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Theorem dmncan1 32321
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
dmncan1  |-  ( ( ( R  e.  Dmn  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X
) )  /\  A  =/=  Z )  ->  (
( A H B )  =  ( A H C )  ->  B  =  C )
)

Proof of Theorem dmncan1
StepHypRef Expression
1 dmnrngo 32302 . . . . . 6  |-  ( R  e.  Dmn  ->  R  e.  RingOps )
2 dmncan.1 . . . . . . 7  |-  G  =  ( 1st `  R
)
3 dmncan.2 . . . . . . 7  |-  H  =  ( 2nd `  R
)
4 dmncan.3 . . . . . . 7  |-  X  =  ran  G
5 eqid 2453 . . . . . . 7  |-  (  /g  `  G )  =  (  /g  `  G )
62, 3, 4, 5rngosubdi 32204 . . . . . 6  |-  ( ( R  e.  RingOps  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( A H ( B (  /g  `  G ) C ) )  =  ( ( A H B ) (  /g  `  G ) ( A H C ) ) )
71, 6sylan 474 . . . . 5  |-  ( ( R  e.  Dmn  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X
) )  ->  ( A H ( B (  /g  `  G ) C ) )  =  ( ( A H B ) (  /g  `  G ) ( A H C ) ) )
87adantr 467 . . . 4  |-  ( ( ( R  e.  Dmn  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X
) )  /\  A  =/=  Z )  ->  ( A H ( B (  /g  `  G ) C ) )  =  ( ( A H B ) (  /g  `  G ) ( A H C ) ) )
98eqeq1d 2455 . . 3  |-  ( ( ( R  e.  Dmn  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X
) )  /\  A  =/=  Z )  ->  (
( A H ( B (  /g  `  G
) C ) )  =  Z  <->  ( ( A H B ) (  /g  `  G ) ( A H C ) )  =  Z ) )
102rngogrpo 26130 . . . . . . . . . . . 12  |-  ( R  e.  RingOps  ->  G  e.  GrpOp )
111, 10syl 17 . . . . . . . . . . 11  |-  ( R  e.  Dmn  ->  G  e.  GrpOp )
124, 5grpodivcl 25987 . . . . . . . . . . . 12  |-  ( ( G  e.  GrpOp  /\  B  e.  X  /\  C  e.  X )  ->  ( B (  /g  `  G
) C )  e.  X )
13123expb 1210 . . . . . . . . . . 11  |-  ( ( G  e.  GrpOp  /\  ( B  e.  X  /\  C  e.  X )
)  ->  ( B
(  /g  `  G ) C )  e.  X
)
1411, 13sylan 474 . . . . . . . . . 10  |-  ( ( R  e.  Dmn  /\  ( B  e.  X  /\  C  e.  X
) )  ->  ( B (  /g  `  G
) C )  e.  X )
1514adantlr 722 . . . . . . . . 9  |-  ( ( ( R  e.  Dmn  /\  A  e.  X )  /\  ( B  e.  X  /\  C  e.  X ) )  -> 
( B (  /g  `  G ) C )  e.  X )
16 dmncan.4 . . . . . . . . . . . 12  |-  Z  =  (GId `  G )
172, 3, 4, 16dmnnzd 32320 . . . . . . . . . . 11  |-  ( ( R  e.  Dmn  /\  ( A  e.  X  /\  ( B (  /g  `  G ) C )  e.  X  /\  ( A H ( B (  /g  `  G ) C ) )  =  Z ) )  -> 
( A  =  Z  \/  ( B (  /g  `  G ) C )  =  Z ) )
18173exp2 1228 . . . . . . . . . 10  |-  ( R  e.  Dmn  ->  ( A  e.  X  ->  ( ( B (  /g  `  G ) C )  e.  X  ->  (
( A H ( B (  /g  `  G
) C ) )  =  Z  ->  ( A  =  Z  \/  ( B (  /g  `  G
) C )  =  Z ) ) ) ) )
1918imp31 434 . . . . . . . . 9  |-  ( ( ( R  e.  Dmn  /\  A  e.  X )  /\  ( B (  /g  `  G ) C )  e.  X
)  ->  ( ( A H ( B (  /g  `  G ) C ) )  =  Z  ->  ( A  =  Z  \/  ( B (  /g  `  G
) C )  =  Z ) ) )
2015, 19syldan 473 . . . . . . . 8  |-  ( ( ( R  e.  Dmn  /\  A  e.  X )  /\  ( B  e.  X  /\  C  e.  X ) )  -> 
( ( A H ( B (  /g  `  G ) C ) )  =  Z  -> 
( A  =  Z  \/  ( B (  /g  `  G ) C )  =  Z ) ) )
2120exp43 617 . . . . . . 7  |-  ( R  e.  Dmn  ->  ( A  e.  X  ->  ( B  e.  X  -> 
( C  e.  X  ->  ( ( A H ( B (  /g  `  G ) C ) )  =  Z  -> 
( A  =  Z  \/  ( B (  /g  `  G ) C )  =  Z ) ) ) ) ) )
22213imp2 1225 . . . . . 6  |-  ( ( R  e.  Dmn  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X
) )  ->  (
( A H ( B (  /g  `  G
) C ) )  =  Z  ->  ( A  =  Z  \/  ( B (  /g  `  G
) C )  =  Z ) ) )
23 neor 2717 . . . . . 6  |-  ( ( A  =  Z  \/  ( B (  /g  `  G
) C )  =  Z )  <->  ( A  =/=  Z  ->  ( B
(  /g  `  G ) C )  =  Z ) )
2422, 23syl6ib 230 . . . . 5  |-  ( ( R  e.  Dmn  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X
) )  ->  (
( A H ( B (  /g  `  G
) C ) )  =  Z  ->  ( A  =/=  Z  ->  ( B (  /g  `  G
) C )  =  Z ) ) )
2524com23 81 . . . 4  |-  ( ( R  e.  Dmn  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X
) )  ->  ( A  =/=  Z  ->  (
( A H ( B (  /g  `  G
) C ) )  =  Z  ->  ( B (  /g  `  G
) C )  =  Z ) ) )
2625imp 431 . . 3  |-  ( ( ( R  e.  Dmn  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X
) )  /\  A  =/=  Z )  ->  (
( A H ( B (  /g  `  G
) C ) )  =  Z  ->  ( B (  /g  `  G
) C )  =  Z ) )
279, 26sylbird 239 . 2  |-  ( ( ( R  e.  Dmn  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X
) )  /\  A  =/=  Z )  ->  (
( ( A H B ) (  /g  `  G ) ( A H C ) )  =  Z  ->  ( B (  /g  `  G
) C )  =  Z ) )
2811adantr 467 . . . 4  |-  ( ( R  e.  Dmn  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X
) )  ->  G  e.  GrpOp )
292, 3, 4rngocl 26122 . . . . . 6  |-  ( ( R  e.  RingOps  /\  A  e.  X  /\  B  e.  X )  ->  ( A H B )  e.  X )
30293adant3r3 1220 . . . . 5  |-  ( ( R  e.  RingOps  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( A H B )  e.  X
)
311, 30sylan 474 . . . 4  |-  ( ( R  e.  Dmn  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X
) )  ->  ( A H B )  e.  X )
322, 3, 4rngocl 26122 . . . . . 6  |-  ( ( R  e.  RingOps  /\  A  e.  X  /\  C  e.  X )  ->  ( A H C )  e.  X )
33323adant3r2 1219 . . . . 5  |-  ( ( R  e.  RingOps  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( A H C )  e.  X
)
341, 33sylan 474 . . . 4  |-  ( ( R  e.  Dmn  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X
) )  ->  ( A H C )  e.  X )
354, 16, 5grpoeqdivid 32191 . . . 4  |-  ( ( G  e.  GrpOp  /\  ( A H B )  e.  X  /\  ( A H C )  e.  X )  ->  (
( A H B )  =  ( A H C )  <->  ( ( A H B ) (  /g  `  G ) ( A H C ) )  =  Z ) )
3628, 31, 34, 35syl3anc 1269 . . 3  |-  ( ( R  e.  Dmn  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X
) )  ->  (
( A H B )  =  ( A H C )  <->  ( ( A H B ) (  /g  `  G ) ( A H C ) )  =  Z ) )
3736adantr 467 . 2  |-  ( ( ( R  e.  Dmn  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X
) )  /\  A  =/=  Z )  ->  (
( A H B )  =  ( A H C )  <->  ( ( A H B ) (  /g  `  G ) ( A H C ) )  =  Z ) )
384, 16, 5grpoeqdivid 32191 . . . . . 6  |-  ( ( G  e.  GrpOp  /\  B  e.  X  /\  C  e.  X )  ->  ( B  =  C  <->  ( B
(  /g  `  G ) C )  =  Z ) )
39383expb 1210 . . . . 5  |-  ( ( G  e.  GrpOp  /\  ( B  e.  X  /\  C  e.  X )
)  ->  ( B  =  C  <->  ( B (  /g  `  G ) C )  =  Z ) )
4011, 39sylan 474 . . . 4  |-  ( ( R  e.  Dmn  /\  ( B  e.  X  /\  C  e.  X
) )  ->  ( B  =  C  <->  ( B
(  /g  `  G ) C )  =  Z ) )
41403adantr1 1168 . . 3  |-  ( ( R  e.  Dmn  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X
) )  ->  ( B  =  C  <->  ( B
(  /g  `  G ) C )  =  Z ) )
4241adantr 467 . 2  |-  ( ( ( R  e.  Dmn  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X
) )  /\  A  =/=  Z )  ->  ( B  =  C  <->  ( B
(  /g  `  G ) C )  =  Z ) )
4327, 37, 423imtr4d 272 1  |-  ( ( ( R  e.  Dmn  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X
) )  /\  A  =/=  Z )  ->  (
( A H B )  =  ( A H C )  ->  B  =  C )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 188    \/ wo 370    /\ wa 371    /\ w3a 986    = wceq 1446    e. wcel 1889    =/= wne 2624   ran crn 4838   ` cfv 5585  (class class class)co 6295   1stc1st 6796   2ndc2nd 6797   GrpOpcgr 25926  GIdcgi 25927    /g cgs 25929   RingOpscrngo 26115   Dmncdmn 32292
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-8 1891  ax-9 1898  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093  ax-ext 2433  ax-rep 4518  ax-sep 4528  ax-nul 4537  ax-pow 4584  ax-pr 4642  ax-un 6588
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 987  df-3an 988  df-tru 1449  df-ex 1666  df-nf 1670  df-sb 1800  df-eu 2305  df-mo 2306  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2583  df-ne 2626  df-ral 2744  df-rex 2745  df-reu 2746  df-rmo 2747  df-rab 2748  df-v 3049  df-sbc 3270  df-csb 3366  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-pss 3422  df-nul 3734  df-if 3884  df-pw 3955  df-sn 3971  df-pr 3973  df-tp 3975  df-op 3977  df-uni 4202  df-int 4238  df-iun 4283  df-br 4406  df-opab 4465  df-mpt 4466  df-tr 4501  df-eprel 4748  df-id 4752  df-po 4758  df-so 4759  df-fr 4796  df-we 4798  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-ord 5429  df-on 5430  df-lim 5431  df-suc 5432  df-iota 5549  df-fun 5587  df-fn 5588  df-f 5589  df-f1 5590  df-fo 5591  df-f1o 5592  df-fv 5593  df-riota 6257  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6698  df-1st 6798  df-2nd 6799  df-1o 7187  df-er 7368  df-en 7575  df-dom 7576  df-sdom 7577  df-fin 7578  df-grpo 25931  df-gid 25932  df-ginv 25933  df-gdiv 25934  df-ablo 26022  df-ass 26053  df-exid 26055  df-mgmOLD 26059  df-sgrOLD 26071  df-mndo 26078  df-rngo 26116  df-com2 26151  df-crngo 32241  df-idl 32255  df-pridl 32256  df-prrngo 32293  df-dmn 32294  df-igen 32305
This theorem is referenced by:  dmncan2  32322
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