coe_tsum_of_nonneg — Mathlib

English

If f: α → ℝ is nonnegative, then the NNReal sum of f equals the NNReal sum of its nonnegative values embedded in NNReal.

Русский

Если f: α → ℝ неотрицательная, то сумма NNReal f равна сумме значений f, отображённых в NNReal.

LaTeX

$$$(\\sum\\'\\_{L} a\\, f(a))\\;: \\mathbb{R}_{\\ge 0} \\; =\\; \\sum\\'_{L} a\\, (f(a), hf(a))$$$

Lean4

theorem coe_tsum_of_nonneg {f : α → ℝ} (hf₁ : ∀ n, 0 ≤ f n) :
    (⟨∑'[L] n, f n, tsum_nonneg hf₁⟩ : ℝ≥0) = (∑'[L] n, ⟨f n, hf₁ n⟩ : ℝ≥0) :=
  NNReal.eq <| Eq.symm <| coe_tsum (f := fun x => ⟨f x, hf₁ x⟩)

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