Definition df-clm 24977

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 20503), left modules over such subrings are the same as right modules, see rmodislmod 20802. 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 24976	. 2 class ℂMod
2		vf	. . . . . . . 8 setvar 𝑓
3	2	cv 1533	. . . . . . 7 class 𝑓
4		ccnfld 21266	. . . . . . . 8 class ℂfld
5		vk	. . . . . . . . 9 setvar 𝑘
6	5	cv 1533	. . . . . . . 8 class 𝑘
7		cress 17200	. . . . . . . 8 class ↾s
8	4, 6, 7	co 7414	. . . . . . 7 class (ℂfld ↾s 𝑘)
9	3, 8	wceq 1534	. . . . . 6 wff 𝑓 = (ℂfld ↾s 𝑘)
10		csubrg 20495	. . . . . . . 8 class SubRing
11	4, 10	cfv 6542	. . . . . . 7 class (SubRing‘ℂfld)
12	6, 11	wcel 2099	. . . . . 6 wff 𝑘 ∈ (SubRing‘ℂfld)
13	9, 12	wa 395	. . . . 5 wff (𝑓 = (ℂfld ↾s 𝑘) ∧ 𝑘 ∈ (SubRing‘ℂfld))
14		cbs 17171	. . . . . 6 class Base
15	3, 14	cfv 6542	. . . . 5 class (Base‘𝑓)
16	13, 5, 15	wsbc 3774	. . . 4 wff [(Base‘𝑓) / 𝑘](𝑓 = (ℂfld ↾s 𝑘) ∧ 𝑘 ∈ (SubRing‘ℂfld))
17		vw	. . . . . 6 setvar 𝑤
18	17	cv 1533	. . . . 5 class 𝑤
19		csca 17227	. . . . 5 class Scalar
20	18, 19	cfv 6542	. . . 4 class (Scalar‘𝑤)
21	16, 2, 20	wsbc 3774	. . 3 wff [(Scalar‘𝑤) / 𝑓][(Base‘𝑓) / 𝑘](𝑓 = (ℂfld ↾s 𝑘) ∧ 𝑘 ∈ (SubRing‘ℂfld))
22		clmod 20732	. . 3 class LMod
23	21, 17, 22	crab 3427	. 2 class {𝑤 ∈ LMod ∣ [(Scalar‘𝑤) / 𝑓][(Base‘𝑓) / 𝑘](𝑓 = (ℂfld ↾s 𝑘) ∧ 𝑘 ∈ (SubRing‘ℂfld))}
24	1, 23	wceq 1534	1 wff ℂMod = {𝑤 ∈ LMod ∣ [(Scalar‘𝑤) / 𝑓][(Base‘𝑓) / 𝑘](𝑓 = (ℂfld ↾s 𝑘) ∧ 𝑘 ∈ (SubRing‘ℂfld))}

This definition is referenced by:  isclm  24978