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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/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	RMT4/Montel.lean	UniformlyBoundedOn.equicontinuousOn	[32, 1]	[58, 40]	refine ⟨δ ⊓ ε / M, gt_iff_lt.2 (lt_inf_iff.2 ⟨hδ, div_pos hε hMp⟩), λ w hw i => ?_⟩	case mk.intro.intro.intro.intro ι : Type u_1 U K : Set ℂ z✝ : ℂ F : ι → 𝓒 U Q : Set ℂ → Set ℂ h1 : UniformlyBoundedOn F U hU : IsOpen U h2 : ∀ (i : ι), DifferentiableOn ℂ (F i) U hK : K ∈ compacts U z : ℂ hz : z ∈ K δ : ℝ hδ : 0 < δ h : closedBall z δ ⊆ U M : ℝ hMp : M > 0 hM : ∀ (i : ι), MapsTo (_root_.deriv (F i)) (closedBall z δ) (closedBall 0 M) ε : ℝ hε : ε > 0 ⊢ ∃ δ > 0, ∀ (x : ↑K), dist x { val := z, property := hz } < δ → ∀ (i : ι), dist ((restrict K ∘ F) i { val := z, property := hz }) ((restrict K ∘ F) i x) < ε	case mk.intro.intro.intro.intro ι : Type u_1 U K : Set ℂ z✝ : ℂ F : ι → 𝓒 U Q : Set ℂ → Set ℂ h1 : UniformlyBoundedOn F U hU : IsOpen U h2 : ∀ (i : ι), DifferentiableOn ℂ (F i) U hK : K ∈ compacts U z : ℂ hz : z ∈ K δ : ℝ hδ : 0 < δ h : closedBall z δ ⊆ U M : ℝ hMp : M > 0 hM : ∀ (i : ι), MapsTo (_root_.deriv (F i)) (closedBall z δ) (closedBall 0 M) ε : ℝ hε : ε > 0 w : ↑K hw : dist w { val := z, property := hz } < δ ⊓ ε / M i : ι ⊢ dist ((restrict K ∘ F) i { val := z, property := hz }) ((restrict K ∘ F) i w) < ε
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	RMT4/Montel.lean	UniformlyBoundedOn.equicontinuousOn	[32, 1]	[58, 40]	simp	case mk.intro.intro.intro.intro ι : Type u_1 U K : Set ℂ z✝ : ℂ F : ι → 𝓒 U Q : Set ℂ → Set ℂ h1 : UniformlyBoundedOn F U hU : IsOpen U h2 : ∀ (i : ι), DifferentiableOn ℂ (F i) U hK : K ∈ compacts U z : ℂ hz : z ∈ K δ : ℝ hδ : 0 < δ h : closedBall z δ ⊆ U M : ℝ hMp : M > 0 hM : ∀ (i : ι), MapsTo (_root_.deriv (F i)) (closedBall z δ) (closedBall 0 M) ε : ℝ hε : ε > 0 w : ↑K hw : dist w { val := z, property := hz } < δ ⊓ ε / M i : ι ⊢ dist ((restrict K ∘ F) i { val := z, property := hz }) ((restrict K ∘ F) i w) < ε	case mk.intro.intro.intro.intro ι : Type u_1 U K : Set ℂ z✝ : ℂ F : ι → 𝓒 U Q : Set ℂ → Set ℂ h1 : UniformlyBoundedOn F U hU : IsOpen U h2 : ∀ (i : ι), DifferentiableOn ℂ (F i) U hK : K ∈ compacts U z : ℂ hz : z ∈ K δ : ℝ hδ : 0 < δ h : closedBall z δ ⊆ U M : ℝ hMp : M > 0 hM : ∀ (i : ι), MapsTo (_root_.deriv (F i)) (closedBall z δ) (closedBall 0 M) ε : ℝ hε : ε > 0 w : ↑K hw : dist w { val := z, property := hz } < δ ⊓ ε / M i : ι ⊢ dist (F i z) (F i ↑w) < ε
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	RMT4/Montel.lean	UniformlyBoundedOn.equicontinuousOn	[32, 1]	[58, 40]	have e1 : ∀ x ∈ closedBall z δ, DifferentiableAt ℂ (F i) x := λ x hx => (h2 i).differentiableAt (hU.mem_nhds (h hx))	case mk.intro.intro.intro.intro ι : Type u_1 U K : Set ℂ z✝ : ℂ F : ι → 𝓒 U Q : Set ℂ → Set ℂ h1 : UniformlyBoundedOn F U hU : IsOpen U h2 : ∀ (i : ι), DifferentiableOn ℂ (F i) U hK : K ∈ compacts U z : ℂ hz : z ∈ K δ : ℝ hδ : 0 < δ h : closedBall z δ ⊆ U M : ℝ hMp : M > 0 hM : ∀ (i : ι), MapsTo (_root_.deriv (F i)) (closedBall z δ) (closedBall 0 M) ε : ℝ hε : ε > 0 w : ↑K hw : dist w { val := z, property := hz } < δ ⊓ ε / M i : ι ⊢ dist (F i z) (F i ↑w) < ε	case mk.intro.intro.intro.intro ι : Type u_1 U K : Set ℂ z✝ : ℂ F : ι → 𝓒 U Q : Set ℂ → Set ℂ h1 : UniformlyBoundedOn F U hU : IsOpen U h2 : ∀ (i : ι), DifferentiableOn ℂ (F i) U hK : K ∈ compacts U z : ℂ hz : z ∈ K δ : ℝ hδ : 0 < δ h : closedBall z δ ⊆ U M : ℝ hMp : M > 0 hM : ∀ (i : ι), MapsTo (_root_.deriv (F i)) (closedBall z δ) (closedBall 0 M) ε : ℝ hε : ε > 0 w : ↑K hw : dist w { val := z, property := hz } < δ ⊓ ε / M i : ι e1 : ∀ x ∈ closedBall z δ, DifferentiableAt ℂ (F i) x ⊢ dist (F i z) (F i ↑w) < ε
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	RMT4/Montel.lean	UniformlyBoundedOn.equicontinuousOn	[32, 1]	[58, 40]	have e2 : ∀ x ∈ closedBall z δ, ‖_root_.deriv (F i) x‖ ≤ M := by simpa [MapsTo] using hM i	case mk.intro.intro.intro.intro ι : Type u_1 U K : Set ℂ z✝ : ℂ F : ι → 𝓒 U Q : Set ℂ → Set ℂ h1 : UniformlyBoundedOn F U hU : IsOpen U h2 : ∀ (i : ι), DifferentiableOn ℂ (F i) U hK : K ∈ compacts U z : ℂ hz : z ∈ K δ : ℝ hδ : 0 < δ h : closedBall z δ ⊆ U M : ℝ hMp : M > 0 hM : ∀ (i : ι), MapsTo (_root_.deriv (F i)) (closedBall z δ) (closedBall 0 M) ε : ℝ hε : ε > 0 w : ↑K hw : dist w { val := z, property := hz } < δ ⊓ ε / M i : ι e1 : ∀ x ∈ closedBall z δ, DifferentiableAt ℂ (F i) x ⊢ dist (F i z) (F i ↑w) < ε	case mk.intro.intro.intro.intro ι : Type u_1 U K : Set ℂ z✝ : ℂ F : ι → 𝓒 U Q : Set ℂ → Set ℂ h1 : UniformlyBoundedOn F U hU : IsOpen U h2 : ∀ (i : ι), DifferentiableOn ℂ (F i) U hK : K ∈ compacts U z : ℂ hz : z ∈ K δ : ℝ hδ : 0 < δ h : closedBall z δ ⊆ U M : ℝ hMp : M > 0 hM : ∀ (i : ι), MapsTo (_root_.deriv (F i)) (closedBall z δ) (closedBall 0 M) ε : ℝ hε : ε > 0 w : ↑K hw : dist w { val := z, property := hz } < δ ⊓ ε / M i : ι e1 : ∀ x ∈ closedBall z δ, DifferentiableAt ℂ (F i) x e2 : ∀ x ∈ closedBall z δ, ‖_root_.deriv (F i) x‖ ≤ M ⊢ dist (F i z) (F i ↑w) < ε
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	RMT4/Montel.lean	UniformlyBoundedOn.equicontinuousOn	[32, 1]	[58, 40]	have e3 : z ∈ closedBall z δ := mem_closedBall_self hδ.le	case mk.intro.intro.intro.intro ι : Type u_1 U K : Set ℂ z✝ : ℂ F : ι → 𝓒 U Q : Set ℂ → Set ℂ h1 : UniformlyBoundedOn F U hU : IsOpen U h2 : ∀ (i : ι), DifferentiableOn ℂ (F i) U hK : K ∈ compacts U z : ℂ hz : z ∈ K δ : ℝ hδ : 0 < δ h : closedBall z δ ⊆ U M : ℝ hMp : M > 0 hM : ∀ (i : ι), MapsTo (_root_.deriv (F i)) (closedBall z δ) (closedBall 0 M) ε : ℝ hε : ε > 0 w : ↑K hw : dist w { val := z, property := hz } < δ ⊓ ε / M i : ι e1 : ∀ x ∈ closedBall z δ, DifferentiableAt ℂ (F i) x e2 : ∀ x ∈ closedBall z δ, ‖_root_.deriv (F i) x‖ ≤ M ⊢ dist (F i z) (F i ↑w) < ε	case mk.intro.intro.intro.intro ι : Type u_1 U K : Set ℂ z✝ : ℂ F : ι → 𝓒 U Q : Set ℂ → Set ℂ h1 : UniformlyBoundedOn F U hU : IsOpen U h2 : ∀ (i : ι), DifferentiableOn ℂ (F i) U hK : K ∈ compacts U z : ℂ hz : z ∈ K δ : ℝ hδ : 0 < δ h : closedBall z δ ⊆ U M : ℝ hMp : M > 0 hM : ∀ (i : ι), MapsTo (_root_.deriv (F i)) (closedBall z δ) (closedBall 0 M) ε : ℝ hε : ε > 0 w : ↑K hw : dist w { val := z, property := hz } < δ ⊓ ε / M i : ι e1 : ∀ x ∈ closedBall z δ, DifferentiableAt ℂ (F i) x e2 : ∀ x ∈ closedBall z δ, ‖_root_.deriv (F i) x‖ ≤ M e3 : z ∈ closedBall z δ ⊢ dist (F i z) (F i ↑w) < ε
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	RMT4/Montel.lean	UniformlyBoundedOn.equicontinuousOn	[32, 1]	[58, 40]	have e4 : w.1 ∈ closedBall z δ := by simpa using (lt_inf_iff.1 hw).1.le	case mk.intro.intro.intro.intro ι : Type u_1 U K : Set ℂ z✝ : ℂ F : ι → 𝓒 U Q : Set ℂ → Set ℂ h1 : UniformlyBoundedOn F U hU : IsOpen U h2 : ∀ (i : ι), DifferentiableOn ℂ (F i) U hK : K ∈ compacts U z : ℂ hz : z ∈ K δ : ℝ hδ : 0 < δ h : closedBall z δ ⊆ U M : ℝ hMp : M > 0 hM : ∀ (i : ι), MapsTo (_root_.deriv (F i)) (closedBall z δ) (closedBall 0 M) ε : ℝ hε : ε > 0 w : ↑K hw : dist w { val := z, property := hz } < δ ⊓ ε / M i : ι e1 : ∀ x ∈ closedBall z δ, DifferentiableAt ℂ (F i) x e2 : ∀ x ∈ closedBall z δ, ‖_root_.deriv (F i) x‖ ≤ M e3 : z ∈ closedBall z δ ⊢ dist (F i z) (F i ↑w) < ε	case mk.intro.intro.intro.intro ι : Type u_1 U K : Set ℂ z✝ : ℂ F : ι → 𝓒 U Q : Set ℂ → Set ℂ h1 : UniformlyBoundedOn F U hU : IsOpen U h2 : ∀ (i : ι), DifferentiableOn ℂ (F i) U hK : K ∈ compacts U z : ℂ hz : z ∈ K δ : ℝ hδ : 0 < δ h : closedBall z δ ⊆ U M : ℝ hMp : M > 0 hM : ∀ (i : ι), MapsTo (_root_.deriv (F i)) (closedBall z δ) (closedBall 0 M) ε : ℝ hε : ε > 0 w : ↑K hw : dist w { val := z, property := hz } < δ ⊓ ε / M i : ι e1 : ∀ x ∈ closedBall z δ, DifferentiableAt ℂ (F i) x e2 : ∀ x ∈ closedBall z δ, ‖_root_.deriv (F i) x‖ ≤ M e3 : z ∈ closedBall z δ e4 : ↑w ∈ closedBall z δ ⊢ dist (F i z) (F i ↑w) < ε
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	RMT4/Montel.lean	UniformlyBoundedOn.equicontinuousOn	[32, 1]	[58, 40]	rw [dist_eq_norm]	case mk.intro.intro.intro.intro ι : Type u_1 U K : Set ℂ z✝ : ℂ F : ι → 𝓒 U Q : Set ℂ → Set ℂ h1 : UniformlyBoundedOn F U hU : IsOpen U h2 : ∀ (i : ι), DifferentiableOn ℂ (F i) U hK : K ∈ compacts U z : ℂ hz : z ∈ K δ : ℝ hδ : 0 < δ h : closedBall z δ ⊆ U M : ℝ hMp : M > 0 hM : ∀ (i : ι), MapsTo (_root_.deriv (F i)) (closedBall z δ) (closedBall 0 M) ε : ℝ hε : ε > 0 w : ↑K hw : dist w { val := z, property := hz } < δ ⊓ ε / M i : ι e1 : ∀ x ∈ closedBall z δ, DifferentiableAt ℂ (F i) x e2 : ∀ x ∈ closedBall z δ, ‖_root_.deriv (F i) x‖ ≤ M e3 : z ∈ closedBall z δ e4 : ↑w ∈ closedBall z δ ⊢ dist (F i z) (F i ↑w) < ε	case mk.intro.intro.intro.intro ι : Type u_1 U K : Set ℂ z✝ : ℂ F : ι → 𝓒 U Q : Set ℂ → Set ℂ h1 : UniformlyBoundedOn F U hU : IsOpen U h2 : ∀ (i : ι), DifferentiableOn ℂ (F i) U hK : K ∈ compacts U z : ℂ hz : z ∈ K δ : ℝ hδ : 0 < δ h : closedBall z δ ⊆ U M : ℝ hMp : M > 0 hM : ∀ (i : ι), MapsTo (_root_.deriv (F i)) (closedBall z δ) (closedBall 0 M) ε : ℝ hε : ε > 0 w : ↑K hw : dist w { val := z, property := hz } < δ ⊓ ε / M i : ι e1 : ∀ x ∈ closedBall z δ, DifferentiableAt ℂ (F i) x e2 : ∀ x ∈ closedBall z δ, ‖_root_.deriv (F i) x‖ ≤ M e3 : z ∈ closedBall z δ e4 : ↑w ∈ closedBall z δ ⊢ ‖F i z - F i ↑w‖ < ε
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	RMT4/Montel.lean	UniformlyBoundedOn.equicontinuousOn	[32, 1]	[58, 40]	refine ((convex_closedBall _ _).norm_image_sub_le_of_norm_deriv_le e1 e2 e4 e3).trans_lt ?_	case mk.intro.intro.intro.intro ι : Type u_1 U K : Set ℂ z✝ : ℂ F : ι → 𝓒 U Q : Set ℂ → Set ℂ h1 : UniformlyBoundedOn F U hU : IsOpen U h2 : ∀ (i : ι), DifferentiableOn ℂ (F i) U hK : K ∈ compacts U z : ℂ hz : z ∈ K δ : ℝ hδ : 0 < δ h : closedBall z δ ⊆ U M : ℝ hMp : M > 0 hM : ∀ (i : ι), MapsTo (_root_.deriv (F i)) (closedBall z δ) (closedBall 0 M) ε : ℝ hε : ε > 0 w : ↑K hw : dist w { val := z, property := hz } < δ ⊓ ε / M i : ι e1 : ∀ x ∈ closedBall z δ, DifferentiableAt ℂ (F i) x e2 : ∀ x ∈ closedBall z δ, ‖_root_.deriv (F i) x‖ ≤ M e3 : z ∈ closedBall z δ e4 : ↑w ∈ closedBall z δ ⊢ ‖F i z - F i ↑w‖ < ε	case mk.intro.intro.intro.intro ι : Type u_1 U K : Set ℂ z✝ : ℂ F : ι → 𝓒 U Q : Set ℂ → Set ℂ h1 : UniformlyBoundedOn F U hU : IsOpen U h2 : ∀ (i : ι), DifferentiableOn ℂ (F i) U hK : K ∈ compacts U z : ℂ hz : z ∈ K δ : ℝ hδ : 0 < δ h : closedBall z δ ⊆ U M : ℝ hMp : M > 0 hM : ∀ (i : ι), MapsTo (_root_.deriv (F i)) (closedBall z δ) (closedBall 0 M) ε : ℝ hε : ε > 0 w : ↑K hw : dist w { val := z, property := hz } < δ ⊓ ε / M i : ι e1 : ∀ x ∈ closedBall z δ, DifferentiableAt ℂ (F i) x e2 : ∀ x ∈ closedBall z δ, ‖_root_.deriv (F i) x‖ ≤ M e3 : z ∈ closedBall z δ e4 : ↑w ∈ closedBall z δ ⊢ M * ‖z - ↑w‖ < ε
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	RMT4/Montel.lean	UniformlyBoundedOn.equicontinuousOn	[32, 1]	[58, 40]	have : ‖z - w.val‖ < ε / M := by have := (lt_inf_iff.1 hw).2 rwa [dist_comm, Subtype.dist_eq, dist_eq_norm] at this	case mk.intro.intro.intro.intro ι : Type u_1 U K : Set ℂ z✝ : ℂ F : ι → 𝓒 U Q : Set ℂ → Set ℂ h1 : UniformlyBoundedOn F U hU : IsOpen U h2 : ∀ (i : ι), DifferentiableOn ℂ (F i) U hK : K ∈ compacts U z : ℂ hz : z ∈ K δ : ℝ hδ : 0 < δ h : closedBall z δ ⊆ U M : ℝ hMp : M > 0 hM : ∀ (i : ι), MapsTo (_root_.deriv (F i)) (closedBall z δ) (closedBall 0 M) ε : ℝ hε : ε > 0 w : ↑K hw : dist w { val := z, property := hz } < δ ⊓ ε / M i : ι e1 : ∀ x ∈ closedBall z δ, DifferentiableAt ℂ (F i) x e2 : ∀ x ∈ closedBall z δ, ‖_root_.deriv (F i) x‖ ≤ M e3 : z ∈ closedBall z δ e4 : ↑w ∈ closedBall z δ ⊢ M * ‖z - ↑w‖ < ε	case mk.intro.intro.intro.intro ι : Type u_1 U K : Set ℂ z✝ : ℂ F : ι → 𝓒 U Q : Set ℂ → Set ℂ h1 : UniformlyBoundedOn F U hU : IsOpen U h2 : ∀ (i : ι), DifferentiableOn ℂ (F i) U hK : K ∈ compacts U z : ℂ hz : z ∈ K δ : ℝ hδ : 0 < δ h : closedBall z δ ⊆ U M : ℝ hMp : M > 0 hM : ∀ (i : ι), MapsTo (_root_.deriv (F i)) (closedBall z δ) (closedBall 0 M) ε : ℝ hε : ε > 0 w : ↑K hw : dist w { val := z, property := hz } < δ ⊓ ε / M i : ι e1 : ∀ x ∈ closedBall z δ, DifferentiableAt ℂ (F i) x e2 : ∀ x ∈ closedBall z δ, ‖_root_.deriv (F i) x‖ ≤ M e3 : z ∈ closedBall z δ e4 : ↑w ∈ closedBall z δ this : ‖z - ↑w‖ < ε / M ⊢ M * ‖z - ↑w‖ < ε
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	RMT4/Montel.lean	UniformlyBoundedOn.equicontinuousOn	[32, 1]	[58, 40]	convert mul_lt_mul' le_rfl this (norm_nonneg _) hMp	case mk.intro.intro.intro.intro ι : Type u_1 U K : Set ℂ z✝ : ℂ F : ι → 𝓒 U Q : Set ℂ → Set ℂ h1 : UniformlyBoundedOn F U hU : IsOpen U h2 : ∀ (i : ι), DifferentiableOn ℂ (F i) U hK : K ∈ compacts U z : ℂ hz : z ∈ K δ : ℝ hδ : 0 < δ h : closedBall z δ ⊆ U M : ℝ hMp : M > 0 hM : ∀ (i : ι), MapsTo (_root_.deriv (F i)) (closedBall z δ) (closedBall 0 M) ε : ℝ hε : ε > 0 w : ↑K hw : dist w { val := z, property := hz } < δ ⊓ ε / M i : ι e1 : ∀ x ∈ closedBall z δ, DifferentiableAt ℂ (F i) x e2 : ∀ x ∈ closedBall z δ, ‖_root_.deriv (F i) x‖ ≤ M e3 : z ∈ closedBall z δ e4 : ↑w ∈ closedBall z δ this : ‖z - ↑w‖ < ε / M ⊢ M * ‖z - ↑w‖ < ε	case h.e'_4 ι : Type u_1 U K : Set ℂ z✝ : ℂ F : ι → 𝓒 U Q : Set ℂ → Set ℂ h1 : UniformlyBoundedOn F U hU : IsOpen U h2 : ∀ (i : ι), DifferentiableOn ℂ (F i) U hK : K ∈ compacts U z : ℂ hz : z ∈ K δ : ℝ hδ : 0 < δ h : closedBall z δ ⊆ U M : ℝ hMp : M > 0 hM : ∀ (i : ι), MapsTo (_root_.deriv (F i)) (closedBall z δ) (closedBall 0 M) ε : ℝ hε : ε > 0 w : ↑K hw : dist w { val := z, property := hz } < δ ⊓ ε / M i : ι e1 : ∀ x ∈ closedBall z δ, DifferentiableAt ℂ (F i) x e2 : ∀ x ∈ closedBall z δ, ‖_root_.deriv (F i) x‖ ≤ M e3 : z ∈ closedBall z δ e4 : ↑w ∈ closedBall z δ this : ‖z - ↑w‖ < ε / M ⊢ ε = M * (ε / M)
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	RMT4/Montel.lean	UniformlyBoundedOn.equicontinuousOn	[32, 1]	[58, 40]	field_simp [hMp.lt.ne.symm, mul_comm]	case h.e'_4 ι : Type u_1 U K : Set ℂ z✝ : ℂ F : ι → 𝓒 U Q : Set ℂ → Set ℂ h1 : UniformlyBoundedOn F U hU : IsOpen U h2 : ∀ (i : ι), DifferentiableOn ℂ (F i) U hK : K ∈ compacts U z : ℂ hz : z ∈ K δ : ℝ hδ : 0 < δ h : closedBall z δ ⊆ U M : ℝ hMp : M > 0 hM : ∀ (i : ι), MapsTo (_root_.deriv (F i)) (closedBall z δ) (closedBall 0 M) ε : ℝ hε : ε > 0 w : ↑K hw : dist w { val := z, property := hz } < δ ⊓ ε / M i : ι e1 : ∀ x ∈ closedBall z δ, DifferentiableAt ℂ (F i) x e2 : ∀ x ∈ closedBall z δ, ‖_root_.deriv (F i) x‖ ≤ M e3 : z ∈ closedBall z δ e4 : ↑w ∈ closedBall z δ this : ‖z - ↑w‖ < ε / M ⊢ ε = M * (ε / M)	no goals
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	RMT4/Montel.lean	UniformlyBoundedOn.equicontinuousOn	[32, 1]	[58, 40]	obtain ⟨Q, hQ1, hQ2⟩ := h1.deriv hU h2 (closedBall z δ) ⟨h, isCompact_closedBall _ _⟩	ι : Type u_1 U K : Set ℂ z✝ : ℂ F : ι → 𝓒 U Q : Set ℂ → Set ℂ h1 : UniformlyBoundedOn F U hU : IsOpen U h2 : ∀ (i : ι), DifferentiableOn ℂ (F i) U hK : K ∈ compacts U z : ℂ hz : z ∈ K δ : ℝ hδ : 0 < δ h : closedBall z δ ⊆ U ⊢ ∃ M > 0, ∀ (i : ι), MapsTo (_root_.deriv (F i)) (closedBall z δ) (closedBall 0 M)	case intro.intro ι : Type u_1 U K : Set ℂ z✝ : ℂ F : ι → 𝓒 U Q✝ : Set ℂ → Set ℂ h1 : UniformlyBoundedOn F U hU : IsOpen U h2 : ∀ (i : ι), DifferentiableOn ℂ (F i) U hK : K ∈ compacts U z : ℂ hz : z ∈ K δ : ℝ hδ : 0 < δ h : closedBall z δ ⊆ U Q : Set ℂ hQ1 : IsCompact Q hQ2 : ∀ (i : ι), MapsTo ((_root_.deriv ∘ F) i) (closedBall z δ) Q ⊢ ∃ M > 0, ∀ (i : ι), MapsTo (_root_.deriv (F i)) (closedBall z δ) (closedBall 0 M)
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	RMT4/Montel.lean	UniformlyBoundedOn.equicontinuousOn	[32, 1]	[58, 40]	obtain ⟨M, hM⟩ := hQ1.isBounded.subset_closedBall 0	case intro.intro ι : Type u_1 U K : Set ℂ z✝ : ℂ F : ι → 𝓒 U Q✝ : Set ℂ → Set ℂ h1 : UniformlyBoundedOn F U hU : IsOpen U h2 : ∀ (i : ι), DifferentiableOn ℂ (F i) U hK : K ∈ compacts U z : ℂ hz : z ∈ K δ : ℝ hδ : 0 < δ h : closedBall z δ ⊆ U Q : Set ℂ hQ1 : IsCompact Q hQ2 : ∀ (i : ι), MapsTo ((_root_.deriv ∘ F) i) (closedBall z δ) Q ⊢ ∃ M > 0, ∀ (i : ι), MapsTo (_root_.deriv (F i)) (closedBall z δ) (closedBall 0 M)	case intro.intro.intro ι : Type u_1 U K : Set ℂ z✝ : ℂ F : ι → 𝓒 U Q✝ : Set ℂ → Set ℂ h1 : UniformlyBoundedOn F U hU : IsOpen U h2 : ∀ (i : ι), DifferentiableOn ℂ (F i) U hK : K ∈ compacts U z : ℂ hz : z ∈ K δ : ℝ hδ : 0 < δ h : closedBall z δ ⊆ U Q : Set ℂ hQ1 : IsCompact Q hQ2 : ∀ (i : ι), MapsTo ((_root_.deriv ∘ F) i) (closedBall z δ) Q M : ℝ hM : Q ⊆ closedBall 0 M ⊢ ∃ M > 0, ∀ (i : ι), MapsTo (_root_.deriv (F i)) (closedBall z δ) (closedBall 0 M)
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	RMT4/Montel.lean	UniformlyBoundedOn.equicontinuousOn	[32, 1]	[58, 40]	refine ⟨M ⊔ 1, by simp, fun i => ?_⟩	case intro.intro.intro ι : Type u_1 U K : Set ℂ z✝ : ℂ F : ι → 𝓒 U Q✝ : Set ℂ → Set ℂ h1 : UniformlyBoundedOn F U hU : IsOpen U h2 : ∀ (i : ι), DifferentiableOn ℂ (F i) U hK : K ∈ compacts U z : ℂ hz : z ∈ K δ : ℝ hδ : 0 < δ h : closedBall z δ ⊆ U Q : Set ℂ hQ1 : IsCompact Q hQ2 : ∀ (i : ι), MapsTo ((_root_.deriv ∘ F) i) (closedBall z δ) Q M : ℝ hM : Q ⊆ closedBall 0 M ⊢ ∃ M > 0, ∀ (i : ι), MapsTo (_root_.deriv (F i)) (closedBall z δ) (closedBall 0 M)	case intro.intro.intro ι : Type u_1 U K : Set ℂ z✝ : ℂ F : ι → 𝓒 U Q✝ : Set ℂ → Set ℂ h1 : UniformlyBoundedOn F U hU : IsOpen U h2 : ∀ (i : ι), DifferentiableOn ℂ (F i) U hK : K ∈ compacts U z : ℂ hz : z ∈ K δ : ℝ hδ : 0 < δ h : closedBall z δ ⊆ U Q : Set ℂ hQ1 : IsCompact Q hQ2 : ∀ (i : ι), MapsTo ((_root_.deriv ∘ F) i) (closedBall z δ) Q M : ℝ hM : Q ⊆ closedBall 0 M i : ι ⊢ MapsTo (_root_.deriv (F i)) (closedBall z δ) (closedBall 0 (M ⊔ 1))
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	RMT4/Montel.lean	UniformlyBoundedOn.equicontinuousOn	[32, 1]	[58, 40]	exact ((hQ2 i).mono_right hM).mono_right <\| closedBall_subset_closedBall le_sup_left	case intro.intro.intro ι : Type u_1 U K : Set ℂ z✝ : ℂ F : ι → 𝓒 U Q✝ : Set ℂ → Set ℂ h1 : UniformlyBoundedOn F U hU : IsOpen U h2 : ∀ (i : ι), DifferentiableOn ℂ (F i) U hK : K ∈ compacts U z : ℂ hz : z ∈ K δ : ℝ hδ : 0 < δ h : closedBall z δ ⊆ U Q : Set ℂ hQ1 : IsCompact Q hQ2 : ∀ (i : ι), MapsTo ((_root_.deriv ∘ F) i) (closedBall z δ) Q M : ℝ hM : Q ⊆ closedBall 0 M i : ι ⊢ MapsTo (_root_.deriv (F i)) (closedBall z δ) (closedBall 0 (M ⊔ 1))	no goals
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	RMT4/Montel.lean	UniformlyBoundedOn.equicontinuousOn	[32, 1]	[58, 40]	simp	ι : Type u_1 U K : Set ℂ z✝ : ℂ F : ι → 𝓒 U Q✝ : Set ℂ → Set ℂ h1 : UniformlyBoundedOn F U hU : IsOpen U h2 : ∀ (i : ι), DifferentiableOn ℂ (F i) U hK : K ∈ compacts U z : ℂ hz : z ∈ K δ : ℝ hδ : 0 < δ h : closedBall z δ ⊆ U Q : Set ℂ hQ1 : IsCompact Q hQ2 : ∀ (i : ι), MapsTo ((_root_.deriv ∘ F) i) (closedBall z δ) Q M : ℝ hM : Q ⊆ closedBall 0 M ⊢ M ⊔ 1 > 0	no goals
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	RMT4/Montel.lean	UniformlyBoundedOn.equicontinuousOn	[32, 1]	[58, 40]	simpa [MapsTo] using hM i	ι : Type u_1 U K : Set ℂ z✝ : ℂ F : ι → 𝓒 U Q : Set ℂ → Set ℂ h1 : UniformlyBoundedOn F U hU : IsOpen U h2 : ∀ (i : ι), DifferentiableOn ℂ (F i) U hK : K ∈ compacts U z : ℂ hz : z ∈ K δ : ℝ hδ : 0 < δ h : closedBall z δ ⊆ U M : ℝ hMp : M > 0 hM : ∀ (i : ι), MapsTo (_root_.deriv (F i)) (closedBall z δ) (closedBall 0 M) ε : ℝ hε : ε > 0 w : ↑K hw : dist w { val := z, property := hz } < δ ⊓ ε / M i : ι e1 : ∀ x ∈ closedBall z δ, DifferentiableAt ℂ (F i) x ⊢ ∀ x ∈ closedBall z δ, ‖_root_.deriv (F i) x‖ ≤ M	no goals
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	RMT4/Montel.lean	UniformlyBoundedOn.equicontinuousOn	[32, 1]	[58, 40]	simpa using (lt_inf_iff.1 hw).1.le	ι : Type u_1 U K : Set ℂ z✝ : ℂ F : ι → 𝓒 U Q : Set ℂ → Set ℂ h1 : UniformlyBoundedOn F U hU : IsOpen U h2 : ∀ (i : ι), DifferentiableOn ℂ (F i) U hK : K ∈ compacts U z : ℂ hz : z ∈ K δ : ℝ hδ : 0 < δ h : closedBall z δ ⊆ U M : ℝ hMp : M > 0 hM : ∀ (i : ι), MapsTo (_root_.deriv (F i)) (closedBall z δ) (closedBall 0 M) ε : ℝ hε : ε > 0 w : ↑K hw : dist w { val := z, property := hz } < δ ⊓ ε / M i : ι e1 : ∀ x ∈ closedBall z δ, DifferentiableAt ℂ (F i) x e2 : ∀ x ∈ closedBall z δ, ‖_root_.deriv (F i) x‖ ≤ M e3 : z ∈ closedBall z δ ⊢ ↑w ∈ closedBall z δ	no goals
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	RMT4/Montel.lean	UniformlyBoundedOn.equicontinuousOn	[32, 1]	[58, 40]	have := (lt_inf_iff.1 hw).2	ι : Type u_1 U K : Set ℂ z✝ : ℂ F : ι → 𝓒 U Q : Set ℂ → Set ℂ h1 : UniformlyBoundedOn F U hU : IsOpen U h2 : ∀ (i : ι), DifferentiableOn ℂ (F i) U hK : K ∈ compacts U z : ℂ hz : z ∈ K δ : ℝ hδ : 0 < δ h : closedBall z δ ⊆ U M : ℝ hMp : M > 0 hM : ∀ (i : ι), MapsTo (_root_.deriv (F i)) (closedBall z δ) (closedBall 0 M) ε : ℝ hε : ε > 0 w : ↑K hw : dist w { val := z, property := hz } < δ ⊓ ε / M i : ι e1 : ∀ x ∈ closedBall z δ, DifferentiableAt ℂ (F i) x e2 : ∀ x ∈ closedBall z δ, ‖_root_.deriv (F i) x‖ ≤ M e3 : z ∈ closedBall z δ e4 : ↑w ∈ closedBall z δ ⊢ ‖z - ↑w‖ < ε / M	ι : Type u_1 U K : Set ℂ z✝ : ℂ F : ι → 𝓒 U Q : Set ℂ → Set ℂ h1 : UniformlyBoundedOn F U hU : IsOpen U h2 : ∀ (i : ι), DifferentiableOn ℂ (F i) U hK : K ∈ compacts U z : ℂ hz : z ∈ K δ : ℝ hδ : 0 < δ h : closedBall z δ ⊆ U M : ℝ hMp : M > 0 hM : ∀ (i : ι), MapsTo (_root_.deriv (F i)) (closedBall z δ) (closedBall 0 M) ε : ℝ hε : ε > 0 w : ↑K hw : dist w { val := z, property := hz } < δ ⊓ ε / M i : ι e1 : ∀ x ∈ closedBall z δ, DifferentiableAt ℂ (F i) x e2 : ∀ x ∈ closedBall z δ, ‖_root_.deriv (F i) x‖ ≤ M e3 : z ∈ closedBall z δ e4 : ↑w ∈ closedBall z δ this : dist w { val := z, property := hz } < ε / M ⊢ ‖z - ↑w‖ < ε / M
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	RMT4/Montel.lean	UniformlyBoundedOn.equicontinuousOn	[32, 1]	[58, 40]	rwa [dist_comm, Subtype.dist_eq, dist_eq_norm] at this	ι : Type u_1 U K : Set ℂ z✝ : ℂ F : ι → 𝓒 U Q : Set ℂ → Set ℂ h1 : UniformlyBoundedOn F U hU : IsOpen U h2 : ∀ (i : ι), DifferentiableOn ℂ (F i) U hK : K ∈ compacts U z : ℂ hz : z ∈ K δ : ℝ hδ : 0 < δ h : closedBall z δ ⊆ U M : ℝ hMp : M > 0 hM : ∀ (i : ι), MapsTo (_root_.deriv (F i)) (closedBall z δ) (closedBall 0 M) ε : ℝ hε : ε > 0 w : ↑K hw : dist w { val := z, property := hz } < δ ⊓ ε / M i : ι e1 : ∀ x ∈ closedBall z δ, DifferentiableAt ℂ (F i) x e2 : ∀ x ∈ closedBall z δ, ‖_root_.deriv (F i) x‖ ≤ M e3 : z ∈ closedBall z δ e4 : ↑w ∈ closedBall z δ this : dist w { val := z, property := hz } < ε / M ⊢ ‖z - ↑w‖ < ε / M	no goals
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	RMT4/Montel.lean	uniformlyBoundedOn_𝓑	[60, 1]	[62, 56]	exact fun K hK => ⟨Q K, hQ K hK, fun f => f.2.2 K hK⟩	ι : Type u_1 U K : Set ℂ z : ℂ F : ι → 𝓒 U Q : Set ℂ → Set ℂ hQ : ∀ K ∈ compacts U, IsCompact (Q K) ⊢ UniformlyBoundedOn Subtype.val U	no goals
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	RMT4/Montel.lean	isCompact_𝓑	[64, 1]	[73, 59]	have l1 (K) (hK : K ∈ compacts U) : EquicontinuousOn ((↑) : 𝓑 U Q → 𝓒 U) K := (uniformlyBoundedOn_𝓑 hQ).equicontinuousOn hU (fun f => f.2.1) hK	ι : Type u_1 U K : Set ℂ z : ℂ F : ι → 𝓒 U Q : Set ℂ → Set ℂ hU : IsOpen U hQ : ∀ K ∈ compacts U, IsCompact (Q K) ⊢ IsCompact (𝓑 U Q)	ι : Type u_1 U K : Set ℂ z : ℂ F : ι → 𝓒 U Q : Set ℂ → Set ℂ hU : IsOpen U hQ : ∀ K ∈ compacts U, IsCompact (Q K) l1 : ∀ K ∈ compacts U, EquicontinuousOn Subtype.val K ⊢ IsCompact (𝓑 U Q)
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	RMT4/Montel.lean	isCompact_𝓑	[64, 1]	[73, 59]	have l2 (K) (hK : K ∈ compacts U) (x) (hx : x ∈ K) : ∃ L, IsCompact L ∧ ∀ i : 𝓑 U Q, i.1 x ∈ L := ⟨Q K, hQ K hK, fun f => f.2.2 K hK hx⟩	ι : Type u_1 U K : Set ℂ z : ℂ F : ι → 𝓒 U Q : Set ℂ → Set ℂ hU : IsOpen U hQ : ∀ K ∈ compacts U, IsCompact (Q K) l1 : ∀ K ∈ compacts U, EquicontinuousOn Subtype.val K ⊢ IsCompact (𝓑 U Q)	ι : Type u_1 U K : Set ℂ z : ℂ F : ι → 𝓒 U Q : Set ℂ → Set ℂ hU : IsOpen U hQ : ∀ K ∈ compacts U, IsCompact (Q K) l1 : ∀ K ∈ compacts U, EquicontinuousOn Subtype.val K l2 : ∀ K ∈ compacts U, ∀ x ∈ K, ∃ L, IsCompact L ∧ ∀ (i : ↑(𝓑 U Q)), ↑i x ∈ L ⊢ IsCompact (𝓑 U Q)
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	RMT4/Montel.lean	isCompact_𝓑	[64, 1]	[73, 59]	rw [isCompact_iff_compactSpace]	ι : Type u_1 U K : Set ℂ z : ℂ F : ι → 𝓒 U Q : Set ℂ → Set ℂ hU : IsOpen U hQ : ∀ K ∈ compacts U, IsCompact (Q K) l1 : ∀ K ∈ compacts U, EquicontinuousOn Subtype.val K l2 : ∀ K ∈ compacts U, ∀ x ∈ K, ∃ L, IsCompact L ∧ ∀ (i : ↑(𝓑 U Q)), ↑i x ∈ L ⊢ IsCompact (𝓑 U Q)	ι : Type u_1 U K : Set ℂ z : ℂ F : ι → 𝓒 U Q : Set ℂ → Set ℂ hU : IsOpen U hQ : ∀ K ∈ compacts U, IsCompact (Q K) l1 : ∀ K ∈ compacts U, EquicontinuousOn Subtype.val K l2 : ∀ K ∈ compacts U, ∀ x ∈ K, ∃ L, IsCompact L ∧ ∀ (i : ↑(𝓑 U Q)), ↑i x ∈ L ⊢ CompactSpace ↑(𝓑 U Q)
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	RMT4/Montel.lean	isCompact_𝓑	[64, 1]	[73, 59]	refine ArzelaAscoli.compactSpace_of_closedEmbedding (fun K hK => hK.2) ?_ l1 l2	ι : Type u_1 U K : Set ℂ z : ℂ F : ι → 𝓒 U Q : Set ℂ → Set ℂ hU : IsOpen U hQ : ∀ K ∈ compacts U, IsCompact (Q K) l1 : ∀ K ∈ compacts U, EquicontinuousOn Subtype.val K l2 : ∀ K ∈ compacts U, ∀ x ∈ K, ∃ L, IsCompact L ∧ ∀ (i : ↑(𝓑 U Q)), ↑i x ∈ L ⊢ CompactSpace ↑(𝓑 U Q)	ι : Type u_1 U K : Set ℂ z : ℂ F : ι → 𝓒 U Q : Set ℂ → Set ℂ hU : IsOpen U hQ : ∀ K ∈ compacts U, IsCompact (Q K) l1 : ∀ K ∈ compacts U, EquicontinuousOn Subtype.val K l2 : ∀ K ∈ compacts U, ∀ x ∈ K, ∃ L, IsCompact L ∧ ∀ (i : ↑(𝓑 U Q)), ↑i x ∈ L ⊢ ClosedEmbedding (⇑(UniformOnFun.ofFun (compacts U)) ∘ Subtype.val)
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	RMT4/Montel.lean	isCompact_𝓑	[64, 1]	[73, 59]	refine ⟨⟨by tauto, fun f g => Subtype.ext⟩, ?_⟩	ι : Type u_1 U K : Set ℂ z : ℂ F : ι → 𝓒 U Q : Set ℂ → Set ℂ hU : IsOpen U hQ : ∀ K ∈ compacts U, IsCompact (Q K) l1 : ∀ K ∈ compacts U, EquicontinuousOn Subtype.val K l2 : ∀ K ∈ compacts U, ∀ x ∈ K, ∃ L, IsCompact L ∧ ∀ (i : ↑(𝓑 U Q)), ↑i x ∈ L ⊢ ClosedEmbedding (⇑(UniformOnFun.ofFun (compacts U)) ∘ Subtype.val)	ι : Type u_1 U K : Set ℂ z : ℂ F : ι → 𝓒 U Q : Set ℂ → Set ℂ hU : IsOpen U hQ : ∀ K ∈ compacts U, IsCompact (Q K) l1 : ∀ K ∈ compacts U, EquicontinuousOn Subtype.val K l2 : ∀ K ∈ compacts U, ∀ x ∈ K, ∃ L, IsCompact L ∧ ∀ (i : ↑(𝓑 U Q)), ↑i x ∈ L ⊢ IsClosed (range (⇑(UniformOnFun.ofFun (compacts U)) ∘ Subtype.val))
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	RMT4/Montel.lean	isCompact_𝓑	[64, 1]	[73, 59]	simpa [range, UniformOnFun.ofFun] using isClosed_𝓑 hU hQ	ι : Type u_1 U K : Set ℂ z : ℂ F : ι → 𝓒 U Q : Set ℂ → Set ℂ hU : IsOpen U hQ : ∀ K ∈ compacts U, IsCompact (Q K) l1 : ∀ K ∈ compacts U, EquicontinuousOn Subtype.val K l2 : ∀ K ∈ compacts U, ∀ x ∈ K, ∃ L, IsCompact L ∧ ∀ (i : ↑(𝓑 U Q)), ↑i x ∈ L ⊢ IsClosed (range (⇑(UniformOnFun.ofFun (compacts U)) ∘ Subtype.val))	no goals
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	RMT4/Montel.lean	isCompact_𝓑	[64, 1]	[73, 59]	tauto	ι : Type u_1 U K : Set ℂ z : ℂ F : ι → 𝓒 U Q : Set ℂ → Set ℂ hU : IsOpen U hQ : ∀ K ∈ compacts U, IsCompact (Q K) l1 : ∀ K ∈ compacts U, EquicontinuousOn Subtype.val K l2 : ∀ K ∈ compacts U, ∀ x ∈ K, ∃ L, IsCompact L ∧ ∀ (i : ↑(𝓑 U Q)), ↑i x ∈ L ⊢ Inducing (⇑(UniformOnFun.ofFun (compacts U)) ∘ Subtype.val)	no goals
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	RMT4/Montel.lean	montel	[75, 1]	[79, 72]	choose! Q hQ1 hQ2 using h1	ι : Type u_1 U K : Set ℂ z : ℂ F : ι → 𝓒 U Q : Set ℂ → Set ℂ hU : IsOpen U h1 : UniformlyBoundedOn F U h2 : ∀ (i : ι), DifferentiableOn ℂ (F i) U ⊢ TotallyBounded (range F)	ι : Type u_1 U K : Set ℂ z : ℂ F : ι → 𝓒 U Q✝ : Set ℂ → Set ℂ hU : IsOpen U h2 : ∀ (i : ι), DifferentiableOn ℂ (F i) U Q : Set ℂ → Set ℂ hQ1 : ∀ K ∈ compacts U, IsCompact (Q K) hQ2 : ∀ K ∈ compacts U, ∀ (i : ι), MapsTo (F i) K (Q K) ⊢ TotallyBounded (range F)
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	RMT4/Montel.lean	montel	[75, 1]	[79, 72]	have l1 : range F ⊆ 𝓑 U Q := by rintro f ⟨i, rfl⟩ ; exact ⟨h2 i, fun K hK => hQ2 K hK i⟩	ι : Type u_1 U K : Set ℂ z : ℂ F : ι → 𝓒 U Q✝ : Set ℂ → Set ℂ hU : IsOpen U h2 : ∀ (i : ι), DifferentiableOn ℂ (F i) U Q : Set ℂ → Set ℂ hQ1 : ∀ K ∈ compacts U, IsCompact (Q K) hQ2 : ∀ K ∈ compacts U, ∀ (i : ι), MapsTo (F i) K (Q K) ⊢ TotallyBounded (range F)	ι : Type u_1 U K : Set ℂ z : ℂ F : ι → 𝓒 U Q✝ : Set ℂ → Set ℂ hU : IsOpen U h2 : ∀ (i : ι), DifferentiableOn ℂ (F i) U Q : Set ℂ → Set ℂ hQ1 : ∀ K ∈ compacts U, IsCompact (Q K) hQ2 : ∀ K ∈ compacts U, ∀ (i : ι), MapsTo (F i) K (Q K) l1 : range F ⊆ 𝓑 U Q ⊢ TotallyBounded (range F)
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	RMT4/Montel.lean	montel	[75, 1]	[79, 72]	exact totallyBounded_subset l1 <\| (isCompact_𝓑 hU hQ1).totallyBounded	ι : Type u_1 U K : Set ℂ z : ℂ F : ι → 𝓒 U Q✝ : Set ℂ → Set ℂ hU : IsOpen U h2 : ∀ (i : ι), DifferentiableOn ℂ (F i) U Q : Set ℂ → Set ℂ hQ1 : ∀ K ∈ compacts U, IsCompact (Q K) hQ2 : ∀ K ∈ compacts U, ∀ (i : ι), MapsTo (F i) K (Q K) l1 : range F ⊆ 𝓑 U Q ⊢ TotallyBounded (range F)	no goals
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	RMT4/Montel.lean	montel	[75, 1]	[79, 72]	rintro f ⟨i, rfl⟩	ι : Type u_1 U K : Set ℂ z : ℂ F : ι → 𝓒 U Q✝ : Set ℂ → Set ℂ hU : IsOpen U h2 : ∀ (i : ι), DifferentiableOn ℂ (F i) U Q : Set ℂ → Set ℂ hQ1 : ∀ K ∈ compacts U, IsCompact (Q K) hQ2 : ∀ K ∈ compacts U, ∀ (i : ι), MapsTo (F i) K (Q K) ⊢ range F ⊆ 𝓑 U Q	case intro ι : Type u_1 U K : Set ℂ z : ℂ F : ι → 𝓒 U Q✝ : Set ℂ → Set ℂ hU : IsOpen U h2 : ∀ (i : ι), DifferentiableOn ℂ (F i) U Q : Set ℂ → Set ℂ hQ1 : ∀ K ∈ compacts U, IsCompact (Q K) hQ2 : ∀ K ∈ compacts U, ∀ (i : ι), MapsTo (F i) K (Q K) i : ι ⊢ F i ∈ 𝓑 U Q
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	RMT4/Montel.lean	montel	[75, 1]	[79, 72]	exact ⟨h2 i, fun K hK => hQ2 K hK i⟩	case intro ι : Type u_1 U K : Set ℂ z : ℂ F : ι → 𝓒 U Q✝ : Set ℂ → Set ℂ hU : IsOpen U h2 : ∀ (i : ι), DifferentiableOn ℂ (F i) U Q : Set ℂ → Set ℂ hQ1 : ∀ K ∈ compacts U, IsCompact (Q K) hQ2 : ∀ K ∈ compacts U, ∀ (i : ι), MapsTo (F i) K (Q K) i : ι ⊢ F i ∈ 𝓑 U Q	no goals
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	RMT4/Montel.lean	isCompact_𝓜	[83, 1]	[84, 82]	simpa only [𝓑_const] using isCompact_𝓑 hU (fun _ _ => isCompact_closedBall 0 1)	ι : Type u_1 U K : Set ℂ z : ℂ F : ι → 𝓒 U Q : Set ℂ → Set ℂ hU : IsOpen U ⊢ IsCompact (𝓜 U)	no goals
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	RMT4/defs.lean	neg_in_𝔻	[19, 1]	[20, 11]	simp [𝔻]	u : ℂ U V W : Set ℂ ⊢ u ∈ 𝔻 → -u ∈ 𝔻	no goals
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	RMT4/defs.lean	sqrt_𝔻_eq_𝔻	[22, 1]	[23, 17]	simp [𝔻, ball]	u : ℂ U V W : Set ℂ ⊢ {z \| z ^ 2 ∈ 𝔻} = 𝔻	no goals
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	RMT4/defs.lean	ne_center_of_not_mem_closed_ball	[69, 1]	[72, 16]	contrapose! hz	u : ℂ U V W : Set ℂ w : ℂ r : ℝ hr : 0 ≤ r z : ℂ hz : z ∈ (closedBall w r)ᶜ ⊢ z ≠ w	u : ℂ U V W : Set ℂ w : ℂ r : ℝ hr : 0 ≤ r z : ℂ hz : z = w ⊢ z ∉ (closedBall w r)ᶜ
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	RMT4/defs.lean	ne_center_of_not_mem_closed_ball	[69, 1]	[72, 16]	simp [hz, hr]	u : ℂ U V W : Set ℂ w : ℂ r : ℝ hr : 0 ≤ r z : ℂ hz : z = w ⊢ z ∉ (closedBall w r)ᶜ	no goals
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	RMT4/etape2.lean	one_sub_mul_conj_ne_zero	[9, 1]	[12, 65]	rw [mem_𝔻_iff] at hu hz	z u z₀ : ℂ U : Set ℂ inst✝ : good_domain U hu : u ∈ 𝔻 hz : z ∈ 𝔻 ⊢ 1 - z * (starRingEnd ℂ) u ≠ 0	z u z₀ : ℂ U : Set ℂ inst✝ : good_domain U hu : ‖u‖ < 1 hz : ‖z‖ < 1 ⊢ 1 - z * (starRingEnd ℂ) u ≠ 0
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	RMT4/etape2.lean	one_sub_mul_conj_ne_zero	[9, 1]	[12, 65]	refine sub_ne_zero.mpr (mt (congr_arg Complex.abs) (ne_comm.mp (ne_of_lt ?_)))	z u z₀ : ℂ U : Set ℂ inst✝ : good_domain U hu : ‖u‖ < 1 hz : ‖z‖ < 1 ⊢ 1 - z * (starRingEnd ℂ) u ≠ 0	z u z₀ : ℂ U : Set ℂ inst✝ : good_domain U hu : ‖u‖ < 1 hz : ‖z‖ < 1 ⊢ Complex.abs (z * (starRingEnd ℂ) u) < Complex.abs 1
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	RMT4/etape2.lean	one_sub_mul_conj_ne_zero	[9, 1]	[12, 65]	simpa using mul_lt_mul'' hz hu (norm_nonneg z) (norm_nonneg u)	z u z₀ : ℂ U : Set ℂ inst✝ : good_domain U hu : ‖u‖ < 1 hz : ‖z‖ < 1 ⊢ Complex.abs (z * (starRingEnd ℂ) u) < Complex.abs 1	no goals
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	RMT4/etape2.lean	one_sub_mul_conj_add_mul_conj_ne_zero	[14, 1]	[18, 22]	have h1 := one_sub_mul_conj_ne_zero hu hu	z u z₀ : ℂ U : Set ℂ inst✝ : good_domain U hu : u ∈ 𝔻 ⊢ 1 - z * (starRingEnd ℂ) u + (z - u) * (starRingEnd ℂ) u ≠ 0	z u z₀ : ℂ U : Set ℂ inst✝ : good_domain U hu : u ∈ 𝔻 h1 : 1 - u * (starRingEnd ℂ) u ≠ 0 ⊢ 1 - z * (starRingEnd ℂ) u + (z - u) * (starRingEnd ℂ) u ≠ 0
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	RMT4/etape2.lean	one_sub_mul_conj_add_mul_conj_ne_zero	[14, 1]	[18, 22]	ring_nf	z u z₀ : ℂ U : Set ℂ inst✝ : good_domain U hu : u ∈ 𝔻 h1 : 1 - u * (starRingEnd ℂ) u ≠ 0 ⊢ 1 - z * (starRingEnd ℂ) u + (z - u) * (starRingEnd ℂ) u ≠ 0	z u z₀ : ℂ U : Set ℂ inst✝ : good_domain U hu : u ∈ 𝔻 h1 : 1 - u * (starRingEnd ℂ) u ≠ 0 ⊢ 1 - (starRingEnd ℂ) u * u ≠ 0
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	RMT4/etape2.lean	one_sub_mul_conj_add_mul_conj_ne_zero	[14, 1]	[18, 22]	simp [h1, mul_comm]	z u z₀ : ℂ U : Set ℂ inst✝ : good_domain U hu : u ∈ 𝔻 h1 : 1 - u * (starRingEnd ℂ) u ≠ 0 ⊢ 1 - (starRingEnd ℂ) u * u ≠ 0	no goals
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	RMT4/etape2.lean	normSq_sub_normSq	[20, 1]	[21, 59]	field_simp [← ofReal_inj, normSq_eq_conj_mul_self]	z u z₀ : ℂ U : Set ℂ inst✝ : good_domain U ⊢ normSq (z - u) - normSq (1 - z * (starRingEnd ℂ) u) = (normSq z - 1) * (1 - normSq u)	z u z₀ : ℂ U : Set ℂ inst✝ : good_domain U ⊢ ((starRingEnd ℂ) z - (starRingEnd ℂ) u) * (z - u) - (1 - (starRingEnd ℂ) z * u) * (1 - z * (starRingEnd ℂ) u) = ((starRingEnd ℂ) z * z - 1) * (1 - (starRingEnd ℂ) u * u)
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	RMT4/etape2.lean	normSq_sub_normSq	[20, 1]	[21, 59]	ring	z u z₀ : ℂ U : Set ℂ inst✝ : good_domain U ⊢ ((starRingEnd ℂ) z - (starRingEnd ℂ) u) * (z - u) - (1 - (starRingEnd ℂ) z * u) * (1 - z * (starRingEnd ℂ) u) = ((starRingEnd ℂ) z * z - 1) * (1 - (starRingEnd ℂ) u * u)	no goals
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	RMT4/etape2.lean	pre_φ_inv	[25, 1]	[28, 7]	rintro z hz	z u z₀ : ℂ U : Set ℂ inst✝ : good_domain U hu : u ∈ 𝔻 ⊢ LeftInvOn (pre_φ (-u)) (pre_φ u) 𝔻	z✝ u z₀ : ℂ U : Set ℂ inst✝ : good_domain U hu : u ∈ 𝔻 z : ℂ hz : z ∈ 𝔻 ⊢ pre_φ (-u) (pre_φ u z) = z
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	RMT4/etape2.lean	pre_φ_inv	[25, 1]	[28, 7]	field_simp [pre_φ, one_sub_mul_conj_ne_zero hu hz, one_sub_mul_conj_add_mul_conj_ne_zero hu]	z✝ u z₀ : ℂ U : Set ℂ inst✝ : good_domain U hu : u ∈ 𝔻 z : ℂ hz : z ∈ 𝔻 ⊢ pre_φ (-u) (pre_φ u z) = z	z✝ u z₀ : ℂ U : Set ℂ inst✝ : good_domain U hu : u ∈ 𝔻 z : ℂ hz : z ∈ 𝔻 ⊢ z - u + (1 - z * (starRingEnd ℂ) u) * u = z * (1 - z * (starRingEnd ℂ) u + (z - u) * (starRingEnd ℂ) u)
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	RMT4/etape2.lean	pre_φ_inv	[25, 1]	[28, 7]	ring	z✝ u z₀ : ℂ U : Set ℂ inst✝ : good_domain U hu : u ∈ 𝔻 z : ℂ hz : z ∈ 𝔻 ⊢ z - u + (1 - z * (starRingEnd ℂ) u) * u = z * (1 - z * (starRingEnd ℂ) u + (z - u) * (starRingEnd ℂ) u)	no goals
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	RMT4/etape2.lean	φ_deriv	[50, 1]	[57, 24]	have h1 : DifferentiableAt ℂ (fun z => z - u) z := by simp	z u z₀ : ℂ U : Set ℂ inst✝ : good_domain U hu : u ∈ 𝔻 hz : z ∈ 𝔻 ⊢ deriv (φ hu).to_fun z = (1 - u * (starRingEnd ℂ) u) / (1 - z * (starRingEnd ℂ) u) ^ 2	z u z₀ : ℂ U : Set ℂ inst✝ : good_domain U hu : u ∈ 𝔻 hz : z ∈ 𝔻 h1 : DifferentiableAt ℂ (fun z => z - u) z ⊢ deriv (φ hu).to_fun z = (1 - u * (starRingEnd ℂ) u) / (1 - z * (starRingEnd ℂ) u) ^ 2
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	RMT4/etape2.lean	φ_deriv	[50, 1]	[57, 24]	have h2 : DifferentiableAt ℂ (fun z => 1 - z * conj u) z := by simp [DifferentiableAt.mul_const]	z u z₀ : ℂ U : Set ℂ inst✝ : good_domain U hu : u ∈ 𝔻 hz : z ∈ 𝔻 h1 : DifferentiableAt ℂ (fun z => z - u) z ⊢ deriv (φ hu).to_fun z = (1 - u * (starRingEnd ℂ) u) / (1 - z * (starRingEnd ℂ) u) ^ 2	z u z₀ : ℂ U : Set ℂ inst✝ : good_domain U hu : u ∈ 𝔻 hz : z ∈ 𝔻 h1 : DifferentiableAt ℂ (fun z => z - u) z h2 : DifferentiableAt ℂ (fun z => 1 - z * (starRingEnd ℂ) u) z ⊢ deriv (φ hu).to_fun z = (1 - u * (starRingEnd ℂ) u) / (1 - z * (starRingEnd ℂ) u) ^ 2
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	RMT4/etape2.lean	φ_deriv	[50, 1]	[57, 24]	have h3 : 1 - z * conj u ≠ 0 := one_sub_mul_conj_ne_zero hu hz	z u z₀ : ℂ U : Set ℂ inst✝ : good_domain U hu : u ∈ 𝔻 hz : z ∈ 𝔻 h1 : DifferentiableAt ℂ (fun z => z - u) z h2 : DifferentiableAt ℂ (fun z => 1 - z * (starRingEnd ℂ) u) z ⊢ deriv (φ hu).to_fun z = (1 - u * (starRingEnd ℂ) u) / (1 - z * (starRingEnd ℂ) u) ^ 2	z u z₀ : ℂ U : Set ℂ inst✝ : good_domain U hu : u ∈ 𝔻 hz : z ∈ 𝔻 h1 : DifferentiableAt ℂ (fun z => z - u) z h2 : DifferentiableAt ℂ (fun z => 1 - z * (starRingEnd ℂ) u) z h3 : 1 - z * (starRingEnd ℂ) u ≠ 0 ⊢ deriv (φ hu).to_fun z = (1 - u * (starRingEnd ℂ) u) / (1 - z * (starRingEnd ℂ) u) ^ 2
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	RMT4/etape2.lean	φ_deriv	[50, 1]	[57, 24]	have h4 : deriv (fun z => z - u) z = 1 := by simp [deriv_sub_const]	z u z₀ : ℂ U : Set ℂ inst✝ : good_domain U hu : u ∈ 𝔻 hz : z ∈ 𝔻 h1 : DifferentiableAt ℂ (fun z => z - u) z h2 : DifferentiableAt ℂ (fun z => 1 - z * (starRingEnd ℂ) u) z h3 : 1 - z * (starRingEnd ℂ) u ≠ 0 ⊢ deriv (φ hu).to_fun z = (1 - u * (starRingEnd ℂ) u) / (1 - z * (starRingEnd ℂ) u) ^ 2	z u z₀ : ℂ U : Set ℂ inst✝ : good_domain U hu : u ∈ 𝔻 hz : z ∈ 𝔻 h1 : DifferentiableAt ℂ (fun z => z - u) z h2 : DifferentiableAt ℂ (fun z => 1 - z * (starRingEnd ℂ) u) z h3 : 1 - z * (starRingEnd ℂ) u ≠ 0 h4 : deriv (fun z => z - u) z = 1 ⊢ deriv (φ hu).to_fun z = (1 - u * (starRingEnd ℂ) u) / (1 - z * (starRingEnd ℂ) u) ^ 2
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	RMT4/etape2.lean	φ_deriv	[50, 1]	[57, 24]	have h5 : deriv (fun z => 1 - z * conj u) z = - conj u := by simp [deriv_const_sub]	z u z₀ : ℂ U : Set ℂ inst✝ : good_domain U hu : u ∈ 𝔻 hz : z ∈ 𝔻 h1 : DifferentiableAt ℂ (fun z => z - u) z h2 : DifferentiableAt ℂ (fun z => 1 - z * (starRingEnd ℂ) u) z h3 : 1 - z * (starRingEnd ℂ) u ≠ 0 h4 : deriv (fun z => z - u) z = 1 ⊢ deriv (φ hu).to_fun z = (1 - u * (starRingEnd ℂ) u) / (1 - z * (starRingEnd ℂ) u) ^ 2	z u z₀ : ℂ U : Set ℂ inst✝ : good_domain U hu : u ∈ 𝔻 hz : z ∈ 𝔻 h1 : DifferentiableAt ℂ (fun z => z - u) z h2 : DifferentiableAt ℂ (fun z => 1 - z * (starRingEnd ℂ) u) z h3 : 1 - z * (starRingEnd ℂ) u ≠ 0 h4 : deriv (fun z => z - u) z = 1 h5 : deriv (fun z => 1 - z * (starRingEnd ℂ) u) z = -(starRingEnd ℂ) u ⊢ deriv (φ hu).to_fun z = (1 - u * (starRingEnd ℂ) u) / (1 - z * (starRingEnd ℂ) u) ^ 2
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	RMT4/etape2.lean	φ_deriv	[50, 1]	[57, 24]	simp [φ, deriv_div h1 h2 h3, h4, h5]	z u z₀ : ℂ U : Set ℂ inst✝ : good_domain U hu : u ∈ 𝔻 hz : z ∈ 𝔻 h1 : DifferentiableAt ℂ (fun z => z - u) z h2 : DifferentiableAt ℂ (fun z => 1 - z * (starRingEnd ℂ) u) z h3 : 1 - z * (starRingEnd ℂ) u ≠ 0 h4 : deriv (fun z => z - u) z = 1 h5 : deriv (fun z => 1 - z * (starRingEnd ℂ) u) z = -(starRingEnd ℂ) u ⊢ deriv (φ hu).to_fun z = (1 - u * (starRingEnd ℂ) u) / (1 - z * (starRingEnd ℂ) u) ^ 2	z u z₀ : ℂ U : Set ℂ inst✝ : good_domain U hu : u ∈ 𝔻 hz : z ∈ 𝔻 h1 : DifferentiableAt ℂ (fun z => z - u) z h2 : DifferentiableAt ℂ (fun z => 1 - z * (starRingEnd ℂ) u) z h3 : 1 - z * (starRingEnd ℂ) u ≠ 0 h4 : deriv (fun z => z - u) z = 1 h5 : deriv (fun z => 1 - z * (starRingEnd ℂ) u) z = -(starRingEnd ℂ) u ⊢ (1 - z * (starRingEnd ℂ) u + (z - u) * (starRingEnd ℂ) u) / (1 - z * (starRingEnd ℂ) u) ^ 2 = (1 - u * (starRingEnd ℂ) u) / (1 - z * (starRingEnd ℂ) u) ^ 2
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	RMT4/etape2.lean	φ_deriv	[50, 1]	[57, 24]	field_simp [h3]	z u z₀ : ℂ U : Set ℂ inst✝ : good_domain U hu : u ∈ 𝔻 hz : z ∈ 𝔻 h1 : DifferentiableAt ℂ (fun z => z - u) z h2 : DifferentiableAt ℂ (fun z => 1 - z * (starRingEnd ℂ) u) z h3 : 1 - z * (starRingEnd ℂ) u ≠ 0 h4 : deriv (fun z => z - u) z = 1 h5 : deriv (fun z => 1 - z * (starRingEnd ℂ) u) z = -(starRingEnd ℂ) u ⊢ (1 - z * (starRingEnd ℂ) u + (z - u) * (starRingEnd ℂ) u) / (1 - z * (starRingEnd ℂ) u) ^ 2 = (1 - u * (starRingEnd ℂ) u) / (1 - z * (starRingEnd ℂ) u) ^ 2	z u z₀ : ℂ U : Set ℂ inst✝ : good_domain U hu : u ∈ 𝔻 hz : z ∈ 𝔻 h1 : DifferentiableAt ℂ (fun z => z - u) z h2 : DifferentiableAt ℂ (fun z => 1 - z * (starRingEnd ℂ) u) z h3 : 1 - z * (starRingEnd ℂ) u ≠ 0 h4 : deriv (fun z => z - u) z = 1 h5 : deriv (fun z => 1 - z * (starRingEnd ℂ) u) z = -(starRingEnd ℂ) u ⊢ 1 - z * (starRingEnd ℂ) u + (z - u) * (starRingEnd ℂ) u = 1 - u * (starRingEnd ℂ) u
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	RMT4/etape2.lean	φ_deriv	[50, 1]	[57, 24]	ring	z u z₀ : ℂ U : Set ℂ inst✝ : good_domain U hu : u ∈ 𝔻 hz : z ∈ 𝔻 h1 : DifferentiableAt ℂ (fun z => z - u) z h2 : DifferentiableAt ℂ (fun z => 1 - z * (starRingEnd ℂ) u) z h3 : 1 - z * (starRingEnd ℂ) u ≠ 0 h4 : deriv (fun z => z - u) z = 1 h5 : deriv (fun z => 1 - z * (starRingEnd ℂ) u) z = -(starRingEnd ℂ) u ⊢ 1 - z * (starRingEnd ℂ) u + (z - u) * (starRingEnd ℂ) u = 1 - u * (starRingEnd ℂ) u	no goals
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	RMT4/etape2.lean	φ_deriv	[50, 1]	[57, 24]	simp	z u z₀ : ℂ U : Set ℂ inst✝ : good_domain U hu : u ∈ 𝔻 hz : z ∈ 𝔻 ⊢ DifferentiableAt ℂ (fun z => z - u) z	no goals
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	RMT4/etape2.lean	φ_deriv	[50, 1]	[57, 24]	simp [DifferentiableAt.mul_const]	z u z₀ : ℂ U : Set ℂ inst✝ : good_domain U hu : u ∈ 𝔻 hz : z ∈ 𝔻 h1 : DifferentiableAt ℂ (fun z => z - u) z ⊢ DifferentiableAt ℂ (fun z => 1 - z * (starRingEnd ℂ) u) z	no goals
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	RMT4/etape2.lean	φ_deriv	[50, 1]	[57, 24]	simp [deriv_sub_const]	z u z₀ : ℂ U : Set ℂ inst✝ : good_domain U hu : u ∈ 𝔻 hz : z ∈ 𝔻 h1 : DifferentiableAt ℂ (fun z => z - u) z h2 : DifferentiableAt ℂ (fun z => 1 - z * (starRingEnd ℂ) u) z h3 : 1 - z * (starRingEnd ℂ) u ≠ 0 ⊢ deriv (fun z => z - u) z = 1	no goals
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	RMT4/etape2.lean	φ_deriv	[50, 1]	[57, 24]	simp [deriv_const_sub]	z u z₀ : ℂ U : Set ℂ inst✝ : good_domain U hu : u ∈ 𝔻 hz : z ∈ 𝔻 h1 : DifferentiableAt ℂ (fun z => z - u) z h2 : DifferentiableAt ℂ (fun z => 1 - z * (starRingEnd ℂ) u) z h3 : 1 - z * (starRingEnd ℂ) u ≠ 0 h4 : deriv (fun z => z - u) z = 1 ⊢ deriv (fun z => 1 - z * (starRingEnd ℂ) u) z = -(starRingEnd ℂ) u	no goals
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	RMT4/etape2.lean	non_injective_schwarz	[62, 1]	[112, 37]	set u := f 0	z u z₀ : ℂ U : Set ℂ inst✝ : good_domain U f : ℂ → ℂ f_diff : DifferentiableOn ℂ f 𝔻 f_img : MapsTo f 𝔻 𝔻 f_noninj : ¬InjOn f 𝔻 ⊢ ‖deriv f 0‖ < 1	z u✝ z₀ : ℂ U : Set ℂ inst✝ : good_domain U f : ℂ → ℂ f_diff : DifferentiableOn ℂ f 𝔻 f_img : MapsTo f 𝔻 𝔻 f_noninj : ¬InjOn f 𝔻 u : ℂ := f 0 ⊢ ‖deriv f 0‖ < 1
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	RMT4/etape2.lean	non_injective_schwarz	[62, 1]	[112, 37]	have u_in_𝔻 : u ∈ 𝔻 := f_img (mem_ball_self zero_lt_one)	z u✝ z₀ : ℂ U : Set ℂ inst✝ : good_domain U f : ℂ → ℂ f_diff : DifferentiableOn ℂ f 𝔻 f_img : MapsTo f 𝔻 𝔻 f_noninj : ¬InjOn f 𝔻 u : ℂ := f 0 ⊢ ‖deriv f 0‖ < 1	z u✝ z₀ : ℂ U : Set ℂ inst✝ : good_domain U f : ℂ → ℂ f_diff : DifferentiableOn ℂ f 𝔻 f_img : MapsTo f 𝔻 𝔻 f_noninj : ¬InjOn f 𝔻 u : ℂ := f 0 u_in_𝔻 : u ∈ 𝔻 ⊢ ‖deriv f 0‖ < 1
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	RMT4/etape2.lean	non_injective_schwarz	[62, 1]	[112, 37]	let g := φ u_in_𝔻 ∘ f	z u✝ z₀ : ℂ U : Set ℂ inst✝ : good_domain U f : ℂ → ℂ f_diff : DifferentiableOn ℂ f 𝔻 f_img : MapsTo f 𝔻 𝔻 f_noninj : ¬InjOn f 𝔻 u : ℂ := f 0 u_in_𝔻 : u ∈ 𝔻 ⊢ ‖deriv f 0‖ < 1	z u✝ z₀ : ℂ U : Set ℂ inst✝ : good_domain U f : ℂ → ℂ f_diff : DifferentiableOn ℂ f 𝔻 f_img : MapsTo f 𝔻 𝔻 f_noninj : ¬InjOn f 𝔻 u : ℂ := f 0 u_in_𝔻 : u ∈ 𝔻 g : ℂ → ℂ := (φ u_in_𝔻).to_fun ∘ f ⊢ ‖deriv f 0‖ < 1
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	RMT4/etape2.lean	non_injective_schwarz	[62, 1]	[112, 37]	have g_diff : DifferentiableOn ℂ g 𝔻 := (φ u_in_𝔻).is_diff.comp f_diff f_img	z u✝ z₀ : ℂ U : Set ℂ inst✝ : good_domain U f : ℂ → ℂ f_diff : DifferentiableOn ℂ f 𝔻 f_img : MapsTo f 𝔻 𝔻 f_noninj : ¬InjOn f 𝔻 u : ℂ := f 0 u_in_𝔻 : u ∈ 𝔻 g : ℂ → ℂ := (φ u_in_𝔻).to_fun ∘ f ⊢ ‖deriv f 0‖ < 1	z u✝ z₀ : ℂ U : Set ℂ inst✝ : good_domain U f : ℂ → ℂ f_diff : DifferentiableOn ℂ f 𝔻 f_img : MapsTo f 𝔻 𝔻 f_noninj : ¬InjOn f 𝔻 u : ℂ := f 0 u_in_𝔻 : u ∈ 𝔻 g : ℂ → ℂ := (φ u_in_𝔻).to_fun ∘ f g_diff : DifferentiableOn ℂ g 𝔻 ⊢ ‖deriv f 0‖ < 1
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	RMT4/etape2.lean	non_injective_schwarz	[62, 1]	[112, 37]	have g_maps : MapsTo g 𝔻 𝔻 := (φ u_in_𝔻).maps_to.comp f_img	z u✝ z₀ : ℂ U : Set ℂ inst✝ : good_domain U f : ℂ → ℂ f_diff : DifferentiableOn ℂ f 𝔻 f_img : MapsTo f 𝔻 𝔻 f_noninj : ¬InjOn f 𝔻 u : ℂ := f 0 u_in_𝔻 : u ∈ 𝔻 g : ℂ → ℂ := (φ u_in_𝔻).to_fun ∘ f g_diff : DifferentiableOn ℂ g 𝔻 ⊢ ‖deriv f 0‖ < 1	z u✝ z₀ : ℂ U : Set ℂ inst✝ : good_domain U f : ℂ → ℂ f_diff : DifferentiableOn ℂ f 𝔻 f_img : MapsTo f 𝔻 𝔻 f_noninj : ¬InjOn f 𝔻 u : ℂ := f 0 u_in_𝔻 : u ∈ 𝔻 g : ℂ → ℂ := (φ u_in_𝔻).to_fun ∘ f g_diff : DifferentiableOn ℂ g 𝔻 g_maps : MapsTo g 𝔻 𝔻 ⊢ ‖deriv f 0‖ < 1
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	RMT4/etape2.lean	non_injective_schwarz	[62, 1]	[112, 37]	have g_0_eq_0 : g 0 = 0 := by simp [g, φ]	z u✝ z₀ : ℂ U : Set ℂ inst✝ : good_domain U f : ℂ → ℂ f_diff : DifferentiableOn ℂ f 𝔻 f_img : MapsTo f 𝔻 𝔻 f_noninj : ¬InjOn f 𝔻 u : ℂ := f 0 u_in_𝔻 : u ∈ 𝔻 g : ℂ → ℂ := (φ u_in_𝔻).to_fun ∘ f g_diff : DifferentiableOn ℂ g 𝔻 g_maps : MapsTo g 𝔻 𝔻 ⊢ ‖deriv f 0‖ < 1	z u✝ z₀ : ℂ U : Set ℂ inst✝ : good_domain U f : ℂ → ℂ f_diff : DifferentiableOn ℂ f 𝔻 f_img : MapsTo f 𝔻 𝔻 f_noninj : ¬InjOn f 𝔻 u : ℂ := f 0 u_in_𝔻 : u ∈ 𝔻 g : ℂ → ℂ := (φ u_in_𝔻).to_fun ∘ f g_diff : DifferentiableOn ℂ g 𝔻 g_maps : MapsTo g 𝔻 𝔻 g_0_eq_0 : g 0 = 0 ⊢ ‖deriv f 0‖ < 1
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	RMT4/etape2.lean	non_injective_schwarz	[62, 1]	[112, 37]	by_cases h : ‖deriv g 0‖ = 1	z u✝ z₀ : ℂ U : Set ℂ inst✝ : good_domain U f : ℂ → ℂ f_diff : DifferentiableOn ℂ f 𝔻 f_img : MapsTo f 𝔻 𝔻 f_noninj : ¬InjOn f 𝔻 u : ℂ := f 0 u_in_𝔻 : u ∈ 𝔻 g : ℂ → ℂ := (φ u_in_𝔻).to_fun ∘ f g_diff : DifferentiableOn ℂ g 𝔻 g_maps : MapsTo g 𝔻 𝔻 g_0_eq_0 : g 0 = 0 ⊢ ‖deriv f 0‖ < 1	case pos z u✝ z₀ : ℂ U : Set ℂ inst✝ : good_domain U f : ℂ → ℂ f_diff : DifferentiableOn ℂ f 𝔻 f_img : MapsTo f 𝔻 𝔻 f_noninj : ¬InjOn f 𝔻 u : ℂ := f 0 u_in_𝔻 : u ∈ 𝔻 g : ℂ → ℂ := (φ u_in_𝔻).to_fun ∘ f g_diff : DifferentiableOn ℂ g 𝔻 g_maps : MapsTo g 𝔻 𝔻 g_0_eq_0 : g 0 = 0 h : ‖deriv g 0‖ = 1 ⊢ ‖deriv f 0‖ < 1 case neg z u✝ z₀ : ℂ U : Set ℂ inst✝ : good_domain U f : ℂ → ℂ f_diff : DifferentiableOn ℂ f 𝔻 f_img : MapsTo f 𝔻 𝔻 f_noninj : ¬InjOn f 𝔻 u : ℂ := f 0 u_in_𝔻 : u ∈ 𝔻 g : ℂ → ℂ := (φ u_in_𝔻).to_fun ∘ f g_diff : DifferentiableOn ℂ g 𝔻 g_maps : MapsTo g 𝔻 𝔻 g_0_eq_0 : g 0 = 0 h : ¬‖deriv g 0‖ = 1 ⊢ ‖deriv f 0‖ < 1
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	RMT4/etape2.lean	non_injective_schwarz	[62, 1]	[112, 37]	case pos => have g_lin : EqOn g (λ (z : ℂ) => z • deriv g 0) (ball 0 1) := by have h2 : MapsTo g (ball 0 1) (ball (g 0) 1) := by rwa [g_0_eq_0] have h1 : Set.EqOn g (fun z => g 0 + (z - 0) • dslope g 0 0) (Metric.ball 0 1) := by apply affine_of_mapsTo_ball_of_exists_norm_dslope_eq_div g_diff h2 (mem_ball_self zero_lt_one) rwa [dslope_same, div_one] convert h1 using 1 ext1 z rw [g_0_eq_0, zero_add, sub_zero, dslope_same] have g'0_ne_0 : deriv g 0 ≠ 0 := λ h' => by simp [h'] at h have g_inj : InjOn g 𝔻 := λ x hx y hy => by rw [g_lin hx, g_lin hy] simp [g'0_ne_0] cases f_noninj (injOn_of_injOn_comp g_inj)	z u✝ z₀ : ℂ U : Set ℂ inst✝ : good_domain U f : ℂ → ℂ f_diff : DifferentiableOn ℂ f 𝔻 f_img : MapsTo f 𝔻 𝔻 f_noninj : ¬InjOn f 𝔻 u : ℂ := f 0 u_in_𝔻 : u ∈ 𝔻 g : ℂ → ℂ := (φ u_in_𝔻).to_fun ∘ f g_diff : DifferentiableOn ℂ g 𝔻 g_maps : MapsTo g 𝔻 𝔻 g_0_eq_0 : g 0 = 0 h : ‖deriv g 0‖ = 1 ⊢ ‖deriv f 0‖ < 1	no goals
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	RMT4/etape2.lean	non_injective_schwarz	[62, 1]	[112, 37]	case neg => rw [norm_eq_abs] at h have g'0_le_1 := abs_deriv_le_one_of_mapsTo_ball g_diff g_maps g_0_eq_0 zero_lt_one have g'0_lt_1 : abs (deriv g 0) < 1 := Ne.lt_of_le h g'0_le_1 have g'0_eq_mul : deriv g 0 = deriv (φ u_in_𝔻) u * deriv f 0 := deriv.comp 0 ((φ u_in_𝔻).is_diff.differentiableAt (isOpen_ball.mem_nhds u_in_𝔻)) (f_diff.differentiableAt (ball_mem_nhds _ zero_lt_one)) have e1 : 1 - (normSq u : ℂ) ≠ 0 := by simpa [normSq_eq_conj_mul_self, mul_comm] using one_sub_mul_conj_ne_zero u_in_𝔻 u_in_𝔻 have φ'u_u : deriv (φ u_in_𝔻) u = 1 / (1 - normSq u) := by set w := 1 - conj u * u with hw have : w ≠ 0 := by simpa [normSq_eq_conj_mul_self, mul_comm u] using e1 rw [φ_deriv u_in_𝔻 u_in_𝔻, normSq_eq_conj_mul_self, mul_comm u, ← hw] field_simp; ring have e2 : 0 ≤ normSq u := normSq_nonneg _ have e3 : normSq u < 1 := by rw [normSq_eq_abs] have : abs u < 1 := mem_𝔻_iff.mp u_in_𝔻 simp only [sq_lt_one_iff_abs_lt_one, Complex.abs_abs, this] simp [normSq_eq_abs, ← mem_𝔻_iff] simp only [φ'u_u, one_div] at g'0_eq_mul rw [eq_comm, inv_mul_eq_iff_eq_mul₀ e1] at g'0_eq_mul rw [← norm_eq_abs, g'0_eq_mul, norm_mul, mul_comm, ← one_mul (1 : ℝ)] refine mul_lt_mul g'0_lt_1 ?_ (norm_pos_iff.mpr e1) zero_le_one simp at e2 e3 ⊢ norm_cast rw [abs_sub_le_iff] refine ⟨?_, ?_⟩; repeat linarith	z u✝ z₀ : ℂ U : Set ℂ inst✝ : good_domain U f : ℂ → ℂ f_diff : DifferentiableOn ℂ f 𝔻 f_img : MapsTo f 𝔻 𝔻 f_noninj : ¬InjOn f 𝔻 u : ℂ := f 0 u_in_𝔻 : u ∈ 𝔻 g : ℂ → ℂ := (φ u_in_𝔻).to_fun ∘ f g_diff : DifferentiableOn ℂ g 𝔻 g_maps : MapsTo g 𝔻 𝔻 g_0_eq_0 : g 0 = 0 h : ¬‖deriv g 0‖ = 1 ⊢ ‖deriv f 0‖ < 1	no goals
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	RMT4/etape2.lean	non_injective_schwarz	[62, 1]	[112, 37]	simp [g, φ]	z u✝ z₀ : ℂ U : Set ℂ inst✝ : good_domain U f : ℂ → ℂ f_diff : DifferentiableOn ℂ f 𝔻 f_img : MapsTo f 𝔻 𝔻 f_noninj : ¬InjOn f 𝔻 u : ℂ := f 0 u_in_𝔻 : u ∈ 𝔻 g : ℂ → ℂ := (φ u_in_𝔻).to_fun ∘ f g_diff : DifferentiableOn ℂ g 𝔻 g_maps : MapsTo g 𝔻 𝔻 ⊢ g 0 = 0	no goals
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	RMT4/etape2.lean	non_injective_schwarz	[62, 1]	[112, 37]	have g_lin : EqOn g (λ (z : ℂ) => z • deriv g 0) (ball 0 1) := by have h2 : MapsTo g (ball 0 1) (ball (g 0) 1) := by rwa [g_0_eq_0] have h1 : Set.EqOn g (fun z => g 0 + (z - 0) • dslope g 0 0) (Metric.ball 0 1) := by apply affine_of_mapsTo_ball_of_exists_norm_dslope_eq_div g_diff h2 (mem_ball_self zero_lt_one) rwa [dslope_same, div_one] convert h1 using 1 ext1 z rw [g_0_eq_0, zero_add, sub_zero, dslope_same]	z u✝ z₀ : ℂ U : Set ℂ inst✝ : good_domain U f : ℂ → ℂ f_diff : DifferentiableOn ℂ f 𝔻 f_img : MapsTo f 𝔻 𝔻 f_noninj : ¬InjOn f 𝔻 u : ℂ := f 0 u_in_𝔻 : u ∈ 𝔻 g : ℂ → ℂ := (φ u_in_𝔻).to_fun ∘ f g_diff : DifferentiableOn ℂ g 𝔻 g_maps : MapsTo g 𝔻 𝔻 g_0_eq_0 : g 0 = 0 h : ‖deriv g 0‖ = 1 ⊢ ‖deriv f 0‖ < 1	z u✝ z₀ : ℂ U : Set ℂ inst✝ : good_domain U f : ℂ → ℂ f_diff : DifferentiableOn ℂ f 𝔻 f_img : MapsTo f 𝔻 𝔻 f_noninj : ¬InjOn f 𝔻 u : ℂ := f 0 u_in_𝔻 : u ∈ 𝔻 g : ℂ → ℂ := (φ u_in_𝔻).to_fun ∘ f g_diff : DifferentiableOn ℂ g 𝔻 g_maps : MapsTo g 𝔻 𝔻 g_0_eq_0 : g 0 = 0 h : ‖deriv g 0‖ = 1 g_lin : EqOn g (fun z => z • deriv g 0) (ball 0 1) ⊢ ‖deriv f 0‖ < 1
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	RMT4/etape2.lean	non_injective_schwarz	[62, 1]	[112, 37]	have g'0_ne_0 : deriv g 0 ≠ 0 := λ h' => by simp [h'] at h	z u✝ z₀ : ℂ U : Set ℂ inst✝ : good_domain U f : ℂ → ℂ f_diff : DifferentiableOn ℂ f 𝔻 f_img : MapsTo f 𝔻 𝔻 f_noninj : ¬InjOn f 𝔻 u : ℂ := f 0 u_in_𝔻 : u ∈ 𝔻 g : ℂ → ℂ := (φ u_in_𝔻).to_fun ∘ f g_diff : DifferentiableOn ℂ g 𝔻 g_maps : MapsTo g 𝔻 𝔻 g_0_eq_0 : g 0 = 0 h : ‖deriv g 0‖ = 1 g_lin : EqOn g (fun z => z • deriv g 0) (ball 0 1) ⊢ ‖deriv f 0‖ < 1	z u✝ z₀ : ℂ U : Set ℂ inst✝ : good_domain U f : ℂ → ℂ f_diff : DifferentiableOn ℂ f 𝔻 f_img : MapsTo f 𝔻 𝔻 f_noninj : ¬InjOn f 𝔻 u : ℂ := f 0 u_in_𝔻 : u ∈ 𝔻 g : ℂ → ℂ := (φ u_in_𝔻).to_fun ∘ f g_diff : DifferentiableOn ℂ g 𝔻 g_maps : MapsTo g 𝔻 𝔻 g_0_eq_0 : g 0 = 0 h : ‖deriv g 0‖ = 1 g_lin : EqOn g (fun z => z • deriv g 0) (ball 0 1) g'0_ne_0 : deriv g 0 ≠ 0 ⊢ ‖deriv f 0‖ < 1
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	RMT4/etape2.lean	non_injective_schwarz	[62, 1]	[112, 37]	have g_inj : InjOn g 𝔻 := λ x hx y hy => by rw [g_lin hx, g_lin hy] simp [g'0_ne_0]	z u✝ z₀ : ℂ U : Set ℂ inst✝ : good_domain U f : ℂ → ℂ f_diff : DifferentiableOn ℂ f 𝔻 f_img : MapsTo f 𝔻 𝔻 f_noninj : ¬InjOn f 𝔻 u : ℂ := f 0 u_in_𝔻 : u ∈ 𝔻 g : ℂ → ℂ := (φ u_in_𝔻).to_fun ∘ f g_diff : DifferentiableOn ℂ g 𝔻 g_maps : MapsTo g 𝔻 𝔻 g_0_eq_0 : g 0 = 0 h : ‖deriv g 0‖ = 1 g_lin : EqOn g (fun z => z • deriv g 0) (ball 0 1) g'0_ne_0 : deriv g 0 ≠ 0 ⊢ ‖deriv f 0‖ < 1	z u✝ z₀ : ℂ U : Set ℂ inst✝ : good_domain U f : ℂ → ℂ f_diff : DifferentiableOn ℂ f 𝔻 f_img : MapsTo f 𝔻 𝔻 f_noninj : ¬InjOn f 𝔻 u : ℂ := f 0 u_in_𝔻 : u ∈ 𝔻 g : ℂ → ℂ := (φ u_in_𝔻).to_fun ∘ f g_diff : DifferentiableOn ℂ g 𝔻 g_maps : MapsTo g 𝔻 𝔻 g_0_eq_0 : g 0 = 0 h : ‖deriv g 0‖ = 1 g_lin : EqOn g (fun z => z • deriv g 0) (ball 0 1) g'0_ne_0 : deriv g 0 ≠ 0 g_inj : InjOn g 𝔻 ⊢ ‖deriv f 0‖ < 1
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	RMT4/etape2.lean	non_injective_schwarz	[62, 1]	[112, 37]	cases f_noninj (injOn_of_injOn_comp g_inj)	z u✝ z₀ : ℂ U : Set ℂ inst✝ : good_domain U f : ℂ → ℂ f_diff : DifferentiableOn ℂ f 𝔻 f_img : MapsTo f 𝔻 𝔻 f_noninj : ¬InjOn f 𝔻 u : ℂ := f 0 u_in_𝔻 : u ∈ 𝔻 g : ℂ → ℂ := (φ u_in_𝔻).to_fun ∘ f g_diff : DifferentiableOn ℂ g 𝔻 g_maps : MapsTo g 𝔻 𝔻 g_0_eq_0 : g 0 = 0 h : ‖deriv g 0‖ = 1 g_lin : EqOn g (fun z => z • deriv g 0) (ball 0 1) g'0_ne_0 : deriv g 0 ≠ 0 g_inj : InjOn g 𝔻 ⊢ ‖deriv f 0‖ < 1	no goals
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	RMT4/etape2.lean	non_injective_schwarz	[62, 1]	[112, 37]	have h2 : MapsTo g (ball 0 1) (ball (g 0) 1) := by rwa [g_0_eq_0]	z u✝ z₀ : ℂ U : Set ℂ inst✝ : good_domain U f : ℂ → ℂ f_diff : DifferentiableOn ℂ f 𝔻 f_img : MapsTo f 𝔻 𝔻 f_noninj : ¬InjOn f 𝔻 u : ℂ := f 0 u_in_𝔻 : u ∈ 𝔻 g : ℂ → ℂ := (φ u_in_𝔻).to_fun ∘ f g_diff : DifferentiableOn ℂ g 𝔻 g_maps : MapsTo g 𝔻 𝔻 g_0_eq_0 : g 0 = 0 h : ‖deriv g 0‖ = 1 ⊢ EqOn g (fun z => z • deriv g 0) (ball 0 1)	z u✝ z₀ : ℂ U : Set ℂ inst✝ : good_domain U f : ℂ → ℂ f_diff : DifferentiableOn ℂ f 𝔻 f_img : MapsTo f 𝔻 𝔻 f_noninj : ¬InjOn f 𝔻 u : ℂ := f 0 u_in_𝔻 : u ∈ 𝔻 g : ℂ → ℂ := (φ u_in_𝔻).to_fun ∘ f g_diff : DifferentiableOn ℂ g 𝔻 g_maps : MapsTo g 𝔻 𝔻 g_0_eq_0 : g 0 = 0 h : ‖deriv g 0‖ = 1 h2 : MapsTo g (ball 0 1) (ball (g 0) 1) ⊢ EqOn g (fun z => z • deriv g 0) (ball 0 1)
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	RMT4/etape2.lean	non_injective_schwarz	[62, 1]	[112, 37]	have h1 : Set.EqOn g (fun z => g 0 + (z - 0) • dslope g 0 0) (Metric.ball 0 1) := by apply affine_of_mapsTo_ball_of_exists_norm_dslope_eq_div g_diff h2 (mem_ball_self zero_lt_one) rwa [dslope_same, div_one]	z u✝ z₀ : ℂ U : Set ℂ inst✝ : good_domain U f : ℂ → ℂ f_diff : DifferentiableOn ℂ f 𝔻 f_img : MapsTo f 𝔻 𝔻 f_noninj : ¬InjOn f 𝔻 u : ℂ := f 0 u_in_𝔻 : u ∈ 𝔻 g : ℂ → ℂ := (φ u_in_𝔻).to_fun ∘ f g_diff : DifferentiableOn ℂ g 𝔻 g_maps : MapsTo g 𝔻 𝔻 g_0_eq_0 : g 0 = 0 h : ‖deriv g 0‖ = 1 h2 : MapsTo g (ball 0 1) (ball (g 0) 1) ⊢ EqOn g (fun z => z • deriv g 0) (ball 0 1)	z u✝ z₀ : ℂ U : Set ℂ inst✝ : good_domain U f : ℂ → ℂ f_diff : DifferentiableOn ℂ f 𝔻 f_img : MapsTo f 𝔻 𝔻 f_noninj : ¬InjOn f 𝔻 u : ℂ := f 0 u_in_𝔻 : u ∈ 𝔻 g : ℂ → ℂ := (φ u_in_𝔻).to_fun ∘ f g_diff : DifferentiableOn ℂ g 𝔻 g_maps : MapsTo g 𝔻 𝔻 g_0_eq_0 : g 0 = 0 h : ‖deriv g 0‖ = 1 h2 : MapsTo g (ball 0 1) (ball (g 0) 1) h1 : EqOn g (fun z => g 0 + (z - 0) • dslope g 0 0) (ball 0 1) ⊢ EqOn g (fun z => z • deriv g 0) (ball 0 1)
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	RMT4/etape2.lean	non_injective_schwarz	[62, 1]	[112, 37]	convert h1 using 1	z u✝ z₀ : ℂ U : Set ℂ inst✝ : good_domain U f : ℂ → ℂ f_diff : DifferentiableOn ℂ f 𝔻 f_img : MapsTo f 𝔻 𝔻 f_noninj : ¬InjOn f 𝔻 u : ℂ := f 0 u_in_𝔻 : u ∈ 𝔻 g : ℂ → ℂ := (φ u_in_𝔻).to_fun ∘ f g_diff : DifferentiableOn ℂ g 𝔻 g_maps : MapsTo g 𝔻 𝔻 g_0_eq_0 : g 0 = 0 h : ‖deriv g 0‖ = 1 h2 : MapsTo g (ball 0 1) (ball (g 0) 1) h1 : EqOn g (fun z => g 0 + (z - 0) • dslope g 0 0) (ball 0 1) ⊢ EqOn g (fun z => z • deriv g 0) (ball 0 1)	case h.e'_4 z u✝ z₀ : ℂ U : Set ℂ inst✝ : good_domain U f : ℂ → ℂ f_diff : DifferentiableOn ℂ f 𝔻 f_img : MapsTo f 𝔻 𝔻 f_noninj : ¬InjOn f 𝔻 u : ℂ := f 0 u_in_𝔻 : u ∈ 𝔻 g : ℂ → ℂ := (φ u_in_𝔻).to_fun ∘ f g_diff : DifferentiableOn ℂ g 𝔻 g_maps : MapsTo g 𝔻 𝔻 g_0_eq_0 : g 0 = 0 h : ‖deriv g 0‖ = 1 h2 : MapsTo g (ball 0 1) (ball (g 0) 1) h1 : EqOn g (fun z => g 0 + (z - 0) • dslope g 0 0) (ball 0 1) ⊢ (fun z => z • deriv g 0) = fun z => g 0 + (z - 0) • dslope g 0 0
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	RMT4/etape2.lean	non_injective_schwarz	[62, 1]	[112, 37]	ext1 z	case h.e'_4 z u✝ z₀ : ℂ U : Set ℂ inst✝ : good_domain U f : ℂ → ℂ f_diff : DifferentiableOn ℂ f 𝔻 f_img : MapsTo f 𝔻 𝔻 f_noninj : ¬InjOn f 𝔻 u : ℂ := f 0 u_in_𝔻 : u ∈ 𝔻 g : ℂ → ℂ := (φ u_in_𝔻).to_fun ∘ f g_diff : DifferentiableOn ℂ g 𝔻 g_maps : MapsTo g 𝔻 𝔻 g_0_eq_0 : g 0 = 0 h : ‖deriv g 0‖ = 1 h2 : MapsTo g (ball 0 1) (ball (g 0) 1) h1 : EqOn g (fun z => g 0 + (z - 0) • dslope g 0 0) (ball 0 1) ⊢ (fun z => z • deriv g 0) = fun z => g 0 + (z - 0) • dslope g 0 0	case h.e'_4.h z✝ u✝ z₀ : ℂ U : Set ℂ inst✝ : good_domain U f : ℂ → ℂ f_diff : DifferentiableOn ℂ f 𝔻 f_img : MapsTo f 𝔻 𝔻 f_noninj : ¬InjOn f 𝔻 u : ℂ := f 0 u_in_𝔻 : u ∈ 𝔻 g : ℂ → ℂ := (φ u_in_𝔻).to_fun ∘ f g_diff : DifferentiableOn ℂ g 𝔻 g_maps : MapsTo g 𝔻 𝔻 g_0_eq_0 : g 0 = 0 h : ‖deriv g 0‖ = 1 h2 : MapsTo g (ball 0 1) (ball (g 0) 1) h1 : EqOn g (fun z => g 0 + (z - 0) • dslope g 0 0) (ball 0 1) z : ℂ ⊢ z • deriv g 0 = g 0 + (z - 0) • dslope g 0 0
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	RMT4/etape2.lean	non_injective_schwarz	[62, 1]	[112, 37]	rw [g_0_eq_0, zero_add, sub_zero, dslope_same]	case h.e'_4.h z✝ u✝ z₀ : ℂ U : Set ℂ inst✝ : good_domain U f : ℂ → ℂ f_diff : DifferentiableOn ℂ f 𝔻 f_img : MapsTo f 𝔻 𝔻 f_noninj : ¬InjOn f 𝔻 u : ℂ := f 0 u_in_𝔻 : u ∈ 𝔻 g : ℂ → ℂ := (φ u_in_𝔻).to_fun ∘ f g_diff : DifferentiableOn ℂ g 𝔻 g_maps : MapsTo g 𝔻 𝔻 g_0_eq_0 : g 0 = 0 h : ‖deriv g 0‖ = 1 h2 : MapsTo g (ball 0 1) (ball (g 0) 1) h1 : EqOn g (fun z => g 0 + (z - 0) • dslope g 0 0) (ball 0 1) z : ℂ ⊢ z • deriv g 0 = g 0 + (z - 0) • dslope g 0 0	no goals
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	RMT4/etape2.lean	non_injective_schwarz	[62, 1]	[112, 37]	rwa [g_0_eq_0]	z u✝ z₀ : ℂ U : Set ℂ inst✝ : good_domain U f : ℂ → ℂ f_diff : DifferentiableOn ℂ f 𝔻 f_img : MapsTo f 𝔻 𝔻 f_noninj : ¬InjOn f 𝔻 u : ℂ := f 0 u_in_𝔻 : u ∈ 𝔻 g : ℂ → ℂ := (φ u_in_𝔻).to_fun ∘ f g_diff : DifferentiableOn ℂ g 𝔻 g_maps : MapsTo g 𝔻 𝔻 g_0_eq_0 : g 0 = 0 h : ‖deriv g 0‖ = 1 ⊢ MapsTo g (ball 0 1) (ball (g 0) 1)	no goals
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	RMT4/etape2.lean	non_injective_schwarz	[62, 1]	[112, 37]	apply affine_of_mapsTo_ball_of_exists_norm_dslope_eq_div g_diff h2 (mem_ball_self zero_lt_one)	z u✝ z₀ : ℂ U : Set ℂ inst✝ : good_domain U f : ℂ → ℂ f_diff : DifferentiableOn ℂ f 𝔻 f_img : MapsTo f 𝔻 𝔻 f_noninj : ¬InjOn f 𝔻 u : ℂ := f 0 u_in_𝔻 : u ∈ 𝔻 g : ℂ → ℂ := (φ u_in_𝔻).to_fun ∘ f g_diff : DifferentiableOn ℂ g 𝔻 g_maps : MapsTo g 𝔻 𝔻 g_0_eq_0 : g 0 = 0 h : ‖deriv g 0‖ = 1 h2 : MapsTo g (ball 0 1) (ball (g 0) 1) ⊢ EqOn g (fun z => g 0 + (z - 0) • dslope g 0 0) (ball 0 1)	z u✝ z₀ : ℂ U : Set ℂ inst✝ : good_domain U f : ℂ → ℂ f_diff : DifferentiableOn ℂ f 𝔻 f_img : MapsTo f 𝔻 𝔻 f_noninj : ¬InjOn f 𝔻 u : ℂ := f 0 u_in_𝔻 : u ∈ 𝔻 g : ℂ → ℂ := (φ u_in_𝔻).to_fun ∘ f g_diff : DifferentiableOn ℂ g 𝔻 g_maps : MapsTo g 𝔻 𝔻 g_0_eq_0 : g 0 = 0 h : ‖deriv g 0‖ = 1 h2 : MapsTo g (ball 0 1) (ball (g 0) 1) ⊢ ‖dslope g 0 0‖ = 1 / 1
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	RMT4/etape2.lean	non_injective_schwarz	[62, 1]	[112, 37]	rwa [dslope_same, div_one]	z u✝ z₀ : ℂ U : Set ℂ inst✝ : good_domain U f : ℂ → ℂ f_diff : DifferentiableOn ℂ f 𝔻 f_img : MapsTo f 𝔻 𝔻 f_noninj : ¬InjOn f 𝔻 u : ℂ := f 0 u_in_𝔻 : u ∈ 𝔻 g : ℂ → ℂ := (φ u_in_𝔻).to_fun ∘ f g_diff : DifferentiableOn ℂ g 𝔻 g_maps : MapsTo g 𝔻 𝔻 g_0_eq_0 : g 0 = 0 h : ‖deriv g 0‖ = 1 h2 : MapsTo g (ball 0 1) (ball (g 0) 1) ⊢ ‖dslope g 0 0‖ = 1 / 1	no goals
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	RMT4/etape2.lean	non_injective_schwarz	[62, 1]	[112, 37]	simp [h'] at h	z u✝ z₀ : ℂ U : Set ℂ inst✝ : good_domain U f : ℂ → ℂ f_diff : DifferentiableOn ℂ f 𝔻 f_img : MapsTo f 𝔻 𝔻 f_noninj : ¬InjOn f 𝔻 u : ℂ := f 0 u_in_𝔻 : u ∈ 𝔻 g : ℂ → ℂ := (φ u_in_𝔻).to_fun ∘ f g_diff : DifferentiableOn ℂ g 𝔻 g_maps : MapsTo g 𝔻 𝔻 g_0_eq_0 : g 0 = 0 h : ‖deriv g 0‖ = 1 g_lin : EqOn g (fun z => z • deriv g 0) (ball 0 1) h' : deriv g 0 = 0 ⊢ False	no goals
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	RMT4/etape2.lean	non_injective_schwarz	[62, 1]	[112, 37]	rw [g_lin hx, g_lin hy]	z u✝ z₀ : ℂ U : Set ℂ inst✝ : good_domain U f : ℂ → ℂ f_diff : DifferentiableOn ℂ f 𝔻 f_img : MapsTo f 𝔻 𝔻 f_noninj : ¬InjOn f 𝔻 u : ℂ := f 0 u_in_𝔻 : u ∈ 𝔻 g : ℂ → ℂ := (φ u_in_𝔻).to_fun ∘ f g_diff : DifferentiableOn ℂ g 𝔻 g_maps : MapsTo g 𝔻 𝔻 g_0_eq_0 : g 0 = 0 h : ‖deriv g 0‖ = 1 g_lin : EqOn g (fun z => z • deriv g 0) (ball 0 1) g'0_ne_0 : deriv g 0 ≠ 0 x : ℂ hx : x ∈ 𝔻 y : ℂ hy : y ∈ 𝔻 ⊢ g x = g y → x = y	z u✝ z₀ : ℂ U : Set ℂ inst✝ : good_domain U f : ℂ → ℂ f_diff : DifferentiableOn ℂ f 𝔻 f_img : MapsTo f 𝔻 𝔻 f_noninj : ¬InjOn f 𝔻 u : ℂ := f 0 u_in_𝔻 : u ∈ 𝔻 g : ℂ → ℂ := (φ u_in_𝔻).to_fun ∘ f g_diff : DifferentiableOn ℂ g 𝔻 g_maps : MapsTo g 𝔻 𝔻 g_0_eq_0 : g 0 = 0 h : ‖deriv g 0‖ = 1 g_lin : EqOn g (fun z => z • deriv g 0) (ball 0 1) g'0_ne_0 : deriv g 0 ≠ 0 x : ℂ hx : x ∈ 𝔻 y : ℂ hy : y ∈ 𝔻 ⊢ (fun z => z • deriv g 0) x = (fun z => z • deriv g 0) y → x = y
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	RMT4/etape2.lean	non_injective_schwarz	[62, 1]	[112, 37]	simp [g'0_ne_0]	z u✝ z₀ : ℂ U : Set ℂ inst✝ : good_domain U f : ℂ → ℂ f_diff : DifferentiableOn ℂ f 𝔻 f_img : MapsTo f 𝔻 𝔻 f_noninj : ¬InjOn f 𝔻 u : ℂ := f 0 u_in_𝔻 : u ∈ 𝔻 g : ℂ → ℂ := (φ u_in_𝔻).to_fun ∘ f g_diff : DifferentiableOn ℂ g 𝔻 g_maps : MapsTo g 𝔻 𝔻 g_0_eq_0 : g 0 = 0 h : ‖deriv g 0‖ = 1 g_lin : EqOn g (fun z => z • deriv g 0) (ball 0 1) g'0_ne_0 : deriv g 0 ≠ 0 x : ℂ hx : x ∈ 𝔻 y : ℂ hy : y ∈ 𝔻 ⊢ (fun z => z • deriv g 0) x = (fun z => z • deriv g 0) y → x = y	no goals
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	RMT4/etape2.lean	non_injective_schwarz	[62, 1]	[112, 37]	rw [norm_eq_abs] at h	z u✝ z₀ : ℂ U : Set ℂ inst✝ : good_domain U f : ℂ → ℂ f_diff : DifferentiableOn ℂ f 𝔻 f_img : MapsTo f 𝔻 𝔻 f_noninj : ¬InjOn f 𝔻 u : ℂ := f 0 u_in_𝔻 : u ∈ 𝔻 g : ℂ → ℂ := (φ u_in_𝔻).to_fun ∘ f g_diff : DifferentiableOn ℂ g 𝔻 g_maps : MapsTo g 𝔻 𝔻 g_0_eq_0 : g 0 = 0 h : ¬‖deriv g 0‖ = 1 ⊢ ‖deriv f 0‖ < 1	z u✝ z₀ : ℂ U : Set ℂ inst✝ : good_domain U f : ℂ → ℂ f_diff : DifferentiableOn ℂ f 𝔻 f_img : MapsTo f 𝔻 𝔻 f_noninj : ¬InjOn f 𝔻 u : ℂ := f 0 u_in_𝔻 : u ∈ 𝔻 g : ℂ → ℂ := (φ u_in_𝔻).to_fun ∘ f g_diff : DifferentiableOn ℂ g 𝔻 g_maps : MapsTo g 𝔻 𝔻 g_0_eq_0 : g 0 = 0 h : ¬Complex.abs (deriv g 0) = 1 ⊢ ‖deriv f 0‖ < 1
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	RMT4/etape2.lean	non_injective_schwarz	[62, 1]	[112, 37]	have g'0_le_1 := abs_deriv_le_one_of_mapsTo_ball g_diff g_maps g_0_eq_0 zero_lt_one	z u✝ z₀ : ℂ U : Set ℂ inst✝ : good_domain U f : ℂ → ℂ f_diff : DifferentiableOn ℂ f 𝔻 f_img : MapsTo f 𝔻 𝔻 f_noninj : ¬InjOn f 𝔻 u : ℂ := f 0 u_in_𝔻 : u ∈ 𝔻 g : ℂ → ℂ := (φ u_in_𝔻).to_fun ∘ f g_diff : DifferentiableOn ℂ g 𝔻 g_maps : MapsTo g 𝔻 𝔻 g_0_eq_0 : g 0 = 0 h : ¬Complex.abs (deriv g 0) = 1 ⊢ ‖deriv f 0‖ < 1	z u✝ z₀ : ℂ U : Set ℂ inst✝ : good_domain U f : ℂ → ℂ f_diff : DifferentiableOn ℂ f 𝔻 f_img : MapsTo f 𝔻 𝔻 f_noninj : ¬InjOn f 𝔻 u : ℂ := f 0 u_in_𝔻 : u ∈ 𝔻 g : ℂ → ℂ := (φ u_in_𝔻).to_fun ∘ f g_diff : DifferentiableOn ℂ g 𝔻 g_maps : MapsTo g 𝔻 𝔻 g_0_eq_0 : g 0 = 0 h : ¬Complex.abs (deriv g 0) = 1 g'0_le_1 : Complex.abs (deriv g 0) ≤ 1 ⊢ ‖deriv f 0‖ < 1
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	RMT4/etape2.lean	non_injective_schwarz	[62, 1]	[112, 37]	have g'0_lt_1 : abs (deriv g 0) < 1 := Ne.lt_of_le h g'0_le_1	z u✝ z₀ : ℂ U : Set ℂ inst✝ : good_domain U f : ℂ → ℂ f_diff : DifferentiableOn ℂ f 𝔻 f_img : MapsTo f 𝔻 𝔻 f_noninj : ¬InjOn f 𝔻 u : ℂ := f 0 u_in_𝔻 : u ∈ 𝔻 g : ℂ → ℂ := (φ u_in_𝔻).to_fun ∘ f g_diff : DifferentiableOn ℂ g 𝔻 g_maps : MapsTo g 𝔻 𝔻 g_0_eq_0 : g 0 = 0 h : ¬Complex.abs (deriv g 0) = 1 g'0_le_1 : Complex.abs (deriv g 0) ≤ 1 ⊢ ‖deriv f 0‖ < 1	z u✝ z₀ : ℂ U : Set ℂ inst✝ : good_domain U f : ℂ → ℂ f_diff : DifferentiableOn ℂ f 𝔻 f_img : MapsTo f 𝔻 𝔻 f_noninj : ¬InjOn f 𝔻 u : ℂ := f 0 u_in_𝔻 : u ∈ 𝔻 g : ℂ → ℂ := (φ u_in_𝔻).to_fun ∘ f g_diff : DifferentiableOn ℂ g 𝔻 g_maps : MapsTo g 𝔻 𝔻 g_0_eq_0 : g 0 = 0 h : ¬Complex.abs (deriv g 0) = 1 g'0_le_1 : Complex.abs (deriv g 0) ≤ 1 g'0_lt_1 : Complex.abs (deriv g 0) < 1 ⊢ ‖deriv f 0‖ < 1
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	RMT4/etape2.lean	non_injective_schwarz	[62, 1]	[112, 37]	have g'0_eq_mul : deriv g 0 = deriv (φ u_in_𝔻) u * deriv f 0 := deriv.comp 0 ((φ u_in_𝔻).is_diff.differentiableAt (isOpen_ball.mem_nhds u_in_𝔻)) (f_diff.differentiableAt (ball_mem_nhds _ zero_lt_one))	z u✝ z₀ : ℂ U : Set ℂ inst✝ : good_domain U f : ℂ → ℂ f_diff : DifferentiableOn ℂ f 𝔻 f_img : MapsTo f 𝔻 𝔻 f_noninj : ¬InjOn f 𝔻 u : ℂ := f 0 u_in_𝔻 : u ∈ 𝔻 g : ℂ → ℂ := (φ u_in_𝔻).to_fun ∘ f g_diff : DifferentiableOn ℂ g 𝔻 g_maps : MapsTo g 𝔻 𝔻 g_0_eq_0 : g 0 = 0 h : ¬Complex.abs (deriv g 0) = 1 g'0_le_1 : Complex.abs (deriv g 0) ≤ 1 g'0_lt_1 : Complex.abs (deriv g 0) < 1 ⊢ ‖deriv f 0‖ < 1	z u✝ z₀ : ℂ U : Set ℂ inst✝ : good_domain U f : ℂ → ℂ f_diff : DifferentiableOn ℂ f 𝔻 f_img : MapsTo f 𝔻 𝔻 f_noninj : ¬InjOn f 𝔻 u : ℂ := f 0 u_in_𝔻 : u ∈ 𝔻 g : ℂ → ℂ := (φ u_in_𝔻).to_fun ∘ f g_diff : DifferentiableOn ℂ g 𝔻 g_maps : MapsTo g 𝔻 𝔻 g_0_eq_0 : g 0 = 0 h : ¬Complex.abs (deriv g 0) = 1 g'0_le_1 : Complex.abs (deriv g 0) ≤ 1 g'0_lt_1 : Complex.abs (deriv g 0) < 1 g'0_eq_mul : deriv g 0 = deriv (φ u_in_𝔻).to_fun u * deriv f 0 ⊢ ‖deriv f 0‖ < 1
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	RMT4/etape2.lean	non_injective_schwarz	[62, 1]	[112, 37]	have e1 : 1 - (normSq u : ℂ) ≠ 0 := by simpa [normSq_eq_conj_mul_self, mul_comm] using one_sub_mul_conj_ne_zero u_in_𝔻 u_in_𝔻	z u✝ z₀ : ℂ U : Set ℂ inst✝ : good_domain U f : ℂ → ℂ f_diff : DifferentiableOn ℂ f 𝔻 f_img : MapsTo f 𝔻 𝔻 f_noninj : ¬InjOn f 𝔻 u : ℂ := f 0 u_in_𝔻 : u ∈ 𝔻 g : ℂ → ℂ := (φ u_in_𝔻).to_fun ∘ f g_diff : DifferentiableOn ℂ g 𝔻 g_maps : MapsTo g 𝔻 𝔻 g_0_eq_0 : g 0 = 0 h : ¬Complex.abs (deriv g 0) = 1 g'0_le_1 : Complex.abs (deriv g 0) ≤ 1 g'0_lt_1 : Complex.abs (deriv g 0) < 1 g'0_eq_mul : deriv g 0 = deriv (φ u_in_𝔻).to_fun u * deriv f 0 ⊢ ‖deriv f 0‖ < 1	z u✝ z₀ : ℂ U : Set ℂ inst✝ : good_domain U f : ℂ → ℂ f_diff : DifferentiableOn ℂ f 𝔻 f_img : MapsTo f 𝔻 𝔻 f_noninj : ¬InjOn f 𝔻 u : ℂ := f 0 u_in_𝔻 : u ∈ 𝔻 g : ℂ → ℂ := (φ u_in_𝔻).to_fun ∘ f g_diff : DifferentiableOn ℂ g 𝔻 g_maps : MapsTo g 𝔻 𝔻 g_0_eq_0 : g 0 = 0 h : ¬Complex.abs (deriv g 0) = 1 g'0_le_1 : Complex.abs (deriv g 0) ≤ 1 g'0_lt_1 : Complex.abs (deriv g 0) < 1 g'0_eq_mul : deriv g 0 = deriv (φ u_in_𝔻).to_fun u * deriv f 0 e1 : 1 - ↑(normSq u) ≠ 0 ⊢ ‖deriv f 0‖ < 1
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	RMT4/etape2.lean	non_injective_schwarz	[62, 1]	[112, 37]	have φ'u_u : deriv (φ u_in_𝔻) u = 1 / (1 - normSq u) := by set w := 1 - conj u * u with hw have : w ≠ 0 := by simpa [normSq_eq_conj_mul_self, mul_comm u] using e1 rw [φ_deriv u_in_𝔻 u_in_𝔻, normSq_eq_conj_mul_self, mul_comm u, ← hw] field_simp; ring	z u✝ z₀ : ℂ U : Set ℂ inst✝ : good_domain U f : ℂ → ℂ f_diff : DifferentiableOn ℂ f 𝔻 f_img : MapsTo f 𝔻 𝔻 f_noninj : ¬InjOn f 𝔻 u : ℂ := f 0 u_in_𝔻 : u ∈ 𝔻 g : ℂ → ℂ := (φ u_in_𝔻).to_fun ∘ f g_diff : DifferentiableOn ℂ g 𝔻 g_maps : MapsTo g 𝔻 𝔻 g_0_eq_0 : g 0 = 0 h : ¬Complex.abs (deriv g 0) = 1 g'0_le_1 : Complex.abs (deriv g 0) ≤ 1 g'0_lt_1 : Complex.abs (deriv g 0) < 1 g'0_eq_mul : deriv g 0 = deriv (φ u_in_𝔻).to_fun u * deriv f 0 e1 : 1 - ↑(normSq u) ≠ 0 ⊢ ‖deriv f 0‖ < 1	z u✝ z₀ : ℂ U : Set ℂ inst✝ : good_domain U f : ℂ → ℂ f_diff : DifferentiableOn ℂ f 𝔻 f_img : MapsTo f 𝔻 𝔻 f_noninj : ¬InjOn f 𝔻 u : ℂ := f 0 u_in_𝔻 : u ∈ 𝔻 g : ℂ → ℂ := (φ u_in_𝔻).to_fun ∘ f g_diff : DifferentiableOn ℂ g 𝔻 g_maps : MapsTo g 𝔻 𝔻 g_0_eq_0 : g 0 = 0 h : ¬Complex.abs (deriv g 0) = 1 g'0_le_1 : Complex.abs (deriv g 0) ≤ 1 g'0_lt_1 : Complex.abs (deriv g 0) < 1 g'0_eq_mul : deriv g 0 = deriv (φ u_in_𝔻).to_fun u * deriv f 0 e1 : 1 - ↑(normSq u) ≠ 0 φ'u_u : deriv (φ u_in_𝔻).to_fun u = 1 / (1 - ↑(normSq u)) ⊢ ‖deriv f 0‖ < 1
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	RMT4/etape2.lean	non_injective_schwarz	[62, 1]	[112, 37]	have e2 : 0 ≤ normSq u := normSq_nonneg _	z u✝ z₀ : ℂ U : Set ℂ inst✝ : good_domain U f : ℂ → ℂ f_diff : DifferentiableOn ℂ f 𝔻 f_img : MapsTo f 𝔻 𝔻 f_noninj : ¬InjOn f 𝔻 u : ℂ := f 0 u_in_𝔻 : u ∈ 𝔻 g : ℂ → ℂ := (φ u_in_𝔻).to_fun ∘ f g_diff : DifferentiableOn ℂ g 𝔻 g_maps : MapsTo g 𝔻 𝔻 g_0_eq_0 : g 0 = 0 h : ¬Complex.abs (deriv g 0) = 1 g'0_le_1 : Complex.abs (deriv g 0) ≤ 1 g'0_lt_1 : Complex.abs (deriv g 0) < 1 g'0_eq_mul : deriv g 0 = deriv (φ u_in_𝔻).to_fun u * deriv f 0 e1 : 1 - ↑(normSq u) ≠ 0 φ'u_u : deriv (φ u_in_𝔻).to_fun u = 1 / (1 - ↑(normSq u)) ⊢ ‖deriv f 0‖ < 1	z u✝ z₀ : ℂ U : Set ℂ inst✝ : good_domain U f : ℂ → ℂ f_diff : DifferentiableOn ℂ f 𝔻 f_img : MapsTo f 𝔻 𝔻 f_noninj : ¬InjOn f 𝔻 u : ℂ := f 0 u_in_𝔻 : u ∈ 𝔻 g : ℂ → ℂ := (φ u_in_𝔻).to_fun ∘ f g_diff : DifferentiableOn ℂ g 𝔻 g_maps : MapsTo g 𝔻 𝔻 g_0_eq_0 : g 0 = 0 h : ¬Complex.abs (deriv g 0) = 1 g'0_le_1 : Complex.abs (deriv g 0) ≤ 1 g'0_lt_1 : Complex.abs (deriv g 0) < 1 g'0_eq_mul : deriv g 0 = deriv (φ u_in_𝔻).to_fun u * deriv f 0 e1 : 1 - ↑(normSq u) ≠ 0 φ'u_u : deriv (φ u_in_𝔻).to_fun u = 1 / (1 - ↑(normSq u)) e2 : 0 ≤ normSq u ⊢ ‖deriv f 0‖ < 1
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	RMT4/etape2.lean	non_injective_schwarz	[62, 1]	[112, 37]	have e3 : normSq u < 1 := by rw [normSq_eq_abs] have : abs u < 1 := mem_𝔻_iff.mp u_in_𝔻 simp only [sq_lt_one_iff_abs_lt_one, Complex.abs_abs, this]	z u✝ z₀ : ℂ U : Set ℂ inst✝ : good_domain U f : ℂ → ℂ f_diff : DifferentiableOn ℂ f 𝔻 f_img : MapsTo f 𝔻 𝔻 f_noninj : ¬InjOn f 𝔻 u : ℂ := f 0 u_in_𝔻 : u ∈ 𝔻 g : ℂ → ℂ := (φ u_in_𝔻).to_fun ∘ f g_diff : DifferentiableOn ℂ g 𝔻 g_maps : MapsTo g 𝔻 𝔻 g_0_eq_0 : g 0 = 0 h : ¬Complex.abs (deriv g 0) = 1 g'0_le_1 : Complex.abs (deriv g 0) ≤ 1 g'0_lt_1 : Complex.abs (deriv g 0) < 1 g'0_eq_mul : deriv g 0 = deriv (φ u_in_𝔻).to_fun u * deriv f 0 e1 : 1 - ↑(normSq u) ≠ 0 φ'u_u : deriv (φ u_in_𝔻).to_fun u = 1 / (1 - ↑(normSq u)) e2 : 0 ≤ normSq u ⊢ ‖deriv f 0‖ < 1	z u✝ z₀ : ℂ U : Set ℂ inst✝ : good_domain U f : ℂ → ℂ f_diff : DifferentiableOn ℂ f 𝔻 f_img : MapsTo f 𝔻 𝔻 f_noninj : ¬InjOn f 𝔻 u : ℂ := f 0 u_in_𝔻 : u ∈ 𝔻 g : ℂ → ℂ := (φ u_in_𝔻).to_fun ∘ f g_diff : DifferentiableOn ℂ g 𝔻 g_maps : MapsTo g 𝔻 𝔻 g_0_eq_0 : g 0 = 0 h : ¬Complex.abs (deriv g 0) = 1 g'0_le_1 : Complex.abs (deriv g 0) ≤ 1 g'0_lt_1 : Complex.abs (deriv g 0) < 1 g'0_eq_mul : deriv g 0 = deriv (φ u_in_𝔻).to_fun u * deriv f 0 e1 : 1 - ↑(normSq u) ≠ 0 φ'u_u : deriv (φ u_in_𝔻).to_fun u = 1 / (1 - ↑(normSq u)) e2 : 0 ≤ normSq u e3 : normSq u < 1 ⊢ ‖deriv f 0‖ < 1
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	RMT4/etape2.lean	non_injective_schwarz	[62, 1]	[112, 37]	simp [normSq_eq_abs, ← mem_𝔻_iff]	z u✝ z₀ : ℂ U : Set ℂ inst✝ : good_domain U f : ℂ → ℂ f_diff : DifferentiableOn ℂ f 𝔻 f_img : MapsTo f 𝔻 𝔻 f_noninj : ¬InjOn f 𝔻 u : ℂ := f 0 u_in_𝔻 : u ∈ 𝔻 g : ℂ → ℂ := (φ u_in_𝔻).to_fun ∘ f g_diff : DifferentiableOn ℂ g 𝔻 g_maps : MapsTo g 𝔻 𝔻 g_0_eq_0 : g 0 = 0 h : ¬Complex.abs (deriv g 0) = 1 g'0_le_1 : Complex.abs (deriv g 0) ≤ 1 g'0_lt_1 : Complex.abs (deriv g 0) < 1 g'0_eq_mul : deriv g 0 = deriv (φ u_in_𝔻).to_fun u * deriv f 0 e1 : 1 - ↑(normSq u) ≠ 0 φ'u_u : deriv (φ u_in_𝔻).to_fun u = 1 / (1 - ↑(normSq u)) e2 : 0 ≤ normSq u e3 : normSq u < 1 ⊢ ‖deriv f 0‖ < 1	z u✝ z₀ : ℂ U : Set ℂ inst✝ : good_domain U f : ℂ → ℂ f_diff : DifferentiableOn ℂ f 𝔻 f_img : MapsTo f 𝔻 𝔻 f_noninj : ¬InjOn f 𝔻 u : ℂ := f 0 u_in_𝔻 : u ∈ 𝔻 g : ℂ → ℂ := (φ u_in_𝔻).to_fun ∘ f g_diff : DifferentiableOn ℂ g 𝔻 g_maps : MapsTo g 𝔻 𝔻 g_0_eq_0 : g 0 = 0 h : ¬Complex.abs (deriv g 0) = 1 g'0_le_1 : Complex.abs (deriv g 0) ≤ 1 g'0_lt_1 : Complex.abs (deriv g 0) < 1 g'0_eq_mul : deriv g 0 = deriv (φ u_in_𝔻).to_fun u * deriv f 0 e1 : 1 - ↑(normSq u) ≠ 0 φ'u_u : deriv (φ u_in_𝔻).to_fun u = 1 / (1 - ↑(normSq u)) e2 : 0 ≤ normSq u e3 : normSq u < 1 ⊢ Complex.abs (deriv f 0) < 1
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	RMT4/etape2.lean	non_injective_schwarz	[62, 1]	[112, 37]	simp only [φ'u_u, one_div] at g'0_eq_mul	z u✝ z₀ : ℂ U : Set ℂ inst✝ : good_domain U f : ℂ → ℂ f_diff : DifferentiableOn ℂ f 𝔻 f_img : MapsTo f 𝔻 𝔻 f_noninj : ¬InjOn f 𝔻 u : ℂ := f 0 u_in_𝔻 : u ∈ 𝔻 g : ℂ → ℂ := (φ u_in_𝔻).to_fun ∘ f g_diff : DifferentiableOn ℂ g 𝔻 g_maps : MapsTo g 𝔻 𝔻 g_0_eq_0 : g 0 = 0 h : ¬Complex.abs (deriv g 0) = 1 g'0_le_1 : Complex.abs (deriv g 0) ≤ 1 g'0_lt_1 : Complex.abs (deriv g 0) < 1 g'0_eq_mul : deriv g 0 = deriv (φ u_in_𝔻).to_fun u * deriv f 0 e1 : 1 - ↑(normSq u) ≠ 0 φ'u_u : deriv (φ u_in_𝔻).to_fun u = 1 / (1 - ↑(normSq u)) e2 : 0 ≤ normSq u e3 : normSq u < 1 ⊢ Complex.abs (deriv f 0) < 1	z u✝ z₀ : ℂ U : Set ℂ inst✝ : good_domain U f : ℂ → ℂ f_diff : DifferentiableOn ℂ f 𝔻 f_img : MapsTo f 𝔻 𝔻 f_noninj : ¬InjOn f 𝔻 u : ℂ := f 0 u_in_𝔻 : u ∈ 𝔻 g : ℂ → ℂ := (φ u_in_𝔻).to_fun ∘ f g_diff : DifferentiableOn ℂ g 𝔻 g_maps : MapsTo g 𝔻 𝔻 g_0_eq_0 : g 0 = 0 h : ¬Complex.abs (deriv g 0) = 1 g'0_le_1 : Complex.abs (deriv g 0) ≤ 1 g'0_lt_1 : Complex.abs (deriv g 0) < 1 e1 : 1 - ↑(normSq u) ≠ 0 φ'u_u : deriv (φ u_in_𝔻).to_fun u = 1 / (1 - ↑(normSq u)) e2 : 0 ≤ normSq u e3 : normSq u < 1 g'0_eq_mul : deriv g 0 = (1 - ↑(normSq u))⁻¹ * deriv f 0 ⊢ Complex.abs (deriv f 0) < 1
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	RMT4/etape2.lean	non_injective_schwarz	[62, 1]	[112, 37]	rw [eq_comm, inv_mul_eq_iff_eq_mul₀ e1] at g'0_eq_mul	z u✝ z₀ : ℂ U : Set ℂ inst✝ : good_domain U f : ℂ → ℂ f_diff : DifferentiableOn ℂ f 𝔻 f_img : MapsTo f 𝔻 𝔻 f_noninj : ¬InjOn f 𝔻 u : ℂ := f 0 u_in_𝔻 : u ∈ 𝔻 g : ℂ → ℂ := (φ u_in_𝔻).to_fun ∘ f g_diff : DifferentiableOn ℂ g 𝔻 g_maps : MapsTo g 𝔻 𝔻 g_0_eq_0 : g 0 = 0 h : ¬Complex.abs (deriv g 0) = 1 g'0_le_1 : Complex.abs (deriv g 0) ≤ 1 g'0_lt_1 : Complex.abs (deriv g 0) < 1 e1 : 1 - ↑(normSq u) ≠ 0 φ'u_u : deriv (φ u_in_𝔻).to_fun u = 1 / (1 - ↑(normSq u)) e2 : 0 ≤ normSq u e3 : normSq u < 1 g'0_eq_mul : deriv g 0 = (1 - ↑(normSq u))⁻¹ * deriv f 0 ⊢ Complex.abs (deriv f 0) < 1	z u✝ z₀ : ℂ U : Set ℂ inst✝ : good_domain U f : ℂ → ℂ f_diff : DifferentiableOn ℂ f 𝔻 f_img : MapsTo f 𝔻 𝔻 f_noninj : ¬InjOn f 𝔻 u : ℂ := f 0 u_in_𝔻 : u ∈ 𝔻 g : ℂ → ℂ := (φ u_in_𝔻).to_fun ∘ f g_diff : DifferentiableOn ℂ g 𝔻 g_maps : MapsTo g 𝔻 𝔻 g_0_eq_0 : g 0 = 0 h : ¬Complex.abs (deriv g 0) = 1 g'0_le_1 : Complex.abs (deriv g 0) ≤ 1 g'0_lt_1 : Complex.abs (deriv g 0) < 1 e1 : 1 - ↑(normSq u) ≠ 0 φ'u_u : deriv (φ u_in_𝔻).to_fun u = 1 / (1 - ↑(normSq u)) e2 : 0 ≤ normSq u e3 : normSq u < 1 g'0_eq_mul : deriv f 0 = (1 - ↑(normSq u)) * deriv g 0 ⊢ Complex.abs (deriv f 0) < 1
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	RMT4/etape2.lean	non_injective_schwarz	[62, 1]	[112, 37]	rw [← norm_eq_abs, g'0_eq_mul, norm_mul, mul_comm, ← one_mul (1 : ℝ)]	z u✝ z₀ : ℂ U : Set ℂ inst✝ : good_domain U f : ℂ → ℂ f_diff : DifferentiableOn ℂ f 𝔻 f_img : MapsTo f 𝔻 𝔻 f_noninj : ¬InjOn f 𝔻 u : ℂ := f 0 u_in_𝔻 : u ∈ 𝔻 g : ℂ → ℂ := (φ u_in_𝔻).to_fun ∘ f g_diff : DifferentiableOn ℂ g 𝔻 g_maps : MapsTo g 𝔻 𝔻 g_0_eq_0 : g 0 = 0 h : ¬Complex.abs (deriv g 0) = 1 g'0_le_1 : Complex.abs (deriv g 0) ≤ 1 g'0_lt_1 : Complex.abs (deriv g 0) < 1 e1 : 1 - ↑(normSq u) ≠ 0 φ'u_u : deriv (φ u_in_𝔻).to_fun u = 1 / (1 - ↑(normSq u)) e2 : 0 ≤ normSq u e3 : normSq u < 1 g'0_eq_mul : deriv f 0 = (1 - ↑(normSq u)) * deriv g 0 ⊢ Complex.abs (deriv f 0) < 1	z u✝ z₀ : ℂ U : Set ℂ inst✝ : good_domain U f : ℂ → ℂ f_diff : DifferentiableOn ℂ f 𝔻 f_img : MapsTo f 𝔻 𝔻 f_noninj : ¬InjOn f 𝔻 u : ℂ := f 0 u_in_𝔻 : u ∈ 𝔻 g : ℂ → ℂ := (φ u_in_𝔻).to_fun ∘ f g_diff : DifferentiableOn ℂ g 𝔻 g_maps : MapsTo g 𝔻 𝔻 g_0_eq_0 : g 0 = 0 h : ¬Complex.abs (deriv g 0) = 1 g'0_le_1 : Complex.abs (deriv g 0) ≤ 1 g'0_lt_1 : Complex.abs (deriv g 0) < 1 e1 : 1 - ↑(normSq u) ≠ 0 φ'u_u : deriv (φ u_in_𝔻).to_fun u = 1 / (1 - ↑(normSq u)) e2 : 0 ≤ normSq u e3 : normSq u < 1 g'0_eq_mul : deriv f 0 = (1 - ↑(normSq u)) * deriv g 0 ⊢ ‖deriv g 0‖ * ‖1 - ↑(normSq u)‖ < 1 * 1
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	RMT4/etape2.lean	non_injective_schwarz	[62, 1]	[112, 37]	refine mul_lt_mul g'0_lt_1 ?_ (norm_pos_iff.mpr e1) zero_le_one	z u✝ z₀ : ℂ U : Set ℂ inst✝ : good_domain U f : ℂ → ℂ f_diff : DifferentiableOn ℂ f 𝔻 f_img : MapsTo f 𝔻 𝔻 f_noninj : ¬InjOn f 𝔻 u : ℂ := f 0 u_in_𝔻 : u ∈ 𝔻 g : ℂ → ℂ := (φ u_in_𝔻).to_fun ∘ f g_diff : DifferentiableOn ℂ g 𝔻 g_maps : MapsTo g 𝔻 𝔻 g_0_eq_0 : g 0 = 0 h : ¬Complex.abs (deriv g 0) = 1 g'0_le_1 : Complex.abs (deriv g 0) ≤ 1 g'0_lt_1 : Complex.abs (deriv g 0) < 1 e1 : 1 - ↑(normSq u) ≠ 0 φ'u_u : deriv (φ u_in_𝔻).to_fun u = 1 / (1 - ↑(normSq u)) e2 : 0 ≤ normSq u e3 : normSq u < 1 g'0_eq_mul : deriv f 0 = (1 - ↑(normSq u)) * deriv g 0 ⊢ ‖deriv g 0‖ * ‖1 - ↑(normSq u)‖ < 1 * 1	z u✝ z₀ : ℂ U : Set ℂ inst✝ : good_domain U f : ℂ → ℂ f_diff : DifferentiableOn ℂ f 𝔻 f_img : MapsTo f 𝔻 𝔻 f_noninj : ¬InjOn f 𝔻 u : ℂ := f 0 u_in_𝔻 : u ∈ 𝔻 g : ℂ → ℂ := (φ u_in_𝔻).to_fun ∘ f g_diff : DifferentiableOn ℂ g 𝔻 g_maps : MapsTo g 𝔻 𝔻 g_0_eq_0 : g 0 = 0 h : ¬Complex.abs (deriv g 0) = 1 g'0_le_1 : Complex.abs (deriv g 0) ≤ 1 g'0_lt_1 : Complex.abs (deriv g 0) < 1 e1 : 1 - ↑(normSq u) ≠ 0 φ'u_u : deriv (φ u_in_𝔻).to_fun u = 1 / (1 - ↑(normSq u)) e2 : 0 ≤ normSq u e3 : normSq u < 1 g'0_eq_mul : deriv f 0 = (1 - ↑(normSq u)) * deriv g 0 ⊢ ‖1 - ↑(normSq u)‖ ≤ 1
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	RMT4/etape2.lean	non_injective_schwarz	[62, 1]	[112, 37]	simp at e2 e3 ⊢	z u✝ z₀ : ℂ U : Set ℂ inst✝ : good_domain U f : ℂ → ℂ f_diff : DifferentiableOn ℂ f 𝔻 f_img : MapsTo f 𝔻 𝔻 f_noninj : ¬InjOn f 𝔻 u : ℂ := f 0 u_in_𝔻 : u ∈ 𝔻 g : ℂ → ℂ := (φ u_in_𝔻).to_fun ∘ f g_diff : DifferentiableOn ℂ g 𝔻 g_maps : MapsTo g 𝔻 𝔻 g_0_eq_0 : g 0 = 0 h : ¬Complex.abs (deriv g 0) = 1 g'0_le_1 : Complex.abs (deriv g 0) ≤ 1 g'0_lt_1 : Complex.abs (deriv g 0) < 1 e1 : 1 - ↑(normSq u) ≠ 0 φ'u_u : deriv (φ u_in_𝔻).to_fun u = 1 / (1 - ↑(normSq u)) e2 : 0 ≤ normSq u e3 : normSq u < 1 g'0_eq_mul : deriv f 0 = (1 - ↑(normSq u)) * deriv g 0 ⊢ ‖1 - ↑(normSq u)‖ ≤ 1	z u✝ z₀ : ℂ U : Set ℂ inst✝ : good_domain U f : ℂ → ℂ f_diff : DifferentiableOn ℂ f 𝔻 f_img : MapsTo f 𝔻 𝔻 f_noninj : ¬InjOn f 𝔻 u : ℂ := f 0 u_in_𝔻 : u ∈ 𝔻 g : ℂ → ℂ := (φ u_in_𝔻).to_fun ∘ f g_diff : DifferentiableOn ℂ g 𝔻 g_maps : MapsTo g 𝔻 𝔻 g_0_eq_0 : g 0 = 0 h : ¬Complex.abs (deriv g 0) = 1 g'0_le_1 : Complex.abs (deriv g 0) ≤ 1 g'0_lt_1 : Complex.abs (deriv g 0) < 1 e1 : 1 - ↑(normSq u) ≠ 0 φ'u_u : deriv (φ u_in_𝔻).to_fun u = 1 / (1 - ↑(normSq u)) e2 : 0 ≤ normSq u e3 : normSq u < 1 g'0_eq_mul : deriv f 0 = (1 - ↑(normSq u)) * deriv g 0 ⊢ Complex.abs (1 - ↑(normSq u)) ≤ 1

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