MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  df-dvdsr Structured version   Visualization version   GIF version

Definition df-dvdsr 18843
Description: Define the (right) divisibility relation in a ring. Access to the left divisibility relation is available through (∥r‘(oppr𝑅)). (Contributed by Mario Carneiro, 1-Dec-2014.)
Assertion
Ref Expression
df-dvdsr r = (𝑤 ∈ V ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (Base‘𝑤) ∧ ∃𝑧 ∈ (Base‘𝑤)(𝑧(.r𝑤)𝑥) = 𝑦)})
Distinct variable group:   𝑥,𝑤,𝑦,𝑧

Detailed syntax breakdown of Definition df-dvdsr
StepHypRef Expression
1 cdsr 18840 . 2 class r
2 vw . . 3 setvar 𝑤
3 cvv 3391 . . 3 class V
4 vx . . . . . . 7 setvar 𝑥
54cv 1636 . . . . . 6 class 𝑥
62cv 1636 . . . . . . 7 class 𝑤
7 cbs 16068 . . . . . . 7 class Base
86, 7cfv 6101 . . . . . 6 class (Base‘𝑤)
95, 8wcel 2156 . . . . 5 wff 𝑥 ∈ (Base‘𝑤)
10 vz . . . . . . . . 9 setvar 𝑧
1110cv 1636 . . . . . . . 8 class 𝑧
12 cmulr 16154 . . . . . . . . 9 class .r
136, 12cfv 6101 . . . . . . . 8 class (.r𝑤)
1411, 5, 13co 6874 . . . . . . 7 class (𝑧(.r𝑤)𝑥)
15 vy . . . . . . . 8 setvar 𝑦
1615cv 1636 . . . . . . 7 class 𝑦
1714, 16wceq 1637 . . . . . 6 wff (𝑧(.r𝑤)𝑥) = 𝑦
1817, 10, 8wrex 3097 . . . . 5 wff 𝑧 ∈ (Base‘𝑤)(𝑧(.r𝑤)𝑥) = 𝑦
199, 18wa 384 . . . 4 wff (𝑥 ∈ (Base‘𝑤) ∧ ∃𝑧 ∈ (Base‘𝑤)(𝑧(.r𝑤)𝑥) = 𝑦)
2019, 4, 15copab 4906 . . 3 class {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (Base‘𝑤) ∧ ∃𝑧 ∈ (Base‘𝑤)(𝑧(.r𝑤)𝑥) = 𝑦)}
212, 3, 20cmpt 4923 . 2 class (𝑤 ∈ V ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (Base‘𝑤) ∧ ∃𝑧 ∈ (Base‘𝑤)(𝑧(.r𝑤)𝑥) = 𝑦)})
221, 21wceq 1637 1 wff r = (𝑤 ∈ V ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (Base‘𝑤) ∧ ∃𝑧 ∈ (Base‘𝑤)(𝑧(.r𝑤)𝑥) = 𝑦)})
Colors of variables: wff setvar class
This definition is referenced by:  reldvdsr  18846  dvdsrval  18847
  Copyright terms: Public domain W3C validator