Module: Mathlib.Analysis.Complex.Harmonic.Analytic

Goal state before:
pp: case right.refine_1
f : ℂ → ℝ
hf : HarmonicOnNhd f univ
g : ℂ → ℂ := (⇑ofRealCLM ∘ fun x ↦ (fderiv ℝ f x) 1) - I • ⇑ofRealCLM ∘ fun x ↦ (fderiv ℝ f x) I
hg : Differentiable ℂ g
F : ℂ → ℂ
hF : F 0 = ↑(f 0) ∧ ∀ x ∈ univ, HasDerivAt F (g x) x
h₁F : ∀ (z₁ : ℂ), HasDerivAt F (g z₁) z₁
h₂F : Differentiable ℂ F
h₃F : Differentiable ℝ F
x : ℂ
⊢ UniqueDiffOn ℝ univ

pp: case heq
f : ℂ → ℝ
hf : HarmonicOnNhd f univ
g : ℂ → ℂ := (⇑ofRealCLM ∘ fun x ↦ (fderiv ℝ f x) 1) - I • ⇑ofRealCLM ∘ fun x ↦ (fderiv ℝ f x) I
hg : Differentiable ℂ g
F : ℂ → ℂ
hF : F 0 = ↑(f 0) ∧ ∀ x ∈ univ, HasDerivAt F (g x) x
h₁F : ∀ (z₁ : ℂ), HasDerivAt F (g z₁) z₁
h₂F : Differentiable ℂ F
h₃F : Differentiable ℝ F
x : ℂ
⊢ EqOn (fderivWithin ℝ (⇑reCLM ∘ F) univ) (fderivWithin ℝ f univ) univ

pp: case right.refine_2
f : ℂ → ℝ
hf : HarmonicOnNhd f univ
g : ℂ → ℂ := (⇑ofRealCLM ∘ fun x ↦ (fderiv ℝ f x) 1) - I • ⇑ofRealCLM ∘ fun x ↦ (fderiv ℝ f x) I
hg : Differentiable ℂ g
F : ℂ → ℂ
hF : F 0 = ↑(f 0) ∧ ∀ x ∈ univ, HasDerivAt F (g x) x
h₁F : ∀ (z₁ : ℂ), HasDerivAt F (g z₁) z₁
h₂F : Differentiable ℂ F
h₃F : Differentiable ℝ F
x : ℂ
⊢ 0 ∈ univ

pp: case right.refine_3
f : ℂ → ℝ
hf : HarmonicOnNhd f univ
g : ℂ → ℂ := (⇑ofRealCLM ∘ fun x ↦ (fderiv ℝ f x) 1) - I • ⇑ofRealCLM ∘ fun x ↦ (fderiv ℝ f x) I
hg : Differentiable ℂ g
F : ℂ → ℂ
hF : F 0 = ↑(f 0) ∧ ∀ x ∈ univ, HasDerivAt F (g x) x
h₁F : ∀ (z₁ : ℂ), HasDerivAt F (g z₁) z₁
h₂F : Differentiable ℂ F
h₃F : Differentiable ℝ F
x : ℂ
⊢ (⇑reCLM ∘ F) 0 = f 0

pp: case right.refine_4
f : ℂ → ℝ
hf : HarmonicOnNhd f univ
g : ℂ → ℂ := (⇑ofRealCLM ∘ fun x ↦ (fderiv ℝ f x) 1) - I • ⇑ofRealCLM ∘ fun x ↦ (fderiv ℝ f x) I
hg : Differentiable ℂ g
F : ℂ → ℂ
hF : F 0 = ↑(f 0) ∧ ∀ x ∈ univ, HasDerivAt F (g x) x
h₁F : ∀ (z₁ : ℂ), HasDerivAt F (g z₁) z₁
h₂F : Differentiable ℂ F
h₃F : Differentiable ℝ F
x : ℂ
⊢ x ∈ univ

Tactic:
case heq =>
    intro y hy
    simp only [fderivWithin_univ]
    rw [fderiv_comp y (by fun_prop) (by fun_prop)]
    ext x
    trans fderiv ℝ f y (x.re • (1 : ℂ) + x.im • (I : ℂ))
    · simp only [map_smul, map_add]
      simp [(h₁F y).hasFDerivAt.restrictScalars ℝ |>.fderiv, g]
    · simp

Goal state after:
case right.refine_1
f : ℂ → ℝ
hf : HarmonicOnNhd f univ
g : ℂ → ℂ := (⇑ofRealCLM ∘ fun x ↦ (fderiv ℝ f x) 1) - I • ⇑ofRealCLM ∘ fun x ↦ (fderiv ℝ f x) I
hg : Differentiable ℂ g
F : ℂ → ℂ
hF : F 0 = ↑(f 0) ∧ ∀ x ∈ univ, HasDerivAt F (g x) x
h₁F : ∀ (z₁ : ℂ), HasDerivAt F (g z₁) z₁
h₂F : Differentiable ℂ F
h₃F : Differentiable ℝ F
x : ℂ
⊢ UniqueDiffOn ℝ univ

case right.refine_2
f : ℂ → ℝ
hf : HarmonicOnNhd f univ
g : ℂ → ℂ := (⇑ofRealCLM ∘ fun x ↦ (fderiv ℝ f x) 1) - I • ⇑ofRealCLM ∘ fun x ↦ (fderiv ℝ f x) I
hg : Differentiable ℂ g
F : ℂ → ℂ
hF : F 0 = ↑(f 0) ∧ ∀ x ∈ univ, HasDerivAt F (g x) x
h₁F : ∀ (z₁ : ℂ), HasDerivAt F (g z₁) z₁
h₂F : Differentiable ℂ F
h₃F : Differentiable ℝ F
x : ℂ
⊢ 0 ∈ univ

case right.refine_3
f : ℂ → ℝ
hf : HarmonicOnNhd f univ
g : ℂ → ℂ := (⇑ofRealCLM ∘ fun x ↦ (fderiv ℝ f x) 1) - I • ⇑ofRealCLM ∘ fun x ↦ (fderiv ℝ f x) I
hg : Differentiable ℂ g
F : ℂ → ℂ
hF : F 0 = ↑(f 0) ∧ ∀ x ∈ univ, HasDerivAt F (g x) x
h₁F : ∀ (z₁ : ℂ), HasDerivAt F (g z₁) z₁
h₂F : Differentiable ℂ F
h₃F : Differentiable ℝ F
x : ℂ
⊢ (⇑reCLM ∘ F) 0 = f 0

case right.refine_4
f : ℂ → ℝ
hf : HarmonicOnNhd f univ
g : ℂ → ℂ := (⇑ofRealCLM ∘ fun x ↦ (fderiv ℝ f x) 1) - I • ⇑ofRealCLM ∘ fun x ↦ (fderiv ℝ f x) I
hg : Differentiable ℂ g
F : ℂ → ℂ
hF : F 0 = ↑(f 0) ∧ ∀ x ∈ univ, HasDerivAt F (g x) x
h₁F : ∀ (z₁ : ℂ), HasDerivAt F (g z₁) z₁
h₂F : Differentiable ℂ F
h₃F : Differentiable ℝ F
x : ℂ
⊢ x ∈ univ