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Theorem grpodivval 23851
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 23850 . . . 4  |-  ( G  e.  GrpOp  ->  D  =  ( x  e.  X ,  y  e.  X  |->  ( x G ( N `  y ) ) ) )
54oveqd 6193 . . 3  |-  ( G  e.  GrpOp  ->  ( A D B )  =  ( A ( x  e.  X ,  y  e.  X  |->  ( x G ( N `  y
) ) ) B ) )
6 oveq1 6183 . . . 4  |-  ( x  =  A  ->  (
x G ( N `
 y ) )  =  ( A G ( N `  y
) ) )
7 fveq2 5775 . . . . 5  |-  ( y  =  B  ->  ( N `  y )  =  ( N `  B ) )
87oveq2d 6192 . . . 4  |-  ( y  =  B  ->  ( A G ( N `  y ) )  =  ( A G ( N `  B ) ) )
9 eqid 2450 . . . 4  |-  ( x  e.  X ,  y  e.  X  |->  ( x G ( N `  y ) ) )  =  ( x  e.  X ,  y  e.  X  |->  ( x G ( N `  y
) ) )
10 ovex 6201 . . . 4  |-  ( A G ( N `  B ) )  e. 
_V
116, 8, 9, 10ovmpt2 6312 . . 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 2510 . 2  |-  ( ( G  e.  GrpOp  /\  ( A  e.  X  /\  B  e.  X )
)  ->  ( A D B )  =  ( A G ( N `
 B ) ) )
13123impb 1184 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 965    = wceq 1370    e. wcel 1757   ran crn 4925   ` cfv 5502  (class class class)co 6176    |-> cmpt2 6178   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-ov 6179  df-oprab 6180  df-mpt2 6181  df-1st 6663  df-2nd 6664  df-gdiv 23802
This theorem is referenced by:  grpodivinv  23852  grpoinvdiv  23853  grpodivdiv  23856  grpomuldivass  23857  grpodivid  23858  grponpcan  23860  grpopnpcan2  23861  grponnncan2  23862  ablodivdiv4  23899  nvmval  24143  rngosub  28878
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