Theorem: LinearMap.exists_map_addHaar_eq_smul_addHaar'
File: Mathlib/MeasureTheory/Measure/Haar/Disintegration.lean

tactic: have : ProperSpace E := .of_locallyCompactSpace 𝕜
state_before: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹² : NontriviallyNormedField 𝕜
inst✝¹¹ : CompleteSpace 𝕜
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : MeasurableSpace E
inst✝⁸ : BorelSpace E
inst✝⁷ : NormedSpace 𝕜 E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : MeasurableSpace F
inst✝⁴ : BorelSpace F
inst✝³ : NormedSpace 𝕜 F
L : E →ₗ[𝕜] F
μ : Measure E
ν : Measure F
inst✝² : μ.IsAddHaarMeasure
inst✝¹ : ν.IsAddHaarMeasure
inst✝ : LocallyCompactSpace E
h : Function.Surjective ⇑L
⊢ ∃ c, 0 < c ∧ c < ⊤ ∧ map (⇑L) μ = (c * addHaar univ) • ν
state_after: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹² : NontriviallyNormedField 𝕜
inst✝¹¹ : CompleteSpace 𝕜
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : MeasurableSpace E
inst✝⁸ : BorelSpace E
inst✝⁷ : NormedSpace 𝕜 E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : MeasurableSpace F
inst✝⁴ : BorelSpace F
inst✝³ : NormedSpace 𝕜 F
L : E →ₗ[𝕜] F
μ : Measure E
ν : Measure F
inst✝² : μ.IsAddHaarMeasure
inst✝¹ : ν.IsAddHaarMeasure
inst✝ : LocallyCompactSpace E
h : Function.Surjective ⇑L
this : ProperSpace E
⊢ ∃ c, 0 < c ∧ c < ⊤ ∧ map (⇑L) μ = (c * addHaar univ) • ν

tactic: have : FiniteDimensional 𝕜 E := .of_locallyCompactSpace 𝕜
state_before: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹² : NontriviallyNormedField 𝕜
inst✝¹¹ : CompleteSpace 𝕜
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : MeasurableSpace E
inst✝⁸ : BorelSpace E
inst✝⁷ : NormedSpace 𝕜 E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : MeasurableSpace F
inst✝⁴ : BorelSpace F
inst✝³ : NormedSpace 𝕜 F
L : E →ₗ[𝕜] F
μ : Measure E
ν : Measure F
inst✝² : μ.IsAddHaarMeasure
inst✝¹ : ν.IsAddHaarMeasure
inst✝ : LocallyCompactSpace E
h : Function.Surjective ⇑L
this : ProperSpace E
⊢ ∃ c, 0 < c ∧ c < ⊤ ∧ map (⇑L) μ = (c * addHaar univ) • ν
state_after: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹² : NontriviallyNormedField 𝕜
inst✝¹¹ : CompleteSpace 𝕜
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : MeasurableSpace E
inst✝⁸ : BorelSpace E
inst✝⁷ : NormedSpace 𝕜 E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : MeasurableSpace F
inst✝⁴ : BorelSpace F
inst✝³ : NormedSpace 𝕜 F
L : E →ₗ[𝕜] F
μ : Measure E
ν : Measure F
inst✝² : μ.IsAddHaarMeasure
inst✝¹ : ν.IsAddHaarMeasure
inst✝ : LocallyCompactSpace E
h : Function.Surjective ⇑L
this✝ : ProperSpace E
this : FiniteDimensional 𝕜 E
⊢ ∃ c, 0 < c ∧ c < ⊤ ∧ map (⇑L) μ = (c * addHaar univ) • ν

tactic: let S : Submodule 𝕜 E := LinearMap.ker L
state_before: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹² : NontriviallyNormedField 𝕜
inst✝¹¹ : CompleteSpace 𝕜
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : MeasurableSpace E
inst✝⁸ : BorelSpace E
inst✝⁷ : NormedSpace 𝕜 E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : MeasurableSpace F
inst✝⁴ : BorelSpace F
inst✝³ : NormedSpace 𝕜 F
L : E →ₗ[𝕜] F
μ : Measure E
ν : Measure F
inst✝² : μ.IsAddHaarMeasure
inst✝¹ : ν.IsAddHaarMeasure
inst✝ : LocallyCompactSpace E
h : Function.Surjective ⇑L
this✝¹ : ProperSpace E
this✝ : FiniteDimensional 𝕜 E
this : ProperSpace F
⊢ ∃ c, 0 < c ∧ c < ⊤ ∧ map (⇑L) μ = (c * addHaar univ) • ν
state_after: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹² : NontriviallyNormedField 𝕜
inst✝¹¹ : CompleteSpace 𝕜
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : MeasurableSpace E
inst✝⁸ : BorelSpace E
inst✝⁷ : NormedSpace 𝕜 E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : MeasurableSpace F
inst✝⁴ : BorelSpace F
inst✝³ : NormedSpace 𝕜 F
L : E →ₗ[𝕜] F
μ : Measure E
ν : Measure F
inst✝² : μ.IsAddHaarMeasure
inst✝¹ : ν.IsAddHaarMeasure
inst✝ : LocallyCompactSpace E
h : Function.Surjective ⇑L
this✝¹ : ProperSpace E
this✝ : FiniteDimensional 𝕜 E
this : ProperSpace F
S : Submodule 𝕜 E := ker L
⊢ ∃ c, 0 < c ∧ c < ⊤ ∧ map (⇑L) μ = (c * addHaar univ) • ν

tactic: obtain ⟨T, hT⟩ : ∃ T : Submodule 𝕜 E, IsCompl S T := Submodule.exists_isCompl S
state_before: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹² : NontriviallyNormedField 𝕜
inst✝¹¹ : CompleteSpace 𝕜
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : MeasurableSpace E
inst✝⁸ : BorelSpace E
inst✝⁷ : NormedSpace 𝕜 E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : MeasurableSpace F
inst✝⁴ : BorelSpace F
inst✝³ : NormedSpace 𝕜 F
L : E →ₗ[𝕜] F
μ : Measure E
ν : Measure F
inst✝² : μ.IsAddHaarMeasure
inst✝¹ : ν.IsAddHaarMeasure
inst✝ : LocallyCompactSpace E
h : Function.Surjective ⇑L
this✝¹ : ProperSpace E
this✝ : FiniteDimensional 𝕜 E
this : ProperSpace F
S : Submodule 𝕜 E := ker L
⊢ ∃ c, 0 < c ∧ c < ⊤ ∧ map (⇑L) μ = (c * addHaar univ) • ν
state_after: case intro
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹² : NontriviallyNormedField 𝕜
inst✝¹¹ : CompleteSpace 𝕜
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : MeasurableSpace E
inst✝⁸ : BorelSpace E
inst✝⁷ : NormedSpace 𝕜 E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : MeasurableSpace F
inst✝⁴ : BorelSpace F
inst✝³ : NormedSpace 𝕜 F
L : E →ₗ[𝕜] F
μ : Measure E
ν : Measure F
inst✝² : μ.IsAddHaarMeasure
inst✝¹ : ν.IsAddHaarMeasure
inst✝ : LocallyCompactSpace E
h : Function.Surjective ⇑L
this✝¹ : ProperSpace E
this✝ : FiniteDimensional 𝕜 E
this : ProperSpace F
S : Submodule 𝕜 E := ker L
T : Submodule 𝕜 E
hT : IsCompl S T
⊢ ∃ c, 0 < c ∧ c < ⊤ ∧ map (⇑L) μ = (c * addHaar univ) • ν

tactic: let M : (S × T) ≃ₗ[𝕜] E := Submodule.prodEquivOfIsCompl S T hT
state_before: case intro
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹² : NontriviallyNormedField 𝕜
inst✝¹¹ : CompleteSpace 𝕜
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : MeasurableSpace E
inst✝⁸ : BorelSpace E
inst✝⁷ : NormedSpace 𝕜 E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : MeasurableSpace F
inst✝⁴ : BorelSpace F
inst✝³ : NormedSpace 𝕜 F
L : E →ₗ[𝕜] F
μ : Measure E
ν : Measure F
inst✝² : μ.IsAddHaarMeasure
inst✝¹ : ν.IsAddHaarMeasure
inst✝ : LocallyCompactSpace E
h : Function.Surjective ⇑L
this✝¹ : ProperSpace E
this✝ : FiniteDimensional 𝕜 E
this : ProperSpace F
S : Submodule 𝕜 E := ker L
T : Submodule 𝕜 E
hT : IsCompl S T
⊢ ∃ c, 0 < c ∧ c < ⊤ ∧ map (⇑L) μ = (c * addHaar univ) • ν
state_after: case intro
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹² : NontriviallyNormedField 𝕜
inst✝¹¹ : CompleteSpace 𝕜
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : MeasurableSpace E
inst✝⁸ : BorelSpace E
inst✝⁷ : NormedSpace 𝕜 E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : MeasurableSpace F
inst✝⁴ : BorelSpace F
inst✝³ : NormedSpace 𝕜 F
L : E →ₗ[𝕜] F
μ : Measure E
ν : Measure F
inst✝² : μ.IsAddHaarMeasure
inst✝¹ : ν.IsAddHaarMeasure
inst✝ : LocallyCompactSpace E
h : Function.Surjective ⇑L
this✝¹ : ProperSpace E
this✝ : FiniteDimensional 𝕜 E
this : ProperSpace F
S : Submodule 𝕜 E := ker L
T : Submodule 𝕜 E
hT : IsCompl S T
M : (↥S × ↥T) ≃ₗ[𝕜] E := S.prodEquivOfIsCompl T hT
⊢ ∃ c, 0 < c ∧ c < ⊤ ∧ map (⇑L) μ = (c * addHaar univ) • ν

tactic: have M_cont : Continuous M.symm := LinearMap.continuous_of_finiteDimensional _
state_before: case intro
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹² : NontriviallyNormedField 𝕜
inst✝¹¹ : CompleteSpace 𝕜
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : MeasurableSpace E
inst✝⁸ : BorelSpace E
inst✝⁷ : NormedSpace 𝕜 E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : MeasurableSpace F
inst✝⁴ : BorelSpace F
inst✝³ : NormedSpace 𝕜 F
L : E →ₗ[𝕜] F
μ : Measure E
ν : Measure F
inst✝² : μ.IsAddHaarMeasure
inst✝¹ : ν.IsAddHaarMeasure
inst✝ : LocallyCompactSpace E
h : Function.Surjective ⇑L
this✝¹ : ProperSpace E
this✝ : FiniteDimensional 𝕜 E
this : ProperSpace F
S : Submodule 𝕜 E := ker L
T : Submodule 𝕜 E
hT : IsCompl S T
M : (↥S × ↥T) ≃ₗ[𝕜] E := S.prodEquivOfIsCompl T hT
⊢ ∃ c, 0 < c ∧ c < ⊤ ∧ map (⇑L) μ = (c * addHaar univ) • ν
state_after: case intro
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹² : NontriviallyNormedField 𝕜
inst✝¹¹ : CompleteSpace 𝕜
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : MeasurableSpace E
inst✝⁸ : BorelSpace E
inst✝⁷ : NormedSpace 𝕜 E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : MeasurableSpace F
inst✝⁴ : BorelSpace F
inst✝³ : NormedSpace 𝕜 F
L : E →ₗ[𝕜] F
μ : Measure E
ν : Measure F
inst✝² : μ.IsAddHaarMeasure
inst✝¹ : ν.IsAddHaarMeasure
inst✝ : LocallyCompactSpace E
h : Function.Surjective ⇑L
this✝¹ : ProperSpace E
this✝ : FiniteDimensional 𝕜 E
this : ProperSpace F
S : Submodule 𝕜 E := ker L
T : Submodule 𝕜 E
hT : IsCompl S T
M : (↥S × ↥T) ≃ₗ[𝕜] E := S.prodEquivOfIsCompl T hT
M_cont : Continuous ⇑M.symm
⊢ ∃ c, 0 < c ∧ c < ⊤ ∧ map (⇑L) μ = (c * addHaar univ) • ν

tactic: let P : S × T →ₗ[𝕜] T := LinearMap.snd 𝕜 S T
state_before: case intro
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹² : NontriviallyNormedField 𝕜
inst✝¹¹ : CompleteSpace 𝕜
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : MeasurableSpace E
inst✝⁸ : BorelSpace E
inst✝⁷ : NormedSpace 𝕜 E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : MeasurableSpace F
inst✝⁴ : BorelSpace F
inst✝³ : NormedSpace 𝕜 F
L : E →ₗ[𝕜] F
μ : Measure E
ν : Measure F
inst✝² : μ.IsAddHaarMeasure
inst✝¹ : ν.IsAddHaarMeasure
inst✝ : LocallyCompactSpace E
h : Function.Surjective ⇑L
this✝¹ : ProperSpace E
this✝ : FiniteDimensional 𝕜 E
this : ProperSpace F
S : Submodule 𝕜 E := ker L
T : Submodule 𝕜 E
hT : IsCompl S T
M : (↥S × ↥T) ≃ₗ[𝕜] E := S.prodEquivOfIsCompl T hT
M_cont : Continuous ⇑M.symm
⊢ ∃ c, 0 < c ∧ c < ⊤ ∧ map (⇑L) μ = (c * addHaar univ) • ν
state_after: case intro
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹² : NontriviallyNormedField 𝕜
inst✝¹¹ : CompleteSpace 𝕜
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : MeasurableSpace E
inst✝⁸ : BorelSpace E
inst✝⁷ : NormedSpace 𝕜 E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : MeasurableSpace F
inst✝⁴ : BorelSpace F
inst✝³ : NormedSpace 𝕜 F
L : E →ₗ[𝕜] F
μ : Measure E
ν : Measure F
inst✝² : μ.IsAddHaarMeasure
inst✝¹ : ν.IsAddHaarMeasure
inst✝ : LocallyCompactSpace E
h : Function.Surjective ⇑L
this✝¹ : ProperSpace E
this✝ : FiniteDimensional 𝕜 E
this : ProperSpace F
S : Submodule 𝕜 E := ker L
T : Submodule 𝕜 E
hT : IsCompl S T
M : (↥S × ↥T) ≃ₗ[𝕜] E := S.prodEquivOfIsCompl T hT
M_cont : Continuous ⇑M.symm
P : ↥S × ↥T →ₗ[𝕜] ↥T := snd 𝕜 ↥S ↥T
⊢ ∃ c, 0 < c ∧ c < ⊤ ∧ map (⇑L) μ = (c * addHaar univ) • ν

tactic: have P_cont : Continuous P := LinearMap.continuous_of_finiteDimensional _
state_before: case intro
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹² : NontriviallyNormedField 𝕜
inst✝¹¹ : CompleteSpace 𝕜
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : MeasurableSpace E
inst✝⁸ : BorelSpace E
inst✝⁷ : NormedSpace 𝕜 E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : MeasurableSpace F
inst✝⁴ : BorelSpace F
inst✝³ : NormedSpace 𝕜 F
L : E →ₗ[𝕜] F
μ : Measure E
ν : Measure F
inst✝² : μ.IsAddHaarMeasure
inst✝¹ : ν.IsAddHaarMeasure
inst✝ : LocallyCompactSpace E
h : Function.Surjective ⇑L
this✝¹ : ProperSpace E
this✝ : FiniteDimensional 𝕜 E
this : ProperSpace F
S : Submodule 𝕜 E := ker L
T : Submodule 𝕜 E
hT : IsCompl S T
M : (↥S × ↥T) ≃ₗ[𝕜] E := S.prodEquivOfIsCompl T hT
M_cont : Continuous ⇑M.symm
P : ↥S × ↥T →ₗ[𝕜] ↥T := snd 𝕜 ↥S ↥T
⊢ ∃ c, 0 < c ∧ c < ⊤ ∧ map (⇑L) μ = (c * addHaar univ) • ν
state_after: case intro
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹² : NontriviallyNormedField 𝕜
inst✝¹¹ : CompleteSpace 𝕜
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : MeasurableSpace E
inst✝⁸ : BorelSpace E
inst✝⁷ : NormedSpace 𝕜 E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : MeasurableSpace F
inst✝⁴ : BorelSpace F
inst✝³ : NormedSpace 𝕜 F
L : E →ₗ[𝕜] F
μ : Measure E
ν : Measure F
inst✝² : μ.IsAddHaarMeasure
inst✝¹ : ν.IsAddHaarMeasure
inst✝ : LocallyCompactSpace E
h : Function.Surjective ⇑L
this✝¹ : ProperSpace E
this✝ : FiniteDimensional 𝕜 E
this : ProperSpace F
S : Submodule 𝕜 E := ker L
T : Submodule 𝕜 E
hT : IsCompl S T
M : (↥S × ↥T) ≃ₗ[𝕜] E := S.prodEquivOfIsCompl T hT
M_cont : Continuous ⇑M.symm
P : ↥S × ↥T →ₗ[𝕜] ↥T := snd 𝕜 ↥S ↥T
P_cont : Continuous ⇑P
⊢ ∃ c, 0 < c ∧ c < ⊤ ∧ map (⇑L) μ = (c * addHaar univ) • ν

tactic: have I : Function.Bijective (LinearMap.domRestrict L T) :=
  ⟨LinearMap.injective_domRestrict_iff.2 (IsCompl.inf_eq_bot hT.symm),
  (LinearMap.surjective_domRestrict_iff h).2 hT.symm.sup_eq_top⟩
state_before: case intro
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹² : NontriviallyNormedField 𝕜
inst✝¹¹ : CompleteSpace 𝕜
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : MeasurableSpace E
inst✝⁸ : BorelSpace E
inst✝⁷ : NormedSpace 𝕜 E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : MeasurableSpace F
inst✝⁴ : BorelSpace F
inst✝³ : NormedSpace 𝕜 F
L : E →ₗ[𝕜] F
μ : Measure E
ν : Measure F
inst✝² : μ.IsAddHaarMeasure
inst✝¹ : ν.IsAddHaarMeasure
inst✝ : LocallyCompactSpace E
h : Function.Surjective ⇑L
this✝¹ : ProperSpace E
this✝ : FiniteDimensional 𝕜 E
this : ProperSpace F
S : Submodule 𝕜 E := ker L
T : Submodule 𝕜 E
hT : IsCompl S T
M : (↥S × ↥T) ≃ₗ[𝕜] E := S.prodEquivOfIsCompl T hT
M_cont : Continuous ⇑M.symm
P : ↥S × ↥T →ₗ[𝕜] ↥T := snd 𝕜 ↥S ↥T
P_cont : Continuous ⇑P
⊢ ∃ c, 0 < c ∧ c < ⊤ ∧ map (⇑L) μ = (c * addHaar univ) • ν
state_after: case intro
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹² : NontriviallyNormedField 𝕜
inst✝¹¹ : CompleteSpace 𝕜
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : MeasurableSpace E
inst✝⁸ : BorelSpace E
inst✝⁷ : NormedSpace 𝕜 E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : MeasurableSpace F
inst✝⁴ : BorelSpace F
inst✝³ : NormedSpace 𝕜 F
L : E →ₗ[𝕜] F
μ : Measure E
ν : Measure F
inst✝² : μ.IsAddHaarMeasure
inst✝¹ : ν.IsAddHaarMeasure
inst✝ : LocallyCompactSpace E
h : Function.Surjective ⇑L
this✝¹ : ProperSpace E
this✝ : FiniteDimensional 𝕜 E
this : ProperSpace F
S : Submodule 𝕜 E := ker L
T : Submodule 𝕜 E
hT : IsCompl S T
M : (↥S × ↥T) ≃ₗ[𝕜] E := S.prodEquivOfIsCompl T hT
M_cont : Continuous ⇑M.symm
P : ↥S × ↥T →ₗ[𝕜] ↥T := snd 𝕜 ↥S ↥T
P_cont : Continuous ⇑P
I : Function.Bijective ⇑(L.domRestrict T)
⊢ ∃ c, 0 < c ∧ c < ⊤ ∧ map (⇑L) μ = (c * addHaar univ) • ν

tactic: let L' : T ≃ₗ[𝕜] F := LinearEquiv.ofBijective (LinearMap.domRestrict L T) I
state_before: case intro
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹² : NontriviallyNormedField 𝕜
inst✝¹¹ : CompleteSpace 𝕜
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : MeasurableSpace E
inst✝⁸ : BorelSpace E
inst✝⁷ : NormedSpace 𝕜 E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : MeasurableSpace F
inst✝⁴ : BorelSpace F
inst✝³ : NormedSpace 𝕜 F
L : E →ₗ[𝕜] F
μ : Measure E
ν : Measure F
inst✝² : μ.IsAddHaarMeasure
inst✝¹ : ν.IsAddHaarMeasure
inst✝ : LocallyCompactSpace E
h : Function.Surjective ⇑L
this✝¹ : ProperSpace E
this✝ : FiniteDimensional 𝕜 E
this : ProperSpace F
S : Submodule 𝕜 E := ker L
T : Submodule 𝕜 E
hT : IsCompl S T
M : (↥S × ↥T) ≃ₗ[𝕜] E := S.prodEquivOfIsCompl T hT
M_cont : Continuous ⇑M.symm
P : ↥S × ↥T →ₗ[𝕜] ↥T := snd 𝕜 ↥S ↥T
P_cont : Continuous ⇑P
I : Function.Bijective ⇑(L.domRestrict T)
⊢ ∃ c, 0 < c ∧ c < ⊤ ∧ map (⇑L) μ = (c * addHaar univ) • ν
state_after: case intro
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹² : NontriviallyNormedField 𝕜
inst✝¹¹ : CompleteSpace 𝕜
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : MeasurableSpace E
inst✝⁸ : BorelSpace E
inst✝⁷ : NormedSpace 𝕜 E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : MeasurableSpace F
inst✝⁴ : BorelSpace F
inst✝³ : NormedSpace 𝕜 F
L : E →ₗ[𝕜] F
μ : Measure E
ν : Measure F
inst✝² : μ.IsAddHaarMeasure
inst✝¹ : ν.IsAddHaarMeasure
inst✝ : LocallyCompactSpace E
h : Function.Surjective ⇑L
this✝¹ : ProperSpace E
this✝ : FiniteDimensional 𝕜 E
this : ProperSpace F
S : Submodule 𝕜 E := ker L
T : Submodule 𝕜 E
hT : IsCompl S T
M : (↥S × ↥T) ≃ₗ[𝕜] E := S.prodEquivOfIsCompl T hT
M_cont : Continuous ⇑M.symm
P : ↥S × ↥T →ₗ[𝕜] ↥T := snd 𝕜 ↥S ↥T
P_cont : Continuous ⇑P
I : Function.Bijective ⇑(L.domRestrict T)
L' : ↥T ≃ₗ[𝕜] F := LinearEquiv.ofBijective (L.domRestrict T) I
⊢ ∃ c, 0 < c ∧ c < ⊤ ∧ map (⇑L) μ = (c * addHaar univ) • ν

tactic: have L'_cont : Continuous L' := LinearMap.continuous_of_finiteDimensional _
state_before: case intro
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹² : NontriviallyNormedField 𝕜
inst✝¹¹ : CompleteSpace 𝕜
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : MeasurableSpace E
inst✝⁸ : BorelSpace E
inst✝⁷ : NormedSpace 𝕜 E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : MeasurableSpace F
inst✝⁴ : BorelSpace F
inst✝³ : NormedSpace 𝕜 F
L : E →ₗ[𝕜] F
μ : Measure E
ν : Measure F
inst✝² : μ.IsAddHaarMeasure
inst✝¹ : ν.IsAddHaarMeasure
inst✝ : LocallyCompactSpace E
h : Function.Surjective ⇑L
this✝¹ : ProperSpace E
this✝ : FiniteDimensional 𝕜 E
this : ProperSpace F
S : Submodule 𝕜 E := ker L
T : Submodule 𝕜 E
hT : IsCompl S T
M : (↥S × ↥T) ≃ₗ[𝕜] E := S.prodEquivOfIsCompl T hT
M_cont : Continuous ⇑M.symm
P : ↥S × ↥T →ₗ[𝕜] ↥T := snd 𝕜 ↥S ↥T
P_cont : Continuous ⇑P
I : Function.Bijective ⇑(L.domRestrict T)
L' : ↥T ≃ₗ[𝕜] F := LinearEquiv.ofBijective (L.domRestrict T) I
⊢ ∃ c, 0 < c ∧ c < ⊤ ∧ map (⇑L) μ = (c * addHaar univ) • ν
state_after: case intro
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹² : NontriviallyNormedField 𝕜
inst✝¹¹ : CompleteSpace 𝕜
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : MeasurableSpace E
inst✝⁸ : BorelSpace E
inst✝⁷ : NormedSpace 𝕜 E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : MeasurableSpace F
inst✝⁴ : BorelSpace F
inst✝³ : NormedSpace 𝕜 F
L : E →ₗ[𝕜] F
μ : Measure E
ν : Measure F
inst✝² : μ.IsAddHaarMeasure
inst✝¹ : ν.IsAddHaarMeasure
inst✝ : LocallyCompactSpace E
h : Function.Surjective ⇑L
this✝¹ : ProperSpace E
this✝ : FiniteDimensional 𝕜 E
this : ProperSpace F
S : Submodule 𝕜 E := ker L
T : Submodule 𝕜 E
hT : IsCompl S T
M : (↥S × ↥T) ≃ₗ[𝕜] E := S.prodEquivOfIsCompl T hT
M_cont : Continuous ⇑M.symm
P : ↥S × ↥T →ₗ[𝕜] ↥T := snd 𝕜 ↥S ↥T
P_cont : Continuous ⇑P
I : Function.Bijective ⇑(L.domRestrict T)
L' : ↥T ≃ₗ[𝕜] F := LinearEquiv.ofBijective (L.domRestrict T) I
L'_cont : Continuous ⇑L'
⊢ ∃ c, 0 < c ∧ c < ⊤ ∧ map (⇑L) μ = (c * addHaar univ) • ν

tactic: have A : L = (L' : T →ₗ[𝕜] F).comp (P.comp (M.symm : E →ₗ[𝕜] (S × T))) := by
  ext x
  obtain ⟨y, z, hyz⟩ : ∃ (y : S) (z : T), M.symm x = (y, z) := ⟨_, _, rfl⟩
  have : x = M (y, z) := by
    rw [← hyz]; simp only [LinearEquiv.apply_symm_apply]
  simp [L', P, M, this]
state_before: case intro
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹² : NontriviallyNormedField 𝕜
inst✝¹¹ : CompleteSpace 𝕜
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : MeasurableSpace E
inst✝⁸ : BorelSpace E
inst✝⁷ : NormedSpace 𝕜 E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : MeasurableSpace F
inst✝⁴ : BorelSpace F
inst✝³ : NormedSpace 𝕜 F
L : E →ₗ[𝕜] F
μ : Measure E
ν : Measure F
inst✝² : μ.IsAddHaarMeasure
inst✝¹ : ν.IsAddHaarMeasure
inst✝ : LocallyCompactSpace E
h : Function.Surjective ⇑L
this✝¹ : ProperSpace E
this✝ : FiniteDimensional 𝕜 E
this : ProperSpace F
S : Submodule 𝕜 E := ker L
T : Submodule 𝕜 E
hT : IsCompl S T
M : (↥S × ↥T) ≃ₗ[𝕜] E := S.prodEquivOfIsCompl T hT
M_cont : Continuous ⇑M.symm
P : ↥S × ↥T →ₗ[𝕜] ↥T := snd 𝕜 ↥S ↥T
P_cont : Continuous ⇑P
I : Function.Bijective ⇑(L.domRestrict T)
L' : ↥T ≃ₗ[𝕜] F := LinearEquiv.ofBijective (L.domRestrict T) I
L'_cont : Continuous ⇑L'
⊢ ∃ c, 0 < c ∧ c < ⊤ ∧ map (⇑L) μ = (c * addHaar univ) • ν
state_after: case intro
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹² : NontriviallyNormedField 𝕜
inst✝¹¹ : CompleteSpace 𝕜
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : MeasurableSpace E
inst✝⁸ : BorelSpace E
inst✝⁷ : NormedSpace 𝕜 E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : MeasurableSpace F
inst✝⁴ : BorelSpace F
inst✝³ : NormedSpace 𝕜 F
L : E →ₗ[𝕜] F
μ : Measure E
ν : Measure F
inst✝² : μ.IsAddHaarMeasure
inst✝¹ : ν.IsAddHaarMeasure
inst✝ : LocallyCompactSpace E
h : Function.Surjective ⇑L
this✝¹ : ProperSpace E
this✝ : FiniteDimensional 𝕜 E
this : ProperSpace F
S : Submodule 𝕜 E := ker L
T : Submodule 𝕜 E
hT : IsCompl S T
M : (↥S × ↥T) ≃ₗ[𝕜] E := S.prodEquivOfIsCompl T hT
M_cont : Continuous ⇑M.symm
P : ↥S × ↥T →ₗ[𝕜] ↥T := snd 𝕜 ↥S ↥T
P_cont : Continuous ⇑P
I : Function.Bijective ⇑(L.domRestrict T)
L' : ↥T ≃ₗ[𝕜] F := LinearEquiv.ofBijective (L.domRestrict T) I
L'_cont : Continuous ⇑L'
A : L = ↑L' ∘ₗ P ∘ₗ ↑M.symm
⊢ ∃ c, 0 < c ∧ c < ⊤ ∧ map (⇑L) μ = (c * addHaar univ) • ν

tactic: let μS : Measure S := addHaar
state_before: case intro
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹² : NontriviallyNormedField 𝕜
inst✝¹¹ : CompleteSpace 𝕜
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : MeasurableSpace E
inst✝⁸ : BorelSpace E
inst✝⁷ : NormedSpace 𝕜 E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : MeasurableSpace F
inst✝⁴ : BorelSpace F
inst✝³ : NormedSpace 𝕜 F
L : E →ₗ[𝕜] F
μ : Measure E
ν : Measure F
inst✝² : μ.IsAddHaarMeasure
inst✝¹ : ν.IsAddHaarMeasure
inst✝ : LocallyCompactSpace E
h : Function.Surjective ⇑L
this✝¹ : ProperSpace E
this✝ : FiniteDimensional 𝕜 E
this : ProperSpace F
S : Submodule 𝕜 E := ker L
T : Submodule 𝕜 E
hT : IsCompl S T
M : (↥S × ↥T) ≃ₗ[𝕜] E := S.prodEquivOfIsCompl T hT
M_cont : Continuous ⇑M.symm
P : ↥S × ↥T →ₗ[𝕜] ↥T := snd 𝕜 ↥S ↥T
P_cont : Continuous ⇑P
I✝ : Function.Bijective ⇑(L.domRestrict T)
L' : ↥T ≃ₗ[𝕜] F := LinearEquiv.ofBijective (L.domRestrict T) I✝
L'_cont : Continuous ⇑L'
A : L = ↑L' ∘ₗ P ∘ₗ ↑M.symm
I : map (⇑L) μ = map (⇑L') (map (⇑P) (map (⇑M.symm) μ))
⊢ ∃ c, 0 < c ∧ c < ⊤ ∧ map (⇑L) μ = (c * addHaar univ) • ν
state_after: case intro
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹² : NontriviallyNormedField 𝕜
inst✝¹¹ : CompleteSpace 𝕜
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : MeasurableSpace E
inst✝⁸ : BorelSpace E
inst✝⁷ : NormedSpace 𝕜 E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : MeasurableSpace F
inst✝⁴ : BorelSpace F
inst✝³ : NormedSpace 𝕜 F
L : E →ₗ[𝕜] F
μ : Measure E
ν : Measure F
inst✝² : μ.IsAddHaarMeasure
inst✝¹ : ν.IsAddHaarMeasure
inst✝ : LocallyCompactSpace E
h : Function.Surjective ⇑L
this✝¹ : ProperSpace E
this✝ : FiniteDimensional 𝕜 E
this : ProperSpace F
S : Submodule 𝕜 E := ker L
T : Submodule 𝕜 E
hT : IsCompl S T
M : (↥S × ↥T) ≃ₗ[𝕜] E := S.prodEquivOfIsCompl T hT
M_cont : Continuous ⇑M.symm
P : ↥S × ↥T →ₗ[𝕜] ↥T := snd 𝕜 ↥S ↥T
P_cont : Continuous ⇑P
I✝ : Function.Bijective ⇑(L.domRestrict T)
L' : ↥T ≃ₗ[𝕜] F := LinearEquiv.ofBijective (L.domRestrict T) I✝
L'_cont : Continuous ⇑L'
A : L = ↑L' ∘ₗ P ∘ₗ ↑M.symm
I : map (⇑L) μ = map (⇑L') (map (⇑P) (map (⇑M.symm) μ))
μS : Measure ↥S := addHaar
⊢ ∃ c, 0 < c ∧ c < ⊤ ∧ map (⇑L) μ = (c * addHaar univ) • ν

tactic: let μT : Measure T := addHaar
state_before: case intro
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹² : NontriviallyNormedField 𝕜
inst✝¹¹ : CompleteSpace 𝕜
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : MeasurableSpace E
inst✝⁸ : BorelSpace E
inst✝⁷ : NormedSpace 𝕜 E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : MeasurableSpace F
inst✝⁴ : BorelSpace F
inst✝³ : NormedSpace 𝕜 F
L : E →ₗ[𝕜] F
μ : Measure E
ν : Measure F
inst✝² : μ.IsAddHaarMeasure
inst✝¹ : ν.IsAddHaarMeasure
inst✝ : LocallyCompactSpace E
h : Function.Surjective ⇑L
this✝¹ : ProperSpace E
this✝ : FiniteDimensional 𝕜 E
this : ProperSpace F
S : Submodule 𝕜 E := ker L
T : Submodule 𝕜 E
hT : IsCompl S T
M : (↥S × ↥T) ≃ₗ[𝕜] E := S.prodEquivOfIsCompl T hT
M_cont : Continuous ⇑M.symm
P : ↥S × ↥T →ₗ[𝕜] ↥T := snd 𝕜 ↥S ↥T
P_cont : Continuous ⇑P
I✝ : Function.Bijective ⇑(L.domRestrict T)
L' : ↥T ≃ₗ[𝕜] F := LinearEquiv.ofBijective (L.domRestrict T) I✝
L'_cont : Continuous ⇑L'
A : L = ↑L' ∘ₗ P ∘ₗ ↑M.symm
I : map (⇑L) μ = map (⇑L') (map (⇑P) (map (⇑M.symm) μ))
μS : Measure ↥S := addHaar
⊢ ∃ c, 0 < c ∧ c < ⊤ ∧ map (⇑L) μ = (c * addHaar univ) • ν
state_after: case intro
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹² : NontriviallyNormedField 𝕜
inst✝¹¹ : CompleteSpace 𝕜
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : MeasurableSpace E
inst✝⁸ : BorelSpace E
inst✝⁷ : NormedSpace 𝕜 E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : MeasurableSpace F
inst✝⁴ : BorelSpace F
inst✝³ : NormedSpace 𝕜 F
L : E →ₗ[𝕜] F
μ : Measure E
ν : Measure F
inst✝² : μ.IsAddHaarMeasure
inst✝¹ : ν.IsAddHaarMeasure
inst✝ : LocallyCompactSpace E
h : Function.Surjective ⇑L
this✝¹ : ProperSpace E
this✝ : FiniteDimensional 𝕜 E
this : ProperSpace F
S : Submodule 𝕜 E := ker L
T : Submodule 𝕜 E
hT : IsCompl S T
M : (↥S × ↥T) ≃ₗ[𝕜] E := S.prodEquivOfIsCompl T hT
M_cont : Continuous ⇑M.symm
P : ↥S × ↥T →ₗ[𝕜] ↥T := snd 𝕜 ↥S ↥T
P_cont : Continuous ⇑P
I✝ : Function.Bijective ⇑(L.domRestrict T)
L' : ↥T ≃ₗ[𝕜] F := LinearEquiv.ofBijective (L.domRestrict T) I✝
L'_cont : Continuous ⇑L'
A : L = ↑L' ∘ₗ P ∘ₗ ↑M.symm
I : map (⇑L) μ = map (⇑L') (map (⇑P) (map (⇑M.symm) μ))
μS : Measure ↥S := addHaar
μT : Measure ↥T := addHaar
⊢ ∃ c, 0 < c ∧ c < ⊤ ∧ map (⇑L) μ = (c * addHaar univ) • ν

tactic: obtain ⟨c₀, c₀_pos, c₀_fin, h₀⟩ :
    ∃ c₀ : ℝ≥0∞, c₀ ≠ 0 ∧ c₀ ≠ ∞ ∧ μ.map M.symm = c₀ • μS.prod μT := by
  have : IsAddHaarMeasure (μ.map M.symm) :=
    M.toContinuousLinearEquiv.symm.isAddHaarMeasure_map μ
  refine ⟨addHaarScalarFactor (μ.map M.symm) (μS.prod μT), ?_, ENNReal.coe_ne_top,
    isAddLeftInvariant_eq_smul _ _⟩
  simpa only [ne_eq, ENNReal.coe_eq_zero] using
    (addHaarScalarFactor_pos_of_isAddHaarMeasure (μ.map M.symm) (μS.prod μT)).ne'
state_before: case intro
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹² : NontriviallyNormedField 𝕜
inst✝¹¹ : CompleteSpace 𝕜
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : MeasurableSpace E
inst✝⁸ : BorelSpace E
inst✝⁷ : NormedSpace 𝕜 E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : MeasurableSpace F
inst✝⁴ : BorelSpace F
inst✝³ : NormedSpace 𝕜 F
L : E →ₗ[𝕜] F
μ : Measure E
ν : Measure F
inst✝² : μ.IsAddHaarMeasure
inst✝¹ : ν.IsAddHaarMeasure
inst✝ : LocallyCompactSpace E
h : Function.Surjective ⇑L
this✝¹ : ProperSpace E
this✝ : FiniteDimensional 𝕜 E
this : ProperSpace F
S : Submodule 𝕜 E := ker L
T : Submodule 𝕜 E
hT : IsCompl S T
M : (↥S × ↥T) ≃ₗ[𝕜] E := S.prodEquivOfIsCompl T hT
M_cont : Continuous ⇑M.symm
P : ↥S × ↥T →ₗ[𝕜] ↥T := snd 𝕜 ↥S ↥T
P_cont : Continuous ⇑P
I✝ : Function.Bijective ⇑(L.domRestrict T)
L' : ↥T ≃ₗ[𝕜] F := LinearEquiv.ofBijective (L.domRestrict T) I✝
L'_cont : Continuous ⇑L'
A : L = ↑L' ∘ₗ P ∘ₗ ↑M.symm
I : map (⇑L) μ = map (⇑L') (map (⇑P) (map (⇑M.symm) μ))
μS : Measure ↥S := addHaar
μT : Measure ↥T := addHaar
⊢ ∃ c, 0 < c ∧ c < ⊤ ∧ map (⇑L) μ = (c * addHaar univ) • ν
state_after: case intro.intro.intro.intro
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹² : NontriviallyNormedField 𝕜
inst✝¹¹ : CompleteSpace 𝕜
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : MeasurableSpace E
inst✝⁸ : BorelSpace E
inst✝⁷ : NormedSpace 𝕜 E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : MeasurableSpace F
inst✝⁴ : BorelSpace F
inst✝³ : NormedSpace 𝕜 F
L : E →ₗ[𝕜] F
μ : Measure E
ν : Measure F
inst✝² : μ.IsAddHaarMeasure
inst✝¹ : ν.IsAddHaarMeasure
inst✝ : LocallyCompactSpace E
h : Function.Surjective ⇑L
this✝¹ : ProperSpace E
this✝ : FiniteDimensional 𝕜 E
this : ProperSpace F
S : Submodule 𝕜 E := ker L
T : Submodule 𝕜 E
hT : IsCompl S T
M : (↥S × ↥T) ≃ₗ[𝕜] E := S.prodEquivOfIsCompl T hT
M_cont : Continuous ⇑M.symm
P : ↥S × ↥T →ₗ[𝕜] ↥T := snd 𝕜 ↥S ↥T
P_cont : Continuous ⇑P
I✝ : Function.Bijective ⇑(L.domRestrict T)
L' : ↥T ≃ₗ[𝕜] F := LinearEquiv.ofBijective (L.domRestrict T) I✝
L'_cont : Continuous ⇑L'
A : L = ↑L' ∘ₗ P ∘ₗ ↑M.symm
I : map (⇑L) μ = map (⇑L') (map (⇑P) (map (⇑M.symm) μ))
μS : Measure ↥S := addHaar
μT : Measure ↥T := addHaar
c₀ : ℝ≥0∞
c₀_pos : c₀ ≠ 0
c₀_fin : c₀ ≠ ⊤
h₀ : map (⇑M.symm) μ = c₀ • μS.prod μT
⊢ ∃ c, 0 < c ∧ c < ⊤ ∧ map (⇑L) μ = (c * addHaar univ) • ν

tactic: have J : (μS.prod μT).map P = (μS univ) • μT := map_snd_prod
state_before: case intro.intro.intro.intro
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹² : NontriviallyNormedField 𝕜
inst✝¹¹ : CompleteSpace 𝕜
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : MeasurableSpace E
inst✝⁸ : BorelSpace E
inst✝⁷ : NormedSpace 𝕜 E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : MeasurableSpace F
inst✝⁴ : BorelSpace F
inst✝³ : NormedSpace 𝕜 F
L : E →ₗ[𝕜] F
μ : Measure E
ν : Measure F
inst✝² : μ.IsAddHaarMeasure
inst✝¹ : ν.IsAddHaarMeasure
inst✝ : LocallyCompactSpace E
h : Function.Surjective ⇑L
this✝¹ : ProperSpace E
this✝ : FiniteDimensional 𝕜 E
this : ProperSpace F
S : Submodule 𝕜 E := ker L
T : Submodule 𝕜 E
hT : IsCompl S T
M : (↥S × ↥T) ≃ₗ[𝕜] E := S.prodEquivOfIsCompl T hT
M_cont : Continuous ⇑M.symm
P : ↥S × ↥T →ₗ[𝕜] ↥T := snd 𝕜 ↥S ↥T
P_cont : Continuous ⇑P
I✝ : Function.Bijective ⇑(L.domRestrict T)
L' : ↥T ≃ₗ[𝕜] F := LinearEquiv.ofBijective (L.domRestrict T) I✝
L'_cont : Continuous ⇑L'
A : L = ↑L' ∘ₗ P ∘ₗ ↑M.symm
I : map (⇑L) μ = map (⇑L') (map (⇑P) (map (⇑M.symm) μ))
μS : Measure ↥S := addHaar
μT : Measure ↥T := addHaar
c₀ : ℝ≥0∞
c₀_pos : c₀ ≠ 0
c₀_fin : c₀ ≠ ⊤
h₀ : map (⇑M.symm) μ = c₀ • μS.prod μT
⊢ ∃ c, 0 < c ∧ c < ⊤ ∧ map (⇑L) μ = (c * addHaar univ) • ν
state_after: case intro.intro.intro.intro
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹² : NontriviallyNormedField 𝕜
inst✝¹¹ : CompleteSpace 𝕜
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : MeasurableSpace E
inst✝⁸ : BorelSpace E
inst✝⁷ : NormedSpace 𝕜 E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : MeasurableSpace F
inst✝⁴ : BorelSpace F
inst✝³ : NormedSpace 𝕜 F
L : E →ₗ[𝕜] F
μ : Measure E
ν : Measure F
inst✝² : μ.IsAddHaarMeasure
inst✝¹ : ν.IsAddHaarMeasure
inst✝ : LocallyCompactSpace E
h : Function.Surjective ⇑L
this✝¹ : ProperSpace E
this✝ : FiniteDimensional 𝕜 E
this : ProperSpace F
S : Submodule 𝕜 E := ker L
T : Submodule 𝕜 E
hT : IsCompl S T
M : (↥S × ↥T) ≃ₗ[𝕜] E := S.prodEquivOfIsCompl T hT
M_cont : Continuous ⇑M.symm
P : ↥S × ↥T →ₗ[𝕜] ↥T := snd 𝕜 ↥S ↥T
P_cont : Continuous ⇑P
I✝ : Function.Bijective ⇑(L.domRestrict T)
L' : ↥T ≃ₗ[𝕜] F := LinearEquiv.ofBijective (L.domRestrict T) I✝
L'_cont : Continuous ⇑L'
A : L = ↑L' ∘ₗ P ∘ₗ ↑M.symm
I : map (⇑L) μ = map (⇑L') (map (⇑P) (map (⇑M.symm) μ))
μS : Measure ↥S := addHaar
μT : Measure ↥T := addHaar
c₀ : ℝ≥0∞
c₀_pos : c₀ ≠ 0
c₀_fin : c₀ ≠ ⊤
h₀ : map (⇑M.symm) μ = c₀ • μS.prod μT
J : map (⇑P) (μS.prod μT) = μS univ • μT
⊢ ∃ c, 0 < c ∧ c < ⊤ ∧ map (⇑L) μ = (c * addHaar univ) • ν

tactic: obtain ⟨c₁, c₁_pos, c₁_fin, h₁⟩ : ∃ c₁ : ℝ≥0∞, c₁ ≠ 0 ∧ c₁ ≠ ∞ ∧ μT.map L' = c₁ • ν := by
  have : IsAddHaarMeasure (μT.map L') :=
    L'.toContinuousLinearEquiv.isAddHaarMeasure_map μT
  refine ⟨addHaarScalarFactor (μT.map L') ν, ?_, ENNReal.coe_ne_top,
    isAddLeftInvariant_eq_smul _ _⟩
  simpa only [ne_eq, ENNReal.coe_eq_zero] using
    (addHaarScalarFactor_pos_of_isAddHaarMeasure (μT.map L') ν).ne'
state_before: case intro.intro.intro.intro
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹² : NontriviallyNormedField 𝕜
inst✝¹¹ : CompleteSpace 𝕜
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : MeasurableSpace E
inst✝⁸ : BorelSpace E
inst✝⁷ : NormedSpace 𝕜 E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : MeasurableSpace F
inst✝⁴ : BorelSpace F
inst✝³ : NormedSpace 𝕜 F
L : E →ₗ[𝕜] F
μ : Measure E
ν : Measure F
inst✝² : μ.IsAddHaarMeasure
inst✝¹ : ν.IsAddHaarMeasure
inst✝ : LocallyCompactSpace E
h : Function.Surjective ⇑L
this✝¹ : ProperSpace E
this✝ : FiniteDimensional 𝕜 E
this : ProperSpace F
S : Submodule 𝕜 E := ker L
T : Submodule 𝕜 E
hT : IsCompl S T
M : (↥S × ↥T) ≃ₗ[𝕜] E := S.prodEquivOfIsCompl T hT
M_cont : Continuous ⇑M.symm
P : ↥S × ↥T →ₗ[𝕜] ↥T := snd 𝕜 ↥S ↥T
P_cont : Continuous ⇑P
I✝ : Function.Bijective ⇑(L.domRestrict T)
L' : ↥T ≃ₗ[𝕜] F := LinearEquiv.ofBijective (L.domRestrict T) I✝
L'_cont : Continuous ⇑L'
A : L = ↑L' ∘ₗ P ∘ₗ ↑M.symm
I : map (⇑L) μ = map (⇑L') (map (⇑P) (map (⇑M.symm) μ))
μS : Measure ↥S := addHaar
μT : Measure ↥T := addHaar
c₀ : ℝ≥0∞
c₀_pos : c₀ ≠ 0
c₀_fin : c₀ ≠ ⊤
h₀ : map (⇑M.symm) μ = c₀ • μS.prod μT
J : map (⇑P) (μS.prod μT) = μS univ • μT
⊢ ∃ c, 0 < c ∧ c < ⊤ ∧ map (⇑L) μ = (c * addHaar univ) • ν
state_after: case intro.intro.intro.intro.intro.intro.intro
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹² : NontriviallyNormedField 𝕜
inst✝¹¹ : CompleteSpace 𝕜
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : MeasurableSpace E
inst✝⁸ : BorelSpace E
inst✝⁷ : NormedSpace 𝕜 E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : MeasurableSpace F
inst✝⁴ : BorelSpace F
inst✝³ : NormedSpace 𝕜 F
L : E →ₗ[𝕜] F
μ : Measure E
ν : Measure F
inst✝² : μ.IsAddHaarMeasure
inst✝¹ : ν.IsAddHaarMeasure
inst✝ : LocallyCompactSpace E
h : Function.Surjective ⇑L
this✝¹ : ProperSpace E
this✝ : FiniteDimensional 𝕜 E
this : ProperSpace F
S : Submodule 𝕜 E := ker L
T : Submodule 𝕜 E
hT : IsCompl S T
M : (↥S × ↥T) ≃ₗ[𝕜] E := S.prodEquivOfIsCompl T hT
M_cont : Continuous ⇑M.symm
P : ↥S × ↥T →ₗ[𝕜] ↥T := snd 𝕜 ↥S ↥T
P_cont : Continuous ⇑P
I✝ : Function.Bijective ⇑(L.domRestrict T)
L' : ↥T ≃ₗ[𝕜] F := LinearEquiv.ofBijective (L.domRestrict T) I✝
L'_cont : Continuous ⇑L'
A : L = ↑L' ∘ₗ P ∘ₗ ↑M.symm
I : map (⇑L) μ = map (⇑L') (map (⇑P) (map (⇑M.symm) μ))
μS : Measure ↥S := addHaar
μT : Measure ↥T := addHaar
c₀ : ℝ≥0∞
c₀_pos : c₀ ≠ 0
c₀_fin : c₀ ≠ ⊤
h₀ : map (⇑M.symm) μ = c₀ • μS.prod μT
J : map (⇑P) (μS.prod μT) = μS univ • μT
c₁ : ℝ≥0∞
c₁_pos : c₁ ≠ 0
c₁_fin : c₁ ≠ ⊤
h₁ : map (⇑L') μT = c₁ • ν
⊢ ∃ c, 0 < c ∧ c < ⊤ ∧ map (⇑L) μ = (c * addHaar univ) • ν

tactic: refine ⟨c₀ * c₁, by simp [pos_iff_ne_zero, c₀_pos, c₁_pos], ENNReal.mul_lt_top c₀_fin c₁_fin, ?_⟩
state_before: case intro.intro.intro.intro.intro.intro.intro
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹² : NontriviallyNormedField 𝕜
inst✝¹¹ : CompleteSpace 𝕜
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : MeasurableSpace E
inst✝⁸ : BorelSpace E
inst✝⁷ : NormedSpace 𝕜 E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : MeasurableSpace F
inst✝⁴ : BorelSpace F
inst✝³ : NormedSpace 𝕜 F
L : E →ₗ[𝕜] F
μ : Measure E
ν : Measure F
inst✝² : μ.IsAddHaarMeasure
inst✝¹ : ν.IsAddHaarMeasure
inst✝ : LocallyCompactSpace E
h : Function.Surjective ⇑L
this✝¹ : ProperSpace E
this✝ : FiniteDimensional 𝕜 E
this : ProperSpace F
S : Submodule 𝕜 E := ker L
T : Submodule 𝕜 E
hT : IsCompl S T
M : (↥S × ↥T) ≃ₗ[𝕜] E := S.prodEquivOfIsCompl T hT
M_cont : Continuous ⇑M.symm
P : ↥S × ↥T →ₗ[𝕜] ↥T := snd 𝕜 ↥S ↥T
P_cont : Continuous ⇑P
I✝ : Function.Bijective ⇑(L.domRestrict T)
L' : ↥T ≃ₗ[𝕜] F := LinearEquiv.ofBijective (L.domRestrict T) I✝
L'_cont : Continuous ⇑L'
A : L = ↑L' ∘ₗ P ∘ₗ ↑M.symm
I : map (⇑L) μ = map (⇑L') (map (⇑P) (map (⇑M.symm) μ))
μS : Measure ↥S := addHaar
μT : Measure ↥T := addHaar
c₀ : ℝ≥0∞
c₀_pos : c₀ ≠ 0
c₀_fin : c₀ ≠ ⊤
h₀ : map (⇑M.symm) μ = c₀ • μS.prod μT
J : map (⇑P) (μS.prod μT) = μS univ • μT
c₁ : ℝ≥0∞
c₁_pos : c₁ ≠ 0
c₁_fin : c₁ ≠ ⊤
h₁ : map (⇑L') μT = c₁ • ν
⊢ ∃ c, 0 < c ∧ c < ⊤ ∧ map (⇑L) μ = (c * addHaar univ) • ν
state_after: case intro.intro.intro.intro.intro.intro.intro
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹² : NontriviallyNormedField 𝕜
inst✝¹¹ : CompleteSpace 𝕜
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : MeasurableSpace E
inst✝⁸ : BorelSpace E
inst✝⁷ : NormedSpace 𝕜 E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : MeasurableSpace F
inst✝⁴ : BorelSpace F
inst✝³ : NormedSpace 𝕜 F
L : E →ₗ[𝕜] F
μ : Measure E
ν : Measure F
inst✝² : μ.IsAddHaarMeasure
inst✝¹ : ν.IsAddHaarMeasure
inst✝ : LocallyCompactSpace E
h : Function.Surjective ⇑L
this✝¹ : ProperSpace E
this✝ : FiniteDimensional 𝕜 E
this : ProperSpace F
S : Submodule 𝕜 E := ker L
T : Submodule 𝕜 E
hT : IsCompl S T
M : (↥S × ↥T) ≃ₗ[𝕜] E := S.prodEquivOfIsCompl T hT
M_cont : Continuous ⇑M.symm
P : ↥S × ↥T →ₗ[𝕜] ↥T := snd 𝕜 ↥S ↥T
P_cont : Continuous ⇑P
I✝ : Function.Bijective ⇑(L.domRestrict T)
L' : ↥T ≃ₗ[𝕜] F := LinearEquiv.ofBijective (L.domRestrict T) I✝
L'_cont : Continuous ⇑L'
A : L = ↑L' ∘ₗ P ∘ₗ ↑M.symm
I : map (⇑L) μ = map (⇑L') (map (⇑P) (map (⇑M.symm) μ))
μS : Measure ↥S := addHaar
μT : Measure ↥T := addHaar
c₀ : ℝ≥0∞
c₀_pos : c₀ ≠ 0
c₀_fin : c₀ ≠ ⊤
h₀ : map (⇑M.symm) μ = c₀ • μS.prod μT
J : map (⇑P) (μS.prod μT) = μS univ • μT
c₁ : ℝ≥0∞
c₁_pos : c₁ ≠ 0
c₁_fin : c₁ ≠ ⊤
h₁ : map (⇑L') μT = c₁ • ν
⊢ map (⇑L) μ = (c₀ * c₁ * addHaar univ) • ν

tactic: simp only [I, h₀, Measure.map_smul, J, smul_smul, h₁]
state_before: case intro.intro.intro.intro.intro.intro.intro
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹² : NontriviallyNormedField 𝕜
inst✝¹¹ : CompleteSpace 𝕜
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : MeasurableSpace E
inst✝⁸ : BorelSpace E
inst✝⁷ : NormedSpace 𝕜 E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : MeasurableSpace F
inst✝⁴ : BorelSpace F
inst✝³ : NormedSpace 𝕜 F
L : E →ₗ[𝕜] F
μ : Measure E
ν : Measure F
inst✝² : μ.IsAddHaarMeasure
inst✝¹ : ν.IsAddHaarMeasure
inst✝ : LocallyCompactSpace E
h : Function.Surjective ⇑L
this✝¹ : ProperSpace E
this✝ : FiniteDimensional 𝕜 E
this : ProperSpace F
S : Submodule 𝕜 E := ker L
T : Submodule 𝕜 E
hT : IsCompl S T
M : (↥S × ↥T) ≃ₗ[𝕜] E := S.prodEquivOfIsCompl T hT
M_cont : Continuous ⇑M.symm
P : ↥S × ↥T →ₗ[𝕜] ↥T := snd 𝕜 ↥S ↥T
P_cont : Continuous ⇑P
I✝ : Function.Bijective ⇑(L.domRestrict T)
L' : ↥T ≃ₗ[𝕜] F := LinearEquiv.ofBijective (L.domRestrict T) I✝
L'_cont : Continuous ⇑L'
A : L = ↑L' ∘ₗ P ∘ₗ ↑M.symm
I : map (⇑L) μ = map (⇑L') (map (⇑P) (map (⇑M.symm) μ))
μS : Measure ↥S := addHaar
μT : Measure ↥T := addHaar
c₀ : ℝ≥0∞
c₀_pos : c₀ ≠ 0
c₀_fin : c₀ ≠ ⊤
h₀ : map (⇑M.symm) μ = c₀ • μS.prod μT
J : map (⇑P) (μS.prod μT) = μS univ • μT
c₁ : ℝ≥0∞
c₁_pos : c₁ ≠ 0
c₁_fin : c₁ ≠ ⊤
h₁ : map (⇑L') μT = c₁ • ν
⊢ map (⇑L) μ = (c₀ * c₁ * addHaar univ) • ν
state_after: case intro.intro.intro.intro.intro.intro.intro
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹² : NontriviallyNormedField 𝕜
inst✝¹¹ : CompleteSpace 𝕜
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : MeasurableSpace E
inst✝⁸ : BorelSpace E
inst✝⁷ : NormedSpace 𝕜 E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : MeasurableSpace F
inst✝⁴ : BorelSpace F
inst✝³ : NormedSpace 𝕜 F
L : E →ₗ[𝕜] F
μ : Measure E
ν : Measure F
inst✝² : μ.IsAddHaarMeasure
inst✝¹ : ν.IsAddHaarMeasure
inst✝ : LocallyCompactSpace E
h : Function.Surjective ⇑L
this✝¹ : ProperSpace E
this✝ : FiniteDimensional 𝕜 E
this : ProperSpace F
S : Submodule 𝕜 E := ker L
T : Submodule 𝕜 E
hT : IsCompl S T
M : (↥S × ↥T) ≃ₗ[𝕜] E := S.prodEquivOfIsCompl T hT
M_cont : Continuous ⇑M.symm
P : ↥S × ↥T →ₗ[𝕜] ↥T := snd 𝕜 ↥S ↥T
P_cont : Continuous ⇑P
I✝ : Function.Bijective ⇑(L.domRestrict T)
L' : ↥T ≃ₗ[𝕜] F := LinearEquiv.ofBijective (L.domRestrict T) I✝
L'_cont : Continuous ⇑L'
A : L = ↑L' ∘ₗ P ∘ₗ ↑M.symm
I : map (⇑L) μ = map (⇑L') (map (⇑P) (map (⇑M.symm) μ))
μS : Measure ↥S := addHaar
μT : Measure ↥T := addHaar
c₀ : ℝ≥0∞
c₀_pos : c₀ ≠ 0
c₀_fin : c₀ ≠ ⊤
h₀ : map (⇑M.symm) μ = c₀ • μS.prod μT
J : map (⇑P) (μS.prod μT) = μS univ • μT
c₁ : ℝ≥0∞
c₁_pos : c₁ ≠ 0
c₁_fin : c₁ ≠ ⊤
h₁ : map (⇑L') μT = c₁ • ν
⊢ (c₀ * μS univ * c₁) • ν = (c₀ * c₁ * addHaar univ) • ν

tactic: rw [mul_assoc, mul_comm _ c₁, ← mul_assoc]
state_before: case intro.intro.intro.intro.intro.intro.intro
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹² : NontriviallyNormedField 𝕜
inst✝¹¹ : CompleteSpace 𝕜
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : MeasurableSpace E
inst✝⁸ : BorelSpace E
inst✝⁷ : NormedSpace 𝕜 E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : MeasurableSpace F
inst✝⁴ : BorelSpace F
inst✝³ : NormedSpace 𝕜 F
L : E →ₗ[𝕜] F
μ : Measure E
ν : Measure F
inst✝² : μ.IsAddHaarMeasure
inst✝¹ : ν.IsAddHaarMeasure
inst✝ : LocallyCompactSpace E
h : Function.Surjective ⇑L
this✝¹ : ProperSpace E
this✝ : FiniteDimensional 𝕜 E
this : ProperSpace F
S : Submodule 𝕜 E := ker L
T : Submodule 𝕜 E
hT : IsCompl S T
M : (↥S × ↥T) ≃ₗ[𝕜] E := S.prodEquivOfIsCompl T hT
M_cont : Continuous ⇑M.symm
P : ↥S × ↥T →ₗ[𝕜] ↥T := snd 𝕜 ↥S ↥T
P_cont : Continuous ⇑P
I✝ : Function.Bijective ⇑(L.domRestrict T)
L' : ↥T ≃ₗ[𝕜] F := LinearEquiv.ofBijective (L.domRestrict T) I✝
L'_cont : Continuous ⇑L'
A : L = ↑L' ∘ₗ P ∘ₗ ↑M.symm
I : map (⇑L) μ = map (⇑L') (map (⇑P) (map (⇑M.symm) μ))
μS : Measure ↥S := addHaar
μT : Measure ↥T := addHaar
c₀ : ℝ≥0∞
c₀_pos : c₀ ≠ 0
c₀_fin : c₀ ≠ ⊤
h₀ : map (⇑M.symm) μ = c₀ • μS.prod μT
J : map (⇑P) (μS.prod μT) = μS univ • μT
c₁ : ℝ≥0∞
c₁_pos : c₁ ≠ 0
c₁_fin : c₁ ≠ ⊤
h₁ : map (⇑L') μT = c₁ • ν
⊢ (c₀ * μS univ * c₁) • ν = (c₀ * c₁ * addHaar univ) • ν
state_after: no goals

tactic: rcases subsingleton_or_nontrivial E with hE|hE
state_before: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹² : NontriviallyNormedField 𝕜
inst✝¹¹ : CompleteSpace 𝕜
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : MeasurableSpace E
inst✝⁸ : BorelSpace E
inst✝⁷ : NormedSpace 𝕜 E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : MeasurableSpace F
inst✝⁴ : BorelSpace F
inst✝³ : NormedSpace 𝕜 F
L : E →ₗ[𝕜] F
μ : Measure E
ν : Measure F
inst✝² : μ.IsAddHaarMeasure
inst✝¹ : ν.IsAddHaarMeasure
inst✝ : LocallyCompactSpace E
h : Function.Surjective ⇑L
this✝ : ProperSpace E
this : FiniteDimensional 𝕜 E
⊢ ProperSpace F
state_after: case inl
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹² : NontriviallyNormedField 𝕜
inst✝¹¹ : CompleteSpace 𝕜
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : MeasurableSpace E
inst✝⁸ : BorelSpace E
inst✝⁷ : NormedSpace 𝕜 E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : MeasurableSpace F
inst✝⁴ : BorelSpace F
inst✝³ : NormedSpace 𝕜 F
L : E →ₗ[𝕜] F
μ : Measure E
ν : Measure F
inst✝² : μ.IsAddHaarMeasure
inst✝¹ : ν.IsAddHaarMeasure
inst✝ : LocallyCompactSpace E
h : Function.Surjective ⇑L
this✝ : ProperSpace E
this : FiniteDimensional 𝕜 E
hE : Subsingleton E
⊢ ProperSpace F

case inr
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹² : NontriviallyNormedField 𝕜
inst✝¹¹ : CompleteSpace 𝕜
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : MeasurableSpace E
inst✝⁸ : BorelSpace E
inst✝⁷ : NormedSpace 𝕜 E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : MeasurableSpace F
inst✝⁴ : BorelSpace F
inst✝³ : NormedSpace 𝕜 F
L : E →ₗ[𝕜] F
μ : Measure E
ν : Measure F
inst✝² : μ.IsAddHaarMeasure
inst✝¹ : ν.IsAddHaarMeasure
inst✝ : LocallyCompactSpace E
h : Function.Surjective ⇑L
this✝ : ProperSpace E
this : FiniteDimensional 𝕜 E
hE : Nontrivial E
⊢ ProperSpace F

tactic: have : Subsingleton F := Function.Surjective.subsingleton h
state_before: case inl
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹² : NontriviallyNormedField 𝕜
inst✝¹¹ : CompleteSpace 𝕜
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : MeasurableSpace E
inst✝⁸ : BorelSpace E
inst✝⁷ : NormedSpace 𝕜 E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : MeasurableSpace F
inst✝⁴ : BorelSpace F
inst✝³ : NormedSpace 𝕜 F
L : E →ₗ[𝕜] F
μ : Measure E
ν : Measure F
inst✝² : μ.IsAddHaarMeasure
inst✝¹ : ν.IsAddHaarMeasure
inst✝ : LocallyCompactSpace E
h : Function.Surjective ⇑L
this✝ : ProperSpace E
this : FiniteDimensional 𝕜 E
hE : Subsingleton E
⊢ ProperSpace F
state_after: case inl
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹² : NontriviallyNormedField 𝕜
inst✝¹¹ : CompleteSpace 𝕜
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : MeasurableSpace E
inst✝⁸ : BorelSpace E
inst✝⁷ : NormedSpace 𝕜 E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : MeasurableSpace F
inst✝⁴ : BorelSpace F
inst✝³ : NormedSpace 𝕜 F
L : E →ₗ[𝕜] F
μ : Measure E
ν : Measure F
inst✝² : μ.IsAddHaarMeasure
inst✝¹ : ν.IsAddHaarMeasure
inst✝ : LocallyCompactSpace E
h : Function.Surjective ⇑L
this✝¹ : ProperSpace E
this✝ : FiniteDimensional 𝕜 E
hE : Subsingleton E
this : Subsingleton F
⊢ ProperSpace F

tactic: infer_instance
state_before: case inl
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹² : NontriviallyNormedField 𝕜
inst✝¹¹ : CompleteSpace 𝕜
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : MeasurableSpace E
inst✝⁸ : BorelSpace E
inst✝⁷ : NormedSpace 𝕜 E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : MeasurableSpace F
inst✝⁴ : BorelSpace F
inst✝³ : NormedSpace 𝕜 F
L : E →ₗ[𝕜] F
μ : Measure E
ν : Measure F
inst✝² : μ.IsAddHaarMeasure
inst✝¹ : ν.IsAddHaarMeasure
inst✝ : LocallyCompactSpace E
h : Function.Surjective ⇑L
this✝¹ : ProperSpace E
this✝ : FiniteDimensional 𝕜 E
hE : Subsingleton E
this : Subsingleton F
⊢ ProperSpace F
state_after: no goals

tactic: have : ProperSpace 𝕜 := .of_locallyCompact_module 𝕜 E
state_before: case inr
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹² : NontriviallyNormedField 𝕜
inst✝¹¹ : CompleteSpace 𝕜
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : MeasurableSpace E
inst✝⁸ : BorelSpace E
inst✝⁷ : NormedSpace 𝕜 E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : MeasurableSpace F
inst✝⁴ : BorelSpace F
inst✝³ : NormedSpace 𝕜 F
L : E →ₗ[𝕜] F
μ : Measure E
ν : Measure F
inst✝² : μ.IsAddHaarMeasure
inst✝¹ : ν.IsAddHaarMeasure
inst✝ : LocallyCompactSpace E
h : Function.Surjective ⇑L
this✝ : ProperSpace E
this : FiniteDimensional 𝕜 E
hE : Nontrivial E
⊢ ProperSpace F
state_after: case inr
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹² : NontriviallyNormedField 𝕜
inst✝¹¹ : CompleteSpace 𝕜
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : MeasurableSpace E
inst✝⁸ : BorelSpace E
inst✝⁷ : NormedSpace 𝕜 E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : MeasurableSpace F
inst✝⁴ : BorelSpace F
inst✝³ : NormedSpace 𝕜 F
L : E →ₗ[𝕜] F
μ : Measure E
ν : Measure F
inst✝² : μ.IsAddHaarMeasure
inst✝¹ : ν.IsAddHaarMeasure
inst✝ : LocallyCompactSpace E
h : Function.Surjective ⇑L
this✝¹ : ProperSpace E
this✝ : FiniteDimensional 𝕜 E
hE : Nontrivial E
this : ProperSpace 𝕜
⊢ ProperSpace F

tactic: have : FiniteDimensional 𝕜 F := Module.Finite.of_surjective L h
state_before: case inr
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹² : NontriviallyNormedField 𝕜
inst✝¹¹ : CompleteSpace 𝕜
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : MeasurableSpace E
inst✝⁸ : BorelSpace E
inst✝⁷ : NormedSpace 𝕜 E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : MeasurableSpace F
inst✝⁴ : BorelSpace F
inst✝³ : NormedSpace 𝕜 F
L : E →ₗ[𝕜] F
μ : Measure E
ν : Measure F
inst✝² : μ.IsAddHaarMeasure
inst✝¹ : ν.IsAddHaarMeasure
inst✝ : LocallyCompactSpace E
h : Function.Surjective ⇑L
this✝¹ : ProperSpace E
this✝ : FiniteDimensional 𝕜 E
hE : Nontrivial E
this : ProperSpace 𝕜
⊢ ProperSpace F
state_after: case inr
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹² : NontriviallyNormedField 𝕜
inst✝¹¹ : CompleteSpace 𝕜
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : MeasurableSpace E
inst✝⁸ : BorelSpace E
inst✝⁷ : NormedSpace 𝕜 E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : MeasurableSpace F
inst✝⁴ : BorelSpace F
inst✝³ : NormedSpace 𝕜 F
L : E →ₗ[𝕜] F
μ : Measure E
ν : Measure F
inst✝² : μ.IsAddHaarMeasure
inst✝¹ : ν.IsAddHaarMeasure
inst✝ : LocallyCompactSpace E
h : Function.Surjective ⇑L
this✝² : ProperSpace E
this✝¹ : FiniteDimensional 𝕜 E
hE : Nontrivial E
this✝ : ProperSpace 𝕜
this : FiniteDimensional 𝕜 F
⊢ ProperSpace F

tactic: exact FiniteDimensional.proper 𝕜 F
state_before: case inr
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹² : NontriviallyNormedField 𝕜
inst✝¹¹ : CompleteSpace 𝕜
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : MeasurableSpace E
inst✝⁸ : BorelSpace E
inst✝⁷ : NormedSpace 𝕜 E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : MeasurableSpace F
inst✝⁴ : BorelSpace F
inst✝³ : NormedSpace 𝕜 F
L : E →ₗ[𝕜] F
μ : Measure E
ν : Measure F
inst✝² : μ.IsAddHaarMeasure
inst✝¹ : ν.IsAddHaarMeasure
inst✝ : LocallyCompactSpace E
h : Function.Surjective ⇑L
this✝² : ProperSpace E
this✝¹ : FiniteDimensional 𝕜 E
hE : Nontrivial E
this✝ : ProperSpace 𝕜
this : FiniteDimensional 𝕜 F
⊢ ProperSpace F
state_after: no goals

tactic: ext x
state_before: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹² : NontriviallyNormedField 𝕜
inst✝¹¹ : CompleteSpace 𝕜
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : MeasurableSpace E
inst✝⁸ : BorelSpace E
inst✝⁷ : NormedSpace 𝕜 E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : MeasurableSpace F
inst✝⁴ : BorelSpace F
inst✝³ : NormedSpace 𝕜 F
L : E →ₗ[𝕜] F
μ : Measure E
ν : Measure F
inst✝² : μ.IsAddHaarMeasure
inst✝¹ : ν.IsAddHaarMeasure
inst✝ : LocallyCompactSpace E
h : Function.Surjective ⇑L
this✝¹ : ProperSpace E
this✝ : FiniteDimensional 𝕜 E
this : ProperSpace F
S : Submodule 𝕜 E := ker L
T : Submodule 𝕜 E
hT : IsCompl S T
M : (↥S × ↥T) ≃ₗ[𝕜] E := S.prodEquivOfIsCompl T hT
M_cont : Continuous ⇑M.symm
P : ↥S × ↥T →ₗ[𝕜] ↥T := snd 𝕜 ↥S ↥T
P_cont : Continuous ⇑P
I : Function.Bijective ⇑(L.domRestrict T)
L' : ↥T ≃ₗ[𝕜] F := LinearEquiv.ofBijective (L.domRestrict T) I
L'_cont : Continuous ⇑L'
⊢ L = ↑L' ∘ₗ P ∘ₗ ↑M.symm
state_after: case h
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹² : NontriviallyNormedField 𝕜
inst✝¹¹ : CompleteSpace 𝕜
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : MeasurableSpace E
inst✝⁸ : BorelSpace E
inst✝⁷ : NormedSpace 𝕜 E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : MeasurableSpace F
inst✝⁴ : BorelSpace F
inst✝³ : NormedSpace 𝕜 F
L : E →ₗ[𝕜] F
μ : Measure E
ν : Measure F
inst✝² : μ.IsAddHaarMeasure
inst✝¹ : ν.IsAddHaarMeasure
inst✝ : LocallyCompactSpace E
h : Function.Surjective ⇑L
this✝¹ : ProperSpace E
this✝ : FiniteDimensional 𝕜 E
this : ProperSpace F
S : Submodule 𝕜 E := ker L
T : Submodule 𝕜 E
hT : IsCompl S T
M : (↥S × ↥T) ≃ₗ[𝕜] E := S.prodEquivOfIsCompl T hT
M_cont : Continuous ⇑M.symm
P : ↥S × ↥T →ₗ[𝕜] ↥T := snd 𝕜 ↥S ↥T
P_cont : Continuous ⇑P
I : Function.Bijective ⇑(L.domRestrict T)
L' : ↥T ≃ₗ[𝕜] F := LinearEquiv.ofBijective (L.domRestrict T) I
L'_cont : Continuous ⇑L'
x : E
⊢ L x = (↑L' ∘ₗ P ∘ₗ ↑M.symm) x

tactic: obtain ⟨y, z, hyz⟩ : ∃ (y : S) (z : T), M.symm x = (y, z) := ⟨_, _, rfl⟩
state_before: case h
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹² : NontriviallyNormedField 𝕜
inst✝¹¹ : CompleteSpace 𝕜
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : MeasurableSpace E
inst✝⁸ : BorelSpace E
inst✝⁷ : NormedSpace 𝕜 E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : MeasurableSpace F
inst✝⁴ : BorelSpace F
inst✝³ : NormedSpace 𝕜 F
L : E →ₗ[𝕜] F
μ : Measure E
ν : Measure F
inst✝² : μ.IsAddHaarMeasure
inst✝¹ : ν.IsAddHaarMeasure
inst✝ : LocallyCompactSpace E
h : Function.Surjective ⇑L
this✝¹ : ProperSpace E
this✝ : FiniteDimensional 𝕜 E
this : ProperSpace F
S : Submodule 𝕜 E := ker L
T : Submodule 𝕜 E
hT : IsCompl S T
M : (↥S × ↥T) ≃ₗ[𝕜] E := S.prodEquivOfIsCompl T hT
M_cont : Continuous ⇑M.symm
P : ↥S × ↥T →ₗ[𝕜] ↥T := snd 𝕜 ↥S ↥T
P_cont : Continuous ⇑P
I : Function.Bijective ⇑(L.domRestrict T)
L' : ↥T ≃ₗ[𝕜] F := LinearEquiv.ofBijective (L.domRestrict T) I
L'_cont : Continuous ⇑L'
x : E
⊢ L x = (↑L' ∘ₗ P ∘ₗ ↑M.symm) x
state_after: case h.intro.intro
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹² : NontriviallyNormedField 𝕜
inst✝¹¹ : CompleteSpace 𝕜
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : MeasurableSpace E
inst✝⁸ : BorelSpace E
inst✝⁷ : NormedSpace 𝕜 E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : MeasurableSpace F
inst✝⁴ : BorelSpace F
inst✝³ : NormedSpace 𝕜 F
L : E →ₗ[𝕜] F
μ : Measure E
ν : Measure F
inst✝² : μ.IsAddHaarMeasure
inst✝¹ : ν.IsAddHaarMeasure
inst✝ : LocallyCompactSpace E
h : Function.Surjective ⇑L
this✝¹ : ProperSpace E
this✝ : FiniteDimensional 𝕜 E
this : ProperSpace F
S : Submodule 𝕜 E := ker L
T : Submodule 𝕜 E
hT : IsCompl S T
M : (↥S × ↥T) ≃ₗ[𝕜] E := S.prodEquivOfIsCompl T hT
M_cont : Continuous ⇑M.symm
P : ↥S × ↥T →ₗ[𝕜] ↥T := snd 𝕜 ↥S ↥T
P_cont : Continuous ⇑P
I : Function.Bijective ⇑(L.domRestrict T)
L' : ↥T ≃ₗ[𝕜] F := LinearEquiv.ofBijective (L.domRestrict T) I
L'_cont : Continuous ⇑L'
x : E
y : ↥S
z : ↥T
hyz : M.symm x = (y, z)
⊢ L x = (↑L' ∘ₗ P ∘ₗ ↑M.symm) x

tactic: have : x = M (y, z) := by
  rw [← hyz]; simp only [LinearEquiv.apply_symm_apply]
state_before: case h.intro.intro
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹² : NontriviallyNormedField 𝕜
inst✝¹¹ : CompleteSpace 𝕜
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : MeasurableSpace E
inst✝⁸ : BorelSpace E
inst✝⁷ : NormedSpace 𝕜 E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : MeasurableSpace F
inst✝⁴ : BorelSpace F
inst✝³ : NormedSpace 𝕜 F
L : E →ₗ[𝕜] F
μ : Measure E
ν : Measure F
inst✝² : μ.IsAddHaarMeasure
inst✝¹ : ν.IsAddHaarMeasure
inst✝ : LocallyCompactSpace E
h : Function.Surjective ⇑L
this✝¹ : ProperSpace E
this✝ : FiniteDimensional 𝕜 E
this : ProperSpace F
S : Submodule 𝕜 E := ker L
T : Submodule 𝕜 E
hT : IsCompl S T
M : (↥S × ↥T) ≃ₗ[𝕜] E := S.prodEquivOfIsCompl T hT
M_cont : Continuous ⇑M.symm
P : ↥S × ↥T →ₗ[𝕜] ↥T := snd 𝕜 ↥S ↥T
P_cont : Continuous ⇑P
I : Function.Bijective ⇑(L.domRestrict T)
L' : ↥T ≃ₗ[𝕜] F := LinearEquiv.ofBijective (L.domRestrict T) I
L'_cont : Continuous ⇑L'
x : E
y : ↥S
z : ↥T
hyz : M.symm x = (y, z)
⊢ L x = (↑L' ∘ₗ P ∘ₗ ↑M.symm) x
state_after: case h.intro.intro
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹² : NontriviallyNormedField 𝕜
inst✝¹¹ : CompleteSpace 𝕜
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : MeasurableSpace E
inst✝⁸ : BorelSpace E
inst✝⁷ : NormedSpace 𝕜 E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : MeasurableSpace F
inst✝⁴ : BorelSpace F
inst✝³ : NormedSpace 𝕜 F
L : E →ₗ[𝕜] F
μ : Measure E
ν : Measure F
inst✝² : μ.IsAddHaarMeasure
inst✝¹ : ν.IsAddHaarMeasure
inst✝ : LocallyCompactSpace E
h : Function.Surjective ⇑L
this✝² : ProperSpace E
this✝¹ : FiniteDimensional 𝕜 E
this✝ : ProperSpace F
S : Submodule 𝕜 E := ker L
T : Submodule 𝕜 E
hT : IsCompl S T
M : (↥S × ↥T) ≃ₗ[𝕜] E := S.prodEquivOfIsCompl T hT
M_cont : Continuous ⇑M.symm
P : ↥S × ↥T →ₗ[𝕜] ↥T := snd 𝕜 ↥S ↥T
P_cont : Continuous ⇑P
I : Function.Bijective ⇑(L.domRestrict T)
L' : ↥T ≃ₗ[𝕜] F := LinearEquiv.ofBijective (L.domRestrict T) I
L'_cont : Continuous ⇑L'
x : E
y : ↥S
z : ↥T
hyz : M.symm x = (y, z)
this : x = M (y, z)
⊢ L x = (↑L' ∘ₗ P ∘ₗ ↑M.symm) x

tactic: simp [L', P, M, this]
state_before: case h.intro.intro
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹² : NontriviallyNormedField 𝕜
inst✝¹¹ : CompleteSpace 𝕜
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : MeasurableSpace E
inst✝⁸ : BorelSpace E
inst✝⁷ : NormedSpace 𝕜 E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : MeasurableSpace F
inst✝⁴ : BorelSpace F
inst✝³ : NormedSpace 𝕜 F
L : E →ₗ[𝕜] F
μ : Measure E
ν : Measure F
inst✝² : μ.IsAddHaarMeasure
inst✝¹ : ν.IsAddHaarMeasure
inst✝ : LocallyCompactSpace E
h : Function.Surjective ⇑L
this✝² : ProperSpace E
this✝¹ : FiniteDimensional 𝕜 E
this✝ : ProperSpace F
S : Submodule 𝕜 E := ker L
T : Submodule 𝕜 E
hT : IsCompl S T
M : (↥S × ↥T) ≃ₗ[𝕜] E := S.prodEquivOfIsCompl T hT
M_cont : Continuous ⇑M.symm
P : ↥S × ↥T →ₗ[𝕜] ↥T := snd 𝕜 ↥S ↥T
P_cont : Continuous ⇑P
I : Function.Bijective ⇑(L.domRestrict T)
L' : ↥T ≃ₗ[𝕜] F := LinearEquiv.ofBijective (L.domRestrict T) I
L'_cont : Continuous ⇑L'
x : E
y : ↥S
z : ↥T
hyz : M.symm x = (y, z)
this : x = M (y, z)
⊢ L x = (↑L' ∘ₗ P ∘ₗ ↑M.symm) x
state_after: no goals

tactic: rw [← hyz]
state_before: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹² : NontriviallyNormedField 𝕜
inst✝¹¹ : CompleteSpace 𝕜
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : MeasurableSpace E
inst✝⁸ : BorelSpace E
inst✝⁷ : NormedSpace 𝕜 E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : MeasurableSpace F
inst✝⁴ : BorelSpace F
inst✝³ : NormedSpace 𝕜 F
L : E →ₗ[𝕜] F
μ : Measure E
ν : Measure F
inst✝² : μ.IsAddHaarMeasure
inst✝¹ : ν.IsAddHaarMeasure
inst✝ : LocallyCompactSpace E
h : Function.Surjective ⇑L
this✝¹ : ProperSpace E
this✝ : FiniteDimensional 𝕜 E
this : ProperSpace F
S : Submodule 𝕜 E := ker L
T : Submodule 𝕜 E
hT : IsCompl S T
M : (↥S × ↥T) ≃ₗ[𝕜] E := S.prodEquivOfIsCompl T hT
M_cont : Continuous ⇑M.symm
P : ↥S × ↥T →ₗ[𝕜] ↥T := snd 𝕜 ↥S ↥T
P_cont : Continuous ⇑P
I : Function.Bijective ⇑(L.domRestrict T)
L' : ↥T ≃ₗ[𝕜] F := LinearEquiv.ofBijective (L.domRestrict T) I
L'_cont : Continuous ⇑L'
x : E
y : ↥S
z : ↥T
hyz : M.symm x = (y, z)
⊢ x = M (y, z)
state_after: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹² : NontriviallyNormedField 𝕜
inst✝¹¹ : CompleteSpace 𝕜
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : MeasurableSpace E
inst✝⁸ : BorelSpace E
inst✝⁷ : NormedSpace 𝕜 E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : MeasurableSpace F
inst✝⁴ : BorelSpace F
inst✝³ : NormedSpace 𝕜 F
L : E →ₗ[𝕜] F
μ : Measure E
ν : Measure F
inst✝² : μ.IsAddHaarMeasure
inst✝¹ : ν.IsAddHaarMeasure
inst✝ : LocallyCompactSpace E
h : Function.Surjective ⇑L
this✝¹ : ProperSpace E
this✝ : FiniteDimensional 𝕜 E
this : ProperSpace F
S : Submodule 𝕜 E := ker L
T : Submodule 𝕜 E
hT : IsCompl S T
M : (↥S × ↥T) ≃ₗ[𝕜] E := S.prodEquivOfIsCompl T hT
M_cont : Continuous ⇑M.symm
P : ↥S × ↥T →ₗ[𝕜] ↥T := snd 𝕜 ↥S ↥T
P_cont : Continuous ⇑P
I : Function.Bijective ⇑(L.domRestrict T)
L' : ↥T ≃ₗ[𝕜] F := LinearEquiv.ofBijective (L.domRestrict T) I
L'_cont : Continuous ⇑L'
x : E
y : ↥S
z : ↥T
hyz : M.symm x = (y, z)
⊢ x = M (M.symm x)

tactic: simp only [LinearEquiv.apply_symm_apply]
state_before: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹² : NontriviallyNormedField 𝕜
inst✝¹¹ : CompleteSpace 𝕜
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : MeasurableSpace E
inst✝⁸ : BorelSpace E
inst✝⁷ : NormedSpace 𝕜 E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : MeasurableSpace F
inst✝⁴ : BorelSpace F
inst✝³ : NormedSpace 𝕜 F
L : E →ₗ[𝕜] F
μ : Measure E
ν : Measure F
inst✝² : μ.IsAddHaarMeasure
inst✝¹ : ν.IsAddHaarMeasure
inst✝ : LocallyCompactSpace E
h : Function.Surjective ⇑L
this✝¹ : ProperSpace E
this✝ : FiniteDimensional 𝕜 E
this : ProperSpace F
S : Submodule 𝕜 E := ker L
T : Submodule 𝕜 E
hT : IsCompl S T
M : (↥S × ↥T) ≃ₗ[𝕜] E := S.prodEquivOfIsCompl T hT
M_cont : Continuous ⇑M.symm
P : ↥S × ↥T →ₗ[𝕜] ↥T := snd 𝕜 ↥S ↥T
P_cont : Continuous ⇑P
I : Function.Bijective ⇑(L.domRestrict T)
L' : ↥T ≃ₗ[𝕜] F := LinearEquiv.ofBijective (L.domRestrict T) I
L'_cont : Continuous ⇑L'
x : E
y : ↥S
z : ↥T
hyz : M.symm x = (y, z)
⊢ x = M (M.symm x)
state_after: no goals

tactic: rw [Measure.map_map, Measure.map_map, A]
state_before: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹² : NontriviallyNormedField 𝕜
inst✝¹¹ : CompleteSpace 𝕜
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : MeasurableSpace E
inst✝⁸ : BorelSpace E
inst✝⁷ : NormedSpace 𝕜 E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : MeasurableSpace F
inst✝⁴ : BorelSpace F
inst✝³ : NormedSpace 𝕜 F
L : E →ₗ[𝕜] F
μ : Measure E
ν : Measure F
inst✝² : μ.IsAddHaarMeasure
inst✝¹ : ν.IsAddHaarMeasure
inst✝ : LocallyCompactSpace E
h : Function.Surjective ⇑L
this✝¹ : ProperSpace E
this✝ : FiniteDimensional 𝕜 E
this : ProperSpace F
S : Submodule 𝕜 E := ker L
T : Submodule 𝕜 E
hT : IsCompl S T
M : (↥S × ↥T) ≃ₗ[𝕜] E := S.prodEquivOfIsCompl T hT
M_cont : Continuous ⇑M.symm
P : ↥S × ↥T →ₗ[𝕜] ↥T := snd 𝕜 ↥S ↥T
P_cont : Continuous ⇑P
I : Function.Bijective ⇑(L.domRestrict T)
L' : ↥T ≃ₗ[𝕜] F := LinearEquiv.ofBijective (L.domRestrict T) I
L'_cont : Continuous ⇑L'
A : L = ↑L' ∘ₗ P ∘ₗ ↑M.symm
⊢ map (⇑L) μ = map (⇑L') (map (⇑P) (map (⇑M.symm) μ))
state_after: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹² : NontriviallyNormedField 𝕜
inst✝¹¹ : CompleteSpace 𝕜
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : MeasurableSpace E
inst✝⁸ : BorelSpace E
inst✝⁷ : NormedSpace 𝕜 E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : MeasurableSpace F
inst✝⁴ : BorelSpace F
inst✝³ : NormedSpace 𝕜 F
L : E →ₗ[𝕜] F
μ : Measure E
ν : Measure F
inst✝² : μ.IsAddHaarMeasure
inst✝¹ : ν.IsAddHaarMeasure
inst✝ : LocallyCompactSpace E
h : Function.Surjective ⇑L
this✝¹ : ProperSpace E
this✝ : FiniteDimensional 𝕜 E
this : ProperSpace F
S : Submodule 𝕜 E := ker L
T : Submodule 𝕜 E
hT : IsCompl S T
M : (↥S × ↥T) ≃ₗ[𝕜] E := S.prodEquivOfIsCompl T hT
M_cont : Continuous ⇑M.symm
P : ↥S × ↥T →ₗ[𝕜] ↥T := snd 𝕜 ↥S ↥T
P_cont : Continuous ⇑P
I : Function.Bijective ⇑(L.domRestrict T)
L' : ↥T ≃ₗ[𝕜] F := LinearEquiv.ofBijective (L.domRestrict T) I
L'_cont : Continuous ⇑L'
A : L = ↑L' ∘ₗ P ∘ₗ ↑M.symm
⊢ map (⇑(↑L' ∘ₗ P ∘ₗ ↑M.symm)) μ = map ((⇑L' ∘ ⇑P) ∘ ⇑M.symm) μ

case hg
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹² : NontriviallyNormedField 𝕜
inst✝¹¹ : CompleteSpace 𝕜
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : MeasurableSpace E
inst✝⁸ : BorelSpace E
inst✝⁷ : NormedSpace 𝕜 E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : MeasurableSpace F
inst✝⁴ : BorelSpace F
inst✝³ : NormedSpace 𝕜 F
L : E →ₗ[𝕜] F
μ : Measure E
ν : Measure F
inst✝² : μ.IsAddHaarMeasure
inst✝¹ : ν.IsAddHaarMeasure
inst✝ : LocallyCompactSpace E
h : Function.Surjective ⇑L
this✝¹ : ProperSpace E
this✝ : FiniteDimensional 𝕜 E
this : ProperSpace F
S : Submodule 𝕜 E := ker L
T : Submodule 𝕜 E
hT : IsCompl S T
M : (↥S × ↥T) ≃ₗ[𝕜] E := S.prodEquivOfIsCompl T hT
M_cont : Continuous ⇑M.symm
P : ↥S × ↥T →ₗ[𝕜] ↥T := snd 𝕜 ↥S ↥T
P_cont : Continuous ⇑P
I : Function.Bijective ⇑(L.domRestrict T)
L' : ↥T ≃ₗ[𝕜] F := LinearEquiv.ofBijective (L.domRestrict T) I
L'_cont : Continuous ⇑L'
A : L = ↑L' ∘ₗ P ∘ₗ ↑M.symm
⊢ Measurable (⇑L' ∘ ⇑P)

case hf
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹² : NontriviallyNormedField 𝕜
inst✝¹¹ : CompleteSpace 𝕜
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : MeasurableSpace E
inst✝⁸ : BorelSpace E
inst✝⁷ : NormedSpace 𝕜 E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : MeasurableSpace F
inst✝⁴ : BorelSpace F
inst✝³ : NormedSpace 𝕜 F
L : E →ₗ[𝕜] F
μ : Measure E
ν : Measure F
inst✝² : μ.IsAddHaarMeasure
inst✝¹ : ν.IsAddHaarMeasure
inst✝ : LocallyCompactSpace E
h : Function.Surjective ⇑L
this✝¹ : ProperSpace E
this✝ : FiniteDimensional 𝕜 E
this : ProperSpace F
S : Submodule 𝕜 E := ker L
T : Submodule 𝕜 E
hT : IsCompl S T
M : (↥S × ↥T) ≃ₗ[𝕜] E := S.prodEquivOfIsCompl T hT
M_cont : Continuous ⇑M.symm
P : ↥S × ↥T →ₗ[𝕜] ↥T := snd 𝕜 ↥S ↥T
P_cont : Continuous ⇑P
I : Function.Bijective ⇑(L.domRestrict T)
L' : ↥T ≃ₗ[𝕜] F := LinearEquiv.ofBijective (L.domRestrict T) I
L'_cont : Continuous ⇑L'
A : L = ↑L' ∘ₗ P ∘ₗ ↑M.symm
⊢ Measurable ⇑M.symm

case hg
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹² : NontriviallyNormedField 𝕜
inst✝¹¹ : CompleteSpace 𝕜
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : MeasurableSpace E
inst✝⁸ : BorelSpace E
inst✝⁷ : NormedSpace 𝕜 E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : MeasurableSpace F
inst✝⁴ : BorelSpace F
inst✝³ : NormedSpace 𝕜 F
L : E →ₗ[𝕜] F
μ : Measure E
ν : Measure F
inst✝² : μ.IsAddHaarMeasure
inst✝¹ : ν.IsAddHaarMeasure
inst✝ : LocallyCompactSpace E
h : Function.Surjective ⇑L
this✝¹ : ProperSpace E
this✝ : FiniteDimensional 𝕜 E
this : ProperSpace F
S : Submodule 𝕜 E := ker L
T : Submodule 𝕜 E
hT : IsCompl S T
M : (↥S × ↥T) ≃ₗ[𝕜] E := S.prodEquivOfIsCompl T hT
M_cont : Continuous ⇑M.symm
P : ↥S × ↥T →ₗ[𝕜] ↥T := snd 𝕜 ↥S ↥T
P_cont : Continuous ⇑P
I : Function.Bijective ⇑(L.domRestrict T)
L' : ↥T ≃ₗ[𝕜] F := LinearEquiv.ofBijective (L.domRestrict T) I
L'_cont : Continuous ⇑L'
A : L = ↑L' ∘ₗ P ∘ₗ ↑M.symm
⊢ Measurable ⇑L'

case hf
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹² : NontriviallyNormedField 𝕜
inst✝¹¹ : CompleteSpace 𝕜
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : MeasurableSpace E
inst✝⁸ : BorelSpace E
inst✝⁷ : NormedSpace 𝕜 E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : MeasurableSpace F
inst✝⁴ : BorelSpace F
inst✝³ : NormedSpace 𝕜 F
L : E →ₗ[𝕜] F
μ : Measure E
ν : Measure F
inst✝² : μ.IsAddHaarMeasure
inst✝¹ : ν.IsAddHaarMeasure
inst✝ : LocallyCompactSpace E
h : Function.Surjective ⇑L
this✝¹ : ProperSpace E
this✝ : FiniteDimensional 𝕜 E
this : ProperSpace F
S : Submodule 𝕜 E := ker L
T : Submodule 𝕜 E
hT : IsCompl S T
M : (↥S × ↥T) ≃ₗ[𝕜] E := S.prodEquivOfIsCompl T hT
M_cont : Continuous ⇑M.symm
P : ↥S × ↥T →ₗ[𝕜] ↥T := snd 𝕜 ↥S ↥T
P_cont : Continuous ⇑P
I : Function.Bijective ⇑(L.domRestrict T)
L' : ↥T ≃ₗ[𝕜] F := LinearEquiv.ofBijective (L.domRestrict T) I
L'_cont : Continuous ⇑L'
A : L = ↑L' ∘ₗ P ∘ₗ ↑M.symm
⊢ Measurable ⇑P

tactic: rfl
state_before: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹² : NontriviallyNormedField 𝕜
inst✝¹¹ : CompleteSpace 𝕜
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : MeasurableSpace E
inst✝⁸ : BorelSpace E
inst✝⁷ : NormedSpace 𝕜 E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : MeasurableSpace F
inst✝⁴ : BorelSpace F
inst✝³ : NormedSpace 𝕜 F
L : E →ₗ[𝕜] F
μ : Measure E
ν : Measure F
inst✝² : μ.IsAddHaarMeasure
inst✝¹ : ν.IsAddHaarMeasure
inst✝ : LocallyCompactSpace E
h : Function.Surjective ⇑L
this✝¹ : ProperSpace E
this✝ : FiniteDimensional 𝕜 E
this : ProperSpace F
S : Submodule 𝕜 E := ker L
T : Submodule 𝕜 E
hT : IsCompl S T
M : (↥S × ↥T) ≃ₗ[𝕜] E := S.prodEquivOfIsCompl T hT
M_cont : Continuous ⇑M.symm
P : ↥S × ↥T →ₗ[𝕜] ↥T := snd 𝕜 ↥S ↥T
P_cont : Continuous ⇑P
I : Function.Bijective ⇑(L.domRestrict T)
L' : ↥T ≃ₗ[𝕜] F := LinearEquiv.ofBijective (L.domRestrict T) I
L'_cont : Continuous ⇑L'
A : L = ↑L' ∘ₗ P ∘ₗ ↑M.symm
⊢ map (⇑(↑L' ∘ₗ P ∘ₗ ↑M.symm)) μ = map ((⇑L' ∘ ⇑P) ∘ ⇑M.symm) μ
state_after: no goals

tactic: exact L'_cont.measurable.comp P_cont.measurable
state_before: case hg
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹² : NontriviallyNormedField 𝕜
inst✝¹¹ : CompleteSpace 𝕜
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : MeasurableSpace E
inst✝⁸ : BorelSpace E
inst✝⁷ : NormedSpace 𝕜 E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : MeasurableSpace F
inst✝⁴ : BorelSpace F
inst✝³ : NormedSpace 𝕜 F
L : E →ₗ[𝕜] F
μ : Measure E
ν : Measure F
inst✝² : μ.IsAddHaarMeasure
inst✝¹ : ν.IsAddHaarMeasure
inst✝ : LocallyCompactSpace E
h : Function.Surjective ⇑L
this✝¹ : ProperSpace E
this✝ : FiniteDimensional 𝕜 E
this : ProperSpace F
S : Submodule 𝕜 E := ker L
T : Submodule 𝕜 E
hT : IsCompl S T
M : (↥S × ↥T) ≃ₗ[𝕜] E := S.prodEquivOfIsCompl T hT
M_cont : Continuous ⇑M.symm
P : ↥S × ↥T →ₗ[𝕜] ↥T := snd 𝕜 ↥S ↥T
P_cont : Continuous ⇑P
I : Function.Bijective ⇑(L.domRestrict T)
L' : ↥T ≃ₗ[𝕜] F := LinearEquiv.ofBijective (L.domRestrict T) I
L'_cont : Continuous ⇑L'
A : L = ↑L' ∘ₗ P ∘ₗ ↑M.symm
⊢ Measurable (⇑L' ∘ ⇑P)
state_after: no goals

tactic: exact M_cont.measurable
state_before: case hf
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹² : NontriviallyNormedField 𝕜
inst✝¹¹ : CompleteSpace 𝕜
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : MeasurableSpace E
inst✝⁸ : BorelSpace E
inst✝⁷ : NormedSpace 𝕜 E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : MeasurableSpace F
inst✝⁴ : BorelSpace F
inst✝³ : NormedSpace 𝕜 F
L : E →ₗ[𝕜] F
μ : Measure E
ν : Measure F
inst✝² : μ.IsAddHaarMeasure
inst✝¹ : ν.IsAddHaarMeasure
inst✝ : LocallyCompactSpace E
h : Function.Surjective ⇑L
this✝¹ : ProperSpace E
this✝ : FiniteDimensional 𝕜 E
this : ProperSpace F
S : Submodule 𝕜 E := ker L
T : Submodule 𝕜 E
hT : IsCompl S T
M : (↥S × ↥T) ≃ₗ[𝕜] E := S.prodEquivOfIsCompl T hT
M_cont : Continuous ⇑M.symm
P : ↥S × ↥T →ₗ[𝕜] ↥T := snd 𝕜 ↥S ↥T
P_cont : Continuous ⇑P
I : Function.Bijective ⇑(L.domRestrict T)
L' : ↥T ≃ₗ[𝕜] F := LinearEquiv.ofBijective (L.domRestrict T) I
L'_cont : Continuous ⇑L'
A : L = ↑L' ∘ₗ P ∘ₗ ↑M.symm
⊢ Measurable ⇑M.symm
state_after: no goals

tactic: exact L'_cont.measurable
state_before: case hg
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹² : NontriviallyNormedField 𝕜
inst✝¹¹ : CompleteSpace 𝕜
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : MeasurableSpace E
inst✝⁸ : BorelSpace E
inst✝⁷ : NormedSpace 𝕜 E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : MeasurableSpace F
inst✝⁴ : BorelSpace F
inst✝³ : NormedSpace 𝕜 F
L : E →ₗ[𝕜] F
μ : Measure E
ν : Measure F
inst✝² : μ.IsAddHaarMeasure
inst✝¹ : ν.IsAddHaarMeasure
inst✝ : LocallyCompactSpace E
h : Function.Surjective ⇑L
this✝¹ : ProperSpace E
this✝ : FiniteDimensional 𝕜 E
this : ProperSpace F
S : Submodule 𝕜 E := ker L
T : Submodule 𝕜 E
hT : IsCompl S T
M : (↥S × ↥T) ≃ₗ[𝕜] E := S.prodEquivOfIsCompl T hT
M_cont : Continuous ⇑M.symm
P : ↥S × ↥T →ₗ[𝕜] ↥T := snd 𝕜 ↥S ↥T
P_cont : Continuous ⇑P
I : Function.Bijective ⇑(L.domRestrict T)
L' : ↥T ≃ₗ[𝕜] F := LinearEquiv.ofBijective (L.domRestrict T) I
L'_cont : Continuous ⇑L'
A : L = ↑L' ∘ₗ P ∘ₗ ↑M.symm
⊢ Measurable ⇑L'
state_after: no goals

tactic: exact P_cont.measurable
state_before: case hf
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹² : NontriviallyNormedField 𝕜
inst✝¹¹ : CompleteSpace 𝕜
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : MeasurableSpace E
inst✝⁸ : BorelSpace E
inst✝⁷ : NormedSpace 𝕜 E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : MeasurableSpace F
inst✝⁴ : BorelSpace F
inst✝³ : NormedSpace 𝕜 F
L : E →ₗ[𝕜] F
μ : Measure E
ν : Measure F
inst✝² : μ.IsAddHaarMeasure
inst✝¹ : ν.IsAddHaarMeasure
inst✝ : LocallyCompactSpace E
h : Function.Surjective ⇑L
this✝¹ : ProperSpace E
this✝ : FiniteDimensional 𝕜 E
this : ProperSpace F
S : Submodule 𝕜 E := ker L
T : Submodule 𝕜 E
hT : IsCompl S T
M : (↥S × ↥T) ≃ₗ[𝕜] E := S.prodEquivOfIsCompl T hT
M_cont : Continuous ⇑M.symm
P : ↥S × ↥T →ₗ[𝕜] ↥T := snd 𝕜 ↥S ↥T
P_cont : Continuous ⇑P
I : Function.Bijective ⇑(L.domRestrict T)
L' : ↥T ≃ₗ[𝕜] F := LinearEquiv.ofBijective (L.domRestrict T) I
L'_cont : Continuous ⇑L'
A : L = ↑L' ∘ₗ P ∘ₗ ↑M.symm
⊢ Measurable ⇑P
state_after: no goals

tactic: have : IsAddHaarMeasure (μ.map M.symm) :=
  M.toContinuousLinearEquiv.symm.isAddHaarMeasure_map μ
state_before: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹² : NontriviallyNormedField 𝕜
inst✝¹¹ : CompleteSpace 𝕜
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : MeasurableSpace E
inst✝⁸ : BorelSpace E
inst✝⁷ : NormedSpace 𝕜 E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : MeasurableSpace F
inst✝⁴ : BorelSpace F
inst✝³ : NormedSpace 𝕜 F
L : E →ₗ[𝕜] F
μ : Measure E
ν : Measure F
inst✝² : μ.IsAddHaarMeasure
inst✝¹ : ν.IsAddHaarMeasure
inst✝ : LocallyCompactSpace E
h : Function.Surjective ⇑L
this✝¹ : ProperSpace E
this✝ : FiniteDimensional 𝕜 E
this : ProperSpace F
S : Submodule 𝕜 E := ker L
T : Submodule 𝕜 E
hT : IsCompl S T
M : (↥S × ↥T) ≃ₗ[𝕜] E := S.prodEquivOfIsCompl T hT
M_cont : Continuous ⇑M.symm
P : ↥S × ↥T →ₗ[𝕜] ↥T := snd 𝕜 ↥S ↥T
P_cont : Continuous ⇑P
I✝ : Function.Bijective ⇑(L.domRestrict T)
L' : ↥T ≃ₗ[𝕜] F := LinearEquiv.ofBijective (L.domRestrict T) I✝
L'_cont : Continuous ⇑L'
A : L = ↑L' ∘ₗ P ∘ₗ ↑M.symm
I : map (⇑L) μ = map (⇑L') (map (⇑P) (map (⇑M.symm) μ))
μS : Measure ↥S := addHaar
μT : Measure ↥T := addHaar
⊢ ∃ c₀, c₀ ≠ 0 ∧ c₀ ≠ ⊤ ∧ map (⇑M.symm) μ = c₀ • μS.prod μT
state_after: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹² : NontriviallyNormedField 𝕜
inst✝¹¹ : CompleteSpace 𝕜
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : MeasurableSpace E
inst✝⁸ : BorelSpace E
inst✝⁷ : NormedSpace 𝕜 E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : MeasurableSpace F
inst✝⁴ : BorelSpace F
inst✝³ : NormedSpace 𝕜 F
L : E →ₗ[𝕜] F
μ : Measure E
ν : Measure F
inst✝² : μ.IsAddHaarMeasure
inst✝¹ : ν.IsAddHaarMeasure
inst✝ : LocallyCompactSpace E
h : Function.Surjective ⇑L
this✝² : ProperSpace E
this✝¹ : FiniteDimensional 𝕜 E
this✝ : ProperSpace F
S : Submodule 𝕜 E := ker L
T : Submodule 𝕜 E
hT : IsCompl S T
M : (↥S × ↥T) ≃ₗ[𝕜] E := S.prodEquivOfIsCompl T hT
M_cont : Continuous ⇑M.symm
P : ↥S × ↥T →ₗ[𝕜] ↥T := snd 𝕜 ↥S ↥T
P_cont : Continuous ⇑P
I✝ : Function.Bijective ⇑(L.domRestrict T)
L' : ↥T ≃ₗ[𝕜] F := LinearEquiv.ofBijective (L.domRestrict T) I✝
L'_cont : Continuous ⇑L'
A : L = ↑L' ∘ₗ P ∘ₗ ↑M.symm
I : map (⇑L) μ = map (⇑L') (map (⇑P) (map (⇑M.symm) μ))
μS : Measure ↥S := addHaar
μT : Measure ↥T := addHaar
this : (map (⇑M.symm) μ).IsAddHaarMeasure
⊢ ∃ c₀, c₀ ≠ 0 ∧ c₀ ≠ ⊤ ∧ map (⇑M.symm) μ = c₀ • μS.prod μT

tactic: refine ⟨addHaarScalarFactor (μ.map M.symm) (μS.prod μT), ?_, ENNReal.coe_ne_top,
  isAddLeftInvariant_eq_smul _ _⟩
state_before: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹² : NontriviallyNormedField 𝕜
inst✝¹¹ : CompleteSpace 𝕜
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : MeasurableSpace E
inst✝⁸ : BorelSpace E
inst✝⁷ : NormedSpace 𝕜 E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : MeasurableSpace F
inst✝⁴ : BorelSpace F
inst✝³ : NormedSpace 𝕜 F
L : E →ₗ[𝕜] F
μ : Measure E
ν : Measure F
inst✝² : μ.IsAddHaarMeasure
inst✝¹ : ν.IsAddHaarMeasure
inst✝ : LocallyCompactSpace E
h : Function.Surjective ⇑L
this✝² : ProperSpace E
this✝¹ : FiniteDimensional 𝕜 E
this✝ : ProperSpace F
S : Submodule 𝕜 E := ker L
T : Submodule 𝕜 E
hT : IsCompl S T
M : (↥S × ↥T) ≃ₗ[𝕜] E := S.prodEquivOfIsCompl T hT
M_cont : Continuous ⇑M.symm
P : ↥S × ↥T →ₗ[𝕜] ↥T := snd 𝕜 ↥S ↥T
P_cont : Continuous ⇑P
I✝ : Function.Bijective ⇑(L.domRestrict T)
L' : ↥T ≃ₗ[𝕜] F := LinearEquiv.ofBijective (L.domRestrict T) I✝
L'_cont : Continuous ⇑L'
A : L = ↑L' ∘ₗ P ∘ₗ ↑M.symm
I : map (⇑L) μ = map (⇑L') (map (⇑P) (map (⇑M.symm) μ))
μS : Measure ↥S := addHaar
μT : Measure ↥T := addHaar
this : (map (⇑M.symm) μ).IsAddHaarMeasure
⊢ ∃ c₀, c₀ ≠ 0 ∧ c₀ ≠ ⊤ ∧ map (⇑M.symm) μ = c₀ • μS.prod μT
state_after: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹² : NontriviallyNormedField 𝕜
inst✝¹¹ : CompleteSpace 𝕜
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : MeasurableSpace E
inst✝⁸ : BorelSpace E
inst✝⁷ : NormedSpace 𝕜 E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : MeasurableSpace F
inst✝⁴ : BorelSpace F
inst✝³ : NormedSpace 𝕜 F
L : E →ₗ[𝕜] F
μ : Measure E
ν : Measure F
inst✝² : μ.IsAddHaarMeasure
inst✝¹ : ν.IsAddHaarMeasure
inst✝ : LocallyCompactSpace E
h : Function.Surjective ⇑L
this✝² : ProperSpace E
this✝¹ : FiniteDimensional 𝕜 E
this✝ : ProperSpace F
S : Submodule 𝕜 E := ker L
T : Submodule 𝕜 E
hT : IsCompl S T
M : (↥S × ↥T) ≃ₗ[𝕜] E := S.prodEquivOfIsCompl T hT
M_cont : Continuous ⇑M.symm
P : ↥S × ↥T →ₗ[𝕜] ↥T := snd 𝕜 ↥S ↥T
P_cont : Continuous ⇑P
I✝ : Function.Bijective ⇑(L.domRestrict T)
L' : ↥T ≃ₗ[𝕜] F := LinearEquiv.ofBijective (L.domRestrict T) I✝
L'_cont : Continuous ⇑L'
A : L = ↑L' ∘ₗ P ∘ₗ ↑M.symm
I : map (⇑L) μ = map (⇑L') (map (⇑P) (map (⇑M.symm) μ))
μS : Measure ↥S := addHaar
μT : Measure ↥T := addHaar
this : (map (⇑M.symm) μ).IsAddHaarMeasure
⊢ ↑((map (⇑M.symm) μ).addHaarScalarFactor (μS.prod μT)) ≠ 0

tactic: simpa only [ne_eq, ENNReal.coe_eq_zero] using
  (addHaarScalarFactor_pos_of_isAddHaarMeasure (μ.map M.symm) (μS.prod μT)).ne'
state_before: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹² : NontriviallyNormedField 𝕜
inst✝¹¹ : CompleteSpace 𝕜
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : MeasurableSpace E
inst✝⁸ : BorelSpace E
inst✝⁷ : NormedSpace 𝕜 E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : MeasurableSpace F
inst✝⁴ : BorelSpace F
inst✝³ : NormedSpace 𝕜 F
L : E →ₗ[𝕜] F
μ : Measure E
ν : Measure F
inst✝² : μ.IsAddHaarMeasure
inst✝¹ : ν.IsAddHaarMeasure
inst✝ : LocallyCompactSpace E
h : Function.Surjective ⇑L
this✝² : ProperSpace E
this✝¹ : FiniteDimensional 𝕜 E
this✝ : ProperSpace F
S : Submodule 𝕜 E := ker L
T : Submodule 𝕜 E
hT : IsCompl S T
M : (↥S × ↥T) ≃ₗ[𝕜] E := S.prodEquivOfIsCompl T hT
M_cont : Continuous ⇑M.symm
P : ↥S × ↥T →ₗ[𝕜] ↥T := snd 𝕜 ↥S ↥T
P_cont : Continuous ⇑P
I✝ : Function.Bijective ⇑(L.domRestrict T)
L' : ↥T ≃ₗ[𝕜] F := LinearEquiv.ofBijective (L.domRestrict T) I✝
L'_cont : Continuous ⇑L'
A : L = ↑L' ∘ₗ P ∘ₗ ↑M.symm
I : map (⇑L) μ = map (⇑L') (map (⇑P) (map (⇑M.symm) μ))
μS : Measure ↥S := addHaar
μT : Measure ↥T := addHaar
this : (map (⇑M.symm) μ).IsAddHaarMeasure
⊢ ↑((map (⇑M.symm) μ).addHaarScalarFactor (μS.prod μT)) ≠ 0
state_after: no goals

tactic: have : IsAddHaarMeasure (μT.map L') :=
  L'.toContinuousLinearEquiv.isAddHaarMeasure_map μT
state_before: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹² : NontriviallyNormedField 𝕜
inst✝¹¹ : CompleteSpace 𝕜
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : MeasurableSpace E
inst✝⁸ : BorelSpace E
inst✝⁷ : NormedSpace 𝕜 E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : MeasurableSpace F
inst✝⁴ : BorelSpace F
inst✝³ : NormedSpace 𝕜 F
L : E →ₗ[𝕜] F
μ : Measure E
ν : Measure F
inst✝² : μ.IsAddHaarMeasure
inst✝¹ : ν.IsAddHaarMeasure
inst✝ : LocallyCompactSpace E
h : Function.Surjective ⇑L
this✝¹ : ProperSpace E
this✝ : FiniteDimensional 𝕜 E
this : ProperSpace F
S : Submodule 𝕜 E := ker L
T : Submodule 𝕜 E
hT : IsCompl S T
M : (↥S × ↥T) ≃ₗ[𝕜] E := S.prodEquivOfIsCompl T hT
M_cont : Continuous ⇑M.symm
P : ↥S × ↥T →ₗ[𝕜] ↥T := snd 𝕜 ↥S ↥T
P_cont : Continuous ⇑P
I✝ : Function.Bijective ⇑(L.domRestrict T)
L' : ↥T ≃ₗ[𝕜] F := LinearEquiv.ofBijective (L.domRestrict T) I✝
L'_cont : Continuous ⇑L'
A : L = ↑L' ∘ₗ P ∘ₗ ↑M.symm
I : map (⇑L) μ = map (⇑L') (map (⇑P) (map (⇑M.symm) μ))
μS : Measure ↥S := addHaar
μT : Measure ↥T := addHaar
c₀ : ℝ≥0∞
c₀_pos : c₀ ≠ 0
c₀_fin : c₀ ≠ ⊤
h₀ : map (⇑M.symm) μ = c₀ • μS.prod μT
J : map (⇑P) (μS.prod μT) = μS univ • μT
⊢ ∃ c₁, c₁ ≠ 0 ∧ c₁ ≠ ⊤ ∧ map (⇑L') μT = c₁ • ν
state_after: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹² : NontriviallyNormedField 𝕜
inst✝¹¹ : CompleteSpace 𝕜
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : MeasurableSpace E
inst✝⁸ : BorelSpace E
inst✝⁷ : NormedSpace 𝕜 E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : MeasurableSpace F
inst✝⁴ : BorelSpace F
inst✝³ : NormedSpace 𝕜 F
L : E →ₗ[𝕜] F
μ : Measure E
ν : Measure F
inst✝² : μ.IsAddHaarMeasure
inst✝¹ : ν.IsAddHaarMeasure
inst✝ : LocallyCompactSpace E
h : Function.Surjective ⇑L
this✝² : ProperSpace E
this✝¹ : FiniteDimensional 𝕜 E
this✝ : ProperSpace F
S : Submodule 𝕜 E := ker L
T : Submodule 𝕜 E
hT : IsCompl S T
M : (↥S × ↥T) ≃ₗ[𝕜] E := S.prodEquivOfIsCompl T hT
M_cont : Continuous ⇑M.symm
P : ↥S × ↥T →ₗ[𝕜] ↥T := snd 𝕜 ↥S ↥T
P_cont : Continuous ⇑P
I✝ : Function.Bijective ⇑(L.domRestrict T)
L' : ↥T ≃ₗ[𝕜] F := LinearEquiv.ofBijective (L.domRestrict T) I✝
L'_cont : Continuous ⇑L'
A : L = ↑L' ∘ₗ P ∘ₗ ↑M.symm
I : map (⇑L) μ = map (⇑L') (map (⇑P) (map (⇑M.symm) μ))
μS : Measure ↥S := addHaar
μT : Measure ↥T := addHaar
c₀ : ℝ≥0∞
c₀_pos : c₀ ≠ 0
c₀_fin : c₀ ≠ ⊤
h₀ : map (⇑M.symm) μ = c₀ • μS.prod μT
J : map (⇑P) (μS.prod μT) = μS univ • μT
this : (map (⇑L') μT).IsAddHaarMeasure
⊢ ∃ c₁, c₁ ≠ 0 ∧ c₁ ≠ ⊤ ∧ map (⇑L') μT = c₁ • ν

tactic: refine ⟨addHaarScalarFactor (μT.map L') ν, ?_, ENNReal.coe_ne_top,
  isAddLeftInvariant_eq_smul _ _⟩
state_before: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹² : NontriviallyNormedField 𝕜
inst✝¹¹ : CompleteSpace 𝕜
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : MeasurableSpace E
inst✝⁸ : BorelSpace E
inst✝⁷ : NormedSpace 𝕜 E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : MeasurableSpace F
inst✝⁴ : BorelSpace F
inst✝³ : NormedSpace 𝕜 F
L : E →ₗ[𝕜] F
μ : Measure E
ν : Measure F
inst✝² : μ.IsAddHaarMeasure
inst✝¹ : ν.IsAddHaarMeasure
inst✝ : LocallyCompactSpace E
h : Function.Surjective ⇑L
this✝² : ProperSpace E
this✝¹ : FiniteDimensional 𝕜 E
this✝ : ProperSpace F
S : Submodule 𝕜 E := ker L
T : Submodule 𝕜 E
hT : IsCompl S T
M : (↥S × ↥T) ≃ₗ[𝕜] E := S.prodEquivOfIsCompl T hT
M_cont : Continuous ⇑M.symm
P : ↥S × ↥T →ₗ[𝕜] ↥T := snd 𝕜 ↥S ↥T
P_cont : Continuous ⇑P
I✝ : Function.Bijective ⇑(L.domRestrict T)
L' : ↥T ≃ₗ[𝕜] F := LinearEquiv.ofBijective (L.domRestrict T) I✝
L'_cont : Continuous ⇑L'
A : L = ↑L' ∘ₗ P ∘ₗ ↑M.symm
I : map (⇑L) μ = map (⇑L') (map (⇑P) (map (⇑M.symm) μ))
μS : Measure ↥S := addHaar
μT : Measure ↥T := addHaar
c₀ : ℝ≥0∞
c₀_pos : c₀ ≠ 0
c₀_fin : c₀ ≠ ⊤
h₀ : map (⇑M.symm) μ = c₀ • μS.prod μT
J : map (⇑P) (μS.prod μT) = μS univ • μT
this : (map (⇑L') μT).IsAddHaarMeasure
⊢ ∃ c₁, c₁ ≠ 0 ∧ c₁ ≠ ⊤ ∧ map (⇑L') μT = c₁ • ν
state_after: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹² : NontriviallyNormedField 𝕜
inst✝¹¹ : CompleteSpace 𝕜
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : MeasurableSpace E
inst✝⁸ : BorelSpace E
inst✝⁷ : NormedSpace 𝕜 E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : MeasurableSpace F
inst✝⁴ : BorelSpace F
inst✝³ : NormedSpace 𝕜 F
L : E →ₗ[𝕜] F
μ : Measure E
ν : Measure F
inst✝² : μ.IsAddHaarMeasure
inst✝¹ : ν.IsAddHaarMeasure
inst✝ : LocallyCompactSpace E
h : Function.Surjective ⇑L
this✝² : ProperSpace E
this✝¹ : FiniteDimensional 𝕜 E
this✝ : ProperSpace F
S : Submodule 𝕜 E := ker L
T : Submodule 𝕜 E
hT : IsCompl S T
M : (↥S × ↥T) ≃ₗ[𝕜] E := S.prodEquivOfIsCompl T hT
M_cont : Continuous ⇑M.symm
P : ↥S × ↥T →ₗ[𝕜] ↥T := snd 𝕜 ↥S ↥T
P_cont : Continuous ⇑P
I✝ : Function.Bijective ⇑(L.domRestrict T)
L' : ↥T ≃ₗ[𝕜] F := LinearEquiv.ofBijective (L.domRestrict T) I✝
L'_cont : Continuous ⇑L'
A : L = ↑L' ∘ₗ P ∘ₗ ↑M.symm
I : map (⇑L) μ = map (⇑L') (map (⇑P) (map (⇑M.symm) μ))
μS : Measure ↥S := addHaar
μT : Measure ↥T := addHaar
c₀ : ℝ≥0∞
c₀_pos : c₀ ≠ 0
c₀_fin : c₀ ≠ ⊤
h₀ : map (⇑M.symm) μ = c₀ • μS.prod μT
J : map (⇑P) (μS.prod μT) = μS univ • μT
this : (map (⇑L') μT).IsAddHaarMeasure
⊢ ↑((map (⇑L') μT).addHaarScalarFactor ν) ≠ 0

tactic: simpa only [ne_eq, ENNReal.coe_eq_zero] using
  (addHaarScalarFactor_pos_of_isAddHaarMeasure (μT.map L') ν).ne'
state_before: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹² : NontriviallyNormedField 𝕜
inst✝¹¹ : CompleteSpace 𝕜
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : MeasurableSpace E
inst✝⁸ : BorelSpace E
inst✝⁷ : NormedSpace 𝕜 E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : MeasurableSpace F
inst✝⁴ : BorelSpace F
inst✝³ : NormedSpace 𝕜 F
L : E →ₗ[𝕜] F
μ : Measure E
ν : Measure F
inst✝² : μ.IsAddHaarMeasure
inst✝¹ : ν.IsAddHaarMeasure
inst✝ : LocallyCompactSpace E
h : Function.Surjective ⇑L
this✝² : ProperSpace E
this✝¹ : FiniteDimensional 𝕜 E
this✝ : ProperSpace F
S : Submodule 𝕜 E := ker L
T : Submodule 𝕜 E
hT : IsCompl S T
M : (↥S × ↥T) ≃ₗ[𝕜] E := S.prodEquivOfIsCompl T hT
M_cont : Continuous ⇑M.symm
P : ↥S × ↥T →ₗ[𝕜] ↥T := snd 𝕜 ↥S ↥T
P_cont : Continuous ⇑P
I✝ : Function.Bijective ⇑(L.domRestrict T)
L' : ↥T ≃ₗ[𝕜] F := LinearEquiv.ofBijective (L.domRestrict T) I✝
L'_cont : Continuous ⇑L'
A : L = ↑L' ∘ₗ P ∘ₗ ↑M.symm
I : map (⇑L) μ = map (⇑L') (map (⇑P) (map (⇑M.symm) μ))
μS : Measure ↥S := addHaar
μT : Measure ↥T := addHaar
c₀ : ℝ≥0∞
c₀_pos : c₀ ≠ 0
c₀_fin : c₀ ≠ ⊤
h₀ : map (⇑M.symm) μ = c₀ • μS.prod μT
J : map (⇑P) (μS.prod μT) = μS univ • μT
this : (map (⇑L') μT).IsAddHaarMeasure
⊢ ↑((map (⇑L') μT).addHaarScalarFactor ν) ≠ 0
state_after: no goals

tactic: simp [pos_iff_ne_zero, c₀_pos, c₁_pos]
state_before: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹² : NontriviallyNormedField 𝕜
inst✝¹¹ : CompleteSpace 𝕜
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : MeasurableSpace E
inst✝⁸ : BorelSpace E
inst✝⁷ : NormedSpace 𝕜 E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : MeasurableSpace F
inst✝⁴ : BorelSpace F
inst✝³ : NormedSpace 𝕜 F
L : E →ₗ[𝕜] F
μ : Measure E
ν : Measure F
inst✝² : μ.IsAddHaarMeasure
inst✝¹ : ν.IsAddHaarMeasure
inst✝ : LocallyCompactSpace E
h : Function.Surjective ⇑L
this✝¹ : ProperSpace E
this✝ : FiniteDimensional 𝕜 E
this : ProperSpace F
S : Submodule 𝕜 E := ker L
T : Submodule 𝕜 E
hT : IsCompl S T
M : (↥S × ↥T) ≃ₗ[𝕜] E := S.prodEquivOfIsCompl T hT
M_cont : Continuous ⇑M.symm
P : ↥S × ↥T →ₗ[𝕜] ↥T := snd 𝕜 ↥S ↥T
P_cont : Continuous ⇑P
I✝ : Function.Bijective ⇑(L.domRestrict T)
L' : ↥T ≃ₗ[𝕜] F := LinearEquiv.ofBijective (L.domRestrict T) I✝
L'_cont : Continuous ⇑L'
A : L = ↑L' ∘ₗ P ∘ₗ ↑M.symm
I : map (⇑L) μ = map (⇑L') (map (⇑P) (map (⇑M.symm) μ))
μS : Measure ↥S := addHaar
μT : Measure ↥T := addHaar
c₀ : ℝ≥0∞
c₀_pos : c₀ ≠ 0
c₀_fin : c₀ ≠ ⊤
h₀ : map (⇑M.symm) μ = c₀ • μS.prod μT
J : map (⇑P) (μS.prod μT) = μS univ • μT
c₁ : ℝ≥0∞
c₁_pos : c₁ ≠ 0
c₁_fin : c₁ ≠ ⊤
h₁ : map (⇑L') μT = c₁ • ν
⊢ 0 < c₀ * c₁
state_after: no goals