Definition df-clm 24570

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 20359), left modules over such subrings are the same as right modules, see rmodislmod 20532. 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 24569	. 2 class ℂMod
2		vf	. . . . . . . 8 setvar 𝑓
3	2	cv 1540	. . . . . . 7 class 𝑓
4		ccnfld 20936	. . . . . . . 8 class ℂfld
5		vk	. . . . . . . . 9 setvar 𝑘
6	5	cv 1540	. . . . . . . 8 class 𝑘
7		cress 17169	. . . . . . . 8 class ↾s
8	4, 6, 7	co 7405	. . . . . . 7 class (ℂfld ↾s 𝑘)
9	3, 8	wceq 1541	. . . . . 6 wff 𝑓 = (ℂfld ↾s 𝑘)
10		csubrg 20351	. . . . . . . 8 class SubRing
11	4, 10	cfv 6540	. . . . . . 7 class (SubRing‘ℂfld)
12	6, 11	wcel 2106	. . . . . 6 wff 𝑘 ∈ (SubRing‘ℂfld)
13	9, 12	wa 396	. . . . 5 wff (𝑓 = (ℂfld ↾s 𝑘) ∧ 𝑘 ∈ (SubRing‘ℂfld))
14		cbs 17140	. . . . . 6 class Base
15	3, 14	cfv 6540	. . . . 5 class (Base‘𝑓)
16	13, 5, 15	wsbc 3776	. . . 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 17196	. . . . 5 class Scalar
20	18, 19	cfv 6540	. . . 4 class (Scalar‘𝑤)
21	16, 2, 20	wsbc 3776	. . 3 wff [(Scalar‘𝑤) / 𝑓][(Base‘𝑓) / 𝑘](𝑓 = (ℂfld ↾s 𝑘) ∧ 𝑘 ∈ (SubRing‘ℂfld))
22		clmod 20463	. . 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 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  24571