nnreal_add — Mathlib

English

The same continuity property as in 42919 holds for the continuous character space to spectrum map.

Русский

Те же свойство непрерывности сохраняется для отображения характер-пр спектр.

LaTeX

$$continuous_characterSpaceToSpectrum (x) : Continuous (characterSpaceToSpectrum x)$$

Lean4

theorem nnreal_add {a b : A} (ha₁ : IsSelfAdjoint a) (hb₁ : IsSelfAdjoint b)
    (ha₂ : SpectrumRestricts a ContinuousMap.realToNNReal) (hb₂ : SpectrumRestricts b ContinuousMap.realToNNReal) :
    SpectrumRestricts (a + b) ContinuousMap.realToNNReal :=
  by
  rw [SpectrumRestricts.nnreal_iff_nnnorm (ha₁.add hb₁) (nnnorm_add_le a b), NNReal.coe_add, map_add, add_sub_add_comm]
  refine nnnorm_add_le _ _ |>.trans ?_
  gcongr
  all_goals
    rw [← SpectrumRestricts.nnreal_iff_nnnorm] <;>
      first
      | rfl
      | assumption

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