English
The simp lemmas relating range_cfc_nnreal to image and set membership reflect standard set-theoretic identities for NNReal-range of cfc.
Русский
Уточнения лемм относительно range_cfc_nnreal отражают стандартные тождества теории множеств для NNReal-произведения cfc.
LaTeX
$$$\\text{range\\_cfc\\_nnreal} \\Rightarrow \\text{set-theoretic identities on image and membership}$$$
Lean4
theorem range_cfcₙ_nnreal (a : A) (ha : 0 ≤ a) :
Set.range (cfcₙ (R := ℝ≥0) · a) = {x | x ∈ NonUnitalStarAlgebra.elemental ℝ a ∧ 0 ≤ x} :=
by
rw [range_cfcₙ_nnreal_eq_image_cfcₙ_real a ha, Set.setOf_and, SetLike.setOf_mem_eq, ← range_cfcₙ _ ha.isSelfAdjoint,
Set.inter_comm, ← Set.image_preimage_eq_inter_range]
refine Set.Subset.antisymm (Set.image_mono (fun _ ↦ cfcₙ_nonneg)) ?_
rintro _ ⟨f, hf, rfl⟩
simp only [Set.preimage_setOf_eq, Set.mem_setOf_eq, Set.mem_image] at hf ⊢
obtain (⟨h₁, h₂, h₃⟩ | h | h | h) := by
simpa only [not_and_or] using em (ContinuousOn f (quasispectrum ℝ a) ∧ f 0 = 0 ∧ IsSelfAdjoint a)
· refine ⟨f, ?_, rfl⟩
rwa [cfcₙ_nonneg_iff f a] at hf
· exact ⟨0, by simp, by simp [cfcₙ_apply_of_not_continuousOn a h]⟩
· exact ⟨0, by simp, by simp [cfcₙ_apply_of_not_map_zero a h]⟩
· exact ⟨0, by simp, by simp [cfcₙ_apply_of_not_predicate a h]⟩
