Definition df-clm 25115

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 20603), left modules over such subrings are the same as right modules, see rmodislmod 20950. 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 25114	. 2 class ℂMod
2		vf	. . . . . . . 8 setvar 𝑓
3	2	cv 1536	. . . . . . 7 class 𝑓
4		ccnfld 21387	. . . . . . . 8 class ℂfld
5		vk	. . . . . . . . 9 setvar 𝑘
6	5	cv 1536	. . . . . . . 8 class 𝑘
7		cress 17287	. . . . . . . 8 class ↾s
8	4, 6, 7	co 7448	. . . . . . 7 class (ℂfld ↾s 𝑘)
9	3, 8	wceq 1537	. . . . . 6 wff 𝑓 = (ℂfld ↾s 𝑘)
10		csubrg 20595	. . . . . . . 8 class SubRing
11	4, 10	cfv 6573	. . . . . . 7 class (SubRing‘ℂfld)
12	6, 11	wcel 2108	. . . . . 6 wff 𝑘 ∈ (SubRing‘ℂfld)
13	9, 12	wa 395	. . . . 5 wff (𝑓 = (ℂfld ↾s 𝑘) ∧ 𝑘 ∈ (SubRing‘ℂfld))
14		cbs 17258	. . . . . 6 class Base
15	3, 14	cfv 6573	. . . . 5 class (Base‘𝑓)
16	13, 5, 15	wsbc 3804	. . . 4 wff [(Base‘𝑓) / 𝑘](𝑓 = (ℂfld ↾s 𝑘) ∧ 𝑘 ∈ (SubRing‘ℂfld))
17		vw	. . . . . 6 setvar 𝑤
18	17	cv 1536	. . . . 5 class 𝑤
19		csca 17314	. . . . 5 class Scalar
20	18, 19	cfv 6573	. . . 4 class (Scalar‘𝑤)
21	16, 2, 20	wsbc 3804	. . 3 wff [(Scalar‘𝑤) / 𝑓][(Base‘𝑓) / 𝑘](𝑓 = (ℂfld ↾s 𝑘) ∧ 𝑘 ∈ (SubRing‘ℂfld))
22		clmod 20880	. . 3 class LMod
23	21, 17, 22	crab 3443	. 2 class {𝑤 ∈ LMod ∣ [(Scalar‘𝑤) / 𝑓][(Base‘𝑓) / 𝑘](𝑓 = (ℂfld ↾s 𝑘) ∧ 𝑘 ∈ (SubRing‘ℂfld))}
24	1, 23	wceq 1537	1 wff ℂMod = {𝑤 ∈ LMod ∣ [(Scalar‘𝑤) / 𝑓][(Base‘𝑓) / 𝑘](𝑓 = (ℂfld ↾s 𝑘) ∧ 𝑘 ∈ (SubRing‘ℂfld))}

This definition is referenced by:  isclm  25116