smul_left — Mathlib

English

Simp lemmas assert equivalences and congruences for pi-type LittleOTVS statements, simplifying by coordinatewise checks.

Русский

Утверждения лемм simp для pi-типа упрощают проверки через координаты.

LaTeX

$$$ \text{simp_1_3} \ ; \text{isLittleOTVS_π} \; (f,g) \; =\; o .$$

Lean4

protected theorem smul_left (h : f =o[𝕜; l] g) (c : α → 𝕜) : (fun x ↦ c x • f x) =o[𝕜; l] (fun x ↦ c x • g x) :=
  by
  simp only [isLittleOTVS_iff] at *
  peel h with U hU V hV ε hε x hx
  simp only at *
  rw [egauge_smul_right, egauge_smul_right, mul_left_comm]
  · gcongr
  all_goals exact fun _ ↦ Filter.nonempty_of_mem ‹_›

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