Theorem zringeuf

Description: The Euclidean function of the ring of integers. (Contributed by Thierry Arnoux, 9-Nov-2026)

	Ref	Expression
Assertion	zringeuf	$$⊢\mathrm{abs}=\mathrm{EuclF}\left({\mathbb{Z}}_{\mathrm{ring}}\right)$$

Proof

Step	Hyp	Ref	Expression
1		zex	$$⊢\mathbb{Z}\in \mathrm{V}$$
2		df-zring	$$⊢{\mathbb{Z}}_{\mathrm{ring}}={ℂ}_{\mathrm{fld}}{\upharpoonright }_{𝑠}\mathbb{Z}$$
3		cnfldeuf	$$⊢\mathrm{abs}=\mathrm{EuclF}\left({ℂ}_{\mathrm{fld}}\right)$$
4	2 3	resseuf	$$⊢\mathbb{Z}\in \mathrm{V}\to \mathrm{abs}=\mathrm{EuclF}\left({\mathbb{Z}}_{\mathrm{ring}}\right)$$
5	1 4	ax-mp	$$⊢\mathrm{abs}=\mathrm{EuclF}\left({\mathbb{Z}}_{\mathrm{ring}}\right)$$