l3lab/ntp-mathlib huggingface dataset:
/-- **Maximum modulus principle** on a closed ball: if `f : E → F` is continuous on a closed ball,
is complex differentiable on the corresponding open ball, and the norm `‖f w‖` takes its maximum
value on the open ball at its center, then the norm `‖f w‖` is constant on the closed ball.  -/
theorem norm_eqOn_closedBall_of_isMaxOn {f : E → F} {z : E} {r : ℝ}
    (hd : DiffContOnCl ℂ f (ball z r)) (hz : IsMaxOn (norm ∘ f) (ball z r) z) :
    EqOn (norm ∘ f) (const E ‖f z‖) (closedBall z r) := by
  intro w hw
  rw [mem_closedBall, dist_comm] at hw
  rcases eq_or_ne z w with (rfl | hne); · rfl
  set e := (lineMap z w : ℂ → E)
  have hde : Differentiable ℂ e := (differentiable_id.smul_const (w - z)).add_const z
  suffices ‖(f ∘ e) (1 : ℂ)‖ = ‖(f ∘ e) (0 : ℂ)‖ by 

Proof state:
E : Type u
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
F : Type v
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℂ F
f : E → F
z : E
r : ℝ
hd : DiffContOnCl ℂ f (ball z r)
hz : IsMaxOn (norm ∘ f) (ball z r) z
w : E
hw : dist z w ≤ r
hne : z ≠ w
e : ℂ → E := ⇑(lineMap z w)
hde : Differentiable ℂ e
this : ‖(f ∘ e) 1‖ = ‖(f ∘ e) 0‖
⊢ (norm ∘ f) w = Function.const E ‖f z‖ w

Next tactic:
simpa
/-- **Maximum modulus principle** on a closed ball: if `f : E → F` is continuous on a closed ball,
is complex differentiable on the corresponding open ball, and the norm `‖f w‖` takes its maximum
value on the open ball at its center, then the norm `‖f w‖` is constant on the closed ball.  -/
theorem norm_eqOn_closedBall_of_isMaxOn {f : E → F} {z : E} {r : ℝ}
    (hd : DiffContOnCl ℂ f (ball z r)) (hz : IsMaxOn (norm ∘ f) (ball z r) z) :
    EqOn (norm ∘ f) (const E ‖f z‖) (closedBall z r) := by
  intro w hw
  rw [mem_closedBall, dist_comm] at hw
  rcases eq_or_ne z w with (rfl | hne); · rfl
  set e := (lineMap z w : ℂ → E)
  have hde : Differentiable ℂ e := (differentiable_id.smul_const (w - z)).add_const z
  suffices ‖(f ∘ e) (1 : ℂ)‖ = ‖(f ∘ e) (0 : ℂ)‖ by simpa
  

Proof state:
case inr
E : Type u
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
F : Type v
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℂ F
f : E → F
z : E
r : ℝ
hd : DiffContOnCl ℂ f (ball z r)
hz : IsMaxOn (norm ∘ f) (ball z r) z
w : E
hw : dist z w ≤ r
hne : z ≠ w
e : ℂ → E := ⇑(lineMap z w)
hde : Differentiable ℂ e
⊢ ‖(f ∘ e) 1‖ = ‖(f ∘ e) 0‖

Next tactic:
have hr : dist (1 : ℂ) 0 = 1 := by simp
/-- **Maximum modulus principle** on a closed ball: if `f : E → F` is continuous on a closed ball,
is complex differentiable on the corresponding open ball, and the norm `‖f w‖` takes its maximum
value on the open ball at its center, then the norm `‖f w‖` is constant on the closed ball.  -/
theorem norm_eqOn_closedBall_of_isMaxOn {f : E → F} {z : E} {r : ℝ}
    (hd : DiffContOnCl ℂ f (ball z r)) (hz : IsMaxOn (norm ∘ f) (ball z r) z) :
    EqOn (norm ∘ f) (const E ‖f z‖) (closedBall z r) := by
  intro w hw
  rw [mem_closedBall, dist_comm] at hw
  rcases eq_or_ne z w with (rfl | hne); · rfl
  set e := (lineMap z w : ℂ → E)
  have hde : Differentiable ℂ e := (differentiable_id.smul_const (w - z)).add_const z
  suffices ‖(f ∘ e) (1 : ℂ)‖ = ‖(f ∘ e) (0 : ℂ)‖ by simpa
  have hr : dist (1 : ℂ) 0 = 1 := by 

Proof state:
E : Type u
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
F : Type v
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℂ F
f : E → F
z : E
r : ℝ
hd : DiffContOnCl ℂ f (ball z r)
hz : IsMaxOn (norm ∘ f) (ball z r) z
w : E
hw : dist z w ≤ r
hne : z ≠ w
e : ℂ → E := ⇑(lineMap z w)
hde : Differentiable ℂ e
⊢ dist 1 0 = 1

Next tactic:
simp
/-- **Maximum modulus principle** on a closed ball: if `f : E → F` is continuous on a closed ball,
is complex differentiable on the corresponding open ball, and the norm `‖f w‖` takes its maximum
value on the open ball at its center, then the norm `‖f w‖` is constant on the closed ball.  -/
theorem norm_eqOn_closedBall_of_isMaxOn {f : E → F} {z : E} {r : ℝ}
    (hd : DiffContOnCl ℂ f (ball z r)) (hz : IsMaxOn (norm ∘ f) (ball z r) z) :
    EqOn (norm ∘ f) (const E ‖f z‖) (closedBall z r) := by
  intro w hw
  rw [mem_closedBall, dist_comm] at hw
  rcases eq_or_ne z w with (rfl | hne); · rfl
  set e := (lineMap z w : ℂ → E)
  have hde : Differentiable ℂ e := (differentiable_id.smul_const (w - z)).add_const z
  suffices ‖(f ∘ e) (1 : ℂ)‖ = ‖(f ∘ e) (0 : ℂ)‖ by simpa
  have hr : dist (1 : ℂ) 0 = 1 := by simp
  

Proof state:
case inr
E : Type u
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
F : Type v
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℂ F
f : E → F
z : E
r : ℝ
hd : DiffContOnCl ℂ f (ball z r)
hz : IsMaxOn (norm ∘ f) (ball z r) z
w : E
hw : dist z w ≤ r
hne : z ≠ w
e : ℂ → E := ⇑(lineMap z w)
hde : Differentiable ℂ e
hr : dist 1 0 = 1
⊢ ‖(f ∘ e) 1‖ = ‖(f ∘ e) 0‖

Next tactic:
have hball : MapsTo e (ball 0 1) (ball z r) := by
    refine ((lipschitzWith_lineMap z w).mapsTo_ball (mt nndist_eq_zero.1 hne) 0 1).mono
      Subset.rfl ?_
    simpa only [lineMap_apply_zero, mul_one, coe_nndist] using ball_subset_ball hw
/-- **Maximum modulus principle** on a closed ball: if `f : E → F` is continuous on a closed ball,
is complex differentiable on the corresponding open ball, and the norm `‖f w‖` takes its maximum
value on the open ball at its center, then the norm `‖f w‖` is constant on the closed ball.  -/
theorem norm_eqOn_closedBall_of_isMaxOn {f : E → F} {z : E} {r : ℝ}
    (hd : DiffContOnCl ℂ f (ball z r)) (hz : IsMaxOn (norm ∘ f) (ball z r) z) :
    EqOn (norm ∘ f) (const E ‖f z‖) (closedBall z r) := by
  intro w hw
  rw [mem_closedBall, dist_comm] at hw
  rcases eq_or_ne z w with (rfl | hne); · rfl
  set e := (lineMap z w : ℂ → E)
  have hde : Differentiable ℂ e := (differentiable_id.smul_const (w - z)).add_const z
  suffices ‖(f ∘ e) (1 : ℂ)‖ = ‖(f ∘ e) (0 : ℂ)‖ by simpa
  have hr : dist (1 : ℂ) 0 = 1 := by simp
  have hball : MapsTo e (ball 0 1) (ball z r) := by
    

Proof state:
E : Type u
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
F : Type v
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℂ F
f : E → F
z : E
r : ℝ
hd : DiffContOnCl ℂ f (ball z r)
hz : IsMaxOn (norm ∘ f) (ball z r) z
w : E
hw : dist z w ≤ r
hne : z ≠ w
e : ℂ → E := ⇑(lineMap z w)
hde : Differentiable ℂ e
hr : dist 1 0 = 1
⊢ MapsTo e (ball 0 1) (ball z r)

Next tactic:
refine ((lipschitzWith_lineMap z w).mapsTo_ball (mt nndist_eq_zero.1 hne) 0 1).mono
      Subset.rfl ?_
/-- **Maximum modulus principle** on a closed ball: if `f : E → F` is continuous on a closed ball,
is complex differentiable on the corresponding open ball, and the norm `‖f w‖` takes its maximum
value on the open ball at its center, then the norm `‖f w‖` is constant on the closed ball.  -/
theorem norm_eqOn_closedBall_of_isMaxOn {f : E → F} {z : E} {r : ℝ}
    (hd : DiffContOnCl ℂ f (ball z r)) (hz : IsMaxOn (norm ∘ f) (ball z r) z) :
    EqOn (norm ∘ f) (const E ‖f z‖) (closedBall z r) := by
  intro w hw
  rw [mem_closedBall, dist_comm] at hw
  rcases eq_or_ne z w with (rfl | hne); · rfl
  set e := (lineMap z w : ℂ → E)
  have hde : Differentiable ℂ e := (differentiable_id.smul_const (w - z)).add_const z
  suffices ‖(f ∘ e) (1 : ℂ)‖ = ‖(f ∘ e) (0 : ℂ)‖ by simpa
  have hr : dist (1 : ℂ) 0 = 1 := by simp
  have hball : MapsTo e (ball 0 1) (ball z r) := by
    refine ((lipschitzWith_lineMap z w).mapsTo_ball (mt nndist_eq_zero.1 hne) 0 1).mono
      Subset.rfl ?_
    

Proof state:
E : Type u
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
F : Type v
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℂ F
f : E → F
z : E
r : ℝ
hd : DiffContOnCl ℂ f (ball z r)
hz : IsMaxOn (norm ∘ f) (ball z r) z
w : E
hw : dist z w ≤ r
hne : z ≠ w
e : ℂ → E := ⇑(lineMap z w)
hde : Differentiable ℂ e
hr : dist 1 0 = 1
⊢ ball ((lineMap z w) 0) (↑(nndist z w) * 1) ⊆ ball z r

Next tactic:
simpa only [lineMap_apply_zero, mul_one, coe_nndist] using ball_subset_ball hw
/-- **Maximum modulus principle** on a closed ball: if `f : E → F` is continuous on a closed ball,
is complex differentiable on the corresponding open ball, and the norm `‖f w‖` takes its maximum
value on the open ball at its center, then the norm `‖f w‖` is constant on the closed ball.  -/
theorem norm_eqOn_closedBall_of_isMaxOn {f : E → F} {z : E} {r : ℝ}
    (hd : DiffContOnCl ℂ f (ball z r)) (hz : IsMaxOn (norm ∘ f) (ball z r) z) :
    EqOn (norm ∘ f) (const E ‖f z‖) (closedBall z r) := by
  intro w hw
  rw [mem_closedBall, dist_comm] at hw
  rcases eq_or_ne z w with (rfl | hne); · rfl
  set e := (lineMap z w : ℂ → E)
  have hde : Differentiable ℂ e := (differentiable_id.smul_const (w - z)).add_const z
  suffices ‖(f ∘ e) (1 : ℂ)‖ = ‖(f ∘ e) (0 : ℂ)‖ by simpa
  have hr : dist (1 : ℂ) 0 = 1 := by simp
  have hball : MapsTo e (ball 0 1) (ball z r) := by
    refine ((lipschitzWith_lineMap z w).mapsTo_ball (mt nndist_eq_zero.1 hne) 0 1).mono
      Subset.rfl ?_
    simpa only [lineMap_apply_zero, mul_one, coe_nndist] using ball_subset_ball hw
  

Proof state:
case inr
E : Type u
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
F : Type v
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℂ F
f : E → F
z : E
r : ℝ
hd : DiffContOnCl ℂ f (ball z r)
hz : IsMaxOn (norm ∘ f) (ball z r) z
w : E
hw : dist z w ≤ r
hne : z ≠ w
e : ℂ → E := ⇑(lineMap z w)
hde : Differentiable ℂ e
hr : dist 1 0 = 1
hball : MapsTo e (ball 0 1) (ball z r)
⊢ ‖(f ∘ e) 1‖ = ‖(f ∘ e) 0‖

Next tactic:
exact norm_max_aux₃ hr (hd.comp hde.diffContOnCl hball)
      (hz.comp_mapsTo hball (lineMap_apply_zero z w))
/-- **Maximum modulus principle**: if `f : E → F` is complex differentiable in a neighborhood of `c`
and the norm `‖f z‖` has a local maximum at `c`, then `‖f z‖` is locally constant in a neighborhood
of `c`. -/
theorem norm_eventually_eq_of_isLocalMax {f : E → F} {c : E}
    (hd : ∀ᶠ z in 𝓝 c, DifferentiableAt ℂ f z) (hc : IsLocalMax (norm ∘ f) c) :
    ∀ᶠ y in 𝓝 c, ‖f y‖ = ‖f c‖ := by
  

Proof state:
E : Type u
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
F : Type v
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℂ F
f : E → F
c : E
hd : ∀ᶠ (z : E) in 𝓝 c, DifferentiableAt ℂ f z
hc : IsLocalMax (norm ∘ f) c
⊢ ∀ᶠ (y : E) in 𝓝 c, ‖f y‖ = ‖f c‖

Next tactic:
rcases nhds_basis_closedBall.eventually_iff.1 (hd.and hc) with ⟨r, hr₀, hr⟩
/-- **Maximum modulus principle**: if `f : E → F` is complex differentiable in a neighborhood of `c`
and the norm `‖f z‖` has a local maximum at `c`, then `‖f z‖` is locally constant in a neighborhood
of `c`. -/
theorem norm_eventually_eq_of_isLocalMax {f : E → F} {c : E}
    (hd : ∀ᶠ z in 𝓝 c, DifferentiableAt ℂ f z) (hc : IsLocalMax (norm ∘ f) c) :
    ∀ᶠ y in 𝓝 c, ‖f y‖ = ‖f c‖ := by
  rcases nhds_basis_closedBall.eventually_iff.1 (hd.and hc) with ⟨r, hr₀, hr⟩
  

Proof state:
case intro.intro
E : Type u
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
F : Type v
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℂ F
f : E → F
c : E
hd : ∀ᶠ (z : E) in 𝓝 c, DifferentiableAt ℂ f z
hc : IsLocalMax (norm ∘ f) c
r : ℝ
hr₀ : 0 < r
hr : ∀ ⦃x : E⦄, x ∈ closedBall c r → DifferentiableAt ℂ f x ∧ (fun x => (norm ∘ f) x ≤ (norm ∘ f) c) x
⊢ ∀ᶠ (y : E) in 𝓝 c, ‖f y‖ = ‖f c‖

Next tactic:
exact nhds_basis_closedBall.eventually_iff.2
    ⟨r, hr₀, norm_eqOn_closedBall_of_isMaxOn (DifferentiableOn.diffContOnCl fun x hx =>
        (hr <| closure_ball_subset_closedBall hx).1.differentiableWithinAt) fun x hx =>
      (hr <| ball_subset_closedBall hx).2⟩
theorem isOpen_setOf_mem_nhds_and_isMaxOn_norm {f : E → F} {s : Set E}
    (hd : DifferentiableOn ℂ f s) : IsOpen {z | s ∈ 𝓝 z ∧ IsMaxOn (norm ∘ f) s z} := by
  

Proof state:
E : Type u
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
F : Type v
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℂ F
f : E → F
s : Set E
hd : DifferentiableOn ℂ f s
⊢ IsOpen {z | s ∈ 𝓝 z ∧ IsMaxOn (norm ∘ f) s z}

Next tactic:
refine' isOpen_iff_mem_nhds.2 fun z hz => (eventually_eventually_nhds.2 hz.1).and _
theorem isOpen_setOf_mem_nhds_and_isMaxOn_norm {f : E → F} {s : Set E}
    (hd : DifferentiableOn ℂ f s) : IsOpen {z | s ∈ 𝓝 z ∧ IsMaxOn (norm ∘ f) s z} := by
  refine' isOpen_iff_mem_nhds.2 fun z hz => (eventually_eventually_nhds.2 hz.1).and _
  

Proof state:
E : Type u
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
F : Type v
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℂ F
f : E → F
s : Set E
hd : DifferentiableOn ℂ f s
z : E
hz : z ∈ {z | s ∈ 𝓝 z ∧ IsMaxOn (norm ∘ f) s z}
⊢ ∀ᶠ (z : E) in 𝓝 z, IsMaxOn (norm ∘ f) s z

Next tactic:
replace hd : ∀ᶠ w in 𝓝 z, DifferentiableAt ℂ f w
theorem isOpen_setOf_mem_nhds_and_isMaxOn_norm {f : E → F} {s : Set E}
    (hd : DifferentiableOn ℂ f s) : IsOpen {z | s ∈ 𝓝 z ∧ IsMaxOn (norm ∘ f) s z} := by
  refine' isOpen_iff_mem_nhds.2 fun z hz => (eventually_eventually_nhds.2 hz.1).and _
  replace hd : ∀ᶠ w in 𝓝 z, DifferentiableAt ℂ f w; 

Proof state:
case hd
E : Type u
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
F : Type v
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℂ F
f : E → F
s : Set E
hd : DifferentiableOn ℂ f s
z : E
hz : z ∈ {z | s ∈ 𝓝 z ∧ IsMaxOn (norm ∘ f) s z}
⊢ ∀ᶠ (w : E) in 𝓝 z, DifferentiableAt ℂ f w
E : Type u
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
F : Type v
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℂ F
f : E → F
s : Set E
z : E
hz : z ∈ {z | s ∈ 𝓝 z ∧ IsMaxOn (norm ∘ f) s z}
hd : ∀ᶠ (w : E) in 𝓝 z, DifferentiableAt ℂ f w
⊢ ∀ᶠ (z : E) in 𝓝 z, IsMaxOn (norm ∘ f) s z

Next tactic:
exact hd.eventually_differentiableAt hz.1
theorem isOpen_setOf_mem_nhds_and_isMaxOn_norm {f : E → F} {s : Set E}
    (hd : DifferentiableOn ℂ f s) : IsOpen {z | s ∈ 𝓝 z ∧ IsMaxOn (norm ∘ f) s z} := by
  refine' isOpen_iff_mem_nhds.2 fun z hz => (eventually_eventually_nhds.2 hz.1).and _
  replace hd : ∀ᶠ w in 𝓝 z, DifferentiableAt ℂ f w; exact hd.eventually_differentiableAt hz.1
  

Proof state:
E : Type u
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
F : Type v
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℂ F
f : E → F
s : Set E
z : E
hz : z ∈ {z | s ∈ 𝓝 z ∧ IsMaxOn (norm ∘ f) s z}
hd : ∀ᶠ (w : E) in 𝓝 z, DifferentiableAt ℂ f w
⊢ ∀ᶠ (z : E) in 𝓝 z, IsMaxOn (norm ∘ f) s z

Next tactic:
exact (norm_eventually_eq_of_isLocalMax hd <| hz.2.isLocalMax hz.1).mono fun x hx y hy =>
    le_trans (hz.2 hy).out hx.ge
/-- **Maximum modulus principle** on a connected set. Let `U` be a (pre)connected open set in a
complex normed space. Let `f : E → F` be a function that is complex differentiable on `U`. Suppose
that `‖f x‖` takes its maximum value on `U` at `c ∈ U`. Then `‖f x‖ = ‖f c‖` for all `x ∈ U`. -/
theorem norm_eqOn_of_isPreconnected_of_isMaxOn {f : E → F} {U : Set E} {c : E}
    (hc : IsPreconnected U) (ho : IsOpen U) (hd : DifferentiableOn ℂ f U) (hcU : c ∈ U)
    (hm : IsMaxOn (norm ∘ f) U c) : EqOn (norm ∘ f) (const E ‖f c‖) U := by
  

Proof state:
E : Type u
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
F : Type v
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℂ F
f : E → F
U : Set E
c : E
hc : IsPreconnected U
ho : IsOpen U
hd : DifferentiableOn ℂ f U
hcU : c ∈ U
hm : IsMaxOn (norm ∘ f) U c
⊢ EqOn (norm ∘ f) (Function.const E ‖f c‖) U

Next tactic:
set V := U ∩ {z | IsMaxOn (norm ∘ f) U z}
/-- **Maximum modulus principle** on a connected set. Let `U` be a (pre)connected open set in a
complex normed space. Let `f : E → F` be a function that is complex differentiable on `U`. Suppose
that `‖f x‖` takes its maximum value on `U` at `c ∈ U`. Then `‖f x‖ = ‖f c‖` for all `x ∈ U`. -/
theorem norm_eqOn_of_isPreconnected_of_isMaxOn {f : E → F} {U : Set E} {c : E}
    (hc : IsPreconnected U) (ho : IsOpen U) (hd : DifferentiableOn ℂ f U) (hcU : c ∈ U)
    (hm : IsMaxOn (norm ∘ f) U c) : EqOn (norm ∘ f) (const E ‖f c‖) U := by
  set V := U ∩ {z | IsMaxOn (norm ∘ f) U z}
  

Proof state:
E : Type u
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
F : Type v
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℂ F
f : E → F
U : Set E
c : E
hc : IsPreconnected U
ho : IsOpen U
hd : DifferentiableOn ℂ f U
hcU : c ∈ U
hm : IsMaxOn (norm ∘ f) U c
V : Set E := U ∩ {z | IsMaxOn (norm ∘ f) U z}
⊢ EqOn (norm ∘ f) (Function.const E ‖f c‖) U

Next tactic:
have hV : ∀ x ∈ V, ‖f x‖ = ‖f c‖ := fun x hx => le_antisymm (hm hx.1) (hx.2 hcU)
/-- **Maximum modulus principle** on a connected set. Let `U` be a (pre)connected open set in a
complex normed space. Let `f : E → F` be a function that is complex differentiable on `U`. Suppose
that `‖f x‖` takes its maximum value on `U` at `c ∈ U`. Then `‖f x‖ = ‖f c‖` for all `x ∈ U`. -/
theorem norm_eqOn_of_isPreconnected_of_isMaxOn {f : E → F} {U : Set E} {c : E}
    (hc : IsPreconnected U) (ho : IsOpen U) (hd : DifferentiableOn ℂ f U) (hcU : c ∈ U)
    (hm : IsMaxOn (norm ∘ f) U c) : EqOn (norm ∘ f) (const E ‖f c‖) U := by
  set V := U ∩ {z | IsMaxOn (norm ∘ f) U z}
  have hV : ∀ x ∈ V, ‖f x‖ = ‖f c‖ := fun x hx => le_antisymm (hm hx.1) (hx.2 hcU)
  

Proof state:
E : Type u
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
F : Type v
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℂ F
f : E → F
U : Set E
c : E
hc : IsPreconnected U
ho : IsOpen U
hd : DifferentiableOn ℂ f U
hcU : c ∈ U
hm : IsMaxOn (norm ∘ f) U c
V : Set E := U ∩ {z | IsMaxOn (norm ∘ f) U z}
hV : ∀ x ∈ V, ‖f x‖ = ‖f c‖
⊢ EqOn (norm ∘ f) (Function.const E ‖f c‖) U

Next tactic:
suffices : U ⊆ V
/-- **Maximum modulus principle** on a connected set. Let `U` be a (pre)connected open set in a
complex normed space. Let `f : E → F` be a function that is complex differentiable on `U`. Suppose
that `‖f x‖` takes its maximum value on `U` at `c ∈ U`. Then `‖f x‖ = ‖f c‖` for all `x ∈ U`. -/
theorem norm_eqOn_of_isPreconnected_of_isMaxOn {f : E → F} {U : Set E} {c : E}
    (hc : IsPreconnected U) (ho : IsOpen U) (hd : DifferentiableOn ℂ f U) (hcU : c ∈ U)
    (hm : IsMaxOn (norm ∘ f) U c) : EqOn (norm ∘ f) (const E ‖f c‖) U := by
  set V := U ∩ {z | IsMaxOn (norm ∘ f) U z}
  have hV : ∀ x ∈ V, ‖f x‖ = ‖f c‖ := fun x hx => le_antisymm (hm hx.1) (hx.2 hcU)
  suffices : U ⊆ V; 

Proof state:
E : Type u
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
F : Type v
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℂ F
f : E → F
U : Set E
c : E
hc : IsPreconnected U
ho : IsOpen U
hd : DifferentiableOn ℂ f U
hcU : c ∈ U
hm : IsMaxOn (norm ∘ f) U c
V : Set E := U ∩ {z | IsMaxOn (norm ∘ f) U z}
hV : ∀ x ∈ V, ‖f x‖ = ‖f c‖
this : U ⊆ V
⊢ EqOn (norm ∘ f) (Function.const E ‖f c‖) U
case this
E : Type u
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
F : Type v
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℂ F
f : E → F
U : Set E
c : E
hc : IsPreconnected U
ho : IsOpen U
hd : DifferentiableOn ℂ f U
hcU : c ∈ U
hm : IsMaxOn (norm ∘ f) U c
V : Set E := U ∩ {z | IsMaxOn (norm ∘ f) U z}
hV : ∀ x ∈ V, ‖f x‖ = ‖f c‖
⊢ U ⊆ V

Next tactic:
exact fun x hx => hV x (this hx)
/-- **Maximum modulus principle** on a connected set. Let `U` be a (pre)connected open set in a
complex normed space. Let `f : E → F` be a function that is complex differentiable on `U`. Suppose
that `‖f x‖` takes its maximum value on `U` at `c ∈ U`. Then `‖f x‖ = ‖f c‖` for all `x ∈ U`. -/
theorem norm_eqOn_of_isPreconnected_of_isMaxOn {f : E → F} {U : Set E} {c : E}
    (hc : IsPreconnected U) (ho : IsOpen U) (hd : DifferentiableOn ℂ f U) (hcU : c ∈ U)
    (hm : IsMaxOn (norm ∘ f) U c) : EqOn (norm ∘ f) (const E ‖f c‖) U := by
  set V := U ∩ {z | IsMaxOn (norm ∘ f) U z}
  have hV : ∀ x ∈ V, ‖f x‖ = ‖f c‖ := fun x hx => le_antisymm (hm hx.1) (hx.2 hcU)
  suffices : U ⊆ V; exact fun x hx => hV x (this hx)
  

Proof state:
case this
E : Type u
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
F : Type v
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℂ F
f : E → F
U : Set E
c : E
hc : IsPreconnected U
ho : IsOpen U
hd : DifferentiableOn ℂ f U
hcU : c ∈ U
hm : IsMaxOn (norm ∘ f) U c
V : Set E := U ∩ {z | IsMaxOn (norm ∘ f) U z}
hV : ∀ x ∈ V, ‖f x‖ = ‖f c‖
⊢ U ⊆ V

Next tactic:
have hVo : IsOpen V := by
    simpa only [ho.mem_nhds_iff, setOf_and, setOf_mem_eq]
      using isOpen_setOf_mem_nhds_and_isMaxOn_norm hd
/-- **Maximum modulus principle** on a connected set. Let `U` be a (pre)connected open set in a
complex normed space. Let `f : E → F` be a function that is complex differentiable on `U`. Suppose
that `‖f x‖` takes its maximum value on `U` at `c ∈ U`. Then `‖f x‖ = ‖f c‖` for all `x ∈ U`. -/
theorem norm_eqOn_of_isPreconnected_of_isMaxOn {f : E → F} {U : Set E} {c : E}
    (hc : IsPreconnected U) (ho : IsOpen U) (hd : DifferentiableOn ℂ f U) (hcU : c ∈ U)
    (hm : IsMaxOn (norm ∘ f) U c) : EqOn (norm ∘ f) (const E ‖f c‖) U := by
  set V := U ∩ {z | IsMaxOn (norm ∘ f) U z}
  have hV : ∀ x ∈ V, ‖f x‖ = ‖f c‖ := fun x hx => le_antisymm (hm hx.1) (hx.2 hcU)
  suffices : U ⊆ V; exact fun x hx => hV x (this hx)
  have hVo : IsOpen V := by
    

Proof state:
E : Type u
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
F : Type v
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℂ F
f : E → F
U : Set E
c : E
hc : IsPreconnected U
ho : IsOpen U
hd : DifferentiableOn ℂ f U
hcU : c ∈ U
hm : IsMaxOn (norm ∘ f) U c
V : Set E := U ∩ {z | IsMaxOn (norm ∘ f) U z}
hV : ∀ x ∈ V, ‖f x‖ = ‖f c‖
⊢ IsOpen V

Next tactic:
simpa only [ho.mem_nhds_iff, setOf_and, setOf_mem_eq]
      using isOpen_setOf_mem_nhds_and_isMaxOn_norm hd
/-- **Maximum modulus principle** on a connected set. Let `U` be a (pre)connected open set in a
complex normed space. Let `f : E → F` be a function that is complex differentiable on `U`. Suppose
that `‖f x‖` takes its maximum value on `U` at `c ∈ U`. Then `‖f x‖ = ‖f c‖` for all `x ∈ U`. -/
theorem norm_eqOn_of_isPreconnected_of_isMaxOn {f : E → F} {U : Set E} {c : E}
    (hc : IsPreconnected U) (ho : IsOpen U) (hd : DifferentiableOn ℂ f U) (hcU : c ∈ U)
    (hm : IsMaxOn (norm ∘ f) U c) : EqOn (norm ∘ f) (const E ‖f c‖) U := by
  set V := U ∩ {z | IsMaxOn (norm ∘ f) U z}
  have hV : ∀ x ∈ V, ‖f x‖ = ‖f c‖ := fun x hx => le_antisymm (hm hx.1) (hx.2 hcU)
  suffices : U ⊆ V; exact fun x hx => hV x (this hx)
  have hVo : IsOpen V := by
    simpa only [ho.mem_nhds_iff, setOf_and, setOf_mem_eq]
      using isOpen_setOf_mem_nhds_and_isMaxOn_norm hd
  

Proof state:
case this
E : Type u
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
F : Type v
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℂ F
f : E → F
U : Set E
c : E
hc : IsPreconnected U
ho : IsOpen U
hd : DifferentiableOn ℂ f U
hcU : c ∈ U
hm : IsMaxOn (norm ∘ f) U c
V : Set E := U ∩ {z | IsMaxOn (norm ∘ f) U z}
hV : ∀ x ∈ V, ‖f x‖ = ‖f c‖
hVo : IsOpen V
⊢ U ⊆ V

Next tactic:
have hVne : (U ∩ V).Nonempty := ⟨c, hcU, hcU, hm⟩
/-- **Maximum modulus principle** on a connected set. Let `U` be a (pre)connected open set in a
complex normed space. Let `f : E → F` be a function that is complex differentiable on `U`. Suppose
that `‖f x‖` takes its maximum value on `U` at `c ∈ U`. Then `‖f x‖ = ‖f c‖` for all `x ∈ U`. -/
theorem norm_eqOn_of_isPreconnected_of_isMaxOn {f : E → F} {U : Set E} {c : E}
    (hc : IsPreconnected U) (ho : IsOpen U) (hd : DifferentiableOn ℂ f U) (hcU : c ∈ U)
    (hm : IsMaxOn (norm ∘ f) U c) : EqOn (norm ∘ f) (const E ‖f c‖) U := by
  set V := U ∩ {z | IsMaxOn (norm ∘ f) U z}
  have hV : ∀ x ∈ V, ‖f x‖ = ‖f c‖ := fun x hx => le_antisymm (hm hx.1) (hx.2 hcU)
  suffices : U ⊆ V; exact fun x hx => hV x (this hx)
  have hVo : IsOpen V := by
    simpa only [ho.mem_nhds_iff, setOf_and, setOf_mem_eq]
      using isOpen_setOf_mem_nhds_and_isMaxOn_norm hd
  have hVne : (U ∩ V).Nonempty := ⟨c, hcU, hcU, hm⟩
  

Proof state:
case this
E : Type u
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
F : Type v
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℂ F
f : E → F
U : Set E
c : E
hc : IsPreconnected U
ho : IsOpen U
hd : DifferentiableOn ℂ f U
hcU : c ∈ U
hm : IsMaxOn (norm ∘ f) U c
V : Set E := U ∩ {z | IsMaxOn (norm ∘ f) U z}
hV : ∀ x ∈ V, ‖f x‖ = ‖f c‖
hVo : IsOpen V
hVne : Set.Nonempty (U ∩ V)
⊢ U ⊆ V

Next tactic:
set W := U ∩ {z | ‖f z‖ ≠ ‖f c‖}