count_finprod — Mathlib

English

A tautology: Eq (a → a) True.

Русский

Тавтология: Eq (a → a) True.

LaTeX

$$$$ \\mathrm{Eq}(a \\to a, \\mathrm{True}). $$$$

Lean4

/-- If `exps` is finitely supported, then `val_v(∏_w w^{exps w}) = exps v`. -/
theorem count_finprod (exps : HeightOneSpectrum R → ℤ)
    (h_exps : ∀ᶠ v : HeightOneSpectrum R in Filter.cofinite, exps v = 0) :
    count K v (∏ᶠ v : HeightOneSpectrum R, (v.asIdeal : FractionalIdeal R⁰ K) ^ exps v) = exps v :=
  by
  convert count_finsuppProd K v (Finsupp.mk h_exps.toFinset exps (fun _ ↦ h_exps.mem_toFinset))
  rw [finprod_eq_finset_prod_of_mulSupport_subset (s := h_exps.toFinset), Finsupp.prod]
  · rfl
  · rw [Finite.coe_toFinset]
    intro v hv h
    rw [mem_mulSupport, h, zpow_zero] at hv
    exact hv (Eq.refl 1)

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