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	|- ( ph -> R e. CRing )
	mxidlprmALT.2	|- ( ph -> M e. ( MaxIdeal ` R ) )
Assertion	mxidlprmALT	|- ( ph -> M e. ( PrmIdeal ` R ) )

Proof

Step	Hyp	Ref	Expression
1		mxidlprmALT.1	|- ( ph -> R e. CRing )
2		mxidlprmALT.2	|- ( ph -> M e. ( MaxIdeal ` R ) )
3		eqid	|- ( R /s ( R ~QG M ) ) = ( R /s ( R ~QG M ) )
4	1	crngringd	|- ( ph -> R e. Ring )
5		eqid	|- ( Base ` R ) = ( Base ` R )
6	5	mxidlnzr	|- ( ( R e. Ring /\ M e. ( MaxIdeal ` R ) ) -> R e. NzRing )
7	4 2 6	syl2anc	|- ( ph -> R e. NzRing )
8	5	mxidlidl	|- ( ( R e. Ring /\ M e. ( MaxIdeal ` R ) ) -> M e. ( LIdeal ` R ) )
9	4 2 8	syl2anc	|- ( ph -> M e. ( LIdeal ` R ) )
10	3 1 7 9	qsfld	|- ( ph -> ( ( R /s ( R ~QG M ) ) e. Field <-> M e. ( MaxIdeal ` R ) ) )
11	2 10	mpbird	|- ( ph -> ( R /s ( R ~QG M ) ) e. Field )
12		fldidom	|- ( ( R /s ( R ~QG M ) ) e. Field -> ( R /s ( R ~QG M ) ) e. IDomn )
13	11 12	syl	|- ( ph -> ( R /s ( R ~QG M ) ) e. IDomn )
14	3	qsidom	|- ( ( R e. CRing /\ M e. ( LIdeal ` R ) ) -> ( ( R /s ( R ~QG M ) ) e. IDomn <-> M e. ( PrmIdeal ` R ) ) )
15	1 9 14	syl2anc	|- ( ph -> ( ( R /s ( R ~QG M ) ) e. IDomn <-> M e. ( PrmIdeal ` R ) ) )
16	13 15	mpbid	|- ( ph -> M e. ( PrmIdeal ` R ) )