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Definition df-subrg 18543
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 component-wise) 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 18541 . 2 class SubRing
2 vw . . 3 setvar 𝑤
3 crg 18312 . . 3 class Ring
42cv 1473 . . . . . . 7 class 𝑤
5 vs . . . . . . . 8 setvar 𝑠
65cv 1473 . . . . . . 7 class 𝑠
7 cress 15638 . . . . . . 7 class s
84, 6, 7co 6523 . . . . . 6 class (𝑤s 𝑠)
98, 3wcel 1975 . . . . 5 wff (𝑤s 𝑠) ∈ Ring
10 cur 18266 . . . . . . 7 class 1r
114, 10cfv 5786 . . . . . 6 class (1r𝑤)
1211, 6wcel 1975 . . . . 5 wff (1r𝑤) ∈ 𝑠
139, 12wa 382 . . . 4 wff ((𝑤s 𝑠) ∈ Ring ∧ (1r𝑤) ∈ 𝑠)
14 cbs 15637 . . . . . 6 class Base
154, 14cfv 5786 . . . . 5 class (Base‘𝑤)
1615cpw 4103 . . . 4 class 𝒫 (Base‘𝑤)
1713, 5, 16crab 2895 . . 3 class {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ ((𝑤s 𝑠) ∈ Ring ∧ (1r𝑤) ∈ 𝑠)}
182, 3, 17cmpt 4633 . 2 class (𝑤 ∈ Ring ↦ {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ ((𝑤s 𝑠) ∈ Ring ∧ (1r𝑤) ∈ 𝑠)})
191, 18wceq 1474 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  18545
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