support_swap — Mathlib

English

For σ ∈ Perm α and n ∈ Z, the same invariance of support holds: (σ^n).support ≤ σ.support.

Русский

Для σ ∈ Perm α и n ∈ Z опора (σ^n)≤опора(σ).

LaTeX

$$$(\sigma^{n}).\operatorname{support} \subseteq \operatorname{support}(\sigma).$$$

Lean4

@[simp]
theorem support_swap {x y : α} (h : x ≠ y) : support (swap x y) = { x, y } :=
  by
  ext z
  by_cases hx : z = x
  any_goals simpa [hx] using h.symm
  by_cases hy : z = y
  · simpa [swap_apply_of_ne_of_ne, hx, hy] using h
  · simp [swap_apply_of_ne_of_ne, hx, hy]

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