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Definition df-sdrg 19569
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) ∣ (𝑤s 𝑠) ∈ DivRing})
Distinct variable group:   𝑤,𝑠

Detailed syntax breakdown of Definition df-sdrg
StepHypRef Expression
1 csdrg 19568 . 2 class SubDRing
2 vw . . 3 setvar 𝑤
3 cdr 19498 . . 3 class DivRing
42cv 1537 . . . . . 6 class 𝑤
5 vs . . . . . . 7 setvar 𝑠
65cv 1537 . . . . . 6 class 𝑠
7 cress 16479 . . . . . 6 class s
84, 6, 7co 7139 . . . . 5 class (𝑤s 𝑠)
98, 3wcel 2112 . . . 4 wff (𝑤s 𝑠) ∈ DivRing
10 csubrg 19527 . . . . 5 class SubRing
114, 10cfv 6328 . . . 4 class (SubRing‘𝑤)
129, 5, 11crab 3113 . . 3 class {𝑠 ∈ (SubRing‘𝑤) ∣ (𝑤s 𝑠) ∈ DivRing}
132, 3, 12cmpt 5113 . 2 class (𝑤 ∈ DivRing ↦ {𝑠 ∈ (SubRing‘𝑤) ∣ (𝑤s 𝑠) ∈ DivRing})
141, 13wceq 1538 1 wff SubDRing = (𝑤 ∈ DivRing ↦ {𝑠 ∈ (SubRing‘𝑤) ∣ (𝑤s 𝑠) ∈ DivRing})
Colors of variables: wff setvar class
This definition is referenced by:  issdrg  19570
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