Definition df-clm 25019

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 20508), left modules over such subrings are the same as right modules, see rmodislmod 20881. 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 25018	. 2 class ℂMod
2		vf	. . . . . . . 8 setvar 𝑓
3	2	cv 1540	. . . . . . 7 class 𝑓
4		ccnfld 21309	. . . . . . . 8 class ℂfld
5		vk	. . . . . . . . 9 setvar 𝑘
6	5	cv 1540	. . . . . . . 8 class 𝑘
7		cress 17157	. . . . . . . 8 class ↾s
8	4, 6, 7	co 7358	. . . . . . 7 class (ℂfld ↾s 𝑘)
9	3, 8	wceq 1541	. . . . . 6 wff 𝑓 = (ℂfld ↾s 𝑘)
10		csubrg 20502	. . . . . . . 8 class SubRing
11	4, 10	cfv 6492	. . . . . . 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 17136	. . . . . 6 class Base
15	3, 14	cfv 6492	. . . . 5 class (Base‘𝑓)
16	13, 5, 15	wsbc 3740	. . . 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 17180	. . . . 5 class Scalar
20	18, 19	cfv 6492	. . . 4 class (Scalar‘𝑤)
21	16, 2, 20	wsbc 3740	. . 3 wff [(Scalar‘𝑤) / 𝑓][(Base‘𝑓) / 𝑘](𝑓 = (ℂfld ↾s 𝑘) ∧ 𝑘 ∈ (SubRing‘ℂfld))
22		clmod 20811	. . 3 class LMod
23	21, 17, 22	crab 3399	. 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  25020