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Theorem grpodivval 24918
Description: Group division (or subtraction) operation value. (Contributed by NM, 15-Feb-2008.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
grpdiv.1  |-  X  =  ran  G
grpdiv.2  |-  N  =  ( inv `  G
)
grpdiv.3  |-  D  =  (  /g  `  G
)
Assertion
Ref Expression
grpodivval  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  ( A D B )  =  ( A G ( N `  B ) ) )

Proof of Theorem grpodivval
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 grpdiv.1 . . . . 5  |-  X  =  ran  G
2 grpdiv.2 . . . . 5  |-  N  =  ( inv `  G
)
3 grpdiv.3 . . . . 5  |-  D  =  (  /g  `  G
)
41, 2, 3grpodivfval 24917 . . . 4  |-  ( G  e.  GrpOp  ->  D  =  ( x  e.  X ,  y  e.  X  |->  ( x G ( N `  y ) ) ) )
54oveqd 6299 . . 3  |-  ( G  e.  GrpOp  ->  ( A D B )  =  ( A ( x  e.  X ,  y  e.  X  |->  ( x G ( N `  y
) ) ) B ) )
6 oveq1 6289 . . . 4  |-  ( x  =  A  ->  (
x G ( N `
 y ) )  =  ( A G ( N `  y
) ) )
7 fveq2 5864 . . . . 5  |-  ( y  =  B  ->  ( N `  y )  =  ( N `  B ) )
87oveq2d 6298 . . . 4  |-  ( y  =  B  ->  ( A G ( N `  y ) )  =  ( A G ( N `  B ) ) )
9 eqid 2467 . . . 4  |-  ( x  e.  X ,  y  e.  X  |->  ( x G ( N `  y ) ) )  =  ( x  e.  X ,  y  e.  X  |->  ( x G ( N `  y
) ) )
10 ovex 6307 . . . 4  |-  ( A G ( N `  B ) )  e. 
_V
116, 8, 9, 10ovmpt2 6420 . . 3  |-  ( ( A  e.  X  /\  B  e.  X )  ->  ( A ( x  e.  X ,  y  e.  X  |->  ( x G ( N `  y ) ) ) B )  =  ( A G ( N `
 B ) ) )
125, 11sylan9eq 2528 . 2  |-  ( ( G  e.  GrpOp  /\  ( A  e.  X  /\  B  e.  X )
)  ->  ( A D B )  =  ( A G ( N `
 B ) ) )
13123impb 1192 1  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  ( A D B )  =  ( A G ( N `  B ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   ran crn 5000   ` cfv 5586  (class class class)co 6282    |-> cmpt2 6284   GrpOpcgr 24861   invcgn 24863    /g cgs 24864
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-1st 6781  df-2nd 6782  df-gdiv 24869
This theorem is referenced by:  grpodivinv  24919  grpoinvdiv  24920  grpodivdiv  24923  grpomuldivass  24924  grpodivid  24925  grponpcan  24927  grpopnpcan2  24928  grponnncan2  24929  ablodivdiv4  24966  nvmval  25210  rngosub  29952
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