root_eq_neg_iff — Mathlib

English

For i, j ∈ ι and y ∈ M, coroot'(reflection i (y)) = coroot'(reflectionPerm i j) y.

Русский

Для i, j ∈ ι и y ∈ M корлот'(reflection i (y)) = coroot'(reflectionPerm i j) y.

LaTeX

$$$ P.coroot'(P.reflection(i)(y)) = P.coroot'(P.reflectionPerm(i,j))(y) $$$

Lean4

@[simp]
theorem root_eq_neg_iff : P.root i = -P.root j ↔ i = P.reflectionPerm j j :=
  by
  refine ⟨fun h ↦ P.root.injective ?_, fun h ↦ by simp [h]⟩
  rw [root_reflectionPerm, reflection_apply_self, h]

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