Definition df-drng 20731

Description: Define class of all division rings. A division ring is a ring in which the set of units is exactly the nonzero elements of the ring. (Contributed by NM, 18-Oct-2012.)

Assertion

Ref	Expression
df-drng	⊢ DivRing = {𝑟 ∈ Ring ∣ (Unit‘𝑟) = ((Base‘𝑟) ∖ {(0g‘𝑟)})}

Detailed syntax breakdown of Definition df-drng

Step	Hyp	Ref	Expression
1		cdr 20729	. 2 class DivRing
2		vr	. . . . . 6 setvar 𝑟
3	2	cv 1539	. . . . 5 class 𝑟
4		cui 20355	. . . . 5 class Unit
5	3, 4	cfv 6561	. . . 4 class (Unit‘𝑟)
6		cbs 17247	. . . . . 6 class Base
7	3, 6	cfv 6561	. . . . 5 class (Base‘𝑟)
8		c0g 17484	. . . . . . 7 class 0g
9	3, 8	cfv 6561	. . . . . 6 class (0g‘𝑟)
10	9	csn 4626	. . . . 5 class {(0g‘𝑟)}
11	7, 10	cdif 3948	. . . 4 class ((Base‘𝑟) ∖ {(0g‘𝑟)})
12	5, 11	wceq 1540	. . 3 wff (Unit‘𝑟) = ((Base‘𝑟) ∖ {(0g‘𝑟)})
13		crg 20230	. . 3 class Ring
14	12, 2, 13	crab 3436	. 2 class {𝑟 ∈ Ring ∣ (Unit‘𝑟) = ((Base‘𝑟) ∖ {(0g‘𝑟)})}
15	1, 14	wceq 1540	1 wff DivRing = {𝑟 ∈ Ring ∣ (Unit‘𝑟) = ((Base‘𝑟) ∖ {(0g‘𝑟)})}

This definition is referenced by:  isdrng  20733