Definition df-clm 24235

Description: Define the class of subcomplex modules, which are left modules over a subring of the field of complex numbers ℂ_fld, which allows us 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 20037), left modules over such subrings are the same as right modules, see rmodislmod 20200. 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 24234	. 2 class ℂMod
2		vf	. . . . . . . 8 setvar 𝑓
3	2	cv 1538	. . . . . . 7 class 𝑓
4		ccnfld 20606	. . . . . . . 8 class ℂfld
5		vk	. . . . . . . . 9 setvar 𝑘
6	5	cv 1538	. . . . . . . 8 class 𝑘
7		cress 16950	. . . . . . . 8 class ↾s
8	4, 6, 7	co 7284	. . . . . . 7 class (ℂfld ↾s 𝑘)
9	3, 8	wceq 1539	. . . . . 6 wff 𝑓 = (ℂfld ↾s 𝑘)
10		csubrg 20029	. . . . . . . 8 class SubRing
11	4, 10	cfv 6437	. . . . . . 7 class (SubRing‘ℂfld)
12	6, 11	wcel 2107	. . . . . 6 wff 𝑘 ∈ (SubRing‘ℂfld)
13	9, 12	wa 396	. . . . 5 wff (𝑓 = (ℂfld ↾s 𝑘) ∧ 𝑘 ∈ (SubRing‘ℂfld))
14		cbs 16921	. . . . . 6 class Base
15	3, 14	cfv 6437	. . . . 5 class (Base‘𝑓)
16	13, 5, 15	wsbc 3717	. . . 4 wff [(Base‘𝑓) / 𝑘](𝑓 = (ℂfld ↾s 𝑘) ∧ 𝑘 ∈ (SubRing‘ℂfld))
17		vw	. . . . . 6 setvar 𝑤
18	17	cv 1538	. . . . 5 class 𝑤
19		csca 16974	. . . . 5 class Scalar
20	18, 19	cfv 6437	. . . 4 class (Scalar‘𝑤)
21	16, 2, 20	wsbc 3717	. . 3 wff [(Scalar‘𝑤) / 𝑓][(Base‘𝑓) / 𝑘](𝑓 = (ℂfld ↾s 𝑘) ∧ 𝑘 ∈ (SubRing‘ℂfld))
22		clmod 20132	. . 3 class LMod
23	21, 17, 22	crab 3069	. 2 class {𝑤 ∈ LMod ∣ [(Scalar‘𝑤) / 𝑓][(Base‘𝑓) / 𝑘](𝑓 = (ℂfld ↾s 𝑘) ∧ 𝑘 ∈ (SubRing‘ℂfld))}
24	1, 23	wceq 1539	1 wff ℂMod = {𝑤 ∈ LMod ∣ [(Scalar‘𝑤) / 𝑓][(Base‘𝑓) / 𝑘](𝑓 = (ℂfld ↾s 𝑘) ∧ 𝑘 ∈ (SubRing‘ℂfld))}

This definition is referenced by:  isclm  24236