Theorem mxidlmax

Description: A maximal ideal is a maximal proper ideal. (Contributed by Jeff Madsen, 16-Jun-2011) (Revised by Thierry Arnoux, 19-Jan-2024)

	Ref	Expression
Hypothesis	mxidlval.1	⊢ 𝐵 = ( Base ‘ 𝑅 )
Assertion	mxidlmax	⊢ ( ( ( 𝑅 ∈ Ring ∧ 𝑀 ∈ ( MaxIdeal ‘ 𝑅 ) ) ∧ ( 𝐼 ∈ ( LIdeal ‘ 𝑅 ) ∧ 𝑀 ⊆ 𝐼 ) ) → ( 𝐼 = 𝑀 ∨ 𝐼 = 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		mxidlval.1	⊢ 𝐵 = ( Base ‘ 𝑅 )
2		sseq2	⊢ ( 𝑗 = 𝐼 → ( 𝑀 ⊆ 𝑗 ↔ 𝑀 ⊆ 𝐼 ) )
3		eqeq1	⊢ ( 𝑗 = 𝐼 → ( 𝑗 = 𝑀 ↔ 𝐼 = 𝑀 ) )
4		eqeq1	⊢ ( 𝑗 = 𝐼 → ( 𝑗 = 𝐵 ↔ 𝐼 = 𝐵 ) )
5	3 4	orbi12d	⊢ ( 𝑗 = 𝐼 → ( ( 𝑗 = 𝑀 ∨ 𝑗 = 𝐵 ) ↔ ( 𝐼 = 𝑀 ∨ 𝐼 = 𝐵 ) ) )
6	2 5	imbi12d	⊢ ( 𝑗 = 𝐼 → ( ( 𝑀 ⊆ 𝑗 → ( 𝑗 = 𝑀 ∨ 𝑗 = 𝐵 ) ) ↔ ( 𝑀 ⊆ 𝐼 → ( 𝐼 = 𝑀 ∨ 𝐼 = 𝐵 ) ) ) )
7	1	ismxidl	⊢ ( 𝑅 ∈ Ring → ( 𝑀 ∈ ( MaxIdeal ‘ 𝑅 ) ↔ ( 𝑀 ∈ ( LIdeal ‘ 𝑅 ) ∧ 𝑀 ≠ 𝐵 ∧ ∀ 𝑗 ∈ ( LIdeal ‘ 𝑅 ) ( 𝑀 ⊆ 𝑗 → ( 𝑗 = 𝑀 ∨ 𝑗 = 𝐵 ) ) ) ) )
8	7	biimpa	⊢ ( ( 𝑅 ∈ Ring ∧ 𝑀 ∈ ( MaxIdeal ‘ 𝑅 ) ) → ( 𝑀 ∈ ( LIdeal ‘ 𝑅 ) ∧ 𝑀 ≠ 𝐵 ∧ ∀ 𝑗 ∈ ( LIdeal ‘ 𝑅 ) ( 𝑀 ⊆ 𝑗 → ( 𝑗 = 𝑀 ∨ 𝑗 = 𝐵 ) ) ) )
9	8	simp3d	⊢ ( ( 𝑅 ∈ Ring ∧ 𝑀 ∈ ( MaxIdeal ‘ 𝑅 ) ) → ∀ 𝑗 ∈ ( LIdeal ‘ 𝑅 ) ( 𝑀 ⊆ 𝑗 → ( 𝑗 = 𝑀 ∨ 𝑗 = 𝐵 ) ) )
10	9	adantr	⊢ ( ( ( 𝑅 ∈ Ring ∧ 𝑀 ∈ ( MaxIdeal ‘ 𝑅 ) ) ∧ 𝐼 ∈ ( LIdeal ‘ 𝑅 ) ) → ∀ 𝑗 ∈ ( LIdeal ‘ 𝑅 ) ( 𝑀 ⊆ 𝑗 → ( 𝑗 = 𝑀 ∨ 𝑗 = 𝐵 ) ) )
11		simpr	⊢ ( ( ( 𝑅 ∈ Ring ∧ 𝑀 ∈ ( MaxIdeal ‘ 𝑅 ) ) ∧ 𝐼 ∈ ( LIdeal ‘ 𝑅 ) ) → 𝐼 ∈ ( LIdeal ‘ 𝑅 ) )
12	6 10 11	rspcdva	⊢ ( ( ( 𝑅 ∈ Ring ∧ 𝑀 ∈ ( MaxIdeal ‘ 𝑅 ) ) ∧ 𝐼 ∈ ( LIdeal ‘ 𝑅 ) ) → ( 𝑀 ⊆ 𝐼 → ( 𝐼 = 𝑀 ∨ 𝐼 = 𝐵 ) ) )
13	12	impr	⊢ ( ( ( 𝑅 ∈ Ring ∧ 𝑀 ∈ ( MaxIdeal ‘ 𝑅 ) ) ∧ ( 𝐼 ∈ ( LIdeal ‘ 𝑅 ) ∧ 𝑀 ⊆ 𝐼 ) ) → ( 𝐼 = 𝑀 ∨ 𝐼 = 𝐵 ) )