Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  df-trg Structured version   Visualization version   GIF version

Definition df-trg 21873
 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 21869 . 2 class TopRing
2 vr . . . . . 6 setvar 𝑟
32cv 1479 . . . . 5 class 𝑟
4 cmgp 18410 . . . . 5 class mulGrp
53, 4cfv 5847 . . . 4 class (mulGrp‘𝑟)
6 ctmd 21784 . . . 4 class TopMnd
75, 6wcel 1987 . . 3 wff (mulGrp‘𝑟) ∈ TopMnd
8 ctgp 21785 . . . 4 class TopGrp
9 crg 18468 . . . 4 class Ring
108, 9cin 3554 . . 3 class (TopGrp ∩ Ring)
117, 2, 10crab 2911 . 2 class {𝑟 ∈ (TopGrp ∩ Ring) ∣ (mulGrp‘𝑟) ∈ TopMnd}
121, 11wceq 1480 1 wff TopRing = {𝑟 ∈ (TopGrp ∩ Ring) ∣ (mulGrp‘𝑟) ∈ TopMnd}
 Colors of variables: wff setvar class This definition is referenced by:  istrg  21877
 Copyright terms: Public domain W3C validator