File: Mathlib/Analysis/Complex/Circle.lean

full_name: circle
code: /-- The unit circle in `ℂ`, here given the structure of a submonoid of `ℂ`. -/
def circle : Submonoid ℂ 
goal: Submonoid ℂ

full_name: mem_circle_iff_abs
code: @[simp]
theorem mem_circle_iff_abs {z : ℂ} : z ∈ circle ↔ abs z = 1 
goal: z ∈ circle ↔ Complex.abs z = 1

full_name: circle_def
code: theorem circle_def : ↑circle = { z : ℂ | abs z = 1 } 
goal: ↑circle = {z | Complex.abs z = 1}

full_name: abs_coe_circle
code: @[simp]
theorem abs_coe_circle (z : circle) : abs z = 1 
goal: Complex.abs ↑z = 1

full_name: mem_circle_iff_normSq
code: theorem mem_circle_iff_normSq {z : ℂ} : z ∈ circle ↔ normSq z = 1 
goal: z ∈ circle ↔ Complex.normSq z = 1

full_name: normSq_eq_of_mem_circle
code: @[simp]
theorem normSq_eq_of_mem_circle (z : circle) : normSq z = 1 
goal: Complex.normSq ↑z = 1

full_name: ne_zero_of_mem_circle
code: theorem ne_zero_of_mem_circle (z : circle) : (z : ℂ) ≠ 0 
goal: ↑z ≠ 0

full_name: commGroup
code: instance commGroup : CommGroup circle 
goal: CommGroup ↥circle

full_name: coe_inv_circle
code: @[simp]
theorem coe_inv_circle (z : circle) : ↑z⁻¹ = (z : ℂ)⁻¹ 
goal: ↑z⁻¹ = (↑z)⁻¹

full_name: coe_inv_circle_eq_conj
code: theorem coe_inv_circle_eq_conj (z : circle) : ↑z⁻¹ = conj (z : ℂ) 
goal: ↑z⁻¹ = (starRingEnd ℂ) ↑z

full_name: coe_div_circle
code: @[simp]
theorem coe_div_circle (z w : circle) : ↑(z / w) = (z : ℂ) / w 
goal: ↑(z / w) = ↑z / ↑w

full_name: circle.toUnits
code: /-- The elements of the circle embed into the units. -/
def circle.toUnits : circle →* Units ℂ 
goal: ↥circle →* ℂˣ

full_name: circle.toUnits_apply
code: @[simp]
theorem circle.toUnits_apply (z : circle) :
    circle.toUnits z = Units.mk0 ↑z (ne_zero_of_mem_circle z) 
goal: circle.toUnits z = Units.mk0 ↑z ⋯

full_name: instCompactSpaceSubtypeComplexMemSubmonoidCircle
code: instance : CompactSpace circle 
goal: CompactSpace ↥circle

full_name: instTopologicalGroupSubtypeComplexMemSubmonoidCircle
code: instance : TopologicalGroup circle 
goal: TopologicalGroup ↥circle

full_name: instUniformGroupSubtypeComplexMemSubmonoidCircle
code: instance : UniformGroup circle 
goal: UniformGroup ↥circle

full_name: circle.ofConjDivSelf_coe
code: /-- If `z` is a nonzero complex number, then `conj z / z` belongs to the unit circle. -/
@[simps]
def circle.ofConjDivSelf (z : ℂ) (hz : z ≠ 0) : circle 
goal: ↑(circle.ofConjDivSelf z hz) = (starRingEnd ℂ) z / z

full_name: circle.ofConjDivSelf
code: /-- If `z` is a nonzero complex number, then `conj z / z` belongs to the unit circle. -/
@[simps]
def circle.ofConjDivSelf (z : ℂ) (hz : z ≠ 0) : circle 
goal: ↥circle

full_name: expMapCircle
code: /-- The map `fun t => exp (t * I)` from `ℝ` to the unit circle in `ℂ`. -/
def expMapCircle : C(ℝ, circle) 
goal: C(ℝ, ↥circle)

full_name: expMapCircle_apply
code: @[simp]
theorem expMapCircle_apply (t : ℝ) : ↑(expMapCircle t) = Complex.exp (t * Complex.I) 
goal: ↑(expMapCircle t) = cexp (↑t * Complex.I)

full_name: expMapCircle_zero
code: @[simp]
theorem expMapCircle_zero : expMapCircle 0 = 1 
goal: expMapCircle 0 = 1

full_name: expMapCircle_add
code: @[simp]
theorem expMapCircle_add (x y : ℝ) : expMapCircle (x + y) = expMapCircle x * expMapCircle y 
goal: expMapCircle (x + y) = expMapCircle x * expMapCircle y

full_name: expMapCircleHom
code: /-- The map `fun t => exp (t * I)` from `ℝ` to the unit circle in `ℂ`,
considered as a homomorphism of groups. -/
@[simps]
def expMapCircleHom : ℝ →+ Additive circle 
goal: ℝ →+ Additive ↥circle

full_name: expMapCircleHom_apply
code: /-- The map `fun t => exp (t * I)` from `ℝ` to the unit circle in `ℂ`,
considered as a homomorphism of groups. -/
@[simps]
def expMapCircleHom : ℝ →+ Additive circle 
goal: ∀ (a : ℝ), expMapCircleHom a = (⇑Additive.ofMul ∘ ⇑expMapCircle) a

full_name: expMapCircle_sub
code: @[simp]
theorem expMapCircle_sub (x y : ℝ) : expMapCircle (x - y) = expMapCircle x / expMapCircle y 
goal: expMapCircle (x - y) = expMapCircle x / expMapCircle y

full_name: expMapCircle_neg
code: @[simp]
theorem expMapCircle_neg (x : ℝ) : expMapCircle (-x) = (expMapCircle x)⁻¹ 
goal: expMapCircle (-x) = (expMapCircle x)⁻¹

full_name: norm_circle_smul
code: @[simp]
lemma norm_circle_smul {E : Type*} [SeminormedAddCommGroup E] [NormedSpace ℂ E]
    (u : circle) (v : E) :
    ‖u • v‖ = ‖v‖ 
goal: ‖u • v‖ = ‖v‖