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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 13524 | . 2 class SubRing | |
2 | vw | . . 3 setvar 𝑤 | |
3 | crg 13310 | . . 3 class Ring | |
4 | 2 | cv 1362 | . . . . . . 7 class 𝑤 |
5 | vs | . . . . . . . 8 setvar 𝑠 | |
6 | 5 | cv 1362 | . . . . . . 7 class 𝑠 |
7 | cress 12480 | . . . . . . 7 class ↾s | |
8 | 4, 6, 7 | co 5890 | . . . . . 6 class (𝑤 ↾s 𝑠) |
9 | 8, 3 | wcel 2159 | . . . . 5 wff (𝑤 ↾s 𝑠) ∈ Ring |
10 | cur 13273 | . . . . . . 7 class 1r | |
11 | 4, 10 | cfv 5230 | . . . . . 6 class (1r‘𝑤) |
12 | 11, 6 | wcel 2159 | . . . . 5 wff (1r‘𝑤) ∈ 𝑠 |
13 | 9, 12 | wa 104 | . . . 4 wff ((𝑤 ↾s 𝑠) ∈ Ring ∧ (1r‘𝑤) ∈ 𝑠) |
14 | cbs 12479 | . . . . . 6 class Base | |
15 | 4, 14 | cfv 5230 | . . . . 5 class (Base‘𝑤) |
16 | 15 | cpw 3589 | . . . 4 class 𝒫 (Base‘𝑤) |
17 | 13, 5, 16 | crab 2471 | . . 3 class {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ ((𝑤 ↾s 𝑠) ∈ Ring ∧ (1r‘𝑤) ∈ 𝑠)} |
18 | 2, 3, 17 | cmpt 4078 | . 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 13528 |
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