Theorems · Theorem · commutative algebra

MvPowerSeries.weightedOrder_eq_nat

∀ {σ : Type u_1} {R : Type u_2} [inst : Semiring R] (w : σ → ℕ) {f : MvPowerSeries σ R} {n : ℕ},
  MvPowerSeries.weightedOrder w f = ↑n ↔
    (∃ d, (MvPowerSeries.coeff d) f ≠ 0 ∧ (Finsupp.weight w) d = n) ∧
      ∀ (d : σ →₀ ℕ), (Finsupp.weight w) d < n → (MvPowerSeries.coeff d) f = 0

The order of a formal power series is exactly n if and only if some coefficient of weight n is nonzero, and the dth coefficient is 0 for all d such that weight w d < n.

Defined in

Mathlib.RingTheory.MvPowerSeries.Order

Cited by

2 results in Mathlib

Foundations

Depth 85 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

Semiring