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[Merged by Bors] - refactor(Analysis): golf Mathlib/Analysis/Complex/Spectrum #38404

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[Merged by Bors] - refactor(Analysis): golf Mathlib/Analysis/Complex/Spectrum #38404

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6 changes: 1 addition & 5 deletions Mathlib/Analysis/Complex/Spectrum.lean
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Expand Up @@ -17,13 +17,9 @@ public section
namespace SpectrumRestricts
variable {A : Type*} [Ring A]

set_option backward.isDefEq.respectTransparency false in
lemma real_iff [Algebra ℂ A] {a : A} :
SpectrumRestricts a Complex.reCLM ↔ ∀ x ∈ spectrum ℂ a, x = x.re := by
refine ⟨fun h x hx ↦ ?_, fun h ↦ ?_⟩
· obtain ⟨x, -, rfl⟩ := h.algebraMap_image.symm ▸ hx
simp
· exact .of_subset_range_algebraMap Complex.ofReal_re fun x hx ↦ ⟨x.re, (h x hx).symm⟩
simp [spectrumRestricts_iff, Set.LeftInvOn, Function.LeftInverse, eq_comm]

end SpectrumRestricts

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