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Definition df-scaf 19258
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 19256 . 2 class ·sf
2 vg . . 3 setvar 𝑔
3 cvv 3398 . . 3 class V
4 vx . . . 4 setvar 𝑥
5 vy . . . 4 setvar 𝑦
62cv 1600 . . . . . 6 class 𝑔
7 csca 16341 . . . . . 6 class Scalar
86, 7cfv 6135 . . . . 5 class (Scalar‘𝑔)
9 cbs 16255 . . . . 5 class Base
108, 9cfv 6135 . . . 4 class (Base‘(Scalar‘𝑔))
116, 9cfv 6135 . . . 4 class (Base‘𝑔)
124cv 1600 . . . . 5 class 𝑥
135cv 1600 . . . . 5 class 𝑦
14 cvsca 16342 . . . . . 6 class ·𝑠
156, 14cfv 6135 . . . . 5 class ( ·𝑠𝑔)
1612, 13, 15co 6922 . . . 4 class (𝑥( ·𝑠𝑔)𝑦)
174, 5, 10, 11, 16cmpt2 6924 . . 3 class (𝑥 ∈ (Base‘(Scalar‘𝑔)), 𝑦 ∈ (Base‘𝑔) ↦ (𝑥( ·𝑠𝑔)𝑦))
182, 3, 17cmpt 4965 . 2 class (𝑔 ∈ V ↦ (𝑥 ∈ (Base‘(Scalar‘𝑔)), 𝑦 ∈ (Base‘𝑔) ↦ (𝑥( ·𝑠𝑔)𝑦)))
191, 18wceq 1601 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  19273
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