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Metamath Proof Explorer

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 ) ) ) ) )