Theorem: isConformalMap_iff_is_complex_or_conj_linear
File: Mathlib/Analysis/Complex/Conformal.lean

tactic: constructor
state_before: f : ℂ → ℂ
z : ℂ
g : ℂ →L[ℝ] ℂ
⊢ IsConformalMap g ↔ ((∃ map, restrictScalars ℝ map = g) ∨ ∃ map, restrictScalars ℝ map = g.comp ↑conjCLE) ∧ g ≠ 0
state_after: case mp
f : ℂ → ℂ
z : ℂ
g : ℂ →L[ℝ] ℂ
⊢ IsConformalMap g → ((∃ map, restrictScalars ℝ map = g) ∨ ∃ map, restrictScalars ℝ map = g.comp ↑conjCLE) ∧ g ≠ 0

case mpr
f : ℂ → ℂ
z : ℂ
g : ℂ →L[ℝ] ℂ
⊢ ((∃ map, restrictScalars ℝ map = g) ∨ ∃ map, restrictScalars ℝ map = g.comp ↑conjCLE) ∧ g ≠ 0 → IsConformalMap g

tactic: exact fun h => ⟨h.is_complex_or_conj_linear, h.ne_zero⟩
state_before: case mp
f : ℂ → ℂ
z : ℂ
g : ℂ →L[ℝ] ℂ
⊢ IsConformalMap g → ((∃ map, restrictScalars ℝ map = g) ∨ ∃ map, restrictScalars ℝ map = g.comp ↑conjCLE) ∧ g ≠ 0
state_after: no goals

tactic: rintro ⟨⟨map, rfl⟩ | ⟨map, hmap⟩, h₂⟩
state_before: case mpr
f : ℂ → ℂ
z : ℂ
g : ℂ →L[ℝ] ℂ
⊢ ((∃ map, restrictScalars ℝ map = g) ∨ ∃ map, restrictScalars ℝ map = g.comp ↑conjCLE) ∧ g ≠ 0 → IsConformalMap g
state_after: case mpr.intro.inl.intro
f : ℂ → ℂ
z : ℂ
map : ℂ →L[ℂ] ℂ
h₂ : restrictScalars ℝ map ≠ 0
⊢ IsConformalMap (restrictScalars ℝ map)

case mpr.intro.inr.intro
f : ℂ → ℂ
z : ℂ
g : ℂ →L[ℝ] ℂ
h₂ : g ≠ 0
map : ℂ →L[ℂ] ℂ
hmap : restrictScalars ℝ map = g.comp ↑conjCLE
⊢ IsConformalMap g

tactic: refine isConformalMap_complex_linear ?_
state_before: case mpr.intro.inl.intro
f : ℂ → ℂ
z : ℂ
map : ℂ →L[ℂ] ℂ
h₂ : restrictScalars ℝ map ≠ 0
⊢ IsConformalMap (restrictScalars ℝ map)
state_after: case mpr.intro.inl.intro
f : ℂ → ℂ
z : ℂ
map : ℂ →L[ℂ] ℂ
h₂ : restrictScalars ℝ map ≠ 0
⊢ map ≠ 0

tactic: contrapose! h₂ with w
state_before: case mpr.intro.inl.intro
f : ℂ → ℂ
z : ℂ
map : ℂ →L[ℂ] ℂ
h₂ : restrictScalars ℝ map ≠ 0
⊢ map ≠ 0
state_after: case mpr.intro.inl.intro
f : ℂ → ℂ
z : ℂ
map : ℂ →L[ℂ] ℂ
w : map = 0
⊢ restrictScalars ℝ map = 0

tactic: simp only [w, restrictScalars_zero]
state_before: case mpr.intro.inl.intro
f : ℂ → ℂ
z : ℂ
map : ℂ →L[ℂ] ℂ
w : map = 0
⊢ restrictScalars ℝ map = 0
state_after: no goals

tactic: have minor₁ : g = map.restrictScalars ℝ ∘L ↑conjCLE := by
  ext1
  simp only [hmap, coe_comp', ContinuousLinearEquiv.coe_coe, Function.comp_apply,
    conjCLE_apply, starRingEnd_self_apply]
state_before: case mpr.intro.inr.intro
f : ℂ → ℂ
z : ℂ
g : ℂ →L[ℝ] ℂ
h₂ : g ≠ 0
map : ℂ →L[ℂ] ℂ
hmap : restrictScalars ℝ map = g.comp ↑conjCLE
⊢ IsConformalMap g
state_after: case mpr.intro.inr.intro
f : ℂ → ℂ
z : ℂ
g : ℂ →L[ℝ] ℂ
h₂ : g ≠ 0
map : ℂ →L[ℂ] ℂ
hmap : restrictScalars ℝ map = g.comp ↑conjCLE
minor₁ : g = (restrictScalars ℝ map).comp ↑conjCLE
⊢ IsConformalMap g

tactic: rw [minor₁] at h₂ ⊢
state_before: case mpr.intro.inr.intro
f : ℂ → ℂ
z : ℂ
g : ℂ →L[ℝ] ℂ
h₂ : g ≠ 0
map : ℂ →L[ℂ] ℂ
hmap : restrictScalars ℝ map = g.comp ↑conjCLE
minor₁ : g = (restrictScalars ℝ map).comp ↑conjCLE
⊢ IsConformalMap g
state_after: case mpr.intro.inr.intro
f : ℂ → ℂ
z : ℂ
g : ℂ →L[ℝ] ℂ
map : ℂ →L[ℂ] ℂ
h₂ : (restrictScalars ℝ map).comp ↑conjCLE ≠ 0
hmap : restrictScalars ℝ map = g.comp ↑conjCLE
minor₁ : g = (restrictScalars ℝ map).comp ↑conjCLE
⊢ IsConformalMap ((restrictScalars ℝ map).comp ↑conjCLE)

tactic: refine isConformalMap_complex_linear_conj ?_
state_before: case mpr.intro.inr.intro
f : ℂ → ℂ
z : ℂ
g : ℂ →L[ℝ] ℂ
map : ℂ →L[ℂ] ℂ
h₂ : (restrictScalars ℝ map).comp ↑conjCLE ≠ 0
hmap : restrictScalars ℝ map = g.comp ↑conjCLE
minor₁ : g = (restrictScalars ℝ map).comp ↑conjCLE
⊢ IsConformalMap ((restrictScalars ℝ map).comp ↑conjCLE)
state_after: case mpr.intro.inr.intro
f : ℂ → ℂ
z : ℂ
g : ℂ →L[ℝ] ℂ
map : ℂ →L[ℂ] ℂ
h₂ : (restrictScalars ℝ map).comp ↑conjCLE ≠ 0
hmap : restrictScalars ℝ map = g.comp ↑conjCLE
minor₁ : g = (restrictScalars ℝ map).comp ↑conjCLE
⊢ map ≠ 0

tactic: contrapose! h₂ with w
state_before: case mpr.intro.inr.intro
f : ℂ → ℂ
z : ℂ
g : ℂ →L[ℝ] ℂ
map : ℂ →L[ℂ] ℂ
h₂ : (restrictScalars ℝ map).comp ↑conjCLE ≠ 0
hmap : restrictScalars ℝ map = g.comp ↑conjCLE
minor₁ : g = (restrictScalars ℝ map).comp ↑conjCLE
⊢ map ≠ 0
state_after: case mpr.intro.inr.intro
f : ℂ → ℂ
z : ℂ
g : ℂ →L[ℝ] ℂ
map : ℂ →L[ℂ] ℂ
hmap : restrictScalars ℝ map = g.comp ↑conjCLE
minor₁ : g = (restrictScalars ℝ map).comp ↑conjCLE
w : map = 0
⊢ (restrictScalars ℝ map).comp ↑conjCLE = 0

tactic: simp only [w, restrictScalars_zero, zero_comp]
state_before: case mpr.intro.inr.intro
f : ℂ → ℂ
z : ℂ
g : ℂ →L[ℝ] ℂ
map : ℂ →L[ℂ] ℂ
hmap : restrictScalars ℝ map = g.comp ↑conjCLE
minor₁ : g = (restrictScalars ℝ map).comp ↑conjCLE
w : map = 0
⊢ (restrictScalars ℝ map).comp ↑conjCLE = 0
state_after: no goals

tactic: ext1
state_before: f : ℂ → ℂ
z : ℂ
g : ℂ →L[ℝ] ℂ
h₂ : g ≠ 0
map : ℂ →L[ℂ] ℂ
hmap : restrictScalars ℝ map = g.comp ↑conjCLE
⊢ g = (restrictScalars ℝ map).comp ↑conjCLE
state_after: case h
f : ℂ → ℂ
z : ℂ
g : ℂ →L[ℝ] ℂ
h₂ : g ≠ 0
map : ℂ →L[ℂ] ℂ
hmap : restrictScalars ℝ map = g.comp ↑conjCLE
x✝ : ℂ
⊢ g x✝ = ((restrictScalars ℝ map).comp ↑conjCLE) x✝

tactic: simp only [hmap, coe_comp', ContinuousLinearEquiv.coe_coe, Function.comp_apply,
  conjCLE_apply, starRingEnd_self_apply]
state_before: case h
f : ℂ → ℂ
z : ℂ
g : ℂ →L[ℝ] ℂ
h₂ : g ≠ 0
map : ℂ →L[ℂ] ℂ
hmap : restrictScalars ℝ map = g.comp ↑conjCLE
x✝ : ℂ
⊢ g x✝ = ((restrictScalars ℝ map).comp ↑conjCLE) x✝
state_after: no goals