Definition df-dvr 13299

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

Assertion

Ref	Expression
df-dvr	⊢ /r = (𝑟 ∈ V ↦ (𝑥 ∈ (Base‘𝑟), 𝑦 ∈ (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 13298	. 2 class /r
2		vr	. . 3 setvar 𝑟
3		cvv 2737	. . 3 class V
4		vx	. . . 4 setvar 𝑥
5		vy	. . . 4 setvar 𝑦
6	2	cv 1352	. . . . 5 class 𝑟
7		cbs 12461	. . . . 5 class Base
8	6, 7	cfv 5216	. . . 4 class (Base‘𝑟)
9		cui 13254	. . . . 5 class Unit
10	6, 9	cfv 5216	. . . 4 class (Unit‘𝑟)
11	4	cv 1352	. . . . 5 class 𝑥
12	5	cv 1352	. . . . . 6 class 𝑦
13		cinvr 13287	. . . . . . 7 class invr
14	6, 13	cfv 5216	. . . . . 6 class (invr‘𝑟)
15	12, 14	cfv 5216	. . . . 5 class ((invr‘𝑟)‘𝑦)
16		cmulr 12536	. . . . . 6 class .r
17	6, 16	cfv 5216	. . . . 5 class (.r‘𝑟)
18	11, 15, 17	co 5874	. . . 4 class (𝑥(.r‘𝑟)((invr‘𝑟)‘𝑦))
19	4, 5, 8, 10, 18	cmpo 5876	. . 3 class (𝑥 ∈ (Base‘𝑟), 𝑦 ∈ (Unit‘𝑟) ↦ (𝑥(.r‘𝑟)((invr‘𝑟)‘𝑦)))
20	2, 3, 19	cmpt 4064	. 2 class (𝑟 ∈ V ↦ (𝑥 ∈ (Base‘𝑟), 𝑦 ∈ (Unit‘𝑟) ↦ (𝑥(.r‘𝑟)((invr‘𝑟)‘𝑦))))
21	1, 20	wceq 1353	1 wff /r = (𝑟 ∈ V ↦ (𝑥 ∈ (Base‘𝑟), 𝑦 ∈ (Unit‘𝑟) ↦ (𝑥(.r‘𝑟)((invr‘𝑟)‘𝑦))))

This definition is referenced by:  dvrfvald  13300