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train · 219k rows

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/OpenMapping.lean	AnalyticOn.ball_subset_image_closedBall_param	[65, 1]	[101, 101]	linarith	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 ⊢ 0 < s	no goals
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/AnalyticManifold/OpenMapping.lean	AnalyticOn.ball_subset_image_closedBall_param	[65, 1]	[101, 101]	linarith	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 ⊢ 0 < e / 4	no goals
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/AnalyticManifold/OpenMapping.lean	AnalyticOn.ball_subset_image_closedBall_param	[65, 1]	[101, 101]	simp only [mem_ball] at m ⊢	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) ⊢ w ∈ ball (f d z) (e / 2)	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) ∧ dist w (f c z) < e / 4 ⊢ dist w (f d z) < e / 2
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/AnalyticManifold/OpenMapping.lean	AnalyticOn.ball_subset_image_closedBall_param	[65, 1]	[101, 101]	specialize @sh ⟨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‖ 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) ∧ dist w (f c z) < e / 4 ⊢ dist w (f d z) < e / 2	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 d w : ℂ m : (d ∈ u ∧ dist d c < s) ∧ dist w (f c z) < e / 4 sh : dist (d, z) (c, z) < s → dist (uncurry f (d, z)) (uncurry f (c, z)) < e / 4 ⊢ dist w (f d z) < e / 2
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/AnalyticManifold/OpenMapping.lean	AnalyticOn.ball_subset_image_closedBall_param	[65, 1]	[101, 101]	simp only [Prod.dist_eq, dist_self, Function.uncurry] at 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 s : ℝ sp : s > 0 d w : ℂ m : (d ∈ u ∧ dist d c < s) ∧ dist w (f c z) < e / 4 sh : dist (d, z) (c, z) < s → dist (uncurry f (d, z)) (uncurry f (c, z)) < e / 4 ⊢ dist w (f d z) < e / 2	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 d w : ℂ m : (d ∈ u ∧ dist d c < s) ∧ dist w (f c z) < e / 4 sh : max (dist d c) 0 < s → dist (f d z) (f c z) < e / 4 ⊢ dist w (f d z) < e / 2
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/AnalyticManifold/OpenMapping.lean	AnalyticOn.ball_subset_image_closedBall_param	[65, 1]	[101, 101]	specialize sh (max_lt m.1.2 sp)	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 d w : ℂ m : (d ∈ u ∧ dist d c < s) ∧ dist w (f c z) < e / 4 sh : max (dist d c) 0 < s → dist (f d z) (f c z) < e / 4 ⊢ dist w (f d z) < e / 2	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 d w : ℂ m : (d ∈ u ∧ dist d c < s) ∧ dist w (f c z) < e / 4 sh : dist (f d z) (f c z) < e / 4 ⊢ dist w (f d z) < e / 2
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/AnalyticManifold/OpenMapping.lean	AnalyticOn.ball_subset_image_closedBall_param	[65, 1]	[101, 101]	rw [dist_comm] at 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 s : ℝ sp : s > 0 d w : ℂ m : (d ∈ u ∧ dist d c < s) ∧ dist w (f c z) < e / 4 sh : dist (f d z) (f c z) < e / 4 ⊢ dist w (f d z) < e / 2	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 d w : ℂ m : (d ∈ u ∧ dist d c < s) ∧ dist w (f c z) < e / 4 sh : dist (f c z) (f d z) < e / 4 ⊢ dist w (f d z) < e / 2
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/AnalyticManifold/OpenMapping.lean	AnalyticOn.ball_subset_image_closedBall_param	[65, 1]	[101, 101]	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	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 d w : ℂ m : (d ∈ u ∧ dist d c < s) ∧ dist w (f c z) < e / 4 sh : dist (f c z) (f d z) < e / 4 ⊢ dist w (f d z) < e / 2	no goals
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/AnalyticManifold/OpenMapping.lean	AnalyticOn.ball_subset_image_closedBall_param	[65, 1]	[101, 101]	bound	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 d w : ℂ m : (d ∈ u ∧ dist d c < s) ∧ dist w (f c z) < e / 4 sh : dist (f c z) (f d z) < e / 4 ⊢ dist w (f d z) ≤ dist w (f c z) + dist (f c z) (f d z)	no goals
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/AnalyticManifold/OpenMapping.lean	AnalyticOn.ball_subset_image_closedBall_param	[65, 1]	[101, 101]	linarith [m.2]	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 d w : ℂ m : (d ∈ u ∧ dist d c < s) ∧ dist w (f c z) < e / 4 sh : dist (f c z) (f d z) < e / 4 ⊢ dist w (f c z) + dist (f c z) (f d z) < e / 4 + dist (f c z) (f d z)	no goals
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/AnalyticManifold/OpenMapping.lean	AnalyticOn.ball_subset_image_closedBall_param	[65, 1]	[101, 101]	linarith [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 s : ℝ sp : s > 0 d w : ℂ m : (d ∈ u ∧ dist d c < s) ∧ dist w (f c z) < e / 4 sh : dist (f c z) (f d z) < e / 4 ⊢ e / 4 + dist (f c z) (f d z) ≤ e / 4 + e / 4	no goals
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/AnalyticManifold/OpenMapping.lean	AnalyticOn.ball_subset_image_closedBall_param	[65, 1]	[101, 101]	ring	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 d w : ℂ m : (d ∈ u ∧ dist d c < s) ∧ dist w (f c z) < e / 4 sh : dist (f c z) (f d z) < e / 4 ⊢ e / 4 + e / 4 = e / 2	no goals
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/AnalyticManifold/OpenMapping.lean	abs_sub_self_lt	[104, 1]	[105, 44]	simp [sub_self, Complex.abs.map_zero, 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 z : ℂ r : ℝ rp : 0 < r ⊢ Complex.abs (z - z) < r	no goals
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/AnalyticManifold/OpenMapping.lean	NontrivialHolomorphicAt.nhds_le_map_nhds_param'	[109, 1]	[180, 64]	rw [Filter.le_map_iff]	X : Type inst✝⁶ : TopologicalSpace X S : Type inst✝⁵ : TopologicalSpace S inst✝⁴ : ChartedSpace ℂ S cms : AnalyticManifold 𝓘(ℂ, ℂ) S T : Type inst✝³ : TopologicalSpace T inst✝² : ChartedSpace ℂ T cmt : AnalyticManifold 𝓘(ℂ, ℂ) T U : Type inst✝¹ : TopologicalSpace U inst✝ : ChartedSpace ℂ U cmu : AnalyticManifold 𝓘(ℂ, ℂ) U f : ℂ → ℂ → ℂ c z : ℂ n : NontrivialHolomorphicAt (f c) z fa : AnalyticAt ℂ (uncurry f) (c, z) ⊢ 𝓝 (c, f c z) ≤ Filter.map (fun p => (p.1, f p.1 p.2)) (𝓝 (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 n : NontrivialHolomorphicAt (f c) z fa : AnalyticAt ℂ (uncurry f) (c, z) ⊢ ∀ s ∈ 𝓝 (c, z), (fun p => (p.1, f p.1 p.2)) '' s ∈ 𝓝 (c, f c z)
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/AnalyticManifold/OpenMapping.lean	NontrivialHolomorphicAt.nhds_le_map_nhds_param'	[109, 1]	[180, 64]	intro s' sn	X : Type inst✝⁶ : TopologicalSpace X S : Type inst✝⁵ : TopologicalSpace S inst✝⁴ : ChartedSpace ℂ S cms : AnalyticManifold 𝓘(ℂ, ℂ) S T : Type inst✝³ : TopologicalSpace T inst✝² : ChartedSpace ℂ T cmt : AnalyticManifold 𝓘(ℂ, ℂ) T U : Type inst✝¹ : TopologicalSpace U inst✝ : ChartedSpace ℂ U cmu : AnalyticManifold 𝓘(ℂ, ℂ) U f : ℂ → ℂ → ℂ c z : ℂ n : NontrivialHolomorphicAt (f c) z fa : AnalyticAt ℂ (uncurry f) (c, z) ⊢ ∀ s ∈ 𝓝 (c, z), (fun p => (p.1, f p.1 p.2)) '' s ∈ 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 n : NontrivialHolomorphicAt (f c) z fa : AnalyticAt ℂ (uncurry f) (c, z) s' : Set (ℂ × ℂ) sn : s' ∈ 𝓝 (c, z) ⊢ (fun p => (p.1, f p.1 p.2)) '' s' ∈ 𝓝 (c, f c z)
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/AnalyticManifold/OpenMapping.lean	NontrivialHolomorphicAt.nhds_le_map_nhds_param'	[109, 1]	[180, 64]	generalize hs : s' ∩ {p \| AnalyticAt ℂ (uncurry f) p} = s	X : Type inst✝⁶ : TopologicalSpace X S : Type inst✝⁵ : TopologicalSpace S inst✝⁴ : ChartedSpace ℂ S cms : AnalyticManifold 𝓘(ℂ, ℂ) S T : Type inst✝³ : TopologicalSpace T inst✝² : ChartedSpace ℂ T cmt : AnalyticManifold 𝓘(ℂ, ℂ) T U : Type inst✝¹ : TopologicalSpace U inst✝ : ChartedSpace ℂ U cmu : AnalyticManifold 𝓘(ℂ, ℂ) U f : ℂ → ℂ → ℂ c z : ℂ n : NontrivialHolomorphicAt (f c) z fa : AnalyticAt ℂ (uncurry f) (c, z) s' : Set (ℂ × ℂ) sn : s' ∈ 𝓝 (c, z) ⊢ (fun p => (p.1, f p.1 p.2)) '' s' ∈ 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 n : NontrivialHolomorphicAt (f c) z fa : AnalyticAt ℂ (uncurry f) (c, z) s' : Set (ℂ × ℂ) sn : s' ∈ 𝓝 (c, z) s : Set (ℂ × ℂ) hs : s' ∩ {p \| AnalyticAt ℂ (uncurry f) p} = s ⊢ (fun p => (p.1, f p.1 p.2)) '' s' ∈ 𝓝 (c, f c z)
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/AnalyticManifold/OpenMapping.lean	NontrivialHolomorphicAt.nhds_le_map_nhds_param'	[109, 1]	[180, 64]	have ss : s ⊆ s' := by rw [← hs]; apply inter_subset_left	X : Type inst✝⁶ : TopologicalSpace X S : Type inst✝⁵ : TopologicalSpace S inst✝⁴ : ChartedSpace ℂ S cms : AnalyticManifold 𝓘(ℂ, ℂ) S T : Type inst✝³ : TopologicalSpace T inst✝² : ChartedSpace ℂ T cmt : AnalyticManifold 𝓘(ℂ, ℂ) T U : Type inst✝¹ : TopologicalSpace U inst✝ : ChartedSpace ℂ U cmu : AnalyticManifold 𝓘(ℂ, ℂ) U f : ℂ → ℂ → ℂ c z : ℂ n : NontrivialHolomorphicAt (f c) z fa : AnalyticAt ℂ (uncurry f) (c, z) s' : Set (ℂ × ℂ) sn : s' ∈ 𝓝 (c, z) s : Set (ℂ × ℂ) hs : s' ∩ {p \| AnalyticAt ℂ (uncurry f) p} = s ⊢ (fun p => (p.1, f p.1 p.2)) '' s' ∈ 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 n : NontrivialHolomorphicAt (f c) z fa : AnalyticAt ℂ (uncurry f) (c, z) s' : Set (ℂ × ℂ) sn : s' ∈ 𝓝 (c, z) s : Set (ℂ × ℂ) hs : s' ∩ {p \| AnalyticAt ℂ (uncurry f) p} = s ss : s ⊆ s' ⊢ (fun p => (p.1, f p.1 p.2)) '' s' ∈ 𝓝 (c, f c z)
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/AnalyticManifold/OpenMapping.lean	NontrivialHolomorphicAt.nhds_le_map_nhds_param'	[109, 1]	[180, 64]	replace sn : s ∈ 𝓝 (c, z) := by rw [← hs]; exact Filter.inter_mem sn fa.eventually_analyticAt	X : Type inst✝⁶ : TopologicalSpace X S : Type inst✝⁵ : TopologicalSpace S inst✝⁴ : ChartedSpace ℂ S cms : AnalyticManifold 𝓘(ℂ, ℂ) S T : Type inst✝³ : TopologicalSpace T inst✝² : ChartedSpace ℂ T cmt : AnalyticManifold 𝓘(ℂ, ℂ) T U : Type inst✝¹ : TopologicalSpace U inst✝ : ChartedSpace ℂ U cmu : AnalyticManifold 𝓘(ℂ, ℂ) U f : ℂ → ℂ → ℂ c z : ℂ n : NontrivialHolomorphicAt (f c) z fa : AnalyticAt ℂ (uncurry f) (c, z) s' : Set (ℂ × ℂ) sn : s' ∈ 𝓝 (c, z) s : Set (ℂ × ℂ) hs : s' ∩ {p \| AnalyticAt ℂ (uncurry f) p} = s ss : s ⊆ s' ⊢ (fun p => (p.1, f p.1 p.2)) '' s' ∈ 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 n : NontrivialHolomorphicAt (f c) z fa : AnalyticAt ℂ (uncurry f) (c, z) s' s : Set (ℂ × ℂ) hs : s' ∩ {p \| AnalyticAt ℂ (uncurry f) p} = s ss : s ⊆ s' sn : s ∈ 𝓝 (c, z) ⊢ (fun p => (p.1, f p.1 p.2)) '' s' ∈ 𝓝 (c, f c z)
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/AnalyticManifold/OpenMapping.lean	NontrivialHolomorphicAt.nhds_le_map_nhds_param'	[109, 1]	[180, 64]	replace fa : AnalyticOn ℂ (uncurry f) s := by rw [← hs]; apply inter_subset_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 n : NontrivialHolomorphicAt (f c) z fa : AnalyticAt ℂ (uncurry f) (c, z) s' s : Set (ℂ × ℂ) hs : s' ∩ {p \| AnalyticAt ℂ (uncurry f) p} = s ss : s ⊆ s' sn : s ∈ 𝓝 (c, z) ⊢ (fun p => (p.1, f p.1 p.2)) '' s' ∈ 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 n : NontrivialHolomorphicAt (f c) z s' s : Set (ℂ × ℂ) hs : s' ∩ {p \| AnalyticAt ℂ (uncurry f) p} = s ss : s ⊆ s' sn : s ∈ 𝓝 (c, z) fa : AnalyticOn ℂ (uncurry f) s ⊢ (fun p => (p.1, f p.1 p.2)) '' s' ∈ 𝓝 (c, f c z)
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/AnalyticManifold/OpenMapping.lean	NontrivialHolomorphicAt.nhds_le_map_nhds_param'	[109, 1]	[180, 64]	refine Filter.mem_of_superset ?_ (image_subset _ ss)	X : Type inst✝⁶ : TopologicalSpace X S : Type inst✝⁵ : TopologicalSpace S inst✝⁴ : ChartedSpace ℂ S cms : AnalyticManifold 𝓘(ℂ, ℂ) S T : Type inst✝³ : TopologicalSpace T inst✝² : ChartedSpace ℂ T cmt : AnalyticManifold 𝓘(ℂ, ℂ) T U : Type inst✝¹ : TopologicalSpace U inst✝ : ChartedSpace ℂ U cmu : AnalyticManifold 𝓘(ℂ, ℂ) U f : ℂ → ℂ → ℂ c z : ℂ n : NontrivialHolomorphicAt (f c) z s' s : Set (ℂ × ℂ) hs : s' ∩ {p \| AnalyticAt ℂ (uncurry f) p} = s ss : s ⊆ s' sn : s ∈ 𝓝 (c, z) fa : AnalyticOn ℂ (uncurry f) s ⊢ (fun p => (p.1, f p.1 p.2)) '' s' ∈ 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 n : NontrivialHolomorphicAt (f c) z s' s : Set (ℂ × ℂ) hs : s' ∩ {p \| AnalyticAt ℂ (uncurry f) p} = s ss : s ⊆ s' sn : s ∈ 𝓝 (c, z) fa : AnalyticOn ℂ (uncurry f) s ⊢ (fun p => (p.1, f p.1 p.2)) '' s ∈ 𝓝 (c, f c z)
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/AnalyticManifold/OpenMapping.lean	NontrivialHolomorphicAt.nhds_le_map_nhds_param'	[109, 1]	[180, 64]	clear ss hs s'	X : Type inst✝⁶ : TopologicalSpace X S : Type inst✝⁵ : TopologicalSpace S inst✝⁴ : ChartedSpace ℂ S cms : AnalyticManifold 𝓘(ℂ, ℂ) S T : Type inst✝³ : TopologicalSpace T inst✝² : ChartedSpace ℂ T cmt : AnalyticManifold 𝓘(ℂ, ℂ) T U : Type inst✝¹ : TopologicalSpace U inst✝ : ChartedSpace ℂ U cmu : AnalyticManifold 𝓘(ℂ, ℂ) U f : ℂ → ℂ → ℂ c z : ℂ n : NontrivialHolomorphicAt (f c) z s' s : Set (ℂ × ℂ) hs : s' ∩ {p \| AnalyticAt ℂ (uncurry f) p} = s ss : s ⊆ s' sn : s ∈ 𝓝 (c, z) fa : AnalyticOn ℂ (uncurry f) s ⊢ (fun p => (p.1, f p.1 p.2)) '' s ∈ 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 n : NontrivialHolomorphicAt (f c) z s : Set (ℂ × ℂ) sn : s ∈ 𝓝 (c, z) fa : AnalyticOn ℂ (uncurry f) s ⊢ (fun p => (p.1, f p.1 p.2)) '' s ∈ 𝓝 (c, f c z)
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/AnalyticManifold/OpenMapping.lean	NontrivialHolomorphicAt.nhds_le_map_nhds_param'	[109, 1]	[180, 64]	rcases Metric.mem_nhds_iff.mp sn with ⟨e, ep, es⟩	X : Type inst✝⁶ : TopologicalSpace X S : Type inst✝⁵ : TopologicalSpace S inst✝⁴ : ChartedSpace ℂ S cms : AnalyticManifold 𝓘(ℂ, ℂ) S T : Type inst✝³ : TopologicalSpace T inst✝² : ChartedSpace ℂ T cmt : AnalyticManifold 𝓘(ℂ, ℂ) T U : Type inst✝¹ : TopologicalSpace U inst✝ : ChartedSpace ℂ U cmu : AnalyticManifold 𝓘(ℂ, ℂ) U f : ℂ → ℂ → ℂ c z : ℂ n : NontrivialHolomorphicAt (f c) z s : Set (ℂ × ℂ) sn : s ∈ 𝓝 (c, z) fa : AnalyticOn ℂ (uncurry f) s ⊢ (fun p => (p.1, f p.1 p.2)) '' s ∈ 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 n : NontrivialHolomorphicAt (f c) z s : Set (ℂ × ℂ) sn : s ∈ 𝓝 (c, z) fa : AnalyticOn ℂ (uncurry f) s e : ℝ ep : e > 0 es : ball (c, z) e ⊆ s ⊢ (fun p => (p.1, f p.1 p.2)) '' s ∈ 𝓝 (c, f c z)
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/AnalyticManifold/OpenMapping.lean	NontrivialHolomorphicAt.nhds_le_map_nhds_param'	[109, 1]	[180, 64]	rcases er with ⟨r, rp, rs, fr⟩	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 n : NontrivialHolomorphicAt (f c) z s : Set (ℂ × ℂ) sn : s ∈ 𝓝 (c, z) fa : AnalyticOn ℂ (uncurry f) s e : ℝ ep : e > 0 es : ball (c, z) e ⊆ s er : ∃ r, 0 < r ∧ closedBall (c, z) r ⊆ s ∧ f c z ∉ f c '' sphere z r ⊢ (fun p => (p.1, f p.1 p.2)) '' s ∈ 𝓝 (c, f c z)	case intro.intro.intro.intro.intro X : Type inst✝⁶ : TopologicalSpace X S : Type inst✝⁵ : TopologicalSpace S inst✝⁴ : ChartedSpace ℂ S cms : AnalyticManifold 𝓘(ℂ, ℂ) S T : Type inst✝³ : TopologicalSpace T inst✝² : ChartedSpace ℂ T cmt : AnalyticManifold 𝓘(ℂ, ℂ) T U : Type inst✝¹ : TopologicalSpace U inst✝ : ChartedSpace ℂ U cmu : AnalyticManifold 𝓘(ℂ, ℂ) U f : ℂ → ℂ → ℂ c z : ℂ n : NontrivialHolomorphicAt (f c) z s : Set (ℂ × ℂ) sn : s ∈ 𝓝 (c, z) fa : AnalyticOn ℂ (uncurry f) s e : ℝ ep : e > 0 es : ball (c, z) e ⊆ s r : ℝ rp : 0 < r rs : closedBall (c, z) r ⊆ s fr : f c z ∉ f c '' sphere z r ⊢ (fun p => (p.1, f p.1 p.2)) '' s ∈ 𝓝 (c, f c z)
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/AnalyticManifold/OpenMapping.lean	NontrivialHolomorphicAt.nhds_le_map_nhds_param'	[109, 1]	[180, 64]	have fc : ContinuousOn (fun w ↦ ‖f c w - f c z‖) (sphere z r) := by apply ContinuousOn.norm; refine ContinuousOn.sub ?_ continuousOn_const apply fa.along_snd.continuousOn.mono; intro x xs; apply rs simp only [← closedBall_prod_same, mem_prod_eq] use Metric.mem_closedBall_self rp.le, Metric.sphere_subset_closedBall xs	case intro.intro.intro.intro.intro X : Type inst✝⁶ : TopologicalSpace X S : Type inst✝⁵ : TopologicalSpace S inst✝⁴ : ChartedSpace ℂ S cms : AnalyticManifold 𝓘(ℂ, ℂ) S T : Type inst✝³ : TopologicalSpace T inst✝² : ChartedSpace ℂ T cmt : AnalyticManifold 𝓘(ℂ, ℂ) T U : Type inst✝¹ : TopologicalSpace U inst✝ : ChartedSpace ℂ U cmu : AnalyticManifold 𝓘(ℂ, ℂ) U f : ℂ → ℂ → ℂ c z : ℂ n : NontrivialHolomorphicAt (f c) z s : Set (ℂ × ℂ) sn : s ∈ 𝓝 (c, z) fa : AnalyticOn ℂ (uncurry f) s e : ℝ ep : e > 0 es : ball (c, z) e ⊆ s r : ℝ rp : 0 < r rs : closedBall (c, z) r ⊆ s fr : f c z ∉ f c '' sphere z r ⊢ (fun p => (p.1, f p.1 p.2)) '' s ∈ 𝓝 (c, f c z)	case intro.intro.intro.intro.intro X : Type inst✝⁶ : TopologicalSpace X S : Type inst✝⁵ : TopologicalSpace S inst✝⁴ : ChartedSpace ℂ S cms : AnalyticManifold 𝓘(ℂ, ℂ) S T : Type inst✝³ : TopologicalSpace T inst✝² : ChartedSpace ℂ T cmt : AnalyticManifold 𝓘(ℂ, ℂ) T U : Type inst✝¹ : TopologicalSpace U inst✝ : ChartedSpace ℂ U cmu : AnalyticManifold 𝓘(ℂ, ℂ) U f : ℂ → ℂ → ℂ c z : ℂ n : NontrivialHolomorphicAt (f c) z s : Set (ℂ × ℂ) sn : s ∈ 𝓝 (c, z) fa : AnalyticOn ℂ (uncurry f) s e : ℝ ep : e > 0 es : ball (c, z) e ⊆ s r : ℝ rp : 0 < r rs : closedBall (c, z) r ⊆ s fr : f c z ∉ f c '' sphere z r fc : ContinuousOn (fun w => ‖f c w - f c z‖) (sphere z r) ⊢ (fun p => (p.1, f p.1 p.2)) '' s ∈ 𝓝 (c, f c z)
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/AnalyticManifold/OpenMapping.lean	NontrivialHolomorphicAt.nhds_le_map_nhds_param'	[109, 1]	[180, 64]	rcases (isCompact_sphere _ _).exists_isMinOn (NormedSpace.sphere_nonempty.mpr rp.le) fc with ⟨x, xs, xm⟩	case intro.intro.intro.intro.intro X : Type inst✝⁶ : TopologicalSpace X S : Type inst✝⁵ : TopologicalSpace S inst✝⁴ : ChartedSpace ℂ S cms : AnalyticManifold 𝓘(ℂ, ℂ) S T : Type inst✝³ : TopologicalSpace T inst✝² : ChartedSpace ℂ T cmt : AnalyticManifold 𝓘(ℂ, ℂ) T U : Type inst✝¹ : TopologicalSpace U inst✝ : ChartedSpace ℂ U cmu : AnalyticManifold 𝓘(ℂ, ℂ) U f : ℂ → ℂ → ℂ c z : ℂ n : NontrivialHolomorphicAt (f c) z s : Set (ℂ × ℂ) sn : s ∈ 𝓝 (c, z) fa : AnalyticOn ℂ (uncurry f) s e : ℝ ep : e > 0 es : ball (c, z) e ⊆ s r : ℝ rp : 0 < r rs : closedBall (c, z) r ⊆ s fr : f c z ∉ f c '' sphere z r fc : ContinuousOn (fun w => ‖f c w - f c z‖) (sphere z r) ⊢ (fun p => (p.1, f p.1 p.2)) '' s ∈ 𝓝 (c, f c z)	case intro.intro.intro.intro.intro.intro.intro X : Type inst✝⁶ : TopologicalSpace X S : Type inst✝⁵ : TopologicalSpace S inst✝⁴ : ChartedSpace ℂ S cms : AnalyticManifold 𝓘(ℂ, ℂ) S T : Type inst✝³ : TopologicalSpace T inst✝² : ChartedSpace ℂ T cmt : AnalyticManifold 𝓘(ℂ, ℂ) T U : Type inst✝¹ : TopologicalSpace U inst✝ : ChartedSpace ℂ U cmu : AnalyticManifold 𝓘(ℂ, ℂ) U f : ℂ → ℂ → ℂ c z : ℂ n : NontrivialHolomorphicAt (f c) z s : Set (ℂ × ℂ) sn : s ∈ 𝓝 (c, z) fa : AnalyticOn ℂ (uncurry f) s e : ℝ ep : e > 0 es : ball (c, z) e ⊆ s r : ℝ rp : 0 < r rs : closedBall (c, z) r ⊆ s fr : f c z ∉ f c '' sphere z r fc : ContinuousOn (fun w => ‖f c w - f c z‖) (sphere z r) x : ℂ xs : x ∈ sphere z r xm : IsMinOn (fun w => ‖f c w - f c z‖) (sphere z r) x ⊢ (fun p => (p.1, f p.1 p.2)) '' s ∈ 𝓝 (c, f c z)
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/AnalyticManifold/OpenMapping.lean	NontrivialHolomorphicAt.nhds_le_map_nhds_param'	[109, 1]	[180, 64]	generalize he : ‖f c x - f c z‖ = e	case intro.intro.intro.intro.intro.intro.intro X : Type inst✝⁶ : TopologicalSpace X S : Type inst✝⁵ : TopologicalSpace S inst✝⁴ : ChartedSpace ℂ S cms : AnalyticManifold 𝓘(ℂ, ℂ) S T : Type inst✝³ : TopologicalSpace T inst✝² : ChartedSpace ℂ T cmt : AnalyticManifold 𝓘(ℂ, ℂ) T U : Type inst✝¹ : TopologicalSpace U inst✝ : ChartedSpace ℂ U cmu : AnalyticManifold 𝓘(ℂ, ℂ) U f : ℂ → ℂ → ℂ c z : ℂ n : NontrivialHolomorphicAt (f c) z s : Set (ℂ × ℂ) sn : s ∈ 𝓝 (c, z) fa : AnalyticOn ℂ (uncurry f) s e : ℝ ep : e > 0 es : ball (c, z) e ⊆ s r : ℝ rp : 0 < r rs : closedBall (c, z) r ⊆ s fr : f c z ∉ f c '' sphere z r fc : ContinuousOn (fun w => ‖f c w - f c z‖) (sphere z r) x : ℂ xs : x ∈ sphere z r xm : IsMinOn (fun w => ‖f c w - f c z‖) (sphere z r) x ⊢ (fun p => (p.1, f p.1 p.2)) '' s ∈ 𝓝 (c, f c z)	case intro.intro.intro.intro.intro.intro.intro X : Type inst✝⁶ : TopologicalSpace X S : Type inst✝⁵ : TopologicalSpace S inst✝⁴ : ChartedSpace ℂ S cms : AnalyticManifold 𝓘(ℂ, ℂ) S T : Type inst✝³ : TopologicalSpace T inst✝² : ChartedSpace ℂ T cmt : AnalyticManifold 𝓘(ℂ, ℂ) T U : Type inst✝¹ : TopologicalSpace U inst✝ : ChartedSpace ℂ U cmu : AnalyticManifold 𝓘(ℂ, ℂ) U f : ℂ → ℂ → ℂ c z : ℂ n : NontrivialHolomorphicAt (f c) z s : Set (ℂ × ℂ) sn : s ∈ 𝓝 (c, z) fa : AnalyticOn ℂ (uncurry f) s e✝ : ℝ ep : e✝ > 0 es : ball (c, z) e✝ ⊆ s r : ℝ rp : 0 < r rs : closedBall (c, z) r ⊆ s fr : f c z ∉ f c '' sphere z r fc : ContinuousOn (fun w => ‖f c w - f c z‖) (sphere z r) x : ℂ xs : x ∈ sphere z r xm : IsMinOn (fun w => ‖f c w - f c z‖) (sphere z r) x e : ℝ he : ‖f c x - f c z‖ = e ⊢ (fun p => (p.1, f p.1 p.2)) '' s ∈ 𝓝 (c, f c z)
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/AnalyticManifold/OpenMapping.lean	NontrivialHolomorphicAt.nhds_le_map_nhds_param'	[109, 1]	[180, 64]	have ep : 0 < e := by contrapose fr simp only [norm_pos_iff, sub_ne_zero, not_not, mem_image, ← he] at fr ⊢ use x, xs, fr	case intro.intro.intro.intro.intro.intro.intro X : Type inst✝⁶ : TopologicalSpace X S : Type inst✝⁵ : TopologicalSpace S inst✝⁴ : ChartedSpace ℂ S cms : AnalyticManifold 𝓘(ℂ, ℂ) S T : Type inst✝³ : TopologicalSpace T inst✝² : ChartedSpace ℂ T cmt : AnalyticManifold 𝓘(ℂ, ℂ) T U : Type inst✝¹ : TopologicalSpace U inst✝ : ChartedSpace ℂ U cmu : AnalyticManifold 𝓘(ℂ, ℂ) U f : ℂ → ℂ → ℂ c z : ℂ n : NontrivialHolomorphicAt (f c) z s : Set (ℂ × ℂ) sn : s ∈ 𝓝 (c, z) fa : AnalyticOn ℂ (uncurry f) s e✝ : ℝ ep : e✝ > 0 es : ball (c, z) e✝ ⊆ s r : ℝ rp : 0 < r rs : closedBall (c, z) r ⊆ s fr : f c z ∉ f c '' sphere z r fc : ContinuousOn (fun w => ‖f c w - f c z‖) (sphere z r) x : ℂ xs : x ∈ sphere z r xm : IsMinOn (fun w => ‖f c w - f c z‖) (sphere z r) x e : ℝ he : ‖f c x - f c z‖ = e ⊢ (fun p => (p.1, f p.1 p.2)) '' s ∈ 𝓝 (c, f c z)	case intro.intro.intro.intro.intro.intro.intro X : Type inst✝⁶ : TopologicalSpace X S : Type inst✝⁵ : TopologicalSpace S inst✝⁴ : ChartedSpace ℂ S cms : AnalyticManifold 𝓘(ℂ, ℂ) S T : Type inst✝³ : TopologicalSpace T inst✝² : ChartedSpace ℂ T cmt : AnalyticManifold 𝓘(ℂ, ℂ) T U : Type inst✝¹ : TopologicalSpace U inst✝ : ChartedSpace ℂ U cmu : AnalyticManifold 𝓘(ℂ, ℂ) U f : ℂ → ℂ → ℂ c z : ℂ n : NontrivialHolomorphicAt (f c) z s : Set (ℂ × ℂ) sn : s ∈ 𝓝 (c, z) fa : AnalyticOn ℂ (uncurry f) s e✝ : ℝ ep✝ : e✝ > 0 es : ball (c, z) e✝ ⊆ s r : ℝ rp : 0 < r rs : closedBall (c, z) r ⊆ s fr : f c z ∉ f c '' sphere z r fc : ContinuousOn (fun w => ‖f c w - f c z‖) (sphere z r) x : ℂ xs : x ∈ sphere z r xm : IsMinOn (fun w => ‖f c w - f c z‖) (sphere z r) x e : ℝ he : ‖f c x - f c z‖ = e ep : 0 < e ⊢ (fun p => (p.1, f p.1 p.2)) '' s ∈ 𝓝 (c, f c z)
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/AnalyticManifold/OpenMapping.lean	NontrivialHolomorphicAt.nhds_le_map_nhds_param'	[109, 1]	[180, 64]	rcases Metric.uniformContinuousOn_iff.mp ((isCompact_closedBall _ _).uniformContinuousOn_of_continuous (fa.continuousOn.mono rs)) (e / 4) (by linarith) with ⟨t, tp, ft⟩	case intro.intro.intro.intro.intro.intro.intro X : Type inst✝⁶ : TopologicalSpace X S : Type inst✝⁵ : TopologicalSpace S inst✝⁴ : ChartedSpace ℂ S cms : AnalyticManifold 𝓘(ℂ, ℂ) S T : Type inst✝³ : TopologicalSpace T inst✝² : ChartedSpace ℂ T cmt : AnalyticManifold 𝓘(ℂ, ℂ) T U : Type inst✝¹ : TopologicalSpace U inst✝ : ChartedSpace ℂ U cmu : AnalyticManifold 𝓘(ℂ, ℂ) U f : ℂ → ℂ → ℂ c z : ℂ n : NontrivialHolomorphicAt (f c) z s : Set (ℂ × ℂ) sn : s ∈ 𝓝 (c, z) fa : AnalyticOn ℂ (uncurry f) s e✝ : ℝ ep✝ : e✝ > 0 es : ball (c, z) e✝ ⊆ s r : ℝ rp : 0 < r rs : closedBall (c, z) r ⊆ s fr : f c z ∉ f c '' sphere z r fc : ContinuousOn (fun w => ‖f c w - f c z‖) (sphere z r) x : ℂ xs : x ∈ sphere z r xm : IsMinOn (fun w => ‖f c w - f c z‖) (sphere z r) x e : ℝ he : ‖f c x - f c z‖ = e ep : 0 < e ⊢ (fun p => (p.1, f p.1 p.2)) '' s ∈ 𝓝 (c, f c z)	case intro.intro.intro.intro.intro.intro.intro.intro.intro X : Type inst✝⁶ : TopologicalSpace X S : Type inst✝⁵ : TopologicalSpace S inst✝⁴ : ChartedSpace ℂ S cms : AnalyticManifold 𝓘(ℂ, ℂ) S T : Type inst✝³ : TopologicalSpace T inst✝² : ChartedSpace ℂ T cmt : AnalyticManifold 𝓘(ℂ, ℂ) T U : Type inst✝¹ : TopologicalSpace U inst✝ : ChartedSpace ℂ U cmu : AnalyticManifold 𝓘(ℂ, ℂ) U f : ℂ → ℂ → ℂ c z : ℂ n : NontrivialHolomorphicAt (f c) z s : Set (ℂ × ℂ) sn : s ∈ 𝓝 (c, z) fa : AnalyticOn ℂ (uncurry f) s e✝ : ℝ ep✝ : e✝ > 0 es : ball (c, z) e✝ ⊆ s r : ℝ rp : 0 < r rs : closedBall (c, z) r ⊆ s fr : f c z ∉ f c '' sphere z r fc : ContinuousOn (fun w => ‖f c w - f c z‖) (sphere z r) x : ℂ xs : x ∈ sphere z r xm : IsMinOn (fun w => ‖f c w - f c z‖) (sphere z r) x e : ℝ he : ‖f c x - f c z‖ = e ep : 0 < e t : ℝ tp : t > 0 ft : ∀ x ∈ closedBall (c, z) r, ∀ y ∈ closedBall (c, z) r, dist x y < t → dist (uncurry f x) (uncurry f y) < e / 4 ⊢ (fun p => (p.1, f p.1 p.2)) '' s ∈ 𝓝 (c, f c z)
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/AnalyticManifold/OpenMapping.lean	NontrivialHolomorphicAt.nhds_le_map_nhds_param'	[109, 1]	[180, 64]	have ef : ∀ d, d ∈ ball c (min t r) → ∀ w, w ∈ sphere z r → e / 2 ≤ ‖f d w - f d z‖ := by intro d dt w wr; simp only [Complex.norm_eq_abs] simp only [Complex.dist_eq, Prod.forall, mem_closedBall, Prod.dist_eq, max_le_iff, max_lt_iff, Function.uncurry, and_imp] at ft simp only [mem_ball, Complex.dist_eq, lt_min_iff] at dt have a1 : abs (f d w - f c w) ≤ e / 4 := (ft d w dt.2.le (le_of_eq wr) c w (abs_sub_self_lt rp).le (le_of_eq wr) dt.1 (abs_sub_self_lt tp)).le have a2 : abs (f c z - f d z) ≤ e / 4 := by refine (ft c z (abs_sub_self_lt rp).le (abs_sub_self_lt rp).le d z dt.2.le (abs_sub_self_lt rp).le ?_ (abs_sub_self_lt tp)).le rw [← neg_sub, Complex.abs.map_neg]; exact dt.1 calc abs (f d w - f d z) _ = abs (f c w - f c z + (f d w - f c w) + (f c z - f d z)) := by ring_nf _ ≥ abs (f c w - f c z + (f d w - f c w)) - abs (f c z - f d z) := by bound _ ≥ abs (f c w - f c z) - abs (f d w - f c w) - abs (f c z - f d z) := by bound _ ≥ e - e / 4 - e / 4 := by rw [← he] at a1 a2 ⊢; exact sub_le_sub (sub_le_sub (xm wr) a1) a2 _ = e / 2 := by ring	case intro.intro.intro.intro.intro.intro.intro.intro.intro X : Type inst✝⁶ : TopologicalSpace X S : Type inst✝⁵ : TopologicalSpace S inst✝⁴ : ChartedSpace ℂ S cms : AnalyticManifold 𝓘(ℂ, ℂ) S T : Type inst✝³ : TopologicalSpace T inst✝² : ChartedSpace ℂ T cmt : AnalyticManifold 𝓘(ℂ, ℂ) T U : Type inst✝¹ : TopologicalSpace U inst✝ : ChartedSpace ℂ U cmu : AnalyticManifold 𝓘(ℂ, ℂ) U f : ℂ → ℂ → ℂ c z : ℂ n : NontrivialHolomorphicAt (f c) z s : Set (ℂ × ℂ) sn : s ∈ 𝓝 (c, z) fa : AnalyticOn ℂ (uncurry f) s e✝ : ℝ ep✝ : e✝ > 0 es : ball (c, z) e✝ ⊆ s r : ℝ rp : 0 < r rs : closedBall (c, z) r ⊆ s fr : f c z ∉ f c '' sphere z r fc : ContinuousOn (fun w => ‖f c w - f c z‖) (sphere z r) x : ℂ xs : x ∈ sphere z r xm : IsMinOn (fun w => ‖f c w - f c z‖) (sphere z r) x e : ℝ he : ‖f c x - f c z‖ = e ep : 0 < e t : ℝ tp : t > 0 ft : ∀ x ∈ closedBall (c, z) r, ∀ y ∈ closedBall (c, z) r, dist x y < t → dist (uncurry f x) (uncurry f y) < e / 4 ⊢ (fun p => (p.1, f p.1 p.2)) '' s ∈ 𝓝 (c, f c z)	case intro.intro.intro.intro.intro.intro.intro.intro.intro X : Type inst✝⁶ : TopologicalSpace X S : Type inst✝⁵ : TopologicalSpace S inst✝⁴ : ChartedSpace ℂ S cms : AnalyticManifold 𝓘(ℂ, ℂ) S T : Type inst✝³ : TopologicalSpace T inst✝² : ChartedSpace ℂ T cmt : AnalyticManifold 𝓘(ℂ, ℂ) T U : Type inst✝¹ : TopologicalSpace U inst✝ : ChartedSpace ℂ U cmu : AnalyticManifold 𝓘(ℂ, ℂ) U f : ℂ → ℂ → ℂ c z : ℂ n : NontrivialHolomorphicAt (f c) z s : Set (ℂ × ℂ) sn : s ∈ 𝓝 (c, z) fa : AnalyticOn ℂ (uncurry f) s e✝ : ℝ ep✝ : e✝ > 0 es : ball (c, z) e✝ ⊆ s r : ℝ rp : 0 < r rs : closedBall (c, z) r ⊆ s fr : f c z ∉ f c '' sphere z r fc : ContinuousOn (fun w => ‖f c w - f c z‖) (sphere z r) x : ℂ xs : x ∈ sphere z r xm : IsMinOn (fun w => ‖f c w - f c z‖) (sphere z r) x e : ℝ he : ‖f c x - f c z‖ = e ep : 0 < e t : ℝ tp : t > 0 ft : ∀ x ∈ closedBall (c, z) r, ∀ y ∈ closedBall (c, z) r, dist x y < t → dist (uncurry f x) (uncurry f y) < e / 4 ef : ∀ d ∈ ball c (min t r), ∀ w ∈ sphere z r, e / 2 ≤ ‖f d w - f d z‖ ⊢ (fun p => (p.1, f p.1 p.2)) '' s ∈ 𝓝 (c, f c z)
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/AnalyticManifold/OpenMapping.lean	NontrivialHolomorphicAt.nhds_le_map_nhds_param'	[109, 1]	[180, 64]	have ss : ball c (min t r) ×ˢ closedBall z r ⊆ s := by refine _root_.trans ?_ rs; rw [← closedBall_prod_same]; apply prod_mono_left exact _root_.trans (Metric.ball_subset_ball (min_le_right _ _)) Metric.ball_subset_closedBall	case intro.intro.intro.intro.intro.intro.intro.intro.intro X : Type inst✝⁶ : TopologicalSpace X S : Type inst✝⁵ : TopologicalSpace S inst✝⁴ : ChartedSpace ℂ S cms : AnalyticManifold 𝓘(ℂ, ℂ) S T : Type inst✝³ : TopologicalSpace T inst✝² : ChartedSpace ℂ T cmt : AnalyticManifold 𝓘(ℂ, ℂ) T U : Type inst✝¹ : TopologicalSpace U inst✝ : ChartedSpace ℂ U cmu : AnalyticManifold 𝓘(ℂ, ℂ) U f : ℂ → ℂ → ℂ c z : ℂ n : NontrivialHolomorphicAt (f c) z s : Set (ℂ × ℂ) sn : s ∈ 𝓝 (c, z) fa : AnalyticOn ℂ (uncurry f) s e✝ : ℝ ep✝ : e✝ > 0 es : ball (c, z) e✝ ⊆ s r : ℝ rp : 0 < r rs : closedBall (c, z) r ⊆ s fr : f c z ∉ f c '' sphere z r fc : ContinuousOn (fun w => ‖f c w - f c z‖) (sphere z r) x : ℂ xs : x ∈ sphere z r xm : IsMinOn (fun w => ‖f c w - f c z‖) (sphere z r) x e : ℝ he : ‖f c x - f c z‖ = e ep : 0 < e t : ℝ tp : t > 0 ft : ∀ x ∈ closedBall (c, z) r, ∀ y ∈ closedBall (c, z) r, dist x y < t → dist (uncurry f x) (uncurry f y) < e / 4 ef : ∀ d ∈ ball c (min t r), ∀ w ∈ sphere z r, e / 2 ≤ ‖f d w - f d z‖ ⊢ (fun p => (p.1, f p.1 p.2)) '' s ∈ 𝓝 (c, f c z)	case intro.intro.intro.intro.intro.intro.intro.intro.intro X : Type inst✝⁶ : TopologicalSpace X S : Type inst✝⁵ : TopologicalSpace S inst✝⁴ : ChartedSpace ℂ S cms : AnalyticManifold 𝓘(ℂ, ℂ) S T : Type inst✝³ : TopologicalSpace T inst✝² : ChartedSpace ℂ T cmt : AnalyticManifold 𝓘(ℂ, ℂ) T U : Type inst✝¹ : TopologicalSpace U inst✝ : ChartedSpace ℂ U cmu : AnalyticManifold 𝓘(ℂ, ℂ) U f : ℂ → ℂ → ℂ c z : ℂ n : NontrivialHolomorphicAt (f c) z s : Set (ℂ × ℂ) sn : s ∈ 𝓝 (c, z) fa : AnalyticOn ℂ (uncurry f) s e✝ : ℝ ep✝ : e✝ > 0 es : ball (c, z) e✝ ⊆ s r : ℝ rp : 0 < r rs : closedBall (c, z) r ⊆ s fr : f c z ∉ f c '' sphere z r fc : ContinuousOn (fun w => ‖f c w - f c z‖) (sphere z r) x : ℂ xs : x ∈ sphere z r xm : IsMinOn (fun w => ‖f c w - f c z‖) (sphere z r) x e : ℝ he : ‖f c x - f c z‖ = e ep : 0 < e t : ℝ tp : t > 0 ft : ∀ x ∈ closedBall (c, z) r, ∀ y ∈ closedBall (c, z) r, dist x y < t → dist (uncurry f x) (uncurry f y) < e / 4 ef : ∀ d ∈ ball c (min t r), ∀ w ∈ sphere z r, e / 2 ≤ ‖f d w - f d z‖ ss : ball c (min t r) ×ˢ closedBall z r ⊆ s ⊢ (fun p => (p.1, f p.1 p.2)) '' s ∈ 𝓝 (c, f c z)
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/AnalyticManifold/OpenMapping.lean	NontrivialHolomorphicAt.nhds_le_map_nhds_param'	[109, 1]	[180, 64]	exact Filter.mem_of_superset ((fa.mono ss).ball_subset_image_closedBall_param rp (half_pos ep) (Metric.ball_mem_nhds _ (by bound)) ef) (image_subset _ ss)	case intro.intro.intro.intro.intro.intro.intro.intro.intro X : Type inst✝⁶ : TopologicalSpace X S : Type inst✝⁵ : TopologicalSpace S inst✝⁴ : ChartedSpace ℂ S cms : AnalyticManifold 𝓘(ℂ, ℂ) S T : Type inst✝³ : TopologicalSpace T inst✝² : ChartedSpace ℂ T cmt : AnalyticManifold 𝓘(ℂ, ℂ) T U : Type inst✝¹ : TopologicalSpace U inst✝ : ChartedSpace ℂ U cmu : AnalyticManifold 𝓘(ℂ, ℂ) U f : ℂ → ℂ → ℂ c z : ℂ n : NontrivialHolomorphicAt (f c) z s : Set (ℂ × ℂ) sn : s ∈ 𝓝 (c, z) fa : AnalyticOn ℂ (uncurry f) s e✝ : ℝ ep✝ : e✝ > 0 es : ball (c, z) e✝ ⊆ s r : ℝ rp : 0 < r rs : closedBall (c, z) r ⊆ s fr : f c z ∉ f c '' sphere z r fc : ContinuousOn (fun w => ‖f c w - f c z‖) (sphere z r) x : ℂ xs : x ∈ sphere z r xm : IsMinOn (fun w => ‖f c w - f c z‖) (sphere z r) x e : ℝ he : ‖f c x - f c z‖ = e ep : 0 < e t : ℝ tp : t > 0 ft : ∀ x ∈ closedBall (c, z) r, ∀ y ∈ closedBall (c, z) r, dist x y < t → dist (uncurry f x) (uncurry f y) < e / 4 ef : ∀ d ∈ ball c (min t r), ∀ w ∈ sphere z r, e / 2 ≤ ‖f d w - f d z‖ ss : ball c (min t r) ×ˢ closedBall z r ⊆ s ⊢ (fun p => (p.1, f p.1 p.2)) '' s ∈ 𝓝 (c, f c z)	no goals
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/AnalyticManifold/OpenMapping.lean	NontrivialHolomorphicAt.nhds_le_map_nhds_param'	[109, 1]	[180, 64]	rw [← hs]	X : Type inst✝⁶ : TopologicalSpace X S : Type inst✝⁵ : TopologicalSpace S inst✝⁴ : ChartedSpace ℂ S cms : AnalyticManifold 𝓘(ℂ, ℂ) S T : Type inst✝³ : TopologicalSpace T inst✝² : ChartedSpace ℂ T cmt : AnalyticManifold 𝓘(ℂ, ℂ) T U : Type inst✝¹ : TopologicalSpace U inst✝ : ChartedSpace ℂ U cmu : AnalyticManifold 𝓘(ℂ, ℂ) U f : ℂ → ℂ → ℂ c z : ℂ n : NontrivialHolomorphicAt (f c) z fa : AnalyticAt ℂ (uncurry f) (c, z) s' : Set (ℂ × ℂ) sn : s' ∈ 𝓝 (c, z) s : Set (ℂ × ℂ) hs : s' ∩ {p \| AnalyticAt ℂ (uncurry f) p} = s ⊢ s ⊆ s'	X : Type inst✝⁶ : TopologicalSpace X S : Type inst✝⁵ : TopologicalSpace S inst✝⁴ : ChartedSpace ℂ S cms : AnalyticManifold 𝓘(ℂ, ℂ) S T : Type inst✝³ : TopologicalSpace T inst✝² : ChartedSpace ℂ T cmt : AnalyticManifold 𝓘(ℂ, ℂ) T U : Type inst✝¹ : TopologicalSpace U inst✝ : ChartedSpace ℂ U cmu : AnalyticManifold 𝓘(ℂ, ℂ) U f : ℂ → ℂ → ℂ c z : ℂ n : NontrivialHolomorphicAt (f c) z fa : AnalyticAt ℂ (uncurry f) (c, z) s' : Set (ℂ × ℂ) sn : s' ∈ 𝓝 (c, z) s : Set (ℂ × ℂ) hs : s' ∩ {p \| AnalyticAt ℂ (uncurry f) p} = s ⊢ s' ∩ {p \| AnalyticAt ℂ (uncurry f) p} ⊆ s'
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/AnalyticManifold/OpenMapping.lean	NontrivialHolomorphicAt.nhds_le_map_nhds_param'	[109, 1]	[180, 64]	apply inter_subset_left	X : Type inst✝⁶ : TopologicalSpace X S : Type inst✝⁵ : TopologicalSpace S inst✝⁴ : ChartedSpace ℂ S cms : AnalyticManifold 𝓘(ℂ, ℂ) S T : Type inst✝³ : TopologicalSpace T inst✝² : ChartedSpace ℂ T cmt : AnalyticManifold 𝓘(ℂ, ℂ) T U : Type inst✝¹ : TopologicalSpace U inst✝ : ChartedSpace ℂ U cmu : AnalyticManifold 𝓘(ℂ, ℂ) U f : ℂ → ℂ → ℂ c z : ℂ n : NontrivialHolomorphicAt (f c) z fa : AnalyticAt ℂ (uncurry f) (c, z) s' : Set (ℂ × ℂ) sn : s' ∈ 𝓝 (c, z) s : Set (ℂ × ℂ) hs : s' ∩ {p \| AnalyticAt ℂ (uncurry f) p} = s ⊢ s' ∩ {p \| AnalyticAt ℂ (uncurry f) p} ⊆ s'	no goals
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/AnalyticManifold/OpenMapping.lean	NontrivialHolomorphicAt.nhds_le_map_nhds_param'	[109, 1]	[180, 64]	rw [← hs]	X : Type inst✝⁶ : TopologicalSpace X S : Type inst✝⁵ : TopologicalSpace S inst✝⁴ : ChartedSpace ℂ S cms : AnalyticManifold 𝓘(ℂ, ℂ) S T : Type inst✝³ : TopologicalSpace T inst✝² : ChartedSpace ℂ T cmt : AnalyticManifold 𝓘(ℂ, ℂ) T U : Type inst✝¹ : TopologicalSpace U inst✝ : ChartedSpace ℂ U cmu : AnalyticManifold 𝓘(ℂ, ℂ) U f : ℂ → ℂ → ℂ c z : ℂ n : NontrivialHolomorphicAt (f c) z fa : AnalyticAt ℂ (uncurry f) (c, z) s' : Set (ℂ × ℂ) sn : s' ∈ 𝓝 (c, z) s : Set (ℂ × ℂ) hs : s' ∩ {p \| AnalyticAt ℂ (uncurry f) p} = s ss : s ⊆ s' ⊢ s ∈ 𝓝 (c, z)	X : Type inst✝⁶ : TopologicalSpace X S : Type inst✝⁵ : TopologicalSpace S inst✝⁴ : ChartedSpace ℂ S cms : AnalyticManifold 𝓘(ℂ, ℂ) S T : Type inst✝³ : TopologicalSpace T inst✝² : ChartedSpace ℂ T cmt : AnalyticManifold 𝓘(ℂ, ℂ) T U : Type inst✝¹ : TopologicalSpace U inst✝ : ChartedSpace ℂ U cmu : AnalyticManifold 𝓘(ℂ, ℂ) U f : ℂ → ℂ → ℂ c z : ℂ n : NontrivialHolomorphicAt (f c) z fa : AnalyticAt ℂ (uncurry f) (c, z) s' : Set (ℂ × ℂ) sn : s' ∈ 𝓝 (c, z) s : Set (ℂ × ℂ) hs : s' ∩ {p \| AnalyticAt ℂ (uncurry f) p} = s ss : s ⊆ s' ⊢ s' ∩ {p \| AnalyticAt ℂ (uncurry f) p} ∈ 𝓝 (c, z)
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/AnalyticManifold/OpenMapping.lean	NontrivialHolomorphicAt.nhds_le_map_nhds_param'	[109, 1]	[180, 64]	exact Filter.inter_mem sn fa.eventually_analyticAt	X : Type inst✝⁶ : TopologicalSpace X S : Type inst✝⁵ : TopologicalSpace S inst✝⁴ : ChartedSpace ℂ S cms : AnalyticManifold 𝓘(ℂ, ℂ) S T : Type inst✝³ : TopologicalSpace T inst✝² : ChartedSpace ℂ T cmt : AnalyticManifold 𝓘(ℂ, ℂ) T U : Type inst✝¹ : TopologicalSpace U inst✝ : ChartedSpace ℂ U cmu : AnalyticManifold 𝓘(ℂ, ℂ) U f : ℂ → ℂ → ℂ c z : ℂ n : NontrivialHolomorphicAt (f c) z fa : AnalyticAt ℂ (uncurry f) (c, z) s' : Set (ℂ × ℂ) sn : s' ∈ 𝓝 (c, z) s : Set (ℂ × ℂ) hs : s' ∩ {p \| AnalyticAt ℂ (uncurry f) p} = s ss : s ⊆ s' ⊢ s' ∩ {p \| AnalyticAt ℂ (uncurry f) p} ∈ 𝓝 (c, z)	no goals
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/AnalyticManifold/OpenMapping.lean	NontrivialHolomorphicAt.nhds_le_map_nhds_param'	[109, 1]	[180, 64]	rw [← hs]	X : Type inst✝⁶ : TopologicalSpace X S : Type inst✝⁵ : TopologicalSpace S inst✝⁴ : ChartedSpace ℂ S cms : AnalyticManifold 𝓘(ℂ, ℂ) S T : Type inst✝³ : TopologicalSpace T inst✝² : ChartedSpace ℂ T cmt : AnalyticManifold 𝓘(ℂ, ℂ) T U : Type inst✝¹ : TopologicalSpace U inst✝ : ChartedSpace ℂ U cmu : AnalyticManifold 𝓘(ℂ, ℂ) U f : ℂ → ℂ → ℂ c z : ℂ n : NontrivialHolomorphicAt (f c) z fa : AnalyticAt ℂ (uncurry f) (c, z) s' s : Set (ℂ × ℂ) hs : s' ∩ {p \| AnalyticAt ℂ (uncurry f) p} = s ss : s ⊆ s' sn : s ∈ 𝓝 (c, z) ⊢ AnalyticOn ℂ (uncurry f) s	X : Type inst✝⁶ : TopologicalSpace X S : Type inst✝⁵ : TopologicalSpace S inst✝⁴ : ChartedSpace ℂ S cms : AnalyticManifold 𝓘(ℂ, ℂ) S T : Type inst✝³ : TopologicalSpace T inst✝² : ChartedSpace ℂ T cmt : AnalyticManifold 𝓘(ℂ, ℂ) T U : Type inst✝¹ : TopologicalSpace U inst✝ : ChartedSpace ℂ U cmu : AnalyticManifold 𝓘(ℂ, ℂ) U f : ℂ → ℂ → ℂ c z : ℂ n : NontrivialHolomorphicAt (f c) z fa : AnalyticAt ℂ (uncurry f) (c, z) s' s : Set (ℂ × ℂ) hs : s' ∩ {p \| AnalyticAt ℂ (uncurry f) p} = s ss : s ⊆ s' sn : s ∈ 𝓝 (c, z) ⊢ AnalyticOn ℂ (uncurry f) (s' ∩ {p \| AnalyticAt ℂ (uncurry f) p})
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/AnalyticManifold/OpenMapping.lean	NontrivialHolomorphicAt.nhds_le_map_nhds_param'	[109, 1]	[180, 64]	apply inter_subset_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 n : NontrivialHolomorphicAt (f c) z fa : AnalyticAt ℂ (uncurry f) (c, z) s' s : Set (ℂ × ℂ) hs : s' ∩ {p \| AnalyticAt ℂ (uncurry f) p} = s ss : s ⊆ s' sn : s ∈ 𝓝 (c, z) ⊢ AnalyticOn ℂ (uncurry f) (s' ∩ {p \| AnalyticAt ℂ (uncurry f) p})	no goals
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/AnalyticManifold/OpenMapping.lean	NontrivialHolomorphicAt.nhds_le_map_nhds_param'	[109, 1]	[180, 64]	have h := n.eventually_ne	X : Type inst✝⁶ : TopologicalSpace X S : Type inst✝⁵ : TopologicalSpace S inst✝⁴ : ChartedSpace ℂ S cms : AnalyticManifold 𝓘(ℂ, ℂ) S T : Type inst✝³ : TopologicalSpace T inst✝² : ChartedSpace ℂ T cmt : AnalyticManifold 𝓘(ℂ, ℂ) T U : Type inst✝¹ : TopologicalSpace U inst✝ : ChartedSpace ℂ U cmu : AnalyticManifold 𝓘(ℂ, ℂ) U f : ℂ → ℂ → ℂ c z : ℂ n : NontrivialHolomorphicAt (f c) z s : Set (ℂ × ℂ) sn : s ∈ 𝓝 (c, z) fa : AnalyticOn ℂ (uncurry f) s e : ℝ ep : e > 0 es : ball (c, z) e ⊆ s ⊢ ∃ r, 0 < r ∧ closedBall (c, z) r ⊆ s ∧ f c z ∉ f c '' sphere z r	X : Type inst✝⁶ : TopologicalSpace X S : Type inst✝⁵ : TopologicalSpace S inst✝⁴ : ChartedSpace ℂ S cms : AnalyticManifold 𝓘(ℂ, ℂ) S T : Type inst✝³ : TopologicalSpace T inst✝² : ChartedSpace ℂ T cmt : AnalyticManifold 𝓘(ℂ, ℂ) T U : Type inst✝¹ : TopologicalSpace U inst✝ : ChartedSpace ℂ U cmu : AnalyticManifold 𝓘(ℂ, ℂ) U f : ℂ → ℂ → ℂ c z : ℂ n : NontrivialHolomorphicAt (f c) z s : Set (ℂ × ℂ) sn : s ∈ 𝓝 (c, z) fa : AnalyticOn ℂ (uncurry f) s e : ℝ ep : e > 0 es : ball (c, z) e ⊆ s h : ∀ᶠ (w : ℂ) in 𝓝 z, w ≠ z → f c w ≠ f c z ⊢ ∃ r, 0 < r ∧ closedBall (c, z) r ⊆ s ∧ f c z ∉ f c '' sphere z r
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/AnalyticManifold/OpenMapping.lean	NontrivialHolomorphicAt.nhds_le_map_nhds_param'	[109, 1]	[180, 64]	contrapose 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 n : NontrivialHolomorphicAt (f c) z s : Set (ℂ × ℂ) sn : s ∈ 𝓝 (c, z) fa : AnalyticOn ℂ (uncurry f) s e : ℝ ep : e > 0 es : ball (c, z) e ⊆ s h : ∀ᶠ (w : ℂ) in 𝓝 z, w ≠ z → f c w ≠ f c z ⊢ ∃ r, 0 < r ∧ closedBall (c, z) r ⊆ s ∧ f c z ∉ f c '' sphere z r	X : Type inst✝⁶ : TopologicalSpace X S : Type inst✝⁵ : TopologicalSpace S inst✝⁴ : ChartedSpace ℂ S cms : AnalyticManifold 𝓘(ℂ, ℂ) S T : Type inst✝³ : TopologicalSpace T inst✝² : ChartedSpace ℂ T cmt : AnalyticManifold 𝓘(ℂ, ℂ) T U : Type inst✝¹ : TopologicalSpace U inst✝ : ChartedSpace ℂ U cmu : AnalyticManifold 𝓘(ℂ, ℂ) U f : ℂ → ℂ → ℂ c z : ℂ n : NontrivialHolomorphicAt (f c) z s : Set (ℂ × ℂ) sn : s ∈ 𝓝 (c, z) fa : AnalyticOn ℂ (uncurry f) s e : ℝ ep : e > 0 es : ball (c, z) e ⊆ s h : ¬∃ r, 0 < r ∧ closedBall (c, z) r ⊆ s ∧ f c z ∉ f c '' sphere z r ⊢ ¬∀ᶠ (w : ℂ) in 𝓝 z, w ≠ z → f c w ≠ f c z
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/AnalyticManifold/OpenMapping.lean	NontrivialHolomorphicAt.nhds_le_map_nhds_param'	[109, 1]	[180, 64]	simp only [not_exists, Filter.not_frequently, not_not, not_and, not_exists] at 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 n : NontrivialHolomorphicAt (f c) z s : Set (ℂ × ℂ) sn : s ∈ 𝓝 (c, z) fa : AnalyticOn ℂ (uncurry f) s e : ℝ ep : e > 0 es : ball (c, z) e ⊆ s h : ¬∃ r, 0 < r ∧ closedBall (c, z) r ⊆ s ∧ f c z ∉ f c '' sphere z r ⊢ ¬∀ᶠ (w : ℂ) in 𝓝 z, w ≠ z → f c w ≠ f c z	X : Type inst✝⁶ : TopologicalSpace X S : Type inst✝⁵ : TopologicalSpace S inst✝⁴ : ChartedSpace ℂ S cms : AnalyticManifold 𝓘(ℂ, ℂ) S T : Type inst✝³ : TopologicalSpace T inst✝² : ChartedSpace ℂ T cmt : AnalyticManifold 𝓘(ℂ, ℂ) T U : Type inst✝¹ : TopologicalSpace U inst✝ : ChartedSpace ℂ U cmu : AnalyticManifold 𝓘(ℂ, ℂ) U f : ℂ → ℂ → ℂ c z : ℂ n : NontrivialHolomorphicAt (f c) z s : Set (ℂ × ℂ) sn : s ∈ 𝓝 (c, z) fa : AnalyticOn ℂ (uncurry f) s e : ℝ ep : e > 0 es : ball (c, z) e ⊆ s h : ∀ (x : ℝ), 0 < x → closedBall (c, z) x ⊆ s → f c z ∈ f c '' sphere z x ⊢ ¬∀ᶠ (w : ℂ) in 𝓝 z, w ≠ z → f c w ≠ f c z
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/AnalyticManifold/OpenMapping.lean	NontrivialHolomorphicAt.nhds_le_map_nhds_param'	[109, 1]	[180, 64]	simp only [Filter.not_eventually, _root_.not_imp, not_not, Filter.eventually_iff, Metric.mem_nhds_iff, not_exists, not_subset, mem_setOf, not_and]	X : Type inst✝⁶ : TopologicalSpace X S : Type inst✝⁵ : TopologicalSpace S inst✝⁴ : ChartedSpace ℂ S cms : AnalyticManifold 𝓘(ℂ, ℂ) S T : Type inst✝³ : TopologicalSpace T inst✝² : ChartedSpace ℂ T cmt : AnalyticManifold 𝓘(ℂ, ℂ) T U : Type inst✝¹ : TopologicalSpace U inst✝ : ChartedSpace ℂ U cmu : AnalyticManifold 𝓘(ℂ, ℂ) U f : ℂ → ℂ → ℂ c z : ℂ n : NontrivialHolomorphicAt (f c) z s : Set (ℂ × ℂ) sn : s ∈ 𝓝 (c, z) fa : AnalyticOn ℂ (uncurry f) s e : ℝ ep : e > 0 es : ball (c, z) e ⊆ s h : ∀ (x : ℝ), 0 < x → closedBall (c, z) x ⊆ s → f c z ∈ f c '' sphere z x ⊢ ¬∀ᶠ (w : ℂ) in 𝓝 z, w ≠ z → f c w ≠ f c z	X : Type inst✝⁶ : TopologicalSpace X S : Type inst✝⁵ : TopologicalSpace S inst✝⁴ : ChartedSpace ℂ S cms : AnalyticManifold 𝓘(ℂ, ℂ) S T : Type inst✝³ : TopologicalSpace T inst✝² : ChartedSpace ℂ T cmt : AnalyticManifold 𝓘(ℂ, ℂ) T U : Type inst✝¹ : TopologicalSpace U inst✝ : ChartedSpace ℂ U cmu : AnalyticManifold 𝓘(ℂ, ℂ) U f : ℂ → ℂ → ℂ c z : ℂ n : NontrivialHolomorphicAt (f c) z s : Set (ℂ × ℂ) sn : s ∈ 𝓝 (c, z) fa : AnalyticOn ℂ (uncurry f) s e : ℝ ep : e > 0 es : ball (c, z) e ⊆ s h : ∀ (x : ℝ), 0 < x → closedBall (c, z) x ⊆ s → f c z ∈ f c '' sphere z x ⊢ ∀ x > 0, ∃ a ∈ ball z x, a ≠ z ∧ f c a = f c z
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/AnalyticManifold/OpenMapping.lean	NontrivialHolomorphicAt.nhds_le_map_nhds_param'	[109, 1]	[180, 64]	intro r 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 → ℂ c z : ℂ n : NontrivialHolomorphicAt (f c) z s : Set (ℂ × ℂ) sn : s ∈ 𝓝 (c, z) fa : AnalyticOn ℂ (uncurry f) s e : ℝ ep : e > 0 es : ball (c, z) e ⊆ s h : ∀ (x : ℝ), 0 < x → closedBall (c, z) x ⊆ s → f c z ∈ f c '' sphere z x ⊢ ∀ x > 0, ∃ a ∈ ball z x, a ≠ z ∧ f c a = 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 n : NontrivialHolomorphicAt (f c) z s : Set (ℂ × ℂ) sn : s ∈ 𝓝 (c, z) fa : AnalyticOn ℂ (uncurry f) s e : ℝ ep : e > 0 es : ball (c, z) e ⊆ s h : ∀ (x : ℝ), 0 < x → closedBall (c, z) x ⊆ s → f c z ∈ f c '' sphere z x r : ℝ rp : r > 0 ⊢ ∃ a ∈ ball z r, a ≠ z ∧ f c a = f c z
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/AnalyticManifold/OpenMapping.lean	NontrivialHolomorphicAt.nhds_le_map_nhds_param'	[109, 1]	[180, 64]	specialize h (min (e/2) (r/2)) ?_ ?_	X : Type inst✝⁶ : TopologicalSpace X S : Type inst✝⁵ : TopologicalSpace S inst✝⁴ : ChartedSpace ℂ S cms : AnalyticManifold 𝓘(ℂ, ℂ) S T : Type inst✝³ : TopologicalSpace T inst✝² : ChartedSpace ℂ T cmt : AnalyticManifold 𝓘(ℂ, ℂ) T U : Type inst✝¹ : TopologicalSpace U inst✝ : ChartedSpace ℂ U cmu : AnalyticManifold 𝓘(ℂ, ℂ) U f : ℂ → ℂ → ℂ c z : ℂ n : NontrivialHolomorphicAt (f c) z s : Set (ℂ × ℂ) sn : s ∈ 𝓝 (c, z) fa : AnalyticOn ℂ (uncurry f) s e : ℝ ep : e > 0 es : ball (c, z) e ⊆ s h : ∀ (x : ℝ), 0 < x → closedBall (c, z) x ⊆ s → f c z ∈ f c '' sphere z x r : ℝ rp : r > 0 ⊢ ∃ a ∈ ball z r, a ≠ z ∧ f c a = f c z	case specialize_1 X : Type inst✝⁶ : TopologicalSpace X S : Type inst✝⁵ : TopologicalSpace S inst✝⁴ : ChartedSpace ℂ S cms : AnalyticManifold 𝓘(ℂ, ℂ) S T : Type inst✝³ : TopologicalSpace T inst✝² : ChartedSpace ℂ T cmt : AnalyticManifold 𝓘(ℂ, ℂ) T U : Type inst✝¹ : TopologicalSpace U inst✝ : ChartedSpace ℂ U cmu : AnalyticManifold 𝓘(ℂ, ℂ) U f : ℂ → ℂ → ℂ c z : ℂ n : NontrivialHolomorphicAt (f c) z s : Set (ℂ × ℂ) sn : s ∈ 𝓝 (c, z) fa : AnalyticOn ℂ (uncurry f) s e : ℝ ep : e > 0 es : ball (c, z) e ⊆ s h : ∀ (x : ℝ), 0 < x → closedBall (c, z) x ⊆ s → f c z ∈ f c '' sphere z x r : ℝ rp : r > 0 ⊢ 0 < min (e / 2) (r / 2) case specialize_2 X : Type inst✝⁶ : TopologicalSpace X S : Type inst✝⁵ : TopologicalSpace S inst✝⁴ : ChartedSpace ℂ S cms : AnalyticManifold 𝓘(ℂ, ℂ) S T : Type inst✝³ : TopologicalSpace T inst✝² : ChartedSpace ℂ T cmt : AnalyticManifold 𝓘(ℂ, ℂ) T U : Type inst✝¹ : TopologicalSpace U inst✝ : ChartedSpace ℂ U cmu : AnalyticManifold 𝓘(ℂ, ℂ) U f : ℂ → ℂ → ℂ c z : ℂ n : NontrivialHolomorphicAt (f c) z s : Set (ℂ × ℂ) sn : s ∈ 𝓝 (c, z) fa : AnalyticOn ℂ (uncurry f) s e : ℝ ep : e > 0 es : ball (c, z) e ⊆ s h : ∀ (x : ℝ), 0 < x → closedBall (c, z) x ⊆ s → f c z ∈ f c '' sphere z x r : ℝ rp : r > 0 ⊢ closedBall (c, z) (min (e / 2) (r / 2)) ⊆ s X : Type inst✝⁶ : TopologicalSpace X S : Type inst✝⁵ : TopologicalSpace S inst✝⁴ : ChartedSpace ℂ S cms : AnalyticManifold 𝓘(ℂ, ℂ) S T : Type inst✝³ : TopologicalSpace T inst✝² : ChartedSpace ℂ T cmt : AnalyticManifold 𝓘(ℂ, ℂ) T U : Type inst✝¹ : TopologicalSpace U inst✝ : ChartedSpace ℂ U cmu : AnalyticManifold 𝓘(ℂ, ℂ) U f : ℂ → ℂ → ℂ c z : ℂ n : NontrivialHolomorphicAt (f c) z s : Set (ℂ × ℂ) sn : s ∈ 𝓝 (c, z) fa : AnalyticOn ℂ (uncurry f) s e : ℝ ep : e > 0 es : ball (c, z) e ⊆ s r : ℝ rp : r > 0 h : f c z ∈ f c '' sphere z (min (e / 2) (r / 2)) ⊢ ∃ a ∈ ball z r, a ≠ z ∧ f c a = f c z
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/AnalyticManifold/OpenMapping.lean	NontrivialHolomorphicAt.nhds_le_map_nhds_param'	[109, 1]	[180, 64]	bound	case specialize_1 X : Type inst✝⁶ : TopologicalSpace X S : Type inst✝⁵ : TopologicalSpace S inst✝⁴ : ChartedSpace ℂ S cms : AnalyticManifold 𝓘(ℂ, ℂ) S T : Type inst✝³ : TopologicalSpace T inst✝² : ChartedSpace ℂ T cmt : AnalyticManifold 𝓘(ℂ, ℂ) T U : Type inst✝¹ : TopologicalSpace U inst✝ : ChartedSpace ℂ U cmu : AnalyticManifold 𝓘(ℂ, ℂ) U f : ℂ → ℂ → ℂ c z : ℂ n : NontrivialHolomorphicAt (f c) z s : Set (ℂ × ℂ) sn : s ∈ 𝓝 (c, z) fa : AnalyticOn ℂ (uncurry f) s e : ℝ ep : e > 0 es : ball (c, z) e ⊆ s h : ∀ (x : ℝ), 0 < x → closedBall (c, z) x ⊆ s → f c z ∈ f c '' sphere z x r : ℝ rp : r > 0 ⊢ 0 < min (e / 2) (r / 2)	no goals
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/AnalyticManifold/OpenMapping.lean	NontrivialHolomorphicAt.nhds_le_map_nhds_param'	[109, 1]	[180, 64]	exact _root_.trans (Metric.closedBall_subset_ball (lt_of_le_of_lt (min_le_left _ _) (half_lt_self ep))) es	case specialize_2 X : Type inst✝⁶ : TopologicalSpace X S : Type inst✝⁵ : TopologicalSpace S inst✝⁴ : ChartedSpace ℂ S cms : AnalyticManifold 𝓘(ℂ, ℂ) S T : Type inst✝³ : TopologicalSpace T inst✝² : ChartedSpace ℂ T cmt : AnalyticManifold 𝓘(ℂ, ℂ) T U : Type inst✝¹ : TopologicalSpace U inst✝ : ChartedSpace ℂ U cmu : AnalyticManifold 𝓘(ℂ, ℂ) U f : ℂ → ℂ → ℂ c z : ℂ n : NontrivialHolomorphicAt (f c) z s : Set (ℂ × ℂ) sn : s ∈ 𝓝 (c, z) fa : AnalyticOn ℂ (uncurry f) s e : ℝ ep : e > 0 es : ball (c, z) e ⊆ s h : ∀ (x : ℝ), 0 < x → closedBall (c, z) x ⊆ s → f c z ∈ f c '' sphere z x r : ℝ rp : r > 0 ⊢ closedBall (c, z) (min (e / 2) (r / 2)) ⊆ s	no goals
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/AnalyticManifold/OpenMapping.lean	NontrivialHolomorphicAt.nhds_le_map_nhds_param'	[109, 1]	[180, 64]	rcases (mem_image _ _ _).mp h with ⟨w, ws, wz⟩	X : Type inst✝⁶ : TopologicalSpace X S : Type inst✝⁵ : TopologicalSpace S inst✝⁴ : ChartedSpace ℂ S cms : AnalyticManifold 𝓘(ℂ, ℂ) S T : Type inst✝³ : TopologicalSpace T inst✝² : ChartedSpace ℂ T cmt : AnalyticManifold 𝓘(ℂ, ℂ) T U : Type inst✝¹ : TopologicalSpace U inst✝ : ChartedSpace ℂ U cmu : AnalyticManifold 𝓘(ℂ, ℂ) U f : ℂ → ℂ → ℂ c z : ℂ n : NontrivialHolomorphicAt (f c) z s : Set (ℂ × ℂ) sn : s ∈ 𝓝 (c, z) fa : AnalyticOn ℂ (uncurry f) s e : ℝ ep : e > 0 es : ball (c, z) e ⊆ s r : ℝ rp : r > 0 h : f c z ∈ f c '' sphere z (min (e / 2) (r / 2)) ⊢ ∃ a ∈ ball z r, a ≠ z ∧ f c a = 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 n : NontrivialHolomorphicAt (f c) z s : Set (ℂ × ℂ) sn : s ∈ 𝓝 (c, z) fa : AnalyticOn ℂ (uncurry f) s e : ℝ ep : e > 0 es : ball (c, z) e ⊆ s r : ℝ rp : r > 0 h : f c z ∈ f c '' sphere z (min (e / 2) (r / 2)) w : ℂ ws : w ∈ sphere z (min (e / 2) (r / 2)) wz : f c w = f c z ⊢ ∃ a ∈ ball z r, a ≠ z ∧ f c a = f c z
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/AnalyticManifold/OpenMapping.lean	NontrivialHolomorphicAt.nhds_le_map_nhds_param'	[109, 1]	[180, 64]	use w	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 n : NontrivialHolomorphicAt (f c) z s : Set (ℂ × ℂ) sn : s ∈ 𝓝 (c, z) fa : AnalyticOn ℂ (uncurry f) s e : ℝ ep : e > 0 es : ball (c, z) e ⊆ s r : ℝ rp : r > 0 h : f c z ∈ f c '' sphere z (min (e / 2) (r / 2)) w : ℂ ws : w ∈ sphere z (min (e / 2) (r / 2)) wz : f c w = f c z ⊢ ∃ a ∈ ball z r, a ≠ z ∧ f c a = f c z	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 n : NontrivialHolomorphicAt (f c) z s : Set (ℂ × ℂ) sn : s ∈ 𝓝 (c, z) fa : AnalyticOn ℂ (uncurry f) s e : ℝ ep : e > 0 es : ball (c, z) e ⊆ s r : ℝ rp : r > 0 h : f c z ∈ f c '' sphere z (min (e / 2) (r / 2)) w : ℂ ws : w ∈ sphere z (min (e / 2) (r / 2)) wz : f c w = f c z ⊢ w ∈ ball z r ∧ w ≠ z ∧ f c w = f c z
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/AnalyticManifold/OpenMapping.lean	NontrivialHolomorphicAt.nhds_le_map_nhds_param'	[109, 1]	[180, 64]	refine ⟨?_, ?_, wz⟩	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 n : NontrivialHolomorphicAt (f c) z s : Set (ℂ × ℂ) sn : s ∈ 𝓝 (c, z) fa : AnalyticOn ℂ (uncurry f) s e : ℝ ep : e > 0 es : ball (c, z) e ⊆ s r : ℝ rp : r > 0 h : f c z ∈ f c '' sphere z (min (e / 2) (r / 2)) w : ℂ ws : w ∈ sphere z (min (e / 2) (r / 2)) wz : f c w = f c z ⊢ w ∈ ball z r ∧ w ≠ z ∧ f c w = f c z	case h.refine_1 X : Type inst✝⁶ : TopologicalSpace X S : Type inst✝⁵ : TopologicalSpace S inst✝⁴ : ChartedSpace ℂ S cms : AnalyticManifold 𝓘(ℂ, ℂ) S T : Type inst✝³ : TopologicalSpace T inst✝² : ChartedSpace ℂ T cmt : AnalyticManifold 𝓘(ℂ, ℂ) T U : Type inst✝¹ : TopologicalSpace U inst✝ : ChartedSpace ℂ U cmu : AnalyticManifold 𝓘(ℂ, ℂ) U f : ℂ → ℂ → ℂ c z : ℂ n : NontrivialHolomorphicAt (f c) z s : Set (ℂ × ℂ) sn : s ∈ 𝓝 (c, z) fa : AnalyticOn ℂ (uncurry f) s e : ℝ ep : e > 0 es : ball (c, z) e ⊆ s r : ℝ rp : r > 0 h : f c z ∈ f c '' sphere z (min (e / 2) (r / 2)) w : ℂ ws : w ∈ sphere z (min (e / 2) (r / 2)) wz : f c w = f c z ⊢ w ∈ ball z r case h.refine_2 X : Type inst✝⁶ : TopologicalSpace X S : Type inst✝⁵ : TopologicalSpace S inst✝⁴ : ChartedSpace ℂ S cms : AnalyticManifold 𝓘(ℂ, ℂ) S T : Type inst✝³ : TopologicalSpace T inst✝² : ChartedSpace ℂ T cmt : AnalyticManifold 𝓘(ℂ, ℂ) T U : Type inst✝¹ : TopologicalSpace U inst✝ : ChartedSpace ℂ U cmu : AnalyticManifold 𝓘(ℂ, ℂ) U f : ℂ → ℂ → ℂ c z : ℂ n : NontrivialHolomorphicAt (f c) z s : Set (ℂ × ℂ) sn : s ∈ 𝓝 (c, z) fa : AnalyticOn ℂ (uncurry f) s e : ℝ ep : e > 0 es : ball (c, z) e ⊆ s r : ℝ rp : r > 0 h : f c z ∈ f c '' sphere z (min (e / 2) (r / 2)) w : ℂ ws : w ∈ sphere z (min (e / 2) (r / 2)) wz : f c w = f c z ⊢ w ≠ z
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/AnalyticManifold/OpenMapping.lean	NontrivialHolomorphicAt.nhds_le_map_nhds_param'	[109, 1]	[180, 64]	exact Metric.closedBall_subset_ball (lt_of_le_of_lt (min_le_right _ _) (half_lt_self rp)) (Metric.sphere_subset_closedBall ws)	case h.refine_1 X : Type inst✝⁶ : TopologicalSpace X S : Type inst✝⁵ : TopologicalSpace S inst✝⁴ : ChartedSpace ℂ S cms : AnalyticManifold 𝓘(ℂ, ℂ) S T : Type inst✝³ : TopologicalSpace T inst✝² : ChartedSpace ℂ T cmt : AnalyticManifold 𝓘(ℂ, ℂ) T U : Type inst✝¹ : TopologicalSpace U inst✝ : ChartedSpace ℂ U cmu : AnalyticManifold 𝓘(ℂ, ℂ) U f : ℂ → ℂ → ℂ c z : ℂ n : NontrivialHolomorphicAt (f c) z s : Set (ℂ × ℂ) sn : s ∈ 𝓝 (c, z) fa : AnalyticOn ℂ (uncurry f) s e : ℝ ep : e > 0 es : ball (c, z) e ⊆ s r : ℝ rp : r > 0 h : f c z ∈ f c '' sphere z (min (e / 2) (r / 2)) w : ℂ ws : w ∈ sphere z (min (e / 2) (r / 2)) wz : f c w = f c z ⊢ w ∈ ball z r	no goals
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/AnalyticManifold/OpenMapping.lean	NontrivialHolomorphicAt.nhds_le_map_nhds_param'	[109, 1]	[180, 64]	contrapose ws	case h.refine_2 X : Type inst✝⁶ : TopologicalSpace X S : Type inst✝⁵ : TopologicalSpace S inst✝⁴ : ChartedSpace ℂ S cms : AnalyticManifold 𝓘(ℂ, ℂ) S T : Type inst✝³ : TopologicalSpace T inst✝² : ChartedSpace ℂ T cmt : AnalyticManifold 𝓘(ℂ, ℂ) T U : Type inst✝¹ : TopologicalSpace U inst✝ : ChartedSpace ℂ U cmu : AnalyticManifold 𝓘(ℂ, ℂ) U f : ℂ → ℂ → ℂ c z : ℂ n : NontrivialHolomorphicAt (f c) z s : Set (ℂ × ℂ) sn : s ∈ 𝓝 (c, z) fa : AnalyticOn ℂ (uncurry f) s e : ℝ ep : e > 0 es : ball (c, z) e ⊆ s r : ℝ rp : r > 0 h : f c z ∈ f c '' sphere z (min (e / 2) (r / 2)) w : ℂ ws : w ∈ sphere z (min (e / 2) (r / 2)) wz : f c w = f c z ⊢ w ≠ z	case h.refine_2 X : Type inst✝⁶ : TopologicalSpace X S : Type inst✝⁵ : TopologicalSpace S inst✝⁴ : ChartedSpace ℂ S cms : AnalyticManifold 𝓘(ℂ, ℂ) S T : Type inst✝³ : TopologicalSpace T inst✝² : ChartedSpace ℂ T cmt : AnalyticManifold 𝓘(ℂ, ℂ) T U : Type inst✝¹ : TopologicalSpace U inst✝ : ChartedSpace ℂ U cmu : AnalyticManifold 𝓘(ℂ, ℂ) U f : ℂ → ℂ → ℂ c z : ℂ n : NontrivialHolomorphicAt (f c) z s : Set (ℂ × ℂ) sn : s ∈ 𝓝 (c, z) fa : AnalyticOn ℂ (uncurry f) s e : ℝ ep : e > 0 es : ball (c, z) e ⊆ s r : ℝ rp : r > 0 h : f c z ∈ f c '' sphere z (min (e / 2) (r / 2)) w : ℂ wz : f c w = f c z ws : ¬w ≠ z ⊢ w ∉ sphere z (min (e / 2) (r / 2))
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/AnalyticManifold/OpenMapping.lean	NontrivialHolomorphicAt.nhds_le_map_nhds_param'	[109, 1]	[180, 64]	simp only [not_not] at ws	case h.refine_2 X : Type inst✝⁶ : TopologicalSpace X S : Type inst✝⁵ : TopologicalSpace S inst✝⁴ : ChartedSpace ℂ S cms : AnalyticManifold 𝓘(ℂ, ℂ) S T : Type inst✝³ : TopologicalSpace T inst✝² : ChartedSpace ℂ T cmt : AnalyticManifold 𝓘(ℂ, ℂ) T U : Type inst✝¹ : TopologicalSpace U inst✝ : ChartedSpace ℂ U cmu : AnalyticManifold 𝓘(ℂ, ℂ) U f : ℂ → ℂ → ℂ c z : ℂ n : NontrivialHolomorphicAt (f c) z s : Set (ℂ × ℂ) sn : s ∈ 𝓝 (c, z) fa : AnalyticOn ℂ (uncurry f) s e : ℝ ep : e > 0 es : ball (c, z) e ⊆ s r : ℝ rp : r > 0 h : f c z ∈ f c '' sphere z (min (e / 2) (r / 2)) w : ℂ wz : f c w = f c z ws : ¬w ≠ z ⊢ w ∉ sphere z (min (e / 2) (r / 2))	case h.refine_2 X : Type inst✝⁶ : TopologicalSpace X S : Type inst✝⁵ : TopologicalSpace S inst✝⁴ : ChartedSpace ℂ S cms : AnalyticManifold 𝓘(ℂ, ℂ) S T : Type inst✝³ : TopologicalSpace T inst✝² : ChartedSpace ℂ T cmt : AnalyticManifold 𝓘(ℂ, ℂ) T U : Type inst✝¹ : TopologicalSpace U inst✝ : ChartedSpace ℂ U cmu : AnalyticManifold 𝓘(ℂ, ℂ) U f : ℂ → ℂ → ℂ c z : ℂ n : NontrivialHolomorphicAt (f c) z s : Set (ℂ × ℂ) sn : s ∈ 𝓝 (c, z) fa : AnalyticOn ℂ (uncurry f) s e : ℝ ep : e > 0 es : ball (c, z) e ⊆ s r : ℝ rp : r > 0 h : f c z ∈ f c '' sphere z (min (e / 2) (r / 2)) w : ℂ wz : f c w = f c z ws : w = z ⊢ w ∉ sphere z (min (e / 2) (r / 2))
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/AnalyticManifold/OpenMapping.lean	NontrivialHolomorphicAt.nhds_le_map_nhds_param'	[109, 1]	[180, 64]	simp only [ws, Metric.mem_sphere, dist_self]	case h.refine_2 X : Type inst✝⁶ : TopologicalSpace X S : Type inst✝⁵ : TopologicalSpace S inst✝⁴ : ChartedSpace ℂ S cms : AnalyticManifold 𝓘(ℂ, ℂ) S T : Type inst✝³ : TopologicalSpace T inst✝² : ChartedSpace ℂ T cmt : AnalyticManifold 𝓘(ℂ, ℂ) T U : Type inst✝¹ : TopologicalSpace U inst✝ : ChartedSpace ℂ U cmu : AnalyticManifold 𝓘(ℂ, ℂ) U f : ℂ → ℂ → ℂ c z : ℂ n : NontrivialHolomorphicAt (f c) z s : Set (ℂ × ℂ) sn : s ∈ 𝓝 (c, z) fa : AnalyticOn ℂ (uncurry f) s e : ℝ ep : e > 0 es : ball (c, z) e ⊆ s r : ℝ rp : r > 0 h : f c z ∈ f c '' sphere z (min (e / 2) (r / 2)) w : ℂ wz : f c w = f c z ws : w = z ⊢ w ∉ sphere z (min (e / 2) (r / 2))	case h.refine_2 X : Type inst✝⁶ : TopologicalSpace X S : Type inst✝⁵ : TopologicalSpace S inst✝⁴ : ChartedSpace ℂ S cms : AnalyticManifold 𝓘(ℂ, ℂ) S T : Type inst✝³ : TopologicalSpace T inst✝² : ChartedSpace ℂ T cmt : AnalyticManifold 𝓘(ℂ, ℂ) T U : Type inst✝¹ : TopologicalSpace U inst✝ : ChartedSpace ℂ U cmu : AnalyticManifold 𝓘(ℂ, ℂ) U f : ℂ → ℂ → ℂ c z : ℂ n : NontrivialHolomorphicAt (f c) z s : Set (ℂ × ℂ) sn : s ∈ 𝓝 (c, z) fa : AnalyticOn ℂ (uncurry f) s e : ℝ ep : e > 0 es : ball (c, z) e ⊆ s r : ℝ rp : r > 0 h : f c z ∈ f c '' sphere z (min (e / 2) (r / 2)) w : ℂ wz : f c w = f c z ws : w = z ⊢ ¬0 = min (e / 2) (r / 2)
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/AnalyticManifold/OpenMapping.lean	NontrivialHolomorphicAt.nhds_le_map_nhds_param'	[109, 1]	[180, 64]	exact ne_of_lt (by bound)	case h.refine_2 X : Type inst✝⁶ : TopologicalSpace X S : Type inst✝⁵ : TopologicalSpace S inst✝⁴ : ChartedSpace ℂ S cms : AnalyticManifold 𝓘(ℂ, ℂ) S T : Type inst✝³ : TopologicalSpace T inst✝² : ChartedSpace ℂ T cmt : AnalyticManifold 𝓘(ℂ, ℂ) T U : Type inst✝¹ : TopologicalSpace U inst✝ : ChartedSpace ℂ U cmu : AnalyticManifold 𝓘(ℂ, ℂ) U f : ℂ → ℂ → ℂ c z : ℂ n : NontrivialHolomorphicAt (f c) z s : Set (ℂ × ℂ) sn : s ∈ 𝓝 (c, z) fa : AnalyticOn ℂ (uncurry f) s e : ℝ ep : e > 0 es : ball (c, z) e ⊆ s r : ℝ rp : r > 0 h : f c z ∈ f c '' sphere z (min (e / 2) (r / 2)) w : ℂ wz : f c w = f c z ws : w = z ⊢ ¬0 = min (e / 2) (r / 2)	no goals
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/AnalyticManifold/OpenMapping.lean	NontrivialHolomorphicAt.nhds_le_map_nhds_param'	[109, 1]	[180, 64]	bound	X : Type inst✝⁶ : TopologicalSpace X S : Type inst✝⁵ : TopologicalSpace S inst✝⁴ : ChartedSpace ℂ S cms : AnalyticManifold 𝓘(ℂ, ℂ) S T : Type inst✝³ : TopologicalSpace T inst✝² : ChartedSpace ℂ T cmt : AnalyticManifold 𝓘(ℂ, ℂ) T U : Type inst✝¹ : TopologicalSpace U inst✝ : ChartedSpace ℂ U cmu : AnalyticManifold 𝓘(ℂ, ℂ) U f : ℂ → ℂ → ℂ c z : ℂ n : NontrivialHolomorphicAt (f c) z s : Set (ℂ × ℂ) sn : s ∈ 𝓝 (c, z) fa : AnalyticOn ℂ (uncurry f) s e : ℝ ep : e > 0 es : ball (c, z) e ⊆ s r : ℝ rp : r > 0 h : f c z ∈ f c '' sphere z (min (e / 2) (r / 2)) w : ℂ wz : f c w = f c z ws : w = z ⊢ 0 < min (e / 2) (r / 2)	no goals
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/AnalyticManifold/OpenMapping.lean	NontrivialHolomorphicAt.nhds_le_map_nhds_param'	[109, 1]	[180, 64]	apply ContinuousOn.norm	X : Type inst✝⁶ : TopologicalSpace X S : Type inst✝⁵ : TopologicalSpace S inst✝⁴ : ChartedSpace ℂ S cms : AnalyticManifold 𝓘(ℂ, ℂ) S T : Type inst✝³ : TopologicalSpace T inst✝² : ChartedSpace ℂ T cmt : AnalyticManifold 𝓘(ℂ, ℂ) T U : Type inst✝¹ : TopologicalSpace U inst✝ : ChartedSpace ℂ U cmu : AnalyticManifold 𝓘(ℂ, ℂ) U f : ℂ → ℂ → ℂ c z : ℂ n : NontrivialHolomorphicAt (f c) z s : Set (ℂ × ℂ) sn : s ∈ 𝓝 (c, z) fa : AnalyticOn ℂ (uncurry f) s e : ℝ ep : e > 0 es : ball (c, z) e ⊆ s r : ℝ rp : 0 < r rs : closedBall (c, z) r ⊆ s fr : f c z ∉ f c '' sphere z r ⊢ ContinuousOn (fun w => ‖f c w - f c z‖) (sphere 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 n : NontrivialHolomorphicAt (f c) z s : Set (ℂ × ℂ) sn : s ∈ 𝓝 (c, z) fa : AnalyticOn ℂ (uncurry f) s e : ℝ ep : e > 0 es : ball (c, z) e ⊆ s r : ℝ rp : 0 < r rs : closedBall (c, z) r ⊆ s fr : f c z ∉ f c '' sphere z r ⊢ ContinuousOn (fun x => f c x - f c z) (sphere z r)
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/AnalyticManifold/OpenMapping.lean	NontrivialHolomorphicAt.nhds_le_map_nhds_param'	[109, 1]	[180, 64]	refine ContinuousOn.sub ?_ continuousOn_const	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 n : NontrivialHolomorphicAt (f c) z s : Set (ℂ × ℂ) sn : s ∈ 𝓝 (c, z) fa : AnalyticOn ℂ (uncurry f) s e : ℝ ep : e > 0 es : ball (c, z) e ⊆ s r : ℝ rp : 0 < r rs : closedBall (c, z) r ⊆ s fr : f c z ∉ f c '' sphere z r ⊢ ContinuousOn (fun x => f c x - f c z) (sphere 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 n : NontrivialHolomorphicAt (f c) z s : Set (ℂ × ℂ) sn : s ∈ 𝓝 (c, z) fa : AnalyticOn ℂ (uncurry f) s e : ℝ ep : e > 0 es : ball (c, z) e ⊆ s r : ℝ rp : 0 < r rs : closedBall (c, z) r ⊆ s fr : f c z ∉ f c '' sphere z r ⊢ ContinuousOn (fun x => f c x) (sphere z r)
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/AnalyticManifold/OpenMapping.lean	NontrivialHolomorphicAt.nhds_le_map_nhds_param'	[109, 1]	[180, 64]	apply fa.along_snd.continuousOn.mono	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 n : NontrivialHolomorphicAt (f c) z s : Set (ℂ × ℂ) sn : s ∈ 𝓝 (c, z) fa : AnalyticOn ℂ (uncurry f) s e : ℝ ep : e > 0 es : ball (c, z) e ⊆ s r : ℝ rp : 0 < r rs : closedBall (c, z) r ⊆ s fr : f c z ∉ f c '' sphere z r ⊢ ContinuousOn (fun x => f c x) (sphere 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 n : NontrivialHolomorphicAt (f c) z s : Set (ℂ × ℂ) sn : s ∈ 𝓝 (c, z) fa : AnalyticOn ℂ (uncurry f) s e : ℝ ep : e > 0 es : ball (c, z) e ⊆ s r : ℝ rp : 0 < r rs : closedBall (c, z) r ⊆ s fr : f c z ∉ f c '' sphere z r ⊢ sphere z r ⊆ {y \| (c, y) ∈ s}
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/AnalyticManifold/OpenMapping.lean	NontrivialHolomorphicAt.nhds_le_map_nhds_param'	[109, 1]	[180, 64]	intro x xs	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 n : NontrivialHolomorphicAt (f c) z s : Set (ℂ × ℂ) sn : s ∈ 𝓝 (c, z) fa : AnalyticOn ℂ (uncurry f) s e : ℝ ep : e > 0 es : ball (c, z) e ⊆ s r : ℝ rp : 0 < r rs : closedBall (c, z) r ⊆ s fr : f c z ∉ f c '' sphere z r ⊢ sphere z r ⊆ {y \| (c, y) ∈ s}	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 n : NontrivialHolomorphicAt (f c) z s : Set (ℂ × ℂ) sn : s ∈ 𝓝 (c, z) fa : AnalyticOn ℂ (uncurry f) s e : ℝ ep : e > 0 es : ball (c, z) e ⊆ s r : ℝ rp : 0 < r rs : closedBall (c, z) r ⊆ s fr : f c z ∉ f c '' sphere z r x : ℂ xs : x ∈ sphere z r ⊢ x ∈ {y \| (c, y) ∈ s}
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/AnalyticManifold/OpenMapping.lean	NontrivialHolomorphicAt.nhds_le_map_nhds_param'	[109, 1]	[180, 64]	apply rs	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 n : NontrivialHolomorphicAt (f c) z s : Set (ℂ × ℂ) sn : s ∈ 𝓝 (c, z) fa : AnalyticOn ℂ (uncurry f) s e : ℝ ep : e > 0 es : ball (c, z) e ⊆ s r : ℝ rp : 0 < r rs : closedBall (c, z) r ⊆ s fr : f c z ∉ f c '' sphere z r x : ℂ xs : x ∈ sphere z r ⊢ x ∈ {y \| (c, y) ∈ s}	case h.a X : Type inst✝⁶ : TopologicalSpace X S : Type inst✝⁵ : TopologicalSpace S inst✝⁴ : ChartedSpace ℂ S cms : AnalyticManifold 𝓘(ℂ, ℂ) S T : Type inst✝³ : TopologicalSpace T inst✝² : ChartedSpace ℂ T cmt : AnalyticManifold 𝓘(ℂ, ℂ) T U : Type inst✝¹ : TopologicalSpace U inst✝ : ChartedSpace ℂ U cmu : AnalyticManifold 𝓘(ℂ, ℂ) U f : ℂ → ℂ → ℂ c z : ℂ n : NontrivialHolomorphicAt (f c) z s : Set (ℂ × ℂ) sn : s ∈ 𝓝 (c, z) fa : AnalyticOn ℂ (uncurry f) s e : ℝ ep : e > 0 es : ball (c, z) e ⊆ s r : ℝ rp : 0 < r rs : closedBall (c, z) r ⊆ s fr : f c z ∉ f c '' sphere z r x : ℂ xs : x ∈ sphere z r ⊢ (c, x) ∈ closedBall (c, z) r
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/AnalyticManifold/OpenMapping.lean	NontrivialHolomorphicAt.nhds_le_map_nhds_param'	[109, 1]	[180, 64]	simp only [← closedBall_prod_same, mem_prod_eq]	case h.a X : Type inst✝⁶ : TopologicalSpace X S : Type inst✝⁵ : TopologicalSpace S inst✝⁴ : ChartedSpace ℂ S cms : AnalyticManifold 𝓘(ℂ, ℂ) S T : Type inst✝³ : TopologicalSpace T inst✝² : ChartedSpace ℂ T cmt : AnalyticManifold 𝓘(ℂ, ℂ) T U : Type inst✝¹ : TopologicalSpace U inst✝ : ChartedSpace ℂ U cmu : AnalyticManifold 𝓘(ℂ, ℂ) U f : ℂ → ℂ → ℂ c z : ℂ n : NontrivialHolomorphicAt (f c) z s : Set (ℂ × ℂ) sn : s ∈ 𝓝 (c, z) fa : AnalyticOn ℂ (uncurry f) s e : ℝ ep : e > 0 es : ball (c, z) e ⊆ s r : ℝ rp : 0 < r rs : closedBall (c, z) r ⊆ s fr : f c z ∉ f c '' sphere z r x : ℂ xs : x ∈ sphere z r ⊢ (c, x) ∈ closedBall (c, z) r	case h.a X : Type inst✝⁶ : TopologicalSpace X S : Type inst✝⁵ : TopologicalSpace S inst✝⁴ : ChartedSpace ℂ S cms : AnalyticManifold 𝓘(ℂ, ℂ) S T : Type inst✝³ : TopologicalSpace T inst✝² : ChartedSpace ℂ T cmt : AnalyticManifold 𝓘(ℂ, ℂ) T U : Type inst✝¹ : TopologicalSpace U inst✝ : ChartedSpace ℂ U cmu : AnalyticManifold 𝓘(ℂ, ℂ) U f : ℂ → ℂ → ℂ c z : ℂ n : NontrivialHolomorphicAt (f c) z s : Set (ℂ × ℂ) sn : s ∈ 𝓝 (c, z) fa : AnalyticOn ℂ (uncurry f) s e : ℝ ep : e > 0 es : ball (c, z) e ⊆ s r : ℝ rp : 0 < r rs : closedBall (c, z) r ⊆ s fr : f c z ∉ f c '' sphere z r x : ℂ xs : x ∈ sphere z r ⊢ c ∈ closedBall c r ∧ x ∈ closedBall z r
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/AnalyticManifold/OpenMapping.lean	NontrivialHolomorphicAt.nhds_le_map_nhds_param'	[109, 1]	[180, 64]	use Metric.mem_closedBall_self rp.le, Metric.sphere_subset_closedBall xs	case h.a X : Type inst✝⁶ : TopologicalSpace X S : Type inst✝⁵ : TopologicalSpace S inst✝⁴ : ChartedSpace ℂ S cms : AnalyticManifold 𝓘(ℂ, ℂ) S T : Type inst✝³ : TopologicalSpace T inst✝² : ChartedSpace ℂ T cmt : AnalyticManifold 𝓘(ℂ, ℂ) T U : Type inst✝¹ : TopologicalSpace U inst✝ : ChartedSpace ℂ U cmu : AnalyticManifold 𝓘(ℂ, ℂ) U f : ℂ → ℂ → ℂ c z : ℂ n : NontrivialHolomorphicAt (f c) z s : Set (ℂ × ℂ) sn : s ∈ 𝓝 (c, z) fa : AnalyticOn ℂ (uncurry f) s e : ℝ ep : e > 0 es : ball (c, z) e ⊆ s r : ℝ rp : 0 < r rs : closedBall (c, z) r ⊆ s fr : f c z ∉ f c '' sphere z r x : ℂ xs : x ∈ sphere z r ⊢ c ∈ closedBall c r ∧ x ∈ closedBall z r	no goals
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/AnalyticManifold/OpenMapping.lean	NontrivialHolomorphicAt.nhds_le_map_nhds_param'	[109, 1]	[180, 64]	contrapose fr	X : Type inst✝⁶ : TopologicalSpace X S : Type inst✝⁵ : TopologicalSpace S inst✝⁴ : ChartedSpace ℂ S cms : AnalyticManifold 𝓘(ℂ, ℂ) S T : Type inst✝³ : TopologicalSpace T inst✝² : ChartedSpace ℂ T cmt : AnalyticManifold 𝓘(ℂ, ℂ) T U : Type inst✝¹ : TopologicalSpace U inst✝ : ChartedSpace ℂ U cmu : AnalyticManifold 𝓘(ℂ, ℂ) U f : ℂ → ℂ → ℂ c z : ℂ n : NontrivialHolomorphicAt (f c) z s : Set (ℂ × ℂ) sn : s ∈ 𝓝 (c, z) fa : AnalyticOn ℂ (uncurry f) s e✝ : ℝ ep : e✝ > 0 es : ball (c, z) e✝ ⊆ s r : ℝ rp : 0 < r rs : closedBall (c, z) r ⊆ s fr : f c z ∉ f c '' sphere z r fc : ContinuousOn (fun w => ‖f c w - f c z‖) (sphere z r) x : ℂ xs : x ∈ sphere z r xm : IsMinOn (fun w => ‖f c w - f c z‖) (sphere z r) x e : ℝ he : ‖f c x - f c z‖ = e ⊢ 0 < 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 n : NontrivialHolomorphicAt (f c) z s : Set (ℂ × ℂ) sn : s ∈ 𝓝 (c, z) fa : AnalyticOn ℂ (uncurry f) s e✝ : ℝ ep : e✝ > 0 es : ball (c, z) e✝ ⊆ s r : ℝ rp : 0 < r rs : closedBall (c, z) r ⊆ s fc : ContinuousOn (fun w => ‖f c w - f c z‖) (sphere z r) x : ℂ xs : x ∈ sphere z r xm : IsMinOn (fun w => ‖f c w - f c z‖) (sphere z r) x e : ℝ he : ‖f c x - f c z‖ = e fr : ¬0 < e ⊢ ¬f c z ∉ f c '' sphere z r
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/AnalyticManifold/OpenMapping.lean	NontrivialHolomorphicAt.nhds_le_map_nhds_param'	[109, 1]	[180, 64]	simp only [norm_pos_iff, sub_ne_zero, not_not, mem_image, ← he] at fr ⊢	X : Type inst✝⁶ : TopologicalSpace X S : Type inst✝⁵ : TopologicalSpace S inst✝⁴ : ChartedSpace ℂ S cms : AnalyticManifold 𝓘(ℂ, ℂ) S T : Type inst✝³ : TopologicalSpace T inst✝² : ChartedSpace ℂ T cmt : AnalyticManifold 𝓘(ℂ, ℂ) T U : Type inst✝¹ : TopologicalSpace U inst✝ : ChartedSpace ℂ U cmu : AnalyticManifold 𝓘(ℂ, ℂ) U f : ℂ → ℂ → ℂ c z : ℂ n : NontrivialHolomorphicAt (f c) z s : Set (ℂ × ℂ) sn : s ∈ 𝓝 (c, z) fa : AnalyticOn ℂ (uncurry f) s e✝ : ℝ ep : e✝ > 0 es : ball (c, z) e✝ ⊆ s r : ℝ rp : 0 < r rs : closedBall (c, z) r ⊆ s fc : ContinuousOn (fun w => ‖f c w - f c z‖) (sphere z r) x : ℂ xs : x ∈ sphere z r xm : IsMinOn (fun w => ‖f c w - f c z‖) (sphere z r) x e : ℝ he : ‖f c x - f c z‖ = e fr : ¬0 < e ⊢ ¬f c z ∉ f c '' sphere 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 n : NontrivialHolomorphicAt (f c) z s : Set (ℂ × ℂ) sn : s ∈ 𝓝 (c, z) fa : AnalyticOn ℂ (uncurry f) s e✝ : ℝ ep : e✝ > 0 es : ball (c, z) e✝ ⊆ s r : ℝ rp : 0 < r rs : closedBall (c, z) r ⊆ s fc : ContinuousOn (fun w => ‖f c w - f c z‖) (sphere z r) x : ℂ xs : x ∈ sphere z r xm : IsMinOn (fun w => ‖f c w - f c z‖) (sphere z r) x e : ℝ he : ‖f c x - f c z‖ = e fr : f c x = f c z ⊢ ∃ x ∈ sphere z r, f c x = f c z
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/AnalyticManifold/OpenMapping.lean	NontrivialHolomorphicAt.nhds_le_map_nhds_param'	[109, 1]	[180, 64]	use x, xs, fr	X : Type inst✝⁶ : TopologicalSpace X S : Type inst✝⁵ : TopologicalSpace S inst✝⁴ : ChartedSpace ℂ S cms : AnalyticManifold 𝓘(ℂ, ℂ) S T : Type inst✝³ : TopologicalSpace T inst✝² : ChartedSpace ℂ T cmt : AnalyticManifold 𝓘(ℂ, ℂ) T U : Type inst✝¹ : TopologicalSpace U inst✝ : ChartedSpace ℂ U cmu : AnalyticManifold 𝓘(ℂ, ℂ) U f : ℂ → ℂ → ℂ c z : ℂ n : NontrivialHolomorphicAt (f c) z s : Set (ℂ × ℂ) sn : s ∈ 𝓝 (c, z) fa : AnalyticOn ℂ (uncurry f) s e✝ : ℝ ep : e✝ > 0 es : ball (c, z) e✝ ⊆ s r : ℝ rp : 0 < r rs : closedBall (c, z) r ⊆ s fc : ContinuousOn (fun w => ‖f c w - f c z‖) (sphere z r) x : ℂ xs : x ∈ sphere z r xm : IsMinOn (fun w => ‖f c w - f c z‖) (sphere z r) x e : ℝ he : ‖f c x - f c z‖ = e fr : f c x = f c z ⊢ ∃ x ∈ sphere z r, f c x = f c z	no goals
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/AnalyticManifold/OpenMapping.lean	NontrivialHolomorphicAt.nhds_le_map_nhds_param'	[109, 1]	[180, 64]	linarith	X : Type inst✝⁶ : TopologicalSpace X S : Type inst✝⁵ : TopologicalSpace S inst✝⁴ : ChartedSpace ℂ S cms : AnalyticManifold 𝓘(ℂ, ℂ) S T : Type inst✝³ : TopologicalSpace T inst✝² : ChartedSpace ℂ T cmt : AnalyticManifold 𝓘(ℂ, ℂ) T U : Type inst✝¹ : TopologicalSpace U inst✝ : ChartedSpace ℂ U cmu : AnalyticManifold 𝓘(ℂ, ℂ) U f : ℂ → ℂ → ℂ c z : ℂ n : NontrivialHolomorphicAt (f c) z s : Set (ℂ × ℂ) sn : s ∈ 𝓝 (c, z) fa : AnalyticOn ℂ (uncurry f) s e✝ : ℝ ep✝ : e✝ > 0 es : ball (c, z) e✝ ⊆ s r : ℝ rp : 0 < r rs : closedBall (c, z) r ⊆ s fr : f c z ∉ f c '' sphere z r fc : ContinuousOn (fun w => ‖f c w - f c z‖) (sphere z r) x : ℂ xs : x ∈ sphere z r xm : IsMinOn (fun w => ‖f c w - f c z‖) (sphere z r) x e : ℝ he : ‖f c x - f c z‖ = e ep : 0 < e ⊢ e / 4 > 0	no goals
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/AnalyticManifold/OpenMapping.lean	NontrivialHolomorphicAt.nhds_le_map_nhds_param'	[109, 1]	[180, 64]	intro d dt w wr	X : Type inst✝⁶ : TopologicalSpace X S : Type inst✝⁵ : TopologicalSpace S inst✝⁴ : ChartedSpace ℂ S cms : AnalyticManifold 𝓘(ℂ, ℂ) S T : Type inst✝³ : TopologicalSpace T inst✝² : ChartedSpace ℂ T cmt : AnalyticManifold 𝓘(ℂ, ℂ) T U : Type inst✝¹ : TopologicalSpace U inst✝ : ChartedSpace ℂ U cmu : AnalyticManifold 𝓘(ℂ, ℂ) U f : ℂ → ℂ → ℂ c z : ℂ n : NontrivialHolomorphicAt (f c) z s : Set (ℂ × ℂ) sn : s ∈ 𝓝 (c, z) fa : AnalyticOn ℂ (uncurry f) s e✝ : ℝ ep✝ : e✝ > 0 es : ball (c, z) e✝ ⊆ s r : ℝ rp : 0 < r rs : closedBall (c, z) r ⊆ s fr : f c z ∉ f c '' sphere z r fc : ContinuousOn (fun w => ‖f c w - f c z‖) (sphere z r) x : ℂ xs : x ∈ sphere z r xm : IsMinOn (fun w => ‖f c w - f c z‖) (sphere z r) x e : ℝ he : ‖f c x - f c z‖ = e ep : 0 < e t : ℝ tp : t > 0 ft : ∀ x ∈ closedBall (c, z) r, ∀ y ∈ closedBall (c, z) r, dist x y < t → dist (uncurry f x) (uncurry f y) < e / 4 ⊢ ∀ d ∈ ball c (min t r), ∀ w ∈ sphere z r, e / 2 ≤ ‖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 n : NontrivialHolomorphicAt (f c) z s : Set (ℂ × ℂ) sn : s ∈ 𝓝 (c, z) fa : AnalyticOn ℂ (uncurry f) s e✝ : ℝ ep✝ : e✝ > 0 es : ball (c, z) e✝ ⊆ s r : ℝ rp : 0 < r rs : closedBall (c, z) r ⊆ s fr : f c z ∉ f c '' sphere z r fc : ContinuousOn (fun w => ‖f c w - f c z‖) (sphere z r) x : ℂ xs : x ∈ sphere z r xm : IsMinOn (fun w => ‖f c w - f c z‖) (sphere z r) x e : ℝ he : ‖f c x - f c z‖ = e ep : 0 < e t : ℝ tp : t > 0 ft : ∀ x ∈ closedBall (c, z) r, ∀ y ∈ closedBall (c, z) r, dist x y < t → dist (uncurry f x) (uncurry f y) < e / 4 d : ℂ dt : d ∈ ball c (min t r) w : ℂ wr : w ∈ sphere z r ⊢ e / 2 ≤ ‖f d w - f d z‖
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/AnalyticManifold/OpenMapping.lean	NontrivialHolomorphicAt.nhds_le_map_nhds_param'	[109, 1]	[180, 64]	simp only [Complex.norm_eq_abs]	X : Type inst✝⁶ : TopologicalSpace X S : Type inst✝⁵ : TopologicalSpace S inst✝⁴ : ChartedSpace ℂ S cms : AnalyticManifold 𝓘(ℂ, ℂ) S T : Type inst✝³ : TopologicalSpace T inst✝² : ChartedSpace ℂ T cmt : AnalyticManifold 𝓘(ℂ, ℂ) T U : Type inst✝¹ : TopologicalSpace U inst✝ : ChartedSpace ℂ U cmu : AnalyticManifold 𝓘(ℂ, ℂ) U f : ℂ → ℂ → ℂ c z : ℂ n : NontrivialHolomorphicAt (f c) z s : Set (ℂ × ℂ) sn : s ∈ 𝓝 (c, z) fa : AnalyticOn ℂ (uncurry f) s e✝ : ℝ ep✝ : e✝ > 0 es : ball (c, z) e✝ ⊆ s r : ℝ rp : 0 < r rs : closedBall (c, z) r ⊆ s fr : f c z ∉ f c '' sphere z r fc : ContinuousOn (fun w => ‖f c w - f c z‖) (sphere z r) x : ℂ xs : x ∈ sphere z r xm : IsMinOn (fun w => ‖f c w - f c z‖) (sphere z r) x e : ℝ he : ‖f c x - f c z‖ = e ep : 0 < e t : ℝ tp : t > 0 ft : ∀ x ∈ closedBall (c, z) r, ∀ y ∈ closedBall (c, z) r, dist x y < t → dist (uncurry f x) (uncurry f y) < e / 4 d : ℂ dt : d ∈ ball c (min t r) w : ℂ wr : w ∈ sphere z r ⊢ e / 2 ≤ ‖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 n : NontrivialHolomorphicAt (f c) z s : Set (ℂ × ℂ) sn : s ∈ 𝓝 (c, z) fa : AnalyticOn ℂ (uncurry f) s e✝ : ℝ ep✝ : e✝ > 0 es : ball (c, z) e✝ ⊆ s r : ℝ rp : 0 < r rs : closedBall (c, z) r ⊆ s fr : f c z ∉ f c '' sphere z r fc : ContinuousOn (fun w => ‖f c w - f c z‖) (sphere z r) x : ℂ xs : x ∈ sphere z r xm : IsMinOn (fun w => ‖f c w - f c z‖) (sphere z r) x e : ℝ he : ‖f c x - f c z‖ = e ep : 0 < e t : ℝ tp : t > 0 ft : ∀ x ∈ closedBall (c, z) r, ∀ y ∈ closedBall (c, z) r, dist x y < t → dist (uncurry f x) (uncurry f y) < e / 4 d : ℂ dt : d ∈ ball c (min t r) w : ℂ wr : w ∈ sphere z r ⊢ e / 2 ≤ Complex.abs (f d w - f d z)
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/AnalyticManifold/OpenMapping.lean	NontrivialHolomorphicAt.nhds_le_map_nhds_param'	[109, 1]	[180, 64]	simp only [Complex.dist_eq, Prod.forall, mem_closedBall, Prod.dist_eq, max_le_iff, max_lt_iff, Function.uncurry, and_imp] at ft	X : Type inst✝⁶ : TopologicalSpace X S : Type inst✝⁵ : TopologicalSpace S inst✝⁴ : ChartedSpace ℂ S cms : AnalyticManifold 𝓘(ℂ, ℂ) S T : Type inst✝³ : TopologicalSpace T inst✝² : ChartedSpace ℂ T cmt : AnalyticManifold 𝓘(ℂ, ℂ) T U : Type inst✝¹ : TopologicalSpace U inst✝ : ChartedSpace ℂ U cmu : AnalyticManifold 𝓘(ℂ, ℂ) U f : ℂ → ℂ → ℂ c z : ℂ n : NontrivialHolomorphicAt (f c) z s : Set (ℂ × ℂ) sn : s ∈ 𝓝 (c, z) fa : AnalyticOn ℂ (uncurry f) s e✝ : ℝ ep✝ : e✝ > 0 es : ball (c, z) e✝ ⊆ s r : ℝ rp : 0 < r rs : closedBall (c, z) r ⊆ s fr : f c z ∉ f c '' sphere z r fc : ContinuousOn (fun w => ‖f c w - f c z‖) (sphere z r) x : ℂ xs : x ∈ sphere z r xm : IsMinOn (fun w => ‖f c w - f c z‖) (sphere z r) x e : ℝ he : ‖f c x - f c z‖ = e ep : 0 < e t : ℝ tp : t > 0 ft : ∀ x ∈ closedBall (c, z) r, ∀ y ∈ closedBall (c, z) r, dist x y < t → dist (uncurry f x) (uncurry f y) < e / 4 d : ℂ dt : d ∈ ball c (min t r) w : ℂ wr : w ∈ sphere z r ⊢ e / 2 ≤ Complex.abs (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 n : NontrivialHolomorphicAt (f c) z s : Set (ℂ × ℂ) sn : s ∈ 𝓝 (c, z) fa : AnalyticOn ℂ (uncurry f) s e✝ : ℝ ep✝ : e✝ > 0 es : ball (c, z) e✝ ⊆ s r : ℝ rp : 0 < r rs : closedBall (c, z) r ⊆ s fr : f c z ∉ f c '' sphere z r fc : ContinuousOn (fun w => ‖f c w - f c z‖) (sphere z r) x : ℂ xs : x ∈ sphere z r xm : IsMinOn (fun w => ‖f c w - f c z‖) (sphere z r) x e : ℝ he : ‖f c x - f c z‖ = e ep : 0 < e t : ℝ tp : t > 0 d : ℂ dt : d ∈ ball c (min t r) w : ℂ wr : w ∈ sphere z r ft : ∀ (a b : ℂ), Complex.abs (a - c) ≤ r → Complex.abs (b - z) ≤ r → ∀ (a_3 b_1 : ℂ), Complex.abs (a_3 - c) ≤ r → Complex.abs (b_1 - z) ≤ r → Complex.abs (a - a_3) < t → Complex.abs (b - b_1) < t → Complex.abs (f a b - f a_3 b_1) < e / 4 ⊢ e / 2 ≤ Complex.abs (f d w - f d z)
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/AnalyticManifold/OpenMapping.lean	NontrivialHolomorphicAt.nhds_le_map_nhds_param'	[109, 1]	[180, 64]	simp only [mem_ball, Complex.dist_eq, lt_min_iff] at dt	X : Type inst✝⁶ : TopologicalSpace X S : Type inst✝⁵ : TopologicalSpace S inst✝⁴ : ChartedSpace ℂ S cms : AnalyticManifold 𝓘(ℂ, ℂ) S T : Type inst✝³ : TopologicalSpace T inst✝² : ChartedSpace ℂ T cmt : AnalyticManifold 𝓘(ℂ, ℂ) T U : Type inst✝¹ : TopologicalSpace U inst✝ : ChartedSpace ℂ U cmu : AnalyticManifold 𝓘(ℂ, ℂ) U f : ℂ → ℂ → ℂ c z : ℂ n : NontrivialHolomorphicAt (f c) z s : Set (ℂ × ℂ) sn : s ∈ 𝓝 (c, z) fa : AnalyticOn ℂ (uncurry f) s e✝ : ℝ ep✝ : e✝ > 0 es : ball (c, z) e✝ ⊆ s r : ℝ rp : 0 < r rs : closedBall (c, z) r ⊆ s fr : f c z ∉ f c '' sphere z r fc : ContinuousOn (fun w => ‖f c w - f c z‖) (sphere z r) x : ℂ xs : x ∈ sphere z r xm : IsMinOn (fun w => ‖f c w - f c z‖) (sphere z r) x e : ℝ he : ‖f c x - f c z‖ = e ep : 0 < e t : ℝ tp : t > 0 d : ℂ dt : d ∈ ball c (min t r) w : ℂ wr : w ∈ sphere z r ft : ∀ (a b : ℂ), Complex.abs (a - c) ≤ r → Complex.abs (b - z) ≤ r → ∀ (a_3 b_1 : ℂ), Complex.abs (a_3 - c) ≤ r → Complex.abs (b_1 - z) ≤ r → Complex.abs (a - a_3) < t → Complex.abs (b - b_1) < t → Complex.abs (f a b - f a_3 b_1) < e / 4 ⊢ e / 2 ≤ Complex.abs (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 n : NontrivialHolomorphicAt (f c) z s : Set (ℂ × ℂ) sn : s ∈ 𝓝 (c, z) fa : AnalyticOn ℂ (uncurry f) s e✝ : ℝ ep✝ : e✝ > 0 es : ball (c, z) e✝ ⊆ s r : ℝ rp : 0 < r rs : closedBall (c, z) r ⊆ s fr : f c z ∉ f c '' sphere z r fc : ContinuousOn (fun w => ‖f c w - f c z‖) (sphere z r) x : ℂ xs : x ∈ sphere z r xm : IsMinOn (fun w => ‖f c w - f c z‖) (sphere z r) x e : ℝ he : ‖f c x - f c z‖ = e ep : 0 < e t : ℝ tp : t > 0 d w : ℂ wr : w ∈ sphere z r ft : ∀ (a b : ℂ), Complex.abs (a - c) ≤ r → Complex.abs (b - z) ≤ r → ∀ (a_3 b_1 : ℂ), Complex.abs (a_3 - c) ≤ r → Complex.abs (b_1 - z) ≤ r → Complex.abs (a - a_3) < t → Complex.abs (b - b_1) < t → Complex.abs (f a b - f a_3 b_1) < e / 4 dt : Complex.abs (d - c) < t ∧ Complex.abs (d - c) < r ⊢ e / 2 ≤ Complex.abs (f d w - f d z)
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/AnalyticManifold/OpenMapping.lean	NontrivialHolomorphicAt.nhds_le_map_nhds_param'	[109, 1]	[180, 64]	have a1 : abs (f d w - f c w) ≤ e / 4 := (ft d w dt.2.le (le_of_eq wr) c w (abs_sub_self_lt rp).le (le_of_eq wr) dt.1 (abs_sub_self_lt tp)).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 → ℂ c z : ℂ n : NontrivialHolomorphicAt (f c) z s : Set (ℂ × ℂ) sn : s ∈ 𝓝 (c, z) fa : AnalyticOn ℂ (uncurry f) s e✝ : ℝ ep✝ : e✝ > 0 es : ball (c, z) e✝ ⊆ s r : ℝ rp : 0 < r rs : closedBall (c, z) r ⊆ s fr : f c z ∉ f c '' sphere z r fc : ContinuousOn (fun w => ‖f c w - f c z‖) (sphere z r) x : ℂ xs : x ∈ sphere z r xm : IsMinOn (fun w => ‖f c w - f c z‖) (sphere z r) x e : ℝ he : ‖f c x - f c z‖ = e ep : 0 < e t : ℝ tp : t > 0 d w : ℂ wr : w ∈ sphere z r ft : ∀ (a b : ℂ), Complex.abs (a - c) ≤ r → Complex.abs (b - z) ≤ r → ∀ (a_3 b_1 : ℂ), Complex.abs (a_3 - c) ≤ r → Complex.abs (b_1 - z) ≤ r → Complex.abs (a - a_3) < t → Complex.abs (b - b_1) < t → Complex.abs (f a b - f a_3 b_1) < e / 4 dt : Complex.abs (d - c) < t ∧ Complex.abs (d - c) < r ⊢ e / 2 ≤ Complex.abs (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 n : NontrivialHolomorphicAt (f c) z s : Set (ℂ × ℂ) sn : s ∈ 𝓝 (c, z) fa : AnalyticOn ℂ (uncurry f) s e✝ : ℝ ep✝ : e✝ > 0 es : ball (c, z) e✝ ⊆ s r : ℝ rp : 0 < r rs : closedBall (c, z) r ⊆ s fr : f c z ∉ f c '' sphere z r fc : ContinuousOn (fun w => ‖f c w - f c z‖) (sphere z r) x : ℂ xs : x ∈ sphere z r xm : IsMinOn (fun w => ‖f c w - f c z‖) (sphere z r) x e : ℝ he : ‖f c x - f c z‖ = e ep : 0 < e t : ℝ tp : t > 0 d w : ℂ wr : w ∈ sphere z r ft : ∀ (a b : ℂ), Complex.abs (a - c) ≤ r → Complex.abs (b - z) ≤ r → ∀ (a_3 b_1 : ℂ), Complex.abs (a_3 - c) ≤ r → Complex.abs (b_1 - z) ≤ r → Complex.abs (a - a_3) < t → Complex.abs (b - b_1) < t → Complex.abs (f a b - f a_3 b_1) < e / 4 dt : Complex.abs (d - c) < t ∧ Complex.abs (d - c) < r a1 : Complex.abs (f d w - f c w) ≤ e / 4 ⊢ e / 2 ≤ Complex.abs (f d w - f d z)
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/AnalyticManifold/OpenMapping.lean	NontrivialHolomorphicAt.nhds_le_map_nhds_param'	[109, 1]	[180, 64]	have a2 : abs (f c z - f d z) ≤ e / 4 := by refine (ft c z (abs_sub_self_lt rp).le (abs_sub_self_lt rp).le d z dt.2.le (abs_sub_self_lt rp).le ?_ (abs_sub_self_lt tp)).le rw [← neg_sub, Complex.abs.map_neg]; exact dt.1	X : Type inst✝⁶ : TopologicalSpace X S : Type inst✝⁵ : TopologicalSpace S inst✝⁴ : ChartedSpace ℂ S cms : AnalyticManifold 𝓘(ℂ, ℂ) S T : Type inst✝³ : TopologicalSpace T inst✝² : ChartedSpace ℂ T cmt : AnalyticManifold 𝓘(ℂ, ℂ) T U : Type inst✝¹ : TopologicalSpace U inst✝ : ChartedSpace ℂ U cmu : AnalyticManifold 𝓘(ℂ, ℂ) U f : ℂ → ℂ → ℂ c z : ℂ n : NontrivialHolomorphicAt (f c) z s : Set (ℂ × ℂ) sn : s ∈ 𝓝 (c, z) fa : AnalyticOn ℂ (uncurry f) s e✝ : ℝ ep✝ : e✝ > 0 es : ball (c, z) e✝ ⊆ s r : ℝ rp : 0 < r rs : closedBall (c, z) r ⊆ s fr : f c z ∉ f c '' sphere z r fc : ContinuousOn (fun w => ‖f c w - f c z‖) (sphere z r) x : ℂ xs : x ∈ sphere z r xm : IsMinOn (fun w => ‖f c w - f c z‖) (sphere z r) x e : ℝ he : ‖f c x - f c z‖ = e ep : 0 < e t : ℝ tp : t > 0 d w : ℂ wr : w ∈ sphere z r ft : ∀ (a b : ℂ), Complex.abs (a - c) ≤ r → Complex.abs (b - z) ≤ r → ∀ (a_3 b_1 : ℂ), Complex.abs (a_3 - c) ≤ r → Complex.abs (b_1 - z) ≤ r → Complex.abs (a - a_3) < t → Complex.abs (b - b_1) < t → Complex.abs (f a b - f a_3 b_1) < e / 4 dt : Complex.abs (d - c) < t ∧ Complex.abs (d - c) < r a1 : Complex.abs (f d w - f c w) ≤ e / 4 ⊢ e / 2 ≤ Complex.abs (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 n : NontrivialHolomorphicAt (f c) z s : Set (ℂ × ℂ) sn : s ∈ 𝓝 (c, z) fa : AnalyticOn ℂ (uncurry f) s e✝ : ℝ ep✝ : e✝ > 0 es : ball (c, z) e✝ ⊆ s r : ℝ rp : 0 < r rs : closedBall (c, z) r ⊆ s fr : f c z ∉ f c '' sphere z r fc : ContinuousOn (fun w => ‖f c w - f c z‖) (sphere z r) x : ℂ xs : x ∈ sphere z r xm : IsMinOn (fun w => ‖f c w - f c z‖) (sphere z r) x e : ℝ he : ‖f c x - f c z‖ = e ep : 0 < e t : ℝ tp : t > 0 d w : ℂ wr : w ∈ sphere z r ft : ∀ (a b : ℂ), Complex.abs (a - c) ≤ r → Complex.abs (b - z) ≤ r → ∀ (a_3 b_1 : ℂ), Complex.abs (a_3 - c) ≤ r → Complex.abs (b_1 - z) ≤ r → Complex.abs (a - a_3) < t → Complex.abs (b - b_1) < t → Complex.abs (f a b - f a_3 b_1) < e / 4 dt : Complex.abs (d - c) < t ∧ Complex.abs (d - c) < r a1 : Complex.abs (f d w - f c w) ≤ e / 4 a2 : Complex.abs (f c z - f d z) ≤ e / 4 ⊢ e / 2 ≤ Complex.abs (f d w - f d z)
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/AnalyticManifold/OpenMapping.lean	NontrivialHolomorphicAt.nhds_le_map_nhds_param'	[109, 1]	[180, 64]	calc abs (f d w - f d z) _ = abs (f c w - f c z + (f d w - f c w) + (f c z - f d z)) := by ring_nf _ ≥ abs (f c w - f c z + (f d w - f c w)) - abs (f c z - f d z) := by bound _ ≥ abs (f c w - f c z) - abs (f d w - f c w) - abs (f c z - f d z) := by bound _ ≥ e - e / 4 - e / 4 := by rw [← he] at a1 a2 ⊢; exact sub_le_sub (sub_le_sub (xm wr) a1) a2 _ = e / 2 := by ring	X : Type inst✝⁶ : TopologicalSpace X S : Type inst✝⁵ : TopologicalSpace S inst✝⁴ : ChartedSpace ℂ S cms : AnalyticManifold 𝓘(ℂ, ℂ) S T : Type inst✝³ : TopologicalSpace T inst✝² : ChartedSpace ℂ T cmt : AnalyticManifold 𝓘(ℂ, ℂ) T U : Type inst✝¹ : TopologicalSpace U inst✝ : ChartedSpace ℂ U cmu : AnalyticManifold 𝓘(ℂ, ℂ) U f : ℂ → ℂ → ℂ c z : ℂ n : NontrivialHolomorphicAt (f c) z s : Set (ℂ × ℂ) sn : s ∈ 𝓝 (c, z) fa : AnalyticOn ℂ (uncurry f) s e✝ : ℝ ep✝ : e✝ > 0 es : ball (c, z) e✝ ⊆ s r : ℝ rp : 0 < r rs : closedBall (c, z) r ⊆ s fr : f c z ∉ f c '' sphere z r fc : ContinuousOn (fun w => ‖f c w - f c z‖) (sphere z r) x : ℂ xs : x ∈ sphere z r xm : IsMinOn (fun w => ‖f c w - f c z‖) (sphere z r) x e : ℝ he : ‖f c x - f c z‖ = e ep : 0 < e t : ℝ tp : t > 0 d w : ℂ wr : w ∈ sphere z r ft : ∀ (a b : ℂ), Complex.abs (a - c) ≤ r → Complex.abs (b - z) ≤ r → ∀ (a_3 b_1 : ℂ), Complex.abs (a_3 - c) ≤ r → Complex.abs (b_1 - z) ≤ r → Complex.abs (a - a_3) < t → Complex.abs (b - b_1) < t → Complex.abs (f a b - f a_3 b_1) < e / 4 dt : Complex.abs (d - c) < t ∧ Complex.abs (d - c) < r a1 : Complex.abs (f d w - f c w) ≤ e / 4 a2 : Complex.abs (f c z - f d z) ≤ e / 4 ⊢ e / 2 ≤ Complex.abs (f d w - f d z)	no goals
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/AnalyticManifold/OpenMapping.lean	NontrivialHolomorphicAt.nhds_le_map_nhds_param'	[109, 1]	[180, 64]	refine (ft c z (abs_sub_self_lt rp).le (abs_sub_self_lt rp).le d z dt.2.le (abs_sub_self_lt rp).le ?_ (abs_sub_self_lt tp)).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 → ℂ c z : ℂ n : NontrivialHolomorphicAt (f c) z s : Set (ℂ × ℂ) sn : s ∈ 𝓝 (c, z) fa : AnalyticOn ℂ (uncurry f) s e✝ : ℝ ep✝ : e✝ > 0 es : ball (c, z) e✝ ⊆ s r : ℝ rp : 0 < r rs : closedBall (c, z) r ⊆ s fr : f c z ∉ f c '' sphere z r fc : ContinuousOn (fun w => ‖f c w - f c z‖) (sphere z r) x : ℂ xs : x ∈ sphere z r xm : IsMinOn (fun w => ‖f c w - f c z‖) (sphere z r) x e : ℝ he : ‖f c x - f c z‖ = e ep : 0 < e t : ℝ tp : t > 0 d w : ℂ wr : w ∈ sphere z r ft : ∀ (a b : ℂ), Complex.abs (a - c) ≤ r → Complex.abs (b - z) ≤ r → ∀ (a_3 b_1 : ℂ), Complex.abs (a_3 - c) ≤ r → Complex.abs (b_1 - z) ≤ r → Complex.abs (a - a_3) < t → Complex.abs (b - b_1) < t → Complex.abs (f a b - f a_3 b_1) < e / 4 dt : Complex.abs (d - c) < t ∧ Complex.abs (d - c) < r a1 : Complex.abs (f d w - f c w) ≤ e / 4 ⊢ Complex.abs (f c z - f d z) ≤ e / 4	X : Type inst✝⁶ : TopologicalSpace X S : Type inst✝⁵ : TopologicalSpace S inst✝⁴ : ChartedSpace ℂ S cms : AnalyticManifold 𝓘(ℂ, ℂ) S T : Type inst✝³ : TopologicalSpace T inst✝² : ChartedSpace ℂ T cmt : AnalyticManifold 𝓘(ℂ, ℂ) T U : Type inst✝¹ : TopologicalSpace U inst✝ : ChartedSpace ℂ U cmu : AnalyticManifold 𝓘(ℂ, ℂ) U f : ℂ → ℂ → ℂ c z : ℂ n : NontrivialHolomorphicAt (f c) z s : Set (ℂ × ℂ) sn : s ∈ 𝓝 (c, z) fa : AnalyticOn ℂ (uncurry f) s e✝ : ℝ ep✝ : e✝ > 0 es : ball (c, z) e✝ ⊆ s r : ℝ rp : 0 < r rs : closedBall (c, z) r ⊆ s fr : f c z ∉ f c '' sphere z r fc : ContinuousOn (fun w => ‖f c w - f c z‖) (sphere z r) x : ℂ xs : x ∈ sphere z r xm : IsMinOn (fun w => ‖f c w - f c z‖) (sphere z r) x e : ℝ he : ‖f c x - f c z‖ = e ep : 0 < e t : ℝ tp : t > 0 d w : ℂ wr : w ∈ sphere z r ft : ∀ (a b : ℂ), Complex.abs (a - c) ≤ r → Complex.abs (b - z) ≤ r → ∀ (a_3 b_1 : ℂ), Complex.abs (a_3 - c) ≤ r → Complex.abs (b_1 - z) ≤ r → Complex.abs (a - a_3) < t → Complex.abs (b - b_1) < t → Complex.abs (f a b - f a_3 b_1) < e / 4 dt : Complex.abs (d - c) < t ∧ Complex.abs (d - c) < r a1 : Complex.abs (f d w - f c w) ≤ e / 4 ⊢ Complex.abs (c - d) < t
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/AnalyticManifold/OpenMapping.lean	NontrivialHolomorphicAt.nhds_le_map_nhds_param'	[109, 1]	[180, 64]	rw [← neg_sub, Complex.abs.map_neg]	X : Type inst✝⁶ : TopologicalSpace X S : Type inst✝⁵ : TopologicalSpace S inst✝⁴ : ChartedSpace ℂ S cms : AnalyticManifold 𝓘(ℂ, ℂ) S T : Type inst✝³ : TopologicalSpace T inst✝² : ChartedSpace ℂ T cmt : AnalyticManifold 𝓘(ℂ, ℂ) T U : Type inst✝¹ : TopologicalSpace U inst✝ : ChartedSpace ℂ U cmu : AnalyticManifold 𝓘(ℂ, ℂ) U f : ℂ → ℂ → ℂ c z : ℂ n : NontrivialHolomorphicAt (f c) z s : Set (ℂ × ℂ) sn : s ∈ 𝓝 (c, z) fa : AnalyticOn ℂ (uncurry f) s e✝ : ℝ ep✝ : e✝ > 0 es : ball (c, z) e✝ ⊆ s r : ℝ rp : 0 < r rs : closedBall (c, z) r ⊆ s fr : f c z ∉ f c '' sphere z r fc : ContinuousOn (fun w => ‖f c w - f c z‖) (sphere z r) x : ℂ xs : x ∈ sphere z r xm : IsMinOn (fun w => ‖f c w - f c z‖) (sphere z r) x e : ℝ he : ‖f c x - f c z‖ = e ep : 0 < e t : ℝ tp : t > 0 d w : ℂ wr : w ∈ sphere z r ft : ∀ (a b : ℂ), Complex.abs (a - c) ≤ r → Complex.abs (b - z) ≤ r → ∀ (a_3 b_1 : ℂ), Complex.abs (a_3 - c) ≤ r → Complex.abs (b_1 - z) ≤ r → Complex.abs (a - a_3) < t → Complex.abs (b - b_1) < t → Complex.abs (f a b - f a_3 b_1) < e / 4 dt : Complex.abs (d - c) < t ∧ Complex.abs (d - c) < r a1 : Complex.abs (f d w - f c w) ≤ e / 4 ⊢ Complex.abs (c - d) < t	X : Type inst✝⁶ : TopologicalSpace X S : Type inst✝⁵ : TopologicalSpace S inst✝⁴ : ChartedSpace ℂ S cms : AnalyticManifold 𝓘(ℂ, ℂ) S T : Type inst✝³ : TopologicalSpace T inst✝² : ChartedSpace ℂ T cmt : AnalyticManifold 𝓘(ℂ, ℂ) T U : Type inst✝¹ : TopologicalSpace U inst✝ : ChartedSpace ℂ U cmu : AnalyticManifold 𝓘(ℂ, ℂ) U f : ℂ → ℂ → ℂ c z : ℂ n : NontrivialHolomorphicAt (f c) z s : Set (ℂ × ℂ) sn : s ∈ 𝓝 (c, z) fa : AnalyticOn ℂ (uncurry f) s e✝ : ℝ ep✝ : e✝ > 0 es : ball (c, z) e✝ ⊆ s r : ℝ rp : 0 < r rs : closedBall (c, z) r ⊆ s fr : f c z ∉ f c '' sphere z r fc : ContinuousOn (fun w => ‖f c w - f c z‖) (sphere z r) x : ℂ xs : x ∈ sphere z r xm : IsMinOn (fun w => ‖f c w - f c z‖) (sphere z r) x e : ℝ he : ‖f c x - f c z‖ = e ep : 0 < e t : ℝ tp : t > 0 d w : ℂ wr : w ∈ sphere z r ft : ∀ (a b : ℂ), Complex.abs (a - c) ≤ r → Complex.abs (b - z) ≤ r → ∀ (a_3 b_1 : ℂ), Complex.abs (a_3 - c) ≤ r → Complex.abs (b_1 - z) ≤ r → Complex.abs (a - a_3) < t → Complex.abs (b - b_1) < t → Complex.abs (f a b - f a_3 b_1) < e / 4 dt : Complex.abs (d - c) < t ∧ Complex.abs (d - c) < r a1 : Complex.abs (f d w - f c w) ≤ e / 4 ⊢ Complex.abs (d - c) < t
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/AnalyticManifold/OpenMapping.lean	NontrivialHolomorphicAt.nhds_le_map_nhds_param'	[109, 1]	[180, 64]	exact dt.1	X : Type inst✝⁶ : TopologicalSpace X S : Type inst✝⁵ : TopologicalSpace S inst✝⁴ : ChartedSpace ℂ S cms : AnalyticManifold 𝓘(ℂ, ℂ) S T : Type inst✝³ : TopologicalSpace T inst✝² : ChartedSpace ℂ T cmt : AnalyticManifold 𝓘(ℂ, ℂ) T U : Type inst✝¹ : TopologicalSpace U inst✝ : ChartedSpace ℂ U cmu : AnalyticManifold 𝓘(ℂ, ℂ) U f : ℂ → ℂ → ℂ c z : ℂ n : NontrivialHolomorphicAt (f c) z s : Set (ℂ × ℂ) sn : s ∈ 𝓝 (c, z) fa : AnalyticOn ℂ (uncurry f) s e✝ : ℝ ep✝ : e✝ > 0 es : ball (c, z) e✝ ⊆ s r : ℝ rp : 0 < r rs : closedBall (c, z) r ⊆ s fr : f c z ∉ f c '' sphere z r fc : ContinuousOn (fun w => ‖f c w - f c z‖) (sphere z r) x : ℂ xs : x ∈ sphere z r xm : IsMinOn (fun w => ‖f c w - f c z‖) (sphere z r) x e : ℝ he : ‖f c x - f c z‖ = e ep : 0 < e t : ℝ tp : t > 0 d w : ℂ wr : w ∈ sphere z r ft : ∀ (a b : ℂ), Complex.abs (a - c) ≤ r → Complex.abs (b - z) ≤ r → ∀ (a_3 b_1 : ℂ), Complex.abs (a_3 - c) ≤ r → Complex.abs (b_1 - z) ≤ r → Complex.abs (a - a_3) < t → Complex.abs (b - b_1) < t → Complex.abs (f a b - f a_3 b_1) < e / 4 dt : Complex.abs (d - c) < t ∧ Complex.abs (d - c) < r a1 : Complex.abs (f d w - f c w) ≤ e / 4 ⊢ Complex.abs (d - c) < t	no goals
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/AnalyticManifold/OpenMapping.lean	NontrivialHolomorphicAt.nhds_le_map_nhds_param'	[109, 1]	[180, 64]	ring_nf	X : Type inst✝⁶ : TopologicalSpace X S : Type inst✝⁵ : TopologicalSpace S inst✝⁴ : ChartedSpace ℂ S cms : AnalyticManifold 𝓘(ℂ, ℂ) S T : Type inst✝³ : TopologicalSpace T inst✝² : ChartedSpace ℂ T cmt : AnalyticManifold 𝓘(ℂ, ℂ) T U : Type inst✝¹ : TopologicalSpace U inst✝ : ChartedSpace ℂ U cmu : AnalyticManifold 𝓘(ℂ, ℂ) U f : ℂ → ℂ → ℂ c z : ℂ n : NontrivialHolomorphicAt (f c) z s : Set (ℂ × ℂ) sn : s ∈ 𝓝 (c, z) fa : AnalyticOn ℂ (uncurry f) s e✝ : ℝ ep✝ : e✝ > 0 es : ball (c, z) e✝ ⊆ s r : ℝ rp : 0 < r rs : closedBall (c, z) r ⊆ s fr : f c z ∉ f c '' sphere z r fc : ContinuousOn (fun w => ‖f c w - f c z‖) (sphere z r) x : ℂ xs : x ∈ sphere z r xm : IsMinOn (fun w => ‖f c w - f c z‖) (sphere z r) x e : ℝ he : ‖f c x - f c z‖ = e ep : 0 < e t : ℝ tp : t > 0 d w : ℂ wr : w ∈ sphere z r ft : ∀ (a b : ℂ), Complex.abs (a - c) ≤ r → Complex.abs (b - z) ≤ r → ∀ (a_3 b_1 : ℂ), Complex.abs (a_3 - c) ≤ r → Complex.abs (b_1 - z) ≤ r → Complex.abs (a - a_3) < t → Complex.abs (b - b_1) < t → Complex.abs (f a b - f a_3 b_1) < e / 4 dt : Complex.abs (d - c) < t ∧ Complex.abs (d - c) < r a1 : Complex.abs (f d w - f c w) ≤ e / 4 a2 : Complex.abs (f c z - f d z) ≤ e / 4 ⊢ Complex.abs (f d w - f d z) = Complex.abs (f c w - f c z + (f d w - f c w) + (f c z - f d z))	no goals
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/AnalyticManifold/OpenMapping.lean	NontrivialHolomorphicAt.nhds_le_map_nhds_param'	[109, 1]	[180, 64]	bound	X : Type inst✝⁶ : TopologicalSpace X S : Type inst✝⁵ : TopologicalSpace S inst✝⁴ : ChartedSpace ℂ S cms : AnalyticManifold 𝓘(ℂ, ℂ) S T : Type inst✝³ : TopologicalSpace T inst✝² : ChartedSpace ℂ T cmt : AnalyticManifold 𝓘(ℂ, ℂ) T U : Type inst✝¹ : TopologicalSpace U inst✝ : ChartedSpace ℂ U cmu : AnalyticManifold 𝓘(ℂ, ℂ) U f : ℂ → ℂ → ℂ c z : ℂ n : NontrivialHolomorphicAt (f c) z s : Set (ℂ × ℂ) sn : s ∈ 𝓝 (c, z) fa : AnalyticOn ℂ (uncurry f) s e✝ : ℝ ep✝ : e✝ > 0 es : ball (c, z) e✝ ⊆ s r : ℝ rp : 0 < r rs : closedBall (c, z) r ⊆ s fr : f c z ∉ f c '' sphere z r fc : ContinuousOn (fun w => ‖f c w - f c z‖) (sphere z r) x : ℂ xs : x ∈ sphere z r xm : IsMinOn (fun w => ‖f c w - f c z‖) (sphere z r) x e : ℝ he : ‖f c x - f c z‖ = e ep : 0 < e t : ℝ tp : t > 0 d w : ℂ wr : w ∈ sphere z r ft : ∀ (a b : ℂ), Complex.abs (a - c) ≤ r → Complex.abs (b - z) ≤ r → ∀ (a_3 b_1 : ℂ), Complex.abs (a_3 - c) ≤ r → Complex.abs (b_1 - z) ≤ r → Complex.abs (a - a_3) < t → Complex.abs (b - b_1) < t → Complex.abs (f a b - f a_3 b_1) < e / 4 dt : Complex.abs (d - c) < t ∧ Complex.abs (d - c) < r a1 : Complex.abs (f d w - f c w) ≤ e / 4 a2 : Complex.abs (f c z - f d z) ≤ e / 4 ⊢ Complex.abs (f c w - f c z + (f d w - f c w) + (f c z - f d z)) ≥ Complex.abs (f c w - f c z + (f d w - f c w)) - Complex.abs (f c z - f d z)	no goals
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/AnalyticManifold/OpenMapping.lean	NontrivialHolomorphicAt.nhds_le_map_nhds_param'	[109, 1]	[180, 64]	bound	X : Type inst✝⁶ : TopologicalSpace X S : Type inst✝⁵ : TopologicalSpace S inst✝⁴ : ChartedSpace ℂ S cms : AnalyticManifold 𝓘(ℂ, ℂ) S T : Type inst✝³ : TopologicalSpace T inst✝² : ChartedSpace ℂ T cmt : AnalyticManifold 𝓘(ℂ, ℂ) T U : Type inst✝¹ : TopologicalSpace U inst✝ : ChartedSpace ℂ U cmu : AnalyticManifold 𝓘(ℂ, ℂ) U f : ℂ → ℂ → ℂ c z : ℂ n : NontrivialHolomorphicAt (f c) z s : Set (ℂ × ℂ) sn : s ∈ 𝓝 (c, z) fa : AnalyticOn ℂ (uncurry f) s e✝ : ℝ ep✝ : e✝ > 0 es : ball (c, z) e✝ ⊆ s r : ℝ rp : 0 < r rs : closedBall (c, z) r ⊆ s fr : f c z ∉ f c '' sphere z r fc : ContinuousOn (fun w => ‖f c w - f c z‖) (sphere z r) x : ℂ xs : x ∈ sphere z r xm : IsMinOn (fun w => ‖f c w - f c z‖) (sphere z r) x e : ℝ he : ‖f c x - f c z‖ = e ep : 0 < e t : ℝ tp : t > 0 d w : ℂ wr : w ∈ sphere z r ft : ∀ (a b : ℂ), Complex.abs (a - c) ≤ r → Complex.abs (b - z) ≤ r → ∀ (a_3 b_1 : ℂ), Complex.abs (a_3 - c) ≤ r → Complex.abs (b_1 - z) ≤ r → Complex.abs (a - a_3) < t → Complex.abs (b - b_1) < t → Complex.abs (f a b - f a_3 b_1) < e / 4 dt : Complex.abs (d - c) < t ∧ Complex.abs (d - c) < r a1 : Complex.abs (f d w - f c w) ≤ e / 4 a2 : Complex.abs (f c z - f d z) ≤ e / 4 ⊢ Complex.abs (f c w - f c z + (f d w - f c w)) - Complex.abs (f c z - f d z) ≥ Complex.abs (f c w - f c z) - Complex.abs (f d w - f c w) - Complex.abs (f c z - f d z)	no goals
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/AnalyticManifold/OpenMapping.lean	NontrivialHolomorphicAt.nhds_le_map_nhds_param'	[109, 1]	[180, 64]	rw [← he] at a1 a2 ⊢	X : Type inst✝⁶ : TopologicalSpace X S : Type inst✝⁵ : TopologicalSpace S inst✝⁴ : ChartedSpace ℂ S cms : AnalyticManifold 𝓘(ℂ, ℂ) S T : Type inst✝³ : TopologicalSpace T inst✝² : ChartedSpace ℂ T cmt : AnalyticManifold 𝓘(ℂ, ℂ) T U : Type inst✝¹ : TopologicalSpace U inst✝ : ChartedSpace ℂ U cmu : AnalyticManifold 𝓘(ℂ, ℂ) U f : ℂ → ℂ → ℂ c z : ℂ n : NontrivialHolomorphicAt (f c) z s : Set (ℂ × ℂ) sn : s ∈ 𝓝 (c, z) fa : AnalyticOn ℂ (uncurry f) s e✝ : ℝ ep✝ : e✝ > 0 es : ball (c, z) e✝ ⊆ s r : ℝ rp : 0 < r rs : closedBall (c, z) r ⊆ s fr : f c z ∉ f c '' sphere z r fc : ContinuousOn (fun w => ‖f c w - f c z‖) (sphere z r) x : ℂ xs : x ∈ sphere z r xm : IsMinOn (fun w => ‖f c w - f c z‖) (sphere z r) x e : ℝ he : ‖f c x - f c z‖ = e ep : 0 < e t : ℝ tp : t > 0 d w : ℂ wr : w ∈ sphere z r ft : ∀ (a b : ℂ), Complex.abs (a - c) ≤ r → Complex.abs (b - z) ≤ r → ∀ (a_3 b_1 : ℂ), Complex.abs (a_3 - c) ≤ r → Complex.abs (b_1 - z) ≤ r → Complex.abs (a - a_3) < t → Complex.abs (b - b_1) < t → Complex.abs (f a b - f a_3 b_1) < e / 4 dt : Complex.abs (d - c) < t ∧ Complex.abs (d - c) < r a1 : Complex.abs (f d w - f c w) ≤ e / 4 a2 : Complex.abs (f c z - f d z) ≤ e / 4 ⊢ Complex.abs (f c w - f c z) - Complex.abs (f d w - f c w) - Complex.abs (f c z - f d z) ≥ e - e / 4 - e / 4	X : Type inst✝⁶ : TopologicalSpace X S : Type inst✝⁵ : TopologicalSpace S inst✝⁴ : ChartedSpace ℂ S cms : AnalyticManifold 𝓘(ℂ, ℂ) S T : Type inst✝³ : TopologicalSpace T inst✝² : ChartedSpace ℂ T cmt : AnalyticManifold 𝓘(ℂ, ℂ) T U : Type inst✝¹ : TopologicalSpace U inst✝ : ChartedSpace ℂ U cmu : AnalyticManifold 𝓘(ℂ, ℂ) U f : ℂ → ℂ → ℂ c z : ℂ n : NontrivialHolomorphicAt (f c) z s : Set (ℂ × ℂ) sn : s ∈ 𝓝 (c, z) fa : AnalyticOn ℂ (uncurry f) s e✝ : ℝ ep✝ : e✝ > 0 es : ball (c, z) e✝ ⊆ s r : ℝ rp : 0 < r rs : closedBall (c, z) r ⊆ s fr : f c z ∉ f c '' sphere z r fc : ContinuousOn (fun w => ‖f c w - f c z‖) (sphere z r) x : ℂ xs : x ∈ sphere z r xm : IsMinOn (fun w => ‖f c w - f c z‖) (sphere z r) x e : ℝ he : ‖f c x - f c z‖ = e ep : 0 < e t : ℝ tp : t > 0 d w : ℂ wr : w ∈ sphere z r ft : ∀ (a b : ℂ), Complex.abs (a - c) ≤ r → Complex.abs (b - z) ≤ r → ∀ (a_3 b_1 : ℂ), Complex.abs (a_3 - c) ≤ r → Complex.abs (b_1 - z) ≤ r → Complex.abs (a - a_3) < t → Complex.abs (b - b_1) < t → Complex.abs (f a b - f a_3 b_1) < e / 4 dt : Complex.abs (d - c) < t ∧ Complex.abs (d - c) < r a1 : Complex.abs (f d w - f c w) ≤ ‖f c x - f c z‖ / 4 a2 : Complex.abs (f c z - f d z) ≤ ‖f c x - f c z‖ / 4 ⊢ Complex.abs (f c w - f c z) - Complex.abs (f d w - f c w) - Complex.abs (f c z - f d z) ≥ ‖f c x - f c z‖ - ‖f c x - f c z‖ / 4 - ‖f c x - f c z‖ / 4
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/AnalyticManifold/OpenMapping.lean	NontrivialHolomorphicAt.nhds_le_map_nhds_param'	[109, 1]	[180, 64]	exact sub_le_sub (sub_le_sub (xm wr) a1) a2	X : Type inst✝⁶ : TopologicalSpace X S : Type inst✝⁵ : TopologicalSpace S inst✝⁴ : ChartedSpace ℂ S cms : AnalyticManifold 𝓘(ℂ, ℂ) S T : Type inst✝³ : TopologicalSpace T inst✝² : ChartedSpace ℂ T cmt : AnalyticManifold 𝓘(ℂ, ℂ) T U : Type inst✝¹ : TopologicalSpace U inst✝ : ChartedSpace ℂ U cmu : AnalyticManifold 𝓘(ℂ, ℂ) U f : ℂ → ℂ → ℂ c z : ℂ n : NontrivialHolomorphicAt (f c) z s : Set (ℂ × ℂ) sn : s ∈ 𝓝 (c, z) fa : AnalyticOn ℂ (uncurry f) s e✝ : ℝ ep✝ : e✝ > 0 es : ball (c, z) e✝ ⊆ s r : ℝ rp : 0 < r rs : closedBall (c, z) r ⊆ s fr : f c z ∉ f c '' sphere z r fc : ContinuousOn (fun w => ‖f c w - f c z‖) (sphere z r) x : ℂ xs : x ∈ sphere z r xm : IsMinOn (fun w => ‖f c w - f c z‖) (sphere z r) x e : ℝ he : ‖f c x - f c z‖ = e ep : 0 < e t : ℝ tp : t > 0 d w : ℂ wr : w ∈ sphere z r ft : ∀ (a b : ℂ), Complex.abs (a - c) ≤ r → Complex.abs (b - z) ≤ r → ∀ (a_3 b_1 : ℂ), Complex.abs (a_3 - c) ≤ r → Complex.abs (b_1 - z) ≤ r → Complex.abs (a - a_3) < t → Complex.abs (b - b_1) < t → Complex.abs (f a b - f a_3 b_1) < e / 4 dt : Complex.abs (d - c) < t ∧ Complex.abs (d - c) < r a1 : Complex.abs (f d w - f c w) ≤ ‖f c x - f c z‖ / 4 a2 : Complex.abs (f c z - f d z) ≤ ‖f c x - f c z‖ / 4 ⊢ Complex.abs (f c w - f c z) - Complex.abs (f d w - f c w) - Complex.abs (f c z - f d z) ≥ ‖f c x - f c z‖ - ‖f c x - f c z‖ / 4 - ‖f c x - f c z‖ / 4	no goals
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/AnalyticManifold/OpenMapping.lean	NontrivialHolomorphicAt.nhds_le_map_nhds_param'	[109, 1]	[180, 64]	ring	X : Type inst✝⁶ : TopologicalSpace X S : Type inst✝⁵ : TopologicalSpace S inst✝⁴ : ChartedSpace ℂ S cms : AnalyticManifold 𝓘(ℂ, ℂ) S T : Type inst✝³ : TopologicalSpace T inst✝² : ChartedSpace ℂ T cmt : AnalyticManifold 𝓘(ℂ, ℂ) T U : Type inst✝¹ : TopologicalSpace U inst✝ : ChartedSpace ℂ U cmu : AnalyticManifold 𝓘(ℂ, ℂ) U f : ℂ → ℂ → ℂ c z : ℂ n : NontrivialHolomorphicAt (f c) z s : Set (ℂ × ℂ) sn : s ∈ 𝓝 (c, z) fa : AnalyticOn ℂ (uncurry f) s e✝ : ℝ ep✝ : e✝ > 0 es : ball (c, z) e✝ ⊆ s r : ℝ rp : 0 < r rs : closedBall (c, z) r ⊆ s fr : f c z ∉ f c '' sphere z r fc : ContinuousOn (fun w => ‖f c w - f c z‖) (sphere z r) x : ℂ xs : x ∈ sphere z r xm : IsMinOn (fun w => ‖f c w - f c z‖) (sphere z r) x e : ℝ he : ‖f c x - f c z‖ = e ep : 0 < e t : ℝ tp : t > 0 d w : ℂ wr : w ∈ sphere z r ft : ∀ (a b : ℂ), Complex.abs (a - c) ≤ r → Complex.abs (b - z) ≤ r → ∀ (a_3 b_1 : ℂ), Complex.abs (a_3 - c) ≤ r → Complex.abs (b_1 - z) ≤ r → Complex.abs (a - a_3) < t → Complex.abs (b - b_1) < t → Complex.abs (f a b - f a_3 b_1) < e / 4 dt : Complex.abs (d - c) < t ∧ Complex.abs (d - c) < r a1 : Complex.abs (f d w - f c w) ≤ e / 4 a2 : Complex.abs (f c z - f d z) ≤ e / 4 ⊢ e - e / 4 - e / 4 = e / 2	no goals
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/AnalyticManifold/OpenMapping.lean	NontrivialHolomorphicAt.nhds_le_map_nhds_param'	[109, 1]	[180, 64]	refine _root_.trans ?_ rs	X : Type inst✝⁶ : TopologicalSpace X S : Type inst✝⁵ : TopologicalSpace S inst✝⁴ : ChartedSpace ℂ S cms : AnalyticManifold 𝓘(ℂ, ℂ) S T : Type inst✝³ : TopologicalSpace T inst✝² : ChartedSpace ℂ T cmt : AnalyticManifold 𝓘(ℂ, ℂ) T U : Type inst✝¹ : TopologicalSpace U inst✝ : ChartedSpace ℂ U cmu : AnalyticManifold 𝓘(ℂ, ℂ) U f : ℂ → ℂ → ℂ c z : ℂ n : NontrivialHolomorphicAt (f c) z s : Set (ℂ × ℂ) sn : s ∈ 𝓝 (c, z) fa : AnalyticOn ℂ (uncurry f) s e✝ : ℝ ep✝ : e✝ > 0 es : ball (c, z) e✝ ⊆ s r : ℝ rp : 0 < r rs : closedBall (c, z) r ⊆ s fr : f c z ∉ f c '' sphere z r fc : ContinuousOn (fun w => ‖f c w - f c z‖) (sphere z r) x : ℂ xs : x ∈ sphere z r xm : IsMinOn (fun w => ‖f c w - f c z‖) (sphere z r) x e : ℝ he : ‖f c x - f c z‖ = e ep : 0 < e t : ℝ tp : t > 0 ft : ∀ x ∈ closedBall (c, z) r, ∀ y ∈ closedBall (c, z) r, dist x y < t → dist (uncurry f x) (uncurry f y) < e / 4 ef : ∀ d ∈ ball c (min t r), ∀ w ∈ sphere z r, e / 2 ≤ ‖f d w - f d z‖ ⊢ ball c (min t r) ×ˢ closedBall z r ⊆ s	X : Type inst✝⁶ : TopologicalSpace X S : Type inst✝⁵ : TopologicalSpace S inst✝⁴ : ChartedSpace ℂ S cms : AnalyticManifold 𝓘(ℂ, ℂ) S T : Type inst✝³ : TopologicalSpace T inst✝² : ChartedSpace ℂ T cmt : AnalyticManifold 𝓘(ℂ, ℂ) T U : Type inst✝¹ : TopologicalSpace U inst✝ : ChartedSpace ℂ U cmu : AnalyticManifold 𝓘(ℂ, ℂ) U f : ℂ → ℂ → ℂ c z : ℂ n : NontrivialHolomorphicAt (f c) z s : Set (ℂ × ℂ) sn : s ∈ 𝓝 (c, z) fa : AnalyticOn ℂ (uncurry f) s e✝ : ℝ ep✝ : e✝ > 0 es : ball (c, z) e✝ ⊆ s r : ℝ rp : 0 < r rs : closedBall (c, z) r ⊆ s fr : f c z ∉ f c '' sphere z r fc : ContinuousOn (fun w => ‖f c w - f c z‖) (sphere z r) x : ℂ xs : x ∈ sphere z r xm : IsMinOn (fun w => ‖f c w - f c z‖) (sphere z r) x e : ℝ he : ‖f c x - f c z‖ = e ep : 0 < e t : ℝ tp : t > 0 ft : ∀ x ∈ closedBall (c, z) r, ∀ y ∈ closedBall (c, z) r, dist x y < t → dist (uncurry f x) (uncurry f y) < e / 4 ef : ∀ d ∈ ball c (min t r), ∀ w ∈ sphere z r, e / 2 ≤ ‖f d w - f d z‖ ⊢ ball c (min t r) ×ˢ closedBall z r ⊆ closedBall (c, z) r
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/AnalyticManifold/OpenMapping.lean	NontrivialHolomorphicAt.nhds_le_map_nhds_param'	[109, 1]	[180, 64]	rw [← closedBall_prod_same]	X : Type inst✝⁶ : TopologicalSpace X S : Type inst✝⁵ : TopologicalSpace S inst✝⁴ : ChartedSpace ℂ S cms : AnalyticManifold 𝓘(ℂ, ℂ) S T : Type inst✝³ : TopologicalSpace T inst✝² : ChartedSpace ℂ T cmt : AnalyticManifold 𝓘(ℂ, ℂ) T U : Type inst✝¹ : TopologicalSpace U inst✝ : ChartedSpace ℂ U cmu : AnalyticManifold 𝓘(ℂ, ℂ) U f : ℂ → ℂ → ℂ c z : ℂ n : NontrivialHolomorphicAt (f c) z s : Set (ℂ × ℂ) sn : s ∈ 𝓝 (c, z) fa : AnalyticOn ℂ (uncurry f) s e✝ : ℝ ep✝ : e✝ > 0 es : ball (c, z) e✝ ⊆ s r : ℝ rp : 0 < r rs : closedBall (c, z) r ⊆ s fr : f c z ∉ f c '' sphere z r fc : ContinuousOn (fun w => ‖f c w - f c z‖) (sphere z r) x : ℂ xs : x ∈ sphere z r xm : IsMinOn (fun w => ‖f c w - f c z‖) (sphere z r) x e : ℝ he : ‖f c x - f c z‖ = e ep : 0 < e t : ℝ tp : t > 0 ft : ∀ x ∈ closedBall (c, z) r, ∀ y ∈ closedBall (c, z) r, dist x y < t → dist (uncurry f x) (uncurry f y) < e / 4 ef : ∀ d ∈ ball c (min t r), ∀ w ∈ sphere z r, e / 2 ≤ ‖f d w - f d z‖ ⊢ ball c (min t r) ×ˢ closedBall z r ⊆ closedBall (c, 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 n : NontrivialHolomorphicAt (f c) z s : Set (ℂ × ℂ) sn : s ∈ 𝓝 (c, z) fa : AnalyticOn ℂ (uncurry f) s e✝ : ℝ ep✝ : e✝ > 0 es : ball (c, z) e✝ ⊆ s r : ℝ rp : 0 < r rs : closedBall (c, z) r ⊆ s fr : f c z ∉ f c '' sphere z r fc : ContinuousOn (fun w => ‖f c w - f c z‖) (sphere z r) x : ℂ xs : x ∈ sphere z r xm : IsMinOn (fun w => ‖f c w - f c z‖) (sphere z r) x e : ℝ he : ‖f c x - f c z‖ = e ep : 0 < e t : ℝ tp : t > 0 ft : ∀ x ∈ closedBall (c, z) r, ∀ y ∈ closedBall (c, z) r, dist x y < t → dist (uncurry f x) (uncurry f y) < e / 4 ef : ∀ d ∈ ball c (min t r), ∀ w ∈ sphere z r, e / 2 ≤ ‖f d w - f d z‖ ⊢ ball c (min t r) ×ˢ closedBall z r ⊆ closedBall c r ×ˢ closedBall z r
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/AnalyticManifold/OpenMapping.lean	NontrivialHolomorphicAt.nhds_le_map_nhds_param'	[109, 1]	[180, 64]	apply prod_mono_left	X : Type inst✝⁶ : TopologicalSpace X S : Type inst✝⁵ : TopologicalSpace S inst✝⁴ : ChartedSpace ℂ S cms : AnalyticManifold 𝓘(ℂ, ℂ) S T : Type inst✝³ : TopologicalSpace T inst✝² : ChartedSpace ℂ T cmt : AnalyticManifold 𝓘(ℂ, ℂ) T U : Type inst✝¹ : TopologicalSpace U inst✝ : ChartedSpace ℂ U cmu : AnalyticManifold 𝓘(ℂ, ℂ) U f : ℂ → ℂ → ℂ c z : ℂ n : NontrivialHolomorphicAt (f c) z s : Set (ℂ × ℂ) sn : s ∈ 𝓝 (c, z) fa : AnalyticOn ℂ (uncurry f) s e✝ : ℝ ep✝ : e✝ > 0 es : ball (c, z) e✝ ⊆ s r : ℝ rp : 0 < r rs : closedBall (c, z) r ⊆ s fr : f c z ∉ f c '' sphere z r fc : ContinuousOn (fun w => ‖f c w - f c z‖) (sphere z r) x : ℂ xs : x ∈ sphere z r xm : IsMinOn (fun w => ‖f c w - f c z‖) (sphere z r) x e : ℝ he : ‖f c x - f c z‖ = e ep : 0 < e t : ℝ tp : t > 0 ft : ∀ x ∈ closedBall (c, z) r, ∀ y ∈ closedBall (c, z) r, dist x y < t → dist (uncurry f x) (uncurry f y) < e / 4 ef : ∀ d ∈ ball c (min t r), ∀ w ∈ sphere z r, e / 2 ≤ ‖f d w - f d z‖ ⊢ ball c (min t r) ×ˢ closedBall z r ⊆ closedBall c r ×ˢ closedBall z r	case hs X : Type inst✝⁶ : TopologicalSpace X S : Type inst✝⁵ : TopologicalSpace S inst✝⁴ : ChartedSpace ℂ S cms : AnalyticManifold 𝓘(ℂ, ℂ) S T : Type inst✝³ : TopologicalSpace T inst✝² : ChartedSpace ℂ T cmt : AnalyticManifold 𝓘(ℂ, ℂ) T U : Type inst✝¹ : TopologicalSpace U inst✝ : ChartedSpace ℂ U cmu : AnalyticManifold 𝓘(ℂ, ℂ) U f : ℂ → ℂ → ℂ c z : ℂ n : NontrivialHolomorphicAt (f c) z s : Set (ℂ × ℂ) sn : s ∈ 𝓝 (c, z) fa : AnalyticOn ℂ (uncurry f) s e✝ : ℝ ep✝ : e✝ > 0 es : ball (c, z) e✝ ⊆ s r : ℝ rp : 0 < r rs : closedBall (c, z) r ⊆ s fr : f c z ∉ f c '' sphere z r fc : ContinuousOn (fun w => ‖f c w - f c z‖) (sphere z r) x : ℂ xs : x ∈ sphere z r xm : IsMinOn (fun w => ‖f c w - f c z‖) (sphere z r) x e : ℝ he : ‖f c x - f c z‖ = e ep : 0 < e t : ℝ tp : t > 0 ft : ∀ x ∈ closedBall (c, z) r, ∀ y ∈ closedBall (c, z) r, dist x y < t → dist (uncurry f x) (uncurry f y) < e / 4 ef : ∀ d ∈ ball c (min t r), ∀ w ∈ sphere z r, e / 2 ≤ ‖f d w - f d z‖ ⊢ ball c (min t r) ⊆ closedBall c r
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/AnalyticManifold/OpenMapping.lean	NontrivialHolomorphicAt.nhds_le_map_nhds_param'	[109, 1]	[180, 64]	exact _root_.trans (Metric.ball_subset_ball (min_le_right _ _)) Metric.ball_subset_closedBall	case hs X : Type inst✝⁶ : TopologicalSpace X S : Type inst✝⁵ : TopologicalSpace S inst✝⁴ : ChartedSpace ℂ S cms : AnalyticManifold 𝓘(ℂ, ℂ) S T : Type inst✝³ : TopologicalSpace T inst✝² : ChartedSpace ℂ T cmt : AnalyticManifold 𝓘(ℂ, ℂ) T U : Type inst✝¹ : TopologicalSpace U inst✝ : ChartedSpace ℂ U cmu : AnalyticManifold 𝓘(ℂ, ℂ) U f : ℂ → ℂ → ℂ c z : ℂ n : NontrivialHolomorphicAt (f c) z s : Set (ℂ × ℂ) sn : s ∈ 𝓝 (c, z) fa : AnalyticOn ℂ (uncurry f) s e✝ : ℝ ep✝ : e✝ > 0 es : ball (c, z) e✝ ⊆ s r : ℝ rp : 0 < r rs : closedBall (c, z) r ⊆ s fr : f c z ∉ f c '' sphere z r fc : ContinuousOn (fun w => ‖f c w - f c z‖) (sphere z r) x : ℂ xs : x ∈ sphere z r xm : IsMinOn (fun w => ‖f c w - f c z‖) (sphere z r) x e : ℝ he : ‖f c x - f c z‖ = e ep : 0 < e t : ℝ tp : t > 0 ft : ∀ x ∈ closedBall (c, z) r, ∀ y ∈ closedBall (c, z) r, dist x y < t → dist (uncurry f x) (uncurry f y) < e / 4 ef : ∀ d ∈ ball c (min t r), ∀ w ∈ sphere z r, e / 2 ≤ ‖f d w - f d z‖ ⊢ ball c (min t r) ⊆ closedBall c r	no goals
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/AnalyticManifold/OpenMapping.lean	NontrivialHolomorphicAt.nhds_le_map_nhds_param'	[109, 1]	[180, 64]	bound	X : Type inst✝⁶ : TopologicalSpace X S : Type inst✝⁵ : TopologicalSpace S inst✝⁴ : ChartedSpace ℂ S cms : AnalyticManifold 𝓘(ℂ, ℂ) S T : Type inst✝³ : TopologicalSpace T inst✝² : ChartedSpace ℂ T cmt : AnalyticManifold 𝓘(ℂ, ℂ) T U : Type inst✝¹ : TopologicalSpace U inst✝ : ChartedSpace ℂ U cmu : AnalyticManifold 𝓘(ℂ, ℂ) U f : ℂ → ℂ → ℂ c z : ℂ n : NontrivialHolomorphicAt (f c) z s : Set (ℂ × ℂ) sn : s ∈ 𝓝 (c, z) fa : AnalyticOn ℂ (uncurry f) s e✝ : ℝ ep✝ : e✝ > 0 es : ball (c, z) e✝ ⊆ s r : ℝ rp : 0 < r rs : closedBall (c, z) r ⊆ s fr : f c z ∉ f c '' sphere z r fc : ContinuousOn (fun w => ‖f c w - f c z‖) (sphere z r) x : ℂ xs : x ∈ sphere z r xm : IsMinOn (fun w => ‖f c w - f c z‖) (sphere z r) x e : ℝ he : ‖f c x - f c z‖ = e ep : 0 < e t : ℝ tp : t > 0 ft : ∀ x ∈ closedBall (c, z) r, ∀ y ∈ closedBall (c, z) r, dist x y < t → dist (uncurry f x) (uncurry f y) < e / 4 ef : ∀ d ∈ ball c (min t r), ∀ w ∈ sphere z r, e / 2 ≤ ‖f d w - f d z‖ ss : ball c (min t r) ×ˢ closedBall z r ⊆ s ⊢ 0 < min t r	no goals
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/AnalyticManifold/OpenMapping.lean	NontrivialHolomorphicAt.inCharts	[183, 1]	[196, 77]	use n.holomorphicAt.2.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 : S → T z : S n : NontrivialHolomorphicAt f z ⊢ NontrivialHolomorphicAt (fun w => ↑(extChartAt 𝓘(ℂ, ℂ) (f z)) (f (↑(extChartAt 𝓘(ℂ, ℂ) z).symm w))) (↑(extChartAt 𝓘(ℂ, ℂ) z) 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 : S → T z : S n : NontrivialHolomorphicAt f z ⊢ ∃ᶠ (w : ℂ) in 𝓝 (↑(extChartAt 𝓘(ℂ, ℂ) z) z), ↑(extChartAt 𝓘(ℂ, ℂ) (f z)) (f (↑(extChartAt 𝓘(ℂ, ℂ) z).symm w)) ≠ ↑(extChartAt 𝓘(ℂ, ℂ) (f z)) (f (↑(extChartAt 𝓘(ℂ, ℂ) z).symm (↑(extChartAt 𝓘(ℂ, ℂ) z) z)))
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/AnalyticManifold/OpenMapping.lean	NontrivialHolomorphicAt.inCharts	[183, 1]	[196, 77]	have c := n.nonconst	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 : S → T z : S n : NontrivialHolomorphicAt f z ⊢ ∃ᶠ (w : ℂ) in 𝓝 (↑(extChartAt 𝓘(ℂ, ℂ) z) z), ↑(extChartAt 𝓘(ℂ, ℂ) (f z)) (f (↑(extChartAt 𝓘(ℂ, ℂ) z).symm w)) ≠ ↑(extChartAt 𝓘(ℂ, ℂ) (f z)) (f (↑(extChartAt 𝓘(ℂ, ℂ) z).symm (↑(extChartAt 𝓘(ℂ, ℂ) z) 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 : S → T z : S n : NontrivialHolomorphicAt f z c : ∃ᶠ (w : S) in 𝓝 z, f w ≠ f z ⊢ ∃ᶠ (w : ℂ) in 𝓝 (↑(extChartAt 𝓘(ℂ, ℂ) z) z), ↑(extChartAt 𝓘(ℂ, ℂ) (f z)) (f (↑(extChartAt 𝓘(ℂ, ℂ) z).symm w)) ≠ ↑(extChartAt 𝓘(ℂ, ℂ) (f z)) (f (↑(extChartAt 𝓘(ℂ, ℂ) z).symm (↑(extChartAt 𝓘(ℂ, ℂ) z) z)))
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/AnalyticManifold/OpenMapping.lean	NontrivialHolomorphicAt.inCharts	[183, 1]	[196, 77]	contrapose c	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 : S → T z : S n : NontrivialHolomorphicAt f z c : ∃ᶠ (w : S) in 𝓝 z, f w ≠ f z ⊢ ∃ᶠ (w : ℂ) in 𝓝 (↑(extChartAt 𝓘(ℂ, ℂ) z) z), ↑(extChartAt 𝓘(ℂ, ℂ) (f z)) (f (↑(extChartAt 𝓘(ℂ, ℂ) z).symm w)) ≠ ↑(extChartAt 𝓘(ℂ, ℂ) (f z)) (f (↑(extChartAt 𝓘(ℂ, ℂ) z).symm (↑(extChartAt 𝓘(ℂ, ℂ) z) 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 : S → T z : S n : NontrivialHolomorphicAt f z c : ¬∃ᶠ (w : ℂ) in 𝓝 (↑(extChartAt 𝓘(ℂ, ℂ) z) z), ↑(extChartAt 𝓘(ℂ, ℂ) (f z)) (f (↑(extChartAt 𝓘(ℂ, ℂ) z).symm w)) ≠ ↑(extChartAt 𝓘(ℂ, ℂ) (f z)) (f (↑(extChartAt 𝓘(ℂ, ℂ) z).symm (↑(extChartAt 𝓘(ℂ, ℂ) z) z))) ⊢ ¬∃ᶠ (w : S) in 𝓝 z, f w ≠ f z
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/AnalyticManifold/OpenMapping.lean	NontrivialHolomorphicAt.inCharts	[183, 1]	[196, 77]	simp only [Filter.not_frequently, not_not, ← extChartAt_map_nhds' I z, Filter.eventually_map] at c ⊢	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 : S → T z : S n : NontrivialHolomorphicAt f z c : ¬∃ᶠ (w : ℂ) in 𝓝 (↑(extChartAt 𝓘(ℂ, ℂ) z) z), ↑(extChartAt 𝓘(ℂ, ℂ) (f z)) (f (↑(extChartAt 𝓘(ℂ, ℂ) z).symm w)) ≠ ↑(extChartAt 𝓘(ℂ, ℂ) (f z)) (f (↑(extChartAt 𝓘(ℂ, ℂ) z).symm (↑(extChartAt 𝓘(ℂ, ℂ) z) z))) ⊢ ¬∃ᶠ (w : S) 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 : S → T z : S n : NontrivialHolomorphicAt f z c : ∀ᶠ (a : S) in 𝓝 z, ↑(extChartAt 𝓘(ℂ, ℂ) (f z)) (f (↑(extChartAt 𝓘(ℂ, ℂ) z).symm (↑(extChartAt 𝓘(ℂ, ℂ) z) a))) = ↑(extChartAt 𝓘(ℂ, ℂ) (f z)) (f (↑(extChartAt 𝓘(ℂ, ℂ) z).symm (↑(extChartAt 𝓘(ℂ, ℂ) z) z))) ⊢ ∀ᶠ (x : S) in 𝓝 z, f x = f z
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/AnalyticManifold/OpenMapping.lean	NontrivialHolomorphicAt.inCharts	[183, 1]	[196, 77]	apply c.mp	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 : S → T z : S n : NontrivialHolomorphicAt f z c : ∀ᶠ (a : S) in 𝓝 z, ↑(extChartAt 𝓘(ℂ, ℂ) (f z)) (f (↑(extChartAt 𝓘(ℂ, ℂ) z).symm (↑(extChartAt 𝓘(ℂ, ℂ) z) a))) = ↑(extChartAt 𝓘(ℂ, ℂ) (f z)) (f (↑(extChartAt 𝓘(ℂ, ℂ) z).symm (↑(extChartAt 𝓘(ℂ, ℂ) z) z))) ⊢ ∀ᶠ (x : S) in 𝓝 z, f x = 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 : S → T z : S n : NontrivialHolomorphicAt f z c : ∀ᶠ (a : S) in 𝓝 z, ↑(extChartAt 𝓘(ℂ, ℂ) (f z)) (f (↑(extChartAt 𝓘(ℂ, ℂ) z).symm (↑(extChartAt 𝓘(ℂ, ℂ) z) a))) = ↑(extChartAt 𝓘(ℂ, ℂ) (f z)) (f (↑(extChartAt 𝓘(ℂ, ℂ) z).symm (↑(extChartAt 𝓘(ℂ, ℂ) z) z))) ⊢ ∀ᶠ (x : S) in 𝓝 z, ↑(extChartAt 𝓘(ℂ, ℂ) (f z)) (f (↑(extChartAt 𝓘(ℂ, ℂ) z).symm (↑(extChartAt 𝓘(ℂ, ℂ) z) x))) = ↑(extChartAt 𝓘(ℂ, ℂ) (f z)) (f (↑(extChartAt 𝓘(ℂ, ℂ) z).symm (↑(extChartAt 𝓘(ℂ, ℂ) z) z))) → f x = f z
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/AnalyticManifold/OpenMapping.lean	NontrivialHolomorphicAt.inCharts	[183, 1]	[196, 77]	apply ((isOpen_extChartAt_source I z).eventually_mem (mem_extChartAt_source I z)).mp	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 : S → T z : S n : NontrivialHolomorphicAt f z c : ∀ᶠ (a : S) in 𝓝 z, ↑(extChartAt 𝓘(ℂ, ℂ) (f z)) (f (↑(extChartAt 𝓘(ℂ, ℂ) z).symm (↑(extChartAt 𝓘(ℂ, ℂ) z) a))) = ↑(extChartAt 𝓘(ℂ, ℂ) (f z)) (f (↑(extChartAt 𝓘(ℂ, ℂ) z).symm (↑(extChartAt 𝓘(ℂ, ℂ) z) z))) ⊢ ∀ᶠ (x : S) in 𝓝 z, ↑(extChartAt 𝓘(ℂ, ℂ) (f z)) (f (↑(extChartAt 𝓘(ℂ, ℂ) z).symm (↑(extChartAt 𝓘(ℂ, ℂ) z) x))) = ↑(extChartAt 𝓘(ℂ, ℂ) (f z)) (f (↑(extChartAt 𝓘(ℂ, ℂ) z).symm (↑(extChartAt 𝓘(ℂ, ℂ) z) z))) → f x = 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 : S → T z : S n : NontrivialHolomorphicAt f z c : ∀ᶠ (a : S) in 𝓝 z, ↑(extChartAt 𝓘(ℂ, ℂ) (f z)) (f (↑(extChartAt 𝓘(ℂ, ℂ) z).symm (↑(extChartAt 𝓘(ℂ, ℂ) z) a))) = ↑(extChartAt 𝓘(ℂ, ℂ) (f z)) (f (↑(extChartAt 𝓘(ℂ, ℂ) z).symm (↑(extChartAt 𝓘(ℂ, ℂ) z) z))) ⊢ ∀ᶠ (x : S) in 𝓝 z, x ∈ (extChartAt 𝓘(ℂ, ℂ) z).source → ↑(extChartAt 𝓘(ℂ, ℂ) (f z)) (f (↑(extChartAt 𝓘(ℂ, ℂ) z).symm (↑(extChartAt 𝓘(ℂ, ℂ) z) x))) = ↑(extChartAt 𝓘(ℂ, ℂ) (f z)) (f (↑(extChartAt 𝓘(ℂ, ℂ) z).symm (↑(extChartAt 𝓘(ℂ, ℂ) z) z))) → f x = f z
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/AnalyticManifold/OpenMapping.lean	NontrivialHolomorphicAt.inCharts	[183, 1]	[196, 77]	apply (n.holomorphicAt.continuousAt.eventually_mem (extChartAt_source_mem_nhds I (f z))).mp	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 : S → T z : S n : NontrivialHolomorphicAt f z c : ∀ᶠ (a : S) in 𝓝 z, ↑(extChartAt 𝓘(ℂ, ℂ) (f z)) (f (↑(extChartAt 𝓘(ℂ, ℂ) z).symm (↑(extChartAt 𝓘(ℂ, ℂ) z) a))) = ↑(extChartAt 𝓘(ℂ, ℂ) (f z)) (f (↑(extChartAt 𝓘(ℂ, ℂ) z).symm (↑(extChartAt 𝓘(ℂ, ℂ) z) z))) ⊢ ∀ᶠ (x : S) in 𝓝 z, x ∈ (extChartAt 𝓘(ℂ, ℂ) z).source → ↑(extChartAt 𝓘(ℂ, ℂ) (f z)) (f (↑(extChartAt 𝓘(ℂ, ℂ) z).symm (↑(extChartAt 𝓘(ℂ, ℂ) z) x))) = ↑(extChartAt 𝓘(ℂ, ℂ) (f z)) (f (↑(extChartAt 𝓘(ℂ, ℂ) z).symm (↑(extChartAt 𝓘(ℂ, ℂ) z) z))) → f x = 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 : S → T z : S n : NontrivialHolomorphicAt f z c : ∀ᶠ (a : S) in 𝓝 z, ↑(extChartAt 𝓘(ℂ, ℂ) (f z)) (f (↑(extChartAt 𝓘(ℂ, ℂ) z).symm (↑(extChartAt 𝓘(ℂ, ℂ) z) a))) = ↑(extChartAt 𝓘(ℂ, ℂ) (f z)) (f (↑(extChartAt 𝓘(ℂ, ℂ) z).symm (↑(extChartAt 𝓘(ℂ, ℂ) z) z))) ⊢ ∀ᶠ (x : S) in 𝓝 z, f x ∈ (extChartAt 𝓘(ℂ, ℂ) (f z)).source → x ∈ (extChartAt 𝓘(ℂ, ℂ) z).source → ↑(extChartAt 𝓘(ℂ, ℂ) (f z)) (f (↑(extChartAt 𝓘(ℂ, ℂ) z).symm (↑(extChartAt 𝓘(ℂ, ℂ) z) x))) = ↑(extChartAt 𝓘(ℂ, ℂ) (f z)) (f (↑(extChartAt 𝓘(ℂ, ℂ) z).symm (↑(extChartAt 𝓘(ℂ, ℂ) z) z))) → f x = f z
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/AnalyticManifold/OpenMapping.lean	NontrivialHolomorphicAt.inCharts	[183, 1]	[196, 77]	refine eventually_of_forall fun w fm m fn ↦ ?_	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 : S → T z : S n : NontrivialHolomorphicAt f z c : ∀ᶠ (a : S) in 𝓝 z, ↑(extChartAt 𝓘(ℂ, ℂ) (f z)) (f (↑(extChartAt 𝓘(ℂ, ℂ) z).symm (↑(extChartAt 𝓘(ℂ, ℂ) z) a))) = ↑(extChartAt 𝓘(ℂ, ℂ) (f z)) (f (↑(extChartAt 𝓘(ℂ, ℂ) z).symm (↑(extChartAt 𝓘(ℂ, ℂ) z) z))) ⊢ ∀ᶠ (x : S) in 𝓝 z, f x ∈ (extChartAt 𝓘(ℂ, ℂ) (f z)).source → x ∈ (extChartAt 𝓘(ℂ, ℂ) z).source → ↑(extChartAt 𝓘(ℂ, ℂ) (f z)) (f (↑(extChartAt 𝓘(ℂ, ℂ) z).symm (↑(extChartAt 𝓘(ℂ, ℂ) z) x))) = ↑(extChartAt 𝓘(ℂ, ℂ) (f z)) (f (↑(extChartAt 𝓘(ℂ, ℂ) z).symm (↑(extChartAt 𝓘(ℂ, ℂ) z) z))) → f x = 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 : S → T z : S n : NontrivialHolomorphicAt f z c : ∀ᶠ (a : S) in 𝓝 z, ↑(extChartAt 𝓘(ℂ, ℂ) (f z)) (f (↑(extChartAt 𝓘(ℂ, ℂ) z).symm (↑(extChartAt 𝓘(ℂ, ℂ) z) a))) = ↑(extChartAt 𝓘(ℂ, ℂ) (f z)) (f (↑(extChartAt 𝓘(ℂ, ℂ) z).symm (↑(extChartAt 𝓘(ℂ, ℂ) z) z))) w : S fm : f w ∈ (extChartAt 𝓘(ℂ, ℂ) (f z)).source m : w ∈ (extChartAt 𝓘(ℂ, ℂ) z).source fn : ↑(extChartAt 𝓘(ℂ, ℂ) (f z)) (f (↑(extChartAt 𝓘(ℂ, ℂ) z).symm (↑(extChartAt 𝓘(ℂ, ℂ) z) w))) = ↑(extChartAt 𝓘(ℂ, ℂ) (f z)) (f (↑(extChartAt 𝓘(ℂ, ℂ) z).symm (↑(extChartAt 𝓘(ℂ, ℂ) z) z))) ⊢ f w = f z
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/AnalyticManifold/OpenMapping.lean	NontrivialHolomorphicAt.inCharts	[183, 1]	[196, 77]	rw [PartialEquiv.left_inv _ m, PartialEquiv.left_inv _ (mem_extChartAt_source I z)] at fn	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 : S → T z : S n : NontrivialHolomorphicAt f z c : ∀ᶠ (a : S) in 𝓝 z, ↑(extChartAt 𝓘(ℂ, ℂ) (f z)) (f (↑(extChartAt 𝓘(ℂ, ℂ) z).symm (↑(extChartAt 𝓘(ℂ, ℂ) z) a))) = ↑(extChartAt 𝓘(ℂ, ℂ) (f z)) (f (↑(extChartAt 𝓘(ℂ, ℂ) z).symm (↑(extChartAt 𝓘(ℂ, ℂ) z) z))) w : S fm : f w ∈ (extChartAt 𝓘(ℂ, ℂ) (f z)).source m : w ∈ (extChartAt 𝓘(ℂ, ℂ) z).source fn : ↑(extChartAt 𝓘(ℂ, ℂ) (f z)) (f (↑(extChartAt 𝓘(ℂ, ℂ) z).symm (↑(extChartAt 𝓘(ℂ, ℂ) z) w))) = ↑(extChartAt 𝓘(ℂ, ℂ) (f z)) (f (↑(extChartAt 𝓘(ℂ, ℂ) z).symm (↑(extChartAt 𝓘(ℂ, ℂ) z) 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 : S → T z : S n : NontrivialHolomorphicAt f z c : ∀ᶠ (a : S) in 𝓝 z, ↑(extChartAt 𝓘(ℂ, ℂ) (f z)) (f (↑(extChartAt 𝓘(ℂ, ℂ) z).symm (↑(extChartAt 𝓘(ℂ, ℂ) z) a))) = ↑(extChartAt 𝓘(ℂ, ℂ) (f z)) (f (↑(extChartAt 𝓘(ℂ, ℂ) z).symm (↑(extChartAt 𝓘(ℂ, ℂ) z) z))) w : S fm : f w ∈ (extChartAt 𝓘(ℂ, ℂ) (f z)).source m : w ∈ (extChartAt 𝓘(ℂ, ℂ) z).source fn : ↑(extChartAt 𝓘(ℂ, ℂ) (f z)) (f w) = ↑(extChartAt 𝓘(ℂ, ℂ) (f z)) (f z) ⊢ f w = f z
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/AnalyticManifold/OpenMapping.lean	NontrivialHolomorphicAt.inCharts	[183, 1]	[196, 77]	exact ((PartialEquiv.injOn _).eq_iff fm (mem_extChartAt_source _ _)).mp fn	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 : S → T z : S n : NontrivialHolomorphicAt f z c : ∀ᶠ (a : S) in 𝓝 z, ↑(extChartAt 𝓘(ℂ, ℂ) (f z)) (f (↑(extChartAt 𝓘(ℂ, ℂ) z).symm (↑(extChartAt 𝓘(ℂ, ℂ) z) a))) = ↑(extChartAt 𝓘(ℂ, ℂ) (f z)) (f (↑(extChartAt 𝓘(ℂ, ℂ) z).symm (↑(extChartAt 𝓘(ℂ, ℂ) z) z))) w : S fm : f w ∈ (extChartAt 𝓘(ℂ, ℂ) (f z)).source m : w ∈ (extChartAt 𝓘(ℂ, ℂ) z).source fn : ↑(extChartAt 𝓘(ℂ, ℂ) (f z)) (f w) = ↑(extChartAt 𝓘(ℂ, ℂ) (f z)) (f z) ⊢ f w = f z	no goals
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/AnalyticManifold/OpenMapping.lean	NontrivialHolomorphicAt.nhds_eq_map_nhds	[201, 1]	[225, 42]	refine le_antisymm ?_ n.holomorphicAt.continuousAt	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 : S → T z : S n : NontrivialHolomorphicAt f z ⊢ 𝓝 (f z) = Filter.map f (𝓝 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 : S → T z : S n : NontrivialHolomorphicAt f z ⊢ 𝓝 (f z) ≤ Filter.map f (𝓝 z)
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/AnalyticManifold/OpenMapping.lean	NontrivialHolomorphicAt.nhds_eq_map_nhds	[201, 1]	[225, 42]	generalize hg : (fun x ↦ extChartAt I (f z) (f ((extChartAt I z).symm x))) = g	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 : S → T z : S n : NontrivialHolomorphicAt f z ⊢ 𝓝 (f z) ≤ Filter.map f (𝓝 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 : S → T z : S n : NontrivialHolomorphicAt f z g : ℂ → ℂ hg : (fun x => ↑(extChartAt 𝓘(ℂ, ℂ) (f z)) (f (↑(extChartAt 𝓘(ℂ, ℂ) z).symm x))) = g ⊢ 𝓝 (f z) ≤ Filter.map f (𝓝 z)
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/AnalyticManifold/OpenMapping.lean	NontrivialHolomorphicAt.nhds_eq_map_nhds	[201, 1]	[225, 42]	have ga : AnalyticAt ℂ g (extChartAt I z z) := by rw [← hg]; exact n.holomorphicAt.2	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 : S → T z : S n : NontrivialHolomorphicAt f z g : ℂ → ℂ hg : (fun x => ↑(extChartAt 𝓘(ℂ, ℂ) (f z)) (f (↑(extChartAt 𝓘(ℂ, ℂ) z).symm x))) = g ⊢ 𝓝 (f z) ≤ Filter.map f (𝓝 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 : S → T z : S n : NontrivialHolomorphicAt f z g : ℂ → ℂ hg : (fun x => ↑(extChartAt 𝓘(ℂ, ℂ) (f z)) (f (↑(extChartAt 𝓘(ℂ, ℂ) z).symm x))) = g ga : AnalyticAt ℂ g (↑(extChartAt 𝓘(ℂ, ℂ) z) z) ⊢ 𝓝 (f z) ≤ Filter.map f (𝓝 z)
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/AnalyticManifold/OpenMapping.lean	NontrivialHolomorphicAt.nhds_eq_map_nhds	[201, 1]	[225, 42]	cases' ga.eventually_constant_or_nhds_le_map_nhds with h 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 : S → T z : S n : NontrivialHolomorphicAt f z g : ℂ → ℂ hg : (fun x => ↑(extChartAt 𝓘(ℂ, ℂ) (f z)) (f (↑(extChartAt 𝓘(ℂ, ℂ) z).symm x))) = g ga : AnalyticAt ℂ g (↑(extChartAt 𝓘(ℂ, ℂ) z) z) ⊢ 𝓝 (f z) ≤ Filter.map f (𝓝 z)	case inl 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 : S → T z : S n : NontrivialHolomorphicAt f z g : ℂ → ℂ hg : (fun x => ↑(extChartAt 𝓘(ℂ, ℂ) (f z)) (f (↑(extChartAt 𝓘(ℂ, ℂ) z).symm x))) = g ga : AnalyticAt ℂ g (↑(extChartAt 𝓘(ℂ, ℂ) z) z) h : ∀ᶠ (z_1 : ℂ) in 𝓝 (↑(extChartAt 𝓘(ℂ, ℂ) z) z), g z_1 = g (↑(extChartAt 𝓘(ℂ, ℂ) z) z) ⊢ 𝓝 (f z) ≤ Filter.map f (𝓝 z) case inr 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 : S → T z : S n : NontrivialHolomorphicAt f z g : ℂ → ℂ hg : (fun x => ↑(extChartAt 𝓘(ℂ, ℂ) (f z)) (f (↑(extChartAt 𝓘(ℂ, ℂ) z).symm x))) = g ga : AnalyticAt ℂ g (↑(extChartAt 𝓘(ℂ, ℂ) z) z) h : 𝓝 (g (↑(extChartAt 𝓘(ℂ, ℂ) z) z)) ≤ Filter.map g (𝓝 (↑(extChartAt 𝓘(ℂ, ℂ) z) z)) ⊢ 𝓝 (f z) ≤ Filter.map f (𝓝 z)

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