Theorem ufdidom

Description: A nonzero unique factorization domain is an integral domain. (Contributed by Thierry Arnoux, 3-Jun-2025)

	Ref	Expression
Hypothesis	ufdidom.2	$$⊢\phi \to R\in \mathrm{UFD}$$
Assertion	ufdidom	$$⊢\phi \to R\in \mathrm{IDomn}$$

Proof

Step	Hyp	Ref	Expression
1		ufdidom.2	$$⊢\phi \to R\in \mathrm{UFD}$$
2		eqid	$$⊢\mathrm{PrmIdeal}\left(R\right)=\mathrm{PrmIdeal}\left(R\right)$$
3		eqid	$$⊢\mathrm{RPrime}\left(R\right)=\mathrm{RPrime}\left(R\right)$$
4		eqid	$$⊢0_R=0_R$$
5	2 3 4	isufd	$$⊢R\in \mathrm{UFD}\leftrightarrow \left(R\in \mathrm{IDomn}\wedge \forall i\in \left(\mathrm{PrmIdeal}\left(R\right)\setminus \left\{\left\{0_R\right\}\right\}\right)i\cap \mathrm{RPrime}\left(R\right)\ne \varnothing \right)$$
6	5	simplbi	$$⊢R\in \mathrm{UFD}\to R\in \mathrm{IDomn}$$
7	1 6	syl	$$⊢\phi \to R\in \mathrm{IDomn}$$