Definition df-clm 25113

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 20612), left modules over such subrings are the same as right modules, see rmodislmod 20985. 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 25112	. 2 class ℂMod
2		vf	. . . . . . . 8 setvar 𝑓
3	2	cv 1558	. . . . . . 7 class 𝑓
4		ccnfld 21412	. . . . . . . 8 class ℂfld
5		vk	. . . . . . . . 9 setvar 𝑘
6	5	cv 1558	. . . . . . . 8 class 𝑘
7		cress 17257	. . . . . . . 8 class ↾s
8	4, 6, 7	co 7391	. . . . . . 7 class (ℂfld ↾s 𝑘)
9	3, 8	wceq 1559	. . . . . 6 wff 𝑓 = (ℂfld ↾s 𝑘)
10		csubrg 20606	. . . . . . . 8 class SubRing
11	4, 10	cfv 6516	. . . . . . 7 class (SubRing‘ℂfld)
12	6, 11	wcel 2141	. . . . . 6 wff 𝑘 ∈ (SubRing‘ℂfld)
13	9, 12	wa 399	. . . . 5 wff (𝑓 = (ℂfld ↾s 𝑘) ∧ 𝑘 ∈ (SubRing‘ℂfld))
14		cbs 17236	. . . . . 6 class Base
15	3, 14	cfv 6516	. . . . 5 class (Base‘𝑓)
16	13, 5, 15	wsbc 3742	. . . 4 wff [(Base‘𝑓) / 𝑘](𝑓 = (ℂfld ↾s 𝑘) ∧ 𝑘 ∈ (SubRing‘ℂfld))
17		vw	. . . . . 6 setvar 𝑤
18	17	cv 1558	. . . . 5 class 𝑤
19		csca 17280	. . . . 5 class Scalar
20	18, 19	cfv 6516	. . . 4 class (Scalar‘𝑤)
21	16, 2, 20	wsbc 3742	. . 3 wff [(Scalar‘𝑤) / 𝑓][(Base‘𝑓) / 𝑘](𝑓 = (ℂfld ↾s 𝑘) ∧ 𝑘 ∈ (SubRing‘ℂfld))
22		clmod 20915	. . 3 class LMod
23	21, 17, 22	crab 3413	. 2 class {𝑤 ∈ LMod ∣ [(Scalar‘𝑤) / 𝑓][(Base‘𝑓) / 𝑘](𝑓 = (ℂfld ↾s 𝑘) ∧ 𝑘 ∈ (SubRing‘ℂfld))}
24	1, 23	wceq 1559	1 wff ℂMod = {𝑤 ∈ LMod ∣ [(Scalar‘𝑤) / 𝑓][(Base‘𝑓) / 𝑘](𝑓 = (ℂfld ↾s 𝑘) ∧ 𝑘 ∈ (SubRing‘ℂfld))}

This definition is referenced by:  isclm  25114