-- Mathlib/Analysis/Normed/Operator/BoundedLinearMaps.lean
theorem isOpen [CompleteSpace E] : IsOpen (range ((↑) : (E ≃L[𝕜] F) → E →L[𝕜] F))

Proof state:
𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
F : Type u_3
inst✝² : SeminormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : CompleteSpace E
⊢ ∀ (i : E ≃L[𝕜] F), range toContinuousLinearMap ∈ 𝓝 ↑i

Tactic:
refine fun e => IsOpen.mem_nhds ?_ (mem_range_self _)

Result:
𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
F : Type u_3
inst✝² : SeminormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : CompleteSpace E
e : E ≃L[𝕜] F
⊢ IsOpen (range toContinuousLinearMap)

Proof state:
case h.e'_3.h.mp.intro
𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
F : Type u_3
inst✝² : SeminormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : CompleteSpace E
e : E ≃L[𝕜] F
O : (E →L[𝕜] F) → E →L[𝕜] E := fun f => (↑e.symm).comp f
h_O : Continuous O
e' : E ≃L[𝕜] F
⊢ ↑e' ∈ O ⁻¹' {x | IsUnit x}

Tactic:
exact ⟨(e'.trans e.symm).toUnit, rfl⟩

Result:
no goals

Proof state:
case h
𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
F : Type u_3
inst✝² : SeminormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : CompleteSpace E
e : E ≃L[𝕜] F
O : (E →L[𝕜] F) → E →L[𝕜] E := fun f => (↑e.symm).comp f
h_O : Continuous O
f' : E →L[𝕜] F
w : (E →L[𝕜] E)ˣ
hw : ↑w = O f'
⊢ ↑(((unitsEquiv 𝕜 E) w).trans e) = f'

Tactic:
ext x

Result:
case h.h
𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
F : Type u_3
inst✝² : SeminormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : CompleteSpace E
e : E ≃L[𝕜] F
O : (E →L[𝕜] F) → E →L[𝕜] E := fun f => (↑e.symm).comp f
h_O : Continuous O
f' : E →L[𝕜] F
w : (E →L[𝕜] E)ˣ
hw : ↑w = O f'
x : E
⊢ ↑(((unitsEquiv 𝕜 E) w).trans e) x = f' x

Proof state:
𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
F : Type u_3
inst✝² : SeminormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : CompleteSpace E
e : E ≃L[𝕜] F
⊢ IsOpen (range toContinuousLinearMap)

Tactic:
let O : (E →L[𝕜] F) → E →L[𝕜] E := fun f => (e.symm : F →L[𝕜] E).comp f

Result:
𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
F : Type u_3
inst✝² : SeminormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : CompleteSpace E
e : E ≃L[𝕜] F
O : (E →L[𝕜] F) → E →L[𝕜] E := fun f => (↑e.symm).comp f
⊢ IsOpen (range toContinuousLinearMap)

Proof state:
case h.e'_3.h.mpr
𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
F : Type u_3
inst✝² : SeminormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : CompleteSpace E
e : E ≃L[𝕜] F
O : (E →L[𝕜] F) → E →L[𝕜] E := fun f => (↑e.symm).comp f
h_O : Continuous O
f' : E →L[𝕜] F
⊢ f' ∈ O ⁻¹' {x | IsUnit x} → f' ∈ range toContinuousLinearMap

Tactic:
rintro ⟨w, hw⟩

Result:
case h.e'_3.h.mpr.intro
𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
F : Type u_3
inst✝² : SeminormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : CompleteSpace E
e : E ≃L[𝕜] F
O : (E →L[𝕜] F) → E →L[𝕜] E := fun f => (↑e.symm).comp f
h_O : Continuous O
f' : E →L[𝕜] F
w : (E →L[𝕜] E)ˣ
hw : ↑w = O f'
⊢ f' ∈ range toContinuousLinearMap

Proof state:
case h.e'_3.h.mp
𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
F : Type u_3
inst✝² : SeminormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : CompleteSpace E
e : E ≃L[𝕜] F
O : (E →L[𝕜] F) → E →L[𝕜] E := fun f => (↑e.symm).comp f
h_O : Continuous O
f' : E →L[𝕜] F
⊢ f' ∈ range toContinuousLinearMap → f' ∈ O ⁻¹' {x | IsUnit x}

Tactic:
rintro ⟨e', rfl⟩

Result:
case h.e'_3.h.mp.intro
𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
F : Type u_3
inst✝² : SeminormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : CompleteSpace E
e : E ≃L[𝕜] F
O : (E →L[𝕜] F) → E →L[𝕜] E := fun f => (↑e.symm).comp f
h_O : Continuous O
e' : E ≃L[𝕜] F
⊢ ↑e' ∈ O ⁻¹' {x | IsUnit x}

Proof state:
case h.e'_3
𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
F : Type u_3
inst✝² : SeminormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : CompleteSpace E
e : E ≃L[𝕜] F
O : (E →L[𝕜] F) → E →L[𝕜] E := fun f => (↑e.symm).comp f
h_O : Continuous O
⊢ range toContinuousLinearMap = O ⁻¹' {x | IsUnit x}

Tactic:
ext f'

Result:
case h.e'_3.h
𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
F : Type u_3
inst✝² : SeminormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : CompleteSpace E
e : E ≃L[𝕜] F
O : (E →L[𝕜] F) → E →L[𝕜] E := fun f => (↑e.symm).comp f
h_O : Continuous O
f' : E →L[𝕜] F
⊢ f' ∈ range toContinuousLinearMap ↔ f' ∈ O ⁻¹' {x | IsUnit x}

Proof state:
𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
F : Type u_3
inst✝² : SeminormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : CompleteSpace E
e : E ≃L[𝕜] F
O : (E →L[𝕜] F) → E →L[𝕜] E := fun f => (↑e.symm).comp f
⊢ IsOpen (range toContinuousLinearMap)

Tactic:
have h_O : Continuous O := isBoundedBilinearMap_comp.continuous_right

Result:
𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
F : Type u_3
inst✝² : SeminormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : CompleteSpace E
e : E ≃L[𝕜] F
O : (E →L[𝕜] F) → E →L[𝕜] E := fun f => (↑e.symm).comp f
h_O : Continuous O
⊢ IsOpen (range toContinuousLinearMap)

Proof state:
case h.h
𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
F : Type u_3
inst✝² : SeminormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : CompleteSpace E
e : E ≃L[𝕜] F
O : (E →L[𝕜] F) → E →L[𝕜] E := fun f => (↑e.symm).comp f
h_O : Continuous O
f' : E →L[𝕜] F
w : (E →L[𝕜] E)ˣ
hw : ↑w = O f'
x : E
⊢ ↑(((unitsEquiv 𝕜 E) w).trans e) x = f' x

Tactic:
simp [O, hw]

Result:
no goals

Proof state:
𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
F : Type u_3
inst✝² : SeminormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : CompleteSpace E
e : E ≃L[𝕜] F
O : (E →L[𝕜] F) → E →L[𝕜] E := fun f => (↑e.symm).comp f
h_O : Continuous O
⊢ IsOpen (range toContinuousLinearMap)

Tactic:
convert show IsOpen (O ⁻¹' { x | IsUnit x }) from Units.isOpen.preimage h_O using 1

Result:
case h.e'_3
𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
F : Type u_3
inst✝² : SeminormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : CompleteSpace E
e : E ≃L[𝕜] F
O : (E →L[𝕜] F) → E →L[𝕜] E := fun f => (↑e.symm).comp f
h_O : Continuous O
⊢ range toContinuousLinearMap = O ⁻¹' {x | IsUnit x}