Theorem maxidlnr

Description: A maximal ideal is proper. (Contributed by Jeff Madsen, 16-Jun-2011)

	Ref	Expression
Hypotheses	maxidlnr.1	⊢ 𝐺 = ( 1st ‘ 𝑅 )
	maxidlnr.2	⊢ 𝑋 = ran 𝐺
Assertion	maxidlnr	⊢ ( ( 𝑅 ∈ RingOps ∧ 𝑀 ∈ ( MaxIdl ‘ 𝑅 ) ) → 𝑀 ≠ 𝑋 )

Proof

Step	Hyp	Ref	Expression
1		maxidlnr.1	⊢ 𝐺 = ( 1st ‘ 𝑅 )
2		maxidlnr.2	⊢ 𝑋 = ran 𝐺
3	1 2	ismaxidl	⊢ ( 𝑅 ∈ RingOps → ( 𝑀 ∈ ( MaxIdl ‘ 𝑅 ) ↔ ( 𝑀 ∈ ( Idl ‘ 𝑅 ) ∧ 𝑀 ≠ 𝑋 ∧ ∀ 𝑗 ∈ ( Idl ‘ 𝑅 ) ( 𝑀 ⊆ 𝑗 → ( 𝑗 = 𝑀 ∨ 𝑗 = 𝑋 ) ) ) ) )
4	3	biimpa	⊢ ( ( 𝑅 ∈ RingOps ∧ 𝑀 ∈ ( MaxIdl ‘ 𝑅 ) ) → ( 𝑀 ∈ ( Idl ‘ 𝑅 ) ∧ 𝑀 ≠ 𝑋 ∧ ∀ 𝑗 ∈ ( Idl ‘ 𝑅 ) ( 𝑀 ⊆ 𝑗 → ( 𝑗 = 𝑀 ∨ 𝑗 = 𝑋 ) ) ) )
5	4	simp2d	⊢ ( ( 𝑅 ∈ RingOps ∧ 𝑀 ∈ ( MaxIdl ‘ 𝑅 ) ) → 𝑀 ≠ 𝑋 )