Definition df-clm 25030

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 20552), left modules over such subrings are the same as right modules, see rmodislmod 20925. 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 25029	. 2 class ℂMod
2		vf	. . . . . . . 8 setvar 𝑓
3	2	cv 1541	. . . . . . 7 class 𝑓
4		ccnfld 21352	. . . . . . . 8 class ℂfld
5		vk	. . . . . . . . 9 setvar 𝑘
6	5	cv 1541	. . . . . . . 8 class 𝑘
7		cress 17200	. . . . . . . 8 class ↾s
8	4, 6, 7	co 7367	. . . . . . 7 class (ℂfld ↾s 𝑘)
9	3, 8	wceq 1542	. . . . . 6 wff 𝑓 = (ℂfld ↾s 𝑘)
10		csubrg 20546	. . . . . . . 8 class SubRing
11	4, 10	cfv 6498	. . . . . . 7 class (SubRing‘ℂfld)
12	6, 11	wcel 2114	. . . . . 6 wff 𝑘 ∈ (SubRing‘ℂfld)
13	9, 12	wa 395	. . . . 5 wff (𝑓 = (ℂfld ↾s 𝑘) ∧ 𝑘 ∈ (SubRing‘ℂfld))
14		cbs 17179	. . . . . 6 class Base
15	3, 14	cfv 6498	. . . . 5 class (Base‘𝑓)
16	13, 5, 15	wsbc 3728	. . . 4 wff [(Base‘𝑓) / 𝑘](𝑓 = (ℂfld ↾s 𝑘) ∧ 𝑘 ∈ (SubRing‘ℂfld))
17		vw	. . . . . 6 setvar 𝑤
18	17	cv 1541	. . . . 5 class 𝑤
19		csca 17223	. . . . 5 class Scalar
20	18, 19	cfv 6498	. . . 4 class (Scalar‘𝑤)
21	16, 2, 20	wsbc 3728	. . 3 wff [(Scalar‘𝑤) / 𝑓][(Base‘𝑓) / 𝑘](𝑓 = (ℂfld ↾s 𝑘) ∧ 𝑘 ∈ (SubRing‘ℂfld))
22		clmod 20855	. . 3 class LMod
23	21, 17, 22	crab 3389	. 2 class {𝑤 ∈ LMod ∣ [(Scalar‘𝑤) / 𝑓][(Base‘𝑓) / 𝑘](𝑓 = (ℂfld ↾s 𝑘) ∧ 𝑘 ∈ (SubRing‘ℂfld))}
24	1, 23	wceq 1542	1 wff ℂMod = {𝑤 ∈ LMod ∣ [(Scalar‘𝑤) / 𝑓][(Base‘𝑓) / 𝑘](𝑓 = (ℂfld ↾s 𝑘) ∧ 𝑘 ∈ (SubRing‘ℂfld))}

This definition is referenced by:  isclm  25031