mdifferentiableWithinAt_inter — Mathlib

English

If t ∈ nhdsWithin x s, then HS implies DS on t.

Русский

Если t принадлежит nhdsWithin x s, то свойство сохраняется для t.

LaTeX

$$mono_of_mem_nhdsWithin {s t : Set M} {x : M} (hs : UniqueMDiffWithinAt I s x) (ht : t ∈ 𝓝[s] x) : UniqueMDiffWithinAt I t x$$

Lean4

theorem mdifferentiableWithinAt_inter (ht : t ∈ 𝓝 x) :
    MDifferentiableWithinAt I I' f (s ∩ t) x ↔ MDifferentiableWithinAt I I' f s x := by
  rw [MDifferentiableWithinAt, MDifferentiableWithinAt,
    differentiableWithinAt_localInvariantProp.liftPropWithinAt_inter ht]

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