English
Elementary simplification lemmas about nonzero products in monoids with zero, ensuring certain expressions are not equal to zero when factors are nonzero.
Русский
Элементарные леммы упрощения о ненулевых произведениях в моноидах с нулём: если элементы ненулевые и не дают делителей нуля, то их произведение ненулевое.
LaTeX
$$$\forall a,b\; (a \neq 0) \land (b \neq 0) \Rightarrow a \cdot b \neq 0$$$
Lean4
/-- If `f` has formal power series `∑ n, pₙ` on a ball of radius `r`, then for `y, z` in any smaller
ball, the norm of the difference `f y - f z - p 1 (fun _ ↦ y - z)` is bounded above by
`C * (max ‖y - x‖ ‖z - x‖) * ‖y - z‖`. This lemma formulates this property using `IsBigO` and
`Filter.principal` on `E × E`. -/
theorem isBigO_image_sub_image_sub_deriv_principal (hf : HasFPowerSeriesOnBall f p x r) (hr : r' < r) :
(fun y : E × E => f y.1 - f y.2 - p 1 fun _ => y.1 - y.2) =O[𝓟 (EMetric.ball (x, x) r')] fun y =>
‖y - (x, x)‖ * ‖y.1 - y.2‖ :=
by
rw [← hasFPowerSeriesWithinOnBall_univ] at hf
simpa using hf.isBigO_image_sub_image_sub_deriv_principal hr
