Definition df-clm 25187

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 20656), left modules over such subrings are the same as right modules, see rmodislmod 21025. 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 25186	. 2 class ℂMod
2		vf	. . . . . . . 8 setvar 𝑓
3	2	cv 1566	. . . . . . 7 class 𝑓
4		ccnfld 21487	. . . . . . . 8 class ℂfld
5		vk	. . . . . . . . 9 setvar 𝑘
6	5	cv 1566	. . . . . . . 8 class 𝑘
7		cress 17286	. . . . . . . 8 class ↾s
8	4, 6, 7	co 7408	. . . . . . 7 class (ℂfld ↾s 𝑘)
9	3, 8	wceq 1567	. . . . . 6 wff 𝑓 = (ℂfld ↾s 𝑘)
10		csubrg 20650	. . . . . . . 8 class SubRing
11	4, 10	cfv 6534	. . . . . . 7 class (SubRing‘ℂfld)
12	6, 11	wcel 2149	. . . . . 6 wff 𝑘 ∈ (SubRing‘ℂfld)
13	9, 12	wa 400	. . . . 5 wff (𝑓 = (ℂfld ↾s 𝑘) ∧ 𝑘 ∈ (SubRing‘ℂfld))
14		cbs 17265	. . . . . 6 class Base
15	3, 14	cfv 6534	. . . . 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 1566	. . . . 5 class 𝑤
19		csca 17309	. . . . 5 class Scalar
20	18, 19	cfv 6534	. . . 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 20955	. . 3 class LMod
23	21, 17, 22	crab 3423	. 2 class {𝑤 ∈ LMod ∣ [(Scalar‘𝑤) / 𝑓][(Base‘𝑓) / 𝑘](𝑓 = (ℂfld ↾s 𝑘) ∧ 𝑘 ∈ (SubRing‘ℂfld))}
24	1, 23	wceq 1567	1 wff ℂMod = {𝑤 ∈ LMod ∣ [(Scalar‘𝑤) / 𝑓][(Base‘𝑓) / 𝑘](𝑓 = (ℂfld ↾s 𝑘) ∧ 𝑘 ∈ (SubRing‘ℂfld))}

This definition is referenced by:  isclm  25188