nth_stream_fr_nonneg_lt_one — Mathlib

English

In a field direct limit, a nonzero element has a multiplicative inverse.

Русский

В полевом прямом пределе любой ненулевой элемент имеет обратный.

LaTeX

$$$\\forall p \\neq 0, \\exists y: p \\cdot y = 1.$$$

Lean4

/-- Shows that the fractional parts of the stream are in `[0,1)`. -/
theorem nth_stream_fr_nonneg_lt_one {ifp_n : IntFractPair K} (nth_stream_eq : IntFractPair.stream v n = some ifp_n) :
    0 ≤ ifp_n.fr ∧ ifp_n.fr < 1 := by
  cases n with
  | zero =>
    have : IntFractPair.of v = ifp_n := by injection nth_stream_eq
    rw [← this, IntFractPair.of]
    exact ⟨fract_nonneg _, fract_lt_one _⟩
  | succ =>
    rcases succ_nth_stream_eq_some_iff.1 nth_stream_eq with ⟨_, _, _, ifp_of_eq_ifp_n⟩
    rw [← ifp_of_eq_ifp_n, IntFractPair.of]
    exact ⟨fract_nonneg _, fract_lt_one _⟩

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