cfc_neg — Mathlib

English

The simplified form cfc f 1 equals algebraMap at f(1).

Русский

Упрощённая форма: cfc f 1 = algebraMap f(1).

LaTeX

$$$ cfc f 1 = \ algebraMap R A (f 1)$$$

Lean4

theorem cfc_neg : cfc (fun x ↦ -(f x)) a = -(cfc f a) :=
  by
  by_cases h : p a ∧ ContinuousOn f (spectrum R a)
  · obtain ⟨ha, hf⟩ := h
    rw [cfc_apply f a, ← map_neg, cfc_apply ..]
    congr
  · obtain (ha | hf) := not_and_or.mp h
    · simp [cfc_apply_of_not_predicate a ha]
    · rw [cfc_apply_of_not_continuousOn a hf, cfc_apply_of_not_continuousOn, neg_zero]
      exact fun hf_neg ↦ hf <| by simpa using hf_neg.neg

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