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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 19796 | . 2 class SubRing | |
2 | vw | . . 3 setvar 𝑤 | |
3 | crg 19562 | . . 3 class Ring | |
4 | 2 | cv 1542 | . . . . . . 7 class 𝑤 |
5 | vs | . . . . . . . 8 setvar 𝑠 | |
6 | 5 | cv 1542 | . . . . . . 7 class 𝑠 |
7 | cress 16784 | . . . . . . 7 class ↾s | |
8 | 4, 6, 7 | co 7213 | . . . . . 6 class (𝑤 ↾s 𝑠) |
9 | 8, 3 | wcel 2110 | . . . . 5 wff (𝑤 ↾s 𝑠) ∈ Ring |
10 | cur 19516 | . . . . . . 7 class 1r | |
11 | 4, 10 | cfv 6380 | . . . . . 6 class (1r‘𝑤) |
12 | 11, 6 | wcel 2110 | . . . . 5 wff (1r‘𝑤) ∈ 𝑠 |
13 | 9, 12 | wa 399 | . . . 4 wff ((𝑤 ↾s 𝑠) ∈ Ring ∧ (1r‘𝑤) ∈ 𝑠) |
14 | cbs 16760 | . . . . . 6 class Base | |
15 | 4, 14 | cfv 6380 | . . . . 5 class (Base‘𝑤) |
16 | 15 | cpw 4513 | . . . 4 class 𝒫 (Base‘𝑤) |
17 | 13, 5, 16 | crab 3065 | . . 3 class {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ ((𝑤 ↾s 𝑠) ∈ Ring ∧ (1r‘𝑤) ∈ 𝑠)} |
18 | 2, 3, 17 | cmpt 5135 | . 2 class (𝑤 ∈ Ring ↦ {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ ((𝑤 ↾s 𝑠) ∈ Ring ∧ (1r‘𝑤) ∈ 𝑠)}) |
19 | 1, 18 | wceq 1543 | 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 19800 |
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