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Update analyticOn_zeroSet.lean

This commit is contained in:
Stefan Kebekus 2024-09-10 11:21:49 +02:00
parent 5dc437751b
commit cd58c18a78
1 changed files with 9 additions and 9 deletions
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18
Nevanlinna/analyticOn_zeroSet.lean
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@ -285,19 +285,19 @@ theorem finiteZeros
(h₂U : IsCompact U) (h₂U : IsCompact U)
(h₁f : AnalyticOn ℂ f U) (h₁f : AnalyticOn ℂ f U)
(h₂f : ∃ u ∈ U, f u ≠ 0) : (h₂f : ∃ u ∈ U, f u ≠ 0) :
Set.Finite ((U.restrict f)⁻¹' {0}) := by Set.Finite (U.restrict f⁻¹' {0}) := by
have hinter : IsCompact ↑(U ∩ f⁻¹' {0}) := by have closedness : IsClosed (U.restrict f⁻¹' {0}) := by
apply IsCompact.of_isClosed_subset h₂U apply IsClosed.preimage
apply h₁f.continuousOn.preimage_isClosed_of_isClosed apply continuousOn_iff_continuous_restrict.1
exact IsCompact.isClosed h₂U exact h₁f.continuousOn
exact isClosed_singleton exact isClosed_singleton
exact Set.inter_subset_left
apply hinter.finite have : CompactSpace U := by
apply DiscreteTopology.of_subset (s := ↑(U ∩ f⁻¹' {0})) exact isCompact_iff_compactSpace.mp h₂U
apply (IsClosed.isCompact closedness).finite
exact discreteZeros h₁U h₁f h₂f exact discreteZeros h₁U h₁f h₂f
rfl
theorem AnalyticOnCompact.eliminateZeros theorem AnalyticOnCompact.eliminateZeros