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Metamath Proof Explorer

Definition df-sdrg

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 ), 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 .

Ref Expression
Assertion df-sdrg ⊢ SubDRing = w ∈ DivRing ⟼ s ∈ SubRing ⁡ w | w ↾ 𝑠 s ∈ DivRing

Detailed syntax breakdown

Step Hyp Ref Expression
0 csdrg class SubDRing
1 vw setvar w
2 cdr class DivRing
3 vs setvar s
4 csubrg class SubRing
5 1 cv setvar w
6 5 4 cfv class SubRing ⁡ w
7 cress class ↾ 𝑠
8 3 cv setvar s
9 5 8 7 co class w ↾ 𝑠 s
10 9 2 wcel wff w ↾ 𝑠 s ∈ DivRing
11 10 3 6 crab class s ∈ SubRing ⁡ w | w ↾ 𝑠 s ∈ DivRing
12 1 2 11 cmpt class w ∈ DivRing ⟼ s ∈ SubRing ⁡ w | w ↾ 𝑠 s ∈ DivRing
13 0 12 wceq wff SubDRing = w ∈ DivRing ⟼ s ∈ SubRing ⁡ w | w ↾ 𝑠 s ∈ DivRing