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Theorem grpodivfval 21820
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 5123 . . . . 5  |-  ( G  e.  GrpOp  ->  ran  G  e. 
_V )
42, 3syl5eqel 2519 . . . 4  |-  ( G  e.  GrpOp  ->  X  e.  _V )
5 mpt2exga 6416 . . . 4  |-  ( ( X  e.  _V  /\  X  e.  _V )  ->  ( x  e.  X ,  y  e.  X  |->  ( x G ( N `  y ) ) )  e.  _V )
64, 4, 5syl2anc 643 . . 3  |-  ( G  e.  GrpOp  ->  ( x  e.  X ,  y  e.  X  |->  ( x G ( N `  y
) ) )  e. 
_V )
7 rneq 5087 . . . . . 6  |-  ( g  =  G  ->  ran  g  =  ran  G )
87, 2syl6eqr 2485 . . . . 5  |-  ( g  =  G  ->  ran  g  =  X )
9 id 20 . . . . . 6  |-  ( g  =  G  ->  g  =  G )
10 eqidd 2436 . . . . . 6  |-  ( g  =  G  ->  x  =  x )
11 fveq2 5720 . . . . . . . 8  |-  ( g  =  G  ->  ( inv `  g )  =  ( inv `  G
) )
12 grpdiv.2 . . . . . . . 8  |-  N  =  ( inv `  G
)
1311, 12syl6eqr 2485 . . . . . . 7  |-  ( g  =  G  ->  ( inv `  g )  =  N )
1413fveq1d 5722 . . . . . 6  |-  ( g  =  G  ->  (
( inv `  g
) `  y )  =  ( N `  y ) )
159, 10, 14oveq123d 6094 . . . . 5  |-  ( g  =  G  ->  (
x g ( ( inv `  g ) `
 y ) )  =  ( x G ( N `  y
) ) )
168, 8, 15mpt2eq123dv 6128 . . . 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 21772 . . . 4  |-  /g  =  ( g  e.  GrpOp  |->  ( x  e.  ran  g ,  y  e.  ran  g  |->  ( x g ( ( inv `  g ) `  y
) ) ) )
1816, 17fvmptg 5796 . . 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 650 . 2  |-  ( G  e.  GrpOp  ->  (  /g  `  G )  =  ( x  e.  X , 
y  e.  X  |->  ( x G ( N `
 y ) ) ) )
201, 19syl5eq 2479 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 1652    e. wcel 1725   _Vcvv 2948   ran crn 4871   ` cfv 5446  (class class class)co 6073    e. cmpt2 6075   GrpOpcgr 21764   invcgn 21766    /g cgs 21767
This theorem is referenced by:  grpodivval  21821  grpodivf  21824  nvmfval  22115
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-gdiv 21772
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