prod_mem_smoothNumbers — Mathlib

English

Decidability of membership in smoothNumbers(n) holds as a definable predicate.

Русский

Членство в smoothNumbers(n) имеет разрешимость как определимое предикат.

LaTeX

$$$\text{DecidablePred}(\cdot \in \mathrm{smoothNumbers}(n))$$$

Lean4

/-- The product of the prime factors of `n` that are less than `N` is an `N`-smooth number. -/
theorem prod_mem_smoothNumbers (n N : ℕ) : (n.primeFactorsList.filter (· < N)).prod ∈ smoothNumbers N := by
  simp only [smoothNumbers_eq_factoredNumbers, ← Finset.mem_range, prod_mem_factoredNumbers]

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