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Theorem ablonncan 24960
Description: Cancellation law for group division. (nncan 9839 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
ablonncan  |-  ( ( G  e.  AbelOp  /\  A  e.  X  /\  B  e.  X )  ->  ( A D ( A D B ) )  =  B )

Proof of Theorem ablonncan
StepHypRef Expression
1 id 22 . . . . 5  |-  ( ( A  e.  X  /\  A  e.  X  /\  B  e.  X )  ->  ( A  e.  X  /\  A  e.  X  /\  B  e.  X
) )
213anidm12 1280 . . . 4  |-  ( ( A  e.  X  /\  B  e.  X )  ->  ( A  e.  X  /\  A  e.  X  /\  B  e.  X
) )
3 abldiv.1 . . . . 5  |-  X  =  ran  G
4 abldiv.3 . . . . 5  |-  D  =  (  /g  `  G
)
53, 4ablodivdiv 24956 . . . 4  |-  ( ( G  e.  AbelOp  /\  ( A  e.  X  /\  A  e.  X  /\  B  e.  X )
)  ->  ( A D ( A D B ) )  =  ( ( A D A ) G B ) )
62, 5sylan2 474 . . 3  |-  ( ( G  e.  AbelOp  /\  ( A  e.  X  /\  B  e.  X )
)  ->  ( A D ( A D B ) )  =  ( ( A D A ) G B ) )
763impb 1187 . 2  |-  ( ( G  e.  AbelOp  /\  A  e.  X  /\  B  e.  X )  ->  ( A D ( A D B ) )  =  ( ( A D A ) G B ) )
8 ablogrpo 24950 . . . . 5  |-  ( G  e.  AbelOp  ->  G  e.  GrpOp )
9 eqid 2462 . . . . . 6  |-  (GId `  G )  =  (GId
`  G )
103, 4, 9grpodivid 24916 . . . . 5  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  ( A D A )  =  (GId `  G )
)
118, 10sylan 471 . . . 4  |-  ( ( G  e.  AbelOp  /\  A  e.  X )  ->  ( A D A )  =  (GId `  G )
)
12113adant3 1011 . . 3  |-  ( ( G  e.  AbelOp  /\  A  e.  X  /\  B  e.  X )  ->  ( A D A )  =  (GId `  G )
)
1312oveq1d 6292 . 2  |-  ( ( G  e.  AbelOp  /\  A  e.  X  /\  B  e.  X )  ->  (
( A D A ) G B )  =  ( (GId `  G ) G B ) )
143, 9grpolid 24885 . . . 4  |-  ( ( G  e.  GrpOp  /\  B  e.  X )  ->  (
(GId `  G ) G B )  =  B )
158, 14sylan 471 . . 3  |-  ( ( G  e.  AbelOp  /\  B  e.  X )  ->  (
(GId `  G ) G B )  =  B )
16153adant2 1010 . 2  |-  ( ( G  e.  AbelOp  /\  A  e.  X  /\  B  e.  X )  ->  (
(GId `  G ) G B )  =  B )
177, 13, 163eqtrd 2507 1  |-  ( ( G  e.  AbelOp  /\  A  e.  X  /\  B  e.  X )  ->  ( A D ( A D B ) )  =  B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 968    = wceq 1374    e. wcel 1762   ran crn 4995   ` cfv 5581  (class class class)co 6277   GrpOpcgr 24852  GIdcgi 24853    /g cgs 24855   AbelOpcablo 24947
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 1963  ax-ext 2440  ax-rep 4553  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681  ax-un 6569
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 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-ral 2814  df-rex 2815  df-reu 2816  df-rab 2818  df-v 3110  df-sbc 3327  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3781  df-if 3935  df-pw 4007  df-sn 4023  df-pr 4025  df-op 4029  df-uni 4241  df-iun 4322  df-br 4443  df-opab 4501  df-mpt 4502  df-id 4790  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-ima 5007  df-iota 5544  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-1st 6776  df-2nd 6777  df-grpo 24857  df-gid 24858  df-ginv 24859  df-gdiv 24860  df-ablo 24948
This theorem is referenced by:  ablonnncan1  24961
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