Definition df-clm 24961

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 20460), left modules over such subrings are the same as right modules, see rmodislmod 20833. 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 24960	. 2 class ℂMod
2		vf	. . . . . . . 8 setvar 𝑓
3	2	cv 1539	. . . . . . 7 class 𝑓
4		ccnfld 21261	. . . . . . . 8 class ℂfld
5		vk	. . . . . . . . 9 setvar 𝑘
6	5	cv 1539	. . . . . . . 8 class 𝑘
7		cress 17141	. . . . . . . 8 class ↾s
8	4, 6, 7	co 7349	. . . . . . 7 class (ℂfld ↾s 𝑘)
9	3, 8	wceq 1540	. . . . . 6 wff 𝑓 = (ℂfld ↾s 𝑘)
10		csubrg 20454	. . . . . . . 8 class SubRing
11	4, 10	cfv 6482	. . . . . . 7 class (SubRing‘ℂfld)
12	6, 11	wcel 2109	. . . . . 6 wff 𝑘 ∈ (SubRing‘ℂfld)
13	9, 12	wa 395	. . . . 5 wff (𝑓 = (ℂfld ↾s 𝑘) ∧ 𝑘 ∈ (SubRing‘ℂfld))
14		cbs 17120	. . . . . 6 class Base
15	3, 14	cfv 6482	. . . . 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 1539	. . . . 5 class 𝑤
19		csca 17164	. . . . 5 class Scalar
20	18, 19	cfv 6482	. . . 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 20763	. . 3 class LMod
23	21, 17, 22	crab 3394	. 2 class {𝑤 ∈ LMod ∣ [(Scalar‘𝑤) / 𝑓][(Base‘𝑓) / 𝑘](𝑓 = (ℂfld ↾s 𝑘) ∧ 𝑘 ∈ (SubRing‘ℂfld))}
24	1, 23	wceq 1540	1 wff ℂMod = {𝑤 ∈ LMod ∣ [(Scalar‘𝑤) / 𝑓][(Base‘𝑓) / 𝑘](𝑓 = (ℂfld ↾s 𝑘) ∧ 𝑘 ∈ (SubRing‘ℂfld))}

This definition is referenced by:  isclm  24962