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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 19460 | . 2 class SubRing | |
2 | vw | . . 3 setvar 𝑤 | |
3 | crg 19226 | . . 3 class Ring | |
4 | 2 | cv 1527 | . . . . . . 7 class 𝑤 |
5 | vs | . . . . . . . 8 setvar 𝑠 | |
6 | 5 | cv 1527 | . . . . . . 7 class 𝑠 |
7 | cress 16472 | . . . . . . 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 19180 | . . . . . . 7 class 1r | |
11 | 4, 10 | cfv 6348 | . . . . . 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 16471 | . . . . . 6 class Base | |
15 | 4, 14 | cfv 6348 | . . . . 5 class (Base‘𝑤) |
16 | 15 | cpw 4535 | . . . 4 class 𝒫 (Base‘𝑤) |
17 | 13, 5, 16 | crab 3139 | . . 3 class {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ ((𝑤 ↾s 𝑠) ∈ Ring ∧ (1r‘𝑤) ∈ 𝑠)} |
18 | 2, 3, 17 | cmpt 5137 | . 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 19464 |
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