English
Variant of the product-intersection lemma for UniqueDiffWithinAt with restricted neighborhoods.
Русский
Вариант леммы про произведение и пересечение для UniqueDiffWithinAt с ограниченными окрестностями.
LaTeX
$$$\\text{Inter'}(hs, ht) : \\operatorname{UniqueDiffWithinAt}_{\\mathbb{k}}(s \\cap t) x \\iff \\ operatorname{UniqueDiffWithinAt}_{\\mathbb{k}}(s,x)$$$
Lean4
theorem mono_nhds (h : UniqueDiffWithinAt 𝕜 s x) (st : 𝓝[s] x ≤ 𝓝[t] x) : UniqueDiffWithinAt 𝕜 t x :=
by
simp only [uniqueDiffWithinAt_iff] at *
rw [mem_closure_iff_nhdsWithin_neBot] at h ⊢
exact ⟨h.1.mono <| Submodule.span_mono <| tangentConeAt_mono_nhds st, h.2.mono st⟩