Theorem ismxidl

Description: The predicate "is a maximal ideal". (Contributed by Jeff Madsen, 5-Jan-2011) (Revised by Thierry Arnoux, 19-Jan-2024)

	Ref	Expression
Hypothesis	mxidlval.1	⊢ 𝐵 = ( Base ‘ 𝑅 )
Assertion	ismxidl	⊢ ( 𝑅 ∈ Ring → ( 𝑀 ∈ ( MaxIdeal ‘ 𝑅 ) ↔ ( 𝑀 ∈ ( LIdeal ‘ 𝑅 ) ∧ 𝑀 ≠ 𝐵 ∧ ∀ 𝑗 ∈ ( LIdeal ‘ 𝑅 ) ( 𝑀 ⊆ 𝑗 → ( 𝑗 = 𝑀 ∨ 𝑗 = 𝐵 ) ) ) ) )

Proof

Step	Hyp	Ref	Expression
1		mxidlval.1	⊢ 𝐵 = ( Base ‘ 𝑅 )
2	1	mxidlval	⊢ ( 𝑅 ∈ Ring → ( MaxIdeal ‘ 𝑅 ) = { 𝑖 ∈ ( LIdeal ‘ 𝑅 ) ∣ ( 𝑖 ≠ 𝐵 ∧ ∀ 𝑗 ∈ ( LIdeal ‘ 𝑅 ) ( 𝑖 ⊆ 𝑗 → ( 𝑗 = 𝑖 ∨ 𝑗 = 𝐵 ) ) ) } )
3	2	eleq2d	⊢ ( 𝑅 ∈ Ring → ( 𝑀 ∈ ( MaxIdeal ‘ 𝑅 ) ↔ 𝑀 ∈ { 𝑖 ∈ ( LIdeal ‘ 𝑅 ) ∣ ( 𝑖 ≠ 𝐵 ∧ ∀ 𝑗 ∈ ( LIdeal ‘ 𝑅 ) ( 𝑖 ⊆ 𝑗 → ( 𝑗 = 𝑖 ∨ 𝑗 = 𝐵 ) ) ) } ) )
4		neeq1	⊢ ( 𝑖 = 𝑀 → ( 𝑖 ≠ 𝐵 ↔ 𝑀 ≠ 𝐵 ) )
5		sseq1	⊢ ( 𝑖 = 𝑀 → ( 𝑖 ⊆ 𝑗 ↔ 𝑀 ⊆ 𝑗 ) )
6		eqeq2	⊢ ( 𝑖 = 𝑀 → ( 𝑗 = 𝑖 ↔ 𝑗 = 𝑀 ) )
7	6	orbi1d	⊢ ( 𝑖 = 𝑀 → ( ( 𝑗 = 𝑖 ∨ 𝑗 = 𝐵 ) ↔ ( 𝑗 = 𝑀 ∨ 𝑗 = 𝐵 ) ) )
8	5 7	imbi12d	⊢ ( 𝑖 = 𝑀 → ( ( 𝑖 ⊆ 𝑗 → ( 𝑗 = 𝑖 ∨ 𝑗 = 𝐵 ) ) ↔ ( 𝑀 ⊆ 𝑗 → ( 𝑗 = 𝑀 ∨ 𝑗 = 𝐵 ) ) ) )
9	8	ralbidv	⊢ ( 𝑖 = 𝑀 → ( ∀ 𝑗 ∈ ( LIdeal ‘ 𝑅 ) ( 𝑖 ⊆ 𝑗 → ( 𝑗 = 𝑖 ∨ 𝑗 = 𝐵 ) ) ↔ ∀ 𝑗 ∈ ( LIdeal ‘ 𝑅 ) ( 𝑀 ⊆ 𝑗 → ( 𝑗 = 𝑀 ∨ 𝑗 = 𝐵 ) ) ) )
10	4 9	anbi12d	⊢ ( 𝑖 = 𝑀 → ( ( 𝑖 ≠ 𝐵 ∧ ∀ 𝑗 ∈ ( LIdeal ‘ 𝑅 ) ( 𝑖 ⊆ 𝑗 → ( 𝑗 = 𝑖 ∨ 𝑗 = 𝐵 ) ) ) ↔ ( 𝑀 ≠ 𝐵 ∧ ∀ 𝑗 ∈ ( LIdeal ‘ 𝑅 ) ( 𝑀 ⊆ 𝑗 → ( 𝑗 = 𝑀 ∨ 𝑗 = 𝐵 ) ) ) ) )
11	10	elrab	⊢ ( 𝑀 ∈ { 𝑖 ∈ ( LIdeal ‘ 𝑅 ) ∣ ( 𝑖 ≠ 𝐵 ∧ ∀ 𝑗 ∈ ( LIdeal ‘ 𝑅 ) ( 𝑖 ⊆ 𝑗 → ( 𝑗 = 𝑖 ∨ 𝑗 = 𝐵 ) ) ) } ↔ ( 𝑀 ∈ ( LIdeal ‘ 𝑅 ) ∧ ( 𝑀 ≠ 𝐵 ∧ ∀ 𝑗 ∈ ( LIdeal ‘ 𝑅 ) ( 𝑀 ⊆ 𝑗 → ( 𝑗 = 𝑀 ∨ 𝑗 = 𝐵 ) ) ) ) )
12		3anass	⊢ ( ( 𝑀 ∈ ( LIdeal ‘ 𝑅 ) ∧ 𝑀 ≠ 𝐵 ∧ ∀ 𝑗 ∈ ( LIdeal ‘ 𝑅 ) ( 𝑀 ⊆ 𝑗 → ( 𝑗 = 𝑀 ∨ 𝑗 = 𝐵 ) ) ) ↔ ( 𝑀 ∈ ( LIdeal ‘ 𝑅 ) ∧ ( 𝑀 ≠ 𝐵 ∧ ∀ 𝑗 ∈ ( LIdeal ‘ 𝑅 ) ( 𝑀 ⊆ 𝑗 → ( 𝑗 = 𝑀 ∨ 𝑗 = 𝐵 ) ) ) ) )
13	11 12	bitr4i	⊢ ( 𝑀 ∈ { 𝑖 ∈ ( LIdeal ‘ 𝑅 ) ∣ ( 𝑖 ≠ 𝐵 ∧ ∀ 𝑗 ∈ ( LIdeal ‘ 𝑅 ) ( 𝑖 ⊆ 𝑗 → ( 𝑗 = 𝑖 ∨ 𝑗 = 𝐵 ) ) ) } ↔ ( 𝑀 ∈ ( LIdeal ‘ 𝑅 ) ∧ 𝑀 ≠ 𝐵 ∧ ∀ 𝑗 ∈ ( LIdeal ‘ 𝑅 ) ( 𝑀 ⊆ 𝑗 → ( 𝑗 = 𝑀 ∨ 𝑗 = 𝐵 ) ) ) )
14	3 13	bitrdi	⊢ ( 𝑅 ∈ Ring → ( 𝑀 ∈ ( MaxIdeal ‘ 𝑅 ) ↔ ( 𝑀 ∈ ( LIdeal ‘ 𝑅 ) ∧ 𝑀 ≠ 𝐵 ∧ ∀ 𝑗 ∈ ( LIdeal ‘ 𝑅 ) ( 𝑀 ⊆ 𝑗 → ( 𝑗 = 𝑀 ∨ 𝑗 = 𝐵 ) ) ) ) )