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Definition df-subrg 18976
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 18974 . 2 class SubRing
2 vw . . 3 setvar 𝑤
3 crg 18743 . . 3 class Ring
42cv 1636 . . . . . . 7 class 𝑤
5 vs . . . . . . . 8 setvar 𝑠
65cv 1636 . . . . . . 7 class 𝑠
7 cress 16063 . . . . . . 7 class s
84, 6, 7co 6868 . . . . . 6 class (𝑤s 𝑠)
98, 3wcel 2155 . . . . 5 wff (𝑤s 𝑠) ∈ Ring
10 cur 18697 . . . . . . 7 class 1r
114, 10cfv 6095 . . . . . 6 class (1r𝑤)
1211, 6wcel 2155 . . . . 5 wff (1r𝑤) ∈ 𝑠
139, 12wa 384 . . . 4 wff ((𝑤s 𝑠) ∈ Ring ∧ (1r𝑤) ∈ 𝑠)
14 cbs 16062 . . . . . 6 class Base
154, 14cfv 6095 . . . . 5 class (Base‘𝑤)
1615cpw 4345 . . . 4 class 𝒫 (Base‘𝑤)
1713, 5, 16crab 3096 . . 3 class {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ ((𝑤s 𝑠) ∈ Ring ∧ (1r𝑤) ∈ 𝑠)}
182, 3, 17cmpt 4916 . 2 class (𝑤 ∈ Ring ↦ {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ ((𝑤s 𝑠) ∈ Ring ∧ (1r𝑤) ∈ 𝑠)})
191, 18wceq 1637 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  18978
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