Definition df-clm 25048

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 20547), left modules over such subrings are the same as right modules, see rmodislmod 20920. 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 25047	. 2 class ℂMod
2		vf	. . . . . . . 8 setvar 𝑓
3	2	cv 1546	. . . . . . 7 class 𝑓
4		ccnfld 21347	. . . . . . . 8 class ℂfld
5		vk	. . . . . . . . 9 setvar 𝑘
6	5	cv 1546	. . . . . . . 8 class 𝑘
7		cress 17191	. . . . . . . 8 class ↾s
8	4, 6, 7	co 7356	. . . . . . 7 class (ℂfld ↾s 𝑘)
9	3, 8	wceq 1547	. . . . . 6 wff 𝑓 = (ℂfld ↾s 𝑘)
10		csubrg 20541	. . . . . . . 8 class SubRing
11	4, 10	cfv 6485	. . . . . . 7 class (SubRing‘ℂfld)
12	6, 11	wcel 2119	. . . . . 6 wff 𝑘 ∈ (SubRing‘ℂfld)
13	9, 12	wa 396	. . . . 5 wff (𝑓 = (ℂfld ↾s 𝑘) ∧ 𝑘 ∈ (SubRing‘ℂfld))
14		cbs 17170	. . . . . 6 class Base
15	3, 14	cfv 6485	. . . . 5 class (Base‘𝑓)
16	13, 5, 15	wsbc 3723	. . . 4 wff [(Base‘𝑓) / 𝑘](𝑓 = (ℂfld ↾s 𝑘) ∧ 𝑘 ∈ (SubRing‘ℂfld))
17		vw	. . . . . 6 setvar 𝑤
18	17	cv 1546	. . . . 5 class 𝑤
19		csca 17214	. . . . 5 class Scalar
20	18, 19	cfv 6485	. . . 4 class (Scalar‘𝑤)
21	16, 2, 20	wsbc 3723	. . 3 wff [(Scalar‘𝑤) / 𝑓][(Base‘𝑓) / 𝑘](𝑓 = (ℂfld ↾s 𝑘) ∧ 𝑘 ∈ (SubRing‘ℂfld))
22		clmod 20850	. . 3 class LMod
23	21, 17, 22	crab 3391	. 2 class {𝑤 ∈ LMod ∣ [(Scalar‘𝑤) / 𝑓][(Base‘𝑓) / 𝑘](𝑓 = (ℂfld ↾s 𝑘) ∧ 𝑘 ∈ (SubRing‘ℂfld))}
24	1, 23	wceq 1547	1 wff ℂMod = {𝑤 ∈ LMod ∣ [(Scalar‘𝑤) / 𝑓][(Base‘𝑓) / 𝑘](𝑓 = (ℂfld ↾s 𝑘) ∧ 𝑘 ∈ (SubRing‘ℂfld))}

This definition is referenced by:  isclm  25049