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Definition df-scaf 20701
Description: Define the functionalization of the ·𝑠 operator. This restricts the value of ·𝑠 to the stated domain, which is necessary when working with restricted structures, whose operations may be defined on a larger set than the true base. (Contributed by Mario Carneiro, 5-Oct-2015.)
Assertion
Ref Expression
df-scaf ·sf = (𝑔 ∈ V ↦ (𝑥 ∈ (Base‘(Scalar‘𝑔)), 𝑦 ∈ (Base‘𝑔) ↦ (𝑥( ·𝑠𝑔)𝑦)))
Distinct variable group:   𝑥,𝑔,𝑦

Detailed syntax breakdown of Definition df-scaf
StepHypRef Expression
1 cscaf 20699 . 2 class ·sf
2 vg . . 3 setvar 𝑔
3 cvv 3466 . . 3 class V
4 vx . . . 4 setvar 𝑥
5 vy . . . 4 setvar 𝑦
62cv 1532 . . . . . 6 class 𝑔
7 csca 17201 . . . . . 6 class Scalar
86, 7cfv 6534 . . . . 5 class (Scalar‘𝑔)
9 cbs 17145 . . . . 5 class Base
108, 9cfv 6534 . . . 4 class (Base‘(Scalar‘𝑔))
116, 9cfv 6534 . . . 4 class (Base‘𝑔)
124cv 1532 . . . . 5 class 𝑥
135cv 1532 . . . . 5 class 𝑦
14 cvsca 17202 . . . . . 6 class ·𝑠
156, 14cfv 6534 . . . . 5 class ( ·𝑠𝑔)
1612, 13, 15co 7402 . . . 4 class (𝑥( ·𝑠𝑔)𝑦)
174, 5, 10, 11, 16cmpo 7404 . . 3 class (𝑥 ∈ (Base‘(Scalar‘𝑔)), 𝑦 ∈ (Base‘𝑔) ↦ (𝑥( ·𝑠𝑔)𝑦))
182, 3, 17cmpt 5222 . 2 class (𝑔 ∈ V ↦ (𝑥 ∈ (Base‘(Scalar‘𝑔)), 𝑦 ∈ (Base‘𝑔) ↦ (𝑥( ·𝑠𝑔)𝑦)))
191, 18wceq 1533 1 wff ·sf = (𝑔 ∈ V ↦ (𝑥 ∈ (Base‘(Scalar‘𝑔)), 𝑦 ∈ (Base‘𝑔) ↦ (𝑥( ·𝑠𝑔)𝑦)))
Colors of variables: wff setvar class
This definition is referenced by:  scaffval  20718
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