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Definition df-scaf 19568
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 19566 . 2 class ·sf
2 vg . . 3 setvar 𝑔
3 cvv 3495 . . 3 class V
4 vx . . . 4 setvar 𝑥
5 vy . . . 4 setvar 𝑦
62cv 1527 . . . . . 6 class 𝑔
7 csca 16558 . . . . . 6 class Scalar
86, 7cfv 6349 . . . . 5 class (Scalar‘𝑔)
9 cbs 16473 . . . . 5 class Base
108, 9cfv 6349 . . . 4 class (Base‘(Scalar‘𝑔))
116, 9cfv 6349 . . . 4 class (Base‘𝑔)
124cv 1527 . . . . 5 class 𝑥
135cv 1527 . . . . 5 class 𝑦
14 cvsca 16559 . . . . . 6 class ·𝑠
156, 14cfv 6349 . . . . 5 class ( ·𝑠𝑔)
1612, 13, 15co 7145 . . . 4 class (𝑥( ·𝑠𝑔)𝑦)
174, 5, 10, 11, 16cmpo 7147 . . 3 class (𝑥 ∈ (Base‘(Scalar‘𝑔)), 𝑦 ∈ (Base‘𝑔) ↦ (𝑥( ·𝑠𝑔)𝑦))
182, 3, 17cmpt 5138 . 2 class (𝑔 ∈ V ↦ (𝑥 ∈ (Base‘(Scalar‘𝑔)), 𝑦 ∈ (Base‘𝑔) ↦ (𝑥( ·𝑠𝑔)𝑦)))
191, 18wceq 1528 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  19583
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