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Definition df-subrg 13403
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 13401 . 2 class SubRing
2 vw . . 3 setvar 𝑤
3 crg 13233 . . 3 class Ring
42cv 1362 . . . . . . 7 class 𝑤
5 vs . . . . . . . 8 setvar 𝑠
65cv 1362 . . . . . . 7 class 𝑠
7 cress 12476 . . . . . . 7 class s
84, 6, 7co 5888 . . . . . 6 class (𝑤s 𝑠)
98, 3wcel 2158 . . . . 5 wff (𝑤s 𝑠) ∈ Ring
10 cur 13196 . . . . . . 7 class 1r
114, 10cfv 5228 . . . . . 6 class (1r𝑤)
1211, 6wcel 2158 . . . . 5 wff (1r𝑤) ∈ 𝑠
139, 12wa 104 . . . 4 wff ((𝑤s 𝑠) ∈ Ring ∧ (1r𝑤) ∈ 𝑠)
14 cbs 12475 . . . . . 6 class Base
154, 14cfv 5228 . . . . 5 class (Base‘𝑤)
1615cpw 3587 . . . 4 class 𝒫 (Base‘𝑤)
1713, 5, 16crab 2469 . . 3 class {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ ((𝑤s 𝑠) ∈ Ring ∧ (1r𝑤) ∈ 𝑠)}
182, 3, 17cmpt 4076 . 2 class (𝑤 ∈ Ring ↦ {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ ((𝑤s 𝑠) ∈ Ring ∧ (1r𝑤) ∈ 𝑠)})
191, 18wceq 1363 1 wff SubRing = (𝑤 ∈ Ring ↦ {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ ((𝑤s 𝑠) ∈ Ring ∧ (1r𝑤) ∈ 𝑠)})
Colors of variables: wff set class
This definition is referenced by:  issubrg  13405
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