Definition df-clm 25014

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 20535), left modules over such subrings are the same as right modules, see rmodislmod 20887. 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 25013	. 2 class ℂMod
2		vf	. . . . . . . 8 setvar 𝑓
3	2	cv 1539	. . . . . . 7 class 𝑓
4		ccnfld 21315	. . . . . . . 8 class ℂfld
5		vk	. . . . . . . . 9 setvar 𝑘
6	5	cv 1539	. . . . . . . 8 class 𝑘
7		cress 17251	. . . . . . . 8 class ↾s
8	4, 6, 7	co 7405	. . . . . . 7 class (ℂfld ↾s 𝑘)
9	3, 8	wceq 1540	. . . . . 6 wff 𝑓 = (ℂfld ↾s 𝑘)
10		csubrg 20529	. . . . . . . 8 class SubRing
11	4, 10	cfv 6531	. . . . . . 7 class (SubRing‘ℂfld)
12	6, 11	wcel 2108	. . . . . 6 wff 𝑘 ∈ (SubRing‘ℂfld)
13	9, 12	wa 395	. . . . 5 wff (𝑓 = (ℂfld ↾s 𝑘) ∧ 𝑘 ∈ (SubRing‘ℂfld))
14		cbs 17228	. . . . . 6 class Base
15	3, 14	cfv 6531	. . . . 5 class (Base‘𝑓)
16	13, 5, 15	wsbc 3765	. . . 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 17274	. . . . 5 class Scalar
20	18, 19	cfv 6531	. . . 4 class (Scalar‘𝑤)
21	16, 2, 20	wsbc 3765	. . 3 wff [(Scalar‘𝑤) / 𝑓][(Base‘𝑓) / 𝑘](𝑓 = (ℂfld ↾s 𝑘) ∧ 𝑘 ∈ (SubRing‘ℂfld))
22		clmod 20817	. . 3 class LMod
23	21, 17, 22	crab 3415	. 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  25015