exists_integer_multiple — Mathlib

English

Given a finite set s and a function f from indices to S, there is b in M such that for all i in s, (b : R) • f(i) is integer.

Русский

Для конечного множества индексов и функции f существует b ∈ M, чтобы для всех i ∈ s (b)•f(i) целое.

LaTeX

$$exist_integer_multiples(s,f)$$

Lean4

/-- Each element `a : S` has an `M`-multiple which is an integer.

This version multiplies `a` on the left, matching the argument order in the `SMul` instance.
-/
theorem exists_integer_multiple (a : S) : ∃ b : M, IsInteger R ((b : R) • a) :=
  by
  simp_rw [Algebra.smul_def, mul_comm _ a]
  apply exists_integer_multiple'

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