . . . . 5  |-  ( ( R  e. CRingOps  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( B H C )  =  ( C H B ) )
96, 8eqeq12d 2297 . . . 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 457 . . 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 451 . 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 939 . . . 4  |-  ( ( C  e.  X  /\  A  e.  X  /\  B  e.  X )  <->  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)
1312biimpri 197 . . 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 26701 . . 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 632 . 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 206 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 set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    =/= wne 2446   ran crn 4690   ` cfv 5255  (class class class)co 5858   1stc1st 6120   2ndc2nd 6121  GIdcgi 20854  CRingOpsccring 26620   Dmncdmn 26672
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-1o 6479  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-grpo 20858  df-gid 20859  df-ginv 20860  df-gdiv 20861  df-ablo 20949  df-ass 20980  df-exid 20982  df-mgm 20986  df-sgr 20998  df-mndo 21005  df-rngo 21043  df-com2 21078  df-crngo 26621  df-idl 26635  df-pridl 26636  df-prrngo 26673  df-dmn 26674  df-igen 26685
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