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Theorem ablonncan 23780
Description: Cancellation law for group division. (nncan 9637 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 1275 . . . 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 23776 . . . 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 1183 . 2  |-  ( ( G  e.  AbelOp  /\  A  e.  X  /\  B  e.  X )  ->  ( A D ( A D B ) )  =  ( ( A D A ) G B ) )
8 ablogrpo 23770 . . . . 5  |-  ( G  e.  AbelOp  ->  G  e.  GrpOp )
9 eqid 2442 . . . . . 6  |-  (GId `  G )  =  (GId
`  G )
103, 4, 9grpodivid 23736 . . . . 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 1008 . . 3  |-  ( ( G  e.  AbelOp  /\  A  e.  X  /\  B  e.  X )  ->  ( A D A )  =  (GId `  G )
)
1312oveq1d 6105 . 2  |-  ( ( G  e.  AbelOp  /\  A  e.  X  /\  B  e.  X )  ->  (
( A D A ) G B )  =  ( (GId `  G ) G B ) )
143, 9grpolid 23705 . . . 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 1007 . 2  |-  ( ( G  e.  AbelOp  /\  A  e.  X  /\  B  e.  X )  ->  (
(GId `  G ) G B )  =  B )
177, 13, 163eqtrd 2478 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 965    = wceq 1369    e. wcel 1756   ran crn 4840   ` cfv 5417  (class class class)co 6090   GrpOpcgr 23672  GIdcgi 23673    /g cgs 23675   AbelOpcablo 23767
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 4402  ax-sep 4412  ax-nul 4420  ax-pow 4469  ax-pr 4530  ax-un 6371
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 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2719  df-rex 2720  df-reu 2721  df-rab 2723  df-v 2973  df-sbc 3186  df-csb 3288  df-dif 3330  df-un 3332  df-in 3334  df-ss 3341  df-nul 3637  df-if 3791  df-pw 3861  df-sn 3877  df-pr 3879  df-op 3883  df-uni 4091  df-iun 4172  df-br 4292  df-opab 4350  df-mpt 4351  df-id 4635  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5380  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-riota 6051  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-1st 6576  df-2nd 6577  df-grpo 23677  df-gid 23678  df-ginv 23679  df-gdiv 23680  df-ablo 23768
This theorem is referenced by:  ablonnncan1  23781
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