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Definition df-asp 19527
Description: Define the algebraic span of a set of vectors in an algebra. (Contributed by Mario Carneiro, 7-Jan-2015.)
Assertion
Ref Expression
df-asp AlgSpan = (𝑤 ∈ AssAlg ↦ (𝑠 ∈ 𝒫 (Base‘𝑤) ↦ {𝑡 ∈ ((SubRing‘𝑤) ∩ (LSubSp‘𝑤)) ∣ 𝑠𝑡}))
Distinct variable group:   𝑡,𝑠,𝑤

Detailed syntax breakdown of Definition df-asp
StepHypRef Expression
1 casp 19524 . 2 class AlgSpan
2 vw . . 3 setvar 𝑤
3 casa 19523 . . 3 class AssAlg
4 vs . . . 4 setvar 𝑠
52cv 1629 . . . . . 6 class 𝑤
6 cbs 16063 . . . . . 6 class Base
75, 6cfv 6031 . . . . 5 class (Base‘𝑤)
87cpw 4295 . . . 4 class 𝒫 (Base‘𝑤)
94cv 1629 . . . . . . 7 class 𝑠
10 vt . . . . . . . 8 setvar 𝑡
1110cv 1629 . . . . . . 7 class 𝑡
129, 11wss 3721 . . . . . 6 wff 𝑠𝑡
13 csubrg 18985 . . . . . . . 8 class SubRing
145, 13cfv 6031 . . . . . . 7 class (SubRing‘𝑤)
15 clss 19141 . . . . . . . 8 class LSubSp
165, 15cfv 6031 . . . . . . 7 class (LSubSp‘𝑤)
1714, 16cin 3720 . . . . . 6 class ((SubRing‘𝑤) ∩ (LSubSp‘𝑤))
1812, 10, 17crab 3064 . . . . 5 class {𝑡 ∈ ((SubRing‘𝑤) ∩ (LSubSp‘𝑤)) ∣ 𝑠𝑡}
1918cint 4609 . . . 4 class {𝑡 ∈ ((SubRing‘𝑤) ∩ (LSubSp‘𝑤)) ∣ 𝑠𝑡}
204, 8, 19cmpt 4861 . . 3 class (𝑠 ∈ 𝒫 (Base‘𝑤) ↦ {𝑡 ∈ ((SubRing‘𝑤) ∩ (LSubSp‘𝑤)) ∣ 𝑠𝑡})
212, 3, 20cmpt 4861 . 2 class (𝑤 ∈ AssAlg ↦ (𝑠 ∈ 𝒫 (Base‘𝑤) ↦ {𝑡 ∈ ((SubRing‘𝑤) ∩ (LSubSp‘𝑤)) ∣ 𝑠𝑡}))
221, 21wceq 1630 1 wff AlgSpan = (𝑤 ∈ AssAlg ↦ (𝑠 ∈ 𝒫 (Base‘𝑤) ↦ {𝑡 ∈ ((SubRing‘𝑤) ∩ (LSubSp‘𝑤)) ∣ 𝑠𝑡}))
Colors of variables: wff setvar class
This definition is referenced by:  aspval  19542
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