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Definition df-subrg 19464
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 19462 . 2 class SubRing
2 vw . . 3 setvar 𝑤
3 crg 19228 . . 3 class Ring
42cv 1527 . . . . . . 7 class 𝑤
5 vs . . . . . . . 8 setvar 𝑠
65cv 1527 . . . . . . 7 class 𝑠
7 cress 16474 . . . . . . 7 class s
84, 6, 7co 7145 . . . . . 6 class (𝑤s 𝑠)
98, 3wcel 2105 . . . . 5 wff (𝑤s 𝑠) ∈ Ring
10 cur 19182 . . . . . . 7 class 1r
114, 10cfv 6349 . . . . . 6 class (1r𝑤)
1211, 6wcel 2105 . . . . 5 wff (1r𝑤) ∈ 𝑠
139, 12wa 396 . . . 4 wff ((𝑤s 𝑠) ∈ Ring ∧ (1r𝑤) ∈ 𝑠)
14 cbs 16473 . . . . . 6 class Base
154, 14cfv 6349 . . . . 5 class (Base‘𝑤)
1615cpw 4537 . . . 4 class 𝒫 (Base‘𝑤)
1713, 5, 16crab 3142 . . 3 class {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ ((𝑤s 𝑠) ∈ Ring ∧ (1r𝑤) ∈ 𝑠)}
182, 3, 17cmpt 5138 . 2 class (𝑤 ∈ Ring ↦ {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ ((𝑤s 𝑠) ∈ Ring ∧ (1r𝑤) ∈ 𝑠)})
191, 18wceq 1528 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  19466
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