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Definition df-trg 22695
Description: Define a topological ring, which is a ring such that the addition is a topological group operation and the multiplication is continuous. (Contributed by Mario Carneiro, 5-Oct-2015.)
Assertion
Ref Expression
df-trg TopRing = {𝑟 ∈ (TopGrp ∩ Ring) ∣ (mulGrp‘𝑟) ∈ TopMnd}

Detailed syntax breakdown of Definition df-trg
StepHypRef Expression
1 ctrg 22691 . 2 class TopRing
2 vr . . . . . 6 setvar 𝑟
32cv 1527 . . . . 5 class 𝑟
4 cmgp 19168 . . . . 5 class mulGrp
53, 4cfv 6348 . . . 4 class (mulGrp‘𝑟)
6 ctmd 22606 . . . 4 class TopMnd
75, 6wcel 2105 . . 3 wff (mulGrp‘𝑟) ∈ TopMnd
8 ctgp 22607 . . . 4 class TopGrp
9 crg 19226 . . . 4 class Ring
108, 9cin 3932 . . 3 class (TopGrp ∩ Ring)
117, 2, 10crab 3139 . 2 class {𝑟 ∈ (TopGrp ∩ Ring) ∣ (mulGrp‘𝑟) ∈ TopMnd}
121, 11wceq 1528 1 wff TopRing = {𝑟 ∈ (TopGrp ∩ Ring) ∣ (mulGrp‘𝑟) ∈ TopMnd}
Colors of variables: wff setvar class
This definition is referenced by:  istrg  22699
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