internlm/Lean-Github huggingface dataset:
Theorem: NontrivialHolomorphicAt.nhds_le_map_nhds_param'
Proof state before:
case intro.intro.intro.intro.intro
X : Type
inst✝⁶ : TopologicalSpace X
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold 𝓘(ℂ, ℂ) S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold 𝓘(ℂ, ℂ) T
U : Type
inst✝¹ : TopologicalSpace U
inst✝ : ChartedSpace ℂ U
cmu : AnalyticManifold 𝓘(ℂ, ℂ) U
f : ℂ → ℂ → ℂ
c z : ℂ
n : NontrivialHolomorphicAt (f c) z
s : Set (ℂ × ℂ)
sn : s ∈ 𝓝 (c, z)
fa : AnalyticOn ℂ (uncurry f) s
e : ℝ
ep : e > 0
es : ball (c, z) e ⊆ s
r : ℝ
rp : 0 < r
rs : closedBall (c, z) r ⊆ s
fr : f c z ∉ f c '' sphere z r
⊢ (fun p => (p.1, f p.1 p.2)) '' s ∈ 𝓝 (c, f c z)
Tactic:
have fc : ContinuousOn (fun w ↦ ‖f c w - f c z‖) (sphere z r) := by
  apply ContinuousOn.norm; refine ContinuousOn.sub ?_ continuousOn_const
  apply fa.along_snd.continuousOn.mono; intro x xs; apply rs
  simp only [← closedBall_prod_same, mem_prod_eq]
  use Metric.mem_closedBall_self rp.le, Metric.sphere_subset_closedBall xs
Proof state after:
case intro.intro.intro.intro.intro
X : Type
inst✝⁶ : TopologicalSpace X
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold 𝓘(ℂ, ℂ) S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold 𝓘(ℂ, ℂ) T
U : Type
inst✝¹ : TopologicalSpace U
inst✝ : ChartedSpace ℂ U
cmu : AnalyticManifold 𝓘(ℂ, ℂ) U
f : ℂ → ℂ → ℂ
c z : ℂ
n : NontrivialHolomorphicAt (f c) z
s : Set (ℂ × ℂ)
sn : s ∈ 𝓝 (c, z)
fa : AnalyticOn ℂ (uncurry f) s
e : ℝ
ep : e > 0
es : ball (c, z) e ⊆ s
r : ℝ
rp : 0 < r
rs : closedBall (c, z) r ⊆ s
fr : f c z ∉ f c '' sphere z r
fc : ContinuousOn (fun w => ‖f c w - f c z‖) (sphere z r)
⊢ (fun p => (p.1, f p.1 p.2)) '' s ∈ 𝓝 (c, f c z)
Theorem: NontrivialHolomorphicAt.nhds_le_map_nhds_param'
Proof state before:
case intro.intro.intro.intro.intro
X : Type
inst✝⁶ : TopologicalSpace X
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold 𝓘(ℂ, ℂ) S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold 𝓘(ℂ, ℂ) T
U : Type
inst✝¹ : TopologicalSpace U
inst✝ : ChartedSpace ℂ U
cmu : AnalyticManifold 𝓘(ℂ, ℂ) U
f : ℂ → ℂ → ℂ
c z : ℂ
n : NontrivialHolomorphicAt (f c) z
s : Set (ℂ × ℂ)
sn : s ∈ 𝓝 (c, z)
fa : AnalyticOn ℂ (uncurry f) s
e : ℝ
ep : e > 0
es : ball (c, z) e ⊆ s
r : ℝ
rp : 0 < r
rs : closedBall (c, z) r ⊆ s
fr : f c z ∉ f c '' sphere z r
fc : ContinuousOn (fun w => ‖f c w - f c z‖) (sphere z r)
⊢ (fun p => (p.1, f p.1 p.2)) '' s ∈ 𝓝 (c, f c z)
Tactic:
rcases (isCompact_sphere _ _).exists_isMinOn (NormedSpace.sphere_nonempty.mpr rp.le) fc with
  ⟨x, xs, xm⟩
Proof state after:
case intro.intro.intro.intro.intro.intro.intro
X : Type
inst✝⁶ : TopologicalSpace X
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold 𝓘(ℂ, ℂ) S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold 𝓘(ℂ, ℂ) T
U : Type
inst✝¹ : TopologicalSpace U
inst✝ : ChartedSpace ℂ U
cmu : AnalyticManifold 𝓘(ℂ, ℂ) U
f : ℂ → ℂ → ℂ
c z : ℂ
n : NontrivialHolomorphicAt (f c) z
s : Set (ℂ × ℂ)
sn : s ∈ 𝓝 (c, z)
fa : AnalyticOn ℂ (uncurry f) s
e : ℝ
ep : e > 0
es : ball (c, z) e ⊆ s
r : ℝ
rp : 0 < r
rs : closedBall (c, z) r ⊆ s
fr : f c z ∉ f c '' sphere z r
fc : ContinuousOn (fun w => ‖f c w - f c z‖) (sphere z r)
x : ℂ
xs : x ∈ sphere z r
xm : IsMinOn (fun w => ‖f c w - f c z‖) (sphere z r) x
⊢ (fun p => (p.1, f p.1 p.2)) '' s ∈ 𝓝 (c, f c z)
Theorem: NontrivialHolomorphicAt.nhds_le_map_nhds_param'
Proof state before:
case intro.intro.intro.intro.intro.intro.intro
X : Type
inst✝⁶ : TopologicalSpace X
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold 𝓘(ℂ, ℂ) S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold 𝓘(ℂ, ℂ) T
U : Type
inst✝¹ : TopologicalSpace U
inst✝ : ChartedSpace ℂ U
cmu : AnalyticManifold 𝓘(ℂ, ℂ) U
f : ℂ → ℂ → ℂ
c z : ℂ
n : NontrivialHolomorphicAt (f c) z
s : Set (ℂ × ℂ)
sn : s ∈ 𝓝 (c, z)
fa : AnalyticOn ℂ (uncurry f) s
e : ℝ
ep : e > 0
es : ball (c, z) e ⊆ s
r : ℝ
rp : 0 < r
rs : closedBall (c, z) r ⊆ s
fr : f c z ∉ f c '' sphere z r
fc : ContinuousOn (fun w => ‖f c w - f c z‖) (sphere z r)
x : ℂ
xs : x ∈ sphere z r
xm : IsMinOn (fun w => ‖f c w - f c z‖) (sphere z r) x
⊢ (fun p => (p.1, f p.1 p.2)) '' s ∈ 𝓝 (c, f c z)
Tactic:
generalize he : ‖f c x - f c z‖ = e
Proof state after:
case intro.intro.intro.intro.intro.intro.intro
X : Type
inst✝⁶ : TopologicalSpace X
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold 𝓘(ℂ, ℂ) S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold 𝓘(ℂ, ℂ) T
U : Type
inst✝¹ : TopologicalSpace U
inst✝ : ChartedSpace ℂ U
cmu : AnalyticManifold 𝓘(ℂ, ℂ) U
f : ℂ → ℂ → ℂ
c z : ℂ
n : NontrivialHolomorphicAt (f c) z
s : Set (ℂ × ℂ)
sn : s ∈ 𝓝 (c, z)
fa : AnalyticOn ℂ (uncurry f) s
e✝ : ℝ
ep : e✝ > 0
es : ball (c, z) e✝ ⊆ s
r : ℝ
rp : 0 < r
rs : closedBall (c, z) r ⊆ s
fr : f c z ∉ f c '' sphere z r
fc : ContinuousOn (fun w => ‖f c w - f c z‖) (sphere z r)
x : ℂ
xs : x ∈ sphere z r
xm : IsMinOn (fun w => ‖f c w - f c z‖) (sphere z r) x
e : ℝ
he : ‖f c x - f c z‖ = e
⊢ (fun p => (p.1, f p.1 p.2)) '' s ∈ 𝓝 (c, f c z)
Theorem: NontrivialHolomorphicAt.nhds_le_map_nhds_param'
Proof state before:
case intro.intro.intro.intro.intro.intro.intro
X : Type
inst✝⁶ : TopologicalSpace X
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold 𝓘(ℂ, ℂ) S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold 𝓘(ℂ, ℂ) T
U : Type
inst✝¹ : TopologicalSpace U
inst✝ : ChartedSpace ℂ U
cmu : AnalyticManifold 𝓘(ℂ, ℂ) U
f : ℂ → ℂ → ℂ
c z : ℂ
n : NontrivialHolomorphicAt (f c) z
s : Set (ℂ × ℂ)
sn : s ∈ 𝓝 (c, z)
fa : AnalyticOn ℂ (uncurry f) s
e✝ : ℝ
ep : e✝ > 0
es : ball (c, z) e✝ ⊆ s
r : ℝ
rp : 0 < r
rs : closedBall (c, z) r ⊆ s
fr : f c z ∉ f c '' sphere z r
fc : ContinuousOn (fun w => ‖f c w - f c z‖) (sphere z r)
x : ℂ
xs : x ∈ sphere z r
xm : IsMinOn (fun w => ‖f c w - f c z‖) (sphere z r) x
e : ℝ
he : ‖f c x - f c z‖ = e
⊢ (fun p => (p.1, f p.1 p.2)) '' s ∈ 𝓝 (c, f c z)
Tactic:
have ep : 0 < e := by
  contrapose fr
  simp only [norm_pos_iff, sub_ne_zero, not_not, mem_image, ← he] at fr ⊢
  use x, xs, fr
Proof state after:
case intro.intro.intro.intro.intro.intro.intro
X : Type
inst✝⁶ : TopologicalSpace X
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold 𝓘(ℂ, ℂ) S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold 𝓘(ℂ, ℂ) T
U : Type
inst✝¹ : TopologicalSpace U
inst✝ : ChartedSpace ℂ U
cmu : AnalyticManifold 𝓘(ℂ, ℂ) U
f : ℂ → ℂ → ℂ
c z : ℂ
n : NontrivialHolomorphicAt (f c) z
s : Set (ℂ × ℂ)
sn : s ∈ 𝓝 (c, z)
fa : AnalyticOn ℂ (uncurry f) s
e✝ : ℝ
ep✝ : e✝ > 0
es : ball (c, z) e✝ ⊆ s
r : ℝ
rp : 0 < r
rs : closedBall (c, z) r ⊆ s
fr : f c z ∉ f c '' sphere z r
fc : ContinuousOn (fun w => ‖f c w - f c z‖) (sphere z r)
x : ℂ
xs : x ∈ sphere z r
xm : IsMinOn (fun w => ‖f c w - f c z‖) (sphere z r) x
e : ℝ
he : ‖f c x - f c z‖ = e
ep : 0 < e
⊢ (fun p => (p.1, f p.1 p.2)) '' s ∈ 𝓝 (c, f c z)
Theorem: NontrivialHolomorphicAt.nhds_le_map_nhds_param'
Proof state before:
case intro.intro.intro.intro.intro.intro.intro
X : Type
inst✝⁶ : TopologicalSpace X
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold 𝓘(ℂ, ℂ) S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold 𝓘(ℂ, ℂ) T
U : Type
inst✝¹ : TopologicalSpace U
inst✝ : ChartedSpace ℂ U
cmu : AnalyticManifold 𝓘(ℂ, ℂ) U
f : ℂ → ℂ → ℂ
c z : ℂ
n : NontrivialHolomorphicAt (f c) z
s : Set (ℂ × ℂ)
sn : s ∈ 𝓝 (c, z)
fa : AnalyticOn ℂ (uncurry f) s
e✝ : ℝ
ep✝ : e✝ > 0
es : ball (c, z) e✝ ⊆ s
r : ℝ
rp : 0 < r
rs : closedBall (c, z) r ⊆ s
fr : f c z ∉ f c '' sphere z r
fc : ContinuousOn (fun w => ‖f c w - f c z‖) (sphere z r)
x : ℂ
xs : x ∈ sphere z r
xm : IsMinOn (fun w => ‖f c w - f c z‖) (sphere z r) x
e : ℝ
he : ‖f c x - f c z‖ = e
ep : 0 < e
⊢ (fun p => (p.1, f p.1 p.2)) '' s ∈ 𝓝 (c, f c z)
Tactic:
rcases Metric.uniformContinuousOn_iff.mp
    ((isCompact_closedBall _ _).uniformContinuousOn_of_continuous (fa.continuousOn.mono rs))
    (e / 4) (by linarith) with
  ⟨t, tp, ft⟩
Proof state after:
case intro.intro.intro.intro.intro.intro.intro.intro.intro
X : Type
inst✝⁶ : TopologicalSpace X
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold 𝓘(ℂ, ℂ) S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold 𝓘(ℂ, ℂ) T
U : Type
inst✝¹ : TopologicalSpace U
inst✝ : ChartedSpace ℂ U
cmu : AnalyticManifold 𝓘(ℂ, ℂ) U
f : ℂ → ℂ → ℂ
c z : ℂ
n : NontrivialHolomorphicAt (f c) z
s : Set (ℂ × ℂ)
sn : s ∈ 𝓝 (c, z)
fa : AnalyticOn ℂ (uncurry f) s
e✝ : ℝ
ep✝ : e✝ > 0
es : ball (c, z) e✝ ⊆ s
r : ℝ
rp : 0 < r
rs : closedBall (c, z) r ⊆ s
fr : f c z ∉ f c '' sphere z r
fc : ContinuousOn (fun w => ‖f c w - f c z‖) (sphere z r)
x : ℂ
xs : x ∈ sphere z r
xm : IsMinOn (fun w => ‖f c w - f c z‖) (sphere z r) x
e : ℝ
he : ‖f c x - f c z‖ = e
ep : 0 < e
t : ℝ
tp : t > 0
ft : ∀ x ∈ closedBall (c, z) r, ∀ y ∈ closedBall (c, z) r, dist x y < t → dist (uncurry f x) (uncurry f y) < e / 4
⊢ (fun p => (p.1, f p.1 p.2)) '' s ∈ 𝓝 (c, f c z)
Theorem: NontrivialHolomorphicAt.nhds_le_map_nhds_param'
Proof state before:
case intro.intro.intro.intro.intro.intro.intro.intro.intro
X : Type
inst✝⁶ : TopologicalSpace X
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold 𝓘(ℂ, ℂ) S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold 𝓘(ℂ, ℂ) T
U : Type
inst✝¹ : TopologicalSpace U
inst✝ : ChartedSpace ℂ U
cmu : AnalyticManifold 𝓘(ℂ, ℂ) U
f : ℂ → ℂ → ℂ
c z : ℂ
n : NontrivialHolomorphicAt (f c) z
s : Set (ℂ × ℂ)
sn : s ∈ 𝓝 (c, z)
fa : AnalyticOn ℂ (uncurry f) s
e✝ : ℝ
ep✝ : e✝ > 0
es : ball (c, z) e✝ ⊆ s
r : ℝ
rp : 0 < r
rs : closedBall (c, z) r ⊆ s
fr : f c z ∉ f c '' sphere z r
fc : ContinuousOn (fun w => ‖f c w - f c z‖) (sphere z r)
x : ℂ
xs : x ∈ sphere z r
xm : IsMinOn (fun w => ‖f c w - f c z‖) (sphere z r) x
e : ℝ
he : ‖f c x - f c z‖ = e
ep : 0 < e
t : ℝ
tp : t > 0
ft : ∀ x ∈ closedBall (c, z) r, ∀ y ∈ closedBall (c, z) r, dist x y < t → dist (uncurry f x) (uncurry f y) < e / 4
ef : ∀ d ∈ ball c (min t r), ∀ w ∈ sphere z r, e / 2 ≤ ‖f d w - f d z‖
⊢ (fun p => (p.1, f p.1 p.2)) '' s ∈ 𝓝 (c, f c z)
Tactic:
have ss : ball c (min t r) ×ˢ closedBall z r ⊆ s := by
  refine _root_.trans ?_ rs; rw [← closedBall_prod_same]; apply prod_mono_left
  exact _root_.trans (Metric.ball_subset_ball (min_le_right _ _)) Metric.ball_subset_closedBall
Proof state after:
case intro.intro.intro.intro.intro.intro.intro.intro.intro
X : Type
inst✝⁶ : TopologicalSpace X
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold 𝓘(ℂ, ℂ) S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold 𝓘(ℂ, ℂ) T
U : Type
inst✝¹ : TopologicalSpace U
inst✝ : ChartedSpace ℂ U
cmu : AnalyticManifold 𝓘(ℂ, ℂ) U
f : ℂ → ℂ → ℂ
c z : ℂ
n : NontrivialHolomorphicAt (f c) z
s : Set (ℂ × ℂ)
sn : s ∈ 𝓝 (c, z)
fa : AnalyticOn ℂ (uncurry f) s
e✝ : ℝ
ep✝ : e✝ > 0
es : ball (c, z) e✝ ⊆ s
r : ℝ
rp : 0 < r
rs : closedBall (c, z) r ⊆ s
fr : f c z ∉ f c '' sphere z r
fc : ContinuousOn (fun w => ‖f c w - f c z‖) (sphere z r)
x : ℂ
xs : x ∈ sphere z r
xm : IsMinOn (fun w => ‖f c w - f c z‖) (sphere z r) x
e : ℝ
he : ‖f c x - f c z‖ = e
ep : 0 < e
t : ℝ
tp : t > 0
ft : ∀ x ∈ closedBall (c, z) r, ∀ y ∈ closedBall (c, z) r, dist x y < t → dist (uncurry f x) (uncurry f y) < e / 4
ef : ∀ d ∈ ball c (min t r), ∀ w ∈ sphere z r, e / 2 ≤ ‖f d w - f d z‖
ss : ball c (min t r) ×ˢ closedBall z r ⊆ s
⊢ (fun p => (p.1, f p.1 p.2)) '' s ∈ 𝓝 (c, f c z)
Theorem: NontrivialHolomorphicAt.nhds_le_map_nhds_param'
Proof state before:
case intro.intro.intro.intro.intro.intro.intro.intro.intro
X : Type
inst✝⁶ : TopologicalSpace X
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold 𝓘(ℂ, ℂ) S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold 𝓘(ℂ, ℂ) T
U : Type
inst✝¹ : TopologicalSpace U
inst✝ : ChartedSpace ℂ U
cmu : AnalyticManifold 𝓘(ℂ, ℂ) U
f : ℂ → ℂ → ℂ
c z : ℂ
n : NontrivialHolomorphicAt (f c) z
s : Set (ℂ × ℂ)
sn : s ∈ 𝓝 (c, z)
fa : AnalyticOn ℂ (uncurry f) s
e✝ : ℝ
ep✝ : e✝ > 0
es : ball (c, z) e✝ ⊆ s
r : ℝ
rp : 0 < r
rs : closedBall (c, z) r ⊆ s
fr : f c z ∉ f c '' sphere z r
fc : ContinuousOn (fun w => ‖f c w - f c z‖) (sphere z r)
x : ℂ
xs : x ∈ sphere z r
xm : IsMinOn (fun w => ‖f c w - f c z‖) (sphere z r) x
e : ℝ
he : ‖f c x - f c z‖ = e
ep : 0 < e
t : ℝ
tp : t > 0
ft : ∀ x ∈ closedBall (c, z) r, ∀ y ∈ closedBall (c, z) r, dist x y < t → dist (uncurry f x) (uncurry f y) < e / 4
ef : ∀ d ∈ ball c (min t r), ∀ w ∈ sphere z r, e / 2 ≤ ‖f d w - f d z‖
ss : ball c (min t r) ×ˢ closedBall z r ⊆ s
⊢ (fun p => (p.1, f p.1 p.2)) '' s ∈ 𝓝 (c, f c z)
Tactic:
exact Filter.mem_of_superset ((fa.mono ss).ball_subset_image_closedBall_param rp (half_pos ep)
  (Metric.ball_mem_nhds _ (by bound)) ef) (image_subset _ ss)
Proof state after:
no goals
Theorem: NontrivialHolomorphicAt.nhds_le_map_nhds_param'
Proof state before:
X : Type
inst✝⁶ : TopologicalSpace X
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold 𝓘(ℂ, ℂ) S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold 𝓘(ℂ, ℂ) T
U : Type
inst✝¹ : TopologicalSpace U
inst✝ : ChartedSpace ℂ U
cmu : AnalyticManifold 𝓘(ℂ, ℂ) U
f : ℂ → ℂ → ℂ
c z : ℂ
n : NontrivialHolomorphicAt (f c) z
fa : AnalyticAt ℂ (uncurry f) (c, z)
s' : Set (ℂ × ℂ)
sn : s' ∈ 𝓝 (c, z)
s : Set (ℂ × ℂ)
hs : s' ∩ {p | AnalyticAt ℂ (uncurry f) p} = s
⊢ s ⊆ s'
Tactic:
rw [← hs]
Proof state after:
X : Type
inst✝⁶ : TopologicalSpace X
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold 𝓘(ℂ, ℂ) S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold 𝓘(ℂ, ℂ) T
U : Type
inst✝¹ : TopologicalSpace U
inst✝ : ChartedSpace ℂ U
cmu : AnalyticManifold 𝓘(ℂ, ℂ) U
f : ℂ → ℂ → ℂ
c z : ℂ
n : NontrivialHolomorphicAt (f c) z
fa : AnalyticAt ℂ (uncurry f) (c, z)
s' : Set (ℂ × ℂ)
sn : s' ∈ 𝓝 (c, z)
s : Set (ℂ × ℂ)
hs : s' ∩ {p | AnalyticAt ℂ (uncurry f) p} = s
⊢ s' ∩ {p | AnalyticAt ℂ (uncurry f) p} ⊆ s'
Theorem: NontrivialHolomorphicAt.nhds_le_map_nhds_param'
Proof state before:
X : Type
inst✝⁶ : TopologicalSpace X
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold 𝓘(ℂ, ℂ) S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold 𝓘(ℂ, ℂ) T
U : Type
inst✝¹ : TopologicalSpace U
inst✝ : ChartedSpace ℂ U
cmu : AnalyticManifold 𝓘(ℂ, ℂ) U
f : ℂ → ℂ → ℂ
c z : ℂ
n : NontrivialHolomorphicAt (f c) z
fa : AnalyticAt ℂ (uncurry f) (c, z)
s' : Set (ℂ × ℂ)
sn : s' ∈ 𝓝 (c, z)
s : Set (ℂ × ℂ)
hs : s' ∩ {p | AnalyticAt ℂ (uncurry f) p} = s
⊢ s' ∩ {p | AnalyticAt ℂ (uncurry f) p} ⊆ s'
Tactic:
apply inter_subset_left
Proof state after:
no goals
Theorem: NontrivialHolomorphicAt.nhds_le_map_nhds_param'
Proof state before:
X : Type
inst✝⁶ : TopologicalSpace X
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold 𝓘(ℂ, ℂ) S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold 𝓘(ℂ, ℂ) T
U : Type
inst✝¹ : TopologicalSpace U
inst✝ : ChartedSpace ℂ U
cmu : AnalyticManifold 𝓘(ℂ, ℂ) U
f : ℂ → ℂ → ℂ
c z : ℂ
n : NontrivialHolomorphicAt (f c) z
fa : AnalyticAt ℂ (uncurry f) (c, z)
s' : Set (ℂ × ℂ)
sn : s' ∈ 𝓝 (c, z)
s : Set (ℂ × ℂ)
hs : s' ∩ {p | AnalyticAt ℂ (uncurry f) p} = s
ss : s ⊆ s'
⊢ s ∈ 𝓝 (c, z)
Tactic:
rw [← hs]
Proof state after:
X : Type
inst✝⁶ : TopologicalSpace X
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold 𝓘(ℂ, ℂ) S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold 𝓘(ℂ, ℂ) T
U : Type
inst✝¹ : TopologicalSpace U
inst✝ : ChartedSpace ℂ U
cmu : AnalyticManifold 𝓘(ℂ, ℂ) U
f : ℂ → ℂ → ℂ
c z : ℂ
n : NontrivialHolomorphicAt (f c) z
fa : AnalyticAt ℂ (uncurry f) (c, z)
s' : Set (ℂ × ℂ)
sn : s' ∈ 𝓝 (c, z)
s : Set (ℂ × ℂ)
hs : s' ∩ {p | AnalyticAt ℂ (uncurry f) p} = s
ss : s ⊆ s'
⊢ s' ∩ {p | AnalyticAt ℂ (uncurry f) p} ∈ 𝓝 (c, z)
Theorem: NontrivialHolomorphicAt.nhds_le_map_nhds_param'
Proof state before:
X : Type
inst✝⁶ : TopologicalSpace X
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold 𝓘(ℂ, ℂ) S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold 𝓘(ℂ, ℂ) T
U : Type
inst✝¹ : TopologicalSpace U
inst✝ : ChartedSpace ℂ U
cmu : AnalyticManifold 𝓘(ℂ, ℂ) U
f : ℂ → ℂ → ℂ
c z : ℂ
n : NontrivialHolomorphicAt (f c) z
fa : AnalyticAt ℂ (uncurry f) (c, z)
s' : Set (ℂ × ℂ)
sn : s' ∈ 𝓝 (c, z)
s : Set (ℂ × ℂ)
hs : s' ∩ {p | AnalyticAt ℂ (uncurry f) p} = s
ss : s ⊆ s'
⊢ s' ∩ {p | AnalyticAt ℂ (uncurry f) p} ∈ 𝓝 (c, z)
Tactic:
exact Filter.inter_mem sn fa.eventually_analyticAt
Proof state after:
no goals
Theorem: NontrivialHolomorphicAt.nhds_le_map_nhds_param'
Proof state before:
X : Type
inst✝⁶ : TopologicalSpace X
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold 𝓘(ℂ, ℂ) S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold 𝓘(ℂ, ℂ) T
U : Type
inst✝¹ : TopologicalSpace U
inst✝ : ChartedSpace ℂ U
cmu : AnalyticManifold 𝓘(ℂ, ℂ) U
f : ℂ → ℂ → ℂ
c z : ℂ
n : NontrivialHolomorphicAt (f c) z
fa : AnalyticAt ℂ (uncurry f) (c, z)
s' s : Set (ℂ × ℂ)
hs : s' ∩ {p | AnalyticAt ℂ (uncurry f) p} = s
ss : s ⊆ s'
sn : s ∈ 𝓝 (c, z)
⊢ AnalyticOn ℂ (uncurry f) s
Tactic:
rw [← hs]
Proof state after:
X : Type
inst✝⁶ : TopologicalSpace X
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold 𝓘(ℂ, ℂ) S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold 𝓘(ℂ, ℂ) T
U : Type
inst✝¹ : TopologicalSpace U
inst✝ : ChartedSpace ℂ U
cmu : AnalyticManifold 𝓘(ℂ, ℂ) U
f : ℂ → ℂ → ℂ
c z : ℂ
n : NontrivialHolomorphicAt (f c) z
fa : AnalyticAt ℂ (uncurry f) (c, z)
s' s : Set (ℂ × ℂ)
hs : s' ∩ {p | AnalyticAt ℂ (uncurry f) p} = s
ss : s ⊆ s'
sn : s ∈ 𝓝 (c, z)
⊢ AnalyticOn ℂ (uncurry f) (s' ∩ {p | AnalyticAt ℂ (uncurry f) p})
Theorem: NontrivialHolomorphicAt.nhds_le_map_nhds_param'
Proof state before:
X : Type
inst✝⁶ : TopologicalSpace X
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold 𝓘(ℂ, ℂ) S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold 𝓘(ℂ, ℂ) T
U : Type
inst✝¹ : TopologicalSpace U
inst✝ : ChartedSpace ℂ U
cmu : AnalyticManifold 𝓘(ℂ, ℂ) U
f : ℂ → ℂ → ℂ
c z : ℂ
n : NontrivialHolomorphicAt (f c) z
fa : AnalyticAt ℂ (uncurry f) (c, z)
s' s : Set (ℂ × ℂ)
hs : s' ∩ {p | AnalyticAt ℂ (uncurry f) p} = s
ss : s ⊆ s'
sn : s ∈ 𝓝 (c, z)
⊢ AnalyticOn ℂ (uncurry f) (s' ∩ {p | AnalyticAt ℂ (uncurry f) p})
Tactic:
apply inter_subset_right
Proof state after:
no goals
Theorem: NontrivialHolomorphicAt.nhds_le_map_nhds_param'
Proof state before:
X : Type
inst✝⁶ : TopologicalSpace X
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold 𝓘(ℂ, ℂ) S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold 𝓘(ℂ, ℂ) T
U : Type
inst✝¹ : TopologicalSpace U
inst✝ : ChartedSpace ℂ U
cmu : AnalyticManifold 𝓘(ℂ, ℂ) U
f : ℂ → ℂ → ℂ
c z : ℂ
n : NontrivialHolomorphicAt (f c) z
s : Set (ℂ × ℂ)
sn : s ∈ 𝓝 (c, z)
fa : AnalyticOn ℂ (uncurry f) s
e : ℝ
ep : e > 0
es : ball (c, z) e ⊆ s
⊢ ∃ r, 0 < r ∧ closedBall (c, z) r ⊆ s ∧ f c z ∉ f c '' sphere z r
Tactic:
have h := n.eventually_ne
Proof state after:
X : Type
inst✝⁶ : TopologicalSpace X
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold 𝓘(ℂ, ℂ) S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold 𝓘(ℂ, ℂ) T
U : Type
inst✝¹ : TopologicalSpace U
inst✝ : ChartedSpace ℂ U
cmu : AnalyticManifold 𝓘(ℂ, ℂ) U
f : ℂ → ℂ → ℂ
c z : ℂ
n : NontrivialHolomorphicAt (f c) z
s : Set (ℂ × ℂ)
sn : s ∈ 𝓝 (c, z)
fa : AnalyticOn ℂ (uncurry f) s
e : ℝ
ep : e > 0
es : ball (c, z) e ⊆ s
h : ∀ᶠ (w : ℂ) in 𝓝 z, w ≠ z → f c w ≠ f c z
⊢ ∃ r, 0 < r ∧ closedBall (c, z) r ⊆ s ∧ f c z ∉ f c '' sphere z r
Theorem: NontrivialHolomorphicAt.nhds_le_map_nhds_param'
Proof state before:
X : Type
inst✝⁶ : TopologicalSpace X
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold 𝓘(ℂ, ℂ) S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold 𝓘(ℂ, ℂ) T
U : Type
inst✝¹ : TopologicalSpace U
inst✝ : ChartedSpace ℂ U
cmu : AnalyticManifold 𝓘(ℂ, ℂ) U
f : ℂ → ℂ → ℂ
c z : ℂ
n : NontrivialHolomorphicAt (f c) z
s : Set (ℂ × ℂ)
sn : s ∈ 𝓝 (c, z)
fa : AnalyticOn ℂ (uncurry f) s
e : ℝ
ep : e > 0
es : ball (c, z) e ⊆ s
h : ∀ᶠ (w : ℂ) in 𝓝 z, w ≠ z → f c w ≠ f c z
⊢ ∃ r, 0 < r ∧ closedBall (c, z) r ⊆ s ∧ f c z ∉ f c '' sphere z r
Tactic:
contrapose h
Proof state after:
X : Type
inst✝⁶ : TopologicalSpace X
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold 𝓘(ℂ, ℂ) S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold 𝓘(ℂ, ℂ) T
U : Type
inst✝¹ : TopologicalSpace U
inst✝ : ChartedSpace ℂ U
cmu : AnalyticManifold 𝓘(ℂ, ℂ) U
f : ℂ → ℂ → ℂ
c z : ℂ
n : NontrivialHolomorphicAt (f c) z
s : Set (ℂ × ℂ)
sn : s ∈ 𝓝 (c, z)
fa : AnalyticOn ℂ (uncurry f) s
e : ℝ
ep : e > 0
es : ball (c, z) e ⊆ s
h : ¬∃ r, 0 < r ∧ closedBall (c, z) r ⊆ s ∧ f c z ∉ f c '' sphere z r
⊢ ¬∀ᶠ (w : ℂ) in 𝓝 z, w ≠ z → f c w ≠ f c z
Theorem: NontrivialHolomorphicAt.nhds_le_map_nhds_param'
Proof state before:
X : Type
inst✝⁶ : TopologicalSpace X
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold 𝓘(ℂ, ℂ) S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold 𝓘(ℂ, ℂ) T
U : Type
inst✝¹ : TopologicalSpace U
inst✝ : ChartedSpace ℂ U
cmu : AnalyticManifold 𝓘(ℂ, ℂ) U
f : ℂ → ℂ → ℂ
c z : ℂ
n : NontrivialHolomorphicAt (f c) z
s : Set (ℂ × ℂ)
sn : s ∈ 𝓝 (c, z)
fa : AnalyticOn ℂ (uncurry f) s
e : ℝ
ep : e > 0
es : ball (c, z) e ⊆ s
h : ¬∃ r, 0 < r ∧ closedBall (c, z) r ⊆ s ∧ f c z ∉ f c '' sphere z r
⊢ ¬∀ᶠ (w : ℂ) in 𝓝 z, w ≠ z → f c w ≠ f c z
Tactic:
simp only [not_exists, Filter.not_frequently, not_not, not_and, not_exists] at h
Proof state after:
X : Type
inst✝⁶ : TopologicalSpace X
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold 𝓘(ℂ, ℂ) S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold 𝓘(ℂ, ℂ) T
U : Type
inst✝¹ : TopologicalSpace U
inst✝ : ChartedSpace ℂ U
cmu : AnalyticManifold 𝓘(ℂ, ℂ) U
f : ℂ → ℂ → ℂ
c z : ℂ
n : NontrivialHolomorphicAt (f c) z
s : Set (ℂ × ℂ)
sn : s ∈ 𝓝 (c, z)
fa : AnalyticOn ℂ (uncurry f) s
e : ℝ
ep : e > 0
es : ball (c, z) e ⊆ s
h : ∀ (x : ℝ), 0 < x → closedBall (c, z) x ⊆ s → f c z ∈ f c '' sphere z x
⊢ ¬∀ᶠ (w : ℂ) in 𝓝 z, w ≠ z → f c w ≠ f c z