Theorem ezsubrg

Description: The Eisenstein integers form a subring of the complex numbers. (Contributed by AV, 26-Oct-2026)

	Ref	Expression
Assertion	ezsubrg	Could not format assertion : No typesetting found for |- ( Z[]l ` ( bRoU ` 3 ) ) e. ( SubRing ` CCfld ) with typecode |-

Proof

Step	Hyp	Ref	Expression
1		3nn	$$⊢3\in \mathbb{N}$$
2		brucn	Could not format ( 3 e. NN -> ( bRoU ` 3 ) e. CC ) : No typesetting found for |- ( 3 e. NN -> ( bRoU ` 3 ) e. CC ) with typecode |-
3	1 2	ax-mp	Could not format ( bRoU ` 3 ) e. CC : No typesetting found for |- ( bRoU ` 3 ) e. CC with typecode |-
4		lazcn	Could not format ( ( bRoU ` 3 ) e. CC -> ( a e. ( Z[]l ` ( bRoU ` 3 ) ) -> a e. CC ) ) : No typesetting found for |- ( ( bRoU ` 3 ) e. CC -> ( a e. ( Z[]l ` ( bRoU ` 3 ) ) -> a e. CC ) ) with typecode |-
5	3 4	ax-mp	Could not format ( a e. ( Z[]l ` ( bRoU ` 3 ) ) -> a e. CC ) : No typesetting found for |- ( a e. ( Z[]l ` ( bRoU ` 3 ) ) -> a e. CC ) with typecode |-
6		eqid	Could not format ( Z[]l ` ( bRoU ` 3 ) ) = ( Z[]l ` ( bRoU ` 3 ) ) : No typesetting found for |- ( Z[]l ` ( bRoU ` 3 ) ) = ( Z[]l ` ( bRoU ` 3 ) ) with typecode |-
7	6	lazaddcl	Could not format ( ( bRoU ` 3 ) e. CC -> ( ( a e. ( Z[]l ` ( bRoU ` 3 ) ) /\ b e. ( Z[]l ` ( bRoU ` 3 ) ) ) -> ( a + b ) e. ( Z[]l ` ( bRoU ` 3 ) ) ) ) : No typesetting found for |- ( ( bRoU ` 3 ) e. CC -> ( ( a e. ( Z[]l ` ( bRoU ` 3 ) ) /\ b e. ( Z[]l ` ( bRoU ` 3 ) ) ) -> ( a + b ) e. ( Z[]l ` ( bRoU ` 3 ) ) ) ) with typecode |-
8	3 7	ax-mp	Could not format ( ( a e. ( Z[]l ` ( bRoU ` 3 ) ) /\ b e. ( Z[]l ` ( bRoU ` 3 ) ) ) -> ( a + b ) e. ( Z[]l ` ( bRoU ` 3 ) ) ) : No typesetting found for |- ( ( a e. ( Z[]l ` ( bRoU ` 3 ) ) /\ b e. ( Z[]l ` ( bRoU ` 3 ) ) ) -> ( a + b ) e. ( Z[]l ` ( bRoU ` 3 ) ) ) with typecode |-
9	6	laznegcl	Could not format ( ( bRoU ` 3 ) e. CC -> ( a e. ( Z[]l ` ( bRoU ` 3 ) ) -> -u a e. ( Z[]l ` ( bRoU ` 3 ) ) ) ) : No typesetting found for |- ( ( bRoU ` 3 ) e. CC -> ( a e. ( Z[]l ` ( bRoU ` 3 ) ) -> -u a e. ( Z[]l ` ( bRoU ` 3 ) ) ) ) with typecode |-
10	3 9	ax-mp	Could not format ( a e. ( Z[]l ` ( bRoU ` 3 ) ) -> -u a e. ( Z[]l ` ( bRoU ` 3 ) ) ) : No typesetting found for |- ( a e. ( Z[]l ` ( bRoU ` 3 ) ) -> -u a e. ( Z[]l ` ( bRoU ` 3 ) ) ) with typecode |-
11		1z	$$⊢1\in \mathbb{Z}$$
12		zellaz	Could not format ( ( ( bRoU ` 3 ) e. CC /\ 1 e. ZZ ) -> 1 e. ( Z[]l ` ( bRoU ` 3 ) ) ) : No typesetting found for |- ( ( ( bRoU ` 3 ) e. CC /\ 1 e. ZZ ) -> 1 e. ( Z[]l ` ( bRoU ` 3 ) ) ) with typecode |-
13	3 11 12	mp2an	Could not format 1 e. ( Z[]l ` ( bRoU ` 3 ) ) : No typesetting found for |- 1 e. ( Z[]l ` ( bRoU ` 3 ) ) with typecode |-
14		bru3sqez	Could not format ( ( bRoU ` 3 ) ^ 2 ) e. ( Z[]l ` ( bRoU ` 3 ) ) : No typesetting found for |- ( ( bRoU ` 3 ) ^ 2 ) e. ( Z[]l ` ( bRoU ` 3 ) ) with typecode |-
15	6	lazmulcl	Could not format ( ( ( bRoU ` 3 ) e. CC /\ ( ( bRoU ` 3 ) ^ 2 ) e. ( Z[]l ` ( bRoU ` 3 ) ) ) -> ( ( a e. ( Z[]l ` ( bRoU ` 3 ) ) /\ b e. ( Z[]l ` ( bRoU ` 3 ) ) ) -> ( a x. b ) e. ( Z[]l ` ( bRoU ` 3 ) ) ) ) : No typesetting found for |- ( ( ( bRoU ` 3 ) e. CC /\ ( ( bRoU ` 3 ) ^ 2 ) e. ( Z[]l ` ( bRoU ` 3 ) ) ) -> ( ( a e. ( Z[]l ` ( bRoU ` 3 ) ) /\ b e. ( Z[]l ` ( bRoU ` 3 ) ) ) -> ( a x. b ) e. ( Z[]l ` ( bRoU ` 3 ) ) ) ) with typecode |-
16	3 14 15	mp2an	Could not format ( ( a e. ( Z[]l ` ( bRoU ` 3 ) ) /\ b e. ( Z[]l ` ( bRoU ` 3 ) ) ) -> ( a x. b ) e. ( Z[]l ` ( bRoU ` 3 ) ) ) : No typesetting found for |- ( ( a e. ( Z[]l ` ( bRoU ` 3 ) ) /\ b e. ( Z[]l ` ( bRoU ` 3 ) ) ) -> ( a x. b ) e. ( Z[]l ` ( bRoU ` 3 ) ) ) with typecode |-
17	5 8 10 13 16	cnsubrglem	Could not format ( Z[]l ` ( bRoU ` 3 ) ) e. ( SubRing ` CCfld ) : No typesetting found for |- ( Z[]l ` ( bRoU ` 3 ) ) e. ( SubRing ` CCfld ) with typecode |-