Update complexHarmonic.lean

Nevanlinna/complexHarmonic.lean

@@ -5,6 +5,12 @@ import Mathlib.Analysis.Calculus.LineDeriv.Basic
 import Mathlib.Analysis.Calculus.ContDiff.Defs
 import Mathlib.Analysis.Calculus.FDeriv.Basic
 import Mathlib.Analysis.Calculus.FDeriv.Symmetric
+import Mathlib.Data.Complex.Module
+import Mathlib.Data.Complex.Order
+import Mathlib.Data.Complex.Exponential
+import Mathlib.Analysis.RCLike.Basic
+import Mathlib.Topology.Algebra.InfiniteSum.Module
+import Mathlib.Topology.Instances.RealVectorSpace
 import Nevanlinna.cauchyRiemann
 import Nevanlinna.partialDeriv
 
@@ -40,18 +46,35 @@ theorem holomorphic_is_harmonic {f : ℂ → ℂ} (h : Differentiable ℂ f) :
     unfold Complex.laplace
     rw [CauchyRiemann₄ h]
 
-    let l : ℂ →L[ℝ] ℂ := by
-      --
-      sorry --(fun x ↦ Complex.I • x)
-    have : (Complex.I • Real.partialDeriv 1 f) = (l ∘ (Real.partialDeriv 1 f)) := by
-      sorry
+    have : ∀ v, ∀ s : ℂ, ∀ g : ℂ → ℂ, Differentiable ℝ g →  Real.partialDeriv v (s • g) = s • (Real.partialDeriv v g) := by
+      intro v s g hg
+
+      let sMuls : ℂ →L[ℝ] ℂ :=
+      {
+        toFun := fun x ↦ s * x
+        map_add' := by
+          intro x y
+          ring
+        map_smul' := by
+          intro m x
+          simp
+          ring
+      }
+
+      have : s • g = sMuls ∘ g := by rfl
       rw [this]
-    rw [partialDeriv_compContLin]
-
-    --rw [partialDeriv_smul₂ fI_is_real_differentiable]
+      rw [partialDeriv_compContLin hg]
+      rfl
 
+    rw [this]
     rw [partialDeriv_comm f_is_real_C2 Complex.I 1]
     rw [CauchyRiemann₄ h]
-    rw [partialDeriv_smul₂ fI_is_real_differentiable]
+    rw [this]
     rw [← smul_assoc]
     simp
+
+    -- Subgoals coming from the application of 'this'
+    -- Differentiable ℝ (Real.partialDeriv 1 f)
+    exact fI_is_real_differentiable
+    -- Differentiable ℝ (Real.partialDeriv 1 f)
+    exact fI_is_real_differentiable