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Definition df-tdrg 17843
Description: Define a topological division ring (which differs from a topological field only in being potentially noncommutative), which is a division ring and topological ring such that the unit group of the division ring (which is the set of nonzero elements) is a topological group. (Contributed by Mario Carneiro, 5-Oct-2015.)
Assertion
Ref Expression
df-tdrg  |- TopDRing  =  {
r  e.  ( TopRing  i^i  DivRing )  |  ( (mulGrp `  r )s  (Unit `  r )
)  e.  TopGrp }

Detailed syntax breakdown of Definition df-tdrg
StepHypRef Expression
1 ctdrg 17839 . 2  class TopDRing
2 vr . . . . . . 7  set  r
32cv 1622 . . . . . 6  class  r
4 cmgp 15325 . . . . . 6  class mulGrp
53, 4cfv 5255 . . . . 5  class  (mulGrp `  r )
6 cui 15421 . . . . . 6  class Unit
73, 6cfv 5255 . . . . 5  class  (Unit `  r )
8 cress 13149 . . . . 5  classs
95, 7, 8co 5858 . . . 4  class  ( (mulGrp `  r )s  (Unit `  r )
)
10 ctgp 17754 . . . 4  class  TopGrp
119, 10wcel 1684 . . 3  wff  ( (mulGrp `  r )s  (Unit `  r )
)  e.  TopGrp
12 ctrg 17838 . . . 4  class  TopRing
13 cdr 15512 . . . 4  class  DivRing
1412, 13cin 3151 . . 3  class  ( TopRing  i^i  DivRing )
1511, 2, 14crab 2547 . 2  class  { r  e.  ( TopRing  i^i  DivRing )  |  ( (mulGrp `  r
)s  (Unit `  r )
)  e.  TopGrp }
161, 15wceq 1623 1  wff TopDRing  =  {
r  e.  ( TopRing  i^i  DivRing )  |  ( (mulGrp `  r )s  (Unit `  r )
)  e.  TopGrp }
Colors of variables: wff set class
This definition is referenced by:  istdrg  17848
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