Definition df-drng 18026

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 e. Ring | (Unit ` r ) = ( ( Base ` r ) \ { ( 0g ` r ) } ) }

Detailed syntax breakdown of Definition df-drng

Step	Hyp	Ref	Expression
1		cdr 18024	. 2 class DivRing
2		vr	. . . . . 6 setvar r
3	2	cv 1454	. . . . 5 class r
4		cui 17916	. . . . 5 class Unit
5	3, 4	cfv 5601	. . . 4 class (Unit ` r )
6		cbs 15170	. . . . . 6 class Base
7	3, 6	cfv 5601	. . . . 5 class ( Base ` r )
8		c0g 15387	. . . . . . 7 class 0g
9	3, 8	cfv 5601	. . . . . 6 class ( 0g ` r )
10	9	csn 3980	. . . . 5 class { ( 0g ` r ) }
11	7, 10	cdif 3413	. . . 4 class ( ( Base ` r ) \ { ( 0g ` r ) } )
12	5, 11	wceq 1455	. . 3 wff (Unit ` r ) = ( ( Base ` r ) \ { ( 0g ` r ) } )
13		crg 17829	. . 3 class Ring
14	12, 2, 13	crab 2753	. 2 class { r e. Ring | (Unit ` r ) = ( ( Base ` r ) \ { ( 0g ` r ) } ) }
15	1, 14	wceq 1455	1 wff DivRing = { r e. Ring | (Unit ` r ) = ( ( Base ` r ) \ { ( 0g ` r ) } ) }

This definition is referenced by:  isdrng  18028