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Definition df-clm 23084
Description: Define the class of subcomplex modules, which are left modules over a subring of the field of complex numbers 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 19007), left modules over such subrings are the same as right modules, see rmodislmod 19154. 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 = (ℂflds 𝑘) ∧ 𝑘 ∈ (SubRing‘ℂfld))}
Distinct variable group:   𝑓,𝑘,𝑤

Detailed syntax breakdown of Definition df-clm
StepHypRef Expression
1 cclm 23083 . 2 class ℂMod
2 vf . . . . . . . 8 setvar 𝑓
32cv 1631 . . . . . . 7 class 𝑓
4 ccnfld 19969 . . . . . . . 8 class fld
5 vk . . . . . . . . 9 setvar 𝑘
65cv 1631 . . . . . . . 8 class 𝑘
7 cress 16081 . . . . . . . 8 class s
84, 6, 7co 6815 . . . . . . 7 class (ℂflds 𝑘)
93, 8wceq 1632 . . . . . 6 wff 𝑓 = (ℂflds 𝑘)
10 csubrg 18999 . . . . . . . 8 class SubRing
114, 10cfv 6050 . . . . . . 7 class (SubRing‘ℂfld)
126, 11wcel 2140 . . . . . 6 wff 𝑘 ∈ (SubRing‘ℂfld)
139, 12wa 383 . . . . 5 wff (𝑓 = (ℂflds 𝑘) ∧ 𝑘 ∈ (SubRing‘ℂfld))
14 cbs 16080 . . . . . 6 class Base
153, 14cfv 6050 . . . . 5 class (Base‘𝑓)
1613, 5, 15wsbc 3577 . . . 4 wff [(Base‘𝑓) / 𝑘](𝑓 = (ℂflds 𝑘) ∧ 𝑘 ∈ (SubRing‘ℂfld))
17 vw . . . . . 6 setvar 𝑤
1817cv 1631 . . . . 5 class 𝑤
19 csca 16167 . . . . 5 class Scalar
2018, 19cfv 6050 . . . 4 class (Scalar‘𝑤)
2116, 2, 20wsbc 3577 . . 3 wff [(Scalar‘𝑤) / 𝑓][(Base‘𝑓) / 𝑘](𝑓 = (ℂflds 𝑘) ∧ 𝑘 ∈ (SubRing‘ℂfld))
22 clmod 19086 . . 3 class LMod
2321, 17, 22crab 3055 . 2 class {𝑤 ∈ LMod ∣ [(Scalar‘𝑤) / 𝑓][(Base‘𝑓) / 𝑘](𝑓 = (ℂflds 𝑘) ∧ 𝑘 ∈ (SubRing‘ℂfld))}
241, 23wceq 1632 1 wff ℂMod = {𝑤 ∈ LMod ∣ [(Scalar‘𝑤) / 𝑓][(Base‘𝑓) / 𝑘](𝑓 = (ℂflds 𝑘) ∧ 𝑘 ∈ (SubRing‘ℂfld))}
Colors of variables: wff setvar class
This definition is referenced by:  isclm  23085
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