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Definition df-tdrg 22180
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 ∈ (TopRing ∩ DivRing) ∣ ((mulGrp‘𝑟) ↾s (Unit‘𝑟)) ∈ TopGrp}

Detailed syntax breakdown of Definition df-tdrg
StepHypRef Expression
1 ctdrg 22176 . 2 class TopDRing
2 vr . . . . . . 7 setvar 𝑟
32cv 1630 . . . . . 6 class 𝑟
4 cmgp 18693 . . . . . 6 class mulGrp
53, 4cfv 6029 . . . . 5 class (mulGrp‘𝑟)
6 cui 18843 . . . . . 6 class Unit
73, 6cfv 6029 . . . . 5 class (Unit‘𝑟)
8 cress 16061 . . . . 5 class s
95, 7, 8co 6792 . . . 4 class ((mulGrp‘𝑟) ↾s (Unit‘𝑟))
10 ctgp 22091 . . . 4 class TopGrp
119, 10wcel 2145 . . 3 wff ((mulGrp‘𝑟) ↾s (Unit‘𝑟)) ∈ TopGrp
12 ctrg 22175 . . . 4 class TopRing
13 cdr 18953 . . . 4 class DivRing
1412, 13cin 3722 . . 3 class (TopRing ∩ DivRing)
1511, 2, 14crab 3065 . 2 class {𝑟 ∈ (TopRing ∩ DivRing) ∣ ((mulGrp‘𝑟) ↾s (Unit‘𝑟)) ∈ TopGrp}
161, 15wceq 1631 1 wff TopDRing = {𝑟 ∈ (TopRing ∩ DivRing) ∣ ((mulGrp‘𝑟) ↾s (Unit‘𝑟)) ∈ TopGrp}
Colors of variables: wff setvar class
This definition is referenced by:  istdrg  22185
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