Definition df-subrg 14636

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 14634	. 2 class SubRing
2		vw	. . 3 setvar 𝑤
3		crg 14407	. . 3 class Ring
4	2	cv 1401	. . . . . . 7 class 𝑤
5		vs	. . . . . . . 8 setvar 𝑠
6	5	cv 1401	. . . . . . 7 class 𝑠
7		cress 13411	. . . . . . 7 class ↾s
8	4, 6, 7	co 6088	. . . . . 6 class (𝑤 ↾s 𝑠)
9	8, 3	wcel 2209	. . . . 5 wff (𝑤 ↾s 𝑠) ∈ Ring
10		cur 14369	. . . . . . 7 class 1r
11	4, 10	cfv 5380	. . . . . 6 class (1r‘𝑤)
12	11, 6	wcel 2209	. . . . 5 wff (1r‘𝑤) ∈ 𝑠
13	9, 12	wa 104	. . . 4 wff ((𝑤 ↾s 𝑠) ∈ Ring ∧ (1r‘𝑤) ∈ 𝑠)
14		cbs 13410	. . . . . 6 class Base
15	4, 14	cfv 5380	. . . . 5 class (Base‘𝑤)
16	15	cpw 3688	. . . 4 class 𝒫 (Base‘𝑤)
17	13, 5, 16	crab 2532	. . 3 class {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ ((𝑤 ↾s 𝑠) ∈ Ring ∧ (1r‘𝑤) ∈ 𝑠)}
18	2, 3, 17	cmpt 4193	. 2 class (𝑤 ∈ Ring ↦ {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ ((𝑤 ↾s 𝑠) ∈ Ring ∧ (1r‘𝑤) ∈ 𝑠)})
19	1, 18	wceq 1402	1 wff SubRing = (𝑤 ∈ Ring ↦ {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ ((𝑤 ↾s 𝑠) ∈ Ring ∧ (1r‘𝑤) ∈ 𝑠)})

This definition is used by:  issubrg  14638