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Theorem grpodivfval 20909
Description: Group division (or subtraction) operation. (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
grpodivfval  |-  ( G  e.  GrpOp  ->  D  =  ( x  e.  X ,  y  e.  X  |->  ( x G ( N `  y ) ) ) )
Distinct variable groups:    x, y, G    x, N, y    x, X, y
Allowed substitution hints:    D( x, y)

Proof of Theorem grpodivfval
Dummy variable  g is distinct from all other variables.
StepHypRef Expression
1 grpdiv.3 . 2  |-  D  =  (  /g  `  G
)
2 grpdiv.1 . . . . 5  |-  X  =  ran  G
3 rnexg 4940 . . . . 5  |-  ( G  e.  GrpOp  ->  ran  G  e. 
_V )
42, 3syl5eqel 2367 . . . 4  |-  ( G  e.  GrpOp  ->  X  e.  _V )
5 mpt2exga 6197 . . . 4  |-  ( ( X  e.  _V  /\  X  e.  _V )  ->  ( x  e.  X ,  y  e.  X  |->  ( x G ( N `  y ) ) )  e.  _V )
64, 4, 5syl2anc 642 . . 3  |-  ( G  e.  GrpOp  ->  ( x  e.  X ,  y  e.  X  |->  ( x G ( N `  y
) ) )  e. 
_V )
7 rneq 4904 . . . . . 6  |-  ( g  =  G  ->  ran  g  =  ran  G )
87, 2syl6eqr 2333 . . . . 5  |-  ( g  =  G  ->  ran  g  =  X )
9 id 19 . . . . . 6  |-  ( g  =  G  ->  g  =  G )
10 eqidd 2284 . . . . . 6  |-  ( g  =  G  ->  x  =  x )
11 fveq2 5525 . . . . . . . 8  |-  ( g  =  G  ->  ( inv `  g )  =  ( inv `  G
) )
12 grpdiv.2 . . . . . . . 8  |-  N  =  ( inv `  G
)
1311, 12syl6eqr 2333 . . . . . . 7  |-  ( g  =  G  ->  ( inv `  g )  =  N )
1413fveq1d 5527 . . . . . 6  |-  ( g  =  G  ->  (
( inv `  g
) `  y )  =  ( N `  y ) )
159, 10, 14oveq123d 5879 . . . . 5  |-  ( g  =  G  ->  (
x g ( ( inv `  g ) `
 y ) )  =  ( x G ( N `  y
) ) )
168, 8, 15mpt2eq123dv 5910 . . . 4  |-  ( g  =  G  ->  (
x  e.  ran  g ,  y  e.  ran  g  |->  ( x g ( ( inv `  g
) `  y )
) )  =  ( x  e.  X , 
y  e.  X  |->  ( x G ( N `
 y ) ) ) )
17 df-gdiv 20861 . . . 4  |-  /g  =  ( g  e.  GrpOp  |->  ( x  e.  ran  g ,  y  e.  ran  g  |->  ( x g ( ( inv `  g ) `  y
) ) ) )
1816, 17fvmptg 5600 . . 3  |-  ( ( G  e.  GrpOp  /\  (
x  e.  X , 
y  e.  X  |->  ( x G ( N `
 y ) ) )  e.  _V )  ->  (  /g  `  G
)  =  ( x  e.  X ,  y  e.  X  |->  ( x G ( N `  y ) ) ) )
196, 18mpdan 649 . 2  |-  ( G  e.  GrpOp  ->  (  /g  `  G )  =  ( x  e.  X , 
y  e.  X  |->  ( x G ( N `
 y ) ) ) )
201, 19syl5eq 2327 1  |-  ( G  e.  GrpOp  ->  D  =  ( x  e.  X ,  y  e.  X  |->  ( x G ( N `  y ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623    e. wcel 1684   _Vcvv 2788   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:  grpodivval  20910  grpodivf  20913  nvmfval  21202
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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