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	$$⊢\phi \to R\in \mathrm{CRing}$$
	mxidlprmALT.2	No typesetting found for |- ( ph -> M e. ( MaxIdeal ` R ) ) with typecode |-
Assertion	mxidlprmALT	Could not format assertion : No typesetting found for |- ( ph -> M e. ( PrmIdeal ` R ) ) with typecode |-

Proof

Step	Hyp	Ref	Expression
1		mxidlprmALT.1	$$⊢\phi \to R\in \mathrm{CRing}$$
2		mxidlprmALT.2	Could not format ( ph -> M e. ( MaxIdeal ` R ) ) : No typesetting found for |- ( ph -> M e. ( MaxIdeal ` R ) ) with typecode |-
3		eqid	$$⊢R{/}_{𝑠}\left(R{~}_{\mathit{QG}}M\right)=R{/}_{𝑠}\left(R{~}_{\mathit{QG}}M\right)$$
4	1	crngringd	$$⊢\phi \to R\in \mathrm{Ring}$$
5		eqid	$$⊢{\mathrm{Base}}_R={\mathrm{Base}}_R$$
6	5	mxidlnzr	Could not format ( ( R e. Ring /\ M e. ( MaxIdeal ` R ) ) -> R e. NzRing ) : No typesetting found for |- ( ( R e. Ring /\ M e. ( MaxIdeal ` R ) ) -> R e. NzRing ) with typecode |-
7	4 2 6	syl2anc	$$⊢\phi \to R\in \mathrm{NzRing}$$
8	5	mxidlidl	Could not format ( ( R e. Ring /\ M e. ( MaxIdeal ` R ) ) -> M e. ( LIdeal ` R ) ) : No typesetting found for |- ( ( R e. Ring /\ M e. ( MaxIdeal ` R ) ) -> M e. ( LIdeal ` R ) ) with typecode |-
9	4 2 8	syl2anc	$$⊢\phi \to M\in \mathrm{LIdeal}\left(R\right)$$
10	3 1 7 9	qsfld	Could not format ( ph -> ( ( R /s ( R ~QG M ) ) e. Field <-> M e. ( MaxIdeal ` R ) ) ) : No typesetting found for |- ( ph -> ( ( R /s ( R ~QG M ) ) e. Field <-> M e. ( MaxIdeal ` R ) ) ) with typecode |-
11	2 10	mpbird	$$⊢\phi \to R{/}_{𝑠}\left(R{~}_{\mathit{QG}}M\right)\in \mathrm{Field}$$
12		fldidom	$$⊢R{/}_{𝑠}\left(R{~}_{\mathit{QG}}M\right)\in \mathrm{Field}\to R{/}_{𝑠}\left(R{~}_{\mathit{QG}}M\right)\in \mathrm{IDomn}$$
13	11 12	syl	$$⊢\phi \to R{/}_{𝑠}\left(R{~}_{\mathit{QG}}M\right)\in \mathrm{IDomn}$$
14	3	qsidom	Could not format ( ( R e. CRing /\ M e. ( LIdeal ` R ) ) -> ( ( R /s ( R ~QG M ) ) e. IDomn <-> M e. ( PrmIdeal ` R ) ) ) : No typesetting found for |- ( ( R e. CRing /\ M e. ( LIdeal ` R ) ) -> ( ( R /s ( R ~QG M ) ) e. IDomn <-> M e. ( PrmIdeal ` R ) ) ) with typecode |-
15	1 9 14	syl2anc	Could not format ( ph -> ( ( R /s ( R ~QG M ) ) e. IDomn <-> M e. ( PrmIdeal ` R ) ) ) : No typesetting found for |- ( ph -> ( ( R /s ( R ~QG M ) ) e. IDomn <-> M e. ( PrmIdeal ` R ) ) ) with typecode |-
16	13 15	mpbid	Could not format ( ph -> M e. ( PrmIdeal ` R ) ) : No typesetting found for |- ( ph -> M e. ( PrmIdeal ` R ) ) with typecode |-