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Theorem ablonnncan 24968
Description: Cancellation law for group division. (nnncan 9850 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 1002 . . . 4  |-  ( ( G  e.  AbelOp  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  A  e.  X )
2 ablogrpo 24959 . . . . . 6  |-  ( G  e.  AbelOp  ->  G  e.  GrpOp )
3 abldiv.1 . . . . . . 7  |-  X  =  ran  G
4 abldiv.3 . . . . . . 7  |-  D  =  (  /g  `  G
)
53, 4grpodivcl 24922 . . . . . 6  |-  ( ( G  e.  GrpOp  /\  B  e.  X  /\  C  e.  X )  ->  ( B D C )  e.  X )
62, 5syl3an1 1261 . . . . 5  |-  ( ( G  e.  AbelOp  /\  B  e.  X  /\  C  e.  X )  ->  ( B D C )  e.  X )
763adant3r1 1205 . . . 4  |-  ( ( G  e.  AbelOp  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( B D C )  e.  X
)
8 simpr3 1004 . . . 4  |-  ( ( G  e.  AbelOp  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  C  e.  X )
91, 7, 83jca 1176 . . 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 24966 . . 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 24927 . . . . 5  |-  ( ( G  e.  GrpOp  /\  B  e.  X  /\  C  e.  X )  ->  (
( B D C ) G C )  =  B )
132, 12syl3an1 1261 . . . 4  |-  ( ( G  e.  AbelOp  /\  B  e.  X  /\  C  e.  X )  ->  (
( B D C ) G C )  =  B )
14133adant3r1 1205 . . 3  |-  ( ( G  e.  AbelOp  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( ( B D C ) G C )  =  B )
1514oveq2d 6298 . 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 2508 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 973    = wceq 1379    e. wcel 1767   ran crn 5000   ` cfv 5586  (class class class)co 6282   GrpOpcgr 24861    /g cgs 24864   AbelOpcablo 24956
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 6574
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  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-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  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 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-1st 6781  df-2nd 6782  df-grpo 24866  df-gid 24867  df-ginv 24868  df-gdiv 24869  df-ablo 24957
This theorem is referenced by: (None)
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