Definition df-clm 25391

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 20834), left modules over such subrings are the same as right modules, see rmodislmod 21212. 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 25390	. 2 class ℂMod
2		vf	. . . . . . . 8 setvar 𝑓
3	2	cv 1568	. . . . . . 7 class 𝑓
4		ccnfld 21685	. . . . . . . 8 class ℂfld
5		vk	. . . . . . . . 9 setvar 𝑘
6	5	cv 1568	. . . . . . . 8 class 𝑘
7		cress 17415	. . . . . . . 8 class ↾s
8	4, 6, 7	co 7422	. . . . . . 7 class (ℂfld ↾s 𝑘)
9	3, 8	wceq 1569	. . . . . 6 wff 𝑓 = (ℂfld ↾s 𝑘)
10		csubrg 20828	. . . . . . . 8 class SubRing
11	4, 10	cfv 6541	. . . . . . 7 class (SubRing‘ℂfld)
12	6, 11	wcel 2144	. . . . . 6 wff 𝑘 ∈ (SubRing‘ℂfld)
13	9, 12	wa 401	. . . . 5 wff (𝑓 = (ℂfld ↾s 𝑘) ∧ 𝑘 ∈ (SubRing‘ℂfld))
14		cbs 17394	. . . . . 6 class Base
15	3, 14	cfv 6541	. . . . 5 class (Base‘𝑓)
16	13, 5, 15	wsbc 3738	. . . 4 wff [(Base‘𝑓) / 𝑘](𝑓 = (ℂfld ↾s 𝑘) ∧ 𝑘 ∈ (SubRing‘ℂfld))
17		vw	. . . . . 6 setvar 𝑤
18	17	cv 1568	. . . . 5 class 𝑤
19		csca 17438	. . . . 5 class Scalar
20	18, 19	cfv 6541	. . . 4 class (Scalar‘𝑤)
21	16, 2, 20	wsbc 3738	. . 3 wff [(Scalar‘𝑤) / 𝑓][(Base‘𝑓) / 𝑘](𝑓 = (ℂfld ↾s 𝑘) ∧ 𝑘 ∈ (SubRing‘ℂfld))
22		clmod 21142	. . 3 class LMod
23	21, 17, 22	crab 3412	. 2 class {𝑤 ∈ LMod ∣ [(Scalar‘𝑤) / 𝑓][(Base‘𝑓) / 𝑘](𝑓 = (ℂfld ↾s 𝑘) ∧ 𝑘 ∈ (SubRing‘ℂfld))}
24	1, 23	wceq 1569	1 wff ℂMod = {𝑤 ∈ LMod ∣ [(Scalar‘𝑤) / 𝑓][(Base‘𝑓) / 𝑘](𝑓 = (ℂfld ↾s 𝑘) ∧ 𝑘 ∈ (SubRing‘ℂfld))}

This definition is used by:  isclm  25392