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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 13531 | . 2 class SubRing | |
2 | vw | . . 3 setvar 𝑤 | |
3 | crg 13317 | . . 3 class Ring | |
4 | 2 | cv 1363 | . . . . . . 7 class 𝑤 |
5 | vs | . . . . . . . 8 setvar 𝑠 | |
6 | 5 | cv 1363 | . . . . . . 7 class 𝑠 |
7 | cress 12487 | . . . . . . 7 class ↾s | |
8 | 4, 6, 7 | co 5891 | . . . . . 6 class (𝑤 ↾s 𝑠) |
9 | 8, 3 | wcel 2160 | . . . . 5 wff (𝑤 ↾s 𝑠) ∈ Ring |
10 | cur 13280 | . . . . . . 7 class 1r | |
11 | 4, 10 | cfv 5231 | . . . . . 6 class (1r‘𝑤) |
12 | 11, 6 | wcel 2160 | . . . . 5 wff (1r‘𝑤) ∈ 𝑠 |
13 | 9, 12 | wa 104 | . . . 4 wff ((𝑤 ↾s 𝑠) ∈ Ring ∧ (1r‘𝑤) ∈ 𝑠) |
14 | cbs 12486 | . . . . . 6 class Base | |
15 | 4, 14 | cfv 5231 | . . . . 5 class (Base‘𝑤) |
16 | 15 | cpw 3590 | . . . 4 class 𝒫 (Base‘𝑤) |
17 | 13, 5, 16 | crab 2472 | . . 3 class {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ ((𝑤 ↾s 𝑠) ∈ Ring ∧ (1r‘𝑤) ∈ 𝑠)} |
18 | 2, 3, 17 | cmpt 4079 | . 2 class (𝑤 ∈ Ring ↦ {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ ((𝑤 ↾s 𝑠) ∈ Ring ∧ (1r‘𝑤) ∈ 𝑠)}) |
19 | 1, 18 | wceq 1364 | 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 13535 |
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