realContinuousMapZeroOfNNReal_apply_comp_toReal

English

If φ is a continuous NNReal- to-Real algebra homomorphism, then its real-ContinuousMapZeroOfNNReal extension is continuous as a map between function spaces.

Русский

Если φ — непрерывный гомоморфизм NNReal → A, то его продолжение Real-версии также непрерывно в пространстве функций.

LaTeX

$$continuous(φ.realContinuousMapZeroOfNNReal) = true$$

Lean4

@[simp high]
theorem realContinuousMapZeroOfNNReal_apply_comp_toReal (φ : C(X, ℝ≥0)₀ →⋆ₙₐ[ℝ≥0] A) (f : C(X, ℝ≥0)₀) :
    φ.realContinuousMapZeroOfNNReal ((ContinuousMapZero.mk ⟨toReal, continuous_coe⟩ rfl).comp f) = φ f :=
  by
  simp only [realContinuousMapZeroOfNNReal_apply]
  convert_to φ f - φ 0 = φ f using 2
  on_goal -1 => rw [map_zero, sub_zero]
  all_goals
    congr
    ext x
    simp

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