53 lines
2.1 KiB
Plaintext
53 lines
2.1 KiB
Plaintext
import Mathlib.Analysis.Analytic.IsolatedZeros
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import Mathlib.Analysis.Complex.Basic
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theorem AnalyticAt.order_mul
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{f₁ f₂ : ℂ → ℂ}
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{z₀ : ℂ}
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(hf₁ : AnalyticAt ℂ f₁ z₀)
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(hf₂ : AnalyticAt ℂ f₂ z₀) :
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(AnalyticAt.mul hf₁ hf₂).order = hf₁.order + hf₂.order := by
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by_cases h₂f₁ : hf₁.order = ⊤
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· simp [h₂f₁]
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rw [AnalyticAt.order_eq_top_iff, eventually_nhds_iff]
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rw [AnalyticAt.order_eq_top_iff, eventually_nhds_iff] at h₂f₁
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obtain ⟨t, h₁t, h₂t, h₃t⟩ := h₂f₁
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use t
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constructor
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· intro y hy
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rw [h₁t y hy]
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ring
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· exact ⟨h₂t, h₃t⟩
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· by_cases h₂f₂ : hf₂.order = ⊤
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· simp [h₂f₂]
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rw [AnalyticAt.order_eq_top_iff, eventually_nhds_iff]
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rw [AnalyticAt.order_eq_top_iff, eventually_nhds_iff] at h₂f₂
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obtain ⟨t, h₁t, h₂t, h₃t⟩ := h₂f₂
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use t
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constructor
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· intro y hy
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rw [h₁t y hy]
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ring
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· exact ⟨h₂t, h₃t⟩
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· obtain ⟨g₁, h₁g₁, h₂g₁, h₃g₁⟩ := (AnalyticAt.order_eq_nat_iff hf₁ ↑hf₁.order.toNat).1 (eq_comm.1 (ENat.coe_toNat h₂f₁))
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obtain ⟨g₂, h₁g₂, h₂g₂, h₃g₂⟩ := (AnalyticAt.order_eq_nat_iff hf₂ ↑hf₂.order.toNat).1 (eq_comm.1 (ENat.coe_toNat h₂f₂))
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rw [← ENat.coe_toNat h₂f₁, ← ENat.coe_toNat h₂f₂, ← ENat.coe_add]
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rw [AnalyticAt.order_eq_nat_iff (AnalyticAt.mul hf₁ hf₂) ↑(hf₁.order.toNat + hf₂.order.toNat)]
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use g₁ * g₂
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constructor
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· exact AnalyticAt.mul h₁g₁ h₁g₂
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· constructor
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· simp; tauto
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· obtain ⟨t₁, h₁t₁, h₂t₁, h₃t₁⟩ := eventually_nhds_iff.1 h₃g₁
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obtain ⟨t₂, h₁t₂, h₂t₂, h₃t₂⟩ := eventually_nhds_iff.1 h₃g₂
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rw [eventually_nhds_iff]
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use t₁ ∩ t₂
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constructor
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· intro y hy
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rw [h₁t₁ y hy.1, h₁t₂ y hy.2]
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simp; ring
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· constructor
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· exact IsOpen.inter h₂t₁ h₂t₂
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· exact Set.mem_inter h₃t₁ h₃t₂
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