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Theorem grpodivfval 23744
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 6525 . . . . 5  |-  ( G  e.  GrpOp  ->  ran  G  e. 
_V )
42, 3syl5eqel 2527 . . . 4  |-  ( G  e.  GrpOp  ->  X  e.  _V )
5 mpt2exga 6664 . . . 4  |-  ( ( X  e.  _V  /\  X  e.  _V )  ->  ( x  e.  X ,  y  e.  X  |->  ( x G ( N `  y ) ) )  e.  _V )
64, 4, 5syl2anc 661 . . 3  |-  ( G  e.  GrpOp  ->  ( x  e.  X ,  y  e.  X  |->  ( x G ( N `  y
) ) )  e. 
_V )
7 rneq 5080 . . . . . 6  |-  ( g  =  G  ->  ran  g  =  ran  G )
87, 2syl6eqr 2493 . . . . 5  |-  ( g  =  G  ->  ran  g  =  X )
9 id 22 . . . . . 6  |-  ( g  =  G  ->  g  =  G )
10 eqidd 2444 . . . . . 6  |-  ( g  =  G  ->  x  =  x )
11 fveq2 5706 . . . . . . . 8  |-  ( g  =  G  ->  ( inv `  g )  =  ( inv `  G
) )
12 grpdiv.2 . . . . . . . 8  |-  N  =  ( inv `  G
)
1311, 12syl6eqr 2493 . . . . . . 7  |-  ( g  =  G  ->  ( inv `  g )  =  N )
1413fveq1d 5708 . . . . . 6  |-  ( g  =  G  ->  (
( inv `  g
) `  y )  =  ( N `  y ) )
159, 10, 14oveq123d 6127 . . . . 5  |-  ( g  =  G  ->  (
x g ( ( inv `  g ) `
 y ) )  =  ( x G ( N `  y
) ) )
168, 8, 15mpt2eq123dv 6163 . . . 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 23696 . . . 4  |-  /g  =  ( g  e.  GrpOp  |->  ( x  e.  ran  g ,  y  e.  ran  g  |->  ( x g ( ( inv `  g ) `  y
) ) ) )
1816, 17fvmptg 5787 . . 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 668 . 2  |-  ( G  e.  GrpOp  ->  (  /g  `  G )  =  ( x  e.  X , 
y  e.  X  |->  ( x G ( N `
 y ) ) ) )
201, 19syl5eq 2487 1  |-  ( G  e.  GrpOp  ->  D  =  ( x  e.  X ,  y  e.  X  |->  ( x G ( N `  y ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1369    e. wcel 1756   _Vcvv 2987   ran crn 4856   ` cfv 5433  (class class class)co 6106    e. cmpt2 6108   GrpOpcgr 23688   invcgn 23690    /g cgs 23691
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4418  ax-sep 4428  ax-nul 4436  ax-pow 4485  ax-pr 4546  ax-un 6387
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-ral 2735  df-rex 2736  df-reu 2737  df-rab 2739  df-v 2989  df-sbc 3202  df-csb 3304  df-dif 3346  df-un 3348  df-in 3350  df-ss 3357  df-nul 3653  df-if 3807  df-pw 3877  df-sn 3893  df-pr 3895  df-op 3899  df-uni 4107  df-iun 4188  df-br 4308  df-opab 4366  df-mpt 4367  df-id 4651  df-xp 4861  df-rel 4862  df-cnv 4863  df-co 4864  df-dm 4865  df-rn 4866  df-res 4867  df-ima 4868  df-iota 5396  df-fun 5435  df-fn 5436  df-f 5437  df-f1 5438  df-fo 5439  df-f1o 5440  df-fv 5441  df-ov 6109  df-oprab 6110  df-mpt2 6111  df-1st 6592  df-2nd 6593  df-gdiv 23696
This theorem is referenced by:  grpodivval  23745  grpodivf  23748  nvmfval  24039
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