Theorem 2zrng

Description: The ring of integers restricted to the even integers is a (non-unital) ring, the "ring of even integers". Remark: the structure of the complementary subset of the set of integers, the odd integers, is not even a magma, see oddinmgm . (Contributed by AV, 20-Feb-2020)

	Ref	Expression
Hypotheses	2zrng.e	$$⊢E=\left\{z\in \mathbb{Z}|\exists x\in \mathbb{Z}z=2x\right\}$$
	2zlidl.u	$$⊢U=\mathrm{LIdeal}\left({\mathbb{Z}}_{\mathrm{ring}}\right)$$
	2zrng.r	$$⊢R={\mathbb{Z}}_{\mathrm{ring}}{\upharpoonright }_{𝑠}E$$
Assertion	2zrng	$$⊢R\in \mathrm{Rng}$$

Proof

Step	Hyp	Ref	Expression
1		2zrng.e	$$⊢E=\left\{z\in \mathbb{Z}|\exists x\in \mathbb{Z}z=2x\right\}$$
2		2zlidl.u	$$⊢U=\mathrm{LIdeal}\left({\mathbb{Z}}_{\mathrm{ring}}\right)$$
3		2zrng.r	$$⊢R={\mathbb{Z}}_{\mathrm{ring}}{\upharpoonright }_{𝑠}E$$
4		zringring	$$⊢{\mathbb{Z}}_{\mathrm{ring}}\in \mathrm{Ring}$$
5	1 2	2zlidl	$$⊢E\in U$$
6	2 3	lidlrng	$$⊢\left({\mathbb{Z}}_{\mathrm{ring}}\in \mathrm{Ring}\wedge E\in U\right)\to R\in \mathrm{Rng}$$
7	4 5 6	mp2an	$$⊢R\in \mathrm{Rng}$$