coeff_zero_eq_eval_zero — Mathlib

English

For the sum of monomials with coefficients x_i ≠ 0 and k_i, the equality of the card of the support equals the predicted combinatorial expansion given by card_support_eq'.

Русский

Для суммы мономов с ненулевыми коэффициентами x_i и степенями k_i справедливо равенство кардинала опоры, согласующееся с формулой card_support_eq'.

LaTeX

$$$\\#f.{\\text{support}} = \\#\\{i : x_i \\neq 0\\} \\quad\\text{(via } card_support_eq'\\text{)}$$$

Lean4

theorem coeff_zero_eq_eval_zero (p : R[X]) : coeff p 0 = p.eval 0 :=
  calc
    coeff p 0 = coeff p 0 * 0 ^ 0 := by simp
    _ = p.eval 0 := by
      symm
      rw [eval_eq_sum]
      exact Finset.sum_eq_single _ (fun b _ hb => by simp [zero_pow hb]) (by simp)

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