url stringclasses 147
values | commit stringclasses 147
values | file_path stringlengths 7 101 | full_name stringlengths 1 94 | start stringlengths 6 10 | end stringlengths 6 11 | tactic stringlengths 1 11.2k | state_before stringlengths 3 2.09M | state_after stringlengths 6 2.09M |
|---|---|---|---|---|---|---|---|---|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.continuous_lift' | [529, 1] | [535, 49] | apply ContinuousAt.continuousWithinAt | X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : ℂ → ℂ
g : X → ℂ → ℂ
y : 𝕊
x✝ : X
z✝ : ℂ
gc : Continuous (uncurry g)
gi : ∀ (x : X), Tendsto (uncurry g) ((𝓝 x).prod atInf) atInf
x : X
z : 𝕊
a✝ : (x, z) ∈ univ
⊢ ContinuousWithinAt (uncurry (lift' g ∞)) univ (x, z) | case h
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : ℂ → ℂ
g : X → ℂ → ℂ
y : 𝕊
x✝ : X
z✝ : ℂ
gc : Continuous (uncurry g)
gi : ∀ (x : X), Tendsto (uncurry g) ((𝓝 x).prod atInf) atInf
x : X
z : 𝕊
a✝ : (x, z) ∈ univ
⊢ ContinuousAt (uncurry (lift' g ∞)) (x, z) |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.continuous_lift' | [529, 1] | [535, 49] | induction z using OnePoint.rec | case h
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : ℂ → ℂ
g : X → ℂ → ℂ
y : 𝕊
x✝ : X
z✝ : ℂ
gc : Continuous (uncurry g)
gi : ∀ (x : X), Tendsto (uncurry g) ((𝓝 x).prod atInf) atInf
x : X
z : 𝕊
a✝ : (x, z) ∈ univ
⊢ ContinuousAt (uncurry (lift' g ∞)) (x, z) | case h.h₁
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : ℂ → ℂ
g : X → ℂ → ℂ
y : 𝕊
x✝ : X
z : ℂ
gc : Continuous (uncurry g)
gi : ∀ (x : X), Tendsto (uncurry g) ((𝓝 x).prod atInf) atInf
x : X
a✝ : (x, ∞) ∈ univ
⊢ ContinuousAt (uncurry (lift' g ∞)) (x, ∞)
case h.h₂
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : ℂ → ℂ
g : X → ℂ → ℂ
y : 𝕊
x✝¹ : X
z : ℂ
gc : Continuous (uncurry g)
gi : ∀ (x : X), Tendsto (uncurry g) ((𝓝 x).prod atInf) atInf
x : X
x✝ : ℂ
a✝ : (x, ↑x✝) ∈ univ
⊢ ContinuousAt (uncurry (lift' g ∞)) (x, ↑x✝) |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.continuous_lift' | [529, 1] | [535, 49] | exact continuousAt_lift_inf' (gi x) | case h.h₁
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : ℂ → ℂ
g : X → ℂ → ℂ
y : 𝕊
x✝ : X
z : ℂ
gc : Continuous (uncurry g)
gi : ∀ (x : X), Tendsto (uncurry g) ((𝓝 x).prod atInf) atInf
x : X
a✝ : (x, ∞) ∈ univ
⊢ ContinuousAt (uncurry (lift' g ∞)) (x, ∞) | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.continuous_lift' | [529, 1] | [535, 49] | exact continuousAt_lift_coe' gc.continuousAt | case h.h₂
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : ℂ → ℂ
g : X → ℂ → ℂ
y : 𝕊
x✝¹ : X
z : ℂ
gc : Continuous (uncurry g)
gi : ∀ (x : X), Tendsto (uncurry g) ((𝓝 x).prod atInf) atInf
x : X
x✝ : ℂ
a✝ : (x, ↑x✝) ∈ univ
⊢ ContinuousAt (uncurry (lift' g ∞)) (x, ↑x✝) | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.continuousAt_lift_coe | [538, 1] | [541, 90] | refine ContinuousAt.comp fc ?_ | X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : ℂ → ℂ
g : X → ℂ → ℂ
y : 𝕊
x : X
z : ℂ
fc : ContinuousAt f z
⊢ ContinuousAt (uncurry fun x => f) ((), z) | X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : ℂ → ℂ
g : X → ℂ → ℂ
y : 𝕊
x : X
z : ℂ
fc : ContinuousAt f z
⊢ ContinuousAt (fun a => a.2) ((), z) |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.continuousAt_lift_coe | [538, 1] | [541, 90] | exact continuousAt_snd | X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : ℂ → ℂ
g : X → ℂ → ℂ
y : 𝕊
x : X
z : ℂ
fc : ContinuousAt f z
⊢ ContinuousAt (fun a => a.2) ((), z) | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.continuous_lift | [550, 1] | [554, 48] | rw [continuous_iff_continuousAt] | X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : ℂ → ℂ
g : X → ℂ → ℂ
y : 𝕊
x : X
z : ℂ
fc : Continuous f
fi : Tendsto f atInf atInf
⊢ Continuous (lift f ∞) | X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : ℂ → ℂ
g : X → ℂ → ℂ
y : 𝕊
x : X
z : ℂ
fc : Continuous f
fi : Tendsto f atInf atInf
⊢ ∀ (x : 𝕊), ContinuousAt (lift f ∞) x |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.continuous_lift | [550, 1] | [554, 48] | intro z | X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : ℂ → ℂ
g : X → ℂ → ℂ
y : 𝕊
x : X
z : ℂ
fc : Continuous f
fi : Tendsto f atInf atInf
⊢ ∀ (x : 𝕊), ContinuousAt (lift f ∞) x | X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : ℂ → ℂ
g : X → ℂ → ℂ
y : 𝕊
x : X
z✝ : ℂ
fc : Continuous f
fi : Tendsto f atInf atInf
z : 𝕊
⊢ ContinuousAt (lift f ∞) z |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.continuous_lift | [550, 1] | [554, 48] | induction z using OnePoint.rec | X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : ℂ → ℂ
g : X → ℂ → ℂ
y : 𝕊
x : X
z✝ : ℂ
fc : Continuous f
fi : Tendsto f atInf atInf
z : 𝕊
⊢ ContinuousAt (lift f ∞) z | case h₁
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : ℂ → ℂ
g : X → ℂ → ℂ
y : 𝕊
x : X
z : ℂ
fc : Continuous f
fi : Tendsto f atInf atInf
⊢ ContinuousAt (lift f ∞) ∞
case h₂
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : ℂ → ℂ
g : X → ℂ → ℂ
y : 𝕊
x : X
z : ℂ
fc : Continuous f
fi : Tendsto f atInf atInf
x✝ : ℂ
⊢ ContinuousAt (lift f ∞) ↑x✝ |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.continuous_lift | [550, 1] | [554, 48] | exact continuousAt_lift_inf fi | case h₁
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : ℂ → ℂ
g : X → ℂ → ℂ
y : 𝕊
x : X
z : ℂ
fc : Continuous f
fi : Tendsto f atInf atInf
⊢ ContinuousAt (lift f ∞) ∞ | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.continuous_lift | [550, 1] | [554, 48] | exact continuousAt_lift_coe fc.continuousAt | case h₂
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : ℂ → ℂ
g : X → ℂ → ℂ
y : 𝕊
x : X
z : ℂ
fc : Continuous f
fi : Tendsto f atInf atInf
x✝ : ℂ
⊢ ContinuousAt (lift f ∞) ↑x✝ | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.holomorphicAt_lift_coe | [557, 1] | [558, 100] | rw [lift_eq_fill] | X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : ℂ → ℂ
g : X → ℂ → ℂ
y : 𝕊
x : X
z : ℂ
fa : AnalyticAt ℂ f z
⊢ HolomorphicAt I I (lift f y) ↑z | X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : ℂ → ℂ
g : X → ℂ → ℂ
y : 𝕊
x : X
z : ℂ
fa : AnalyticAt ℂ f z
⊢ HolomorphicAt I I (fill (fun z => ↑(f z)) y) ↑z |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.holomorphicAt_lift_coe | [557, 1] | [558, 100] | exact holomorphicAt_fill_coe ((holomorphic_coe _).comp (fa.holomorphicAt I I)) | X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : ℂ → ℂ
g : X → ℂ → ℂ
y : 𝕊
x : X
z : ℂ
fa : AnalyticAt ℂ f z
⊢ HolomorphicAt I I (fill (fun z => ↑(f z)) y) ↑z | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.holomorphicAt_lift_inf | [561, 1] | [565, 32] | rw [lift_eq_fill] | X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : ℂ → ℂ
g : X → ℂ → ℂ
y : 𝕊
x : X
z : ℂ
fa : ∀ᶠ (z : ℂ) in atInf, AnalyticAt ℂ f z
fi : Tendsto f atInf atInf
⊢ HolomorphicAt I I (lift f ∞) ∞ | X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : ℂ → ℂ
g : X → ℂ → ℂ
y : 𝕊
x : X
z : ℂ
fa : ∀ᶠ (z : ℂ) in atInf, AnalyticAt ℂ f z
fi : Tendsto f atInf atInf
⊢ HolomorphicAt I I (fill (fun z => ↑(f z)) ∞) ∞ |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.holomorphicAt_lift_inf | [561, 1] | [565, 32] | apply holomorphicAt_fill_inf | X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : ℂ → ℂ
g : X → ℂ → ℂ
y : 𝕊
x : X
z : ℂ
fa : ∀ᶠ (z : ℂ) in atInf, AnalyticAt ℂ f z
fi : Tendsto f atInf atInf
⊢ HolomorphicAt I I (fill (fun z => ↑(f z)) ∞) ∞ | case fa
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : ℂ → ℂ
g : X → ℂ → ℂ
y : 𝕊
x : X
z : ℂ
fa : ∀ᶠ (z : ℂ) in atInf, AnalyticAt ℂ f z
fi : Tendsto f atInf atInf
⊢ ∀ᶠ (z : ℂ) in atInf, HolomorphicAt I I (fun z => ↑(f z)) z
case fi
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : ℂ → ℂ
g : X → ℂ → ℂ
y : 𝕊
x : X
z : ℂ
fa : ∀ᶠ (z : ℂ) in atInf, AnalyticAt ℂ f z
fi : Tendsto f atInf atInf
⊢ Tendsto (fun z => ↑(f z)) atInf (𝓝 ∞) |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.holomorphicAt_lift_inf | [561, 1] | [565, 32] | exact fa.mp (eventually_of_forall fun z fa ↦ (holomorphic_coe _).comp (fa.holomorphicAt I I)) | case fa
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : ℂ → ℂ
g : X → ℂ → ℂ
y : 𝕊
x : X
z : ℂ
fa : ∀ᶠ (z : ℂ) in atInf, AnalyticAt ℂ f z
fi : Tendsto f atInf atInf
⊢ ∀ᶠ (z : ℂ) in atInf, HolomorphicAt I I (fun z => ↑(f z)) z
case fi
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : ℂ → ℂ
g : X → ℂ → ℂ
y : 𝕊
x : X
z : ℂ
fa : ∀ᶠ (z : ℂ) in atInf, AnalyticAt ℂ f z
fi : Tendsto f atInf atInf
⊢ Tendsto (fun z => ↑(f z)) atInf (𝓝 ∞) | case fi
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : ℂ → ℂ
g : X → ℂ → ℂ
y : 𝕊
x : X
z : ℂ
fa : ∀ᶠ (z : ℂ) in atInf, AnalyticAt ℂ f z
fi : Tendsto f atInf atInf
⊢ Tendsto (fun z => ↑(f z)) atInf (𝓝 ∞) |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.holomorphicAt_lift_inf | [561, 1] | [565, 32] | exact coe_tendsto_inf.comp fi | case fi
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : ℂ → ℂ
g : X → ℂ → ℂ
y : 𝕊
x : X
z : ℂ
fa : ∀ᶠ (z : ℂ) in atInf, AnalyticAt ℂ f z
fi : Tendsto f atInf atInf
⊢ Tendsto (fun z => ↑(f z)) atInf (𝓝 ∞) | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.holomorphic_lift | [568, 1] | [572, 53] | intro z | X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : ℂ → ℂ
g : X → ℂ → ℂ
y : 𝕊
x : X
z : ℂ
fa : AnalyticOn ℂ f univ
fi : Tendsto f atInf atInf
⊢ Holomorphic I I (lift f ∞) | X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : ℂ → ℂ
g : X → ℂ → ℂ
y : 𝕊
x : X
z✝ : ℂ
fa : AnalyticOn ℂ f univ
fi : Tendsto f atInf atInf
z : 𝕊
⊢ HolomorphicAt I I (lift f ∞) z |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.holomorphic_lift | [568, 1] | [572, 53] | induction z using OnePoint.rec | X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : ℂ → ℂ
g : X → ℂ → ℂ
y : 𝕊
x : X
z✝ : ℂ
fa : AnalyticOn ℂ f univ
fi : Tendsto f atInf atInf
z : 𝕊
⊢ HolomorphicAt I I (lift f ∞) z | case h₁
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : ℂ → ℂ
g : X → ℂ → ℂ
y : 𝕊
x : X
z : ℂ
fa : AnalyticOn ℂ f univ
fi : Tendsto f atInf atInf
⊢ HolomorphicAt I I (lift f ∞) ∞
case h₂
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : ℂ → ℂ
g : X → ℂ → ℂ
y : 𝕊
x : X
z : ℂ
fa : AnalyticOn ℂ f univ
fi : Tendsto f atInf atInf
x✝ : ℂ
⊢ HolomorphicAt I I (lift f ∞) ↑x✝ |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.holomorphic_lift | [568, 1] | [572, 53] | exact holomorphicAt_lift_inf (eventually_of_forall fun z ↦ fa z (mem_univ _)) fi | case h₁
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : ℂ → ℂ
g : X → ℂ → ℂ
y : 𝕊
x : X
z : ℂ
fa : AnalyticOn ℂ f univ
fi : Tendsto f atInf atInf
⊢ HolomorphicAt I I (lift f ∞) ∞ | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.holomorphic_lift | [568, 1] | [572, 53] | exact holomorphicAt_lift_coe (fa _ (mem_univ _)) | case h₂
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : ℂ → ℂ
g : X → ℂ → ℂ
y : 𝕊
x : X
z : ℂ
fa : AnalyticOn ℂ f univ
fi : Tendsto f atInf atInf
x✝ : ℂ
⊢ HolomorphicAt I I (lift f ∞) ↑x✝ | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.holomorphicLift' | [575, 1] | [585, 69] | apply osgoodManifold (continuous_lift' fa.continuous fi) | X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f✝ : ℂ → ℂ
g : X → ℂ → ℂ
y : 𝕊
x : X
z : ℂ
f : ℂ → ℂ → ℂ
fa : AnalyticOn ℂ (uncurry f) univ
fi : ∀ (x : ℂ), Tendsto (uncurry f) ((𝓝 x).prod atInf) atInf
⊢ Holomorphic (I.prod I) I (uncurry (lift' f ∞)) | case f0
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f✝ : ℂ → ℂ
g : X → ℂ → ℂ
y : 𝕊
x : X
z : ℂ
f : ℂ → ℂ → ℂ
fa : AnalyticOn ℂ (uncurry f) univ
fi : ∀ (x : ℂ), Tendsto (uncurry f) ((𝓝 x).prod atInf) atInf
⊢ ∀ (x : ℂ) (y : 𝕊), HolomorphicAt I I (fun x => uncurry (lift' f ∞) (x, y)) x
case f1
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f✝ : ℂ → ℂ
g : X → ℂ → ℂ
y : 𝕊
x : X
z : ℂ
f : ℂ → ℂ → ℂ
fa : AnalyticOn ℂ (uncurry f) univ
fi : ∀ (x : ℂ), Tendsto (uncurry f) ((𝓝 x).prod atInf) atInf
⊢ ∀ (x : ℂ) (y : 𝕊), HolomorphicAt I I (fun y => uncurry (lift' f ∞) (x, y)) y |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.holomorphicLift' | [575, 1] | [585, 69] | intro x z | case f0
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f✝ : ℂ → ℂ
g : X → ℂ → ℂ
y : 𝕊
x : X
z : ℂ
f : ℂ → ℂ → ℂ
fa : AnalyticOn ℂ (uncurry f) univ
fi : ∀ (x : ℂ), Tendsto (uncurry f) ((𝓝 x).prod atInf) atInf
⊢ ∀ (x : ℂ) (y : 𝕊), HolomorphicAt I I (fun x => uncurry (lift' f ∞) (x, y)) x | case f0
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f✝ : ℂ → ℂ
g : X → ℂ → ℂ
y : 𝕊
x✝ : X
z✝ : ℂ
f : ℂ → ℂ → ℂ
fa : AnalyticOn ℂ (uncurry f) univ
fi : ∀ (x : ℂ), Tendsto (uncurry f) ((𝓝 x).prod atInf) atInf
x : ℂ
z : 𝕊
⊢ HolomorphicAt I I (fun x => uncurry (lift' f ∞) (x, z)) x |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.holomorphicLift' | [575, 1] | [585, 69] | induction z using OnePoint.rec | case f0
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f✝ : ℂ → ℂ
g : X → ℂ → ℂ
y : 𝕊
x✝ : X
z✝ : ℂ
f : ℂ → ℂ → ℂ
fa : AnalyticOn ℂ (uncurry f) univ
fi : ∀ (x : ℂ), Tendsto (uncurry f) ((𝓝 x).prod atInf) atInf
x : ℂ
z : 𝕊
⊢ HolomorphicAt I I (fun x => uncurry (lift' f ∞) (x, z)) x | case f0.h₁
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f✝ : ℂ → ℂ
g : X → ℂ → ℂ
y : 𝕊
x✝ : X
z : ℂ
f : ℂ → ℂ → ℂ
fa : AnalyticOn ℂ (uncurry f) univ
fi : ∀ (x : ℂ), Tendsto (uncurry f) ((𝓝 x).prod atInf) atInf
x : ℂ
⊢ HolomorphicAt I I (fun x => uncurry (lift' f ∞) (x, ∞)) x
case f0.h₂
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f✝ : ℂ → ℂ
g : X → ℂ → ℂ
y : 𝕊
x✝¹ : X
z : ℂ
f : ℂ → ℂ → ℂ
fa : AnalyticOn ℂ (uncurry f) univ
fi : ∀ (x : ℂ), Tendsto (uncurry f) ((𝓝 x).prod atInf) atInf
x x✝ : ℂ
⊢ HolomorphicAt I I (fun x => uncurry (lift' f ∞) (x, ↑x✝)) x |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.holomorphicLift' | [575, 1] | [585, 69] | simp only [uncurry, lift_inf'] | case f0.h₁
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f✝ : ℂ → ℂ
g : X → ℂ → ℂ
y : 𝕊
x✝ : X
z : ℂ
f : ℂ → ℂ → ℂ
fa : AnalyticOn ℂ (uncurry f) univ
fi : ∀ (x : ℂ), Tendsto (uncurry f) ((𝓝 x).prod atInf) atInf
x : ℂ
⊢ HolomorphicAt I I (fun x => uncurry (lift' f ∞) (x, ∞)) x | case f0.h₁
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f✝ : ℂ → ℂ
g : X → ℂ → ℂ
y : 𝕊
x✝ : X
z : ℂ
f : ℂ → ℂ → ℂ
fa : AnalyticOn ℂ (uncurry f) univ
fi : ∀ (x : ℂ), Tendsto (uncurry f) ((𝓝 x).prod atInf) atInf
x : ℂ
⊢ HolomorphicAt I I (fun x => ∞) x |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.holomorphicLift' | [575, 1] | [585, 69] | exact holomorphicAt_const | case f0.h₁
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f✝ : ℂ → ℂ
g : X → ℂ → ℂ
y : 𝕊
x✝ : X
z : ℂ
f : ℂ → ℂ → ℂ
fa : AnalyticOn ℂ (uncurry f) univ
fi : ∀ (x : ℂ), Tendsto (uncurry f) ((𝓝 x).prod atInf) atInf
x : ℂ
⊢ HolomorphicAt I I (fun x => ∞) x | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.holomorphicLift' | [575, 1] | [585, 69] | exact (holomorphic_coe _).comp ((fa _ (mem_univ ⟨_,_⟩)).along_fst.holomorphicAt _ _) | case f0.h₂
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f✝ : ℂ → ℂ
g : X → ℂ → ℂ
y : 𝕊
x✝¹ : X
z : ℂ
f : ℂ → ℂ → ℂ
fa : AnalyticOn ℂ (uncurry f) univ
fi : ∀ (x : ℂ), Tendsto (uncurry f) ((𝓝 x).prod atInf) atInf
x x✝ : ℂ
⊢ HolomorphicAt I I (fun x => uncurry (lift' f ∞) (x, ↑x✝)) x | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.holomorphicLift' | [575, 1] | [585, 69] | intro x z | case f1
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f✝ : ℂ → ℂ
g : X → ℂ → ℂ
y : 𝕊
x : X
z : ℂ
f : ℂ → ℂ → ℂ
fa : AnalyticOn ℂ (uncurry f) univ
fi : ∀ (x : ℂ), Tendsto (uncurry f) ((𝓝 x).prod atInf) atInf
⊢ ∀ (x : ℂ) (y : 𝕊), HolomorphicAt I I (fun y => uncurry (lift' f ∞) (x, y)) y | case f1
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f✝ : ℂ → ℂ
g : X → ℂ → ℂ
y : 𝕊
x✝ : X
z✝ : ℂ
f : ℂ → ℂ → ℂ
fa : AnalyticOn ℂ (uncurry f) univ
fi : ∀ (x : ℂ), Tendsto (uncurry f) ((𝓝 x).prod atInf) atInf
x : ℂ
z : 𝕊
⊢ HolomorphicAt I I (fun y => uncurry (lift' f ∞) (x, y)) z |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.holomorphicLift' | [575, 1] | [585, 69] | exact holomorphic_lift (fun _ _ ↦ (fa _ (mem_univ ⟨_,_⟩)).along_snd)
((fi x).comp (tendsto_const_nhds.prod_mk Filter.tendsto_id)) z | case f1
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f✝ : ℂ → ℂ
g : X → ℂ → ℂ
y : 𝕊
x✝ : X
z✝ : ℂ
f : ℂ → ℂ → ℂ
fa : AnalyticOn ℂ (uncurry f) univ
fi : ∀ (x : ℂ), Tendsto (uncurry f) ((𝓝 x).prod atInf) atInf
x : ℂ
z : 𝕊
⊢ HolomorphicAt I I (fun y => uncurry (lift' f ∞) (x, y)) z | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/LocalInj.lean | HolomorphicAt.local_inj | [24, 1] | [35, 61] | rcases complex_inverse_fun' fa nc with ⟨g, ga, gf, fg⟩ | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : S → T
z : S
fa : HolomorphicAt I I f z
nc : mfderiv I I f z ≠ 0
⊢ ∀ᶠ (p : S × S) in 𝓝 (z, z), f p.1 = f p.2 → p.1 = p.2 | case intro.intro.intro
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : S → T
z : S
fa : HolomorphicAt I I f z
nc : mfderiv I I f z ≠ 0
g : T → S
ga : HolomorphicAt I I g (f z)
gf : ∀ᶠ (x : S) in 𝓝 z, g (f x) = x
fg : ∀ᶠ (x : T) in 𝓝 (f z), f (g x) = x
⊢ ∀ᶠ (p : S × S) in 𝓝 (z, z), f p.1 = f p.2 → p.1 = p.2 |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/LocalInj.lean | HolomorphicAt.local_inj | [24, 1] | [35, 61] | have n : NontrivialHolomorphicAt g (f z) := by
rw [← gf.self_of_nhds] at fa
refine (NontrivialHolomorphicAt.anti ?_ fa ga).2
exact (nontrivialHolomorphicAt_id _).congr (Filter.EventuallyEq.symm fg) | case intro.intro.intro
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : S → T
z : S
fa : HolomorphicAt I I f z
nc : mfderiv I I f z ≠ 0
g : T → S
ga : HolomorphicAt I I g (f z)
gf : ∀ᶠ (x : S) in 𝓝 z, g (f x) = x
fg : ∀ᶠ (x : T) in 𝓝 (f z), f (g x) = x
⊢ ∀ᶠ (p : S × S) in 𝓝 (z, z), f p.1 = f p.2 → p.1 = p.2 | case intro.intro.intro
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : S → T
z : S
fa : HolomorphicAt I I f z
nc : mfderiv I I f z ≠ 0
g : T → S
ga : HolomorphicAt I I g (f z)
gf : ∀ᶠ (x : S) in 𝓝 z, g (f x) = x
fg : ∀ᶠ (x : T) in 𝓝 (f z), f (g x) = x
n : NontrivialHolomorphicAt g (f z)
⊢ ∀ᶠ (p : S × S) in 𝓝 (z, z), f p.1 = f p.2 → p.1 = p.2 |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/LocalInj.lean | HolomorphicAt.local_inj | [24, 1] | [35, 61] | have o := n.nhds_eq_map_nhds | case intro.intro.intro
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : S → T
z : S
fa : HolomorphicAt I I f z
nc : mfderiv I I f z ≠ 0
g : T → S
ga : HolomorphicAt I I g (f z)
gf : ∀ᶠ (x : S) in 𝓝 z, g (f x) = x
fg : ∀ᶠ (x : T) in 𝓝 (f z), f (g x) = x
n : NontrivialHolomorphicAt g (f z)
⊢ ∀ᶠ (p : S × S) in 𝓝 (z, z), f p.1 = f p.2 → p.1 = p.2 | case intro.intro.intro
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : S → T
z : S
fa : HolomorphicAt I I f z
nc : mfderiv I I f z ≠ 0
g : T → S
ga : HolomorphicAt I I g (f z)
gf : ∀ᶠ (x : S) in 𝓝 z, g (f x) = x
fg : ∀ᶠ (x : T) in 𝓝 (f z), f (g x) = x
n : NontrivialHolomorphicAt g (f z)
o : 𝓝 (g (f z)) = Filter.map g (𝓝 (f z))
⊢ ∀ᶠ (p : S × S) in 𝓝 (z, z), f p.1 = f p.2 → p.1 = p.2 |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/LocalInj.lean | HolomorphicAt.local_inj | [24, 1] | [35, 61] | rw [gf.self_of_nhds] at o | case intro.intro.intro
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : S → T
z : S
fa : HolomorphicAt I I f z
nc : mfderiv I I f z ≠ 0
g : T → S
ga : HolomorphicAt I I g (f z)
gf : ∀ᶠ (x : S) in 𝓝 z, g (f x) = x
fg : ∀ᶠ (x : T) in 𝓝 (f z), f (g x) = x
n : NontrivialHolomorphicAt g (f z)
o : 𝓝 (g (f z)) = Filter.map g (𝓝 (f z))
⊢ ∀ᶠ (p : S × S) in 𝓝 (z, z), f p.1 = f p.2 → p.1 = p.2 | case intro.intro.intro
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : S → T
z : S
fa : HolomorphicAt I I f z
nc : mfderiv I I f z ≠ 0
g : T → S
ga : HolomorphicAt I I g (f z)
gf : ∀ᶠ (x : S) in 𝓝 z, g (f x) = x
fg : ∀ᶠ (x : T) in 𝓝 (f z), f (g x) = x
n : NontrivialHolomorphicAt g (f z)
o : 𝓝 z = Filter.map g (𝓝 (f z))
⊢ ∀ᶠ (p : S × S) in 𝓝 (z, z), f p.1 = f p.2 → p.1 = p.2 |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/LocalInj.lean | HolomorphicAt.local_inj | [24, 1] | [35, 61] | simp only [nhds_prod_eq, o, Filter.prod_map_map_eq, Filter.eventually_map] | case intro.intro.intro
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : S → T
z : S
fa : HolomorphicAt I I f z
nc : mfderiv I I f z ≠ 0
g : T → S
ga : HolomorphicAt I I g (f z)
gf : ∀ᶠ (x : S) in 𝓝 z, g (f x) = x
fg : ∀ᶠ (x : T) in 𝓝 (f z), f (g x) = x
n : NontrivialHolomorphicAt g (f z)
o : 𝓝 z = Filter.map g (𝓝 (f z))
⊢ ∀ᶠ (p : S × S) in 𝓝 (z, z), f p.1 = f p.2 → p.1 = p.2 | case intro.intro.intro
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : S → T
z : S
fa : HolomorphicAt I I f z
nc : mfderiv I I f z ≠ 0
g : T → S
ga : HolomorphicAt I I g (f z)
gf : ∀ᶠ (x : S) in 𝓝 z, g (f x) = x
fg : ∀ᶠ (x : T) in 𝓝 (f z), f (g x) = x
n : NontrivialHolomorphicAt g (f z)
o : 𝓝 z = Filter.map g (𝓝 (f z))
⊢ ∀ᶠ (a : T × T) in 𝓝 (f z) ×ˢ 𝓝 (f z), f (g a.1) = f (g a.2) → g a.1 = g a.2 |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/LocalInj.lean | HolomorphicAt.local_inj | [24, 1] | [35, 61] | refine (fg.prod_mk fg).mp (eventually_of_forall ?_) | case intro.intro.intro
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : S → T
z : S
fa : HolomorphicAt I I f z
nc : mfderiv I I f z ≠ 0
g : T → S
ga : HolomorphicAt I I g (f z)
gf : ∀ᶠ (x : S) in 𝓝 z, g (f x) = x
fg : ∀ᶠ (x : T) in 𝓝 (f z), f (g x) = x
n : NontrivialHolomorphicAt g (f z)
o : 𝓝 z = Filter.map g (𝓝 (f z))
⊢ ∀ᶠ (a : T × T) in 𝓝 (f z) ×ˢ 𝓝 (f z), f (g a.1) = f (g a.2) → g a.1 = g a.2 | case intro.intro.intro
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : S → T
z : S
fa : HolomorphicAt I I f z
nc : mfderiv I I f z ≠ 0
g : T → S
ga : HolomorphicAt I I g (f z)
gf : ∀ᶠ (x : S) in 𝓝 z, g (f x) = x
fg : ∀ᶠ (x : T) in 𝓝 (f z), f (g x) = x
n : NontrivialHolomorphicAt g (f z)
o : 𝓝 z = Filter.map g (𝓝 (f z))
⊢ ∀ (x : T × T), f (g x.1) = x.1 ∧ f (g x.2) = x.2 → f (g x.1) = f (g x.2) → g x.1 = g x.2 |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/LocalInj.lean | HolomorphicAt.local_inj | [24, 1] | [35, 61] | intro ⟨x, y⟩ ⟨ex, ey⟩ h | case intro.intro.intro
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : S → T
z : S
fa : HolomorphicAt I I f z
nc : mfderiv I I f z ≠ 0
g : T → S
ga : HolomorphicAt I I g (f z)
gf : ∀ᶠ (x : S) in 𝓝 z, g (f x) = x
fg : ∀ᶠ (x : T) in 𝓝 (f z), f (g x) = x
n : NontrivialHolomorphicAt g (f z)
o : 𝓝 z = Filter.map g (𝓝 (f z))
⊢ ∀ (x : T × T), f (g x.1) = x.1 ∧ f (g x.2) = x.2 → f (g x.1) = f (g x.2) → g x.1 = g x.2 | case intro.intro.intro
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : S → T
z : S
fa : HolomorphicAt I I f z
nc : mfderiv I I f z ≠ 0
g : T → S
ga : HolomorphicAt I I g (f z)
gf : ∀ᶠ (x : S) in 𝓝 z, g (f x) = x
fg : ∀ᶠ (x : T) in 𝓝 (f z), f (g x) = x
n : NontrivialHolomorphicAt g (f z)
o : 𝓝 z = Filter.map g (𝓝 (f z))
x y : T
ex : f (g (x, y).1) = (x, y).1
ey : f (g (x, y).2) = (x, y).2
h : f (g (x, y).1) = f (g (x, y).2)
⊢ g (x, y).1 = g (x, y).2 |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/LocalInj.lean | HolomorphicAt.local_inj | [24, 1] | [35, 61] | simp only at ex ey | case intro.intro.intro
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : S → T
z : S
fa : HolomorphicAt I I f z
nc : mfderiv I I f z ≠ 0
g : T → S
ga : HolomorphicAt I I g (f z)
gf : ∀ᶠ (x : S) in 𝓝 z, g (f x) = x
fg : ∀ᶠ (x : T) in 𝓝 (f z), f (g x) = x
n : NontrivialHolomorphicAt g (f z)
o : 𝓝 z = Filter.map g (𝓝 (f z))
x y : T
ex : f (g (x, y).1) = (x, y).1
ey : f (g (x, y).2) = (x, y).2
h : f (g (x, y).1) = f (g (x, y).2)
⊢ g (x, y).1 = g (x, y).2 | case intro.intro.intro
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : S → T
z : S
fa : HolomorphicAt I I f z
nc : mfderiv I I f z ≠ 0
g : T → S
ga : HolomorphicAt I I g (f z)
gf : ∀ᶠ (x : S) in 𝓝 z, g (f x) = x
fg : ∀ᶠ (x : T) in 𝓝 (f z), f (g x) = x
n : NontrivialHolomorphicAt g (f z)
o : 𝓝 z = Filter.map g (𝓝 (f z))
x y : T
ex : f (g x) = x
ey : f (g y) = y
h : f (g (x, y).1) = f (g (x, y).2)
⊢ g (x, y).1 = g (x, y).2 |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/LocalInj.lean | HolomorphicAt.local_inj | [24, 1] | [35, 61] | simp only [ex, ey] at h | case intro.intro.intro
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : S → T
z : S
fa : HolomorphicAt I I f z
nc : mfderiv I I f z ≠ 0
g : T → S
ga : HolomorphicAt I I g (f z)
gf : ∀ᶠ (x : S) in 𝓝 z, g (f x) = x
fg : ∀ᶠ (x : T) in 𝓝 (f z), f (g x) = x
n : NontrivialHolomorphicAt g (f z)
o : 𝓝 z = Filter.map g (𝓝 (f z))
x y : T
ex : f (g x) = x
ey : f (g y) = y
h : f (g (x, y).1) = f (g (x, y).2)
⊢ g (x, y).1 = g (x, y).2 | case intro.intro.intro
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : S → T
z : S
fa : HolomorphicAt I I f z
nc : mfderiv I I f z ≠ 0
g : T → S
ga : HolomorphicAt I I g (f z)
gf : ∀ᶠ (x : S) in 𝓝 z, g (f x) = x
fg : ∀ᶠ (x : T) in 𝓝 (f z), f (g x) = x
n : NontrivialHolomorphicAt g (f z)
o : 𝓝 z = Filter.map g (𝓝 (f z))
x y : T
ex : f (g x) = x
ey : f (g y) = y
h : x = y
⊢ g (x, y).1 = g (x, y).2 |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/LocalInj.lean | HolomorphicAt.local_inj | [24, 1] | [35, 61] | simp only [h] | case intro.intro.intro
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : S → T
z : S
fa : HolomorphicAt I I f z
nc : mfderiv I I f z ≠ 0
g : T → S
ga : HolomorphicAt I I g (f z)
gf : ∀ᶠ (x : S) in 𝓝 z, g (f x) = x
fg : ∀ᶠ (x : T) in 𝓝 (f z), f (g x) = x
n : NontrivialHolomorphicAt g (f z)
o : 𝓝 z = Filter.map g (𝓝 (f z))
x y : T
ex : f (g x) = x
ey : f (g y) = y
h : x = y
⊢ g (x, y).1 = g (x, y).2 | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/LocalInj.lean | HolomorphicAt.local_inj | [24, 1] | [35, 61] | rw [← gf.self_of_nhds] at fa | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : S → T
z : S
fa : HolomorphicAt I I f z
nc : mfderiv I I f z ≠ 0
g : T → S
ga : HolomorphicAt I I g (f z)
gf : ∀ᶠ (x : S) in 𝓝 z, g (f x) = x
fg : ∀ᶠ (x : T) in 𝓝 (f z), f (g x) = x
⊢ NontrivialHolomorphicAt g (f z) | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : S → T
z : S
nc : mfderiv I I f z ≠ 0
g : T → S
fa : HolomorphicAt I I f (g (f z))
ga : HolomorphicAt I I g (f z)
gf : ∀ᶠ (x : S) in 𝓝 z, g (f x) = x
fg : ∀ᶠ (x : T) in 𝓝 (f z), f (g x) = x
⊢ NontrivialHolomorphicAt g (f z) |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/LocalInj.lean | HolomorphicAt.local_inj | [24, 1] | [35, 61] | refine (NontrivialHolomorphicAt.anti ?_ fa ga).2 | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : S → T
z : S
nc : mfderiv I I f z ≠ 0
g : T → S
fa : HolomorphicAt I I f (g (f z))
ga : HolomorphicAt I I g (f z)
gf : ∀ᶠ (x : S) in 𝓝 z, g (f x) = x
fg : ∀ᶠ (x : T) in 𝓝 (f z), f (g x) = x
⊢ NontrivialHolomorphicAt g (f z) | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : S → T
z : S
nc : mfderiv I I f z ≠ 0
g : T → S
fa : HolomorphicAt I I f (g (f z))
ga : HolomorphicAt I I g (f z)
gf : ∀ᶠ (x : S) in 𝓝 z, g (f x) = x
fg : ∀ᶠ (x : T) in 𝓝 (f z), f (g x) = x
⊢ NontrivialHolomorphicAt (fun z => f (g z)) (f z) |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/LocalInj.lean | HolomorphicAt.local_inj | [24, 1] | [35, 61] | exact (nontrivialHolomorphicAt_id _).congr (Filter.EventuallyEq.symm fg) | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : S → T
z : S
nc : mfderiv I I f z ≠ 0
g : T → S
fa : HolomorphicAt I I f (g (f z))
ga : HolomorphicAt I I g (f z)
gf : ∀ᶠ (x : S) in 𝓝 z, g (f x) = x
fg : ∀ᶠ (x : T) in 𝓝 (f z), f (g x) = x
⊢ NontrivialHolomorphicAt (fun z => f (g z)) (f z) | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/LocalInj.lean | HolomorphicAt.local_inj'' | [39, 1] | [54, 90] | rcases complex_inverse_fun fa nc with ⟨g, ga, gf, fg⟩ | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
nc : mfderiv I I (f c) z ≠ 0
⊢ ∀ᶠ (p : (ℂ × S) × ℂ × S) in 𝓝 ((c, z), c, z), p.1.1 = p.2.1 → f p.1.1 p.1.2 = f p.2.1 p.2.2 → p.1 = p.2 | case intro.intro.intro
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
nc : mfderiv I I (f c) z ≠ 0
g : ℂ → T → S
ga : HolomorphicAt (I.prod I) I (uncurry g) (c, f c z)
gf : ∀ᶠ (x : ℂ × S) in 𝓝 (c, z), g x.1 (f x.1 x.2) = x.2
fg : ∀ᶠ (x : ℂ × T) in 𝓝 (c, f c z), f x.1 (g x.1 x.2) = x.2
⊢ ∀ᶠ (p : (ℂ × S) × ℂ × S) in 𝓝 ((c, z), c, z), p.1.1 = p.2.1 → f p.1.1 p.1.2 = f p.2.1 p.2.2 → p.1 = p.2 |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/LocalInj.lean | HolomorphicAt.local_inj'' | [39, 1] | [54, 90] | have n : NontrivialHolomorphicAt (g c) (f c z) := by
have e : (c, z) = (c, g c (f c z)) := by rw [gf.self_of_nhds]
rw [e] at fa
refine (NontrivialHolomorphicAt.anti ?_ fa.along_snd ga.along_snd).2
refine (nontrivialHolomorphicAt_id _).congr ?_
refine ((continuousAt_const.prod continuousAt_id).eventually fg).mp (eventually_of_forall ?_)
exact fun _ e ↦ e.symm | case intro.intro.intro
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
nc : mfderiv I I (f c) z ≠ 0
g : ℂ → T → S
ga : HolomorphicAt (I.prod I) I (uncurry g) (c, f c z)
gf : ∀ᶠ (x : ℂ × S) in 𝓝 (c, z), g x.1 (f x.1 x.2) = x.2
fg : ∀ᶠ (x : ℂ × T) in 𝓝 (c, f c z), f x.1 (g x.1 x.2) = x.2
⊢ ∀ᶠ (p : (ℂ × S) × ℂ × S) in 𝓝 ((c, z), c, z), p.1.1 = p.2.1 → f p.1.1 p.1.2 = f p.2.1 p.2.2 → p.1 = p.2 | case intro.intro.intro
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
nc : mfderiv I I (f c) z ≠ 0
g : ℂ → T → S
ga : HolomorphicAt (I.prod I) I (uncurry g) (c, f c z)
gf : ∀ᶠ (x : ℂ × S) in 𝓝 (c, z), g x.1 (f x.1 x.2) = x.2
fg : ∀ᶠ (x : ℂ × T) in 𝓝 (c, f c z), f x.1 (g x.1 x.2) = x.2
n : NontrivialHolomorphicAt (g c) (f c z)
⊢ ∀ᶠ (p : (ℂ × S) × ℂ × S) in 𝓝 ((c, z), c, z), p.1.1 = p.2.1 → f p.1.1 p.1.2 = f p.2.1 p.2.2 → p.1 = p.2 |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/LocalInj.lean | HolomorphicAt.local_inj'' | [39, 1] | [54, 90] | have o := n.nhds_eq_map_nhds_param ga | case intro.intro.intro
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
nc : mfderiv I I (f c) z ≠ 0
g : ℂ → T → S
ga : HolomorphicAt (I.prod I) I (uncurry g) (c, f c z)
gf : ∀ᶠ (x : ℂ × S) in 𝓝 (c, z), g x.1 (f x.1 x.2) = x.2
fg : ∀ᶠ (x : ℂ × T) in 𝓝 (c, f c z), f x.1 (g x.1 x.2) = x.2
n : NontrivialHolomorphicAt (g c) (f c z)
⊢ ∀ᶠ (p : (ℂ × S) × ℂ × S) in 𝓝 ((c, z), c, z), p.1.1 = p.2.1 → f p.1.1 p.1.2 = f p.2.1 p.2.2 → p.1 = p.2 | case intro.intro.intro
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
nc : mfderiv I I (f c) z ≠ 0
g : ℂ → T → S
ga : HolomorphicAt (I.prod I) I (uncurry g) (c, f c z)
gf : ∀ᶠ (x : ℂ × S) in 𝓝 (c, z), g x.1 (f x.1 x.2) = x.2
fg : ∀ᶠ (x : ℂ × T) in 𝓝 (c, f c z), f x.1 (g x.1 x.2) = x.2
n : NontrivialHolomorphicAt (g c) (f c z)
o : 𝓝 (c, g c (f c z)) = Filter.map (fun p => (p.1, g p.1 p.2)) (𝓝 (c, f c z))
⊢ ∀ᶠ (p : (ℂ × S) × ℂ × S) in 𝓝 ((c, z), c, z), p.1.1 = p.2.1 → f p.1.1 p.1.2 = f p.2.1 p.2.2 → p.1 = p.2 |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/LocalInj.lean | HolomorphicAt.local_inj'' | [39, 1] | [54, 90] | rw [gf.self_of_nhds] at o | case intro.intro.intro
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
nc : mfderiv I I (f c) z ≠ 0
g : ℂ → T → S
ga : HolomorphicAt (I.prod I) I (uncurry g) (c, f c z)
gf : ∀ᶠ (x : ℂ × S) in 𝓝 (c, z), g x.1 (f x.1 x.2) = x.2
fg : ∀ᶠ (x : ℂ × T) in 𝓝 (c, f c z), f x.1 (g x.1 x.2) = x.2
n : NontrivialHolomorphicAt (g c) (f c z)
o : 𝓝 (c, g c (f c z)) = Filter.map (fun p => (p.1, g p.1 p.2)) (𝓝 (c, f c z))
⊢ ∀ᶠ (p : (ℂ × S) × ℂ × S) in 𝓝 ((c, z), c, z), p.1.1 = p.2.1 → f p.1.1 p.1.2 = f p.2.1 p.2.2 → p.1 = p.2 | case intro.intro.intro
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
nc : mfderiv I I (f c) z ≠ 0
g : ℂ → T → S
ga : HolomorphicAt (I.prod I) I (uncurry g) (c, f c z)
gf : ∀ᶠ (x : ℂ × S) in 𝓝 (c, z), g x.1 (f x.1 x.2) = x.2
fg : ∀ᶠ (x : ℂ × T) in 𝓝 (c, f c z), f x.1 (g x.1 x.2) = x.2
n : NontrivialHolomorphicAt (g c) (f c z)
o : 𝓝 (c, (c, z).2) = Filter.map (fun p => (p.1, g p.1 p.2)) (𝓝 (c, f c z))
⊢ ∀ᶠ (p : (ℂ × S) × ℂ × S) in 𝓝 ((c, z), c, z), p.1.1 = p.2.1 → f p.1.1 p.1.2 = f p.2.1 p.2.2 → p.1 = p.2 |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/LocalInj.lean | HolomorphicAt.local_inj'' | [39, 1] | [54, 90] | simp only at o | case intro.intro.intro
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
nc : mfderiv I I (f c) z ≠ 0
g : ℂ → T → S
ga : HolomorphicAt (I.prod I) I (uncurry g) (c, f c z)
gf : ∀ᶠ (x : ℂ × S) in 𝓝 (c, z), g x.1 (f x.1 x.2) = x.2
fg : ∀ᶠ (x : ℂ × T) in 𝓝 (c, f c z), f x.1 (g x.1 x.2) = x.2
n : NontrivialHolomorphicAt (g c) (f c z)
o : 𝓝 (c, (c, z).2) = Filter.map (fun p => (p.1, g p.1 p.2)) (𝓝 (c, f c z))
⊢ ∀ᶠ (p : (ℂ × S) × ℂ × S) in 𝓝 ((c, z), c, z), p.1.1 = p.2.1 → f p.1.1 p.1.2 = f p.2.1 p.2.2 → p.1 = p.2 | case intro.intro.intro
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
nc : mfderiv I I (f c) z ≠ 0
g : ℂ → T → S
ga : HolomorphicAt (I.prod I) I (uncurry g) (c, f c z)
gf : ∀ᶠ (x : ℂ × S) in 𝓝 (c, z), g x.1 (f x.1 x.2) = x.2
fg : ∀ᶠ (x : ℂ × T) in 𝓝 (c, f c z), f x.1 (g x.1 x.2) = x.2
n : NontrivialHolomorphicAt (g c) (f c z)
o : 𝓝 (c, z) = Filter.map (fun p => (p.1, g p.1 p.2)) (𝓝 (c, f c z))
⊢ ∀ᶠ (p : (ℂ × S) × ℂ × S) in 𝓝 ((c, z), c, z), p.1.1 = p.2.1 → f p.1.1 p.1.2 = f p.2.1 p.2.2 → p.1 = p.2 |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/LocalInj.lean | HolomorphicAt.local_inj'' | [39, 1] | [54, 90] | rw [nhds_prod_eq, o] | case intro.intro.intro
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
nc : mfderiv I I (f c) z ≠ 0
g : ℂ → T → S
ga : HolomorphicAt (I.prod I) I (uncurry g) (c, f c z)
gf : ∀ᶠ (x : ℂ × S) in 𝓝 (c, z), g x.1 (f x.1 x.2) = x.2
fg : ∀ᶠ (x : ℂ × T) in 𝓝 (c, f c z), f x.1 (g x.1 x.2) = x.2
n : NontrivialHolomorphicAt (g c) (f c z)
o : 𝓝 (c, z) = Filter.map (fun p => (p.1, g p.1 p.2)) (𝓝 (c, f c z))
⊢ ∀ᶠ (p : (ℂ × S) × ℂ × S) in 𝓝 ((c, z), c, z), p.1.1 = p.2.1 → f p.1.1 p.1.2 = f p.2.1 p.2.2 → p.1 = p.2 | case intro.intro.intro
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
nc : mfderiv I I (f c) z ≠ 0
g : ℂ → T → S
ga : HolomorphicAt (I.prod I) I (uncurry g) (c, f c z)
gf : ∀ᶠ (x : ℂ × S) in 𝓝 (c, z), g x.1 (f x.1 x.2) = x.2
fg : ∀ᶠ (x : ℂ × T) in 𝓝 (c, f c z), f x.1 (g x.1 x.2) = x.2
n : NontrivialHolomorphicAt (g c) (f c z)
o : 𝓝 (c, z) = Filter.map (fun p => (p.1, g p.1 p.2)) (𝓝 (c, f c z))
⊢ ∀ᶠ (p : (ℂ × S) × ℂ × S) in
Filter.map (fun p => (p.1, g p.1 p.2)) (𝓝 (c, f c z)) ×ˢ Filter.map (fun p => (p.1, g p.1 p.2)) (𝓝 (c, f c z)),
p.1.1 = p.2.1 → f p.1.1 p.1.2 = f p.2.1 p.2.2 → p.1 = p.2 |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/LocalInj.lean | HolomorphicAt.local_inj'' | [39, 1] | [54, 90] | simp only [Filter.prod_map_map_eq, Filter.eventually_map] | case intro.intro.intro
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
nc : mfderiv I I (f c) z ≠ 0
g : ℂ → T → S
ga : HolomorphicAt (I.prod I) I (uncurry g) (c, f c z)
gf : ∀ᶠ (x : ℂ × S) in 𝓝 (c, z), g x.1 (f x.1 x.2) = x.2
fg : ∀ᶠ (x : ℂ × T) in 𝓝 (c, f c z), f x.1 (g x.1 x.2) = x.2
n : NontrivialHolomorphicAt (g c) (f c z)
o : 𝓝 (c, z) = Filter.map (fun p => (p.1, g p.1 p.2)) (𝓝 (c, f c z))
⊢ ∀ᶠ (p : (ℂ × S) × ℂ × S) in
Filter.map (fun p => (p.1, g p.1 p.2)) (𝓝 (c, f c z)) ×ˢ Filter.map (fun p => (p.1, g p.1 p.2)) (𝓝 (c, f c z)),
p.1.1 = p.2.1 → f p.1.1 p.1.2 = f p.2.1 p.2.2 → p.1 = p.2 | case intro.intro.intro
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
nc : mfderiv I I (f c) z ≠ 0
g : ℂ → T → S
ga : HolomorphicAt (I.prod I) I (uncurry g) (c, f c z)
gf : ∀ᶠ (x : ℂ × S) in 𝓝 (c, z), g x.1 (f x.1 x.2) = x.2
fg : ∀ᶠ (x : ℂ × T) in 𝓝 (c, f c z), f x.1 (g x.1 x.2) = x.2
n : NontrivialHolomorphicAt (g c) (f c z)
o : 𝓝 (c, z) = Filter.map (fun p => (p.1, g p.1 p.2)) (𝓝 (c, f c z))
⊢ ∀ᶠ (a : (ℂ × T) × ℂ × T) in 𝓝 (c, f c z) ×ˢ 𝓝 (c, f c z),
a.1.1 = a.2.1 → f a.1.1 (g a.1.1 a.1.2) = f a.2.1 (g a.2.1 a.2.2) → (a.1.1, g a.1.1 a.1.2) = (a.2.1, g a.2.1 a.2.2) |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/LocalInj.lean | HolomorphicAt.local_inj'' | [39, 1] | [54, 90] | refine (fg.prod_mk fg).mp (eventually_of_forall ?_) | case intro.intro.intro
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
nc : mfderiv I I (f c) z ≠ 0
g : ℂ → T → S
ga : HolomorphicAt (I.prod I) I (uncurry g) (c, f c z)
gf : ∀ᶠ (x : ℂ × S) in 𝓝 (c, z), g x.1 (f x.1 x.2) = x.2
fg : ∀ᶠ (x : ℂ × T) in 𝓝 (c, f c z), f x.1 (g x.1 x.2) = x.2
n : NontrivialHolomorphicAt (g c) (f c z)
o : 𝓝 (c, z) = Filter.map (fun p => (p.1, g p.1 p.2)) (𝓝 (c, f c z))
⊢ ∀ᶠ (a : (ℂ × T) × ℂ × T) in 𝓝 (c, f c z) ×ˢ 𝓝 (c, f c z),
a.1.1 = a.2.1 → f a.1.1 (g a.1.1 a.1.2) = f a.2.1 (g a.2.1 a.2.2) → (a.1.1, g a.1.1 a.1.2) = (a.2.1, g a.2.1 a.2.2) | case intro.intro.intro
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
nc : mfderiv I I (f c) z ≠ 0
g : ℂ → T → S
ga : HolomorphicAt (I.prod I) I (uncurry g) (c, f c z)
gf : ∀ᶠ (x : ℂ × S) in 𝓝 (c, z), g x.1 (f x.1 x.2) = x.2
fg : ∀ᶠ (x : ℂ × T) in 𝓝 (c, f c z), f x.1 (g x.1 x.2) = x.2
n : NontrivialHolomorphicAt (g c) (f c z)
o : 𝓝 (c, z) = Filter.map (fun p => (p.1, g p.1 p.2)) (𝓝 (c, f c z))
⊢ ∀ (x : (ℂ × T) × ℂ × T),
f x.1.1 (g x.1.1 x.1.2) = x.1.2 ∧ f x.2.1 (g x.2.1 x.2.2) = x.2.2 →
x.1.1 = x.2.1 →
f x.1.1 (g x.1.1 x.1.2) = f x.2.1 (g x.2.1 x.2.2) → (x.1.1, g x.1.1 x.1.2) = (x.2.1, g x.2.1 x.2.2) |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/LocalInj.lean | HolomorphicAt.local_inj'' | [39, 1] | [54, 90] | intro ⟨x, y⟩ ⟨ex, ey⟩ h1 h2 | case intro.intro.intro
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
nc : mfderiv I I (f c) z ≠ 0
g : ℂ → T → S
ga : HolomorphicAt (I.prod I) I (uncurry g) (c, f c z)
gf : ∀ᶠ (x : ℂ × S) in 𝓝 (c, z), g x.1 (f x.1 x.2) = x.2
fg : ∀ᶠ (x : ℂ × T) in 𝓝 (c, f c z), f x.1 (g x.1 x.2) = x.2
n : NontrivialHolomorphicAt (g c) (f c z)
o : 𝓝 (c, z) = Filter.map (fun p => (p.1, g p.1 p.2)) (𝓝 (c, f c z))
⊢ ∀ (x : (ℂ × T) × ℂ × T),
f x.1.1 (g x.1.1 x.1.2) = x.1.2 ∧ f x.2.1 (g x.2.1 x.2.2) = x.2.2 →
x.1.1 = x.2.1 →
f x.1.1 (g x.1.1 x.1.2) = f x.2.1 (g x.2.1 x.2.2) → (x.1.1, g x.1.1 x.1.2) = (x.2.1, g x.2.1 x.2.2) | case intro.intro.intro
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
nc : mfderiv I I (f c) z ≠ 0
g : ℂ → T → S
ga : HolomorphicAt (I.prod I) I (uncurry g) (c, f c z)
gf : ∀ᶠ (x : ℂ × S) in 𝓝 (c, z), g x.1 (f x.1 x.2) = x.2
fg : ∀ᶠ (x : ℂ × T) in 𝓝 (c, f c z), f x.1 (g x.1 x.2) = x.2
n : NontrivialHolomorphicAt (g c) (f c z)
o : 𝓝 (c, z) = Filter.map (fun p => (p.1, g p.1 p.2)) (𝓝 (c, f c z))
x y : ℂ × T
ex : f (x, y).1.1 (g (x, y).1.1 (x, y).1.2) = (x, y).1.2
ey : f (x, y).2.1 (g (x, y).2.1 (x, y).2.2) = (x, y).2.2
h1 : (x, y).1.1 = (x, y).2.1
h2 : f (x, y).1.1 (g (x, y).1.1 (x, y).1.2) = f (x, y).2.1 (g (x, y).2.1 (x, y).2.2)
⊢ ((x, y).1.1, g (x, y).1.1 (x, y).1.2) = ((x, y).2.1, g (x, y).2.1 (x, y).2.2) |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/LocalInj.lean | HolomorphicAt.local_inj'' | [39, 1] | [54, 90] | simp only at h1 | case intro.intro.intro
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
nc : mfderiv I I (f c) z ≠ 0
g : ℂ → T → S
ga : HolomorphicAt (I.prod I) I (uncurry g) (c, f c z)
gf : ∀ᶠ (x : ℂ × S) in 𝓝 (c, z), g x.1 (f x.1 x.2) = x.2
fg : ∀ᶠ (x : ℂ × T) in 𝓝 (c, f c z), f x.1 (g x.1 x.2) = x.2
n : NontrivialHolomorphicAt (g c) (f c z)
o : 𝓝 (c, z) = Filter.map (fun p => (p.1, g p.1 p.2)) (𝓝 (c, f c z))
x y : ℂ × T
ex : f (x, y).1.1 (g (x, y).1.1 (x, y).1.2) = (x, y).1.2
ey : f (x, y).2.1 (g (x, y).2.1 (x, y).2.2) = (x, y).2.2
h1 : (x, y).1.1 = (x, y).2.1
h2 : f (x, y).1.1 (g (x, y).1.1 (x, y).1.2) = f (x, y).2.1 (g (x, y).2.1 (x, y).2.2)
⊢ ((x, y).1.1, g (x, y).1.1 (x, y).1.2) = ((x, y).2.1, g (x, y).2.1 (x, y).2.2) | case intro.intro.intro
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
nc : mfderiv I I (f c) z ≠ 0
g : ℂ → T → S
ga : HolomorphicAt (I.prod I) I (uncurry g) (c, f c z)
gf : ∀ᶠ (x : ℂ × S) in 𝓝 (c, z), g x.1 (f x.1 x.2) = x.2
fg : ∀ᶠ (x : ℂ × T) in 𝓝 (c, f c z), f x.1 (g x.1 x.2) = x.2
n : NontrivialHolomorphicAt (g c) (f c z)
o : 𝓝 (c, z) = Filter.map (fun p => (p.1, g p.1 p.2)) (𝓝 (c, f c z))
x y : ℂ × T
ex : f (x, y).1.1 (g (x, y).1.1 (x, y).1.2) = (x, y).1.2
ey : f (x, y).2.1 (g (x, y).2.1 (x, y).2.2) = (x, y).2.2
h1 : x.1 = y.1
h2 : f (x, y).1.1 (g (x, y).1.1 (x, y).1.2) = f (x, y).2.1 (g (x, y).2.1 (x, y).2.2)
⊢ ((x, y).1.1, g (x, y).1.1 (x, y).1.2) = ((x, y).2.1, g (x, y).2.1 (x, y).2.2) |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/LocalInj.lean | HolomorphicAt.local_inj'' | [39, 1] | [54, 90] | simp only [h1] at ex ey h2 ⊢ | case intro.intro.intro
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
nc : mfderiv I I (f c) z ≠ 0
g : ℂ → T → S
ga : HolomorphicAt (I.prod I) I (uncurry g) (c, f c z)
gf : ∀ᶠ (x : ℂ × S) in 𝓝 (c, z), g x.1 (f x.1 x.2) = x.2
fg : ∀ᶠ (x : ℂ × T) in 𝓝 (c, f c z), f x.1 (g x.1 x.2) = x.2
n : NontrivialHolomorphicAt (g c) (f c z)
o : 𝓝 (c, z) = Filter.map (fun p => (p.1, g p.1 p.2)) (𝓝 (c, f c z))
x y : ℂ × T
ex : f (x, y).1.1 (g (x, y).1.1 (x, y).1.2) = (x, y).1.2
ey : f (x, y).2.1 (g (x, y).2.1 (x, y).2.2) = (x, y).2.2
h1 : x.1 = y.1
h2 : f (x, y).1.1 (g (x, y).1.1 (x, y).1.2) = f (x, y).2.1 (g (x, y).2.1 (x, y).2.2)
⊢ ((x, y).1.1, g (x, y).1.1 (x, y).1.2) = ((x, y).2.1, g (x, y).2.1 (x, y).2.2) | case intro.intro.intro
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
nc : mfderiv I I (f c) z ≠ 0
g : ℂ → T → S
ga : HolomorphicAt (I.prod I) I (uncurry g) (c, f c z)
gf : ∀ᶠ (x : ℂ × S) in 𝓝 (c, z), g x.1 (f x.1 x.2) = x.2
fg : ∀ᶠ (x : ℂ × T) in 𝓝 (c, f c z), f x.1 (g x.1 x.2) = x.2
n : NontrivialHolomorphicAt (g c) (f c z)
o : 𝓝 (c, z) = Filter.map (fun p => (p.1, g p.1 p.2)) (𝓝 (c, f c z))
x y : ℂ × T
ey : f y.1 (g y.1 y.2) = y.2
h1 : x.1 = y.1
ex : f y.1 (g y.1 x.2) = x.2
h2 : f y.1 (g y.1 x.2) = f y.1 (g y.1 y.2)
⊢ (y.1, g y.1 x.2) = (y.1, g y.1 y.2) |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/LocalInj.lean | HolomorphicAt.local_inj'' | [39, 1] | [54, 90] | simp only [ex, ey] at h2 | case intro.intro.intro
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
nc : mfderiv I I (f c) z ≠ 0
g : ℂ → T → S
ga : HolomorphicAt (I.prod I) I (uncurry g) (c, f c z)
gf : ∀ᶠ (x : ℂ × S) in 𝓝 (c, z), g x.1 (f x.1 x.2) = x.2
fg : ∀ᶠ (x : ℂ × T) in 𝓝 (c, f c z), f x.1 (g x.1 x.2) = x.2
n : NontrivialHolomorphicAt (g c) (f c z)
o : 𝓝 (c, z) = Filter.map (fun p => (p.1, g p.1 p.2)) (𝓝 (c, f c z))
x y : ℂ × T
ey : f y.1 (g y.1 y.2) = y.2
h1 : x.1 = y.1
ex : f y.1 (g y.1 x.2) = x.2
h2 : f y.1 (g y.1 x.2) = f y.1 (g y.1 y.2)
⊢ (y.1, g y.1 x.2) = (y.1, g y.1 y.2) | case intro.intro.intro
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
nc : mfderiv I I (f c) z ≠ 0
g : ℂ → T → S
ga : HolomorphicAt (I.prod I) I (uncurry g) (c, f c z)
gf : ∀ᶠ (x : ℂ × S) in 𝓝 (c, z), g x.1 (f x.1 x.2) = x.2
fg : ∀ᶠ (x : ℂ × T) in 𝓝 (c, f c z), f x.1 (g x.1 x.2) = x.2
n : NontrivialHolomorphicAt (g c) (f c z)
o : 𝓝 (c, z) = Filter.map (fun p => (p.1, g p.1 p.2)) (𝓝 (c, f c z))
x y : ℂ × T
ey : f y.1 (g y.1 y.2) = y.2
h1 : x.1 = y.1
ex : f y.1 (g y.1 x.2) = x.2
h2 : x.2 = y.2
⊢ (y.1, g y.1 x.2) = (y.1, g y.1 y.2) |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/LocalInj.lean | HolomorphicAt.local_inj'' | [39, 1] | [54, 90] | simp only [h2] | case intro.intro.intro
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
nc : mfderiv I I (f c) z ≠ 0
g : ℂ → T → S
ga : HolomorphicAt (I.prod I) I (uncurry g) (c, f c z)
gf : ∀ᶠ (x : ℂ × S) in 𝓝 (c, z), g x.1 (f x.1 x.2) = x.2
fg : ∀ᶠ (x : ℂ × T) in 𝓝 (c, f c z), f x.1 (g x.1 x.2) = x.2
n : NontrivialHolomorphicAt (g c) (f c z)
o : 𝓝 (c, z) = Filter.map (fun p => (p.1, g p.1 p.2)) (𝓝 (c, f c z))
x y : ℂ × T
ey : f y.1 (g y.1 y.2) = y.2
h1 : x.1 = y.1
ex : f y.1 (g y.1 x.2) = x.2
h2 : x.2 = y.2
⊢ (y.1, g y.1 x.2) = (y.1, g y.1 y.2) | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/LocalInj.lean | HolomorphicAt.local_inj'' | [39, 1] | [54, 90] | have e : (c, z) = (c, g c (f c z)) := by rw [gf.self_of_nhds] | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
nc : mfderiv I I (f c) z ≠ 0
g : ℂ → T → S
ga : HolomorphicAt (I.prod I) I (uncurry g) (c, f c z)
gf : ∀ᶠ (x : ℂ × S) in 𝓝 (c, z), g x.1 (f x.1 x.2) = x.2
fg : ∀ᶠ (x : ℂ × T) in 𝓝 (c, f c z), f x.1 (g x.1 x.2) = x.2
⊢ NontrivialHolomorphicAt (g c) (f c z) | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
nc : mfderiv I I (f c) z ≠ 0
g : ℂ → T → S
ga : HolomorphicAt (I.prod I) I (uncurry g) (c, f c z)
gf : ∀ᶠ (x : ℂ × S) in 𝓝 (c, z), g x.1 (f x.1 x.2) = x.2
fg : ∀ᶠ (x : ℂ × T) in 𝓝 (c, f c z), f x.1 (g x.1 x.2) = x.2
e : (c, z) = (c, g c (f c z))
⊢ NontrivialHolomorphicAt (g c) (f c z) |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/LocalInj.lean | HolomorphicAt.local_inj'' | [39, 1] | [54, 90] | rw [e] at fa | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
nc : mfderiv I I (f c) z ≠ 0
g : ℂ → T → S
ga : HolomorphicAt (I.prod I) I (uncurry g) (c, f c z)
gf : ∀ᶠ (x : ℂ × S) in 𝓝 (c, z), g x.1 (f x.1 x.2) = x.2
fg : ∀ᶠ (x : ℂ × T) in 𝓝 (c, f c z), f x.1 (g x.1 x.2) = x.2
e : (c, z) = (c, g c (f c z))
⊢ NontrivialHolomorphicAt (g c) (f c z) | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
nc : mfderiv I I (f c) z ≠ 0
g : ℂ → T → S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, g c (f c z))
ga : HolomorphicAt (I.prod I) I (uncurry g) (c, f c z)
gf : ∀ᶠ (x : ℂ × S) in 𝓝 (c, z), g x.1 (f x.1 x.2) = x.2
fg : ∀ᶠ (x : ℂ × T) in 𝓝 (c, f c z), f x.1 (g x.1 x.2) = x.2
e : (c, z) = (c, g c (f c z))
⊢ NontrivialHolomorphicAt (g c) (f c z) |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/LocalInj.lean | HolomorphicAt.local_inj'' | [39, 1] | [54, 90] | refine (NontrivialHolomorphicAt.anti ?_ fa.along_snd ga.along_snd).2 | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
nc : mfderiv I I (f c) z ≠ 0
g : ℂ → T → S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, g c (f c z))
ga : HolomorphicAt (I.prod I) I (uncurry g) (c, f c z)
gf : ∀ᶠ (x : ℂ × S) in 𝓝 (c, z), g x.1 (f x.1 x.2) = x.2
fg : ∀ᶠ (x : ℂ × T) in 𝓝 (c, f c z), f x.1 (g x.1 x.2) = x.2
e : (c, z) = (c, g c (f c z))
⊢ NontrivialHolomorphicAt (g c) (f c z) | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
nc : mfderiv I I (f c) z ≠ 0
g : ℂ → T → S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, g c (f c z))
ga : HolomorphicAt (I.prod I) I (uncurry g) (c, f c z)
gf : ∀ᶠ (x : ℂ × S) in 𝓝 (c, z), g x.1 (f x.1 x.2) = x.2
fg : ∀ᶠ (x : ℂ × T) in 𝓝 (c, f c z), f x.1 (g x.1 x.2) = x.2
e : (c, z) = (c, g c (f c z))
⊢ NontrivialHolomorphicAt (fun z => f c (g c z)) (f c z) |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/LocalInj.lean | HolomorphicAt.local_inj'' | [39, 1] | [54, 90] | refine (nontrivialHolomorphicAt_id _).congr ?_ | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
nc : mfderiv I I (f c) z ≠ 0
g : ℂ → T → S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, g c (f c z))
ga : HolomorphicAt (I.prod I) I (uncurry g) (c, f c z)
gf : ∀ᶠ (x : ℂ × S) in 𝓝 (c, z), g x.1 (f x.1 x.2) = x.2
fg : ∀ᶠ (x : ℂ × T) in 𝓝 (c, f c z), f x.1 (g x.1 x.2) = x.2
e : (c, z) = (c, g c (f c z))
⊢ NontrivialHolomorphicAt (fun z => f c (g c z)) (f c z) | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
nc : mfderiv I I (f c) z ≠ 0
g : ℂ → T → S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, g c (f c z))
ga : HolomorphicAt (I.prod I) I (uncurry g) (c, f c z)
gf : ∀ᶠ (x : ℂ × S) in 𝓝 (c, z), g x.1 (f x.1 x.2) = x.2
fg : ∀ᶠ (x : ℂ × T) in 𝓝 (c, f c z), f x.1 (g x.1 x.2) = x.2
e : (c, z) = (c, g c (f c z))
⊢ (𝓝 (f c z)).EventuallyEq (fun w => w) fun z => f c (g c z) |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/LocalInj.lean | HolomorphicAt.local_inj'' | [39, 1] | [54, 90] | refine ((continuousAt_const.prod continuousAt_id).eventually fg).mp (eventually_of_forall ?_) | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
nc : mfderiv I I (f c) z ≠ 0
g : ℂ → T → S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, g c (f c z))
ga : HolomorphicAt (I.prod I) I (uncurry g) (c, f c z)
gf : ∀ᶠ (x : ℂ × S) in 𝓝 (c, z), g x.1 (f x.1 x.2) = x.2
fg : ∀ᶠ (x : ℂ × T) in 𝓝 (c, f c z), f x.1 (g x.1 x.2) = x.2
e : (c, z) = (c, g c (f c z))
⊢ (𝓝 (f c z)).EventuallyEq (fun w => w) fun z => f c (g c z) | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
nc : mfderiv I I (f c) z ≠ 0
g : ℂ → T → S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, g c (f c z))
ga : HolomorphicAt (I.prod I) I (uncurry g) (c, f c z)
gf : ∀ᶠ (x : ℂ × S) in 𝓝 (c, z), g x.1 (f x.1 x.2) = x.2
fg : ∀ᶠ (x : ℂ × T) in 𝓝 (c, f c z), f x.1 (g x.1 x.2) = x.2
e : (c, z) = (c, g c (f c z))
⊢ ∀ (x : T), f (c, id x).1 (g (c, id x).1 (c, id x).2) = (c, id x).2 → (fun w => w) x = (fun z => f c (g c z)) x |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/LocalInj.lean | HolomorphicAt.local_inj'' | [39, 1] | [54, 90] | exact fun _ e ↦ e.symm | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
nc : mfderiv I I (f c) z ≠ 0
g : ℂ → T → S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, g c (f c z))
ga : HolomorphicAt (I.prod I) I (uncurry g) (c, f c z)
gf : ∀ᶠ (x : ℂ × S) in 𝓝 (c, z), g x.1 (f x.1 x.2) = x.2
fg : ∀ᶠ (x : ℂ × T) in 𝓝 (c, f c z), f x.1 (g x.1 x.2) = x.2
e : (c, z) = (c, g c (f c z))
⊢ ∀ (x : T), f (c, id x).1 (g (c, id x).1 (c, id x).2) = (c, id x).2 → (fun w => w) x = (fun z => f c (g c z)) x | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/LocalInj.lean | HolomorphicAt.local_inj'' | [39, 1] | [54, 90] | rw [gf.self_of_nhds] | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
nc : mfderiv I I (f c) z ≠ 0
g : ℂ → T → S
ga : HolomorphicAt (I.prod I) I (uncurry g) (c, f c z)
gf : ∀ᶠ (x : ℂ × S) in 𝓝 (c, z), g x.1 (f x.1 x.2) = x.2
fg : ∀ᶠ (x : ℂ × T) in 𝓝 (c, f c z), f x.1 (g x.1 x.2) = x.2
⊢ (c, z) = (c, g c (f c z)) | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/LocalInj.lean | HolomorphicAt.local_inj' | [58, 1] | [67, 65] | set g : ℂ × S × S → (ℂ × S) × ℂ × S := fun p ↦ ((p.1, p.2.1), (p.1, p.2.2)) | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
nc : mfderiv I I (f c) z ≠ 0
⊢ ∀ᶠ (p : ℂ × S × S) in 𝓝 (c, z, z), f p.1 p.2.1 = f p.1 p.2.2 → p.2.1 = p.2.2 | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
nc : mfderiv I I (f c) z ≠ 0
g : ℂ × S × S → (ℂ × S) × ℂ × S := fun p => ((p.1, p.2.1), p.1, p.2.2)
⊢ ∀ᶠ (p : ℂ × S × S) in 𝓝 (c, z, z), f p.1 p.2.1 = f p.1 p.2.2 → p.2.1 = p.2.2 |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/LocalInj.lean | HolomorphicAt.local_inj' | [58, 1] | [67, 65] | refine (t.eventually (fa.local_inj'' nc)).mp (eventually_of_forall ?_) | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
nc : mfderiv I I (f c) z ≠ 0
g : ℂ × S × S → (ℂ × S) × ℂ × S := fun p => ((p.1, p.2.1), p.1, p.2.2)
t : Tendsto g (𝓝 (c, z, z)) (𝓝 ((c, z), c, z))
⊢ ∀ᶠ (p : ℂ × S × S) in 𝓝 (c, z, z), f p.1 p.2.1 = f p.1 p.2.2 → p.2.1 = p.2.2 | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
nc : mfderiv I I (f c) z ≠ 0
g : ℂ × S × S → (ℂ × S) × ℂ × S := fun p => ((p.1, p.2.1), p.1, p.2.2)
t : Tendsto g (𝓝 (c, z, z)) (𝓝 ((c, z), c, z))
⊢ ∀ (x : ℂ × S × S),
((g x).1.1 = (g x).2.1 → f (g x).1.1 (g x).1.2 = f (g x).2.1 (g x).2.2 → (g x).1 = (g x).2) →
f x.1 x.2.1 = f x.1 x.2.2 → x.2.1 = x.2.2 |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/LocalInj.lean | HolomorphicAt.local_inj' | [58, 1] | [67, 65] | intro ⟨e, x, y⟩ inj fe | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
nc : mfderiv I I (f c) z ≠ 0
g : ℂ × S × S → (ℂ × S) × ℂ × S := fun p => ((p.1, p.2.1), p.1, p.2.2)
t : Tendsto g (𝓝 (c, z, z)) (𝓝 ((c, z), c, z))
⊢ ∀ (x : ℂ × S × S),
((g x).1.1 = (g x).2.1 → f (g x).1.1 (g x).1.2 = f (g x).2.1 (g x).2.2 → (g x).1 = (g x).2) →
f x.1 x.2.1 = f x.1 x.2.2 → x.2.1 = x.2.2 | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
nc : mfderiv I I (f c) z ≠ 0
g : ℂ × S × S → (ℂ × S) × ℂ × S := fun p => ((p.1, p.2.1), p.1, p.2.2)
t : Tendsto g (𝓝 (c, z, z)) (𝓝 ((c, z), c, z))
e : ℂ
x y : S
inj :
(g (e, x, y)).1.1 = (g (e, x, y)).2.1 →
f (g (e, x, y)).1.1 (g (e, x, y)).1.2 = f (g (e, x, y)).2.1 (g (e, x, y)).2.2 → (g (e, x, y)).1 = (g (e, x, y)).2
fe : f (e, x, y).1 (e, x, y).2.1 = f (e, x, y).1 (e, x, y).2.2
⊢ (e, x, y).2.1 = (e, x, y).2.2 |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/LocalInj.lean | HolomorphicAt.local_inj' | [58, 1] | [67, 65] | exact (Prod.ext_iff.mp (inj rfl fe)).2 | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
nc : mfderiv I I (f c) z ≠ 0
g : ℂ × S × S → (ℂ × S) × ℂ × S := fun p => ((p.1, p.2.1), p.1, p.2.2)
t : Tendsto g (𝓝 (c, z, z)) (𝓝 ((c, z), c, z))
e : ℂ
x y : S
inj :
(g (e, x, y)).1.1 = (g (e, x, y)).2.1 →
f (g (e, x, y)).1.1 (g (e, x, y)).1.2 = f (g (e, x, y)).2.1 (g (e, x, y)).2.2 → (g (e, x, y)).1 = (g (e, x, y)).2
fe : f (e, x, y).1 (e, x, y).2.1 = f (e, x, y).1 (e, x, y).2.2
⊢ (e, x, y).2.1 = (e, x, y).2.2 | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/LocalInj.lean | HolomorphicAt.local_inj' | [58, 1] | [67, 65] | apply Continuous.continuousAt | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
nc : mfderiv I I (f c) z ≠ 0
g : ℂ × S × S → (ℂ × S) × ℂ × S := fun p => ((p.1, p.2.1), p.1, p.2.2)
⊢ Tendsto g (𝓝 (c, z, z)) (𝓝 ((c, z), c, z)) | case h
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
nc : mfderiv I I (f c) z ≠ 0
g : ℂ × S × S → (ℂ × S) × ℂ × S := fun p => ((p.1, p.2.1), p.1, p.2.2)
⊢ Continuous g |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/LocalInj.lean | HolomorphicAt.local_inj' | [58, 1] | [67, 65] | apply Continuous.prod_mk | case h
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
nc : mfderiv I I (f c) z ≠ 0
g : ℂ × S × S → (ℂ × S) × ℂ × S := fun p => ((p.1, p.2.1), p.1, p.2.2)
⊢ Continuous g | case h.hf
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
nc : mfderiv I I (f c) z ≠ 0
g : ℂ × S × S → (ℂ × S) × ℂ × S := fun p => ((p.1, p.2.1), p.1, p.2.2)
⊢ Continuous fun x => (x.1, x.2.1)
case h.hg
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
nc : mfderiv I I (f c) z ≠ 0
g : ℂ × S × S → (ℂ × S) × ℂ × S := fun p => ((p.1, p.2.1), p.1, p.2.2)
⊢ Continuous fun x => (x.1, x.2.2) |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/LocalInj.lean | HolomorphicAt.local_inj' | [58, 1] | [67, 65] | exact continuous_fst.prod_mk (continuous_fst.comp continuous_snd) | case h.hf
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
nc : mfderiv I I (f c) z ≠ 0
g : ℂ × S × S → (ℂ × S) × ℂ × S := fun p => ((p.1, p.2.1), p.1, p.2.2)
⊢ Continuous fun x => (x.1, x.2.1) | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/LocalInj.lean | HolomorphicAt.local_inj' | [58, 1] | [67, 65] | exact continuous_fst.prod_mk (continuous_snd.comp continuous_snd) | case h.hg
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
nc : mfderiv I I (f c) z ≠ 0
g : ℂ × S × S → (ℂ × S) × ℂ × S := fun p => ((p.1, p.2.1), p.1, p.2.2)
⊢ Continuous fun x => (x.1, x.2.2) | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OpenMapping.lean | nontrivial_local_of_global | [46, 1] | [61, 29] | have fh : HolomorphicOn I I f (closedBall z r) := fun _ m ↦ (fa _ m).holomorphicAt I I | 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
z : ℂ
e r : ℝ
fa : AnalyticOn ℂ f (closedBall z r)
rp : 0 < r
ep : 0 < e
ef : ∀ w ∈ sphere z r, e ≤ ‖f w - f z‖
⊢ NontrivialHolomorphicAt f z | 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
z : ℂ
e r : ℝ
fa : AnalyticOn ℂ f (closedBall z r)
rp : 0 < r
ep : 0 < e
ef : ∀ w ∈ sphere z r, e ≤ ‖f w - f z‖
fh : HolomorphicOn 𝓘(ℂ, ℂ) 𝓘(ℂ, ℂ) f (closedBall z r)
⊢ NontrivialHolomorphicAt f z |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OpenMapping.lean | nontrivial_local_of_global | [46, 1] | [61, 29] | have zs : z ∈ closedBall z r := mem_closedBall_self rp.le | 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
z : ℂ
e r : ℝ
fa : AnalyticOn ℂ f (closedBall z r)
rp : 0 < r
ep : 0 < e
ef : ∀ w ∈ sphere z r, e ≤ ‖f w - f z‖
fh : HolomorphicOn 𝓘(ℂ, ℂ) 𝓘(ℂ, ℂ) f (closedBall z r)
⊢ NontrivialHolomorphicAt f z | 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
z : ℂ
e r : ℝ
fa : AnalyticOn ℂ f (closedBall z r)
rp : 0 < r
ep : 0 < e
ef : ∀ w ∈ sphere z r, e ≤ ‖f w - f z‖
fh : HolomorphicOn 𝓘(ℂ, ℂ) 𝓘(ℂ, ℂ) f (closedBall z r)
zs : z ∈ closedBall z r
⊢ NontrivialHolomorphicAt f z |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OpenMapping.lean | nontrivial_local_of_global | [46, 1] | [61, 29] | use fh _ zs | 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
z : ℂ
e r : ℝ
fa : AnalyticOn ℂ f (closedBall z r)
rp : 0 < r
ep : 0 < e
ef : ∀ w ∈ sphere z r, e ≤ ‖f w - f z‖
fh : HolomorphicOn 𝓘(ℂ, ℂ) 𝓘(ℂ, ℂ) f (closedBall z r)
zs : z ∈ closedBall z r
⊢ NontrivialHolomorphicAt f z | case nonconst
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
z : ℂ
e r : ℝ
fa : AnalyticOn ℂ f (closedBall z r)
rp : 0 < r
ep : 0 < e
ef : ∀ w ∈ sphere z r, e ≤ ‖f w - f z‖
fh : HolomorphicOn 𝓘(ℂ, ℂ) 𝓘(ℂ, ℂ) f (closedBall z r)
zs : z ∈ closedBall z r
⊢ ∃ᶠ (w : ℂ) in 𝓝 z, f w ≠ f z |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OpenMapping.lean | nontrivial_local_of_global | [46, 1] | [61, 29] | contrapose ef | case nonconst
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
z : ℂ
e r : ℝ
fa : AnalyticOn ℂ f (closedBall z r)
rp : 0 < r
ep : 0 < e
ef : ∀ w ∈ sphere z r, e ≤ ‖f w - f z‖
fh : HolomorphicOn 𝓘(ℂ, ℂ) 𝓘(ℂ, ℂ) f (closedBall z r)
zs : z ∈ closedBall z r
⊢ ∃ᶠ (w : ℂ) in 𝓝 z, f w ≠ f z | case nonconst
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
z : ℂ
e r : ℝ
fa : AnalyticOn ℂ f (closedBall z r)
rp : 0 < r
ep : 0 < e
fh : HolomorphicOn 𝓘(ℂ, ℂ) 𝓘(ℂ, ℂ) f (closedBall z r)
zs : z ∈ closedBall z r
ef : ¬∃ᶠ (w : ℂ) in 𝓝 z, f w ≠ f z
⊢ ¬∀ w ∈ sphere z r, e ≤ ‖f w - f z‖ |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OpenMapping.lean | nontrivial_local_of_global | [46, 1] | [61, 29] | simp only [Filter.not_frequently, not_not] at ef | case nonconst
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
z : ℂ
e r : ℝ
fa : AnalyticOn ℂ f (closedBall z r)
rp : 0 < r
ep : 0 < e
fh : HolomorphicOn 𝓘(ℂ, ℂ) 𝓘(ℂ, ℂ) f (closedBall z r)
zs : z ∈ closedBall z r
ef : ¬∃ᶠ (w : ℂ) in 𝓝 z, f w ≠ f z
⊢ ¬∀ w ∈ sphere z r, e ≤ ‖f w - f z‖ | case nonconst
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
z : ℂ
e r : ℝ
fa : AnalyticOn ℂ f (closedBall z r)
rp : 0 < r
ep : 0 < e
fh : HolomorphicOn 𝓘(ℂ, ℂ) 𝓘(ℂ, ℂ) f (closedBall z r)
zs : z ∈ closedBall z r
ef : ∀ᶠ (x : ℂ) in 𝓝 z, f x = f z
⊢ ¬∀ w ∈ sphere z r, e ≤ ‖f w - f z‖ |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OpenMapping.lean | nontrivial_local_of_global | [46, 1] | [61, 29] | simp only [not_forall, not_le] | case nonconst
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
z : ℂ
e r : ℝ
fa : AnalyticOn ℂ f (closedBall z r)
rp : 0 < r
ep : 0 < e
fh : HolomorphicOn 𝓘(ℂ, ℂ) 𝓘(ℂ, ℂ) f (closedBall z r)
zs : z ∈ closedBall z r
ef : ∀ᶠ (x : ℂ) in 𝓝 z, f x = f z
⊢ ¬∀ w ∈ sphere z r, e ≤ ‖f w - f z‖ | case nonconst
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
z : ℂ
e r : ℝ
fa : AnalyticOn ℂ f (closedBall z r)
rp : 0 < r
ep : 0 < e
fh : HolomorphicOn 𝓘(ℂ, ℂ) 𝓘(ℂ, ℂ) f (closedBall z r)
zs : z ∈ closedBall z r
ef : ∀ᶠ (x : ℂ) in 𝓝 z, f x = f z
⊢ ∃ x, ∃ (_ : x ∈ sphere z r), ‖f x - f z‖ < e |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OpenMapping.lean | nontrivial_local_of_global | [46, 1] | [61, 29] | have zrs : z + r ∈ sphere z r := by
simp only [mem_sphere, Complex.dist_eq, add_sub_cancel_left, Complex.abs_ofReal, abs_of_pos rp] | case nonconst
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
z : ℂ
e r : ℝ
fa : AnalyticOn ℂ f (closedBall z r)
rp : 0 < r
ep : 0 < e
fh : HolomorphicOn 𝓘(ℂ, ℂ) 𝓘(ℂ, ℂ) f (closedBall z r)
zs : z ∈ closedBall z r
ef : ∀ᶠ (x : ℂ) in 𝓝 z, f x = f z
⊢ ∃ x, ∃ (_ : x ∈ sphere z r), ‖f x - f z‖ < e | case nonconst
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
z : ℂ
e r : ℝ
fa : AnalyticOn ℂ f (closedBall z r)
rp : 0 < r
ep : 0 < e
fh : HolomorphicOn 𝓘(ℂ, ℂ) 𝓘(ℂ, ℂ) f (closedBall z r)
zs : z ∈ closedBall z r
ef : ∀ᶠ (x : ℂ) in 𝓝 z, f x = f z
zrs : z + ↑r ∈ sphere z r
⊢ ∃ x, ∃ (_ : x ∈ sphere z r), ‖f x - f z‖ < e |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OpenMapping.lean | nontrivial_local_of_global | [46, 1] | [61, 29] | use z + r, zrs | case nonconst
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
z : ℂ
e r : ℝ
fa : AnalyticOn ℂ f (closedBall z r)
rp : 0 < r
ep : 0 < e
fh : HolomorphicOn 𝓘(ℂ, ℂ) 𝓘(ℂ, ℂ) f (closedBall z r)
zs : z ∈ closedBall z r
ef : ∀ᶠ (x : ℂ) in 𝓝 z, f x = f z
zrs : z + ↑r ∈ sphere z r
⊢ ∃ x, ∃ (_ : x ∈ sphere z r), ‖f x - f z‖ < e | case h
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
z : ℂ
e r : ℝ
fa : AnalyticOn ℂ f (closedBall z r)
rp : 0 < r
ep : 0 < e
fh : HolomorphicOn 𝓘(ℂ, ℂ) 𝓘(ℂ, ℂ) f (closedBall z r)
zs : z ∈ closedBall z r
ef : ∀ᶠ (x : ℂ) in 𝓝 z, f x = f z
zrs : z + ↑r ∈ sphere z r
⊢ ‖f (z + ↑r) - f z‖ < e |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OpenMapping.lean | nontrivial_local_of_global | [46, 1] | [61, 29] | simp only [fh.const_of_locally_const' zs (convex_closedBall z r).isPreconnected ef (z + r)
(Metric.sphere_subset_closedBall zrs),
sub_self, norm_zero, ep] | case h
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
z : ℂ
e r : ℝ
fa : AnalyticOn ℂ f (closedBall z r)
rp : 0 < r
ep : 0 < e
fh : HolomorphicOn 𝓘(ℂ, ℂ) 𝓘(ℂ, ℂ) f (closedBall z r)
zs : z ∈ closedBall z r
ef : ∀ᶠ (x : ℂ) in 𝓝 z, f x = f z
zrs : z + ↑r ∈ sphere z r
⊢ ‖f (z + ↑r) - f z‖ < e | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OpenMapping.lean | nontrivial_local_of_global | [46, 1] | [61, 29] | simp only [mem_sphere, Complex.dist_eq, add_sub_cancel_left, Complex.abs_ofReal, abs_of_pos rp] | 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
z : ℂ
e r : ℝ
fa : AnalyticOn ℂ f (closedBall z r)
rp : 0 < r
ep : 0 < e
fh : HolomorphicOn 𝓘(ℂ, ℂ) 𝓘(ℂ, ℂ) f (closedBall z r)
zs : z ∈ closedBall z r
ef : ∀ᶠ (x : ℂ) in 𝓝 z, f x = f z
⊢ z + ↑r ∈ sphere z r | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OpenMapping.lean | AnalyticOn.ball_subset_image_closedBall_param | [65, 1] | [101, 101] | have fn : ∀ d, d ∈ u → ∃ᶠ w in 𝓝 z, f d w ≠ f d z := by
refine fun d m ↦ (nontrivial_local_of_global (fa.along_snd.mono ?_) rp ep (ef d m)).nonconst
simp only [← closedBall_prod_same, mem_prod_eq, setOf_mem_eq, iff_true_iff.mpr m,
true_and_iff, subset_refl] | 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
e r : ℝ
u : Set ℂ
fa : AnalyticOn ℂ (uncurry f) (u ×ˢ closedBall z r)
rp : 0 < r
ep : 0 < e
un : u ∈ 𝓝 c
ef : ∀ d ∈ u, ∀ w ∈ sphere z r, e ≤ ‖f d w - f d z‖
⊢ (fun p => (p.1, f p.1 p.2)) '' u ×ˢ closedBall z r ∈ 𝓝 (c, f c z) | 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
e r : ℝ
u : Set ℂ
fa : AnalyticOn ℂ (uncurry f) (u ×ˢ closedBall z r)
rp : 0 < r
ep : 0 < e
un : u ∈ 𝓝 c
ef : ∀ d ∈ u, ∀ w ∈ sphere z r, e ≤ ‖f d w - f d z‖
fn : ∀ d ∈ u, ∃ᶠ (w : ℂ) in 𝓝 z, f d w ≠ f d z
⊢ (fun p => (p.1, f p.1 p.2)) '' u ×ˢ closedBall z r ∈ 𝓝 (c, f c z) |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OpenMapping.lean | AnalyticOn.ball_subset_image_closedBall_param | [65, 1] | [101, 101] | have op : ∀ d, d ∈ u → ball (f d z) (e / 2) ⊆ f d '' closedBall z r := by
intro d du; refine DiffContOnCl.ball_subset_image_closedBall ?_ rp (ef d du) (fn d du)
have e : f d = uncurry f ∘ fun w ↦ (d, w) := rfl
rw [e]; apply DifferentiableOn.diffContOnCl; apply AnalyticOn.differentiableOn
refine fa.comp (analyticOn_const.prod (analyticOn_id _)) ?_
intro w wr; simp only [closure_ball _ rp.ne'] at wr
simp only [← closedBall_prod_same, mem_prod_eq, du, wr, true_and_iff, du] | 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
e r : ℝ
u : Set ℂ
fa : AnalyticOn ℂ (uncurry f) (u ×ˢ closedBall z r)
rp : 0 < r
ep : 0 < e
un : u ∈ 𝓝 c
ef : ∀ d ∈ u, ∀ w ∈ sphere z r, e ≤ ‖f d w - f d z‖
fn : ∀ d ∈ u, ∃ᶠ (w : ℂ) in 𝓝 z, f d w ≠ f d z
⊢ (fun p => (p.1, f p.1 p.2)) '' u ×ˢ closedBall z r ∈ 𝓝 (c, f c z) | 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
e r : ℝ
u : Set ℂ
fa : AnalyticOn ℂ (uncurry f) (u ×ˢ closedBall z r)
rp : 0 < r
ep : 0 < e
un : u ∈ 𝓝 c
ef : ∀ d ∈ u, ∀ w ∈ sphere z r, e ≤ ‖f d w - f d z‖
fn : ∀ d ∈ u, ∃ᶠ (w : ℂ) in 𝓝 z, f d w ≠ f d z
op : ∀ d ∈ u, ball (f d z) (e / 2) ⊆ f d '' closedBall z r
⊢ (fun p => (p.1, f p.1 p.2)) '' u ×ˢ closedBall z r ∈ 𝓝 (c, f c z) |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OpenMapping.lean | AnalyticOn.ball_subset_image_closedBall_param | [65, 1] | [101, 101] | rcases Metric.continuousAt_iff.mp
(fa (c, z) (mk_mem_prod (mem_of_mem_nhds un) (mem_closedBall_self rp.le))).continuousAt
(e / 4) (by linarith) with
⟨s, sp, sh⟩ | 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
e r : ℝ
u : Set ℂ
fa : AnalyticOn ℂ (uncurry f) (u ×ˢ closedBall z r)
rp : 0 < r
ep : 0 < e
un : u ∈ 𝓝 c
ef : ∀ d ∈ u, ∀ w ∈ sphere z r, e ≤ ‖f d w - f d z‖
fn : ∀ d ∈ u, ∃ᶠ (w : ℂ) in 𝓝 z, f d w ≠ f d z
op : ∀ d ∈ u, ball (f d z) (e / 2) ⊆ f d '' closedBall z r
⊢ (fun p => (p.1, f p.1 p.2)) '' u ×ˢ closedBall z r ∈ 𝓝 (c, f c z) | case 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
e r : ℝ
u : Set ℂ
fa : AnalyticOn ℂ (uncurry f) (u ×ˢ closedBall z r)
rp : 0 < r
ep : 0 < e
un : u ∈ 𝓝 c
ef : ∀ d ∈ u, ∀ w ∈ sphere z r, e ≤ ‖f d w - f d z‖
fn : ∀ d ∈ u, ∃ᶠ (w : ℂ) in 𝓝 z, f d w ≠ f d z
op : ∀ d ∈ u, ball (f d z) (e / 2) ⊆ f d '' closedBall z r
s : ℝ
sp : s > 0
sh : ∀ {x : ℂ × ℂ}, dist x (c, z) < s → dist (uncurry f x) (uncurry f (c, z)) < e / 4
⊢ (fun p => (p.1, f p.1 p.2)) '' u ×ˢ closedBall z r ∈ 𝓝 (c, f c z) |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OpenMapping.lean | AnalyticOn.ball_subset_image_closedBall_param | [65, 1] | [101, 101] | rw [mem_nhds_prod_iff] | case 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
e r : ℝ
u : Set ℂ
fa : AnalyticOn ℂ (uncurry f) (u ×ˢ closedBall z r)
rp : 0 < r
ep : 0 < e
un : u ∈ 𝓝 c
ef : ∀ d ∈ u, ∀ w ∈ sphere z r, e ≤ ‖f d w - f d z‖
fn : ∀ d ∈ u, ∃ᶠ (w : ℂ) in 𝓝 z, f d w ≠ f d z
op : ∀ d ∈ u, ball (f d z) (e / 2) ⊆ f d '' closedBall z r
s : ℝ
sp : s > 0
sh : ∀ {x : ℂ × ℂ}, dist x (c, z) < s → dist (uncurry f x) (uncurry f (c, z)) < e / 4
⊢ (fun p => (p.1, f p.1 p.2)) '' u ×ˢ closedBall z r ∈ 𝓝 (c, f c z) | case 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
e r : ℝ
u : Set ℂ
fa : AnalyticOn ℂ (uncurry f) (u ×ˢ closedBall z r)
rp : 0 < r
ep : 0 < e
un : u ∈ 𝓝 c
ef : ∀ d ∈ u, ∀ w ∈ sphere z r, e ≤ ‖f d w - f d z‖
fn : ∀ d ∈ u, ∃ᶠ (w : ℂ) in 𝓝 z, f d w ≠ f d z
op : ∀ d ∈ u, ball (f d z) (e / 2) ⊆ f d '' closedBall z r
s : ℝ
sp : s > 0
sh : ∀ {x : ℂ × ℂ}, dist x (c, z) < s → dist (uncurry f x) (uncurry f (c, z)) < e / 4
⊢ ∃ u_1 ∈ 𝓝 c, ∃ v ∈ 𝓝 (f c z), u_1 ×ˢ v ⊆ (fun p => (p.1, f p.1 p.2)) '' u ×ˢ closedBall z r |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OpenMapping.lean | AnalyticOn.ball_subset_image_closedBall_param | [65, 1] | [101, 101] | refine ⟨u ∩ ball c s, Filter.inter_mem un (Metric.ball_mem_nhds c (by linarith)), ?_⟩ | case 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
e r : ℝ
u : Set ℂ
fa : AnalyticOn ℂ (uncurry f) (u ×ˢ closedBall z r)
rp : 0 < r
ep : 0 < e
un : u ∈ 𝓝 c
ef : ∀ d ∈ u, ∀ w ∈ sphere z r, e ≤ ‖f d w - f d z‖
fn : ∀ d ∈ u, ∃ᶠ (w : ℂ) in 𝓝 z, f d w ≠ f d z
op : ∀ d ∈ u, ball (f d z) (e / 2) ⊆ f d '' closedBall z r
s : ℝ
sp : s > 0
sh : ∀ {x : ℂ × ℂ}, dist x (c, z) < s → dist (uncurry f x) (uncurry f (c, z)) < e / 4
⊢ ∃ u_1 ∈ 𝓝 c, ∃ v ∈ 𝓝 (f c z), u_1 ×ˢ v ⊆ (fun p => (p.1, f p.1 p.2)) '' u ×ˢ closedBall z r | case 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
e r : ℝ
u : Set ℂ
fa : AnalyticOn ℂ (uncurry f) (u ×ˢ closedBall z r)
rp : 0 < r
ep : 0 < e
un : u ∈ 𝓝 c
ef : ∀ d ∈ u, ∀ w ∈ sphere z r, e ≤ ‖f d w - f d z‖
fn : ∀ d ∈ u, ∃ᶠ (w : ℂ) in 𝓝 z, f d w ≠ f d z
op : ∀ d ∈ u, ball (f d z) (e / 2) ⊆ f d '' closedBall z r
s : ℝ
sp : s > 0
sh : ∀ {x : ℂ × ℂ}, dist x (c, z) < s → dist (uncurry f x) (uncurry f (c, z)) < e / 4
⊢ ∃ v ∈ 𝓝 (f c z), (u ∩ ball c s) ×ˢ v ⊆ (fun p => (p.1, f p.1 p.2)) '' u ×ˢ closedBall z r |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OpenMapping.lean | AnalyticOn.ball_subset_image_closedBall_param | [65, 1] | [101, 101] | use ball (f c z) (e / 4), Metric.ball_mem_nhds _ (by linarith) | case 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
e r : ℝ
u : Set ℂ
fa : AnalyticOn ℂ (uncurry f) (u ×ˢ closedBall z r)
rp : 0 < r
ep : 0 < e
un : u ∈ 𝓝 c
ef : ∀ d ∈ u, ∀ w ∈ sphere z r, e ≤ ‖f d w - f d z‖
fn : ∀ d ∈ u, ∃ᶠ (w : ℂ) in 𝓝 z, f d w ≠ f d z
op : ∀ d ∈ u, ball (f d z) (e / 2) ⊆ f d '' closedBall z r
s : ℝ
sp : s > 0
sh : ∀ {x : ℂ × ℂ}, dist x (c, z) < s → dist (uncurry f x) (uncurry f (c, z)) < e / 4
⊢ ∃ v ∈ 𝓝 (f c z), (u ∩ ball c s) ×ˢ v ⊆ (fun p => (p.1, f p.1 p.2)) '' u ×ˢ closedBall z r | case right
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
e r : ℝ
u : Set ℂ
fa : AnalyticOn ℂ (uncurry f) (u ×ˢ closedBall z r)
rp : 0 < r
ep : 0 < e
un : u ∈ 𝓝 c
ef : ∀ d ∈ u, ∀ w ∈ sphere z r, e ≤ ‖f d w - f d z‖
fn : ∀ d ∈ u, ∃ᶠ (w : ℂ) in 𝓝 z, f d w ≠ f d z
op : ∀ d ∈ u, ball (f d z) (e / 2) ⊆ f d '' closedBall z r
s : ℝ
sp : s > 0
sh : ∀ {x : ℂ × ℂ}, dist x (c, z) < s → dist (uncurry f x) (uncurry f (c, z)) < e / 4
⊢ (u ∩ ball c s) ×ˢ ball (f c z) (e / 4) ⊆ (fun p => (p.1, f p.1 p.2)) '' u ×ˢ closedBall z r |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OpenMapping.lean | AnalyticOn.ball_subset_image_closedBall_param | [65, 1] | [101, 101] | intro ⟨d, w⟩ m | case right
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
e r : ℝ
u : Set ℂ
fa : AnalyticOn ℂ (uncurry f) (u ×ˢ closedBall z r)
rp : 0 < r
ep : 0 < e
un : u ∈ 𝓝 c
ef : ∀ d ∈ u, ∀ w ∈ sphere z r, e ≤ ‖f d w - f d z‖
fn : ∀ d ∈ u, ∃ᶠ (w : ℂ) in 𝓝 z, f d w ≠ f d z
op : ∀ d ∈ u, ball (f d z) (e / 2) ⊆ f d '' closedBall z r
s : ℝ
sp : s > 0
sh : ∀ {x : ℂ × ℂ}, dist x (c, z) < s → dist (uncurry f x) (uncurry f (c, z)) < e / 4
⊢ (u ∩ ball c s) ×ˢ ball (f c z) (e / 4) ⊆ (fun p => (p.1, f p.1 p.2)) '' u ×ˢ closedBall z r | case right
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
e r : ℝ
u : Set ℂ
fa : AnalyticOn ℂ (uncurry f) (u ×ˢ closedBall z r)
rp : 0 < r
ep : 0 < e
un : u ∈ 𝓝 c
ef : ∀ d ∈ u, ∀ w ∈ sphere z r, e ≤ ‖f d w - f d z‖
fn : ∀ d ∈ u, ∃ᶠ (w : ℂ) in 𝓝 z, f d w ≠ f d z
op : ∀ d ∈ u, ball (f d z) (e / 2) ⊆ f d '' closedBall z r
s : ℝ
sp : s > 0
sh : ∀ {x : ℂ × ℂ}, dist x (c, z) < s → dist (uncurry f x) (uncurry f (c, z)) < e / 4
d w : ℂ
m : (d, w) ∈ (u ∩ ball c s) ×ˢ ball (f c z) (e / 4)
⊢ (d, w) ∈ (fun p => (p.1, f p.1 p.2)) '' u ×ˢ closedBall z r |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OpenMapping.lean | AnalyticOn.ball_subset_image_closedBall_param | [65, 1] | [101, 101] | simp only [mem_inter_iff, mem_prod_eq, mem_image, @mem_ball _ _ c, lt_min_iff] at m op ⊢ | case right
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
e r : ℝ
u : Set ℂ
fa : AnalyticOn ℂ (uncurry f) (u ×ˢ closedBall z r)
rp : 0 < r
ep : 0 < e
un : u ∈ 𝓝 c
ef : ∀ d ∈ u, ∀ w ∈ sphere z r, e ≤ ‖f d w - f d z‖
fn : ∀ d ∈ u, ∃ᶠ (w : ℂ) in 𝓝 z, f d w ≠ f d z
op : ∀ d ∈ u, ball (f d z) (e / 2) ⊆ f d '' closedBall z r
s : ℝ
sp : s > 0
sh : ∀ {x : ℂ × ℂ}, dist x (c, z) < s → dist (uncurry f x) (uncurry f (c, z)) < e / 4
d w : ℂ
m : (d, w) ∈ (u ∩ ball c s) ×ˢ ball (f c z) (e / 4)
⊢ (d, w) ∈ (fun p => (p.1, f p.1 p.2)) '' u ×ˢ closedBall z r | case right
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
e r : ℝ
u : Set ℂ
fa : AnalyticOn ℂ (uncurry f) (u ×ˢ closedBall z r)
rp : 0 < r
ep : 0 < e
un : u ∈ 𝓝 c
ef : ∀ d ∈ u, ∀ w ∈ sphere z r, e ≤ ‖f d w - f d z‖
fn : ∀ d ∈ u, ∃ᶠ (w : ℂ) in 𝓝 z, f d w ≠ f d z
op : ∀ d ∈ u, ball (f d z) (e / 2) ⊆ f d '' closedBall z r
s : ℝ
sp : s > 0
sh : ∀ {x : ℂ × ℂ}, dist x (c, z) < s → dist (uncurry f x) (uncurry f (c, z)) < e / 4
d w : ℂ
m : (d ∈ u ∧ dist d c < s) ∧ w ∈ ball (f c z) (e / 4)
⊢ ∃ x, (x.1 ∈ u ∧ x.2 ∈ closedBall z r) ∧ (x.1, f x.1 x.2) = (d, w) |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OpenMapping.lean | AnalyticOn.ball_subset_image_closedBall_param | [65, 1] | [101, 101] | have wm : w ∈ ball (f d z) (e / 2) := by
simp only [mem_ball] at m ⊢
specialize @sh ⟨d, z⟩; simp only [Prod.dist_eq, dist_self, Function.uncurry] at sh
specialize sh (max_lt m.1.2 sp); rw [dist_comm] at sh
calc dist w (f d z)
_ ≤ dist w (f c z) + dist (f c z) (f d z) := by bound
_ < e / 4 + dist (f c z) (f d z) := by linarith [m.2]
_ ≤ e / 4 + e / 4 := by linarith [sh]
_ = e / 2 := by ring | case right
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
e r : ℝ
u : Set ℂ
fa : AnalyticOn ℂ (uncurry f) (u ×ˢ closedBall z r)
rp : 0 < r
ep : 0 < e
un : u ∈ 𝓝 c
ef : ∀ d ∈ u, ∀ w ∈ sphere z r, e ≤ ‖f d w - f d z‖
fn : ∀ d ∈ u, ∃ᶠ (w : ℂ) in 𝓝 z, f d w ≠ f d z
op : ∀ d ∈ u, ball (f d z) (e / 2) ⊆ f d '' closedBall z r
s : ℝ
sp : s > 0
sh : ∀ {x : ℂ × ℂ}, dist x (c, z) < s → dist (uncurry f x) (uncurry f (c, z)) < e / 4
d w : ℂ
m : (d ∈ u ∧ dist d c < s) ∧ w ∈ ball (f c z) (e / 4)
⊢ ∃ x, (x.1 ∈ u ∧ x.2 ∈ closedBall z r) ∧ (x.1, f x.1 x.2) = (d, w) | case right
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
e r : ℝ
u : Set ℂ
fa : AnalyticOn ℂ (uncurry f) (u ×ˢ closedBall z r)
rp : 0 < r
ep : 0 < e
un : u ∈ 𝓝 c
ef : ∀ d ∈ u, ∀ w ∈ sphere z r, e ≤ ‖f d w - f d z‖
fn : ∀ d ∈ u, ∃ᶠ (w : ℂ) in 𝓝 z, f d w ≠ f d z
op : ∀ d ∈ u, ball (f d z) (e / 2) ⊆ f d '' closedBall z r
s : ℝ
sp : s > 0
sh : ∀ {x : ℂ × ℂ}, dist x (c, z) < s → dist (uncurry f x) (uncurry f (c, z)) < e / 4
d w : ℂ
m : (d ∈ u ∧ dist d c < s) ∧ w ∈ ball (f c z) (e / 4)
wm : w ∈ ball (f d z) (e / 2)
⊢ ∃ x, (x.1 ∈ u ∧ x.2 ∈ closedBall z r) ∧ (x.1, f x.1 x.2) = (d, w) |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OpenMapping.lean | AnalyticOn.ball_subset_image_closedBall_param | [65, 1] | [101, 101] | specialize op d m.1.1 wm | case right
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
e r : ℝ
u : Set ℂ
fa : AnalyticOn ℂ (uncurry f) (u ×ˢ closedBall z r)
rp : 0 < r
ep : 0 < e
un : u ∈ 𝓝 c
ef : ∀ d ∈ u, ∀ w ∈ sphere z r, e ≤ ‖f d w - f d z‖
fn : ∀ d ∈ u, ∃ᶠ (w : ℂ) in 𝓝 z, f d w ≠ f d z
op : ∀ d ∈ u, ball (f d z) (e / 2) ⊆ f d '' closedBall z r
s : ℝ
sp : s > 0
sh : ∀ {x : ℂ × ℂ}, dist x (c, z) < s → dist (uncurry f x) (uncurry f (c, z)) < e / 4
d w : ℂ
m : (d ∈ u ∧ dist d c < s) ∧ w ∈ ball (f c z) (e / 4)
wm : w ∈ ball (f d z) (e / 2)
⊢ ∃ x, (x.1 ∈ u ∧ x.2 ∈ closedBall z r) ∧ (x.1, f x.1 x.2) = (d, w) | case right
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
e r : ℝ
u : Set ℂ
fa : AnalyticOn ℂ (uncurry f) (u ×ˢ closedBall z r)
rp : 0 < r
ep : 0 < e
un : u ∈ 𝓝 c
ef : ∀ d ∈ u, ∀ w ∈ sphere z r, e ≤ ‖f d w - f d z‖
fn : ∀ d ∈ u, ∃ᶠ (w : ℂ) in 𝓝 z, f d w ≠ f d z
s : ℝ
sp : s > 0
sh : ∀ {x : ℂ × ℂ}, dist x (c, z) < s → dist (uncurry f x) (uncurry f (c, z)) < e / 4
d w : ℂ
m : (d ∈ u ∧ dist d c < s) ∧ w ∈ ball (f c z) (e / 4)
wm : w ∈ ball (f d z) (e / 2)
op : w ∈ f d '' closedBall z r
⊢ ∃ x, (x.1 ∈ u ∧ x.2 ∈ closedBall z r) ∧ (x.1, f x.1 x.2) = (d, w) |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OpenMapping.lean | AnalyticOn.ball_subset_image_closedBall_param | [65, 1] | [101, 101] | rcases (mem_image _ _ _).mp op with ⟨y, yr, yw⟩ | case right
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
e r : ℝ
u : Set ℂ
fa : AnalyticOn ℂ (uncurry f) (u ×ˢ closedBall z r)
rp : 0 < r
ep : 0 < e
un : u ∈ 𝓝 c
ef : ∀ d ∈ u, ∀ w ∈ sphere z r, e ≤ ‖f d w - f d z‖
fn : ∀ d ∈ u, ∃ᶠ (w : ℂ) in 𝓝 z, f d w ≠ f d z
s : ℝ
sp : s > 0
sh : ∀ {x : ℂ × ℂ}, dist x (c, z) < s → dist (uncurry f x) (uncurry f (c, z)) < e / 4
d w : ℂ
m : (d ∈ u ∧ dist d c < s) ∧ w ∈ ball (f c z) (e / 4)
wm : w ∈ ball (f d z) (e / 2)
op : w ∈ f d '' closedBall z r
⊢ ∃ x, (x.1 ∈ u ∧ x.2 ∈ closedBall z r) ∧ (x.1, f x.1 x.2) = (d, w) | case right.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
e r : ℝ
u : Set ℂ
fa : AnalyticOn ℂ (uncurry f) (u ×ˢ closedBall z r)
rp : 0 < r
ep : 0 < e
un : u ∈ 𝓝 c
ef : ∀ d ∈ u, ∀ w ∈ sphere z r, e ≤ ‖f d w - f d z‖
fn : ∀ d ∈ u, ∃ᶠ (w : ℂ) in 𝓝 z, f d w ≠ f d z
s : ℝ
sp : s > 0
sh : ∀ {x : ℂ × ℂ}, dist x (c, z) < s → dist (uncurry f x) (uncurry f (c, z)) < e / 4
d w : ℂ
m : (d ∈ u ∧ dist d c < s) ∧ w ∈ ball (f c z) (e / 4)
wm : w ∈ ball (f d z) (e / 2)
op : w ∈ f d '' closedBall z r
y : ℂ
yr : y ∈ closedBall z r
yw : f d y = w
⊢ ∃ x, (x.1 ∈ u ∧ x.2 ∈ closedBall z r) ∧ (x.1, f x.1 x.2) = (d, w) |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OpenMapping.lean | AnalyticOn.ball_subset_image_closedBall_param | [65, 1] | [101, 101] | use⟨d, y⟩ | case right.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
e r : ℝ
u : Set ℂ
fa : AnalyticOn ℂ (uncurry f) (u ×ˢ closedBall z r)
rp : 0 < r
ep : 0 < e
un : u ∈ 𝓝 c
ef : ∀ d ∈ u, ∀ w ∈ sphere z r, e ≤ ‖f d w - f d z‖
fn : ∀ d ∈ u, ∃ᶠ (w : ℂ) in 𝓝 z, f d w ≠ f d z
s : ℝ
sp : s > 0
sh : ∀ {x : ℂ × ℂ}, dist x (c, z) < s → dist (uncurry f x) (uncurry f (c, z)) < e / 4
d w : ℂ
m : (d ∈ u ∧ dist d c < s) ∧ w ∈ ball (f c z) (e / 4)
wm : w ∈ ball (f d z) (e / 2)
op : w ∈ f d '' closedBall z r
y : ℂ
yr : y ∈ closedBall z r
yw : f d y = w
⊢ ∃ x, (x.1 ∈ u ∧ x.2 ∈ closedBall z r) ∧ (x.1, f x.1 x.2) = (d, w) | case h
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
e r : ℝ
u : Set ℂ
fa : AnalyticOn ℂ (uncurry f) (u ×ˢ closedBall z r)
rp : 0 < r
ep : 0 < e
un : u ∈ 𝓝 c
ef : ∀ d ∈ u, ∀ w ∈ sphere z r, e ≤ ‖f d w - f d z‖
fn : ∀ d ∈ u, ∃ᶠ (w : ℂ) in 𝓝 z, f d w ≠ f d z
s : ℝ
sp : s > 0
sh : ∀ {x : ℂ × ℂ}, dist x (c, z) < s → dist (uncurry f x) (uncurry f (c, z)) < e / 4
d w : ℂ
m : (d ∈ u ∧ dist d c < s) ∧ w ∈ ball (f c z) (e / 4)
wm : w ∈ ball (f d z) (e / 2)
op : w ∈ f d '' closedBall z r
y : ℂ
yr : y ∈ closedBall z r
yw : f d y = w
⊢ ((d, y).1 ∈ u ∧ (d, y).2 ∈ closedBall z r) ∧ ((d, y).1, f (d, y).1 (d, y).2) = (d, w) |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OpenMapping.lean | AnalyticOn.ball_subset_image_closedBall_param | [65, 1] | [101, 101] | simp only [mem_prod_eq, Prod.ext_iff, yw, and_true_iff, eq_self_iff_true, true_and_iff, yr, m.1.1] | case h
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
e r : ℝ
u : Set ℂ
fa : AnalyticOn ℂ (uncurry f) (u ×ˢ closedBall z r)
rp : 0 < r
ep : 0 < e
un : u ∈ 𝓝 c
ef : ∀ d ∈ u, ∀ w ∈ sphere z r, e ≤ ‖f d w - f d z‖
fn : ∀ d ∈ u, ∃ᶠ (w : ℂ) in 𝓝 z, f d w ≠ f d z
s : ℝ
sp : s > 0
sh : ∀ {x : ℂ × ℂ}, dist x (c, z) < s → dist (uncurry f x) (uncurry f (c, z)) < e / 4
d w : ℂ
m : (d ∈ u ∧ dist d c < s) ∧ w ∈ ball (f c z) (e / 4)
wm : w ∈ ball (f d z) (e / 2)
op : w ∈ f d '' closedBall z r
y : ℂ
yr : y ∈ closedBall z r
yw : f d y = w
⊢ ((d, y).1 ∈ u ∧ (d, y).2 ∈ closedBall z r) ∧ ((d, y).1, f (d, y).1 (d, y).2) = (d, w) | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OpenMapping.lean | AnalyticOn.ball_subset_image_closedBall_param | [65, 1] | [101, 101] | refine fun d m ↦ (nontrivial_local_of_global (fa.along_snd.mono ?_) rp ep (ef d m)).nonconst | 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
e r : ℝ
u : Set ℂ
fa : AnalyticOn ℂ (uncurry f) (u ×ˢ closedBall z r)
rp : 0 < r
ep : 0 < e
un : u ∈ 𝓝 c
ef : ∀ d ∈ u, ∀ w ∈ sphere z r, e ≤ ‖f d w - f d z‖
⊢ ∀ d ∈ u, ∃ᶠ (w : ℂ) in 𝓝 z, f d w ≠ f d z | 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
e r : ℝ
u : Set ℂ
fa : AnalyticOn ℂ (uncurry f) (u ×ˢ closedBall z r)
rp : 0 < r
ep : 0 < e
un : u ∈ 𝓝 c
ef : ∀ d ∈ u, ∀ w ∈ sphere z r, e ≤ ‖f d w - f d z‖
d : ℂ
m : d ∈ u
⊢ closedBall z r ⊆ {y | (d, y) ∈ u ×ˢ closedBall z r} |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OpenMapping.lean | AnalyticOn.ball_subset_image_closedBall_param | [65, 1] | [101, 101] | simp only [← closedBall_prod_same, mem_prod_eq, setOf_mem_eq, iff_true_iff.mpr m,
true_and_iff, subset_refl] | 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
e r : ℝ
u : Set ℂ
fa : AnalyticOn ℂ (uncurry f) (u ×ˢ closedBall z r)
rp : 0 < r
ep : 0 < e
un : u ∈ 𝓝 c
ef : ∀ d ∈ u, ∀ w ∈ sphere z r, e ≤ ‖f d w - f d z‖
d : ℂ
m : d ∈ u
⊢ closedBall z r ⊆ {y | (d, y) ∈ u ×ˢ closedBall z r} | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OpenMapping.lean | AnalyticOn.ball_subset_image_closedBall_param | [65, 1] | [101, 101] | intro d du | 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
e r : ℝ
u : Set ℂ
fa : AnalyticOn ℂ (uncurry f) (u ×ˢ closedBall z r)
rp : 0 < r
ep : 0 < e
un : u ∈ 𝓝 c
ef : ∀ d ∈ u, ∀ w ∈ sphere z r, e ≤ ‖f d w - f d z‖
fn : ∀ d ∈ u, ∃ᶠ (w : ℂ) in 𝓝 z, f d w ≠ f d z
⊢ ∀ d ∈ u, ball (f d z) (e / 2) ⊆ f d '' closedBall z r | 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
e r : ℝ
u : Set ℂ
fa : AnalyticOn ℂ (uncurry f) (u ×ˢ closedBall z r)
rp : 0 < r
ep : 0 < e
un : u ∈ 𝓝 c
ef : ∀ d ∈ u, ∀ w ∈ sphere z r, e ≤ ‖f d w - f d z‖
fn : ∀ d ∈ u, ∃ᶠ (w : ℂ) in 𝓝 z, f d w ≠ f d z
d : ℂ
du : d ∈ u
⊢ ball (f d z) (e / 2) ⊆ f d '' closedBall z r |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OpenMapping.lean | AnalyticOn.ball_subset_image_closedBall_param | [65, 1] | [101, 101] | refine DiffContOnCl.ball_subset_image_closedBall ?_ rp (ef d du) (fn d du) | 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
e r : ℝ
u : Set ℂ
fa : AnalyticOn ℂ (uncurry f) (u ×ˢ closedBall z r)
rp : 0 < r
ep : 0 < e
un : u ∈ 𝓝 c
ef : ∀ d ∈ u, ∀ w ∈ sphere z r, e ≤ ‖f d w - f d z‖
fn : ∀ d ∈ u, ∃ᶠ (w : ℂ) in 𝓝 z, f d w ≠ f d z
d : ℂ
du : d ∈ u
⊢ ball (f d z) (e / 2) ⊆ f d '' closedBall z r | 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
e r : ℝ
u : Set ℂ
fa : AnalyticOn ℂ (uncurry f) (u ×ˢ closedBall z r)
rp : 0 < r
ep : 0 < e
un : u ∈ 𝓝 c
ef : ∀ d ∈ u, ∀ w ∈ sphere z r, e ≤ ‖f d w - f d z‖
fn : ∀ d ∈ u, ∃ᶠ (w : ℂ) in 𝓝 z, f d w ≠ f d z
d : ℂ
du : d ∈ u
⊢ DiffContOnCl ℂ (f d) (ball z r) |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OpenMapping.lean | AnalyticOn.ball_subset_image_closedBall_param | [65, 1] | [101, 101] | have e : f d = uncurry f ∘ fun w ↦ (d, w) := rfl | 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
e r : ℝ
u : Set ℂ
fa : AnalyticOn ℂ (uncurry f) (u ×ˢ closedBall z r)
rp : 0 < r
ep : 0 < e
un : u ∈ 𝓝 c
ef : ∀ d ∈ u, ∀ w ∈ sphere z r, e ≤ ‖f d w - f d z‖
fn : ∀ d ∈ u, ∃ᶠ (w : ℂ) in 𝓝 z, f d w ≠ f d z
d : ℂ
du : d ∈ u
⊢ DiffContOnCl ℂ (f d) (ball z r) | 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
e✝ r : ℝ
u : Set ℂ
fa : AnalyticOn ℂ (uncurry f) (u ×ˢ closedBall z r)
rp : 0 < r
ep : 0 < e✝
un : u ∈ 𝓝 c
ef : ∀ d ∈ u, ∀ w ∈ sphere z r, e✝ ≤ ‖f d w - f d z‖
fn : ∀ d ∈ u, ∃ᶠ (w : ℂ) in 𝓝 z, f d w ≠ f d z
d : ℂ
du : d ∈ u
e : f d = uncurry f ∘ fun w => (d, w)
⊢ DiffContOnCl ℂ (f d) (ball z r) |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OpenMapping.lean | AnalyticOn.ball_subset_image_closedBall_param | [65, 1] | [101, 101] | rw [e] | 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
e✝ r : ℝ
u : Set ℂ
fa : AnalyticOn ℂ (uncurry f) (u ×ˢ closedBall z r)
rp : 0 < r
ep : 0 < e✝
un : u ∈ 𝓝 c
ef : ∀ d ∈ u, ∀ w ∈ sphere z r, e✝ ≤ ‖f d w - f d z‖
fn : ∀ d ∈ u, ∃ᶠ (w : ℂ) in 𝓝 z, f d w ≠ f d z
d : ℂ
du : d ∈ u
e : f d = uncurry f ∘ fun w => (d, w)
⊢ DiffContOnCl ℂ (f d) (ball z r) | 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
e✝ r : ℝ
u : Set ℂ
fa : AnalyticOn ℂ (uncurry f) (u ×ˢ closedBall z r)
rp : 0 < r
ep : 0 < e✝
un : u ∈ 𝓝 c
ef : ∀ d ∈ u, ∀ w ∈ sphere z r, e✝ ≤ ‖f d w - f d z‖
fn : ∀ d ∈ u, ∃ᶠ (w : ℂ) in 𝓝 z, f d w ≠ f d z
d : ℂ
du : d ∈ u
e : f d = uncurry f ∘ fun w => (d, w)
⊢ DiffContOnCl ℂ (uncurry f ∘ fun w => (d, w)) (ball z r) |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OpenMapping.lean | AnalyticOn.ball_subset_image_closedBall_param | [65, 1] | [101, 101] | apply DifferentiableOn.diffContOnCl | 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
e✝ r : ℝ
u : Set ℂ
fa : AnalyticOn ℂ (uncurry f) (u ×ˢ closedBall z r)
rp : 0 < r
ep : 0 < e✝
un : u ∈ 𝓝 c
ef : ∀ d ∈ u, ∀ w ∈ sphere z r, e✝ ≤ ‖f d w - f d z‖
fn : ∀ d ∈ u, ∃ᶠ (w : ℂ) in 𝓝 z, f d w ≠ f d z
d : ℂ
du : d ∈ u
e : f d = uncurry f ∘ fun w => (d, w)
⊢ DiffContOnCl ℂ (uncurry f ∘ fun w => (d, w)) (ball z r) | case h
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
e✝ r : ℝ
u : Set ℂ
fa : AnalyticOn ℂ (uncurry f) (u ×ˢ closedBall z r)
rp : 0 < r
ep : 0 < e✝
un : u ∈ 𝓝 c
ef : ∀ d ∈ u, ∀ w ∈ sphere z r, e✝ ≤ ‖f d w - f d z‖
fn : ∀ d ∈ u, ∃ᶠ (w : ℂ) in 𝓝 z, f d w ≠ f d z
d : ℂ
du : d ∈ u
e : f d = uncurry f ∘ fun w => (d, w)
⊢ DifferentiableOn ℂ (uncurry f ∘ fun w => (d, w)) (closure (ball z r)) |
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