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Theorem ablonnncan1 24959
Description: Cancellation law for group division. (nnncan1 9844 analog.) (Contributed by NM, 7-Mar-2008.) (New usage is discouraged.)
Hypotheses
Ref Expression
abldiv.1  |-  X  =  ran  G
abldiv.3  |-  D  =  (  /g  `  G
)
Assertion
Ref Expression
ablonnncan1  |-  ( ( G  e.  AbelOp  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( ( A D B ) D ( A D C ) )  =  ( C D B ) )

Proof of Theorem ablonnncan1
StepHypRef Expression
1 simpr1 997 . . . 4  |-  ( ( G  e.  AbelOp  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  A  e.  X )
2 simpr2 998 . . . 4  |-  ( ( G  e.  AbelOp  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  B  e.  X )
3 ablogrpo 24948 . . . . . 6  |-  ( G  e.  AbelOp  ->  G  e.  GrpOp )
4 abldiv.1 . . . . . . 7  |-  X  =  ran  G
5 abldiv.3 . . . . . . 7  |-  D  =  (  /g  `  G
)
64, 5grpodivcl 24911 . . . . . 6  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  C  e.  X )  ->  ( A D C )  e.  X )
73, 6syl3an1 1256 . . . . 5  |-  ( ( G  e.  AbelOp  /\  A  e.  X  /\  C  e.  X )  ->  ( A D C )  e.  X )
873adant3r2 1201 . . . 4  |-  ( ( G  e.  AbelOp  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( A D C )  e.  X
)
91, 2, 83jca 1171 . . 3  |-  ( ( G  e.  AbelOp  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( A  e.  X  /\  B  e.  X  /\  ( A D C )  e.  X ) )
104, 5ablodiv32 24956 . . 3  |-  ( ( G  e.  AbelOp  /\  ( A  e.  X  /\  B  e.  X  /\  ( A D C )  e.  X ) )  ->  ( ( A D B ) D ( A D C ) )  =  ( ( A D ( A D C ) ) D B ) )
119, 10syldan 470 . 2  |-  ( ( G  e.  AbelOp  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( ( A D B ) D ( A D C ) )  =  ( ( A D ( A D C ) ) D B ) )
124, 5ablonncan 24958 . . . 4  |-  ( ( G  e.  AbelOp  /\  A  e.  X  /\  C  e.  X )  ->  ( A D ( A D C ) )  =  C )
13123adant3r2 1201 . . 3  |-  ( ( G  e.  AbelOp  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( A D ( A D C ) )  =  C )
1413oveq1d 6290 . 2  |-  ( ( G  e.  AbelOp  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( ( A D ( A D C ) ) D B )  =  ( C D B ) )
1511, 14eqtrd 2501 1  |-  ( ( G  e.  AbelOp  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( ( A D B ) D ( A D C ) )  =  ( C D B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 968    = wceq 1374    e. wcel 1762   ran crn 4993   ` cfv 5579  (class class class)co 6275   GrpOpcgr 24850    /g cgs 24853   AbelOpcablo 24945
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-id 4788  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-riota 6236  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-1st 6774  df-2nd 6775  df-grpo 24855  df-gid 24856  df-ginv 24857  df-gdiv 24858  df-ablo 24946
This theorem is referenced by:  nvnnncan1  25205
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