Definition df-dvdsr 20357

Description: Define the (right) divisibility relation in a ring. Access to the left divisibility relation is available through (∥_r‘(opp_r‘𝑅)). (Contributed by Mario Carneiro, 1-Dec-2014.)

Assertion

Ref	Expression
df-dvdsr	⊢ ∥r = (𝑤 ∈ V ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (Base‘𝑤) ∧ ∃𝑧 ∈ (Base‘𝑤)(𝑧(.r‘𝑤)𝑥) = 𝑦)})

Distinct variable group:   𝑥,𝑤,𝑦,𝑧

Detailed syntax breakdown of Definition df-dvdsr

Step	Hyp	Ref	Expression
1		cdsr 20354	. 2 class ∥r
2		vw	. . 3 setvar 𝑤
3		cvv 3480	. . 3 class V
4		vx	. . . . . . 7 setvar 𝑥
5	4	cv 1539	. . . . . 6 class 𝑥
6	2	cv 1539	. . . . . . 7 class 𝑤
7		cbs 17247	. . . . . . 7 class Base
8	6, 7	cfv 6561	. . . . . 6 class (Base‘𝑤)
9	5, 8	wcel 2108	. . . . 5 wff 𝑥 ∈ (Base‘𝑤)
10		vz	. . . . . . . . 9 setvar 𝑧
11	10	cv 1539	. . . . . . . 8 class 𝑧
12		cmulr 17298	. . . . . . . . 9 class .r
13	6, 12	cfv 6561	. . . . . . . 8 class (.r‘𝑤)
14	11, 5, 13	co 7431	. . . . . . 7 class (𝑧(.r‘𝑤)𝑥)
15		vy	. . . . . . . 8 setvar 𝑦
16	15	cv 1539	. . . . . . 7 class 𝑦
17	14, 16	wceq 1540	. . . . . 6 wff (𝑧(.r‘𝑤)𝑥) = 𝑦
18	17, 10, 8	wrex 3070	. . . . 5 wff ∃𝑧 ∈ (Base‘𝑤)(𝑧(.r‘𝑤)𝑥) = 𝑦
19	9, 18	wa 395	. . . 4 wff (𝑥 ∈ (Base‘𝑤) ∧ ∃𝑧 ∈ (Base‘𝑤)(𝑧(.r‘𝑤)𝑥) = 𝑦)
20	19, 4, 15	copab 5205	. . 3 class {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (Base‘𝑤) ∧ ∃𝑧 ∈ (Base‘𝑤)(𝑧(.r‘𝑤)𝑥) = 𝑦)}
21	2, 3, 20	cmpt 5225	. 2 class (𝑤 ∈ V ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (Base‘𝑤) ∧ ∃𝑧 ∈ (Base‘𝑤)(𝑧(.r‘𝑤)𝑥) = 𝑦)})
22	1, 21	wceq 1540	1 wff ∥r = (𝑤 ∈ V ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (Base‘𝑤) ∧ ∃𝑧 ∈ (Base‘𝑤)(𝑧(.r‘𝑤)𝑥) = 𝑦)})

This definition is referenced by:  reldvdsr  20360  dvdsrval  20361