coeff_mem_coeffs — Mathlib

English

If a nonzero coefficient c occurs at index m, then c is a member of the coefficient set p.coeffs.

Русский

Если c непуcтицируется к коэффициенту p в индексе m, то c ∈ p.coeffs.

LaTeX

$$$ {p : \\mathrm{MvPolynomial}(\\sigma, R)} \\; (m : \\sigma \\to \\mathbb{N}) \\; (h : p.\\mathrm{coeff} m \\neq 0) \\\\Rightarrow p.\\mathrm{coeff} m \\in p.\\mathrm{coeffs}. $$$

Lean4

theorem coeff_mem_coeffs {p : MvPolynomial σ R} (m : σ →₀ ℕ) (h : p.coeff m ≠ 0) : p.coeff m ∈ p.coeffs :=
  letI := Classical.decEq R
  Finset.mem_image_of_mem p.coeff (mem_support_iff.mpr h)

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