Definition df-clm 25109

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 20591), left modules over such subrings are the same as right modules, see rmodislmod 20944. 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 25108	. 2 class ℂMod
2		vf	. . . . . . . 8 setvar 𝑓
3	2	cv 1535	. . . . . . 7 class 𝑓
4		ccnfld 21381	. . . . . . . 8 class ℂfld
5		vk	. . . . . . . . 9 setvar 𝑘
6	5	cv 1535	. . . . . . . 8 class 𝑘
7		cress 17273	. . . . . . . 8 class ↾s
8	4, 6, 7	co 7430	. . . . . . 7 class (ℂfld ↾s 𝑘)
9	3, 8	wceq 1536	. . . . . 6 wff 𝑓 = (ℂfld ↾s 𝑘)
10		csubrg 20585	. . . . . . . 8 class SubRing
11	4, 10	cfv 6562	. . . . . . 7 class (SubRing‘ℂfld)
12	6, 11	wcel 2105	. . . . . 6 wff 𝑘 ∈ (SubRing‘ℂfld)
13	9, 12	wa 395	. . . . 5 wff (𝑓 = (ℂfld ↾s 𝑘) ∧ 𝑘 ∈ (SubRing‘ℂfld))
14		cbs 17244	. . . . . 6 class Base
15	3, 14	cfv 6562	. . . . 5 class (Base‘𝑓)
16	13, 5, 15	wsbc 3790	. . . 4 wff [(Base‘𝑓) / 𝑘](𝑓 = (ℂfld ↾s 𝑘) ∧ 𝑘 ∈ (SubRing‘ℂfld))
17		vw	. . . . . 6 setvar 𝑤
18	17	cv 1535	. . . . 5 class 𝑤
19		csca 17300	. . . . 5 class Scalar
20	18, 19	cfv 6562	. . . 4 class (Scalar‘𝑤)
21	16, 2, 20	wsbc 3790	. . 3 wff [(Scalar‘𝑤) / 𝑓][(Base‘𝑓) / 𝑘](𝑓 = (ℂfld ↾s 𝑘) ∧ 𝑘 ∈ (SubRing‘ℂfld))
22		clmod 20874	. . 3 class LMod
23	21, 17, 22	crab 3432	. 2 class {𝑤 ∈ LMod ∣ [(Scalar‘𝑤) / 𝑓][(Base‘𝑓) / 𝑘](𝑓 = (ℂfld ↾s 𝑘) ∧ 𝑘 ∈ (SubRing‘ℂfld))}
24	1, 23	wceq 1536	1 wff ℂMod = {𝑤 ∈ LMod ∣ [(Scalar‘𝑤) / 𝑓][(Base‘𝑓) / 𝑘](𝑓 = (ℂfld ↾s 𝑘) ∧ 𝑘 ∈ (SubRing‘ℂfld))}

This definition is referenced by:  isclm  25110