Definition df-clm 24803

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 20465), left modules over such subrings are the same as right modules, see rmodislmod 20684. 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 24802	. 2 class ℂMod
2		vf	. . . . . . . 8 setvar 𝑓
3	2	cv 1540	. . . . . . 7 class 𝑓
4		ccnfld 21144	. . . . . . . 8 class ℂfld
5		vk	. . . . . . . . 9 setvar 𝑘
6	5	cv 1540	. . . . . . . 8 class 𝑘
7		cress 17177	. . . . . . . 8 class ↾s
8	4, 6, 7	co 7411	. . . . . . 7 class (ℂfld ↾s 𝑘)
9	3, 8	wceq 1541	. . . . . 6 wff 𝑓 = (ℂfld ↾s 𝑘)
10		csubrg 20457	. . . . . . . 8 class SubRing
11	4, 10	cfv 6543	. . . . . . 7 class (SubRing‘ℂfld)
12	6, 11	wcel 2106	. . . . . 6 wff 𝑘 ∈ (SubRing‘ℂfld)
13	9, 12	wa 396	. . . . 5 wff (𝑓 = (ℂfld ↾s 𝑘) ∧ 𝑘 ∈ (SubRing‘ℂfld))
14		cbs 17148	. . . . . 6 class Base
15	3, 14	cfv 6543	. . . . 5 class (Base‘𝑓)
16	13, 5, 15	wsbc 3777	. . . 4 wff [(Base‘𝑓) / 𝑘](𝑓 = (ℂfld ↾s 𝑘) ∧ 𝑘 ∈ (SubRing‘ℂfld))
17		vw	. . . . . 6 setvar 𝑤
18	17	cv 1540	. . . . 5 class 𝑤
19		csca 17204	. . . . 5 class Scalar
20	18, 19	cfv 6543	. . . 4 class (Scalar‘𝑤)
21	16, 2, 20	wsbc 3777	. . 3 wff [(Scalar‘𝑤) / 𝑓][(Base‘𝑓) / 𝑘](𝑓 = (ℂfld ↾s 𝑘) ∧ 𝑘 ∈ (SubRing‘ℂfld))
22		clmod 20614	. . 3 class LMod
23	21, 17, 22	crab 3432	. 2 class {𝑤 ∈ LMod ∣ [(Scalar‘𝑤) / 𝑓][(Base‘𝑓) / 𝑘](𝑓 = (ℂfld ↾s 𝑘) ∧ 𝑘 ∈ (SubRing‘ℂfld))}
24	1, 23	wceq 1541	1 wff ℂMod = {𝑤 ∈ LMod ∣ [(Scalar‘𝑤) / 𝑓][(Base‘𝑓) / 𝑘](𝑓 = (ℂfld ↾s 𝑘) ∧ 𝑘 ∈ (SubRing‘ℂfld))}

This definition is referenced by:  isclm  24804