Definition df-sdrg 38725

Description: A sub-division-ring is a subset of a division ring's set which is a division ring under the induced operation. If the overring is commutative this is a field; no special consideration is made of the fields in the center of a skew field. (Contributed by Stefan O'Rear, 3-Oct-2015.)

Assertion

Ref	Expression
df-sdrg	⊢ SubDRing = (𝑤 ∈ DivRing ↦ {𝑠 ∈ (SubRing‘𝑤) ∣ (𝑤 ↾s 𝑠) ∈ DivRing})

Distinct variable group:   𝑤,𝑠

Detailed syntax breakdown of Definition df-sdrg

Step	Hyp	Ref	Expression
1		csdrg 38724	. 2 class SubDRing
2		vw	. . 3 setvar 𝑤
3		cdr 19139	. . 3 class DivRing
4	2	cv 1600	. . . . . 6 class 𝑤
5		vs	. . . . . . 7 setvar 𝑠
6	5	cv 1600	. . . . . 6 class 𝑠
7		cress 16256	. . . . . 6 class ↾s
8	4, 6, 7	co 6922	. . . . 5 class (𝑤 ↾s 𝑠)
9	8, 3	wcel 2107	. . . 4 wff (𝑤 ↾s 𝑠) ∈ DivRing
10		csubrg 19168	. . . . 5 class SubRing
11	4, 10	cfv 6135	. . . 4 class (SubRing‘𝑤)
12	9, 5, 11	crab 3094	. . 3 class {𝑠 ∈ (SubRing‘𝑤) ∣ (𝑤 ↾s 𝑠) ∈ DivRing}
13	2, 3, 12	cmpt 4965	. 2 class (𝑤 ∈ DivRing ↦ {𝑠 ∈ (SubRing‘𝑤) ∣ (𝑤 ↾s 𝑠) ∈ DivRing})
14	1, 13	wceq 1601	1 wff SubDRing = (𝑤 ∈ DivRing ↦ {𝑠 ∈ (SubRing‘𝑤) ∣ (𝑤 ↾s 𝑠) ∈ DivRing})

This definition is referenced by:  issdrg  38726