Definition df-dvr 13301

Description: Define ring division. (Contributed by Mario Carneiro, 2-Jul-2014.)

Assertion

Ref	Expression
df-dvr	|- /r = ( r e. _V |-> ( x e. ( Base ` r ) , y e. (Unit ` r ) |-> ( x ( .r ` r ) ( ( invr ` r ) ` y ) ) ) )

Distinct variable group:   x, r, y

Detailed syntax breakdown of Definition df-dvr

Step	Hyp	Ref	Expression
1		cdvr 13300	. 2 class /r
2		vr	. . 3 setvar r
3		cvv 2738	. . 3 class _V
4		vx	. . . 4 setvar x
5		vy	. . . 4 setvar y
6	2	cv 1352	. . . . 5 class r
7		cbs 12462	. . . . 5 class Base
8	6, 7	cfv 5217	. . . 4 class ( Base ` r )
9		cui 13256	. . . . 5 class Unit
10	6, 9	cfv 5217	. . . 4 class (Unit ` r )
11	4	cv 1352	. . . . 5 class x
12	5	cv 1352	. . . . . 6 class y
13		cinvr 13289	. . . . . . 7 class invr
14	6, 13	cfv 5217	. . . . . 6 class ( invr ` r )
15	12, 14	cfv 5217	. . . . 5 class ( ( invr ` r ) ` y )
16		cmulr 12537	. . . . . 6 class .r
17	6, 16	cfv 5217	. . . . 5 class ( .r ` r )
18	11, 15, 17	co 5875	. . . 4 class ( x ( .r ` r ) ( ( invr ` r ) ` y ) )
19	4, 5, 8, 10, 18	cmpo 5877	. . 3 class ( x e. ( Base ` r ) , y e. (Unit ` r ) |-> ( x ( .r ` r ) ( ( invr ` r ) ` y ) ) )
20	2, 3, 19	cmpt 4065	. 2 class ( r e. _V |-> ( x e. ( Base ` r ) , y e. (Unit ` r ) |-> ( x ( .r ` r ) ( ( invr ` r ) ` y ) ) ) )
21	1, 20	wceq 1353	1 wff /r = ( r e. _V |-> ( x e. ( Base ` r ) , y e. (Unit ` r ) |-> ( x ( .r ` r ) ( ( invr ` r ) ` y ) ) ) )

This definition is referenced by:  dvrfvald  13302