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Definition df-subrg 18826
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 18824 . 2 class SubRing
2 vw . . 3 setvar 𝑤
3 crg 18593 . . 3 class Ring
42cv 1522 . . . . . . 7 class 𝑤
5 vs . . . . . . . 8 setvar 𝑠
65cv 1522 . . . . . . 7 class 𝑠
7 cress 15905 . . . . . . 7 class s
84, 6, 7co 6690 . . . . . 6 class (𝑤s 𝑠)
98, 3wcel 2030 . . . . 5 wff (𝑤s 𝑠) ∈ Ring
10 cur 18547 . . . . . . 7 class 1r
114, 10cfv 5926 . . . . . 6 class (1r𝑤)
1211, 6wcel 2030 . . . . 5 wff (1r𝑤) ∈ 𝑠
139, 12wa 383 . . . 4 wff ((𝑤s 𝑠) ∈ Ring ∧ (1r𝑤) ∈ 𝑠)
14 cbs 15904 . . . . . 6 class Base
154, 14cfv 5926 . . . . 5 class (Base‘𝑤)
1615cpw 4191 . . . 4 class 𝒫 (Base‘𝑤)
1713, 5, 16crab 2945 . . 3 class {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ ((𝑤s 𝑠) ∈ Ring ∧ (1r𝑤) ∈ 𝑠)}
182, 3, 17cmpt 4762 . 2 class (𝑤 ∈ Ring ↦ {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ ((𝑤s 𝑠) ∈ Ring ∧ (1r𝑤) ∈ 𝑠)})
191, 18wceq 1523 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  18828
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