Theorem maxidlidl

Description: A maximal ideal is an ideal. (Contributed by Jeff Madsen, 5-Jan-2011)

	Ref	Expression
Assertion	maxidlidl	⊢ ( ( 𝑅 ∈ RingOps ∧ 𝑀 ∈ ( MaxIdl ‘ 𝑅 ) ) → 𝑀 ∈ ( Idl ‘ 𝑅 ) )

Proof

Step	Hyp	Ref	Expression
1		eqid	⊢ ( 1st ‘ 𝑅 ) = ( 1st ‘ 𝑅 )
2		eqid	⊢ ran ( 1st ‘ 𝑅 ) = ran ( 1st ‘ 𝑅 )
3	1 2	ismaxidl	⊢ ( 𝑅 ∈ RingOps → ( 𝑀 ∈ ( MaxIdl ‘ 𝑅 ) ↔ ( 𝑀 ∈ ( Idl ‘ 𝑅 ) ∧ 𝑀 ≠ ran ( 1st ‘ 𝑅 ) ∧ ∀ 𝑗 ∈ ( Idl ‘ 𝑅 ) ( 𝑀 ⊆ 𝑗 → ( 𝑗 = 𝑀 ∨ 𝑗 = ran ( 1st ‘ 𝑅 ) ) ) ) ) )
4		3anass	⊢ ( ( 𝑀 ∈ ( Idl ‘ 𝑅 ) ∧ 𝑀 ≠ ran ( 1st ‘ 𝑅 ) ∧ ∀ 𝑗 ∈ ( Idl ‘ 𝑅 ) ( 𝑀 ⊆ 𝑗 → ( 𝑗 = 𝑀 ∨ 𝑗 = ran ( 1st ‘ 𝑅 ) ) ) ) ↔ ( 𝑀 ∈ ( Idl ‘ 𝑅 ) ∧ ( 𝑀 ≠ ran ( 1st ‘ 𝑅 ) ∧ ∀ 𝑗 ∈ ( Idl ‘ 𝑅 ) ( 𝑀 ⊆ 𝑗 → ( 𝑗 = 𝑀 ∨ 𝑗 = ran ( 1st ‘ 𝑅 ) ) ) ) ) )
5	3 4	bitrdi	⊢ ( 𝑅 ∈ RingOps → ( 𝑀 ∈ ( MaxIdl ‘ 𝑅 ) ↔ ( 𝑀 ∈ ( Idl ‘ 𝑅 ) ∧ ( 𝑀 ≠ ran ( 1st ‘ 𝑅 ) ∧ ∀ 𝑗 ∈ ( Idl ‘ 𝑅 ) ( 𝑀 ⊆ 𝑗 → ( 𝑗 = 𝑀 ∨ 𝑗 = ran ( 1st ‘ 𝑅 ) ) ) ) ) ) )
6	5	simprbda	⊢ ( ( 𝑅 ∈ RingOps ∧ 𝑀 ∈ ( MaxIdl ‘ 𝑅 ) ) → 𝑀 ∈ ( Idl ‘ 𝑅 ) )