Theorem: IsConformalMap.is_complex_or_conj_linear
File: Mathlib/Analysis/Complex/Conformal.lean

tactic: rcases h with ⟨c, -, li, rfl⟩
state_before: f : ℂ → ℂ
z : ℂ
g : ℂ →L[ℝ] ℂ
h : IsConformalMap g
⊢ (∃ map, restrictScalars ℝ map = g) ∨ ∃ map, restrictScalars ℝ map = g.comp ↑conjCLE
state_after: case intro.intro.intro
f : ℂ → ℂ
z : ℂ
c : ℝ
li : ℂ →ₗᵢ[ℝ] ℂ
⊢ (∃ map, restrictScalars ℝ map = c • li.toContinuousLinearMap) ∨
    ∃ map, restrictScalars ℝ map = (c • li.toContinuousLinearMap).comp ↑conjCLE

tactic: obtain ⟨li, rfl⟩ : ∃ li' : ℂ ≃ₗᵢ[ℝ] ℂ, li'.toLinearIsometry = li :=
  ⟨li.toLinearIsometryEquiv rfl, by ext1; rfl⟩
state_before: case intro.intro.intro
f : ℂ → ℂ
z : ℂ
c : ℝ
li : ℂ →ₗᵢ[ℝ] ℂ
⊢ (∃ map, restrictScalars ℝ map = c • li.toContinuousLinearMap) ∨
    ∃ map, restrictScalars ℝ map = (c • li.toContinuousLinearMap).comp ↑conjCLE
state_after: case intro.intro.intro.intro
f : ℂ → ℂ
z : ℂ
c : ℝ
li : ℂ ≃ₗᵢ[ℝ] ℂ
⊢ (∃ map, restrictScalars ℝ map = c • li.toLinearIsometry.toContinuousLinearMap) ∨
    ∃ map, restrictScalars ℝ map = (c • li.toLinearIsometry.toContinuousLinearMap).comp ↑conjCLE

tactic: rcases linear_isometry_complex li with ⟨a, rfl | rfl⟩
state_before: case intro.intro.intro.intro
f : ℂ → ℂ
z : ℂ
c : ℝ
li : ℂ ≃ₗᵢ[ℝ] ℂ
⊢ (∃ map, restrictScalars ℝ map = c • li.toLinearIsometry.toContinuousLinearMap) ∨
    ∃ map, restrictScalars ℝ map = (c • li.toLinearIsometry.toContinuousLinearMap).comp ↑conjCLE
state_after: case intro.intro.intro.intro.intro.inl
f : ℂ → ℂ
z : ℂ
c : ℝ
a : ↥circle
⊢ (∃ map, restrictScalars ℝ map = c • (rotation a).toLinearIsometry.toContinuousLinearMap) ∨
    ∃ map, restrictScalars ℝ map = (c • (rotation a).toLinearIsometry.toContinuousLinearMap).comp ↑conjCLE

case intro.intro.intro.intro.intro.inr
f : ℂ → ℂ
z : ℂ
c : ℝ
a : ↥circle
⊢ (∃ map, restrictScalars ℝ map = c • (conjLIE.trans (rotation a)).toLinearIsometry.toContinuousLinearMap) ∨
    ∃ map,
      restrictScalars ℝ map = (c • (conjLIE.trans (rotation a)).toLinearIsometry.toContinuousLinearMap).comp ↑conjCLE

tactic: ext1
state_before: f : ℂ → ℂ
z : ℂ
c : ℝ
li : ℂ →ₗᵢ[ℝ] ℂ
⊢ (li.toLinearIsometryEquiv ⋯).toLinearIsometry = li
state_after: case h
f : ℂ → ℂ
z : ℂ
c : ℝ
li : ℂ →ₗᵢ[ℝ] ℂ
x✝ : ℂ
⊢ (li.toLinearIsometryEquiv ⋯).toLinearIsometry x✝ = li x✝

tactic: rfl
state_before: case h
f : ℂ → ℂ
z : ℂ
c : ℝ
li : ℂ →ₗᵢ[ℝ] ℂ
x✝ : ℂ
⊢ (li.toLinearIsometryEquiv ⋯).toLinearIsometry x✝ = li x✝
state_after: no goals

tactic: refine Or.inl ⟨c • (a : ℂ) • ContinuousLinearMap.id ℂ ℂ, ?_⟩
state_before: case intro.intro.intro.intro.intro.inl
f : ℂ → ℂ
z : ℂ
c : ℝ
a : ↥circle
⊢ (∃ map, restrictScalars ℝ map = c • (rotation a).toLinearIsometry.toContinuousLinearMap) ∨
    ∃ map, restrictScalars ℝ map = (c • (rotation a).toLinearIsometry.toContinuousLinearMap).comp ↑conjCLE
state_after: case intro.intro.intro.intro.intro.inl
f : ℂ → ℂ
z : ℂ
c : ℝ
a : ↥circle
⊢ restrictScalars ℝ (c • ↑a • ContinuousLinearMap.id ℂ ℂ) = c • (rotation a).toLinearIsometry.toContinuousLinearMap

tactic: ext1
state_before: case intro.intro.intro.intro.intro.inl
f : ℂ → ℂ
z : ℂ
c : ℝ
a : ↥circle
⊢ restrictScalars ℝ (c • ↑a • ContinuousLinearMap.id ℂ ℂ) = c • (rotation a).toLinearIsometry.toContinuousLinearMap
state_after: case intro.intro.intro.intro.intro.inl.h
f : ℂ → ℂ
z : ℂ
c : ℝ
a : ↥circle
x✝ : ℂ
⊢ (restrictScalars ℝ (c • ↑a • ContinuousLinearMap.id ℂ ℂ)) x✝ =
    (c • (rotation a).toLinearIsometry.toContinuousLinearMap) x✝

tactic: simp
state_before: case intro.intro.intro.intro.intro.inl.h
f : ℂ → ℂ
z : ℂ
c : ℝ
a : ↥circle
x✝ : ℂ
⊢ (restrictScalars ℝ (c • ↑a • ContinuousLinearMap.id ℂ ℂ)) x✝ =
    (c • (rotation a).toLinearIsometry.toContinuousLinearMap) x✝
state_after: no goals

tactic: refine Or.inr ⟨c • (a : ℂ) • ContinuousLinearMap.id ℂ ℂ, ?_⟩
state_before: case intro.intro.intro.intro.intro.inr
f : ℂ → ℂ
z : ℂ
c : ℝ
a : ↥circle
⊢ (∃ map, restrictScalars ℝ map = c • (conjLIE.trans (rotation a)).toLinearIsometry.toContinuousLinearMap) ∨
    ∃ map,
      restrictScalars ℝ map = (c • (conjLIE.trans (rotation a)).toLinearIsometry.toContinuousLinearMap).comp ↑conjCLE
state_after: case intro.intro.intro.intro.intro.inr
f : ℂ → ℂ
z : ℂ
c : ℝ
a : ↥circle
⊢ restrictScalars ℝ (c • ↑a • ContinuousLinearMap.id ℂ ℂ) =
    (c • (conjLIE.trans (rotation a)).toLinearIsometry.toContinuousLinearMap).comp ↑conjCLE

tactic: ext1
state_before: case intro.intro.intro.intro.intro.inr
f : ℂ → ℂ
z : ℂ
c : ℝ
a : ↥circle
⊢ restrictScalars ℝ (c • ↑a • ContinuousLinearMap.id ℂ ℂ) =
    (c • (conjLIE.trans (rotation a)).toLinearIsometry.toContinuousLinearMap).comp ↑conjCLE
state_after: case intro.intro.intro.intro.intro.inr.h
f : ℂ → ℂ
z : ℂ
c : ℝ
a : ↥circle
x✝ : ℂ
⊢ (restrictScalars ℝ (c • ↑a • ContinuousLinearMap.id ℂ ℂ)) x✝ =
    ((c • (conjLIE.trans (rotation a)).toLinearIsometry.toContinuousLinearMap).comp ↑conjCLE) x✝

tactic: simp
state_before: case intro.intro.intro.intro.intro.inr.h
f : ℂ → ℂ
z : ℂ
c : ℝ
a : ↥circle
x✝ : ℂ
⊢ (restrictScalars ℝ (c • ↑a • ContinuousLinearMap.id ℂ ℂ)) x✝ =
    ((c • (conjLIE.trans (rotation a)).toLinearIsometry.toContinuousLinearMap).comp ↑conjCLE) x✝
state_after: no goals