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Definition df-drng 15514
Description: Define class of all division rings. A division ring is a ring in which the set of units is exactly the nonzero elements of the ring. (Contributed by NM, 18-Oct-2012.)
Assertion
Ref Expression
df-drng  |-  DivRing  =  {
r  e.  Ring  |  (Unit `  r )  =  ( ( Base `  r
)  \  { ( 0g `  r ) } ) }

Detailed syntax breakdown of Definition df-drng
StepHypRef Expression
1 cdr 15512 . 2  class  DivRing
2 vr . . . . . 6  set  r
32cv 1622 . . . . 5  class  r
4 cui 15421 . . . . 5  class Unit
53, 4cfv 5255 . . . 4  class  (Unit `  r )
6 cbs 13148 . . . . . 6  class  Base
73, 6cfv 5255 . . . . 5  class  ( Base `  r )
8 c0g 13400 . . . . . . 7  class  0g
93, 8cfv 5255 . . . . . 6  class  ( 0g
`  r )
109csn 3640 . . . . 5  class  { ( 0g `  r ) }
117, 10cdif 3149 . . . 4  class  ( (
Base `  r )  \  { ( 0g `  r ) } )
125, 11wceq 1623 . . 3  wff  (Unit `  r )  =  ( ( Base `  r
)  \  { ( 0g `  r ) } )
13 crg 15337 . . 3  class  Ring
1412, 2, 13crab 2547 . 2  class  { r  e.  Ring  |  (Unit `  r )  =  ( ( Base `  r
)  \  { ( 0g `  r ) } ) }
151, 14wceq 1623 1  wff  DivRing  =  {
r  e.  Ring  |  (Unit `  r )  =  ( ( Base `  r
)  \  { ( 0g `  r ) } ) }
Colors of variables: wff set class
This definition is referenced by:  isdrng  15516
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