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Copyright 2025 The Formal Conjectures Authors.
Licensed under the Apache License, Version 2.0 (the "License");
you may not use this file except in compliance with the License.
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-/
module
public import Mathlib.Analysis.InnerProductSpace.PiL2
public import Mathlib.LinearAlgebra.Orientation@[expose] public sectionscoped[EuclideanGeometry] notation "ℝ³" => EuclideanSpace ℝ (Fin 3)open scoped EuclideanGeometry
The standard basis gives us a preferred orientation in ℝ³.
Note: when upstreaming this to Mathlib (and generalizing to Fin n) one
must be careful to avoid an instance diamond with IsEmpty.Orientation.
Presumably this can be avoided by assuming [NeZero n].
noncomputable instance Module.orientedEuclideanSpaceFinThree : Module.Oriented ℝ ℝ³ (Fin 3) :=
⟨Basis.orientation <| PiLp.basisFun ..⟩Three dimensional euclidean space is three-dimensional.
instance fact_finrank_euclideanSpace_fin_three : Fact (Module.finrank ℝ ℝ³ = 3) :=
⟨finrank_euclideanSpace_fin⟩