Definition df-trg 21963

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

Step	Hyp	Ref	Expression
1		ctrg 21959	. 2 class TopRing
2		vr	. . . . . 6 setvar 𝑟
3	2	cv 1482	. . . . 5 class 𝑟
4		cmgp 18489	. . . . 5 class mulGrp
5	3, 4	cfv 5888	. . . 4 class (mulGrp‘𝑟)
6		ctmd 21874	. . . 4 class TopMnd
7	5, 6	wcel 1990	. . 3 wff (mulGrp‘𝑟) ∈ TopMnd
8		ctgp 21875	. . . 4 class TopGrp
9		crg 18547	. . . 4 class Ring
10	8, 9	cin 3573	. . 3 class (TopGrp ∩ Ring)
11	7, 2, 10	crab 2916	. 2 class {𝑟 ∈ (TopGrp ∩ Ring) ∣ (mulGrp‘𝑟) ∈ TopMnd}
12	1, 11	wceq 1483	1 wff TopRing = {𝑟 ∈ (TopGrp ∩ Ring) ∣ (mulGrp‘𝑟) ∈ TopMnd}

This definition is referenced by:  istrg  21967