Definition df-trg 23219

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 23215	. 2 class TopRing
2		vr	. . . . . 6 setvar 𝑟
3	2	cv 1538	. . . . 5 class 𝑟
4		cmgp 19635	. . . . 5 class mulGrp
5	3, 4	cfv 6418	. . . 4 class (mulGrp‘𝑟)
6		ctmd 23129	. . . 4 class TopMnd
7	5, 6	wcel 2108	. . 3 wff (mulGrp‘𝑟) ∈ TopMnd
8		ctgp 23130	. . . 4 class TopGrp
9		crg 19698	. . . 4 class Ring
10	8, 9	cin 3882	. . 3 class (TopGrp ∩ Ring)
11	7, 2, 10	crab 3067	. 2 class {𝑟 ∈ (TopGrp ∩ Ring) ∣ (mulGrp‘𝑟) ∈ TopMnd}
12	1, 11	wceq 1539	1 wff TopRing = {𝑟 ∈ (TopGrp ∩ Ring) ∣ (mulGrp‘𝑟) ∈ TopMnd}

This definition is referenced by:  istrg  23223