Definition df-mxidl 30991

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)(𝑖 ⊆ 𝑗 → (𝑗 = 𝑖 ∨ 𝑗 = (Base‘𝑟))))})

Distinct variable group:   𝑖,𝑟,𝑗

Detailed syntax breakdown of Definition df-mxidl

Step	Hyp	Ref	Expression
1		cmxidl 30990	. 2 class MaxIdeal
2		vr	. . 3 setvar 𝑟
3		crg 19290	. . 3 class Ring
4		vi	. . . . . . 7 setvar 𝑖
5	4	cv 1535	. . . . . 6 class 𝑖
6	2	cv 1535	. . . . . . 7 class 𝑟
7		cbs 16476	. . . . . . 7 class Base
8	6, 7	cfv 6348	. . . . . 6 class (Base‘𝑟)
9	5, 8	wne 3015	. . . . 5 wff 𝑖 ≠ (Base‘𝑟)
10		vj	. . . . . . . . 9 setvar 𝑗
11	10	cv 1535	. . . . . . . 8 class 𝑗
12	5, 11	wss 3929	. . . . . . 7 wff 𝑖 ⊆ 𝑗
13	10, 4	weq 1963	. . . . . . . 8 wff 𝑗 = 𝑖
14	11, 8	wceq 1536	. . . . . . . 8 wff 𝑗 = (Base‘𝑟)
15	13, 14	wo 843	. . . . . . 7 wff (𝑗 = 𝑖 ∨ 𝑗 = (Base‘𝑟))
16	12, 15	wi 4	. . . . . 6 wff (𝑖 ⊆ 𝑗 → (𝑗 = 𝑖 ∨ 𝑗 = (Base‘𝑟)))
17		clidl 19935	. . . . . . 7 class LIdeal
18	6, 17	cfv 6348	. . . . . 6 class (LIdeal‘𝑟)
19	16, 10, 18	wral 3137	. . . . 5 wff ∀𝑗 ∈ (LIdeal‘𝑟)(𝑖 ⊆ 𝑗 → (𝑗 = 𝑖 ∨ 𝑗 = (Base‘𝑟)))
20	9, 19	wa 398	. . . 4 wff (𝑖 ≠ (Base‘𝑟) ∧ ∀𝑗 ∈ (LIdeal‘𝑟)(𝑖 ⊆ 𝑗 → (𝑗 = 𝑖 ∨ 𝑗 = (Base‘𝑟))))
21	20, 4, 18	crab 3141	. . 3 class {𝑖 ∈ (LIdeal‘𝑟) ∣ (𝑖 ≠ (Base‘𝑟) ∧ ∀𝑗 ∈ (LIdeal‘𝑟)(𝑖 ⊆ 𝑗 → (𝑗 = 𝑖 ∨ 𝑗 = (Base‘𝑟))))}
22	2, 3, 21	cmpt 5139	. 2 class (𝑟 ∈ Ring ↦ {𝑖 ∈ (LIdeal‘𝑟) ∣ (𝑖 ≠ (Base‘𝑟) ∧ ∀𝑗 ∈ (LIdeal‘𝑟)(𝑖 ⊆ 𝑗 → (𝑗 = 𝑖 ∨ 𝑗 = (Base‘𝑟))))})
23	1, 22	wceq 1536	1 wff MaxIdeal = (𝑟 ∈ Ring ↦ {𝑖 ∈ (LIdeal‘𝑟) ∣ (𝑖 ≠ (Base‘𝑟) ∧ ∀𝑗 ∈ (LIdeal‘𝑟)(𝑖 ⊆ 𝑗 → (𝑗 = 𝑖 ∨ 𝑗 = (Base‘𝑟))))})

This definition is referenced by:  mxidlval  30992