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Theorem grponnncan2 23862
Description: Cancellation law for group division. (nnncan2 9733 analog.) (Contributed by NM, 15-Feb-2008.) (New usage is discouraged.)
Hypotheses
Ref Expression
grpdivf.1  |-  X  =  ran  G
grpdivf.3  |-  D  =  (  /g  `  G
)
Assertion
Ref Expression
grponnncan2  |-  ( ( G  e.  GrpOp  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( ( A D C ) D ( B D C ) )  =  ( A D B ) )

Proof of Theorem grponnncan2
StepHypRef Expression
1 grpdivf.1 . . . . 5  |-  X  =  ran  G
2 eqid 2450 . . . . 5  |-  ( inv `  G )  =  ( inv `  G )
3 grpdivf.3 . . . . 5  |-  D  =  (  /g  `  G
)
41, 2, 3grpodivval 23851 . . . 4  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  C  e.  X )  ->  ( A D C )  =  ( A G ( ( inv `  G
) `  C )
) )
543adant3r2 1198 . . 3  |-  ( ( G  e.  GrpOp  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( A D C )  =  ( A G ( ( inv `  G ) `
 C ) ) )
61, 2, 3grpodivval 23851 . . . 4  |-  ( ( G  e.  GrpOp  /\  B  e.  X  /\  C  e.  X )  ->  ( B D C )  =  ( B G ( ( inv `  G
) `  C )
) )
763adant3r1 1197 . . 3  |-  ( ( G  e.  GrpOp  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( B D C )  =  ( B G ( ( inv `  G ) `
 C ) ) )
85, 7oveq12d 6194 . 2  |-  ( ( G  e.  GrpOp  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( ( A D C ) D ( B D C ) )  =  ( ( A G ( ( inv `  G
) `  C )
) D ( B G ( ( inv `  G ) `  C
) ) ) )
9 idd 24 . . . . 5  |-  ( G  e.  GrpOp  ->  ( A  e.  X  ->  A  e.  X ) )
10 idd 24 . . . . 5  |-  ( G  e.  GrpOp  ->  ( B  e.  X  ->  B  e.  X ) )
111, 2grpoinvcl 23834 . . . . . 6  |-  ( ( G  e.  GrpOp  /\  C  e.  X )  ->  (
( inv `  G
) `  C )  e.  X )
1211ex 434 . . . . 5  |-  ( G  e.  GrpOp  ->  ( C  e.  X  ->  ( ( inv `  G ) `
 C )  e.  X ) )
139, 10, 123anim123d 1297 . . . 4  |-  ( G  e.  GrpOp  ->  ( ( A  e.  X  /\  B  e.  X  /\  C  e.  X )  ->  ( A  e.  X  /\  B  e.  X  /\  ( ( inv `  G
) `  C )  e.  X ) ) )
1413imp 429 . . 3  |-  ( ( G  e.  GrpOp  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( A  e.  X  /\  B  e.  X  /\  ( ( inv `  G ) `
 C )  e.  X ) )
151, 3grpopnpcan2 23861 . . 3  |-  ( ( G  e.  GrpOp  /\  ( A  e.  X  /\  B  e.  X  /\  ( ( inv `  G
) `  C )  e.  X ) )  -> 
( ( A G ( ( inv `  G
) `  C )
) D ( B G ( ( inv `  G ) `  C
) ) )  =  ( A D B ) )
1614, 15syldan 470 . 2  |-  ( ( G  e.  GrpOp  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( ( A G ( ( inv `  G ) `  C
) ) D ( B G ( ( inv `  G ) `
 C ) ) )  =  ( A D B ) )
178, 16eqtrd 2490 1  |-  ( ( G  e.  GrpOp  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( ( A D C ) D ( B D C ) )  =  ( A D B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1757   ran crn 4925   ` cfv 5502  (class class class)co 6176   GrpOpcgr 23794   invcgn 23796    /g cgs 23797
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1709  ax-7 1729  ax-8 1759  ax-9 1761  ax-10 1776  ax-11 1781  ax-12 1793  ax-13 1944  ax-ext 2429  ax-rep 4487  ax-sep 4497  ax-nul 4505  ax-pow 4554  ax-pr 4615  ax-un 6458
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1702  df-eu 2263  df-mo 2264  df-clab 2436  df-cleq 2442  df-clel 2445  df-nfc 2598  df-ne 2643  df-ral 2797  df-rex 2798  df-reu 2799  df-rab 2801  df-v 3056  df-sbc 3271  df-csb 3373  df-dif 3415  df-un 3417  df-in 3419  df-ss 3426  df-nul 3722  df-if 3876  df-pw 3946  df-sn 3962  df-pr 3964  df-op 3968  df-uni 4176  df-iun 4257  df-br 4377  df-opab 4435  df-mpt 4436  df-id 4720  df-xp 4930  df-rel 4931  df-cnv 4932  df-co 4933  df-dm 4934  df-rn 4935  df-res 4936  df-ima 4937  df-iota 5465  df-fun 5504  df-fn 5505  df-f 5506  df-f1 5507  df-fo 5508  df-f1o 5509  df-fv 5510  df-riota 6137  df-ov 6179  df-oprab 6180  df-mpt2 6181  df-1st 6663  df-2nd 6664  df-grpo 23799  df-gid 23800  df-ginv 23801  df-gdiv 23802
This theorem is referenced by:  nvnnncan2  24150
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