leanprover-community / leanprover-community/mathlib4

Orthogonality for affine subspaces

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#5,539 5 comments 0 reactions 1 assignee View on GitHub

@MithicSpirit is already working on this.

Since Jul 11, 2023.

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Description

  • Define AffineSubspace.IsOrtho s t as s.direction ⟂ t.direction and develop basic API (mostly copy from
    Submodule.IsOrtho). Additions specific to affine subspaces
    include lemmas like IsOrtho.trans_parallel.

  • Define AffineSubspace.orthogonal, e.g., as

    /-- Orthogonal complement to an affine subspace passing through a given point. -/
    def orthogonal (s : AffineSubspace ℝ P) (b : P) : AffineSubspace ℝ P :=
    .comap (AffineMap.id ℝ P -ᵥ AffineMap.const ℝ P b) (s.directionᗮ).toAffineSubspace
    

    and prove lemmas like "orthogonal complements through different
    points are parallel", "orthogonal complements to parallel affine
    subspaces are parallel" (+ an iff version assuming that the
    subspaces are closed).

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