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Definition df-subrg 20317
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 20315 . 2 class SubRing
2 vw . . 3 setvar 𝑤
3 crg 20056 . . 3 class Ring
42cv 1541 . . . . . . 7 class 𝑤
5 vs . . . . . . . 8 setvar 𝑠
65cv 1541 . . . . . . 7 class 𝑠
7 cress 17173 . . . . . . 7 class s
84, 6, 7co 7409 . . . . . 6 class (𝑤s 𝑠)
98, 3wcel 2107 . . . . 5 wff (𝑤s 𝑠) ∈ Ring
10 cur 20004 . . . . . . 7 class 1r
114, 10cfv 6544 . . . . . 6 class (1r𝑤)
1211, 6wcel 2107 . . . . 5 wff (1r𝑤) ∈ 𝑠
139, 12wa 397 . . . 4 wff ((𝑤s 𝑠) ∈ Ring ∧ (1r𝑤) ∈ 𝑠)
14 cbs 17144 . . . . . 6 class Base
154, 14cfv 6544 . . . . 5 class (Base‘𝑤)
1615cpw 4603 . . . 4 class 𝒫 (Base‘𝑤)
1713, 5, 16crab 3433 . . 3 class {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ ((𝑤s 𝑠) ∈ Ring ∧ (1r𝑤) ∈ 𝑠)}
182, 3, 17cmpt 5232 . 2 class (𝑤 ∈ Ring ↦ {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ ((𝑤s 𝑠) ∈ Ring ∧ (1r𝑤) ∈ 𝑠)})
191, 18wceq 1542 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  20319
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