Definition df-zlm 14377

Description: Augment an abelian group with vector space operations to turn it into a ℤ-module. (Contributed by Mario Carneiro, 2-Oct-2015.) (Revised by AV, 12-Jun-2019.)

Assertion

Ref	Expression
df-zlm	⊢ ℤMod = (𝑔 ∈ V ↦ ((𝑔 sSet ⟨(Scalar‘ndx), ℤring⟩) sSet ⟨( ·𝑠 ‘ndx), (.g‘𝑔)⟩))

Detailed syntax breakdown of Definition df-zlm

Step	Hyp	Ref	Expression
1		czlm 14374	. 2 class ℤMod
2		vg	. . 3 setvar 𝑔
3		cvv 2772	. . 3 class V
4	2	cv 1372	. . . . 5 class 𝑔
5		cnx 12829	. . . . . . 7 class ndx
6		csca 12912	. . . . . . 7 class Scalar
7	5, 6	cfv 5271	. . . . . 6 class (Scalar‘ndx)
8		czring 14352	. . . . . 6 class ℤring
9	7, 8	cop 3636	. . . . 5 class ⟨(Scalar‘ndx), ℤring⟩
10		csts 12830	. . . . 5 class sSet
11	4, 9, 10	co 5944	. . . 4 class (𝑔 sSet ⟨(Scalar‘ndx), ℤring⟩)
12		cvsca 12913	. . . . . 6 class ·𝑠
13	5, 12	cfv 5271	. . . . 5 class ( ·𝑠 ‘ndx)
14		cmg 13455	. . . . . 6 class .g
15	4, 14	cfv 5271	. . . . 5 class (.g‘𝑔)
16	13, 15	cop 3636	. . . 4 class ⟨( ·𝑠 ‘ndx), (.g‘𝑔)⟩
17	11, 16, 10	co 5944	. . 3 class ((𝑔 sSet ⟨(Scalar‘ndx), ℤring⟩) sSet ⟨( ·𝑠 ‘ndx), (.g‘𝑔)⟩)
18	2, 3, 17	cmpt 4105	. 2 class (𝑔 ∈ V ↦ ((𝑔 sSet ⟨(Scalar‘ndx), ℤring⟩) sSet ⟨( ·𝑠 ‘ndx), (.g‘𝑔)⟩))
19	1, 18	wceq 1373	1 wff ℤMod = (𝑔 ∈ V ↦ ((𝑔 sSet ⟨(Scalar‘ndx), ℤring⟩) sSet ⟨( ·𝑠 ‘ndx), (.g‘𝑔)⟩))

This definition is referenced by:  zlmval  14389