leanprover-community / leanprover-community/mathlib4
Orthogonality for affine subspaces
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@MithicSpirit is already working on this.
Since Jul 11, 2023.
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Description
-
Define
AffineSubspace.IsOrtho s tass.direction ⟂ t.directionand develop basic API (mostly copy from
Submodule.IsOrtho). Additions specific to affine subspaces
include lemmas likeIsOrtho.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).toAffineSubspaceand prove lemmas like "orthogonal complements through different
points are parallel", "orthogonal complements to parallel affine
subspaces are parallel" (+ aniffversion assuming that the
subspaces are closed).
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