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Theorem ablonnncan1 25591
Description: Cancellation law for group division. (nnncan1 9811 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 1003 . . . 4  |-  ( ( G  e.  AbelOp  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  A  e.  X )
2 simpr2 1004 . . . 4  |-  ( ( G  e.  AbelOp  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  B  e.  X )
3 ablogrpo 25580 . . . . . 6  |-  ( G  e.  AbelOp  ->  G  e.  GrpOp )
4 abldiv.1 . . . . . . 7  |-  X  =  ran  G
5 abldiv.3 . . . . . . 7  |-  D  =  (  /g  `  G
)
64, 5grpodivcl 25543 . . . . . 6  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  C  e.  X )  ->  ( A D C )  e.  X )
73, 6syl3an1 1263 . . . . 5  |-  ( ( G  e.  AbelOp  /\  A  e.  X  /\  C  e.  X )  ->  ( A D C )  e.  X )
873adant3r2 1207 . . . 4  |-  ( ( G  e.  AbelOp  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( A D C )  e.  X
)
91, 2, 83jca 1177 . . 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 25588 . . 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 468 . 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 25590 . . . 4  |-  ( ( G  e.  AbelOp  /\  A  e.  X  /\  C  e.  X )  ->  ( A D ( A D C ) )  =  C )
13123adant3r2 1207 . . 3  |-  ( ( G  e.  AbelOp  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( A D ( A D C ) )  =  C )
1413oveq1d 6249 . 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 2443 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 367    /\ w3a 974    = wceq 1405    e. wcel 1842   ran crn 4943   ` cfv 5525  (class class class)co 6234   GrpOpcgr 25482    /g cgs 25485   AbelOpcablo 25577
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6530
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-reu 2760  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-iun 4272  df-br 4395  df-opab 4453  df-mpt 4454  df-id 4737  df-xp 4948  df-rel 4949  df-cnv 4950  df-co 4951  df-dm 4952  df-rn 4953  df-res 4954  df-ima 4955  df-iota 5489  df-fun 5527  df-fn 5528  df-f 5529  df-f1 5530  df-fo 5531  df-f1o 5532  df-fv 5533  df-riota 6196  df-ov 6237  df-oprab 6238  df-mpt2 6239  df-1st 6738  df-2nd 6739  df-grpo 25487  df-gid 25488  df-ginv 25489  df-gdiv 25490  df-ablo 25578
This theorem is referenced by:  nvnnncan1  25837
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