Users' Mathboxes Mathbox for Jeff Madsen < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  dmncan1 Structured version   Unicode version

Theorem dmncan1 26688
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 26669 . . . . . 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 2438 . . . . . . 7  |-  (  /g  `  G )  =  (  /g  `  G )
62, 3, 4, 5rngosubdi 26571 . . . . . 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 459 . . . . 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 453 . . . 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 2446 . . 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 21980 . . . . . . . . . . . 12  |-  ( R  e.  RingOps  ->  G  e.  GrpOp )
111, 10syl 16 . . . . . . . . . . 11  |-  ( R  e.  Dmn  ->  G  e.  GrpOp )
124, 5grpodivcl 21837 . . . . . . . . . . . 12  |-  ( ( G  e.  GrpOp  /\  B  e.  X  /\  C  e.  X )  ->  ( B (  /g  `  G
) C )  e.  X )
13123expb 1155 . . . . . . . . . . 11  |-  ( ( G  e.  GrpOp  /\  ( B  e.  X  /\  C  e.  X )
)  ->  ( B
(  /g  `  G ) C )  e.  X
)
1411, 13sylan 459 . . . . . . . . . 10  |-  ( ( R  e.  Dmn  /\  ( B  e.  X  /\  C  e.  X
) )  ->  ( B (  /g  `  G
) C )  e.  X )
1514adantlr 697 . . . . . . . . 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 26687 . . . . . . . . . . 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 1172 . . . . . . . . . 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 423 . . . . . . . . 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 458 . . . . . . . 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 597 . . . . . . 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 1169 . . . . . 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 2690 . . . . . 6  |-  ( ( A  =  Z  \/  ( B (  /g  `  G
) C )  =  Z )  <->  ( A  =/=  Z  ->  ( B
(  /g  `  G ) C )  =  Z ) )
2422, 23syl6ib 219 . . . . 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 75 . . . 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 420 . . 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 228 . 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 453 . . . 4  |-  ( ( R  e.  Dmn  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X
) )  ->  G  e.  GrpOp )
292, 3, 4rngocl 21972 . . . . . 6  |-  ( ( R  e.  RingOps  /\  A  e.  X  /\  B  e.  X )  ->  ( A H B )  e.  X )
30293adant3r3 1165 . . . . 5  |-  ( ( R  e.  RingOps  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( A H B )  e.  X
)
311, 30sylan 459 . . . 4  |-  ( ( R  e.  Dmn  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X
) )  ->  ( A H B )  e.  X )
322, 3, 4rngocl 21972 . . . . . 6  |-  ( ( R  e.  RingOps  /\  A  e.  X  /\  C  e.  X )  ->  ( A H C )  e.  X )
33323adant3r2 1164 . . . . 5  |-  ( ( R  e.  RingOps  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( A H C )  e.  X
)
341, 33sylan 459 . . . 4  |-  ( ( R  e.  Dmn  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X
) )  ->  ( A H C )  e.  X )
354, 16, 5grpoeqdivid 26558 . . . 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 1185 . . 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 453 . 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 26558 . . . . . 6  |-  ( ( G  e.  GrpOp  /\  B  e.  X  /\  C  e.  X )  ->  ( B  =  C  <->  ( B
(  /g  `  G ) C )  =  Z ) )
39383expb 1155 . . . . 5  |-  ( ( G  e.  GrpOp  /\  ( B  e.  X  /\  C  e.  X )
)  ->  ( B  =  C  <->  ( B (  /g  `  G ) C )  =  Z ) )
4011, 39sylan 459 . . . 4  |-  ( ( R  e.  Dmn  /\  ( B  e.  X  /\  C  e.  X
) )  ->  ( B  =  C  <->  ( B
(  /g  `  G ) C )  =  Z ) )
41403adantr1 1117 . . 3  |-  ( ( R  e.  Dmn  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X
) )  ->  ( B  =  C  <->  ( B
(  /g  `  G ) C )  =  Z ) )
4241adantr 453 . 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 261 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 set class
Syntax hints:    -> wi 4    <-> wb 178    \/ wo 359    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726    =/= wne 2601   ran crn 4881   ` cfv 5456  (class class class)co 6083   1stc1st 6349   2ndc2nd 6350   GrpOpcgr 21776  GIdcgi 21777    /g cgs 21779   RingOpscrngo 21965   Dmncdmn 26659
This theorem is referenced by:  dmncan2  26689
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-1st 6351  df-2nd 6352  df-riota 6551  df-1o 6726  df-er 6907  df-en 7112  df-dom 7113  df-sdom 7114  df-fin 7115  df-grpo 21781  df-gid 21782  df-ginv 21783  df-gdiv 21784  df-ablo 21872  df-ass 21903  df-exid 21905  df-mgm 21909  df-sgr 21921  df-mndo 21928  df-rngo 21966  df-com2 22001  df-crngo 26608  df-idl 26622  df-pridl 26623  df-prrngo 26660  df-dmn 26661  df-igen 26672
  Copyright terms: Public domain W3C validator