Definition df-subrg 13533

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 13531	. 2 class SubRing
2		vw	. . 3 setvar 𝑤
3		crg 13317	. . 3 class Ring
4	2	cv 1363	. . . . . . 7 class 𝑤
5		vs	. . . . . . . 8 setvar 𝑠
6	5	cv 1363	. . . . . . 7 class 𝑠
7		cress 12487	. . . . . . 7 class ↾s
8	4, 6, 7	co 5891	. . . . . 6 class (𝑤 ↾s 𝑠)
9	8, 3	wcel 2160	. . . . 5 wff (𝑤 ↾s 𝑠) ∈ Ring
10		cur 13280	. . . . . . 7 class 1r
11	4, 10	cfv 5231	. . . . . 6 class (1r‘𝑤)
12	11, 6	wcel 2160	. . . . 5 wff (1r‘𝑤) ∈ 𝑠
13	9, 12	wa 104	. . . 4 wff ((𝑤 ↾s 𝑠) ∈ Ring ∧ (1r‘𝑤) ∈ 𝑠)
14		cbs 12486	. . . . . 6 class Base
15	4, 14	cfv 5231	. . . . 5 class (Base‘𝑤)
16	15	cpw 3590	. . . 4 class 𝒫 (Base‘𝑤)
17	13, 5, 16	crab 2472	. . 3 class {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ ((𝑤 ↾s 𝑠) ∈ Ring ∧ (1r‘𝑤) ∈ 𝑠)}
18	2, 3, 17	cmpt 4079	. 2 class (𝑤 ∈ Ring ↦ {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ ((𝑤 ↾s 𝑠) ∈ Ring ∧ (1r‘𝑤) ∈ 𝑠)})
19	1, 18	wceq 1364	1 wff SubRing = (𝑤 ∈ Ring ↦ {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ ((𝑤 ↾s 𝑠) ∈ Ring ∧ (1r‘𝑤) ∈ 𝑠)})

This definition is referenced by:  issubrg  13535