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Definition df-sbg 15538
Description: Define group subtraction (also called division for multiplicative groups). (Contributed by NM, 31-Mar-2014.)
Assertion
Ref Expression
df-sbg  |-  -g  =  ( g  e.  _V  |->  ( x  e.  ( Base `  g ) ,  y  e.  ( Base `  g )  |->  ( x ( +g  `  g
) ( ( invg `  g ) `
 y ) ) ) )
Distinct variable group:    x, g, y

Detailed syntax breakdown of Definition df-sbg
StepHypRef Expression
1 csg 15405 . 2  class  -g
2 vg . . 3  setvar  g
3 cvv 2967 . . 3  class  _V
4 vx . . . 4  setvar  x
5 vy . . . 4  setvar  y
62cv 1368 . . . . 5  class  g
7 cbs 14166 . . . . 5  class  Base
86, 7cfv 5413 . . . 4  class  ( Base `  g )
94cv 1368 . . . . 5  class  x
105cv 1368 . . . . . 6  class  y
11 cminusg 15403 . . . . . . 7  class  invg
126, 11cfv 5413 . . . . . 6  class  ( invg `  g )
1310, 12cfv 5413 . . . . 5  class  ( ( invg `  g
) `  y )
14 cplusg 14230 . . . . . 6  class  +g
156, 14cfv 5413 . . . . 5  class  ( +g  `  g )
169, 13, 15co 6086 . . . 4  class  ( x ( +g  `  g
) ( ( invg `  g ) `
 y ) )
174, 5, 8, 8, 16cmpt2 6088 . . 3  class  ( x  e.  ( Base `  g
) ,  y  e.  ( Base `  g
)  |->  ( x ( +g  `  g ) ( ( invg `  g ) `  y
) ) )
182, 3, 17cmpt 4345 . 2  class  ( g  e.  _V  |->  ( x  e.  ( Base `  g
) ,  y  e.  ( Base `  g
)  |->  ( x ( +g  `  g ) ( ( invg `  g ) `  y
) ) ) )
191, 18wceq 1369 1  wff  -g  =  ( g  e.  _V  |->  ( x  e.  ( Base `  g ) ,  y  e.  ( Base `  g )  |->  ( x ( +g  `  g
) ( ( invg `  g ) `
 y ) ) ) )
Colors of variables: wff setvar class
This definition is referenced by:  grpsubfval  15571
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