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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 19462 | . 2 class SubRing | |
2 | vw | . . 3 setvar 𝑤 | |
3 | crg 19228 | . . 3 class Ring | |
4 | 2 | cv 1527 | . . . . . . 7 class 𝑤 |
5 | vs | . . . . . . . 8 setvar 𝑠 | |
6 | 5 | cv 1527 | . . . . . . 7 class 𝑠 |
7 | cress 16474 | . . . . . . 7 class ↾s | |
8 | 4, 6, 7 | co 7145 | . . . . . 6 class (𝑤 ↾s 𝑠) |
9 | 8, 3 | wcel 2105 | . . . . 5 wff (𝑤 ↾s 𝑠) ∈ Ring |
10 | cur 19182 | . . . . . . 7 class 1r | |
11 | 4, 10 | cfv 6349 | . . . . . 6 class (1r‘𝑤) |
12 | 11, 6 | wcel 2105 | . . . . 5 wff (1r‘𝑤) ∈ 𝑠 |
13 | 9, 12 | wa 396 | . . . 4 wff ((𝑤 ↾s 𝑠) ∈ Ring ∧ (1r‘𝑤) ∈ 𝑠) |
14 | cbs 16473 | . . . . . 6 class Base | |
15 | 4, 14 | cfv 6349 | . . . . 5 class (Base‘𝑤) |
16 | 15 | cpw 4537 | . . . 4 class 𝒫 (Base‘𝑤) |
17 | 13, 5, 16 | crab 3142 | . . 3 class {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ ((𝑤 ↾s 𝑠) ∈ Ring ∧ (1r‘𝑤) ∈ 𝑠)} |
18 | 2, 3, 17 | cmpt 5138 | . 2 class (𝑤 ∈ Ring ↦ {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ ((𝑤 ↾s 𝑠) ∈ Ring ∧ (1r‘𝑤) ∈ 𝑠)}) |
19 | 1, 18 | wceq 1528 | 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 19466 |
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