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Definition df-scaf 20474
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 Β·sf = (𝑔 ∈ V ↦ (π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘”)), 𝑦 ∈ (Baseβ€˜π‘”) ↦ (π‘₯( ·𝑠 β€˜π‘”)𝑦)))
Distinct variable group:   π‘₯,𝑔,𝑦

Detailed syntax breakdown of Definition df-scaf
StepHypRef Expression
1 cscaf 20472 . 2 class Β·sf
2 vg . . 3 setvar 𝑔
3 cvv 3475 . . 3 class V
4 vx . . . 4 setvar π‘₯
5 vy . . . 4 setvar 𝑦
62cv 1541 . . . . . 6 class 𝑔
7 csca 17200 . . . . . 6 class Scalar
86, 7cfv 6544 . . . . 5 class (Scalarβ€˜π‘”)
9 cbs 17144 . . . . 5 class Base
108, 9cfv 6544 . . . 4 class (Baseβ€˜(Scalarβ€˜π‘”))
116, 9cfv 6544 . . . 4 class (Baseβ€˜π‘”)
124cv 1541 . . . . 5 class π‘₯
135cv 1541 . . . . 5 class 𝑦
14 cvsca 17201 . . . . . 6 class ·𝑠
156, 14cfv 6544 . . . . 5 class ( ·𝑠 β€˜π‘”)
1612, 13, 15co 7409 . . . 4 class (π‘₯( ·𝑠 β€˜π‘”)𝑦)
174, 5, 10, 11, 16cmpo 7411 . . 3 class (π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘”)), 𝑦 ∈ (Baseβ€˜π‘”) ↦ (π‘₯( ·𝑠 β€˜π‘”)𝑦))
182, 3, 17cmpt 5232 . 2 class (𝑔 ∈ V ↦ (π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘”)), 𝑦 ∈ (Baseβ€˜π‘”) ↦ (π‘₯( ·𝑠 β€˜π‘”)𝑦)))
191, 18wceq 1542 1 wff Β·sf = (𝑔 ∈ V ↦ (π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘”)), 𝑦 ∈ (Baseβ€˜π‘”) ↦ (π‘₯( ·𝑠 β€˜π‘”)𝑦)))
Colors of variables: wff setvar class
This definition is referenced by:  scaffval  20490
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