Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  df-resv Structured version   Visualization version   GIF version

Definition df-resv 30825
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 30824 . 2 class v
2 vw . . 3 setvar 𝑤
3 vx . . 3 setvar 𝑥
4 cvv 3492 . . 3 class V
52cv 1527 . . . . . . 7 class 𝑤
6 csca 16556 . . . . . . 7 class Scalar
75, 6cfv 6348 . . . . . 6 class (Scalar‘𝑤)
8 cbs 16471 . . . . . 6 class Base
97, 8cfv 6348 . . . . 5 class (Base‘(Scalar‘𝑤))
103cv 1527 . . . . 5 class 𝑥
119, 10wss 3933 . . . 4 wff (Base‘(Scalar‘𝑤)) ⊆ 𝑥
12 cnx 16468 . . . . . . 7 class ndx
1312, 6cfv 6348 . . . . . 6 class (Scalar‘ndx)
14 cress 16472 . . . . . . 7 class s
157, 10, 14co 7145 . . . . . 6 class ((Scalar‘𝑤) ↾s 𝑥)
1613, 15cop 4563 . . . . 5 class ⟨(Scalar‘ndx), ((Scalar‘𝑤) ↾s 𝑥)⟩
17 csts 16469 . . . . 5 class sSet
185, 16, 17co 7145 . . . 4 class (𝑤 sSet ⟨(Scalar‘ndx), ((Scalar‘𝑤) ↾s 𝑥)⟩)
1911, 5, 18cif 4463 . . 3 class if((Base‘(Scalar‘𝑤)) ⊆ 𝑥, 𝑤, (𝑤 sSet ⟨(Scalar‘ndx), ((Scalar‘𝑤) ↾s 𝑥)⟩))
202, 3, 4, 4, 19cmpo 7147 . 2 class (𝑤 ∈ V, 𝑥 ∈ V ↦ if((Base‘(Scalar‘𝑤)) ⊆ 𝑥, 𝑤, (𝑤 sSet ⟨(Scalar‘ndx), ((Scalar‘𝑤) ↾s 𝑥)⟩)))
211, 20wceq 1528 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  30826  resvval  30827
  Copyright terms: Public domain W3C validator