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Theorem dmncan1 30092
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 30073 . . . . . 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 2467 . . . . . . 7  |-  (  /g  `  G )  =  (  /g  `  G )
62, 3, 4, 5rngosubdi 29975 . . . . . 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 471 . . . . 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 465 . . . 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 2469 . . 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 25084 . . . . . . . . . . . 12  |-  ( R  e.  RingOps  ->  G  e.  GrpOp )
111, 10syl 16 . . . . . . . . . . 11  |-  ( R  e.  Dmn  ->  G  e.  GrpOp )
124, 5grpodivcl 24941 . . . . . . . . . . . 12  |-  ( ( G  e.  GrpOp  /\  B  e.  X  /\  C  e.  X )  ->  ( B (  /g  `  G
) C )  e.  X )
13123expb 1197 . . . . . . . . . . 11  |-  ( ( G  e.  GrpOp  /\  ( B  e.  X  /\  C  e.  X )
)  ->  ( B
(  /g  `  G ) C )  e.  X
)
1411, 13sylan 471 . . . . . . . . . 10  |-  ( ( R  e.  Dmn  /\  ( B  e.  X  /\  C  e.  X
) )  ->  ( B (  /g  `  G
) C )  e.  X )
1514adantlr 714 . . . . . . . . 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 30091 . . . . . . . . . . 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 1214 . . . . . . . . . 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 432 . . . . . . . . 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 470 . . . . . . . 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 612 . . . . . . 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 1211 . . . . . 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 2791 . . . . . 6  |-  ( ( A  =  Z  \/  ( B (  /g  `  G
) C )  =  Z )  <->  ( A  =/=  Z  ->  ( B
(  /g  `  G ) C )  =  Z ) )
2422, 23syl6ib 226 . . . . 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 78 . . . 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 429 . . 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 235 . 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 465 . . . 4  |-  ( ( R  e.  Dmn  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X
) )  ->  G  e.  GrpOp )
292, 3, 4rngocl 25076 . . . . . 6  |-  ( ( R  e.  RingOps  /\  A  e.  X  /\  B  e.  X )  ->  ( A H B )  e.  X )
30293adant3r3 1207 . . . . 5  |-  ( ( R  e.  RingOps  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( A H B )  e.  X
)
311, 30sylan 471 . . . 4  |-  ( ( R  e.  Dmn  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X
) )  ->  ( A H B )  e.  X )
322, 3, 4rngocl 25076 . . . . . 6  |-  ( ( R  e.  RingOps  /\  A  e.  X  /\  C  e.  X )  ->  ( A H C )  e.  X )
33323adant3r2 1206 . . . . 5  |-  ( ( R  e.  RingOps  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( A H C )  e.  X
)
341, 33sylan 471 . . . 4  |-  ( ( R  e.  Dmn  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X
) )  ->  ( A H C )  e.  X )
354, 16, 5grpoeqdivid 29962 . . . 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 1228 . . 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 465 . 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 29962 . . . . . 6  |-  ( ( G  e.  GrpOp  /\  B  e.  X  /\  C  e.  X )  ->  ( B  =  C  <->  ( B
(  /g  `  G ) C )  =  Z ) )
39383expb 1197 . . . . 5  |-  ( ( G  e.  GrpOp  /\  ( B  e.  X  /\  C  e.  X )
)  ->  ( B  =  C  <->  ( B (  /g  `  G ) C )  =  Z ) )
4011, 39sylan 471 . . . 4  |-  ( ( R  e.  Dmn  /\  ( B  e.  X  /\  C  e.  X
) )  ->  ( B  =  C  <->  ( B
(  /g  `  G ) C )  =  Z ) )
41403adantr1 1155 . . 3  |-  ( ( R  e.  Dmn  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X
) )  ->  ( B  =  C  <->  ( B
(  /g  `  G ) C )  =  Z ) )
4241adantr 465 . 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 268 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 184    \/ wo 368    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767    =/= wne 2662   ran crn 5000   ` cfv 5587  (class class class)co 6283   1stc1st 6782   2ndc2nd 6783   GrpOpcgr 24880  GIdcgi 24881    /g cgs 24883   RingOpscrngo 25069   Dmncdmn 30063
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 6575
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 5550  df-fun 5589  df-fn 5590  df-f 5591  df-f1 5592  df-fo 5593  df-f1o 5594  df-fv 5595  df-riota 6244  df-ov 6286  df-oprab 6287  df-mpt2 6288  df-om 6680  df-1st 6784  df-2nd 6785  df-1o 7130  df-er 7311  df-en 7517  df-dom 7518  df-sdom 7519  df-fin 7520  df-grpo 24885  df-gid 24886  df-ginv 24887  df-gdiv 24888  df-ablo 24976  df-ass 25007  df-exid 25009  df-mgm 25013  df-sgr 25025  df-mndo 25032  df-rngo 25070  df-com2 25105  df-crngo 30012  df-idl 30026  df-pridl 30027  df-prrngo 30064  df-dmn 30065  df-igen 30076
This theorem is referenced by:  dmncan2  30093
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