congr_set — Mathlib

English

If f and f1 agree on s via a pointwise constraint and x ∈ s, then ContDiffWithinAt 𝕜 n f1 s x iff ContDiffWithinAt 𝕜 n f s x.

Русский

Если f1 и f совпадают на s в точечном виде и x ∈ s, то ContDiffWithinAt 𝕜 n f1 s x эквивалентно ContDiffWithinAt 𝕜 n f s x.

LaTeX

$$$ ContDiffWithinAt 𝕜 n f s x \\Rightarrow (∀ y ∈ s, f1 y = f y) \\Rightarrow ContDiffWithinAt 𝕜 n f1 s x \\iff ContDiffWithinAt 𝕜 n f s x$$

Lean4

theorem congr_set (h : ContDiffWithinAt 𝕜 n f s x) {t : Set E} (hst : s =ᶠ[𝓝 x] t) : ContDiffWithinAt 𝕜 n f t x :=
  by
  rw [← nhdsWithin_eq_iff_eventuallyEq] at hst
  apply h.mono_of_mem_nhdsWithin <| hst ▸ self_mem_nhdsWithin

Похожие утверждения