Theorem crngprmringidom

Description: A commutative ring is a prime ring if and only if it is an integral domain. (Contributed by AV, 27-Jun-2026)

	Ref	Expression
Assertion	crngprmringidom	Could not format assertion : No typesetting found for |- ( R e. CRing -> ( R e. PrmRing <-> R e. IDomn ) ) with typecode |-

Proof

Step	Hyp	Ref	Expression
1		crngring	$$⊢R\in \mathrm{CRing}\to R\in \mathrm{Ring}$$
2		eqid	$$⊢0_R=0_R$$
3		eqid	$$⊢\mathrm{PrmIdeal}\left(R\right)=\mathrm{PrmIdeal}\left(R\right)$$
4	2 3	isprmrng	Could not format ( R e. PrmRing <-> ( R e. Ring /\ { ( 0g ` R ) } e. ( PrmIdeal ` R ) ) ) : No typesetting found for |- ( R e. PrmRing <-> ( R e. Ring /\ { ( 0g ` R ) } e. ( PrmIdeal ` R ) ) ) with typecode |-
5	4	a1i	Could not format ( R e. CRing -> ( R e. PrmRing <-> ( R e. Ring /\ { ( 0g ` R ) } e. ( PrmIdeal ` R ) ) ) ) : No typesetting found for |- ( R e. CRing -> ( R e. PrmRing <-> ( R e. Ring /\ { ( 0g ` R ) } e. ( PrmIdeal ` R ) ) ) ) with typecode |-
6	1 5	mpbirand	Could not format ( R e. CRing -> ( R e. PrmRing <-> { ( 0g ` R ) } e. ( PrmIdeal ` R ) ) ) : No typesetting found for |- ( R e. CRing -> ( R e. PrmRing <-> { ( 0g ` R ) } e. ( PrmIdeal ` R ) ) ) with typecode |-
7		ibar	$$⊢R\in \mathrm{CRing}\to \left(\left\{0_R\right\}\in \mathrm{PrmIdeal}\left(R\right)\leftrightarrow \left(R\in \mathrm{CRing}\wedge \left\{0_R\right\}\in \mathrm{PrmIdeal}\left(R\right)\right)\right)$$
8	2	prmidl0	$$⊢\left(R\in \mathrm{CRing}\wedge \left\{0_R\right\}\in \mathrm{PrmIdeal}\left(R\right)\right)\leftrightarrow R\in \mathrm{IDomn}$$
9	8	a1i	$$⊢R\in \mathrm{CRing}\to \left(\left(R\in \mathrm{CRing}\wedge \left\{0_R\right\}\in \mathrm{PrmIdeal}\left(R\right)\right)\leftrightarrow R\in \mathrm{IDomn}\right)$$
10	6 7 9	3bitrd	Could not format ( R e. CRing -> ( R e. PrmRing <-> R e. IDomn ) ) : No typesetting found for |- ( R e. CRing -> ( R e. PrmRing <-> R e. IDomn ) ) with typecode |-