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Definition df-scaf 19060
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 19058 . 2 class ·sf
2 vg . . 3 setvar 𝑔
3 cvv 3332 . . 3 class V
4 vx . . . 4 setvar 𝑥
5 vy . . . 4 setvar 𝑦
62cv 1623 . . . . . 6 class 𝑔
7 csca 16138 . . . . . 6 class Scalar
86, 7cfv 6041 . . . . 5 class (Scalar‘𝑔)
9 cbs 16051 . . . . 5 class Base
108, 9cfv 6041 . . . 4 class (Base‘(Scalar‘𝑔))
116, 9cfv 6041 . . . 4 class (Base‘𝑔)
124cv 1623 . . . . 5 class 𝑥
135cv 1623 . . . . 5 class 𝑦
14 cvsca 16139 . . . . . 6 class ·𝑠
156, 14cfv 6041 . . . . 5 class ( ·𝑠𝑔)
1612, 13, 15co 6805 . . . 4 class (𝑥( ·𝑠𝑔)𝑦)
174, 5, 10, 11, 16cmpt2 6807 . . 3 class (𝑥 ∈ (Base‘(Scalar‘𝑔)), 𝑦 ∈ (Base‘𝑔) ↦ (𝑥( ·𝑠𝑔)𝑦))
182, 3, 17cmpt 4873 . 2 class (𝑔 ∈ V ↦ (𝑥 ∈ (Base‘(Scalar‘𝑔)), 𝑦 ∈ (Base‘𝑔) ↦ (𝑥( ·𝑠𝑔)𝑦)))
191, 18wceq 1624 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  19075
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