Definition df-subrg 20119

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) ∣ ((𝑤 ↾s 𝑠) ∈ Ring ∧ (1r‘𝑤) ∈ 𝑠)})

Distinct variable group:   𝑤,𝑠

Detailed syntax breakdown of Definition df-subrg

Step	Hyp	Ref	Expression
1		csubrg 20117	. 2 class SubRing
2		vw	. . 3 setvar 𝑤
3		crg 19870	. . 3 class Ring
4	2	cv 1539	. . . . . . 7 class 𝑤
5		vs	. . . . . . . 8 setvar 𝑠
6	5	cv 1539	. . . . . . 7 class 𝑠
7		cress 17030	. . . . . . 7 class ↾s
8	4, 6, 7	co 7329	. . . . . 6 class (𝑤 ↾s 𝑠)
9	8, 3	wcel 2105	. . . . 5 wff (𝑤 ↾s 𝑠) ∈ Ring
10		cur 19824	. . . . . . 7 class 1r
11	4, 10	cfv 6473	. . . . . 6 class (1r‘𝑤)
12	11, 6	wcel 2105	. . . . 5 wff (1r‘𝑤) ∈ 𝑠
13	9, 12	wa 396	. . . 4 wff ((𝑤 ↾s 𝑠) ∈ Ring ∧ (1r‘𝑤) ∈ 𝑠)
14		cbs 17001	. . . . . 6 class Base
15	4, 14	cfv 6473	. . . . 5 class (Base‘𝑤)
16	15	cpw 4546	. . . 4 class 𝒫 (Base‘𝑤)
17	13, 5, 16	crab 3403	. . 3 class {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ ((𝑤 ↾s 𝑠) ∈ Ring ∧ (1r‘𝑤) ∈ 𝑠)}
18	2, 3, 17	cmpt 5172	. 2 class (𝑤 ∈ Ring ↦ {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ ((𝑤 ↾s 𝑠) ∈ Ring ∧ (1r‘𝑤) ∈ 𝑠)})
19	1, 18	wceq 1540	1 wff SubRing = (𝑤 ∈ Ring ↦ {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ ((𝑤 ↾s 𝑠) ∈ Ring ∧ (1r‘𝑤) ∈ 𝑠)})

This definition is referenced by:  issubrg  20121