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Definition df-clm 23231
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 19139), left modules over such subrings are the same as right modules, see rmodislmod 19286. 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 23230 . 2 class ℂMod
2 vf . . . . . . . 8 setvar 𝑓
32cv 1657 . . . . . . 7 class 𝑓
4 ccnfld 20105 . . . . . . . 8 class fld
5 vk . . . . . . . . 9 setvar 𝑘
65cv 1657 . . . . . . . 8 class 𝑘
7 cress 16222 . . . . . . . 8 class s
84, 6, 7co 6904 . . . . . . 7 class (ℂflds 𝑘)
93, 8wceq 1658 . . . . . 6 wff 𝑓 = (ℂflds 𝑘)
10 csubrg 19131 . . . . . . . 8 class SubRing
114, 10cfv 6122 . . . . . . 7 class (SubRing‘ℂfld)
126, 11wcel 2166 . . . . . 6 wff 𝑘 ∈ (SubRing‘ℂfld)
139, 12wa 386 . . . . 5 wff (𝑓 = (ℂflds 𝑘) ∧ 𝑘 ∈ (SubRing‘ℂfld))
14 cbs 16221 . . . . . 6 class Base
153, 14cfv 6122 . . . . 5 class (Base‘𝑓)
1613, 5, 15wsbc 3661 . . . 4 wff [(Base‘𝑓) / 𝑘](𝑓 = (ℂflds 𝑘) ∧ 𝑘 ∈ (SubRing‘ℂfld))
17 vw . . . . . 6 setvar 𝑤
1817cv 1657 . . . . 5 class 𝑤
19 csca 16307 . . . . 5 class Scalar
2018, 19cfv 6122 . . . 4 class (Scalar‘𝑤)
2116, 2, 20wsbc 3661 . . 3 wff [(Scalar‘𝑤) / 𝑓][(Base‘𝑓) / 𝑘](𝑓 = (ℂflds 𝑘) ∧ 𝑘 ∈ (SubRing‘ℂfld))
22 clmod 19218 . . 3 class LMod
2321, 17, 22crab 3120 . 2 class {𝑤 ∈ LMod ∣ [(Scalar‘𝑤) / 𝑓][(Base‘𝑓) / 𝑘](𝑓 = (ℂflds 𝑘) ∧ 𝑘 ∈ (SubRing‘ℂfld))}
241, 23wceq 1658 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  23232
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