Definition df-clm 25222

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 20674), left modules over such subrings are the same as right modules, see rmodislmod 21051. 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 25221	. 2 class ℂMod
2		vf	. . . . . . . 8 setvar 𝑓
3	2	cv 1569	. . . . . . 7 class 𝑓
4		ccnfld 21522	. . . . . . . 8 class ℂfld
5		vk	. . . . . . . . 9 setvar 𝑘
6	5	cv 1569	. . . . . . . 8 class 𝑘
7		cress 17285	. . . . . . . 8 class ↾s
8	4, 6, 7	co 7410	. . . . . . 7 class (ℂfld ↾s 𝑘)
9	3, 8	wceq 1570	. . . . . 6 wff 𝑓 = (ℂfld ↾s 𝑘)
10		csubrg 20668	. . . . . . . 8 class SubRing
11	4, 10	cfv 6536	. . . . . . 7 class (SubRing‘ℂfld)
12	6, 11	wcel 2143	. . . . . 6 wff 𝑘 ∈ (SubRing‘ℂfld)
13	9, 12	wa 400	. . . . 5 wff (𝑓 = (ℂfld ↾s 𝑘) ∧ 𝑘 ∈ (SubRing‘ℂfld))
14		cbs 17264	. . . . . 6 class Base
15	3, 14	cfv 6536	. . . . 5 class (Base‘𝑓)
16	13, 5, 15	wsbc 3744	. . . 4 wff [(Base‘𝑓) / 𝑘](𝑓 = (ℂfld ↾s 𝑘) ∧ 𝑘 ∈ (SubRing‘ℂfld))
17		vw	. . . . . 6 setvar 𝑤
18	17	cv 1569	. . . . 5 class 𝑤
19		csca 17308	. . . . 5 class Scalar
20	18, 19	cfv 6536	. . . 4 class (Scalar‘𝑤)
21	16, 2, 20	wsbc 3744	. . 3 wff [(Scalar‘𝑤) / 𝑓][(Base‘𝑓) / 𝑘](𝑓 = (ℂfld ↾s 𝑘) ∧ 𝑘 ∈ (SubRing‘ℂfld))
22		clmod 20981	. . 3 class LMod
23	21, 17, 22	crab 3416	. 2 class {𝑤 ∈ LMod ∣ [(Scalar‘𝑤) / 𝑓][(Base‘𝑓) / 𝑘](𝑓 = (ℂfld ↾s 𝑘) ∧ 𝑘 ∈ (SubRing‘ℂfld))}
24	1, 23	wceq 1570	1 wff ℂMod = {𝑤 ∈ LMod ∣ [(Scalar‘𝑤) / 𝑓][(Base‘𝑓) / 𝑘](𝑓 = (ℂfld ↾s 𝑘) ∧ 𝑘 ∈ (SubRing‘ℂfld))}

This definition is referenced by:  isclm  25223