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 18772
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 18770 . 2 class SubRing
2 vw . . 3 setvar 𝑤
3 crg 18541 . . 3 class Ring
42cv 1481 . . . . . . 7 class 𝑤
5 vs . . . . . . . 8 setvar 𝑠
65cv 1481 . . . . . . 7 class 𝑠
7 cress 15852 . . . . . . 7 class s
84, 6, 7co 6647 . . . . . 6 class (𝑤s 𝑠)
98, 3wcel 1989 . . . . 5 wff (𝑤s 𝑠) ∈ Ring
10 cur 18495 . . . . . . 7 class 1r
114, 10cfv 5886 . . . . . 6 class (1r𝑤)
1211, 6wcel 1989 . . . . 5 wff (1r𝑤) ∈ 𝑠
139, 12wa 384 . . . 4 wff ((𝑤s 𝑠) ∈ Ring ∧ (1r𝑤) ∈ 𝑠)
14 cbs 15851 . . . . . 6 class Base
154, 14cfv 5886 . . . . 5 class (Base‘𝑤)
1615cpw 4156 . . . 4 class 𝒫 (Base‘𝑤)
1713, 5, 16crab 2915 . . 3 class {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ ((𝑤s 𝑠) ∈ Ring ∧ (1r𝑤) ∈ 𝑠)}
182, 3, 17cmpt 4727 . 2 class (𝑤 ∈ Ring ↦ {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ ((𝑤s 𝑠) ∈ Ring ∧ (1r𝑤) ∈ 𝑠)})
191, 18wceq 1482 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  18774
  Copyright terms: Public domain W3C validator