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Definition df-sdrg 20634
Description: Define the function associating with a ring the set of its sub-division-rings. A sub-division-ring of a ring is a subset of its base set which is a division ring when equipped with the induced structure (sum, multiplication, zero, and unity). If a ring is commutative (resp., a field), then its sub-division-rings are commutative (resp., are fields) (fldsdrgfld 20645), so we do not make a specific definition for subfields. (Contributed by Stefan O'Rear, 3-Oct-2015.) TODO: extend this definition to a function with domain V or at least Ring and not only DivRing.
Assertion
Ref Expression
df-sdrg SubDRing = (𝑤 ∈ DivRing ↦ {𝑠 ∈ (SubRing‘𝑤) ∣ (𝑤s 𝑠) ∈ DivRing})
Distinct variable group:   𝑤,𝑠

Detailed syntax breakdown of Definition df-sdrg
StepHypRef Expression
1 csdrg 20633 . 2 class SubDRing
2 vw . . 3 setvar 𝑤
3 cdr 20583 . . 3 class DivRing
42cv 1539 . . . . . 6 class 𝑤
5 vs . . . . . . 7 setvar 𝑠
65cv 1539 . . . . . 6 class 𝑠
7 cress 17180 . . . . . 6 class s
84, 6, 7co 7412 . . . . 5 class (𝑤s 𝑠)
98, 3wcel 2105 . . . 4 wff (𝑤s 𝑠) ∈ DivRing
10 csubrg 20465 . . . . 5 class SubRing
114, 10cfv 6543 . . . 4 class (SubRing‘𝑤)
129, 5, 11crab 3431 . . 3 class {𝑠 ∈ (SubRing‘𝑤) ∣ (𝑤s 𝑠) ∈ DivRing}
132, 3, 12cmpt 5231 . 2 class (𝑤 ∈ DivRing ↦ {𝑠 ∈ (SubRing‘𝑤) ∣ (𝑤s 𝑠) ∈ DivRing})
141, 13wceq 1540 1 wff SubDRing = (𝑤 ∈ DivRing ↦ {𝑠 ∈ (SubRing‘𝑤) ∣ (𝑤s 𝑠) ∈ DivRing})
Colors of variables: wff setvar class
This definition is referenced by:  issdrg  20635
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