![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > df-subrg | Structured version Visualization version GIF version |
Description: Define a subring of a
ring as a set of elements that is a ring in its
own right and contains the multiplicative identity.
The additional constraint is necessary because the multiplicative identity of a ring, unlike the additive identity of a ring/group or the multiplicative identity of a field, cannot be identified by a local property. Thus, it is possible for a subset of a ring to be a ring while not containing the true identity if it contains a false identity. For instance, the subset (ℤ × {0}) of (ℤ × ℤ) (where multiplication is componentwise) contains the false identity 〈1, 0〉 which preserves every element of the subset and thus appears to be the identity of the subset, but is not the identity of the larger ring. (Contributed by Stefan O'Rear, 27-Nov-2014.) |
Ref | Expression |
---|---|
df-subrg | ⊢ SubRing = (𝑤 ∈ Ring ↦ {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ ((𝑤 ↾s 𝑠) ∈ Ring ∧ (1r‘𝑤) ∈ 𝑠)}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | csubrg 20585 | . 2 class SubRing | |
2 | vw | . . 3 setvar 𝑤 | |
3 | crg 20250 | . . 3 class Ring | |
4 | 2 | cv 1535 | . . . . . . 7 class 𝑤 |
5 | vs | . . . . . . . 8 setvar 𝑠 | |
6 | 5 | cv 1535 | . . . . . . 7 class 𝑠 |
7 | cress 17273 | . . . . . . 7 class ↾s | |
8 | 4, 6, 7 | co 7430 | . . . . . 6 class (𝑤 ↾s 𝑠) |
9 | 8, 3 | wcel 2105 | . . . . 5 wff (𝑤 ↾s 𝑠) ∈ Ring |
10 | cur 20198 | . . . . . . 7 class 1r | |
11 | 4, 10 | cfv 6562 | . . . . . 6 class (1r‘𝑤) |
12 | 11, 6 | wcel 2105 | . . . . 5 wff (1r‘𝑤) ∈ 𝑠 |
13 | 9, 12 | wa 395 | . . . 4 wff ((𝑤 ↾s 𝑠) ∈ Ring ∧ (1r‘𝑤) ∈ 𝑠) |
14 | cbs 17244 | . . . . . 6 class Base | |
15 | 4, 14 | cfv 6562 | . . . . 5 class (Base‘𝑤) |
16 | 15 | cpw 4604 | . . . 4 class 𝒫 (Base‘𝑤) |
17 | 13, 5, 16 | crab 3432 | . . 3 class {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ ((𝑤 ↾s 𝑠) ∈ Ring ∧ (1r‘𝑤) ∈ 𝑠)} |
18 | 2, 3, 17 | cmpt 5230 | . 2 class (𝑤 ∈ Ring ↦ {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ ((𝑤 ↾s 𝑠) ∈ Ring ∧ (1r‘𝑤) ∈ 𝑠)}) |
19 | 1, 18 | wceq 1536 | 1 wff SubRing = (𝑤 ∈ Ring ↦ {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ ((𝑤 ↾s 𝑠) ∈ Ring ∧ (1r‘𝑤) ∈ 𝑠)}) |
Colors of variables: wff setvar class |
This definition is referenced by: issubrg 20587 |
Copyright terms: Public domain | W3C validator |