MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ablonnncan Structured version   Unicode version

Theorem ablonnncan 23780
Description: Cancellation law for group division. (nnncan 9644 analog.) (Contributed by NM, 29-Feb-2008.) (New usage is discouraged.)
Hypotheses
Ref Expression
abldiv.1  |-  X  =  ran  G
abldiv.3  |-  D  =  (  /g  `  G
)
Assertion
Ref Expression
ablonnncan  |-  ( ( G  e.  AbelOp  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( ( A D ( B D C ) ) D C )  =  ( A D B ) )

Proof of Theorem ablonnncan
StepHypRef Expression
1 simpr1 994 . . . 4  |-  ( ( G  e.  AbelOp  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  A  e.  X )
2 ablogrpo 23771 . . . . . 6  |-  ( G  e.  AbelOp  ->  G  e.  GrpOp )
3 abldiv.1 . . . . . . 7  |-  X  =  ran  G
4 abldiv.3 . . . . . . 7  |-  D  =  (  /g  `  G
)
53, 4grpodivcl 23734 . . . . . 6  |-  ( ( G  e.  GrpOp  /\  B  e.  X  /\  C  e.  X )  ->  ( B D C )  e.  X )
62, 5syl3an1 1251 . . . . 5  |-  ( ( G  e.  AbelOp  /\  B  e.  X  /\  C  e.  X )  ->  ( B D C )  e.  X )
763adant3r1 1196 . . . 4  |-  ( ( G  e.  AbelOp  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( B D C )  e.  X
)
8 simpr3 996 . . . 4  |-  ( ( G  e.  AbelOp  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  C  e.  X )
91, 7, 83jca 1168 . . 3  |-  ( ( G  e.  AbelOp  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( A  e.  X  /\  ( B D C )  e.  X  /\  C  e.  X ) )
103, 4ablodivdiv4 23778 . . 3  |-  ( ( G  e.  AbelOp  /\  ( A  e.  X  /\  ( B D C )  e.  X  /\  C  e.  X ) )  -> 
( ( A D ( B D C ) ) D C )  =  ( A D ( ( B D C ) G C ) ) )
119, 10syldan 470 . 2  |-  ( ( G  e.  AbelOp  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( ( A D ( B D C ) ) D C )  =  ( A D ( ( B D C ) G C ) ) )
123, 4grponpcan 23739 . . . . 5  |-  ( ( G  e.  GrpOp  /\  B  e.  X  /\  C  e.  X )  ->  (
( B D C ) G C )  =  B )
132, 12syl3an1 1251 . . . 4  |-  ( ( G  e.  AbelOp  /\  B  e.  X  /\  C  e.  X )  ->  (
( B D C ) G C )  =  B )
14133adant3r1 1196 . . 3  |-  ( ( G  e.  AbelOp  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( ( B D C ) G C )  =  B )
1514oveq2d 6107 . 2  |-  ( ( G  e.  AbelOp  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( A D ( ( B D C ) G C ) )  =  ( A D B ) )
1611, 15eqtrd 2475 1  |-  ( ( G  e.  AbelOp  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( ( A D ( B D C ) ) D C )  =  ( A D B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   ran crn 4841   ` cfv 5418  (class class class)co 6091   GrpOpcgr 23673    /g cgs 23676   AbelOpcablo 23768
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-reu 2722  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-op 3884  df-uni 4092  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-id 4636  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-riota 6052  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-1st 6577  df-2nd 6578  df-grpo 23678  df-gid 23679  df-ginv 23680  df-gdiv 23681  df-ablo 23769
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator