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Theorem dmncan2 25850
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 25829 . . . 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 25774 . . . . . 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 25774 . . . . . 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 2330 . . . 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 25849 . . 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 1633    e. wcel 1701    =/= wne 2479   ran crn 4727   ` cfv 5292  (class class class)co 5900   1stc1st 6162   2ndc2nd 6163  GIdcgi 20907  CRingOpsccring 25768   Dmncdmn 25820
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-rep 4168  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251  ax-un 4549
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 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-ral 2582  df-rex 2583  df-reu 2584  df-rmo 2585  df-rab 2586  df-v 2824  df-sbc 3026  df-csb 3116  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-pss 3202  df-nul 3490  df-if 3600  df-pw 3661  df-sn 3680  df-pr 3681  df-tp 3682  df-op 3683  df-uni 3865  df-int 3900  df-iun 3944  df-br 4061  df-opab 4115  df-mpt 4116  df-tr 4151  df-eprel 4342  df-id 4346  df-po 4351  df-so 4352  df-fr 4389  df-we 4391  df-ord 4432  df-on 4433  df-lim 4434  df-suc 4435  df-om 4694  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-f1 5297  df-fo 5298  df-f1o 5299  df-fv 5300  df-ov 5903  df-oprab 5904  df-mpt2 5905  df-1st 6164  df-2nd 6165  df-riota 6346  df-1o 6521  df-er 6702  df-en 6907  df-dom 6908  df-sdom 6909  df-fin 6910  df-grpo 20911  df-gid 20912  df-ginv 20913  df-gdiv 20914  df-ablo 21002  df-ass 21033  df-exid 21035  df-mgm 21039  df-sgr 21051  df-mndo 21058  df-rngo 21096  df-com2 21131  df-crngo 25769  df-idl 25783  df-pridl 25784  df-prrngo 25821  df-dmn 25822  df-igen 25833
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