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Mirrors > Home > ILE Home > Th. List > df-subrg | 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 13401 | . 2 class SubRing | |
2 | vw | . . 3 setvar 𝑤 | |
3 | crg 13233 | . . 3 class Ring | |
4 | 2 | cv 1362 | . . . . . . 7 class 𝑤 |
5 | vs | . . . . . . . 8 setvar 𝑠 | |
6 | 5 | cv 1362 | . . . . . . 7 class 𝑠 |
7 | cress 12476 | . . . . . . 7 class ↾s | |
8 | 4, 6, 7 | co 5888 | . . . . . 6 class (𝑤 ↾s 𝑠) |
9 | 8, 3 | wcel 2158 | . . . . 5 wff (𝑤 ↾s 𝑠) ∈ Ring |
10 | cur 13196 | . . . . . . 7 class 1r | |
11 | 4, 10 | cfv 5228 | . . . . . 6 class (1r‘𝑤) |
12 | 11, 6 | wcel 2158 | . . . . 5 wff (1r‘𝑤) ∈ 𝑠 |
13 | 9, 12 | wa 104 | . . . 4 wff ((𝑤 ↾s 𝑠) ∈ Ring ∧ (1r‘𝑤) ∈ 𝑠) |
14 | cbs 12475 | . . . . . 6 class Base | |
15 | 4, 14 | cfv 5228 | . . . . 5 class (Base‘𝑤) |
16 | 15 | cpw 3587 | . . . 4 class 𝒫 (Base‘𝑤) |
17 | 13, 5, 16 | crab 2469 | . . 3 class {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ ((𝑤 ↾s 𝑠) ∈ Ring ∧ (1r‘𝑤) ∈ 𝑠)} |
18 | 2, 3, 17 | cmpt 4076 | . 2 class (𝑤 ∈ Ring ↦ {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ ((𝑤 ↾s 𝑠) ∈ Ring ∧ (1r‘𝑤) ∈ 𝑠)}) |
19 | 1, 18 | wceq 1363 | 1 wff SubRing = (𝑤 ∈ Ring ↦ {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ ((𝑤 ↾s 𝑠) ∈ Ring ∧ (1r‘𝑤) ∈ 𝑠)}) |
Colors of variables: wff set class |
This definition is referenced by: issubrg 13405 |
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