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Definition df-subrg 20586
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.)

Assertion
Ref Expression
df-subrg SubRing = (𝑤 ∈ Ring ↦ {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ ((𝑤s 𝑠) ∈ Ring ∧ (1r𝑤) ∈ 𝑠)})
Distinct variable group:   𝑤,𝑠

Detailed syntax breakdown of Definition df-subrg
StepHypRef Expression
1 csubrg 20585 . 2 class SubRing
2 vw . . 3 setvar 𝑤
3 crg 20250 . . 3 class Ring
42cv 1535 . . . . . . 7 class 𝑤
5 vs . . . . . . . 8 setvar 𝑠
65cv 1535 . . . . . . 7 class 𝑠
7 cress 17273 . . . . . . 7 class s
84, 6, 7co 7430 . . . . . 6 class (𝑤s 𝑠)
98, 3wcel 2105 . . . . 5 wff (𝑤s 𝑠) ∈ Ring
10 cur 20198 . . . . . . 7 class 1r
114, 10cfv 6562 . . . . . 6 class (1r𝑤)
1211, 6wcel 2105 . . . . 5 wff (1r𝑤) ∈ 𝑠
139, 12wa 395 . . . 4 wff ((𝑤s 𝑠) ∈ Ring ∧ (1r𝑤) ∈ 𝑠)
14 cbs 17244 . . . . . 6 class Base
154, 14cfv 6562 . . . . 5 class (Base‘𝑤)
1615cpw 4604 . . . 4 class 𝒫 (Base‘𝑤)
1713, 5, 16crab 3432 . . 3 class {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ ((𝑤s 𝑠) ∈ Ring ∧ (1r𝑤) ∈ 𝑠)}
182, 3, 17cmpt 5230 . 2 class (𝑤 ∈ Ring ↦ {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ ((𝑤s 𝑠) ∈ Ring ∧ (1r𝑤) ∈ 𝑠)})
191, 18wceq 1536 1 wff SubRing = (𝑤 ∈ Ring ↦ {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ ((𝑤s 𝑠) ∈ Ring ∧ (1r𝑤) ∈ 𝑠)})
Colors of variables: wff setvar class
This definition is referenced by:  issubrg  20587
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