Definition df-clm 23667

Description: Define the class of subcomplex modules, which are left modules over a subring of the field of complex numbers ℂ_fld, which allows us 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 19539), left modules over such subrings are the same as right modules, see rmodislmod 19702. 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 23666	. 2 class ℂMod
2		vf	. . . . . . . 8 setvar 𝑓
3	2	cv 1536	. . . . . . 7 class 𝑓
4		ccnfld 20545	. . . . . . . 8 class ℂfld
5		vk	. . . . . . . . 9 setvar 𝑘
6	5	cv 1536	. . . . . . . 8 class 𝑘
7		cress 16484	. . . . . . . 8 class ↾s
8	4, 6, 7	co 7156	. . . . . . 7 class (ℂfld ↾s 𝑘)
9	3, 8	wceq 1537	. . . . . 6 wff 𝑓 = (ℂfld ↾s 𝑘)
10		csubrg 19531	. . . . . . . 8 class SubRing
11	4, 10	cfv 6355	. . . . . . 7 class (SubRing‘ℂfld)
12	6, 11	wcel 2114	. . . . . 6 wff 𝑘 ∈ (SubRing‘ℂfld)
13	9, 12	wa 398	. . . . 5 wff (𝑓 = (ℂfld ↾s 𝑘) ∧ 𝑘 ∈ (SubRing‘ℂfld))
14		cbs 16483	. . . . . 6 class Base
15	3, 14	cfv 6355	. . . . 5 class (Base‘𝑓)
16	13, 5, 15	wsbc 3772	. . . 4 wff [(Base‘𝑓) / 𝑘](𝑓 = (ℂfld ↾s 𝑘) ∧ 𝑘 ∈ (SubRing‘ℂfld))
17		vw	. . . . . 6 setvar 𝑤
18	17	cv 1536	. . . . 5 class 𝑤
19		csca 16568	. . . . 5 class Scalar
20	18, 19	cfv 6355	. . . 4 class (Scalar‘𝑤)
21	16, 2, 20	wsbc 3772	. . 3 wff [(Scalar‘𝑤) / 𝑓][(Base‘𝑓) / 𝑘](𝑓 = (ℂfld ↾s 𝑘) ∧ 𝑘 ∈ (SubRing‘ℂfld))
22		clmod 19634	. . 3 class LMod
23	21, 17, 22	crab 3142	. 2 class {𝑤 ∈ LMod ∣ [(Scalar‘𝑤) / 𝑓][(Base‘𝑓) / 𝑘](𝑓 = (ℂfld ↾s 𝑘) ∧ 𝑘 ∈ (SubRing‘ℂfld))}
24	1, 23	wceq 1537	1 wff ℂMod = {𝑤 ∈ LMod ∣ [(Scalar‘𝑤) / 𝑓][(Base‘𝑓) / 𝑘](𝑓 = (ℂfld ↾s 𝑘) ∧ 𝑘 ∈ (SubRing‘ℂfld))}

This definition is referenced by:  isclm  23668