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Definition df-subrg 20459
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 20457 . 2 class SubRing
2 vw . . 3 setvar 𝑤
3 crg 20127 . . 3 class Ring
42cv 1538 . . . . . . 7 class 𝑤
5 vs . . . . . . . 8 setvar 𝑠
65cv 1538 . . . . . . 7 class 𝑠
7 cress 17177 . . . . . . 7 class s
84, 6, 7co 7411 . . . . . 6 class (𝑤s 𝑠)
98, 3wcel 2104 . . . . 5 wff (𝑤s 𝑠) ∈ Ring
10 cur 20075 . . . . . . 7 class 1r
114, 10cfv 6542 . . . . . 6 class (1r𝑤)
1211, 6wcel 2104 . . . . 5 wff (1r𝑤) ∈ 𝑠
139, 12wa 394 . . . 4 wff ((𝑤s 𝑠) ∈ Ring ∧ (1r𝑤) ∈ 𝑠)
14 cbs 17148 . . . . . 6 class Base
154, 14cfv 6542 . . . . 5 class (Base‘𝑤)
1615cpw 4601 . . . 4 class 𝒫 (Base‘𝑤)
1713, 5, 16crab 3430 . . 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 1539 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  20461
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