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Definition df-scaf 20473
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 20471 . 2 class Β·sf
2 vg . . 3 setvar 𝑔
3 cvv 3474 . . 3 class V
4 vx . . . 4 setvar π‘₯
5 vy . . . 4 setvar 𝑦
62cv 1540 . . . . . 6 class 𝑔
7 csca 17199 . . . . . 6 class Scalar
86, 7cfv 6543 . . . . 5 class (Scalarβ€˜π‘”)
9 cbs 17143 . . . . 5 class Base
108, 9cfv 6543 . . . 4 class (Baseβ€˜(Scalarβ€˜π‘”))
116, 9cfv 6543 . . . 4 class (Baseβ€˜π‘”)
124cv 1540 . . . . 5 class π‘₯
135cv 1540 . . . . 5 class 𝑦
14 cvsca 17200 . . . . . 6 class ·𝑠
156, 14cfv 6543 . . . . 5 class ( ·𝑠 β€˜π‘”)
1612, 13, 15co 7408 . . . 4 class (π‘₯( ·𝑠 β€˜π‘”)𝑦)
174, 5, 10, 11, 16cmpo 7410 . . 3 class (π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘”)), 𝑦 ∈ (Baseβ€˜π‘”) ↦ (π‘₯( ·𝑠 β€˜π‘”)𝑦))
182, 3, 17cmpt 5231 . 2 class (𝑔 ∈ V ↦ (π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘”)), 𝑦 ∈ (Baseβ€˜π‘”) ↦ (π‘₯( ·𝑠 β€˜π‘”)𝑦)))
191, 18wceq 1541 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  20489
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