Definition df-clm 24963

Description: Define the class of subcomplex modules, which are left modules over a subring of the field of complex numbers ℂ_fld, which allows to use the complex addition, multiplication, etc. in theorems about subcomplex modules. Since the field of complex numbers is commutative and so are its subrings (see subrgcrng 20484), left modules over such subrings are the same as right modules, see rmodislmod 20836. Therefore, we drop the word "left" from "subcomplex left module". (Contributed by Mario Carneiro, 16-Oct-2015.)

Assertion

Ref	Expression
df-clm	⊢ ℂMod = {𝑤 ∈ LMod ∣ [(Scalar‘𝑤) / 𝑓][(Base‘𝑓) / 𝑘](𝑓 = (ℂfld ↾s 𝑘) ∧ 𝑘 ∈ (SubRing‘ℂfld))}

Distinct variable group:   𝑓,𝑘,𝑤

Detailed syntax breakdown of Definition df-clm

Step	Hyp	Ref	Expression
1		cclm 24962	. 2 class ℂMod
2		vf	. . . . . . . 8 setvar 𝑓
3	2	cv 1539	. . . . . . 7 class 𝑓
4		ccnfld 21264	. . . . . . . 8 class ℂfld
5		vk	. . . . . . . . 9 setvar 𝑘
6	5	cv 1539	. . . . . . . 8 class 𝑘
7		cress 17200	. . . . . . . 8 class ↾s
8	4, 6, 7	co 7387	. . . . . . 7 class (ℂfld ↾s 𝑘)
9	3, 8	wceq 1540	. . . . . 6 wff 𝑓 = (ℂfld ↾s 𝑘)
10		csubrg 20478	. . . . . . . 8 class SubRing
11	4, 10	cfv 6511	. . . . . . 7 class (SubRing‘ℂfld)
12	6, 11	wcel 2109	. . . . . 6 wff 𝑘 ∈ (SubRing‘ℂfld)
13	9, 12	wa 395	. . . . 5 wff (𝑓 = (ℂfld ↾s 𝑘) ∧ 𝑘 ∈ (SubRing‘ℂfld))
14		cbs 17179	. . . . . 6 class Base
15	3, 14	cfv 6511	. . . . 5 class (Base‘𝑓)
16	13, 5, 15	wsbc 3753	. . . 4 wff [(Base‘𝑓) / 𝑘](𝑓 = (ℂfld ↾s 𝑘) ∧ 𝑘 ∈ (SubRing‘ℂfld))
17		vw	. . . . . 6 setvar 𝑤
18	17	cv 1539	. . . . 5 class 𝑤
19		csca 17223	. . . . 5 class Scalar
20	18, 19	cfv 6511	. . . 4 class (Scalar‘𝑤)
21	16, 2, 20	wsbc 3753	. . 3 wff [(Scalar‘𝑤) / 𝑓][(Base‘𝑓) / 𝑘](𝑓 = (ℂfld ↾s 𝑘) ∧ 𝑘 ∈ (SubRing‘ℂfld))
22		clmod 20766	. . 3 class LMod
23	21, 17, 22	crab 3405	. 2 class {𝑤 ∈ LMod ∣ [(Scalar‘𝑤) / 𝑓][(Base‘𝑓) / 𝑘](𝑓 = (ℂfld ↾s 𝑘) ∧ 𝑘 ∈ (SubRing‘ℂfld))}
24	1, 23	wceq 1540	1 wff ℂMod = {𝑤 ∈ LMod ∣ [(Scalar‘𝑤) / 𝑓][(Base‘𝑓) / 𝑘](𝑓 = (ℂfld ↾s 𝑘) ∧ 𝑘 ∈ (SubRing‘ℂfld))}

This definition is referenced by:  isclm  24964