ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  df-subrg GIF version

Definition df-subrg 13278
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 13276 . 2 class SubRing
2 vw . . 3 setvar 𝑤
3 crg 13110 . . 3 class Ring
42cv 1352 . . . . . . 7 class 𝑤
5 vs . . . . . . . 8 setvar 𝑠
65cv 1352 . . . . . . 7 class 𝑠
7 cress 12455 . . . . . . 7 class s
84, 6, 7co 5872 . . . . . 6 class (𝑤s 𝑠)
98, 3wcel 2148 . . . . 5 wff (𝑤s 𝑠) ∈ Ring
10 cur 13073 . . . . . . 7 class 1r
114, 10cfv 5215 . . . . . 6 class (1r𝑤)
1211, 6wcel 2148 . . . . 5 wff (1r𝑤) ∈ 𝑠
139, 12wa 104 . . . 4 wff ((𝑤s 𝑠) ∈ Ring ∧ (1r𝑤) ∈ 𝑠)
14 cbs 12454 . . . . . 6 class Base
154, 14cfv 5215 . . . . 5 class (Base‘𝑤)
1615cpw 3575 . . . 4 class 𝒫 (Base‘𝑤)
1713, 5, 16crab 2459 . . 3 class {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ ((𝑤s 𝑠) ∈ Ring ∧ (1r𝑤) ∈ 𝑠)}
182, 3, 17cmpt 4063 . 2 class (𝑤 ∈ Ring ↦ {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ ((𝑤s 𝑠) ∈ Ring ∧ (1r𝑤) ∈ 𝑠)})
191, 18wceq 1353 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  13280
  Copyright terms: Public domain W3C validator