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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 20457 | . 2 class SubRing | |
2 | vw | . . 3 setvar 𝑤 | |
3 | crg 20127 | . . 3 class Ring | |
4 | 2 | cv 1538 | . . . . . . 7 class 𝑤 |
5 | vs | . . . . . . . 8 setvar 𝑠 | |
6 | 5 | cv 1538 | . . . . . . 7 class 𝑠 |
7 | cress 17177 | . . . . . . 7 class ↾s | |
8 | 4, 6, 7 | co 7411 | . . . . . 6 class (𝑤 ↾s 𝑠) |
9 | 8, 3 | wcel 2104 | . . . . 5 wff (𝑤 ↾s 𝑠) ∈ Ring |
10 | cur 20075 | . . . . . . 7 class 1r | |
11 | 4, 10 | cfv 6542 | . . . . . 6 class (1r‘𝑤) |
12 | 11, 6 | wcel 2104 | . . . . 5 wff (1r‘𝑤) ∈ 𝑠 |
13 | 9, 12 | wa 394 | . . . 4 wff ((𝑤 ↾s 𝑠) ∈ Ring ∧ (1r‘𝑤) ∈ 𝑠) |
14 | cbs 17148 | . . . . . 6 class Base | |
15 | 4, 14 | cfv 6542 | . . . . 5 class (Base‘𝑤) |
16 | 15 | cpw 4601 | . . . 4 class 𝒫 (Base‘𝑤) |
17 | 13, 5, 16 | crab 3430 | . . 3 class {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ ((𝑤 ↾s 𝑠) ∈ Ring ∧ (1r‘𝑤) ∈ 𝑠)} |
18 | 2, 3, 17 | cmpt 5230 | . 2 class (𝑤 ∈ Ring ↦ {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ ((𝑤 ↾s 𝑠) ∈ Ring ∧ (1r‘𝑤) ∈ 𝑠)}) |
19 | 1, 18 | wceq 1539 | 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 20461 |
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