Theorem mxidlprmALT

Description: Every maximal ideal is prime - alternative proof. (Contributed by Thierry Arnoux, 15-Mar-2025) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Hypotheses	mxidlprmALT.1	⊢ ( 𝜑 → 𝑅 ∈ CRing )
	mxidlprmALT.2	⊢ ( 𝜑 → 𝑀 ∈ ( MaxIdeal ‘ 𝑅 ) )
Assertion	mxidlprmALT	⊢ ( 𝜑 → 𝑀 ∈ ( PrmIdeal ‘ 𝑅 ) )

Proof

Step	Hyp	Ref	Expression
1		mxidlprmALT.1	⊢ ( 𝜑 → 𝑅 ∈ CRing )
2		mxidlprmALT.2	⊢ ( 𝜑 → 𝑀 ∈ ( MaxIdeal ‘ 𝑅 ) )
3		eqid	⊢ ( 𝑅 /s ( 𝑅 ~QG 𝑀 ) ) = ( 𝑅 /s ( 𝑅 ~QG 𝑀 ) )
4	1	crngringd	⊢ ( 𝜑 → 𝑅 ∈ Ring )
5		eqid	⊢ ( Base ‘ 𝑅 ) = ( Base ‘ 𝑅 )
6	5	mxidlnzr	⊢ ( ( 𝑅 ∈ Ring ∧ 𝑀 ∈ ( MaxIdeal ‘ 𝑅 ) ) → 𝑅 ∈ NzRing )
7	4 2 6	syl2anc	⊢ ( 𝜑 → 𝑅 ∈ NzRing )
8	5	mxidlidl	⊢ ( ( 𝑅 ∈ Ring ∧ 𝑀 ∈ ( MaxIdeal ‘ 𝑅 ) ) → 𝑀 ∈ ( LIdeal ‘ 𝑅 ) )
9	4 2 8	syl2anc	⊢ ( 𝜑 → 𝑀 ∈ ( LIdeal ‘ 𝑅 ) )
10	3 1 7 9	qsfld	⊢ ( 𝜑 → ( ( 𝑅 /s ( 𝑅 ~QG 𝑀 ) ) ∈ Field ↔ 𝑀 ∈ ( MaxIdeal ‘ 𝑅 ) ) )
11	2 10	mpbird	⊢ ( 𝜑 → ( 𝑅 /s ( 𝑅 ~QG 𝑀 ) ) ∈ Field )
12		fldidom	⊢ ( ( 𝑅 /s ( 𝑅 ~QG 𝑀 ) ) ∈ Field → ( 𝑅 /s ( 𝑅 ~QG 𝑀 ) ) ∈ IDomn )
13	11 12	syl	⊢ ( 𝜑 → ( 𝑅 /s ( 𝑅 ~QG 𝑀 ) ) ∈ IDomn )
14	3	qsidom	⊢ ( ( 𝑅 ∈ CRing ∧ 𝑀 ∈ ( LIdeal ‘ 𝑅 ) ) → ( ( 𝑅 /s ( 𝑅 ~QG 𝑀 ) ) ∈ IDomn ↔ 𝑀 ∈ ( PrmIdeal ‘ 𝑅 ) ) )
15	1 9 14	syl2anc	⊢ ( 𝜑 → ( ( 𝑅 /s ( 𝑅 ~QG 𝑀 ) ) ∈ IDomn ↔ 𝑀 ∈ ( PrmIdeal ‘ 𝑅 ) ) )
16	13 15	mpbid	⊢ ( 𝜑 → 𝑀 ∈ ( PrmIdeal ‘ 𝑅 ) )