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Definition df-clm 23647
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 19515), left modules over such subrings are the same as right modules, see rmodislmod 19678. 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 23646 . 2 class ℂMod
2 vf . . . . . . . 8 setvar 𝑓
32cv 1536 . . . . . . 7 class 𝑓
4 ccnfld 20521 . . . . . . . 8 class fld
5 vk . . . . . . . . 9 setvar 𝑘
65cv 1536 . . . . . . . 8 class 𝑘
7 cress 16463 . . . . . . . 8 class s
84, 6, 7co 7133 . . . . . . 7 class (ℂflds 𝑘)
93, 8wceq 1537 . . . . . 6 wff 𝑓 = (ℂflds 𝑘)
10 csubrg 19507 . . . . . . . 8 class SubRing
114, 10cfv 6331 . . . . . . 7 class (SubRing‘ℂfld)
126, 11wcel 2114 . . . . . 6 wff 𝑘 ∈ (SubRing‘ℂfld)
139, 12wa 398 . . . . 5 wff (𝑓 = (ℂflds 𝑘) ∧ 𝑘 ∈ (SubRing‘ℂfld))
14 cbs 16462 . . . . . 6 class Base
153, 14cfv 6331 . . . . 5 class (Base‘𝑓)
1613, 5, 15wsbc 3752 . . . 4 wff [(Base‘𝑓) / 𝑘](𝑓 = (ℂflds 𝑘) ∧ 𝑘 ∈ (SubRing‘ℂfld))
17 vw . . . . . 6 setvar 𝑤
1817cv 1536 . . . . 5 class 𝑤
19 csca 16547 . . . . 5 class Scalar
2018, 19cfv 6331 . . . 4 class (Scalar‘𝑤)
2116, 2, 20wsbc 3752 . . 3 wff [(Scalar‘𝑤) / 𝑓][(Base‘𝑓) / 𝑘](𝑓 = (ℂflds 𝑘) ∧ 𝑘 ∈ (SubRing‘ℂfld))
22 clmod 19610 . . 3 class LMod
2321, 17, 22crab 3129 . 2 class {𝑤 ∈ LMod ∣ [(Scalar‘𝑤) / 𝑓][(Base‘𝑓) / 𝑘](𝑓 = (ℂflds 𝑘) ∧ 𝑘 ∈ (SubRing‘ℂfld))}
241, 23wceq 1537 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  23648
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