commute_monomial — Mathlib

English

The product of a unit with φ is φ.

Русский

Произведение единицы на φ даёт φ.

LaTeX

$$$$ (1 : MvPowerSeries \\sigma \\; R) * \\phi = \\phi $$$$

Lean4

@[simp]
theorem commute_monomial {a : R} {n} : Commute φ (monomial n a) ↔ ∀ m, Commute (coeff m φ) a :=
  by
  rw [commute_iff_eq, MvPowerSeries.ext_iff]
  refine ⟨fun h m => ?_, fun h m => ?_⟩
  · have := h (m + n)
    rwa [coeff_add_mul_monomial, add_comm, coeff_add_monomial_mul] at this
  · rw [coeff_mul_monomial, coeff_monomial_mul]
    split_ifs <;> [apply h; rfl]

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