sameRay_nonneg_smul_right — Mathlib

English

A refinement of exists_pos with more specific hypotheses.

Русский

Уточнение существования положительных коэффициентов.

LaTeX

$$exists_pos (h) (hx) (hy) : ∃ r1 r2 > 0, r1 x = r2 y$$

Lean4

/-- A vector is in the same ray as a nonnegative multiple of itself. -/
theorem sameRay_nonneg_smul_right (v : M) (h : 0 ≤ a) : SameRay R v (a • v) :=
  by
  obtain h | h := (algebraMap_nonneg R h).eq_or_lt'
  · rw [← algebraMap_smul R a v, h, zero_smul]
    exact zero_right _
  · refine Or.inr <| Or.inr ⟨algebraMap S R a, 1, h, by nontriviality R; exact zero_lt_one, ?_⟩
    module

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