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Theorem grpodivval 20910
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 20909 . . . 4  |-  ( G  e.  GrpOp  ->  D  =  ( x  e.  X ,  y  e.  X  |->  ( x G ( N `  y ) ) ) )
54oveqd 5875 . . 3  |-  ( G  e.  GrpOp  ->  ( A D B )  =  ( A ( x  e.  X ,  y  e.  X  |->  ( x G ( N `  y
) ) ) B ) )
6 oveq1 5865 . . . 4  |-  ( x  =  A  ->  (
x G ( N `
 y ) )  =  ( A G ( N `  y
) ) )
7 fveq2 5525 . . . . 5  |-  ( y  =  B  ->  ( N `  y )  =  ( N `  B ) )
87oveq2d 5874 . . . 4  |-  ( y  =  B  ->  ( A G ( N `  y ) )  =  ( A G ( N `  B ) ) )
9 eqid 2283 . . . 4  |-  ( x  e.  X ,  y  e.  X  |->  ( x G ( N `  y ) ) )  =  ( x  e.  X ,  y  e.  X  |->  ( x G ( N `  y
) ) )
10 ovex 5883 . . . 4  |-  ( A G ( N `  B ) )  e. 
_V
116, 8, 9, 10ovmpt2 5983 . . 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 2335 . 2  |-  ( ( G  e.  GrpOp  /\  ( A  e.  X  /\  B  e.  X )
)  ->  ( A D B )  =  ( A G ( N `
 B ) ) )
13123impb 1147 1  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  ( A D B )  =  ( A G ( N `  B ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   ran crn 4690   ` cfv 5255  (class class class)co 5858    e. cmpt2 5860   GrpOpcgr 20853   invcgn 20855    /g cgs 20856
This theorem is referenced by:  grpodivinv  20911  grpoinvdiv  20912  grpodivdiv  20915  grpomuldivass  20916  grpodivid  20917  grponpcan  20919  grpopnpcan2  20920  grponnncan2  20921  ablodivdiv4  20958  nvmval  21200  grpodivone  25373  grpodrcan  25375  grpodlcan  25376  grpodivzer  25377  sub2vec  25472  muldisc  25481  rngosub  26579
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-gdiv 20861
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