File: Mathlib/Analysis/Complex/UpperHalfPlane/Topology.lean

full_name: UpperHalfPlane.instTopologicalSpace
code: instance : TopologicalSpace ℍ 
goal: TopologicalSpace ℍ

full_name: UpperHalfPlane.openEmbedding_coe
code: theorem openEmbedding_coe : OpenEmbedding ((↑) : ℍ → ℂ) 
goal: OpenEmbedding UpperHalfPlane.coe

full_name: UpperHalfPlane.embedding_coe
code: theorem embedding_coe : Embedding ((↑) : ℍ → ℂ) 
goal: Embedding UpperHalfPlane.coe

full_name: UpperHalfPlane.continuous_coe
code: theorem continuous_coe : Continuous ((↑) : ℍ → ℂ) 
goal: Continuous UpperHalfPlane.coe

full_name: UpperHalfPlane.continuous_re
code: theorem continuous_re : Continuous re 
goal: Continuous UpperHalfPlane.re

full_name: UpperHalfPlane.continuous_im
code: theorem continuous_im : Continuous im 
goal: Continuous UpperHalfPlane.im

full_name: UpperHalfPlane.instSecondCountableTopology
code: instance : SecondCountableTopology ℍ 
goal: SecondCountableTopology ℍ

full_name: UpperHalfPlane.instT3Space
code: instance : T3Space ℍ 
goal: T3Space ℍ

full_name: UpperHalfPlane.instT4Space
code: instance : T4Space ℍ 
goal: T4Space ℍ

full_name: UpperHalfPlane.instContractibleSpace
code: instance : ContractibleSpace ℍ 
goal: ContractibleSpace ℍ

full_name: UpperHalfPlane.instLocPathConnectedSpace
code: instance : LocPathConnectedSpace ℍ 
goal: LocPathConnectedSpace ℍ

full_name: UpperHalfPlane.instNoncompactSpace
code: instance : NoncompactSpace ℍ 
goal: NoncompactSpace ℍ

full_name: UpperHalfPlane.instLocallyCompactSpace
code: instance : LocallyCompactSpace ℍ 
goal: LocallyCompactSpace ℍ

full_name: UpperHalfPlane.verticalStrip
code: /-- The vertical strip of width `A` and height `B`, defined by elements whose real part has absolute
value less than or equal to `A` and imaginary part is at least `B`. -/
def verticalStrip (A B : ℝ) 
goal: Set ℍ

full_name: UpperHalfPlane.mem_verticalStrip_iff
code: theorem mem_verticalStrip_iff (A B : ℝ) (z : ℍ) : z ∈ verticalStrip A B ↔ |z.re| ≤ A ∧ B ≤ z.im 
goal: z ∈ UpperHalfPlane.verticalStrip A B ↔ |z.re| ≤ A ∧ B ≤ z.im

full_name: UpperHalfPlane.verticalStrip_mono
code: @[gcongr]
lemma verticalStrip_mono {A B A' B' : ℝ} (hA : A ≤ A') (hB : B' ≤ B) :
    verticalStrip A B ⊆ verticalStrip A' B' 
goal: UpperHalfPlane.verticalStrip A B ⊆ UpperHalfPlane.verticalStrip A' B'

full_name: UpperHalfPlane.verticalStrip_mono_left
code: @[gcongr]
lemma verticalStrip_mono_left {A A'} (h : A ≤ A') (B) : verticalStrip A B ⊆ verticalStrip A' B 
goal: UpperHalfPlane.verticalStrip A B ⊆ UpperHalfPlane.verticalStrip A' B

full_name: UpperHalfPlane.verticalStrip_anti_right
code: @[gcongr]
lemma verticalStrip_anti_right (A) {B B'} (h : B' ≤ B) : verticalStrip A B ⊆ verticalStrip A B' 
goal: UpperHalfPlane.verticalStrip A B ⊆ UpperHalfPlane.verticalStrip A B'

full_name: UpperHalfPlane.subset_verticalStrip_of_isCompact
code: lemma subset_verticalStrip_of_isCompact {K : Set ℍ} (hK : IsCompact K) :
    ∃ A B : ℝ, 0 < B ∧ K ⊆ verticalStrip A B 
goal: ∃ A B, 0 < B ∧ K ⊆ UpperHalfPlane.verticalStrip A B

full_name: UpperHalfPlane.ModularGroup_T_zpow_mem_verticalStrip
code: theorem ModularGroup_T_zpow_mem_verticalStrip (z : ℍ) {N : ℕ} (hn : 0 < N) :
    ∃ n : ℤ, ModularGroup.T ^ (N * n) • z ∈ verticalStrip N z.im 
goal: ∃ n, ModularGroup.T ^ (↑N * n) • z ∈ UpperHalfPlane.verticalStrip (↑N) z.im

full_name: UpperHalfPlane.ofComplex
code: /-- A continuous section `ℂ → ℍ` of the natural inclusion map, bundled as a `PartialHomeomorph`. -/
def ofComplex : PartialHomeomorph ℂ ℍ 
goal: PartialHomeomorph ℂ ℍ

full_name: UpperHalfPlane.«_aux_Mathlib_Analysis_Complex_UpperHalfPlane_Topology___macroRules_UpperHalfPlane_term↑ₕ__1»
code: /-- Extend a function on `ℍ` arbitrarily to a function on all of `ℂ`. -/
scoped notation "↑ₕ" f => f ∘ ofComplex
goal: Lean.Macro

full_name: UpperHalfPlane.ofComplex_apply
code: lemma ofComplex_apply (z : ℍ) : ofComplex (z : ℂ) = z 
goal: ↑UpperHalfPlane.ofComplex ↑z = z

full_name: UpperHalfPlane.comp_ofComplex
code: lemma comp_ofComplex (f : ℍ → ℂ) (z : ℍ) : (↑ₕ f) z = f z 
goal: (f ∘ ↑UpperHalfPlane.ofComplex) ↑z = f z