English
If σ, τ ∈ range_toPermHom'(g), then the corresponding a.ofPermHom functors compose accordingly: a.ofPermHomFun (σ ∘ τ) x = a.ofPermHomFun σ (a.ofPermHomFun τ x).
Русский
Если σ, τ ⊂ range_toPermHom'(g), то полученные отображения поBasis соответствуют композиции: a.ofPermHomFun (σ ∘ τ) x = a.ofPermHomFun σ (a.ofPermHomFun τ x).
LaTeX
$$a.ofPermHom (instHMul.hMul σ τ) x = a.ofPermHomFun σ (a.ofPermHomFun τ x)$$
Lean4
theorem ofPermHomFun_mul (σ τ : range_toPermHom' g) (x) :
ofPermHomFun a (σ * τ) x = (ofPermHomFun a σ) (ofPermHomFun a τ x) :=
by
rcases mem_fixedPoints_or_exists_zpow_eq a x with (hx | ⟨c, hc, m, hm⟩)
· simp only [ofPermHomFun_apply_of_mem_fixedPoints a _ hx]
· simp only [ofPermHomFun_apply_of_cycleOf_mem a _ hc hm]
rw [ofPermHomFun_apply_of_cycleOf_mem a _ _ rfl]
· rfl
· rw [zpow_apply_mem_support_of_mem_cycleFactorsFinset_iff]
apply mem_support_self
