orthogonalProjectionFn_vsub_mem_direction_orthogonal

English

If s1 = s2 and p1 = p2, then orthogonalProjection s1 p1 = orthogonalProjection s2 p2.

Русский

Если s1 = s2 и p1 = p2, то ортогональная проекция совпадает.

LaTeX

$$orthogonalProjection s1 p1 = orthogonalProjection s2 p2$$

Lean4

/-- Subtracting `p` from its `orthogonalProjectionFn` produces a
result in the orthogonal direction. This lemma is only intended for
use in setting up the bundled version and should not be used once that
is defined. -/
theorem orthogonalProjectionFn_vsub_mem_direction_orthogonal {s : AffineSubspace ℝ P} [Nonempty s]
    [s.direction.HasOrthogonalProjection] (p : P) : orthogonalProjectionFn s p -ᵥ p ∈ s.directionᗮ :=
  direction_mk' p s.directionᗮ ▸ vsub_mem_direction (orthogonalProjectionFn_mem_orthogonal p) (self_mem_mk' _ _)

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