cfc_nonneg — Mathlib

English

Restriction of cfcHomSuperset to the restricted id yields the original element a.

Русский

Ограничение cfcHomSuperset к id возвращает исходный элемент a.

LaTeX

$$cfcHomSuperset ha hs (.restrict s (\text{id} R)) = a$$

Lean4

theorem cfc_nonneg {f : R → R} {a : A} (h : ∀ x ∈ spectrum R a, 0 ≤ f x) : 0 ≤ cfc f a :=
  by
  by_cases hf : ContinuousOn f (spectrum R a)
  · simpa using cfc_mono h
  · simp only [cfc_apply_of_not_continuousOn _ hf, le_rfl]

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