Update cauchyRiemann.lean

nevanlinna/cauchyRiemann.lean

@@ -1,15 +1,4 @@
---import Mathlib.Analysis.Calculus.Conformal.NormedSpace
---import Mathlib.Analysis.Calculus.ContDiff.Basic
---import Mathlib.Analysis.Calculus.ContDiff.Defs
---import Mathlib.Analysis.Calculus.Deriv.Linear
---import Mathlib.Analysis.Calculus.FDeriv.Basic
---import Mathlib.Analysis.Calculus.FDeriv.Comp
---import Mathlib.Analysis.Calculus.FDeriv.Linear
---import Mathlib.Analysis.Calculus.FDeriv.RestrictScalars
 import Mathlib.Analysis.Calculus.LineDeriv.Basic
---import Mathlib.Analysis.Complex.Basic
---import Mathlib.Analysis.Complex.CauchyIntegral
---import Mathlib.Analysis.Complex.Conformal
 import Mathlib.Analysis.Complex.RealDeriv
 
 variable {z : ℂ} {f : ℂ → ℂ}
@@ -17,37 +6,15 @@ variable {z : ℂ} {f : ℂ → ℂ}
 theorem CauchyRiemann₁ : (DifferentiableAt ℂ f z)
   → (fderiv ℝ f z) Complex.I = Complex.I * (fderiv ℝ f z) 1 := by
   intro h
-  let A := fderiv ℂ f z
-  have t₁ : A (Complex.I • 1) = Complex.I • (A 1) := by
-    exact ContinuousLinearMap.map_smul_of_tower A Complex.I 1
-  let AR := (ContinuousLinearMap.restrictScalars ℝ (fderiv ℂ f z))
-  have t₂ : AR (Complex.I • 1) = Complex.I • (AR 1) := by
-    exact t₁
-  have t₂a : Complex.I * (AR 1) = Complex.I • (AR 1) := by
-    exact rfl
-  rw [← t₂a] at t₂
-  have t₂c : AR Complex.I = Complex.I • (AR 1) := by
-    simp at t₂
-    exact t₂
-  have t₃ : fderiv ℝ f z = ContinuousLinearMap.restrictScalars ℝ (fderiv ℂ f z) := by
-    exact DifferentiableAt.fderiv_restrictScalars ℝ h
-  rw [t₃]
-  exact t₂c
+  rw [DifferentiableAt.fderiv_restrictScalars ℝ h]
+  nth_rewrite 1 [← mul_one Complex.I]
+  exact ContinuousLinearMap.map_smul_of_tower (fderiv ℂ f z) Complex.I 1
 
 theorem CauchyRiemann₂ : (DifferentiableAt ℂ f z)
   → lineDeriv ℝ f z Complex.I = Complex.I * lineDeriv ℝ f z 1 := by
   intro h
-
-  have : lineDeriv ℝ f z Complex.I = (fderiv ℝ f z) Complex.I := by
-    apply DifferentiableAt.lineDeriv_eq_fderiv
-    apply h.restrictScalars ℝ
-  rw [this]
-
-  have : lineDeriv ℝ f z 1 = (fderiv ℝ f z) 1 := by
-    apply DifferentiableAt.lineDeriv_eq_fderiv
-    apply h.restrictScalars ℝ
-  rw [this]
-
+  rw [DifferentiableAt.lineDeriv_eq_fderiv (h.restrictScalars ℝ)]
+  rw [DifferentiableAt.lineDeriv_eq_fderiv (h.restrictScalars ℝ)]
   exact CauchyRiemann₁ h
 
 theorem CauchyRiemann₃ : (DifferentiableAt ℂ f z)