File: Mathlib/Analysis/Complex/UnitDisc/Basic.lean

full_name: _aux_Mathlib_Analysis_Complex_UnitDisc_Basic___unexpand_starRingEnd_1
code: local notation "conj'" => starRingEnd ℂ
goal: Lean.PrettyPrinter.Unexpander

full_name: _aux_Mathlib_Analysis_Complex_UnitDisc_Basic___macroRules_termConj'_1
code: local notation "conj'" => starRingEnd ℂ
goal: Lean.Macro

full_name: Complex.UnitDisc
code: /-- Complex unit disc. -/
def UnitDisc : Type 
goal: Type

full_name: UnitDisc._aux_Mathlib_Analysis_Complex_UnitDisc_Basic___macroRules_UnitDisc_term𝔻_1
code: @[inherit_doc] scoped[UnitDisc] notation "𝔻" => Complex.UnitDisc
goal: Lean.Macro

full_name: UnitDisc._aux_Mathlib_Analysis_Complex_UnitDisc_Basic___unexpand_Complex_UnitDisc_1
code: @[inherit_doc] scoped[UnitDisc] notation "𝔻" => Complex.UnitDisc
goal: Lean.PrettyPrinter.Unexpander

full_name: Complex.UnitDisc.instCommSemigroup
code: instance instCommSemigroup : CommSemigroup UnitDisc 
goal: CommSemigroup Complex.UnitDisc

full_name: Complex.UnitDisc.instHasDistribNeg
code: instance instHasDistribNeg : HasDistribNeg UnitDisc 
goal: HasDistribNeg Complex.UnitDisc

full_name: Complex.UnitDisc.instCoe
code: instance instCoe : Coe UnitDisc ℂ 
goal: Coe Complex.UnitDisc ℂ

full_name: Complex.UnitDisc.coe_injective
code: theorem coe_injective : Injective ((↑) : 𝔻 → ℂ) 
goal: Function.Injective Subtype.val

full_name: Complex.UnitDisc.abs_lt_one
code: theorem abs_lt_one (z : 𝔻) : abs (z : ℂ) < 1 
goal: Complex.abs ↑z < 1

full_name: Complex.UnitDisc.abs_ne_one
code: theorem abs_ne_one (z : 𝔻) : abs (z : ℂ) ≠ 1 
goal: Complex.abs ↑z ≠ 1

full_name: Complex.UnitDisc.normSq_lt_one
code: theorem normSq_lt_one (z : 𝔻) : normSq z < 1 
goal: Complex.normSq ↑z < 1

full_name: Complex.UnitDisc.coe_ne_one
code: theorem coe_ne_one (z : 𝔻) : (z : ℂ) ≠ 1 
goal: ↑z ≠ 1

full_name: Complex.UnitDisc.coe_ne_neg_one
code: theorem coe_ne_neg_one (z : 𝔻) : (z : ℂ) ≠ -1 
goal: ↑z ≠ -1

full_name: Complex.UnitDisc.one_add_coe_ne_zero
code: theorem one_add_coe_ne_zero (z : 𝔻) : (1 + z : ℂ) ≠ 0 
goal: 1 + ↑z ≠ 0

full_name: Complex.UnitDisc.coe_mul
code: @[simp, norm_cast]
theorem coe_mul (z w : 𝔻) : ↑(z * w) = (z * w : ℂ) 
goal: ↑(z * w) = ↑z * ↑w

full_name: Complex.UnitDisc.mk
code: /-- A constructor that assumes `abs z < 1` instead of `dist z 0 < 1` and returns an element
of `𝔻` instead of `↥Metric.ball (0 : ℂ) 1`. -/
def mk (z : ℂ) (hz : abs z < 1) : 𝔻 
goal: Complex.UnitDisc

full_name: Complex.UnitDisc.coe_mk
code: @[simp]
theorem coe_mk (z : ℂ) (hz : abs z < 1) : (mk z hz : ℂ) = z 
goal: ↑(Complex.UnitDisc.mk z hz) = z

full_name: Complex.UnitDisc.mk_coe
code: @[simp]
theorem mk_coe (z : 𝔻) (hz : abs (z : ℂ) < 1 := z.abs_lt_one) : mk z hz = z 
goal: Complex.UnitDisc.mk (↑z) hz = z

full_name: Complex.UnitDisc.mk_neg
code: @[simp]
theorem mk_neg (z : ℂ) (hz : abs (-z) < 1) : mk (-z) hz = -mk z (abs.map_neg z ▸ hz) 
goal: Complex.UnitDisc.mk (-z) hz = -Complex.UnitDisc.mk z ⋯

full_name: Complex.UnitDisc.instSemigroupWithZero
code: instance : SemigroupWithZero 𝔻 
goal: SemigroupWithZero Complex.UnitDisc

full_name: Complex.UnitDisc.coe_zero
code: @[simp]
theorem coe_zero : ((0 : 𝔻) : ℂ) = 0 
goal: ↑0 = 0

full_name: Complex.UnitDisc.coe_eq_zero
code: @[simp]
theorem coe_eq_zero {z : 𝔻} : (z : ℂ) = 0 ↔ z = 0 
goal: ↑z = 0 ↔ z = 0

full_name: Complex.UnitDisc.instInhabited
code: instance : Inhabited 𝔻 
goal: Inhabited Complex.UnitDisc

full_name: Complex.UnitDisc.circleAction
code: instance circleAction : MulAction circle 𝔻 
goal: MulAction (↥circle) Complex.UnitDisc

full_name: Complex.UnitDisc.isScalarTower_circle_circle
code: instance isScalarTower_circle_circle : IsScalarTower circle circle 𝔻 
goal: IsScalarTower (↥circle) (↥circle) Complex.UnitDisc

full_name: Complex.UnitDisc.isScalarTower_circle
code: instance isScalarTower_circle : IsScalarTower circle 𝔻 𝔻 
goal: IsScalarTower (↥circle) Complex.UnitDisc Complex.UnitDisc

full_name: Complex.UnitDisc.instSMulCommClass_circle
code: instance instSMulCommClass_circle : SMulCommClass circle 𝔻 𝔻 
goal: SMulCommClass (↥circle) Complex.UnitDisc Complex.UnitDisc

full_name: Complex.UnitDisc.instSMulCommClass_circle'
code: instance instSMulCommClass_circle' : SMulCommClass 𝔻 circle 𝔻 
goal: SMulCommClass Complex.UnitDisc (↥circle) Complex.UnitDisc

full_name: Complex.UnitDisc.coe_smul_circle
code: @[simp, norm_cast]
theorem coe_smul_circle (z : circle) (w : 𝔻) : ↑(z • w) = (z * w : ℂ) 
goal: ↑(z • w) = ↑z * ↑w

full_name: Complex.UnitDisc.closedBallAction
code: instance closedBallAction : MulAction (closedBall (0 : ℂ) 1) 𝔻 
goal: MulAction (↑(Metric.closedBall 0 1)) Complex.UnitDisc

full_name: Complex.UnitDisc.isScalarTower_closedBall_closedBall
code: instance isScalarTower_closedBall_closedBall :
    IsScalarTower (closedBall (0 : ℂ) 1) (closedBall (0 : ℂ) 1) 𝔻 
goal: IsScalarTower (↑(Metric.closedBall 0 1)) (↑(Metric.closedBall 0 1)) Complex.UnitDisc

full_name: Complex.UnitDisc.isScalarTower_closedBall
code: instance isScalarTower_closedBall : IsScalarTower (closedBall (0 : ℂ) 1) 𝔻 𝔻 
goal: IsScalarTower (↑(Metric.closedBall 0 1)) Complex.UnitDisc Complex.UnitDisc

full_name: Complex.UnitDisc.instSMulCommClass_closedBall
code: instance instSMulCommClass_closedBall : SMulCommClass (closedBall (0 : ℂ) 1) 𝔻 𝔻 
goal: SMulCommClass (↑(Metric.closedBall 0 1)) Complex.UnitDisc Complex.UnitDisc

full_name: Complex.UnitDisc.instSMulCommClass_closedBall'
code: instance instSMulCommClass_closedBall' : SMulCommClass 𝔻 (closedBall (0 : ℂ) 1) 𝔻 
goal: SMulCommClass Complex.UnitDisc (↑(Metric.closedBall 0 1)) Complex.UnitDisc

full_name: Complex.UnitDisc.instSMulCommClass_circle_closedBall
code: instance instSMulCommClass_circle_closedBall : SMulCommClass circle (closedBall (0 : ℂ) 1) 𝔻 
goal: SMulCommClass (↥circle) (↑(Metric.closedBall 0 1)) Complex.UnitDisc

full_name: Complex.UnitDisc.instSMulCommClass_closedBall_circle
code: instance instSMulCommClass_closedBall_circle : SMulCommClass (closedBall (0 : ℂ) 1) circle 𝔻 
goal: SMulCommClass (↑(Metric.closedBall 0 1)) (↥circle) Complex.UnitDisc

full_name: Complex.UnitDisc.coe_smul_closedBall
code: @[simp, norm_cast]
theorem coe_smul_closedBall (z : closedBall (0 : ℂ) 1) (w : 𝔻) : ↑(z • w) = (z * w : ℂ) 
goal: ↑(z • w) = ↑z * ↑w

full_name: Complex.UnitDisc.re
code: /-- Real part of a point of the unit disc. -/
def re (z : 𝔻) : ℝ 
goal: ℝ

full_name: Complex.UnitDisc.im
code: /-- Imaginary part of a point of the unit disc. -/
def im (z : 𝔻) : ℝ 
goal: ℝ

full_name: Complex.UnitDisc.re_coe
code: @[simp, norm_cast]
theorem re_coe (z : 𝔻) : (z : ℂ).re = z.re 
goal: (↑z).re = z.re

full_name: Complex.UnitDisc.im_coe
code: @[simp, norm_cast]
theorem im_coe (z : 𝔻) : (z : ℂ).im = z.im 
goal: (↑z).im = z.im

full_name: Complex.UnitDisc.re_neg
code: @[simp]
theorem re_neg (z : 𝔻) : (-z).re = -z.re 
goal: (-z).re = -z.re

full_name: Complex.UnitDisc.im_neg
code: @[simp]
theorem im_neg (z : 𝔻) : (-z).im = -z.im 
goal: (-z).im = -z.im

full_name: Complex.UnitDisc.conj
code: /-- Conjugate point of the unit disc. -/
def conj (z : 𝔻) : 𝔻 
goal: Complex.UnitDisc

full_name: Complex.UnitDisc.coe_conj
code: @[simp]
theorem coe_conj (z : 𝔻) : (z.conj : ℂ) = conj' ↑z 
goal: ↑z.conj = conj' ↑z

full_name: Complex.UnitDisc.conj_zero
code: @[simp]
theorem conj_zero : conj 0 = 0 
goal: Complex.UnitDisc.conj 0 = 0

full_name: Complex.UnitDisc.conj_conj
code: @[simp]
theorem conj_conj (z : 𝔻) : conj (conj z) = z 
goal: z.conj.conj = z

full_name: Complex.UnitDisc.conj_neg
code: @[simp]
theorem conj_neg (z : 𝔻) : (-z).conj = -z.conj 
goal: (-z).conj = -z.conj

full_name: Complex.UnitDisc.re_conj
code: @[simp]
theorem re_conj (z : 𝔻) : z.conj.re = z.re 
goal: z.conj.re = z.re

full_name: Complex.UnitDisc.im_conj
code: @[simp]
theorem im_conj (z : 𝔻) : z.conj.im = -z.im 
goal: z.conj.im = -z.im

full_name: Complex.UnitDisc.conj_mul
code: @[simp]
theorem conj_mul (z w : 𝔻) : (z * w).conj = z.conj * w.conj 
goal: (z * w).conj = z.conj * w.conj