exists_coeff_ne_zero_and_order — Mathlib

English

If the order of f is finite, there exists a coefficient nonzero whose degree equals the order.

Русский

Если порядок f конечен, существует коэффициент не нулевой с степенью, равной порядку.

LaTeX

$$∃ d, coeff d f ≠ 0 ∧ degree d = f.order$$

Lean4

/-- If the order of a formal power series `f` is finite,
then some coefficient of degree the order of `f` is nonzero. -/
theorem exists_coeff_ne_zero_and_order (h : f.order.toNat = f.order) :
    ∃ d : σ →₀ ℕ, coeff d f ≠ 0 ∧ degree d = f.order :=
  by
  simp_rw [degree_eq_weight_one]
  exact exists_coeff_ne_zero_and_weightedOrder _ h

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