MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  df-subrg Structured version   Visualization version   GIF version

Definition df-subrg 19455
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 19453 . 2 class SubRing
2 vw . . 3 setvar 𝑤
3 crg 19219 . . 3 class Ring
42cv 1529 . . . . . . 7 class 𝑤
5 vs . . . . . . . 8 setvar 𝑠
65cv 1529 . . . . . . 7 class 𝑠
7 cress 16476 . . . . . . 7 class s
84, 6, 7co 7151 . . . . . 6 class (𝑤s 𝑠)
98, 3wcel 2107 . . . . 5 wff (𝑤s 𝑠) ∈ Ring
10 cur 19173 . . . . . . 7 class 1r
114, 10cfv 6351 . . . . . 6 class (1r𝑤)
1211, 6wcel 2107 . . . . 5 wff (1r𝑤) ∈ 𝑠
139, 12wa 396 . . . 4 wff ((𝑤s 𝑠) ∈ Ring ∧ (1r𝑤) ∈ 𝑠)
14 cbs 16475 . . . . . 6 class Base
154, 14cfv 6351 . . . . 5 class (Base‘𝑤)
1615cpw 4541 . . . 4 class 𝒫 (Base‘𝑤)
1713, 5, 16crab 3146 . . 3 class {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ ((𝑤s 𝑠) ∈ Ring ∧ (1r𝑤) ∈ 𝑠)}
182, 3, 17cmpt 5142 . 2 class (𝑤 ∈ Ring ↦ {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ ((𝑤s 𝑠) ∈ Ring ∧ (1r𝑤) ∈ 𝑠)})
191, 18wceq 1530 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  19457
  Copyright terms: Public domain W3C validator