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Definition df-scaf 20877
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 20875 . 2 class ·sf
2 vg . . 3 setvar 𝑔
3 cvv 3477 . . 3 class V
4 vx . . . 4 setvar 𝑥
5 vy . . . 4 setvar 𝑦
62cv 1535 . . . . . 6 class 𝑔
7 csca 17300 . . . . . 6 class Scalar
86, 7cfv 6562 . . . . 5 class (Scalar‘𝑔)
9 cbs 17244 . . . . 5 class Base
108, 9cfv 6562 . . . 4 class (Base‘(Scalar‘𝑔))
116, 9cfv 6562 . . . 4 class (Base‘𝑔)
124cv 1535 . . . . 5 class 𝑥
135cv 1535 . . . . 5 class 𝑦
14 cvsca 17301 . . . . . 6 class ·𝑠
156, 14cfv 6562 . . . . 5 class ( ·𝑠𝑔)
1612, 13, 15co 7430 . . . 4 class (𝑥( ·𝑠𝑔)𝑦)
174, 5, 10, 11, 16cmpo 7432 . . 3 class (𝑥 ∈ (Base‘(Scalar‘𝑔)), 𝑦 ∈ (Base‘𝑔) ↦ (𝑥( ·𝑠𝑔)𝑦))
182, 3, 17cmpt 5230 . 2 class (𝑔 ∈ V ↦ (𝑥 ∈ (Base‘(Scalar‘𝑔)), 𝑦 ∈ (Base‘𝑔) ↦ (𝑥( ·𝑠𝑔)𝑦)))
191, 18wceq 1536 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  20894
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