Definition df-clm 24132

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 19943), left modules over such subrings are the same as right modules, see rmodislmod 20106. 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 24131	. 2 class ℂMod
2		vf	. . . . . . . 8 setvar 𝑓
3	2	cv 1538	. . . . . . 7 class 𝑓
4		ccnfld 20510	. . . . . . . 8 class ℂfld
5		vk	. . . . . . . . 9 setvar 𝑘
6	5	cv 1538	. . . . . . . 8 class 𝑘
7		cress 16867	. . . . . . . 8 class ↾s
8	4, 6, 7	co 7255	. . . . . . 7 class (ℂfld ↾s 𝑘)
9	3, 8	wceq 1539	. . . . . 6 wff 𝑓 = (ℂfld ↾s 𝑘)
10		csubrg 19935	. . . . . . . 8 class SubRing
11	4, 10	cfv 6418	. . . . . . 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 16840	. . . . . 6 class Base
15	3, 14	cfv 6418	. . . . 5 class (Base‘𝑓)
16	13, 5, 15	wsbc 3711	. . . 4 wff [(Base‘𝑓) / 𝑘](𝑓 = (ℂfld ↾s 𝑘) ∧ 𝑘 ∈ (SubRing‘ℂfld))
17		vw	. . . . . 6 setvar 𝑤
18	17	cv 1538	. . . . 5 class 𝑤
19		csca 16891	. . . . 5 class Scalar
20	18, 19	cfv 6418	. . . 4 class (Scalar‘𝑤)
21	16, 2, 20	wsbc 3711	. . 3 wff [(Scalar‘𝑤) / 𝑓][(Base‘𝑓) / 𝑘](𝑓 = (ℂfld ↾s 𝑘) ∧ 𝑘 ∈ (SubRing‘ℂfld))
22		clmod 20038	. . 3 class LMod
23	21, 17, 22	crab 3067	. 2 class {𝑤 ∈ LMod ∣ [(Scalar‘𝑤) / 𝑓][(Base‘𝑓) / 𝑘](𝑓 = (ℂfld ↾s 𝑘) ∧ 𝑘 ∈ (SubRing‘ℂfld))}
24	1, 23	wceq 1539	1 wff ℂMod = {𝑤 ∈ LMod ∣ [(Scalar‘𝑤) / 𝑓][(Base‘𝑓) / 𝑘](𝑓 = (ℂfld ↾s 𝑘) ∧ 𝑘 ∈ (SubRing‘ℂfld))}

This definition is referenced by:  isclm  24133