Definition df-clm 25017

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 20506), left modules over such subrings are the same as right modules, see rmodislmod 20879. 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 25016	. 2 class ℂMod
2		vf	. . . . . . . 8 setvar 𝑓
3	2	cv 1540	. . . . . . 7 class 𝑓
4		ccnfld 21307	. . . . . . . 8 class ℂfld
5		vk	. . . . . . . . 9 setvar 𝑘
6	5	cv 1540	. . . . . . . 8 class 𝑘
7		cress 17155	. . . . . . . 8 class ↾s
8	4, 6, 7	co 7356	. . . . . . 7 class (ℂfld ↾s 𝑘)
9	3, 8	wceq 1541	. . . . . 6 wff 𝑓 = (ℂfld ↾s 𝑘)
10		csubrg 20500	. . . . . . . 8 class SubRing
11	4, 10	cfv 6490	. . . . . . 7 class (SubRing‘ℂfld)
12	6, 11	wcel 2113	. . . . . 6 wff 𝑘 ∈ (SubRing‘ℂfld)
13	9, 12	wa 395	. . . . 5 wff (𝑓 = (ℂfld ↾s 𝑘) ∧ 𝑘 ∈ (SubRing‘ℂfld))
14		cbs 17134	. . . . . 6 class Base
15	3, 14	cfv 6490	. . . . 5 class (Base‘𝑓)
16	13, 5, 15	wsbc 3738	. . . 4 wff [(Base‘𝑓) / 𝑘](𝑓 = (ℂfld ↾s 𝑘) ∧ 𝑘 ∈ (SubRing‘ℂfld))
17		vw	. . . . . 6 setvar 𝑤
18	17	cv 1540	. . . . 5 class 𝑤
19		csca 17178	. . . . 5 class Scalar
20	18, 19	cfv 6490	. . . 4 class (Scalar‘𝑤)
21	16, 2, 20	wsbc 3738	. . 3 wff [(Scalar‘𝑤) / 𝑓][(Base‘𝑓) / 𝑘](𝑓 = (ℂfld ↾s 𝑘) ∧ 𝑘 ∈ (SubRing‘ℂfld))
22		clmod 20809	. . 3 class LMod
23	21, 17, 22	crab 3397	. 2 class {𝑤 ∈ LMod ∣ [(Scalar‘𝑤) / 𝑓][(Base‘𝑓) / 𝑘](𝑓 = (ℂfld ↾s 𝑘) ∧ 𝑘 ∈ (SubRing‘ℂfld))}
24	1, 23	wceq 1541	1 wff ℂMod = {𝑤 ∈ LMod ∣ [(Scalar‘𝑤) / 𝑓][(Base‘𝑓) / 𝑘](𝑓 = (ℂfld ↾s 𝑘) ∧ 𝑘 ∈ (SubRing‘ℂfld))}

This definition is referenced by:  isclm  25018