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Definition df-rgspn 18827
Description: The ring-span of a set of elements in a ring is the smallest subring which contains all of them. (Contributed by Stefan O'Rear, 7-Dec-2014.)
Assertion
Ref Expression
df-rgspn RingSpan = (𝑤 ∈ V ↦ (𝑠 ∈ 𝒫 (Base‘𝑤) ↦ {𝑡 ∈ (SubRing‘𝑤) ∣ 𝑠𝑡}))
Distinct variable group:   𝑤,𝑠,𝑡

Detailed syntax breakdown of Definition df-rgspn
StepHypRef Expression
1 crgspn 18825 . 2 class RingSpan
2 vw . . 3 setvar 𝑤
3 cvv 3231 . . 3 class V
4 vs . . . 4 setvar 𝑠
52cv 1522 . . . . . 6 class 𝑤
6 cbs 15904 . . . . . 6 class Base
75, 6cfv 5926 . . . . 5 class (Base‘𝑤)
87cpw 4191 . . . 4 class 𝒫 (Base‘𝑤)
94cv 1522 . . . . . . 7 class 𝑠
10 vt . . . . . . . 8 setvar 𝑡
1110cv 1522 . . . . . . 7 class 𝑡
129, 11wss 3607 . . . . . 6 wff 𝑠𝑡
13 csubrg 18824 . . . . . . 7 class SubRing
145, 13cfv 5926 . . . . . 6 class (SubRing‘𝑤)
1512, 10, 14crab 2945 . . . . 5 class {𝑡 ∈ (SubRing‘𝑤) ∣ 𝑠𝑡}
1615cint 4507 . . . 4 class {𝑡 ∈ (SubRing‘𝑤) ∣ 𝑠𝑡}
174, 8, 16cmpt 4762 . . 3 class (𝑠 ∈ 𝒫 (Base‘𝑤) ↦ {𝑡 ∈ (SubRing‘𝑤) ∣ 𝑠𝑡})
182, 3, 17cmpt 4762 . 2 class (𝑤 ∈ V ↦ (𝑠 ∈ 𝒫 (Base‘𝑤) ↦ {𝑡 ∈ (SubRing‘𝑤) ∣ 𝑠𝑡}))
191, 18wceq 1523 1 wff RingSpan = (𝑤 ∈ V ↦ (𝑠 ∈ 𝒫 (Base‘𝑤) ↦ {𝑡 ∈ (SubRing‘𝑤) ∣ 𝑠𝑡}))
Colors of variables: wff setvar class
This definition is referenced by:  rgspnval  38055
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