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Theorem grpodivfval 25445
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 6705 . . . . 5  |-  ( G  e.  GrpOp  ->  ran  G  e. 
_V )
42, 3syl5eqel 2546 . . . 4  |-  ( G  e.  GrpOp  ->  X  e.  _V )
5 mpt2exga 6849 . . . 4  |-  ( ( X  e.  _V  /\  X  e.  _V )  ->  ( x  e.  X ,  y  e.  X  |->  ( x G ( N `  y ) ) )  e.  _V )
64, 4, 5syl2anc 659 . . 3  |-  ( G  e.  GrpOp  ->  ( x  e.  X ,  y  e.  X  |->  ( x G ( N `  y
) ) )  e. 
_V )
7 rneq 5217 . . . . . 6  |-  ( g  =  G  ->  ran  g  =  ran  G )
87, 2syl6eqr 2513 . . . . 5  |-  ( g  =  G  ->  ran  g  =  X )
9 id 22 . . . . . 6  |-  ( g  =  G  ->  g  =  G )
10 eqidd 2455 . . . . . 6  |-  ( g  =  G  ->  x  =  x )
11 fveq2 5848 . . . . . . . 8  |-  ( g  =  G  ->  ( inv `  g )  =  ( inv `  G
) )
12 grpdiv.2 . . . . . . . 8  |-  N  =  ( inv `  G
)
1311, 12syl6eqr 2513 . . . . . . 7  |-  ( g  =  G  ->  ( inv `  g )  =  N )
1413fveq1d 5850 . . . . . 6  |-  ( g  =  G  ->  (
( inv `  g
) `  y )  =  ( N `  y ) )
159, 10, 14oveq123d 6291 . . . . 5  |-  ( g  =  G  ->  (
x g ( ( inv `  g ) `
 y ) )  =  ( x G ( N `  y
) ) )
168, 8, 15mpt2eq123dv 6332 . . . 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 25397 . . . 4  |-  /g  =  ( g  e.  GrpOp  |->  ( x  e.  ran  g ,  y  e.  ran  g  |->  ( x g ( ( inv `  g ) `  y
) ) ) )
1816, 17fvmptg 5929 . . 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 666 . 2  |-  ( G  e.  GrpOp  ->  (  /g  `  G )  =  ( x  e.  X , 
y  e.  X  |->  ( x G ( N `
 y ) ) ) )
201, 19syl5eq 2507 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 1398    e. wcel 1823   _Vcvv 3106   ran crn 4989   ` cfv 5570  (class class class)co 6270    |-> cmpt2 6272   GrpOpcgr 25389   invcgn 25391    /g cgs 25392
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-1st 6773  df-2nd 6774  df-gdiv 25397
This theorem is referenced by:  grpodivval  25446  grpodivf  25449  nvmfval  25740
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