one_minus_single_neg_mul — Mathlib

English

Reiterates the embDomain shift relation for smul with powers under a positivity assumption on orderTop.

Русский

Повторяет соотношение embDomain для умножения на powers при условии положительности orderTop.

LaTeX

$$$ (x \cdot powers x).embDomain ⟨Nat.succ, Nat.succ_injective⟩ = powers x - ofFinsupp (Finsupp.single 0 1) $$$

Lean4

theorem one_minus_single_neg_mul {x y : HahnSeries Γ R} {r : R} (hr : r * x.leadingCoeff = 1)
    (hxy : x = y + single x.order x.leadingCoeff) (oinv : Γ) (hxo : oinv + x.order = 0) :
    1 - single oinv r * x = -(single oinv r * y) := by
  nth_rw 1 [hxy]
  rw [mul_add, single_mul_single, hr, hxo, sub_add_eq_sub_sub_swap, sub_eq_neg_self,
    sub_eq_zero_of_eq single_zero_one.symm]

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