Definition df-clm 24985

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 20485), left modules over such subrings are the same as right modules, see rmodislmod 20858. 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 24984	. 2 class ℂMod
2		vf	. . . . . . . 8 setvar 𝑓
3	2	cv 1540	. . . . . . 7 class 𝑓
4		ccnfld 21286	. . . . . . . 8 class ℂfld
5		vk	. . . . . . . . 9 setvar 𝑘
6	5	cv 1540	. . . . . . . 8 class 𝑘
7		cress 17136	. . . . . . . 8 class ↾s
8	4, 6, 7	co 7341	. . . . . . 7 class (ℂfld ↾s 𝑘)
9	3, 8	wceq 1541	. . . . . 6 wff 𝑓 = (ℂfld ↾s 𝑘)
10		csubrg 20479	. . . . . . . 8 class SubRing
11	4, 10	cfv 6476	. . . . . . 7 class (SubRing‘ℂfld)
12	6, 11	wcel 2111	. . . . . 6 wff 𝑘 ∈ (SubRing‘ℂfld)
13	9, 12	wa 395	. . . . 5 wff (𝑓 = (ℂfld ↾s 𝑘) ∧ 𝑘 ∈ (SubRing‘ℂfld))
14		cbs 17115	. . . . . 6 class Base
15	3, 14	cfv 6476	. . . . 5 class (Base‘𝑓)
16	13, 5, 15	wsbc 3736	. . . 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 17159	. . . . 5 class Scalar
20	18, 19	cfv 6476	. . . 4 class (Scalar‘𝑤)
21	16, 2, 20	wsbc 3736	. . 3 wff [(Scalar‘𝑤) / 𝑓][(Base‘𝑓) / 𝑘](𝑓 = (ℂfld ↾s 𝑘) ∧ 𝑘 ∈ (SubRing‘ℂfld))
22		clmod 20788	. . 3 class LMod
23	21, 17, 22	crab 3395	. 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  24986