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Definition df-resv 30165
Description: Define an operator to restrict the scalar field component of an extended structure. (Contributed by Thierry Arnoux, 5-Sep-2018.)
Assertion
Ref Expression
df-resv v = (𝑤 ∈ V, 𝑥 ∈ V ↦ if((Base‘(Scalar‘𝑤)) ⊆ 𝑥, 𝑤, (𝑤 sSet ⟨(Scalar‘ndx), ((Scalar‘𝑤) ↾s 𝑥)⟩)))
Distinct variable group:   𝑥,𝑤

Detailed syntax breakdown of Definition df-resv
StepHypRef Expression
1 cresv 30164 . 2 class v
2 vw . . 3 setvar 𝑤
3 vx . . 3 setvar 𝑥
4 cvv 3351 . . 3 class V
52cv 1630 . . . . . . 7 class 𝑤
6 csca 16152 . . . . . . 7 class Scalar
75, 6cfv 6030 . . . . . 6 class (Scalar‘𝑤)
8 cbs 16064 . . . . . 6 class Base
97, 8cfv 6030 . . . . 5 class (Base‘(Scalar‘𝑤))
103cv 1630 . . . . 5 class 𝑥
119, 10wss 3723 . . . 4 wff (Base‘(Scalar‘𝑤)) ⊆ 𝑥
12 cnx 16061 . . . . . . 7 class ndx
1312, 6cfv 6030 . . . . . 6 class (Scalar‘ndx)
14 cress 16065 . . . . . . 7 class s
157, 10, 14co 6796 . . . . . 6 class ((Scalar‘𝑤) ↾s 𝑥)
1613, 15cop 4323 . . . . 5 class ⟨(Scalar‘ndx), ((Scalar‘𝑤) ↾s 𝑥)⟩
17 csts 16062 . . . . 5 class sSet
185, 16, 17co 6796 . . . 4 class (𝑤 sSet ⟨(Scalar‘ndx), ((Scalar‘𝑤) ↾s 𝑥)⟩)
1911, 5, 18cif 4226 . . 3 class if((Base‘(Scalar‘𝑤)) ⊆ 𝑥, 𝑤, (𝑤 sSet ⟨(Scalar‘ndx), ((Scalar‘𝑤) ↾s 𝑥)⟩))
202, 3, 4, 4, 19cmpt2 6798 . 2 class (𝑤 ∈ V, 𝑥 ∈ V ↦ if((Base‘(Scalar‘𝑤)) ⊆ 𝑥, 𝑤, (𝑤 sSet ⟨(Scalar‘ndx), ((Scalar‘𝑤) ↾s 𝑥)⟩)))
211, 20wceq 1631 1 wff v = (𝑤 ∈ V, 𝑥 ∈ V ↦ if((Base‘(Scalar‘𝑤)) ⊆ 𝑥, 𝑤, (𝑤 sSet ⟨(Scalar‘ndx), ((Scalar‘𝑤) ↾s 𝑥)⟩)))
Colors of variables: wff setvar class
This definition is referenced by:  reldmresv  30166  resvval  30167
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