continuous_reorderRestrictProd — Mathlib

English

If V is preirreducible, f is an open map, and fibers are preirreducible, then the preimage f⁻¹'(V) is preirreducible.

Русский

Если V преднеразложимо, f — открытое отображение, и фибры преднеразложимы, то f⁻¹'(V) — преднеразложимо.

LaTeX

$$$\mathrm{IsPreirreducible}(V) \rightarrow (f:X\to Y) \rightarrow \mathrm{IsOpenMap}(f) \rightarrow (\forall x, \mathrm{IsPreirreducible}(f^{-1}({x}))) \rightarrow \mathrm{IsPreirreducible}(f^{-1}(V))$$$

Lean4

theorem continuous_reorderRestrictProd [∀ i, TopologicalSpace (α i)] : Continuous (reorderRestrictProd S s) :=
  by
  refine continuous_pi fun j ↦ ?_
  simp only [reorderRestrictProd]
  split_ifs with h
  · fun_prop
  · exact ((continuous_apply _).comp continuous_subtype_val).comp continuous_fst

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