ofPermHomFun_one — Mathlib

English

If σ, τ are from range_toPermHom'(g), then a.ofPermHom commutes with multiplication: a.ofPermHom (σ τ) = a.ofPermHom σ ∘ a.ofPermHom τ.

Русский

Если σ, τ из range_toPermHom'(g), то a.ofPermHom сохраняет произведение: a.ofPermHom (σ τ) = a.ofPermHom σ ∘ a.ofPermHom τ.

LaTeX

$$a.ofPermHom (instHMul.hMul σ τ) = MonoidHom.func_like$$

Lean4

theorem ofPermHomFun_one (x : α) : (ofPermHomFun a 1) x = x :=
  by
  rcases mem_fixedPoints_or_exists_zpow_eq a x with (hx | ⟨c, hc, m, hm⟩)
  · rw [ofPermHomFun_apply_of_mem_fixedPoints a _ hx]
  · rw [ofPermHomFun_apply_of_cycleOf_mem a _ hc hm, OneMemClass.coe_one, coe_one, id_eq, hm]

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