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Definition df-mxidl 32277
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 = (π‘Ÿ ∈ Ring ↦ {𝑖 ∈ (LIdealβ€˜π‘Ÿ) ∣ (𝑖 β‰  (Baseβ€˜π‘Ÿ) ∧ βˆ€π‘— ∈ (LIdealβ€˜π‘Ÿ)(𝑖 βŠ† 𝑗 β†’ (𝑗 = 𝑖 ∨ 𝑗 = (Baseβ€˜π‘Ÿ))))})
Distinct variable group:   𝑖,π‘Ÿ,𝑗

Detailed syntax breakdown of Definition df-mxidl
StepHypRef Expression
1 cmxidl 32276 . 2 class MaxIdeal
2 vr . . 3 setvar π‘Ÿ
3 crg 19969 . . 3 class Ring
4 vi . . . . . . 7 setvar 𝑖
54cv 1541 . . . . . 6 class 𝑖
62cv 1541 . . . . . . 7 class π‘Ÿ
7 cbs 17088 . . . . . . 7 class Base
86, 7cfv 6497 . . . . . 6 class (Baseβ€˜π‘Ÿ)
95, 8wne 2940 . . . . 5 wff 𝑖 β‰  (Baseβ€˜π‘Ÿ)
10 vj . . . . . . . . 9 setvar 𝑗
1110cv 1541 . . . . . . . 8 class 𝑗
125, 11wss 3911 . . . . . . 7 wff 𝑖 βŠ† 𝑗
1310, 4weq 1967 . . . . . . . 8 wff 𝑗 = 𝑖
1411, 8wceq 1542 . . . . . . . 8 wff 𝑗 = (Baseβ€˜π‘Ÿ)
1513, 14wo 846 . . . . . . 7 wff (𝑗 = 𝑖 ∨ 𝑗 = (Baseβ€˜π‘Ÿ))
1612, 15wi 4 . . . . . 6 wff (𝑖 βŠ† 𝑗 β†’ (𝑗 = 𝑖 ∨ 𝑗 = (Baseβ€˜π‘Ÿ)))
17 clidl 20647 . . . . . . 7 class LIdeal
186, 17cfv 6497 . . . . . 6 class (LIdealβ€˜π‘Ÿ)
1916, 10, 18wral 3061 . . . . 5 wff βˆ€π‘— ∈ (LIdealβ€˜π‘Ÿ)(𝑖 βŠ† 𝑗 β†’ (𝑗 = 𝑖 ∨ 𝑗 = (Baseβ€˜π‘Ÿ)))
209, 19wa 397 . . . 4 wff (𝑖 β‰  (Baseβ€˜π‘Ÿ) ∧ βˆ€π‘— ∈ (LIdealβ€˜π‘Ÿ)(𝑖 βŠ† 𝑗 β†’ (𝑗 = 𝑖 ∨ 𝑗 = (Baseβ€˜π‘Ÿ))))
2120, 4, 18crab 3406 . . 3 class {𝑖 ∈ (LIdealβ€˜π‘Ÿ) ∣ (𝑖 β‰  (Baseβ€˜π‘Ÿ) ∧ βˆ€π‘— ∈ (LIdealβ€˜π‘Ÿ)(𝑖 βŠ† 𝑗 β†’ (𝑗 = 𝑖 ∨ 𝑗 = (Baseβ€˜π‘Ÿ))))}
222, 3, 21cmpt 5189 . 2 class (π‘Ÿ ∈ Ring ↦ {𝑖 ∈ (LIdealβ€˜π‘Ÿ) ∣ (𝑖 β‰  (Baseβ€˜π‘Ÿ) ∧ βˆ€π‘— ∈ (LIdealβ€˜π‘Ÿ)(𝑖 βŠ† 𝑗 β†’ (𝑗 = 𝑖 ∨ 𝑗 = (Baseβ€˜π‘Ÿ))))})
231, 22wceq 1542 1 wff MaxIdeal = (π‘Ÿ ∈ Ring ↦ {𝑖 ∈ (LIdealβ€˜π‘Ÿ) ∣ (𝑖 β‰  (Baseβ€˜π‘Ÿ) ∧ βˆ€π‘— ∈ (LIdealβ€˜π‘Ÿ)(𝑖 βŠ† 𝑗 β†’ (𝑗 = 𝑖 ∨ 𝑗 = (Baseβ€˜π‘Ÿ))))})
Colors of variables: wff setvar class
This definition is referenced by:  mxidlval  32278
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