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Theorem grpodivval 25371
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 25370 . . . 4  |-  ( G  e.  GrpOp  ->  D  =  ( x  e.  X ,  y  e.  X  |->  ( x G ( N `  y ) ) ) )
54oveqd 6313 . . 3  |-  ( G  e.  GrpOp  ->  ( A D B )  =  ( A ( x  e.  X ,  y  e.  X  |->  ( x G ( N `  y
) ) ) B ) )
6 oveq1 6303 . . . 4  |-  ( x  =  A  ->  (
x G ( N `
 y ) )  =  ( A G ( N `  y
) ) )
7 fveq2 5872 . . . . 5  |-  ( y  =  B  ->  ( N `  y )  =  ( N `  B ) )
87oveq2d 6312 . . . 4  |-  ( y  =  B  ->  ( A G ( N `  y ) )  =  ( A G ( N `  B ) ) )
9 eqid 2457 . . . 4  |-  ( x  e.  X ,  y  e.  X  |->  ( x G ( N `  y ) ) )  =  ( x  e.  X ,  y  e.  X  |->  ( x G ( N `  y
) ) )
10 ovex 6324 . . . 4  |-  ( A G ( N `  B ) )  e. 
_V
116, 8, 9, 10ovmpt2 6437 . . 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 2518 . 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 1395    e. wcel 1819   ran crn 5009   ` cfv 5594  (class class class)co 6296    |-> cmpt2 6298   GrpOpcgr 25314   invcgn 25316    /g cgs 25317
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-1st 6799  df-2nd 6800  df-gdiv 25322
This theorem is referenced by:  grpodivinv  25372  grpoinvdiv  25373  grpodivdiv  25376  grpomuldivass  25377  grpodivid  25378  grponpcan  25380  grpopnpcan2  25381  grponnncan2  25382  ablodivdiv4  25419  nvmval  25663  rngosub  30513
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