Theorem: MeasureTheory.det_one_smulRight
File: Mathlib/MeasureTheory/Function/Jacobian.lean

tactic: have : (1 : 𝕜 →L[𝕜] 𝕜).smulRight v = v • (1 : 𝕜 →L[𝕜] 𝕜) := by
  ext1
  simp only [ContinuousLinearMap.smulRight_apply, ContinuousLinearMap.one_apply,
    Algebra.id.smul_eq_mul, one_mul, ContinuousLinearMap.coe_smul', Pi.smul_apply, mul_one]
state_before: E : Type ?u.1032031
F : Type ?u.1032034
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
inst✝⁶ : FiniteDimensional ℝ E
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℝ F
s : Set E
f : E → E
f' : E → E →L[ℝ] E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
μ : Measure E
inst✝¹ : IsAddHaarMeasure μ
𝕜 : Type u_1
inst✝ : NormedField 𝕜
v : 𝕜
⊢ ContinuousLinearMap.det (ContinuousLinearMap.smulRight 1 v) = v

tactic: rw [this, ContinuousLinearMap.det, ContinuousLinearMap.coe_smul]
state_before: E : Type ?u.1032031
F : Type ?u.1032034
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
inst✝⁶ : FiniteDimensional ℝ E
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℝ F
s : Set E
f : E → E
f' : E → E →L[ℝ] E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
μ : Measure E
inst✝¹ : IsAddHaarMeasure μ
𝕜 : Type u_1
inst✝ : NormedField 𝕜
v : 𝕜
this : ContinuousLinearMap.smulRight 1 v = v • 1
⊢ ContinuousLinearMap.det (ContinuousLinearMap.smulRight 1 v) = v

tactic: rw [show ((1 : 𝕜 →L[𝕜] 𝕜) : 𝕜 →ₗ[𝕜] 𝕜) = LinearMap.id from rfl]
state_before: E : Type ?u.1032031
F : Type ?u.1032034
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
inst✝⁶ : FiniteDimensional ℝ E
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℝ F
s : Set E
f : E → E
f' : E → E →L[ℝ] E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
μ : Measure E
inst✝¹ : IsAddHaarMeasure μ
𝕜 : Type u_1
inst✝ : NormedField 𝕜
v : 𝕜
this : ContinuousLinearMap.smulRight 1 v = v • 1
⊢ ↑LinearMap.det (v • ↑1) = v

tactic: rw [LinearMap.det_smul, FiniteDimensional.finrank_self, LinearMap.det_id, pow_one, mul_one]
state_before: E : Type ?u.1032031
F : Type ?u.1032034
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
inst✝⁶ : FiniteDimensional ℝ E
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℝ F
s : Set E
f : E → E
f' : E → E →L[ℝ] E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
μ : Measure E
inst✝¹ : IsAddHaarMeasure μ
𝕜 : Type u_1
inst✝ : NormedField 𝕜
v : 𝕜
this : ContinuousLinearMap.smulRight 1 v = v • 1
⊢ ↑LinearMap.det (v • LinearMap.id) = v

tactic: ext1
state_before: E : Type ?u.1032031
F : Type ?u.1032034
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
inst✝⁶ : FiniteDimensional ℝ E
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℝ F
s : Set E
f : E → E
f' : E → E →L[ℝ] E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
μ : Measure E
inst✝¹ : IsAddHaarMeasure μ
𝕜 : Type u_1
inst✝ : NormedField 𝕜
v : 𝕜
⊢ ContinuousLinearMap.smulRight 1 v = v • 1

tactic: simp only [ContinuousLinearMap.smulRight_apply, ContinuousLinearMap.one_apply,
  Algebra.id.smul_eq_mul, one_mul, ContinuousLinearMap.coe_smul', Pi.smul_apply, mul_one]
state_before: case h
E : Type ?u.1032031
F : Type ?u.1032034
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
inst✝⁶ : FiniteDimensional ℝ E
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℝ F
s : Set E
f : E → E
f' : E → E →L[ℝ] E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
μ : Measure E
inst✝¹ : IsAddHaarMeasure μ
𝕜 : Type u_1
inst✝ : NormedField 𝕜
v : 𝕜
⊢ ↑(ContinuousLinearMap.smulRight 1 v) 1 = ↑(v • 1) 1