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Definition df-mxidl 31013
Description: Define the class of maximal ideals of a ring 𝑅. A proper ideal is called maximal if it is maximal with respect to inclusion among proper ideals. (Contributed by Jeff Madsen, 5-Jan-2011.) (Revised by Thierry Arnoux, 19-Jan-2024.)
Assertion
Ref Expression
df-mxidl MaxIdeal = (𝑟 ∈ Ring ↦ {𝑖 ∈ (LIdeal‘𝑟) ∣ (𝑖 ≠ (Base‘𝑟) ∧ ∀𝑗 ∈ (LIdeal‘𝑟)(𝑖𝑗 → (𝑗 = 𝑖𝑗 = (Base‘𝑟))))})
Distinct variable group:   𝑖,𝑟,𝑗

Detailed syntax breakdown of Definition df-mxidl
StepHypRef Expression
1 cmxidl 31012 . 2 class MaxIdeal
2 vr . . 3 setvar 𝑟
3 crg 19297 . . 3 class Ring
4 vi . . . . . . 7 setvar 𝑖
54cv 1537 . . . . . 6 class 𝑖
62cv 1537 . . . . . . 7 class 𝑟
7 cbs 16483 . . . . . . 7 class Base
86, 7cfv 6343 . . . . . 6 class (Base‘𝑟)
95, 8wne 3014 . . . . 5 wff 𝑖 ≠ (Base‘𝑟)
10 vj . . . . . . . . 9 setvar 𝑗
1110cv 1537 . . . . . . . 8 class 𝑗
125, 11wss 3919 . . . . . . 7 wff 𝑖𝑗
1310, 4weq 1965 . . . . . . . 8 wff 𝑗 = 𝑖
1411, 8wceq 1538 . . . . . . . 8 wff 𝑗 = (Base‘𝑟)
1513, 14wo 844 . . . . . . 7 wff (𝑗 = 𝑖𝑗 = (Base‘𝑟))
1612, 15wi 4 . . . . . 6 wff (𝑖𝑗 → (𝑗 = 𝑖𝑗 = (Base‘𝑟)))
17 clidl 19942 . . . . . . 7 class LIdeal
186, 17cfv 6343 . . . . . 6 class (LIdeal‘𝑟)
1916, 10, 18wral 3133 . . . . 5 wff 𝑗 ∈ (LIdeal‘𝑟)(𝑖𝑗 → (𝑗 = 𝑖𝑗 = (Base‘𝑟)))
209, 19wa 399 . . . 4 wff (𝑖 ≠ (Base‘𝑟) ∧ ∀𝑗 ∈ (LIdeal‘𝑟)(𝑖𝑗 → (𝑗 = 𝑖𝑗 = (Base‘𝑟))))
2120, 4, 18crab 3137 . . 3 class {𝑖 ∈ (LIdeal‘𝑟) ∣ (𝑖 ≠ (Base‘𝑟) ∧ ∀𝑗 ∈ (LIdeal‘𝑟)(𝑖𝑗 → (𝑗 = 𝑖𝑗 = (Base‘𝑟))))}
222, 3, 21cmpt 5132 . 2 class (𝑟 ∈ Ring ↦ {𝑖 ∈ (LIdeal‘𝑟) ∣ (𝑖 ≠ (Base‘𝑟) ∧ ∀𝑗 ∈ (LIdeal‘𝑟)(𝑖𝑗 → (𝑗 = 𝑖𝑗 = (Base‘𝑟))))})
231, 22wceq 1538 1 wff MaxIdeal = (𝑟 ∈ Ring ↦ {𝑖 ∈ (LIdeal‘𝑟) ∣ (𝑖 ≠ (Base‘𝑟) ∧ ∀𝑗 ∈ (LIdeal‘𝑟)(𝑖𝑗 → (𝑗 = 𝑖𝑗 = (Base‘𝑟))))})
Colors of variables: wff setvar class
This definition is referenced by:  mxidlval  31014
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