min_lexOrder_le — Mathlib

English

For any scalar r ∈ k and any φ in the multivariate power series ring, (r • φ)⁻¹ = r⁻¹ • φ⁻¹.

Русский

Для любого скалярa r и любого ряда φ выполняется (r • φ)⁻¹ = r⁻¹ • φ⁻¹.

LaTeX

$$$\forall r \in k, \forall \phi \in \mathrm{MvPowerSeries}(\sigma,k), (r \cdot \phi)^{-1} = r^{-1} \cdot \phi^{-1}$$$

Lean4

theorem min_lexOrder_le {φ ψ : MvPowerSeries σ R} : min (lexOrder φ) (lexOrder ψ) ≤ lexOrder (φ + ψ) :=
  by
  rw [le_lexOrder_iff]
  intro d hd
  simp only [lt_min_iff] at hd
  rw [map_add, coeff_eq_zero_of_lt_lexOrder hd.1, coeff_eq_zero_of_lt_lexOrder hd.2, add_zero]

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