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Theorem grponnncan2 25454
Description: Cancellation law for group division. (nnncan2 9847 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 2454 . . . . 5  |-  ( inv `  G )  =  ( inv `  G )
3 grpdivf.3 . . . . 5  |-  D  =  (  /g  `  G
)
41, 2, 3grpodivval 25443 . . . 4  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  C  e.  X )  ->  ( A D C )  =  ( A G ( ( inv `  G
) `  C )
) )
543adant3r2 1204 . . 3  |-  ( ( G  e.  GrpOp  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( A D C )  =  ( A G ( ( inv `  G ) `
 C ) ) )
61, 2, 3grpodivval 25443 . . . 4  |-  ( ( G  e.  GrpOp  /\  B  e.  X  /\  C  e.  X )  ->  ( B D C )  =  ( B G ( ( inv `  G
) `  C )
) )
763adant3r1 1203 . . 3  |-  ( ( G  e.  GrpOp  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( B D C )  =  ( B G ( ( inv `  G ) `
 C ) ) )
85, 7oveq12d 6288 . 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 25426 . . . . . 6  |-  ( ( G  e.  GrpOp  /\  C  e.  X )  ->  (
( inv `  G
) `  C )  e.  X )
1211ex 432 . . . . 5  |-  ( G  e.  GrpOp  ->  ( C  e.  X  ->  ( ( inv `  G ) `
 C )  e.  X ) )
139, 10, 123anim123d 1304 . . . 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 427 . . 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 25453 . . 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 468 . 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 2495 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 367    /\ w3a 971    = wceq 1398    e. wcel 1823   ran crn 4989   ` cfv 5570  (class class class)co 6270   GrpOpcgr 25386   invcgn 25388    /g cgs 25389
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-1st 6773  df-2nd 6774  df-grpo 25391  df-gid 25392  df-ginv 25393  df-gdiv 25394
This theorem is referenced by:  nvnnncan2  25742
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