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Mirrors > Home > MPE Home > Th. List > df-clm | Structured version Visualization version Unicode version |
Description: Define the class of subcomplex modules, which are left modules over a subring of the field of complex numbers ℂfld, 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 18784), left modules over such subrings are the same as right modules, see rmodislmod 18931. Therefore, we drop the word "left" from "subcomplex left module". (Contributed by Mario Carneiro, 16-Oct-2015.) |
Ref | Expression |
---|---|
df-clm | CMod Scalar ℂfld ↾s SubRingℂfld |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cclm 22862 | . 2 CMod | |
2 | vf | . . . . . . . 8 | |
3 | 2 | cv 1482 | . . . . . . 7 |
4 | ccnfld 19746 | . . . . . . . 8 ℂfld | |
5 | vk | . . . . . . . . 9 | |
6 | 5 | cv 1482 | . . . . . . . 8 |
7 | cress 15858 | . . . . . . . 8 ↾s | |
8 | 4, 6, 7 | co 6650 | . . . . . . 7 ℂfld ↾s |
9 | 3, 8 | wceq 1483 | . . . . . 6 ℂfld ↾s |
10 | csubrg 18776 | . . . . . . . 8 SubRing | |
11 | 4, 10 | cfv 5888 | . . . . . . 7 SubRingℂfld |
12 | 6, 11 | wcel 1990 | . . . . . 6 SubRingℂfld |
13 | 9, 12 | wa 384 | . . . . 5 ℂfld ↾s SubRingℂfld |
14 | cbs 15857 | . . . . . 6 | |
15 | 3, 14 | cfv 5888 | . . . . 5 |
16 | 13, 5, 15 | wsbc 3435 | . . . 4 ℂfld ↾s SubRingℂfld |
17 | vw | . . . . . 6 | |
18 | 17 | cv 1482 | . . . . 5 |
19 | csca 15944 | . . . . 5 Scalar | |
20 | 18, 19 | cfv 5888 | . . . 4 Scalar |
21 | 16, 2, 20 | wsbc 3435 | . . 3 Scalar ℂfld ↾s SubRingℂfld |
22 | clmod 18863 | . . 3 | |
23 | 21, 17, 22 | crab 2916 | . 2 Scalar ℂfld ↾s SubRingℂfld |
24 | 1, 23 | wceq 1483 | 1 CMod Scalar ℂfld ↾s SubRingℂfld |
Colors of variables: wff setvar class |
This definition is referenced by: isclm 22864 |
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