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Algebra.intTrace_eq_trace | A : Type u_1
K : Type u_2
L : Type u_3
B : Type u_4
inst✝³⁰ : CommRing A
inst✝²⁹ : CommRing B
inst✝²⁸ : Algebra A B
inst✝²⁷ : Field K
inst✝²⁶ : Field L
inst✝²⁵ : Algebra A K
inst✝²⁴ : IsFractionRing A K
inst✝²³ : Algebra B L
inst✝²² : Algebra K L
inst✝²¹ : Algebra A L
inst✝²⁰ : IsScalarTower A B L
inst✝¹⁹ : IsScalarTower A K L
inst✝¹⁸ : IsIntegralClosure B A L
inst✝¹⁷ : FiniteDimensional K L
Aₘ : Type ?u.319125
Bₘ : Type ?u.319128
inst✝¹⁶ : CommRing Aₘ
inst✝¹⁵ : CommRing Bₘ
inst✝¹⁴ : Algebra Aₘ Bₘ
inst✝¹³ : Algebra A Aₘ
inst✝¹² : Algebra B Bₘ
inst✝¹¹ : Algebra A Bₘ
inst✝¹⁰ : IsScalarTower A Aₘ Bₘ
inst✝⁹ : IsScalarTower A B Bₘ
M : Submonoid A
inst✝⁸ : IsLocalization M Aₘ
inst✝⁷ : IsLocalization (algebraMapSubmonoid B M) Bₘ
inst✝⁶ : IsDomain A
inst✝⁵ : IsIntegrallyClosed A
inst✝⁴ : IsDomain B
inst✝³ : IsIntegrallyClosed B
inst✝² : Module.Finite A B
inst✝¹ : NoZeroSMulDivisors A B
inst✝ : Module.Free A B
⊢ intTrace A B = trace A B | ext x | case h
A : Type u_1
K : Type u_2
L : Type u_3
B : Type u_4
inst✝³⁰ : CommRing A
inst✝²⁹ : CommRing B
inst✝²⁸ : Algebra A B
inst✝²⁷ : Field K
inst✝²⁶ : Field L
inst✝²⁵ : Algebra A K
inst✝²⁴ : IsFractionRing A K
inst✝²³ : Algebra B L
inst✝²² : Algebra K L
inst✝²¹ : Algebra A L
inst✝²⁰ : IsScalarTower A B L
inst✝¹⁹ : IsScalarTower A K L
inst✝¹⁸ : IsIntegralClosure B A L
inst✝¹⁷ : FiniteDimensional K L
Aₘ : Type ?u.319125
Bₘ : Type ?u.319128
inst✝¹⁶ : CommRing Aₘ
inst✝¹⁵ : CommRing Bₘ
inst✝¹⁴ : Algebra Aₘ Bₘ
inst✝¹³ : Algebra A Aₘ
inst✝¹² : Algebra B Bₘ
inst✝¹¹ : Algebra A Bₘ
inst✝¹⁰ : IsScalarTower A Aₘ Bₘ
inst✝⁹ : IsScalarTower A B Bₘ
M : Submonoid A
inst✝⁸ : IsLocalization M Aₘ
inst✝⁷ : IsLocalization (algebraMapSubmonoid B M) Bₘ
inst✝⁶ : IsDomain A
inst✝⁵ : IsIntegrallyClosed A
inst✝⁴ : IsDomain B
inst✝³ : IsIntegrallyClosed B
inst✝² : Module.Finite A B
inst✝¹ : NoZeroSMulDivisors A B
inst✝ : Module.Free A B
x : B
⊢ (intTrace A B) x = (trace A B) x | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/RingTheory/IntegralRestrict.lean |
Algebra.intTrace_eq_trace | case h
A : Type u_1
K : Type u_2
L : Type u_3
B : Type u_4
inst✝³⁰ : CommRing A
inst✝²⁹ : CommRing B
inst✝²⁸ : Algebra A B
inst✝²⁷ : Field K
inst✝²⁶ : Field L
inst✝²⁵ : Algebra A K
inst✝²⁴ : IsFractionRing A K
inst✝²³ : Algebra B L
inst✝²² : Algebra K L
inst✝²¹ : Algebra A L
inst✝²⁰ : IsScalarTower A B L
inst✝¹⁹ : IsScalarTower A K L
inst✝¹⁸ : IsIntegralClosure B A L
inst✝¹⁷ : FiniteDimensional K L
Aₘ : Type ?u.319125
Bₘ : Type ?u.319128
inst✝¹⁶ : CommRing Aₘ
inst✝¹⁵ : CommRing Bₘ
inst✝¹⁴ : Algebra Aₘ Bₘ
inst✝¹³ : Algebra A Aₘ
inst✝¹² : Algebra B Bₘ
inst✝¹¹ : Algebra A Bₘ
inst✝¹⁰ : IsScalarTower A Aₘ Bₘ
inst✝⁹ : IsScalarTower A B Bₘ
M : Submonoid A
inst✝⁸ : IsLocalization M Aₘ
inst✝⁷ : IsLocalization (algebraMapSubmonoid B M) Bₘ
inst✝⁶ : IsDomain A
inst✝⁵ : IsIntegrallyClosed A
inst✝⁴ : IsDomain B
inst✝³ : IsIntegrallyClosed B
inst✝² : Module.Finite A B
inst✝¹ : NoZeroSMulDivisors A B
inst✝ : Module.Free A B
x : B
⊢ (intTrace A B) x = (trace A B) x | haveI : <a>IsIntegralClosure</a> B A (<a>FractionRing</a> B) := <a>IsIntegralClosure.of_isIntegrallyClosed</a> _ _ _ | case h
A : Type u_1
K : Type u_2
L : Type u_3
B : Type u_4
inst✝³⁰ : CommRing A
inst✝²⁹ : CommRing B
inst✝²⁸ : Algebra A B
inst✝²⁷ : Field K
inst✝²⁶ : Field L
inst✝²⁵ : Algebra A K
inst✝²⁴ : IsFractionRing A K
inst✝²³ : Algebra B L
inst✝²² : Algebra K L
inst✝²¹ : Algebra A L
inst✝²⁰ : IsScalarTower A B L
inst✝¹⁹ : IsScalarTower A K L
inst✝¹⁸ : IsIntegralClosure B A L
inst✝¹⁷ : FiniteDimensional K L
Aₘ : Type ?u.319125
Bₘ : Type ?u.319128
inst✝¹⁶ : CommRing Aₘ
inst✝¹⁵ : CommRing Bₘ
inst✝¹⁴ : Algebra Aₘ Bₘ
inst✝¹³ : Algebra A Aₘ
inst✝¹² : Algebra B Bₘ
inst✝¹¹ : Algebra A Bₘ
inst✝¹⁰ : IsScalarTower A Aₘ Bₘ
inst✝⁹ : IsScalarTower A B Bₘ
M : Submonoid A
inst✝⁸ : IsLocalization M Aₘ
inst✝⁷ : IsLocalization (algebraMapSubmonoid B M) Bₘ
inst✝⁶ : IsDomain A
inst✝⁵ : IsIntegrallyClosed A
inst✝⁴ : IsDomain B
inst✝³ : IsIntegrallyClosed B
inst✝² : Module.Finite A B
inst✝¹ : NoZeroSMulDivisors A B
inst✝ : Module.Free A B
x : B
this : IsIntegralClosure B A (FractionRing B)
⊢ (intTrace A B) x = (trace A B) x | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/RingTheory/IntegralRestrict.lean |
Algebra.intTrace_eq_trace | case h
A : Type u_1
K : Type u_2
L : Type u_3
B : Type u_4
inst✝³⁰ : CommRing A
inst✝²⁹ : CommRing B
inst✝²⁸ : Algebra A B
inst✝²⁷ : Field K
inst✝²⁶ : Field L
inst✝²⁵ : Algebra A K
inst✝²⁴ : IsFractionRing A K
inst✝²³ : Algebra B L
inst✝²² : Algebra K L
inst✝²¹ : Algebra A L
inst✝²⁰ : IsScalarTower A B L
inst✝¹⁹ : IsScalarTower A K L
inst✝¹⁸ : IsIntegralClosure B A L
inst✝¹⁷ : FiniteDimensional K L
Aₘ : Type ?u.319125
Bₘ : Type ?u.319128
inst✝¹⁶ : CommRing Aₘ
inst✝¹⁵ : CommRing Bₘ
inst✝¹⁴ : Algebra Aₘ Bₘ
inst✝¹³ : Algebra A Aₘ
inst✝¹² : Algebra B Bₘ
inst✝¹¹ : Algebra A Bₘ
inst✝¹⁰ : IsScalarTower A Aₘ Bₘ
inst✝⁹ : IsScalarTower A B Bₘ
M : Submonoid A
inst✝⁸ : IsLocalization M Aₘ
inst✝⁷ : IsLocalization (algebraMapSubmonoid B M) Bₘ
inst✝⁶ : IsDomain A
inst✝⁵ : IsIntegrallyClosed A
inst✝⁴ : IsDomain B
inst✝³ : IsIntegrallyClosed B
inst✝² : Module.Finite A B
inst✝¹ : NoZeroSMulDivisors A B
inst✝ : Module.Free A B
x : B
this : IsIntegralClosure B A (FractionRing B)
⊢ (intTrace A B) x = (trace A B) x | haveI : <a>IsLocalization</a> (<a>Algebra.algebraMapSubmonoid</a> B A⁰) (<a>FractionRing</a> B) := <a>IsIntegralClosure.isLocalization</a> _ (<a>FractionRing</a> A) _ _ | case h
A : Type u_1
K : Type u_2
L : Type u_3
B : Type u_4
inst✝³⁰ : CommRing A
inst✝²⁹ : CommRing B
inst✝²⁸ : Algebra A B
inst✝²⁷ : Field K
inst✝²⁶ : Field L
inst✝²⁵ : Algebra A K
inst✝²⁴ : IsFractionRing A K
inst✝²³ : Algebra B L
inst✝²² : Algebra K L
inst✝²¹ : Algebra A L
inst✝²⁰ : IsScalarTower A B L
inst✝¹⁹ : IsScalarTower A K L
inst✝¹⁸ : IsIntegralClosure B A L
inst✝¹⁷ : FiniteDimensional K L
Aₘ : Type ?u.319125
Bₘ : Type ?u.319128
inst✝¹⁶ : CommRing Aₘ
inst✝¹⁵ : CommRing Bₘ
inst✝¹⁴ : Algebra Aₘ Bₘ
inst✝¹³ : Algebra A Aₘ
inst✝¹² : Algebra B Bₘ
inst✝¹¹ : Algebra A Bₘ
inst✝¹⁰ : IsScalarTower A Aₘ Bₘ
inst✝⁹ : IsScalarTower A B Bₘ
M : Submonoid A
inst✝⁸ : IsLocalization M Aₘ
inst✝⁷ : IsLocalization (algebraMapSubmonoid B M) Bₘ
inst✝⁶ : IsDomain A
inst✝⁵ : IsIntegrallyClosed A
inst✝⁴ : IsDomain B
inst✝³ : IsIntegrallyClosed B
inst✝² : Module.Finite A B
inst✝¹ : NoZeroSMulDivisors A B
inst✝ : Module.Free A B
x : B
this✝ : IsIntegralClosure B A (FractionRing B)
this : IsLocalization (algebraMapSubmonoid B A⁰) (FractionRing B)
⊢ (intTrace A B) x = (trace A B) x | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/RingTheory/IntegralRestrict.lean |
Algebra.intTrace_eq_trace | case h
A : Type u_1
K : Type u_2
L : Type u_3
B : Type u_4
inst✝³⁰ : CommRing A
inst✝²⁹ : CommRing B
inst✝²⁸ : Algebra A B
inst✝²⁷ : Field K
inst✝²⁶ : Field L
inst✝²⁵ : Algebra A K
inst✝²⁴ : IsFractionRing A K
inst✝²³ : Algebra B L
inst✝²² : Algebra K L
inst✝²¹ : Algebra A L
inst✝²⁰ : IsScalarTower A B L
inst✝¹⁹ : IsScalarTower A K L
inst✝¹⁸ : IsIntegralClosure B A L
inst✝¹⁷ : FiniteDimensional K L
Aₘ : Type ?u.319125
Bₘ : Type ?u.319128
inst✝¹⁶ : CommRing Aₘ
inst✝¹⁵ : CommRing Bₘ
inst✝¹⁴ : Algebra Aₘ Bₘ
inst✝¹³ : Algebra A Aₘ
inst✝¹² : Algebra B Bₘ
inst✝¹¹ : Algebra A Bₘ
inst✝¹⁰ : IsScalarTower A Aₘ Bₘ
inst✝⁹ : IsScalarTower A B Bₘ
M : Submonoid A
inst✝⁸ : IsLocalization M Aₘ
inst✝⁷ : IsLocalization (algebraMapSubmonoid B M) Bₘ
inst✝⁶ : IsDomain A
inst✝⁵ : IsIntegrallyClosed A
inst✝⁴ : IsDomain B
inst✝³ : IsIntegrallyClosed B
inst✝² : Module.Finite A B
inst✝¹ : NoZeroSMulDivisors A B
inst✝ : Module.Free A B
x : B
this✝ : IsIntegralClosure B A (FractionRing B)
this : IsLocalization (algebraMapSubmonoid B A⁰) (FractionRing B)
⊢ (intTrace A B) x = (trace A B) x | apply <a>IsFractionRing.injective</a> A (<a>FractionRing</a> A) | case h.a
A : Type u_1
K : Type u_2
L : Type u_3
B : Type u_4
inst✝³⁰ : CommRing A
inst✝²⁹ : CommRing B
inst✝²⁸ : Algebra A B
inst✝²⁷ : Field K
inst✝²⁶ : Field L
inst✝²⁵ : Algebra A K
inst✝²⁴ : IsFractionRing A K
inst✝²³ : Algebra B L
inst✝²² : Algebra K L
inst✝²¹ : Algebra A L
inst✝²⁰ : IsScalarTower A B L
inst✝¹⁹ : IsScalarTower A K L
inst✝¹⁸ : IsIntegralClosure B A L
inst✝¹⁷ : FiniteDimensional K L
Aₘ : Type ?u.319125
Bₘ : Type ?u.319128
inst✝¹⁶ : CommRing Aₘ
inst✝¹⁵ : CommRing Bₘ
inst✝¹⁴ : Algebra Aₘ Bₘ
inst✝¹³ : Algebra A Aₘ
inst✝¹² : Algebra B Bₘ
inst✝¹¹ : Algebra A Bₘ
inst✝¹⁰ : IsScalarTower A Aₘ Bₘ
inst✝⁹ : IsScalarTower A B Bₘ
M : Submonoid A
inst✝⁸ : IsLocalization M Aₘ
inst✝⁷ : IsLocalization (algebraMapSubmonoid B M) Bₘ
inst✝⁶ : IsDomain A
inst✝⁵ : IsIntegrallyClosed A
inst✝⁴ : IsDomain B
inst✝³ : IsIntegrallyClosed B
inst✝² : Module.Finite A B
inst✝¹ : NoZeroSMulDivisors A B
inst✝ : Module.Free A B
x : B
this✝ : IsIntegralClosure B A (FractionRing B)
this : IsLocalization (algebraMapSubmonoid B A⁰) (FractionRing B)
⊢ (algebraMap A (FractionRing A)) ((intTrace A B) x) = (algebraMap A (FractionRing A)) ((trace A B) x) | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/RingTheory/IntegralRestrict.lean |
Algebra.intTrace_eq_trace | case h.a
A : Type u_1
K : Type u_2
L : Type u_3
B : Type u_4
inst✝³⁰ : CommRing A
inst✝²⁹ : CommRing B
inst✝²⁸ : Algebra A B
inst✝²⁷ : Field K
inst✝²⁶ : Field L
inst✝²⁵ : Algebra A K
inst✝²⁴ : IsFractionRing A K
inst✝²³ : Algebra B L
inst✝²² : Algebra K L
inst✝²¹ : Algebra A L
inst✝²⁰ : IsScalarTower A B L
inst✝¹⁹ : IsScalarTower A K L
inst✝¹⁸ : IsIntegralClosure B A L
inst✝¹⁷ : FiniteDimensional K L
Aₘ : Type ?u.319125
Bₘ : Type ?u.319128
inst✝¹⁶ : CommRing Aₘ
inst✝¹⁵ : CommRing Bₘ
inst✝¹⁴ : Algebra Aₘ Bₘ
inst✝¹³ : Algebra A Aₘ
inst✝¹² : Algebra B Bₘ
inst✝¹¹ : Algebra A Bₘ
inst✝¹⁰ : IsScalarTower A Aₘ Bₘ
inst✝⁹ : IsScalarTower A B Bₘ
M : Submonoid A
inst✝⁸ : IsLocalization M Aₘ
inst✝⁷ : IsLocalization (algebraMapSubmonoid B M) Bₘ
inst✝⁶ : IsDomain A
inst✝⁵ : IsIntegrallyClosed A
inst✝⁴ : IsDomain B
inst✝³ : IsIntegrallyClosed B
inst✝² : Module.Finite A B
inst✝¹ : NoZeroSMulDivisors A B
inst✝ : Module.Free A B
x : B
this✝ : IsIntegralClosure B A (FractionRing B)
this : IsLocalization (algebraMapSubmonoid B A⁰) (FractionRing B)
⊢ (algebraMap A (FractionRing A)) ((intTrace A B) x) = (algebraMap A (FractionRing A)) ((trace A B) x) | rw [<a>Algebra.algebraMap_intTrace_fractionRing</a>, <a>Algebra.trace_localization</a> A A⁰] | no goals | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/RingTheory/IntegralRestrict.lean |
EisensteinSeries.vecMul_SL2_mem_gammaSet | N : ℕ
a : Fin 2 → ZMod N
v : Fin 2 → ℤ
hv : v ∈ gammaSet N a
γ : SL(2, ℤ)
⊢ v ᵥ* ↑γ ∈ gammaSet N (a ᵥ* ↑((SpecialLinearGroup.map (Int.castRingHom (ZMod N))) γ)) | refine ⟨?_, hv.2.<a>IsCoprime.vecMulSL</a> γ⟩ | N : ℕ
a : Fin 2 → ZMod N
v : Fin 2 → ℤ
hv : v ∈ gammaSet N a
γ : SL(2, ℤ)
⊢ Int.cast ∘ (v ᵥ* ↑γ) = a ᵥ* ↑((SpecialLinearGroup.map (Int.castRingHom (ZMod N))) γ) | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/NumberTheory/ModularForms/EisensteinSeries/Basic.lean |
EisensteinSeries.vecMul_SL2_mem_gammaSet | N : ℕ
a : Fin 2 → ZMod N
v : Fin 2 → ℤ
hv : v ∈ gammaSet N a
γ : SL(2, ℤ)
⊢ Int.cast ∘ (v ᵥ* ↑γ) = a ᵥ* ↑((SpecialLinearGroup.map (Int.castRingHom (ZMod N))) γ) | have := <a>RingHom.map_vecMul</a> (m := <a>Fin</a> 2) (n := <a>Fin</a> 2) (<a>Int.castRingHom</a> (<a>ZMod</a> N)) γ v | N : ℕ
a : Fin 2 → ZMod N
v : Fin 2 → ℤ
hv : v ∈ gammaSet N a
γ : SL(2, ℤ)
this :
∀ (i : Fin 2),
(Int.castRingHom (ZMod N)) ((v ᵥ* ↑γ) i) =
(⇑(Int.castRingHom (ZMod N)) ∘ v ᵥ* (↑γ).map ⇑(Int.castRingHom (ZMod N))) i
⊢ Int.cast ∘ (v ᵥ* ↑γ) = a ᵥ* ↑((SpecialLinearGroup.map (Int.castRingHom (ZMod N))) γ) | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/NumberTheory/ModularForms/EisensteinSeries/Basic.lean |
EisensteinSeries.vecMul_SL2_mem_gammaSet | N : ℕ
a : Fin 2 → ZMod N
v : Fin 2 → ℤ
hv : v ∈ gammaSet N a
γ : SL(2, ℤ)
this :
∀ (i : Fin 2),
(Int.castRingHom (ZMod N)) ((v ᵥ* ↑γ) i) =
(⇑(Int.castRingHom (ZMod N)) ∘ v ᵥ* (↑γ).map ⇑(Int.castRingHom (ZMod N))) i
⊢ Int.cast ∘ (v ᵥ* ↑γ) = a ᵥ* ↑((SpecialLinearGroup.map (Int.castRingHom (ZMod N))) γ) | simp only [<a>eq_intCast</a>, <a>Int.coe_castRingHom</a>] at this | N : ℕ
a : Fin 2 → ZMod N
v : Fin 2 → ℤ
hv : v ∈ gammaSet N a
γ : SL(2, ℤ)
this : ∀ (i : Fin 2), ↑((v ᵥ* ↑γ) i) = ((fun x => ↑x) ∘ v ᵥ* (↑γ).map fun x => ↑x) i
⊢ Int.cast ∘ (v ᵥ* ↑γ) = a ᵥ* ↑((SpecialLinearGroup.map (Int.castRingHom (ZMod N))) γ) | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/NumberTheory/ModularForms/EisensteinSeries/Basic.lean |
EisensteinSeries.vecMul_SL2_mem_gammaSet | N : ℕ
a : Fin 2 → ZMod N
v : Fin 2 → ℤ
hv : v ∈ gammaSet N a
γ : SL(2, ℤ)
this : ∀ (i : Fin 2), ↑((v ᵥ* ↑γ) i) = ((fun x => ↑x) ∘ v ᵥ* (↑γ).map fun x => ↑x) i
⊢ Int.cast ∘ (v ᵥ* ↑γ) = a ᵥ* ↑((SpecialLinearGroup.map (Int.castRingHom (ZMod N))) γ) | simp_rw [<a>Function.comp</a>, this, hv.1] | N : ℕ
a : Fin 2 → ZMod N
v : Fin 2 → ℤ
hv : v ∈ gammaSet N a
γ : SL(2, ℤ)
this : ∀ (i : Fin 2), ↑((v ᵥ* ↑γ) i) = ((fun x => ↑x) ∘ v ᵥ* (↑γ).map fun x => ↑x) i
⊢ (fun x => (a ᵥ* (↑γ).map fun x => ↑x) x) = a ᵥ* ↑((SpecialLinearGroup.map (Int.castRingHom (ZMod N))) γ) | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/NumberTheory/ModularForms/EisensteinSeries/Basic.lean |
EisensteinSeries.vecMul_SL2_mem_gammaSet | N : ℕ
a : Fin 2 → ZMod N
v : Fin 2 → ℤ
hv : v ∈ gammaSet N a
γ : SL(2, ℤ)
this : ∀ (i : Fin 2), ↑((v ᵥ* ↑γ) i) = ((fun x => ↑x) ∘ v ᵥ* (↑γ).map fun x => ↑x) i
⊢ (fun x => (a ᵥ* (↑γ).map fun x => ↑x) x) = a ᵥ* ↑((SpecialLinearGroup.map (Int.castRingHom (ZMod N))) γ) | simp | no goals | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/NumberTheory/ModularForms/EisensteinSeries/Basic.lean |
Matrix.toBlocks₂₂_diagonal | l : Type u_1
m : Type u_2
n : Type u_3
o : Type u_4
p : Type u_5
q : Type u_6
m' : o → Type u_7
n' : o → Type u_8
p' : o → Type u_9
R : Type u_10
S : Type u_11
α : Type u_12
β : Type u_13
inst✝² : DecidableEq l
inst✝¹ : DecidableEq m
inst✝ : Zero α
v : l ⊕ m → α
⊢ (diagonal v).toBlocks₂₂ = diagonal fun i => v (Sum.inr i) | unfold <a>Matrix.toBlocks₂₂</a> | l : Type u_1
m : Type u_2
n : Type u_3
o : Type u_4
p : Type u_5
q : Type u_6
m' : o → Type u_7
n' : o → Type u_8
p' : o → Type u_9
R : Type u_10
S : Type u_11
α : Type u_12
β : Type u_13
inst✝² : DecidableEq l
inst✝¹ : DecidableEq m
inst✝ : Zero α
v : l ⊕ m → α
⊢ (of fun i j => diagonal v (Sum.inr i) (Sum.inr j)) = diagonal fun i => v (Sum.inr i) | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/Data/Matrix/Block.lean |
Matrix.toBlocks₂₂_diagonal | l : Type u_1
m : Type u_2
n : Type u_3
o : Type u_4
p : Type u_5
q : Type u_6
m' : o → Type u_7
n' : o → Type u_8
p' : o → Type u_9
R : Type u_10
S : Type u_11
α : Type u_12
β : Type u_13
inst✝² : DecidableEq l
inst✝¹ : DecidableEq m
inst✝ : Zero α
v : l ⊕ m → α
⊢ (of fun i j => diagonal v (Sum.inr i) (Sum.inr j)) = diagonal fun i => v (Sum.inr i) | funext i j | case h.h
l : Type u_1
m : Type u_2
n : Type u_3
o : Type u_4
p : Type u_5
q : Type u_6
m' : o → Type u_7
n' : o → Type u_8
p' : o → Type u_9
R : Type u_10
S : Type u_11
α : Type u_12
β : Type u_13
inst✝² : DecidableEq l
inst✝¹ : DecidableEq m
inst✝ : Zero α
v : l ⊕ m → α
i j : m
⊢ of (fun i j => diagonal v (Sum.inr i) (Sum.inr j)) i j = diagonal (fun i => v (Sum.inr i)) i j | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/Data/Matrix/Block.lean |
Matrix.toBlocks₂₂_diagonal | case h.h
l : Type u_1
m : Type u_2
n : Type u_3
o : Type u_4
p : Type u_5
q : Type u_6
m' : o → Type u_7
n' : o → Type u_8
p' : o → Type u_9
R : Type u_10
S : Type u_11
α : Type u_12
β : Type u_13
inst✝² : DecidableEq l
inst✝¹ : DecidableEq m
inst✝ : Zero α
v : l ⊕ m → α
i j : m
⊢ of (fun i j => diagonal v (Sum.inr i) (Sum.inr j)) i j = diagonal (fun i => v (Sum.inr i)) i j | simp only [<a>ne_eq</a>, Sum.inr.injEq, <a>Matrix.of_apply</a>, <a>Matrix.diagonal_apply</a>] | no goals | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/Data/Matrix/Block.lean |
CategoryTheory.Idempotents.DoldKan.hη | C : Type u_1
inst✝³ : Category.{u_2, u_1} C
inst✝² : Preadditive C
inst✝¹ : IsIdempotentComplete C
inst✝ : HasFiniteCoproducts C
⊢ Compatibility.τ₀ = Compatibility.τ₁ isoN₁ isoΓ₀ N₁Γ₀ | ext K : 3 | case w.w.h
C : Type u_1
inst✝³ : Category.{u_2, u_1} C
inst✝² : Preadditive C
inst✝¹ : IsIdempotentComplete C
inst✝ : HasFiniteCoproducts C
K : ChainComplex C ℕ
⊢ Compatibility.τ₀.hom.app K = (Compatibility.τ₁ isoN₁ isoΓ₀ N₁Γ₀).hom.app K | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/AlgebraicTopology/DoldKan/EquivalencePseudoabelian.lean |
CategoryTheory.Idempotents.DoldKan.hη | case w.w.h
C : Type u_1
inst✝³ : Category.{u_2, u_1} C
inst✝² : Preadditive C
inst✝¹ : IsIdempotentComplete C
inst✝ : HasFiniteCoproducts C
K : ChainComplex C ℕ
⊢ Compatibility.τ₀.hom.app K = (Compatibility.τ₁ isoN₁ isoΓ₀ N₁Γ₀).hom.app K | simp only [<a>AlgebraicTopology.DoldKan.Compatibility.τ₀_hom_app</a>, <a>AlgebraicTopology.DoldKan.Compatibility.τ₁_hom_app</a>] | case w.w.h
C : Type u_1
inst✝³ : Category.{u_2, u_1} C
inst✝² : Preadditive C
inst✝¹ : IsIdempotentComplete C
inst✝ : HasFiniteCoproducts C
K : ChainComplex C ℕ
⊢ Preadditive.DoldKan.equivalence.counitIso.hom.app ((toKaroubiEquivalence (ChainComplex C ℕ)).functor.obj K) =
Preadditive.DoldKan.equivalence.functor.map (isoΓ₀.hom.app K) ≫ isoN₁.hom.app (Γ.obj K) ≫ N₁Γ₀.hom.app K | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/AlgebraicTopology/DoldKan/EquivalencePseudoabelian.lean |
CategoryTheory.Idempotents.DoldKan.hη | case w.w.h
C : Type u_1
inst✝³ : Category.{u_2, u_1} C
inst✝² : Preadditive C
inst✝¹ : IsIdempotentComplete C
inst✝ : HasFiniteCoproducts C
K : ChainComplex C ℕ
⊢ Preadditive.DoldKan.equivalence.counitIso.hom.app ((toKaroubiEquivalence (ChainComplex C ℕ)).functor.obj K) =
Preadditive.DoldKan.equivalence.functor.map (isoΓ₀.hom.app K) ≫ isoN₁.hom.app (Γ.obj K) ≫ N₁Γ₀.hom.app K | exact (<a>AlgebraicTopology.DoldKan.N₂Γ₂_compatible_with_N₁Γ₀</a> K).<a>Eq.trans</a> (by simp ) | no goals | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/AlgebraicTopology/DoldKan/EquivalencePseudoabelian.lean |
CategoryTheory.Idempotents.DoldKan.hη | C : Type u_1
inst✝³ : Category.{u_2, u_1} C
inst✝² : Preadditive C
inst✝¹ : IsIdempotentComplete C
inst✝ : HasFiniteCoproducts C
K : ChainComplex C ℕ
⊢ N₂Γ₂ToKaroubiIso.hom.app K ≫ N₁Γ₀.hom.app K =
Preadditive.DoldKan.equivalence.functor.map (isoΓ₀.hom.app K) ≫ isoN₁.hom.app (Γ.obj K) ≫ N₁Γ₀.hom.app K | simp | no goals | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/AlgebraicTopology/DoldKan/EquivalencePseudoabelian.lean |
Algebra.algebraMap_intTrace_fractionRing | A : Type u_1
K : Type u_2
L : Type u_3
B : Type u_4
inst✝²⁹ : CommRing A
inst✝²⁸ : CommRing B
inst✝²⁷ : Algebra A B
inst✝²⁶ : Field K
inst✝²⁵ : Field L
inst✝²⁴ : Algebra A K
inst✝²³ : IsFractionRing A K
inst✝²² : Algebra B L
inst✝²¹ : Algebra K L
inst✝²⁰ : Algebra A L
inst✝¹⁹ : IsScalarTower A B L
inst✝¹⁸ : IsScalarTower A K L
inst✝¹⁷ : IsIntegralClosure B A L
inst✝¹⁶ : FiniteDimensional K L
Aₘ : Type ?u.292271
Bₘ : Type ?u.292274
inst✝¹⁵ : CommRing Aₘ
inst✝¹⁴ : CommRing Bₘ
inst✝¹³ : Algebra Aₘ Bₘ
inst✝¹² : Algebra A Aₘ
inst✝¹¹ : Algebra B Bₘ
inst✝¹⁰ : Algebra A Bₘ
inst✝⁹ : IsScalarTower A Aₘ Bₘ
inst✝⁸ : IsScalarTower A B Bₘ
M : Submonoid A
inst✝⁷ : IsLocalization M Aₘ
inst✝⁶ : IsLocalization (algebraMapSubmonoid B M) Bₘ
inst✝⁵ : IsDomain A
inst✝⁴ : IsIntegrallyClosed A
inst✝³ : IsDomain B
inst✝² : IsIntegrallyClosed B
inst✝¹ : Module.Finite A B
inst✝ : NoZeroSMulDivisors A B
x : B
⊢ (algebraMap A (FractionRing A)) ((intTrace A B) x) =
(trace (FractionRing A) (FractionRing B)) ((algebraMap B (FractionRing B)) x) | haveI : <a>IsIntegralClosure</a> B A (<a>FractionRing</a> B) := <a>IsIntegralClosure.of_isIntegrallyClosed</a> _ _ _ | A : Type u_1
K : Type u_2
L : Type u_3
B : Type u_4
inst✝²⁹ : CommRing A
inst✝²⁸ : CommRing B
inst✝²⁷ : Algebra A B
inst✝²⁶ : Field K
inst✝²⁵ : Field L
inst✝²⁴ : Algebra A K
inst✝²³ : IsFractionRing A K
inst✝²² : Algebra B L
inst✝²¹ : Algebra K L
inst✝²⁰ : Algebra A L
inst✝¹⁹ : IsScalarTower A B L
inst✝¹⁸ : IsScalarTower A K L
inst✝¹⁷ : IsIntegralClosure B A L
inst✝¹⁶ : FiniteDimensional K L
Aₘ : Type ?u.292271
Bₘ : Type ?u.292274
inst✝¹⁵ : CommRing Aₘ
inst✝¹⁴ : CommRing Bₘ
inst✝¹³ : Algebra Aₘ Bₘ
inst✝¹² : Algebra A Aₘ
inst✝¹¹ : Algebra B Bₘ
inst✝¹⁰ : Algebra A Bₘ
inst✝⁹ : IsScalarTower A Aₘ Bₘ
inst✝⁸ : IsScalarTower A B Bₘ
M : Submonoid A
inst✝⁷ : IsLocalization M Aₘ
inst✝⁶ : IsLocalization (algebraMapSubmonoid B M) Bₘ
inst✝⁵ : IsDomain A
inst✝⁴ : IsIntegrallyClosed A
inst✝³ : IsDomain B
inst✝² : IsIntegrallyClosed B
inst✝¹ : Module.Finite A B
inst✝ : NoZeroSMulDivisors A B
x : B
this : IsIntegralClosure B A (FractionRing B)
⊢ (algebraMap A (FractionRing A)) ((intTrace A B) x) =
(trace (FractionRing A) (FractionRing B)) ((algebraMap B (FractionRing B)) x) | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/RingTheory/IntegralRestrict.lean |
Algebra.algebraMap_intTrace_fractionRing | A : Type u_1
K : Type u_2
L : Type u_3
B : Type u_4
inst✝²⁹ : CommRing A
inst✝²⁸ : CommRing B
inst✝²⁷ : Algebra A B
inst✝²⁶ : Field K
inst✝²⁵ : Field L
inst✝²⁴ : Algebra A K
inst✝²³ : IsFractionRing A K
inst✝²² : Algebra B L
inst✝²¹ : Algebra K L
inst✝²⁰ : Algebra A L
inst✝¹⁹ : IsScalarTower A B L
inst✝¹⁸ : IsScalarTower A K L
inst✝¹⁷ : IsIntegralClosure B A L
inst✝¹⁶ : FiniteDimensional K L
Aₘ : Type ?u.292271
Bₘ : Type ?u.292274
inst✝¹⁵ : CommRing Aₘ
inst✝¹⁴ : CommRing Bₘ
inst✝¹³ : Algebra Aₘ Bₘ
inst✝¹² : Algebra A Aₘ
inst✝¹¹ : Algebra B Bₘ
inst✝¹⁰ : Algebra A Bₘ
inst✝⁹ : IsScalarTower A Aₘ Bₘ
inst✝⁸ : IsScalarTower A B Bₘ
M : Submonoid A
inst✝⁷ : IsLocalization M Aₘ
inst✝⁶ : IsLocalization (algebraMapSubmonoid B M) Bₘ
inst✝⁵ : IsDomain A
inst✝⁴ : IsIntegrallyClosed A
inst✝³ : IsDomain B
inst✝² : IsIntegrallyClosed B
inst✝¹ : Module.Finite A B
inst✝ : NoZeroSMulDivisors A B
x : B
this : IsIntegralClosure B A (FractionRing B)
⊢ (algebraMap A (FractionRing A)) ((intTrace A B) x) =
(trace (FractionRing A) (FractionRing B)) ((algebraMap B (FractionRing B)) x) | haveI : <a>IsLocalization</a> (<a>Algebra.algebraMapSubmonoid</a> B A⁰) (<a>FractionRing</a> B) := <a>IsIntegralClosure.isLocalization</a> _ (<a>FractionRing</a> A) _ _ | A : Type u_1
K : Type u_2
L : Type u_3
B : Type u_4
inst✝²⁹ : CommRing A
inst✝²⁸ : CommRing B
inst✝²⁷ : Algebra A B
inst✝²⁶ : Field K
inst✝²⁵ : Field L
inst✝²⁴ : Algebra A K
inst✝²³ : IsFractionRing A K
inst✝²² : Algebra B L
inst✝²¹ : Algebra K L
inst✝²⁰ : Algebra A L
inst✝¹⁹ : IsScalarTower A B L
inst✝¹⁸ : IsScalarTower A K L
inst✝¹⁷ : IsIntegralClosure B A L
inst✝¹⁶ : FiniteDimensional K L
Aₘ : Type ?u.292271
Bₘ : Type ?u.292274
inst✝¹⁵ : CommRing Aₘ
inst✝¹⁴ : CommRing Bₘ
inst✝¹³ : Algebra Aₘ Bₘ
inst✝¹² : Algebra A Aₘ
inst✝¹¹ : Algebra B Bₘ
inst✝¹⁰ : Algebra A Bₘ
inst✝⁹ : IsScalarTower A Aₘ Bₘ
inst✝⁸ : IsScalarTower A B Bₘ
M : Submonoid A
inst✝⁷ : IsLocalization M Aₘ
inst✝⁶ : IsLocalization (algebraMapSubmonoid B M) Bₘ
inst✝⁵ : IsDomain A
inst✝⁴ : IsIntegrallyClosed A
inst✝³ : IsDomain B
inst✝² : IsIntegrallyClosed B
inst✝¹ : Module.Finite A B
inst✝ : NoZeroSMulDivisors A B
x : B
this✝ : IsIntegralClosure B A (FractionRing B)
this : IsLocalization (algebraMapSubmonoid B A⁰) (FractionRing B)
⊢ (algebraMap A (FractionRing A)) ((intTrace A B) x) =
(trace (FractionRing A) (FractionRing B)) ((algebraMap B (FractionRing B)) x) | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/RingTheory/IntegralRestrict.lean |
Algebra.algebraMap_intTrace_fractionRing | A : Type u_1
K : Type u_2
L : Type u_3
B : Type u_4
inst✝²⁹ : CommRing A
inst✝²⁸ : CommRing B
inst✝²⁷ : Algebra A B
inst✝²⁶ : Field K
inst✝²⁵ : Field L
inst✝²⁴ : Algebra A K
inst✝²³ : IsFractionRing A K
inst✝²² : Algebra B L
inst✝²¹ : Algebra K L
inst✝²⁰ : Algebra A L
inst✝¹⁹ : IsScalarTower A B L
inst✝¹⁸ : IsScalarTower A K L
inst✝¹⁷ : IsIntegralClosure B A L
inst✝¹⁶ : FiniteDimensional K L
Aₘ : Type ?u.292271
Bₘ : Type ?u.292274
inst✝¹⁵ : CommRing Aₘ
inst✝¹⁴ : CommRing Bₘ
inst✝¹³ : Algebra Aₘ Bₘ
inst✝¹² : Algebra A Aₘ
inst✝¹¹ : Algebra B Bₘ
inst✝¹⁰ : Algebra A Bₘ
inst✝⁹ : IsScalarTower A Aₘ Bₘ
inst✝⁸ : IsScalarTower A B Bₘ
M : Submonoid A
inst✝⁷ : IsLocalization M Aₘ
inst✝⁶ : IsLocalization (algebraMapSubmonoid B M) Bₘ
inst✝⁵ : IsDomain A
inst✝⁴ : IsIntegrallyClosed A
inst✝³ : IsDomain B
inst✝² : IsIntegrallyClosed B
inst✝¹ : Module.Finite A B
inst✝ : NoZeroSMulDivisors A B
x : B
this✝ : IsIntegralClosure B A (FractionRing B)
this : IsLocalization (algebraMapSubmonoid B A⁰) (FractionRing B)
⊢ (algebraMap A (FractionRing A)) ((intTrace A B) x) =
(trace (FractionRing A) (FractionRing B)) ((algebraMap B (FractionRing B)) x) | haveI : <a>FiniteDimensional</a> (<a>FractionRing</a> A) (<a>FractionRing</a> B) := <a>Module.Finite_of_isLocalization</a> A B _ _ A⁰ | A : Type u_1
K : Type u_2
L : Type u_3
B : Type u_4
inst✝²⁹ : CommRing A
inst✝²⁸ : CommRing B
inst✝²⁷ : Algebra A B
inst✝²⁶ : Field K
inst✝²⁵ : Field L
inst✝²⁴ : Algebra A K
inst✝²³ : IsFractionRing A K
inst✝²² : Algebra B L
inst✝²¹ : Algebra K L
inst✝²⁰ : Algebra A L
inst✝¹⁹ : IsScalarTower A B L
inst✝¹⁸ : IsScalarTower A K L
inst✝¹⁷ : IsIntegralClosure B A L
inst✝¹⁶ : FiniteDimensional K L
Aₘ : Type ?u.292271
Bₘ : Type ?u.292274
inst✝¹⁵ : CommRing Aₘ
inst✝¹⁴ : CommRing Bₘ
inst✝¹³ : Algebra Aₘ Bₘ
inst✝¹² : Algebra A Aₘ
inst✝¹¹ : Algebra B Bₘ
inst✝¹⁰ : Algebra A Bₘ
inst✝⁹ : IsScalarTower A Aₘ Bₘ
inst✝⁸ : IsScalarTower A B Bₘ
M : Submonoid A
inst✝⁷ : IsLocalization M Aₘ
inst✝⁶ : IsLocalization (algebraMapSubmonoid B M) Bₘ
inst✝⁵ : IsDomain A
inst✝⁴ : IsIntegrallyClosed A
inst✝³ : IsDomain B
inst✝² : IsIntegrallyClosed B
inst✝¹ : Module.Finite A B
inst✝ : NoZeroSMulDivisors A B
x : B
this✝¹ : IsIntegralClosure B A (FractionRing B)
this✝ : IsLocalization (algebraMapSubmonoid B A⁰) (FractionRing B)
this : FiniteDimensional (FractionRing A) (FractionRing B)
⊢ (algebraMap A (FractionRing A)) ((intTrace A B) x) =
(trace (FractionRing A) (FractionRing B)) ((algebraMap B (FractionRing B)) x) | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/RingTheory/IntegralRestrict.lean |
Algebra.algebraMap_intTrace_fractionRing | A : Type u_1
K : Type u_2
L : Type u_3
B : Type u_4
inst✝²⁹ : CommRing A
inst✝²⁸ : CommRing B
inst✝²⁷ : Algebra A B
inst✝²⁶ : Field K
inst✝²⁵ : Field L
inst✝²⁴ : Algebra A K
inst✝²³ : IsFractionRing A K
inst✝²² : Algebra B L
inst✝²¹ : Algebra K L
inst✝²⁰ : Algebra A L
inst✝¹⁹ : IsScalarTower A B L
inst✝¹⁸ : IsScalarTower A K L
inst✝¹⁷ : IsIntegralClosure B A L
inst✝¹⁶ : FiniteDimensional K L
Aₘ : Type ?u.292271
Bₘ : Type ?u.292274
inst✝¹⁵ : CommRing Aₘ
inst✝¹⁴ : CommRing Bₘ
inst✝¹³ : Algebra Aₘ Bₘ
inst✝¹² : Algebra A Aₘ
inst✝¹¹ : Algebra B Bₘ
inst✝¹⁰ : Algebra A Bₘ
inst✝⁹ : IsScalarTower A Aₘ Bₘ
inst✝⁸ : IsScalarTower A B Bₘ
M : Submonoid A
inst✝⁷ : IsLocalization M Aₘ
inst✝⁶ : IsLocalization (algebraMapSubmonoid B M) Bₘ
inst✝⁵ : IsDomain A
inst✝⁴ : IsIntegrallyClosed A
inst✝³ : IsDomain B
inst✝² : IsIntegrallyClosed B
inst✝¹ : Module.Finite A B
inst✝ : NoZeroSMulDivisors A B
x : B
this✝¹ : IsIntegralClosure B A (FractionRing B)
this✝ : IsLocalization (algebraMapSubmonoid B A⁰) (FractionRing B)
this : FiniteDimensional (FractionRing A) (FractionRing B)
⊢ (algebraMap A (FractionRing A)) ((intTrace A B) x) =
(trace (FractionRing A) (FractionRing B)) ((algebraMap B (FractionRing B)) x) | exact <a>Algebra.map_intTraceAux</a> x | no goals | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/RingTheory/IntegralRestrict.lean |
edist_vsub_vsub_le | α : Type u_1
V : Type u_2
P : Type u_3
W : Type u_4
Q : Type u_5
inst✝⁵ : SeminormedAddCommGroup V
inst✝⁴ : PseudoMetricSpace P
inst✝³ : NormedAddTorsor V P
inst✝² : NormedAddCommGroup W
inst✝¹ : MetricSpace Q
inst✝ : NormedAddTorsor W Q
p₁ p₂ p₃ p₄ : P
⊢ edist (p₁ -ᵥ p₂) (p₃ -ᵥ p₄) ≤ edist p₁ p₃ + edist p₂ p₄ | simp only [<a>edist_nndist</a>] | α : Type u_1
V : Type u_2
P : Type u_3
W : Type u_4
Q : Type u_5
inst✝⁵ : SeminormedAddCommGroup V
inst✝⁴ : PseudoMetricSpace P
inst✝³ : NormedAddTorsor V P
inst✝² : NormedAddCommGroup W
inst✝¹ : MetricSpace Q
inst✝ : NormedAddTorsor W Q
p₁ p₂ p₃ p₄ : P
⊢ ↑(nndist (p₁ -ᵥ p₂) (p₃ -ᵥ p₄)) ≤ ↑(nndist p₁ p₃) + ↑(nndist p₂ p₄) | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/Analysis/Normed/Group/AddTorsor.lean |
edist_vsub_vsub_le | α : Type u_1
V : Type u_2
P : Type u_3
W : Type u_4
Q : Type u_5
inst✝⁵ : SeminormedAddCommGroup V
inst✝⁴ : PseudoMetricSpace P
inst✝³ : NormedAddTorsor V P
inst✝² : NormedAddCommGroup W
inst✝¹ : MetricSpace Q
inst✝ : NormedAddTorsor W Q
p₁ p₂ p₃ p₄ : P
⊢ ↑(nndist (p₁ -ᵥ p₂) (p₃ -ᵥ p₄)) ≤ ↑(nndist p₁ p₃) + ↑(nndist p₂ p₄) | norm_cast | α : Type u_1
V : Type u_2
P : Type u_3
W : Type u_4
Q : Type u_5
inst✝⁵ : SeminormedAddCommGroup V
inst✝⁴ : PseudoMetricSpace P
inst✝³ : NormedAddTorsor V P
inst✝² : NormedAddCommGroup W
inst✝¹ : MetricSpace Q
inst✝ : NormedAddTorsor W Q
p₁ p₂ p₃ p₄ : P
⊢ nndist (p₁ -ᵥ p₂) (p₃ -ᵥ p₄) ≤ nndist p₁ p₃ + nndist p₂ p₄ | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/Analysis/Normed/Group/AddTorsor.lean |
edist_vsub_vsub_le | α : Type u_1
V : Type u_2
P : Type u_3
W : Type u_4
Q : Type u_5
inst✝⁵ : SeminormedAddCommGroup V
inst✝⁴ : PseudoMetricSpace P
inst✝³ : NormedAddTorsor V P
inst✝² : NormedAddCommGroup W
inst✝¹ : MetricSpace Q
inst✝ : NormedAddTorsor W Q
p₁ p₂ p₃ p₄ : P
⊢ nndist (p₁ -ᵥ p₂) (p₃ -ᵥ p₄) ≤ nndist p₁ p₃ + nndist p₂ p₄ | apply <a>dist_vsub_vsub_le</a> | no goals | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/Analysis/Normed/Group/AddTorsor.lean |
Nat.mul_add_mod' | a✝ b✝ c✝ d m n k : ℕ
p q : ℕ → Prop
a b c : ℕ
⊢ (a * b + c) % b = c % b | rw [<a>Nat.mul_comm</a>, <a>Nat.mul_add_mod</a>] | no goals | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/Data/Nat/Defs.lean |
TopologicalSpace.isTopologicalBasis_of_subbasis | α : Type u
β : Type u_1
t : TopologicalSpace α
B : Set (Set α)
s✝ : Set α
s : Set (Set α)
hs : t = generateFrom s
⊢ IsTopologicalBasis ((fun f => ⋂₀ f) '' {f | f.Finite ∧ f ⊆ s}) | subst t | α : Type u
β : Type u_1
B : Set (Set α)
s✝ : Set α
s : Set (Set α)
⊢ IsTopologicalBasis ((fun f => ⋂₀ f) '' {f | f.Finite ∧ f ⊆ s}) | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/Topology/Bases.lean |
TopologicalSpace.isTopologicalBasis_of_subbasis | α : Type u
β : Type u_1
B : Set (Set α)
s✝ : Set α
s : Set (Set α)
⊢ IsTopologicalBasis ((fun f => ⋂₀ f) '' {f | f.Finite ∧ f ⊆ s}) | letI := <a>TopologicalSpace.generateFrom</a> s | α : Type u
β : Type u_1
B : Set (Set α)
s✝ : Set α
s : Set (Set α)
this : TopologicalSpace α := generateFrom s
⊢ IsTopologicalBasis ((fun f => ⋂₀ f) '' {f | f.Finite ∧ f ⊆ s}) | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/Topology/Bases.lean |
TopologicalSpace.isTopologicalBasis_of_subbasis | α : Type u
β : Type u_1
B : Set (Set α)
s✝ : Set α
s : Set (Set α)
this : TopologicalSpace α := generateFrom s
⊢ IsTopologicalBasis ((fun f => ⋂₀ f) '' {f | f.Finite ∧ f ⊆ s}) | refine ⟨?_, ?_, <a>le_antisymm</a> (<a>le_generateFrom</a> ?_) <| <a>TopologicalSpace.generateFrom_anti</a> fun t ht => ?_⟩ | case refine_1
α : Type u
β : Type u_1
B : Set (Set α)
s✝ : Set α
s : Set (Set α)
this : TopologicalSpace α := generateFrom s
⊢ ∀ t₁ ∈ (fun f => ⋂₀ f) '' {f | f.Finite ∧ f ⊆ s},
∀ t₂ ∈ (fun f => ⋂₀ f) '' {f | f.Finite ∧ f ⊆ s},
∀ x ∈ t₁ ∩ t₂, ∃ t₃ ∈ (fun f => ⋂₀ f) '' {f | f.Finite ∧ f ⊆ s}, x ∈ t₃ ∧ t₃ ⊆ t₁ ∩ t₂
case refine_2
α : Type u
β : Type u_1
B : Set (Set α)
s✝ : Set α
s : Set (Set α)
this : TopologicalSpace α := generateFrom s
⊢ ⋃₀ ((fun f => ⋂₀ f) '' {f | f.Finite ∧ f ⊆ s}) = univ
case refine_3
α : Type u
β : Type u_1
B : Set (Set α)
s✝ : Set α
s : Set (Set α)
this : TopologicalSpace α := generateFrom s
⊢ ∀ s_1 ∈ (fun f => ⋂₀ f) '' {f | f.Finite ∧ f ⊆ s}, IsOpen s_1
case refine_4
α : Type u
β : Type u_1
B : Set (Set α)
s✝ : Set α
s : Set (Set α)
this : TopologicalSpace α := generateFrom s
t : Set α
ht : t ∈ s
⊢ t ∈ (fun f => ⋂₀ f) '' {f | f.Finite ∧ f ⊆ s} | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/Topology/Bases.lean |
TopologicalSpace.isTopologicalBasis_of_subbasis | case refine_1
α : Type u
β : Type u_1
B : Set (Set α)
s✝ : Set α
s : Set (Set α)
this : TopologicalSpace α := generateFrom s
⊢ ∀ t₁ ∈ (fun f => ⋂₀ f) '' {f | f.Finite ∧ f ⊆ s},
∀ t₂ ∈ (fun f => ⋂₀ f) '' {f | f.Finite ∧ f ⊆ s},
∀ x ∈ t₁ ∩ t₂, ∃ t₃ ∈ (fun f => ⋂₀ f) '' {f | f.Finite ∧ f ⊆ s}, x ∈ t₃ ∧ t₃ ⊆ t₁ ∩ t₂ | rintro _ ⟨t₁, ⟨hft₁, ht₁b⟩, rfl⟩ _ ⟨t₂, ⟨hft₂, ht₂b⟩, rfl⟩ x h | case refine_1.intro.intro.intro.intro.intro.intro
α : Type u
β : Type u_1
B : Set (Set α)
s✝ : Set α
s : Set (Set α)
this : TopologicalSpace α := generateFrom s
t₁ : Set (Set α)
hft₁ : t₁.Finite
ht₁b : t₁ ⊆ s
t₂ : Set (Set α)
hft₂ : t₂.Finite
ht₂b : t₂ ⊆ s
x : α
h : x ∈ (fun f => ⋂₀ f) t₁ ∩ (fun f => ⋂₀ f) t₂
⊢ ∃ t₃ ∈ (fun f => ⋂₀ f) '' {f | f.Finite ∧ f ⊆ s}, x ∈ t₃ ∧ t₃ ⊆ (fun f => ⋂₀ f) t₁ ∩ (fun f => ⋂₀ f) t₂ | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/Topology/Bases.lean |
TopologicalSpace.isTopologicalBasis_of_subbasis | case refine_1.intro.intro.intro.intro.intro.intro
α : Type u
β : Type u_1
B : Set (Set α)
s✝ : Set α
s : Set (Set α)
this : TopologicalSpace α := generateFrom s
t₁ : Set (Set α)
hft₁ : t₁.Finite
ht₁b : t₁ ⊆ s
t₂ : Set (Set α)
hft₂ : t₂.Finite
ht₂b : t₂ ⊆ s
x : α
h : x ∈ (fun f => ⋂₀ f) t₁ ∩ (fun f => ⋂₀ f) t₂
⊢ ∃ t₃ ∈ (fun f => ⋂₀ f) '' {f | f.Finite ∧ f ⊆ s}, x ∈ t₃ ∧ t₃ ⊆ (fun f => ⋂₀ f) t₁ ∩ (fun f => ⋂₀ f) t₂ | exact ⟨_, ⟨_, ⟨hft₁.union hft₂, <a>Set.union_subset</a> ht₁b ht₂b⟩, <a>Set.sInter_union</a> t₁ t₂⟩, h, <a>Set.Subset.rfl</a>⟩ | no goals | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/Topology/Bases.lean |
TopologicalSpace.isTopologicalBasis_of_subbasis | case refine_2
α : Type u
β : Type u_1
B : Set (Set α)
s✝ : Set α
s : Set (Set α)
this : TopologicalSpace α := generateFrom s
⊢ ⋃₀ ((fun f => ⋂₀ f) '' {f | f.Finite ∧ f ⊆ s}) = univ | rw [<a>Set.sUnion_image</a>, <a>Set.iUnion₂_eq_univ_iff</a>] | case refine_2
α : Type u
β : Type u_1
B : Set (Set α)
s✝ : Set α
s : Set (Set α)
this : TopologicalSpace α := generateFrom s
⊢ ∀ (a : α), ∃ i, ∃ (_ : i ∈ {f | f.Finite ∧ f ⊆ s}), a ∈ ⋂₀ i | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/Topology/Bases.lean |
TopologicalSpace.isTopologicalBasis_of_subbasis | case refine_2
α : Type u
β : Type u_1
B : Set (Set α)
s✝ : Set α
s : Set (Set α)
this : TopologicalSpace α := generateFrom s
⊢ ∀ (a : α), ∃ i, ∃ (_ : i ∈ {f | f.Finite ∧ f ⊆ s}), a ∈ ⋂₀ i | exact fun x => ⟨∅, ⟨<a>Set.finite_empty</a>, <a>Set.empty_subset</a> _⟩, sInter_empty.substr <| <a>Set.mem_univ</a> x⟩ | no goals | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/Topology/Bases.lean |
TopologicalSpace.isTopologicalBasis_of_subbasis | case refine_3
α : Type u
β : Type u_1
B : Set (Set α)
s✝ : Set α
s : Set (Set α)
this : TopologicalSpace α := generateFrom s
⊢ ∀ s_1 ∈ (fun f => ⋂₀ f) '' {f | f.Finite ∧ f ⊆ s}, IsOpen s_1 | rintro _ ⟨t, ⟨hft, htb⟩, rfl⟩ | case refine_3.intro.intro.intro
α : Type u
β : Type u_1
B : Set (Set α)
s✝ : Set α
s : Set (Set α)
this : TopologicalSpace α := generateFrom s
t : Set (Set α)
hft : t.Finite
htb : t ⊆ s
⊢ IsOpen ((fun f => ⋂₀ f) t) | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/Topology/Bases.lean |
TopologicalSpace.isTopologicalBasis_of_subbasis | case refine_3.intro.intro.intro
α : Type u
β : Type u_1
B : Set (Set α)
s✝ : Set α
s : Set (Set α)
this : TopologicalSpace α := generateFrom s
t : Set (Set α)
hft : t.Finite
htb : t ⊆ s
⊢ IsOpen ((fun f => ⋂₀ f) t) | exact hft.isOpen_sInter fun s hs ↦ <a>TopologicalSpace.GenerateOpen.basic</a> _ <| htb hs | no goals | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/Topology/Bases.lean |
TopologicalSpace.isTopologicalBasis_of_subbasis | case refine_4
α : Type u
β : Type u_1
B : Set (Set α)
s✝ : Set α
s : Set (Set α)
this : TopologicalSpace α := generateFrom s
t : Set α
ht : t ∈ s
⊢ t ∈ (fun f => ⋂₀ f) '' {f | f.Finite ∧ f ⊆ s} | rw [← <a>Set.sInter_singleton</a> t] | case refine_4
α : Type u
β : Type u_1
B : Set (Set α)
s✝ : Set α
s : Set (Set α)
this : TopologicalSpace α := generateFrom s
t : Set α
ht : t ∈ s
⊢ ⋂₀ {t} ∈ (fun f => ⋂₀ f) '' {f | f.Finite ∧ f ⊆ s} | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/Topology/Bases.lean |
TopologicalSpace.isTopologicalBasis_of_subbasis | case refine_4
α : Type u
β : Type u_1
B : Set (Set α)
s✝ : Set α
s : Set (Set α)
this : TopologicalSpace α := generateFrom s
t : Set α
ht : t ∈ s
⊢ ⋂₀ {t} ∈ (fun f => ⋂₀ f) '' {f | f.Finite ∧ f ⊆ s} | exact ⟨{t}, ⟨<a>Set.finite_singleton</a> t, <a>Set.singleton_subset_iff</a>.2 ht⟩, <a>rfl</a>⟩ | no goals | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/Topology/Bases.lean |
MeasureTheory.OuterMeasure.restrict_biInf | α : Type u_1
ι : Type u_2
I : Set ι
hI : I.Nonempty
s : Set α
m : ι → OuterMeasure α
⊢ (restrict s) (⨅ i ∈ I, m i) = ⨅ i ∈ I, (restrict s) (m i) | haveI := hI.to_subtype | α : Type u_1
ι : Type u_2
I : Set ι
hI : I.Nonempty
s : Set α
m : ι → OuterMeasure α
this : Nonempty ↑I
⊢ (restrict s) (⨅ i ∈ I, m i) = ⨅ i ∈ I, (restrict s) (m i) | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/MeasureTheory/OuterMeasure/OfFunction.lean |
MeasureTheory.OuterMeasure.restrict_biInf | α : Type u_1
ι : Type u_2
I : Set ι
hI : I.Nonempty
s : Set α
m : ι → OuterMeasure α
this : Nonempty ↑I
⊢ (restrict s) (⨅ i ∈ I, m i) = ⨅ i ∈ I, (restrict s) (m i) | rw [← <a>iInf_subtype''</a>, ← <a>iInf_subtype''</a>] | α : Type u_1
ι : Type u_2
I : Set ι
hI : I.Nonempty
s : Set α
m : ι → OuterMeasure α
this : Nonempty ↑I
⊢ (restrict s) (⨅ i, m ↑i) = ⨅ i, (restrict s) (m ↑i) | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/MeasureTheory/OuterMeasure/OfFunction.lean |
MeasureTheory.OuterMeasure.restrict_biInf | α : Type u_1
ι : Type u_2
I : Set ι
hI : I.Nonempty
s : Set α
m : ι → OuterMeasure α
this : Nonempty ↑I
⊢ (restrict s) (⨅ i, m ↑i) = ⨅ i, (restrict s) (m ↑i) | exact <a>MeasureTheory.OuterMeasure.restrict_iInf</a> _ _ | no goals | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/MeasureTheory/OuterMeasure/OfFunction.lean |
Finset.disjoint_insert_right | α : Type u_1
β : Type u_2
γ : Type u_3
inst✝ : DecidableEq α
s t u v : Finset α
a b : α
⊢ _root_.Disjoint (insert a t) s ↔ a ∉ s ∧ _root_.Disjoint s t | rw [<a>Finset.disjoint_insert_left</a>, <a>disjoint_comm</a>] | no goals | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/Data/Finset/Basic.lean |
isNoetherian_of_fg_of_noetherian | R : Type u_1
M : Type u_2
inst✝² : Ring R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
N : Submodule R M
I : IsNoetherianRing R
hN : N.FG
⊢ IsNoetherian R ↥N | let ⟨s, hs⟩ := hN | R : Type u_1
M : Type u_2
inst✝² : Ring R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
N : Submodule R M
I : IsNoetherianRing R
hN : N.FG
s : Finset M
hs : span R ↑s = N
⊢ IsNoetherian R ↥N | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/RingTheory/Noetherian.lean |
isNoetherian_of_fg_of_noetherian | R : Type u_1
M : Type u_2
inst✝² : Ring R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
N : Submodule R M
I : IsNoetherianRing R
hN : N.FG
s : Finset M
hs : span R ↑s = N
⊢ IsNoetherian R ↥N | haveI := <a>Classical.decEq</a> M | R : Type u_1
M : Type u_2
inst✝² : Ring R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
N : Submodule R M
I : IsNoetherianRing R
hN : N.FG
s : Finset M
hs : span R ↑s = N
this : DecidableEq M
⊢ IsNoetherian R ↥N | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/RingTheory/Noetherian.lean |
isNoetherian_of_fg_of_noetherian | R : Type u_1
M : Type u_2
inst✝² : Ring R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
N : Submodule R M
I : IsNoetherianRing R
hN : N.FG
s : Finset M
hs : span R ↑s = N
this : DecidableEq M
⊢ IsNoetherian R ↥N | haveI := <a>Classical.decEq</a> R | R : Type u_1
M : Type u_2
inst✝² : Ring R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
N : Submodule R M
I : IsNoetherianRing R
hN : N.FG
s : Finset M
hs : span R ↑s = N
this✝ : DecidableEq M
this : DecidableEq R
⊢ IsNoetherian R ↥N | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/RingTheory/Noetherian.lean |
isNoetherian_of_fg_of_noetherian | R : Type u_1
M : Type u_2
inst✝² : Ring R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
N : Submodule R M
I : IsNoetherianRing R
hN : N.FG
s : Finset M
hs : span R ↑s = N
this✝ : DecidableEq M
this : DecidableEq R
⊢ IsNoetherian R ↥N | have : ∀ x ∈ s, x ∈ N := fun x hx => hs ▸ <a>Submodule.subset_span</a> hx | R : Type u_1
M : Type u_2
inst✝² : Ring R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
N : Submodule R M
I : IsNoetherianRing R
hN : N.FG
s : Finset M
hs : span R ↑s = N
this✝¹ : DecidableEq M
this✝ : DecidableEq R
this : ∀ x ∈ s, x ∈ N
⊢ IsNoetherian R ↥N | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/RingTheory/Noetherian.lean |
isNoetherian_of_fg_of_noetherian | R : Type u_1
M : Type u_2
inst✝² : Ring R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
N : Submodule R M
I : IsNoetherianRing R
hN : N.FG
s : Finset M
hs : span R ↑s = N
this✝¹ : DecidableEq M
this✝ : DecidableEq R
this : ∀ x ∈ s, x ∈ N
⊢ IsNoetherian R ↥N | refine @<a>isNoetherian_of_surjective</a> R ((↑s : <a>Set</a> M) → R) N _ _ _ (<a>Pi.module</a> _ _ _) _ ?_ ?_ <a>isNoetherian_pi</a> | case refine_1
R : Type u_1
M : Type u_2
inst✝² : Ring R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
N : Submodule R M
I : IsNoetherianRing R
hN : N.FG
s : Finset M
hs : span R ↑s = N
this✝¹ : DecidableEq M
this✝ : DecidableEq R
this : ∀ x ∈ s, x ∈ N
⊢ (↑↑s → R) →ₗ[R] ↥N
case refine_2
R : Type u_1
M : Type u_2
inst✝² : Ring R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
N : Submodule R M
I : IsNoetherianRing R
hN : N.FG
s : Finset M
hs : span R ↑s = N
this✝¹ : DecidableEq M
this✝ : DecidableEq R
this : ∀ x ∈ s, x ∈ N
⊢ LinearMap.range ?refine_1 = ⊤ | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/RingTheory/Noetherian.lean |
isNoetherian_of_fg_of_noetherian | case refine_1
R : Type u_1
M : Type u_2
inst✝² : Ring R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
N : Submodule R M
I : IsNoetherianRing R
hN : N.FG
s : Finset M
hs : span R ↑s = N
this✝¹ : DecidableEq M
this✝ : DecidableEq R
this : ∀ x ∈ s, x ∈ N
⊢ (↑↑s → R) →ₗ[R] ↥N | fapply <a>LinearMap.mk</a> | case refine_1.toAddHom
R : Type u_1
M : Type u_2
inst✝² : Ring R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
N : Submodule R M
I : IsNoetherianRing R
hN : N.FG
s : Finset M
hs : span R ↑s = N
this✝¹ : DecidableEq M
this✝ : DecidableEq R
this : ∀ x ∈ s, x ∈ N
⊢ AddHom (↑↑s → R) ↥N
case refine_1.map_smul'
R : Type u_1
M : Type u_2
inst✝² : Ring R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
N : Submodule R M
I : IsNoetherianRing R
hN : N.FG
s : Finset M
hs : span R ↑s = N
this✝¹ : DecidableEq M
this✝ : DecidableEq R
this : ∀ x ∈ s, x ∈ N
⊢ ∀ (m : R) (x : ↑↑s → R), ?refine_1.toAddHom.toFun (m • x) = (RingHom.id R) m • ?refine_1.toAddHom.toFun x | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/RingTheory/Noetherian.lean |
isNoetherian_of_fg_of_noetherian | case refine_1.toAddHom
R : Type u_1
M : Type u_2
inst✝² : Ring R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
N : Submodule R M
I : IsNoetherianRing R
hN : N.FG
s : Finset M
hs : span R ↑s = N
this✝¹ : DecidableEq M
this✝ : DecidableEq R
this : ∀ x ∈ s, x ∈ N
⊢ AddHom (↑↑s → R) ↥N | fapply <a>AddHom.mk</a> | case refine_1.toAddHom.toFun
R : Type u_1
M : Type u_2
inst✝² : Ring R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
N : Submodule R M
I : IsNoetherianRing R
hN : N.FG
s : Finset M
hs : span R ↑s = N
this✝¹ : DecidableEq M
this✝ : DecidableEq R
this : ∀ x ∈ s, x ∈ N
⊢ (↑↑s → R) → ↥N
case refine_1.toAddHom.map_add'
R : Type u_1
M : Type u_2
inst✝² : Ring R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
N : Submodule R M
I : IsNoetherianRing R
hN : N.FG
s : Finset M
hs : span R ↑s = N
this✝¹ : DecidableEq M
this✝ : DecidableEq R
this : ∀ x ∈ s, x ∈ N
⊢ ∀ (x y : ↑↑s → R), ?refine_1.toAddHom.toFun (x + y) = ?refine_1.toAddHom.toFun x + ?refine_1.toAddHom.toFun y | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/RingTheory/Noetherian.lean |
isNoetherian_of_fg_of_noetherian | case refine_1.toAddHom.toFun
R : Type u_1
M : Type u_2
inst✝² : Ring R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
N : Submodule R M
I : IsNoetherianRing R
hN : N.FG
s : Finset M
hs : span R ↑s = N
this✝¹ : DecidableEq M
this✝ : DecidableEq R
this : ∀ x ∈ s, x ∈ N
⊢ (↑↑s → R) → ↥N | exact fun f => ⟨∑ i ∈ s.attach, f i • i.1, N.sum_mem fun c _ => N.smul_mem _ <| this _ c.2⟩ | no goals | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/RingTheory/Noetherian.lean |
isNoetherian_of_fg_of_noetherian | case refine_1.toAddHom.map_add'
R : Type u_1
M : Type u_2
inst✝² : Ring R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
N : Submodule R M
I : IsNoetherianRing R
hN : N.FG
s : Finset M
hs : span R ↑s = N
this✝¹ : DecidableEq M
this✝ : DecidableEq R
this : ∀ x ∈ s, x ∈ N
⊢ ∀ (x y : ↑↑s → R), ⟨∑ i ∈ s.attach, (x + y) i • ↑i, ⋯⟩ = ⟨∑ i ∈ s.attach, x i • ↑i, ⋯⟩ + ⟨∑ i ∈ s.attach, y i • ↑i, ⋯⟩ | intro f g | case refine_1.toAddHom.map_add'
R : Type u_1
M : Type u_2
inst✝² : Ring R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
N : Submodule R M
I : IsNoetherianRing R
hN : N.FG
s : Finset M
hs : span R ↑s = N
this✝¹ : DecidableEq M
this✝ : DecidableEq R
this : ∀ x ∈ s, x ∈ N
f g : ↑↑s → R
⊢ ⟨∑ i ∈ s.attach, (f + g) i • ↑i, ⋯⟩ = ⟨∑ i ∈ s.attach, f i • ↑i, ⋯⟩ + ⟨∑ i ∈ s.attach, g i • ↑i, ⋯⟩ | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/RingTheory/Noetherian.lean |
isNoetherian_of_fg_of_noetherian | case refine_1.toAddHom.map_add'
R : Type u_1
M : Type u_2
inst✝² : Ring R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
N : Submodule R M
I : IsNoetherianRing R
hN : N.FG
s : Finset M
hs : span R ↑s = N
this✝¹ : DecidableEq M
this✝ : DecidableEq R
this : ∀ x ∈ s, x ∈ N
f g : ↑↑s → R
⊢ ⟨∑ i ∈ s.attach, (f + g) i • ↑i, ⋯⟩ = ⟨∑ i ∈ s.attach, f i • ↑i, ⋯⟩ + ⟨∑ i ∈ s.attach, g i • ↑i, ⋯⟩ | apply <a>Subtype.eq</a> | case refine_1.toAddHom.map_add'.a
R : Type u_1
M : Type u_2
inst✝² : Ring R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
N : Submodule R M
I : IsNoetherianRing R
hN : N.FG
s : Finset M
hs : span R ↑s = N
this✝¹ : DecidableEq M
this✝ : DecidableEq R
this : ∀ x ∈ s, x ∈ N
f g : ↑↑s → R
⊢ ↑⟨∑ i ∈ s.attach, (f + g) i • ↑i, ⋯⟩ = ↑(⟨∑ i ∈ s.attach, f i • ↑i, ⋯⟩ + ⟨∑ i ∈ s.attach, g i • ↑i, ⋯⟩) | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/RingTheory/Noetherian.lean |
isNoetherian_of_fg_of_noetherian | case refine_1.toAddHom.map_add'.a
R : Type u_1
M : Type u_2
inst✝² : Ring R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
N : Submodule R M
I : IsNoetherianRing R
hN : N.FG
s : Finset M
hs : span R ↑s = N
this✝¹ : DecidableEq M
this✝ : DecidableEq R
this : ∀ x ∈ s, x ∈ N
f g : ↑↑s → R
⊢ ↑⟨∑ i ∈ s.attach, (f + g) i • ↑i, ⋯⟩ = ↑(⟨∑ i ∈ s.attach, f i • ↑i, ⋯⟩ + ⟨∑ i ∈ s.attach, g i • ↑i, ⋯⟩) | change (∑ i ∈ s.attach, (f i + g i) • _) = _ | case refine_1.toAddHom.map_add'.a
R : Type u_1
M : Type u_2
inst✝² : Ring R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
N : Submodule R M
I : IsNoetherianRing R
hN : N.FG
s : Finset M
hs : span R ↑s = N
this✝¹ : DecidableEq M
this✝ : DecidableEq R
this : ∀ x ∈ s, x ∈ N
f g : ↑↑s → R
⊢ ∑ i ∈ s.attach, (f i + g i) • ↑i = ↑(⟨∑ i ∈ s.attach, f i • ↑i, ⋯⟩ + ⟨∑ i ∈ s.attach, g i • ↑i, ⋯⟩) | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/RingTheory/Noetherian.lean |
isNoetherian_of_fg_of_noetherian | case refine_1.toAddHom.map_add'.a
R : Type u_1
M : Type u_2
inst✝² : Ring R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
N : Submodule R M
I : IsNoetherianRing R
hN : N.FG
s : Finset M
hs : span R ↑s = N
this✝¹ : DecidableEq M
this✝ : DecidableEq R
this : ∀ x ∈ s, x ∈ N
f g : ↑↑s → R
⊢ ∑ i ∈ s.attach, (f i + g i) • ↑i = ↑(⟨∑ i ∈ s.attach, f i • ↑i, ⋯⟩ + ⟨∑ i ∈ s.attach, g i • ↑i, ⋯⟩) | simp only [<a>add_smul</a>, <a>Finset.sum_add_distrib</a>] | case refine_1.toAddHom.map_add'.a
R : Type u_1
M : Type u_2
inst✝² : Ring R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
N : Submodule R M
I : IsNoetherianRing R
hN : N.FG
s : Finset M
hs : span R ↑s = N
this✝¹ : DecidableEq M
this✝ : DecidableEq R
this : ∀ x ∈ s, x ∈ N
f g : ↑↑s → R
⊢ ∑ x ∈ s.attach, f x • ↑x + ∑ x ∈ s.attach, g x • ↑x = ↑(⟨∑ i ∈ s.attach, f i • ↑i, ⋯⟩ + ⟨∑ i ∈ s.attach, g i • ↑i, ⋯⟩) | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/RingTheory/Noetherian.lean |
isNoetherian_of_fg_of_noetherian | case refine_1.toAddHom.map_add'.a
R : Type u_1
M : Type u_2
inst✝² : Ring R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
N : Submodule R M
I : IsNoetherianRing R
hN : N.FG
s : Finset M
hs : span R ↑s = N
this✝¹ : DecidableEq M
this✝ : DecidableEq R
this : ∀ x ∈ s, x ∈ N
f g : ↑↑s → R
⊢ ∑ x ∈ s.attach, f x • ↑x + ∑ x ∈ s.attach, g x • ↑x = ↑(⟨∑ i ∈ s.attach, f i • ↑i, ⋯⟩ + ⟨∑ i ∈ s.attach, g i • ↑i, ⋯⟩) | rfl | no goals | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/RingTheory/Noetherian.lean |
isNoetherian_of_fg_of_noetherian | case refine_1.map_smul'
R : Type u_1
M : Type u_2
inst✝² : Ring R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
N : Submodule R M
I : IsNoetherianRing R
hN : N.FG
s : Finset M
hs : span R ↑s = N
this✝¹ : DecidableEq M
this✝ : DecidableEq R
this : ∀ x ∈ s, x ∈ N
⊢ ∀ (m : R) (x : ↑↑s → R),
{ toFun := fun f => ⟨∑ i ∈ s.attach, f i • ↑i, ⋯⟩, map_add' := ⋯ }.toFun (m • x) =
(RingHom.id R) m • { toFun := fun f => ⟨∑ i ∈ s.attach, f i • ↑i, ⋯⟩, map_add' := ⋯ }.toFun x | intro c f | case refine_1.map_smul'
R : Type u_1
M : Type u_2
inst✝² : Ring R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
N : Submodule R M
I : IsNoetherianRing R
hN : N.FG
s : Finset M
hs : span R ↑s = N
this✝¹ : DecidableEq M
this✝ : DecidableEq R
this : ∀ x ∈ s, x ∈ N
c : R
f : ↑↑s → R
⊢ { toFun := fun f => ⟨∑ i ∈ s.attach, f i • ↑i, ⋯⟩, map_add' := ⋯ }.toFun (c • f) =
(RingHom.id R) c • { toFun := fun f => ⟨∑ i ∈ s.attach, f i • ↑i, ⋯⟩, map_add' := ⋯ }.toFun f | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/RingTheory/Noetherian.lean |
isNoetherian_of_fg_of_noetherian | case refine_1.map_smul'
R : Type u_1
M : Type u_2
inst✝² : Ring R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
N : Submodule R M
I : IsNoetherianRing R
hN : N.FG
s : Finset M
hs : span R ↑s = N
this✝¹ : DecidableEq M
this✝ : DecidableEq R
this : ∀ x ∈ s, x ∈ N
c : R
f : ↑↑s → R
⊢ { toFun := fun f => ⟨∑ i ∈ s.attach, f i • ↑i, ⋯⟩, map_add' := ⋯ }.toFun (c • f) =
(RingHom.id R) c • { toFun := fun f => ⟨∑ i ∈ s.attach, f i • ↑i, ⋯⟩, map_add' := ⋯ }.toFun f | apply <a>Subtype.eq</a> | case refine_1.map_smul'.a
R : Type u_1
M : Type u_2
inst✝² : Ring R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
N : Submodule R M
I : IsNoetherianRing R
hN : N.FG
s : Finset M
hs : span R ↑s = N
this✝¹ : DecidableEq M
this✝ : DecidableEq R
this : ∀ x ∈ s, x ∈ N
c : R
f : ↑↑s → R
⊢ ↑({ toFun := fun f => ⟨∑ i ∈ s.attach, f i • ↑i, ⋯⟩, map_add' := ⋯ }.toFun (c • f)) =
↑((RingHom.id R) c • { toFun := fun f => ⟨∑ i ∈ s.attach, f i • ↑i, ⋯⟩, map_add' := ⋯ }.toFun f) | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/RingTheory/Noetherian.lean |
isNoetherian_of_fg_of_noetherian | case refine_1.map_smul'.a
R : Type u_1
M : Type u_2
inst✝² : Ring R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
N : Submodule R M
I : IsNoetherianRing R
hN : N.FG
s : Finset M
hs : span R ↑s = N
this✝¹ : DecidableEq M
this✝ : DecidableEq R
this : ∀ x ∈ s, x ∈ N
c : R
f : ↑↑s → R
⊢ ↑({ toFun := fun f => ⟨∑ i ∈ s.attach, f i • ↑i, ⋯⟩, map_add' := ⋯ }.toFun (c • f)) =
↑((RingHom.id R) c • { toFun := fun f => ⟨∑ i ∈ s.attach, f i • ↑i, ⋯⟩, map_add' := ⋯ }.toFun f) | change (∑ i ∈ s.attach, (c • f i) • _) = _ | case refine_1.map_smul'.a
R : Type u_1
M : Type u_2
inst✝² : Ring R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
N : Submodule R M
I : IsNoetherianRing R
hN : N.FG
s : Finset M
hs : span R ↑s = N
this✝¹ : DecidableEq M
this✝ : DecidableEq R
this : ∀ x ∈ s, x ∈ N
c : R
f : ↑↑s → R
⊢ ∑ i ∈ s.attach, (c • f i) • ↑i =
↑((RingHom.id R) c • { toFun := fun f => ⟨∑ i ∈ s.attach, f i • ↑i, ⋯⟩, map_add' := ⋯ }.toFun f) | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/RingTheory/Noetherian.lean |
isNoetherian_of_fg_of_noetherian | case refine_1.map_smul'.a
R : Type u_1
M : Type u_2
inst✝² : Ring R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
N : Submodule R M
I : IsNoetherianRing R
hN : N.FG
s : Finset M
hs : span R ↑s = N
this✝¹ : DecidableEq M
this✝ : DecidableEq R
this : ∀ x ∈ s, x ∈ N
c : R
f : ↑↑s → R
⊢ ∑ i ∈ s.attach, (c • f i) • ↑i =
↑((RingHom.id R) c • { toFun := fun f => ⟨∑ i ∈ s.attach, f i • ↑i, ⋯⟩, map_add' := ⋯ }.toFun f) | simp only [<a>smul_eq_mul</a>, <a>MulAction.mul_smul</a>] | case refine_1.map_smul'.a
R : Type u_1
M : Type u_2
inst✝² : Ring R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
N : Submodule R M
I : IsNoetherianRing R
hN : N.FG
s : Finset M
hs : span R ↑s = N
this✝¹ : DecidableEq M
this✝ : DecidableEq R
this : ∀ x ∈ s, x ∈ N
c : R
f : ↑↑s → R
⊢ ∑ x ∈ s.attach, c • f x • ↑x = ↑((RingHom.id R) c • ⟨∑ i ∈ s.attach, f i • ↑i, ⋯⟩) | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/RingTheory/Noetherian.lean |
isNoetherian_of_fg_of_noetherian | case refine_1.map_smul'.a
R : Type u_1
M : Type u_2
inst✝² : Ring R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
N : Submodule R M
I : IsNoetherianRing R
hN : N.FG
s : Finset M
hs : span R ↑s = N
this✝¹ : DecidableEq M
this✝ : DecidableEq R
this : ∀ x ∈ s, x ∈ N
c : R
f : ↑↑s → R
⊢ ∑ x ∈ s.attach, c • f x • ↑x = ↑((RingHom.id R) c • ⟨∑ i ∈ s.attach, f i • ↑i, ⋯⟩) | exact Finset.smul_sum.symm | no goals | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/RingTheory/Noetherian.lean |
isNoetherian_of_fg_of_noetherian | case refine_2
R : Type u_1
M : Type u_2
inst✝² : Ring R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
N : Submodule R M
I : IsNoetherianRing R
hN : N.FG
s : Finset M
hs : span R ↑s = N
this✝¹ : DecidableEq M
this✝ : DecidableEq R
this : ∀ x ∈ s, x ∈ N
⊢ LinearMap.range { toFun := fun f => ⟨∑ i ∈ s.attach, f i • ↑i, ⋯⟩, map_add' := ⋯, map_smul' := ⋯ } = ⊤ | rw [<a>LinearMap.range_eq_top</a>] | case refine_2
R : Type u_1
M : Type u_2
inst✝² : Ring R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
N : Submodule R M
I : IsNoetherianRing R
hN : N.FG
s : Finset M
hs : span R ↑s = N
this✝¹ : DecidableEq M
this✝ : DecidableEq R
this : ∀ x ∈ s, x ∈ N
⊢ Surjective ⇑{ toFun := fun f => ⟨∑ i ∈ s.attach, f i • ↑i, ⋯⟩, map_add' := ⋯, map_smul' := ⋯ } | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/RingTheory/Noetherian.lean |
isNoetherian_of_fg_of_noetherian | case refine_2
R : Type u_1
M : Type u_2
inst✝² : Ring R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
N : Submodule R M
I : IsNoetherianRing R
hN : N.FG
s : Finset M
hs : span R ↑s = N
this✝¹ : DecidableEq M
this✝ : DecidableEq R
this : ∀ x ∈ s, x ∈ N
⊢ Surjective ⇑{ toFun := fun f => ⟨∑ i ∈ s.attach, f i • ↑i, ⋯⟩, map_add' := ⋯, map_smul' := ⋯ } | rintro ⟨n, hn⟩ | case refine_2.mk
R : Type u_1
M : Type u_2
inst✝² : Ring R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
N : Submodule R M
I : IsNoetherianRing R
hN : N.FG
s : Finset M
hs : span R ↑s = N
this✝¹ : DecidableEq M
this✝ : DecidableEq R
this : ∀ x ∈ s, x ∈ N
n : M
hn : n ∈ N
⊢ ∃ a, { toFun := fun f => ⟨∑ i ∈ s.attach, f i • ↑i, ⋯⟩, map_add' := ⋯, map_smul' := ⋯ } a = ⟨n, hn⟩ | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/RingTheory/Noetherian.lean |
isNoetherian_of_fg_of_noetherian | case refine_2.mk
R : Type u_1
M : Type u_2
inst✝² : Ring R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
N : Submodule R M
I : IsNoetherianRing R
hN : N.FG
s : Finset M
hs : span R ↑s = N
this✝¹ : DecidableEq M
this✝ : DecidableEq R
this : ∀ x ∈ s, x ∈ N
n : M
hn : n ∈ N
⊢ ∃ a, { toFun := fun f => ⟨∑ i ∈ s.attach, f i • ↑i, ⋯⟩, map_add' := ⋯, map_smul' := ⋯ } a = ⟨n, hn⟩ | rw [← hs, ← <a>Set.image_id</a> (s : <a>Set</a> M), <a>Finsupp.mem_span_image_iff_total</a>] at hn | case refine_2.mk
R : Type u_1
M : Type u_2
inst✝² : Ring R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
N : Submodule R M
I : IsNoetherianRing R
hN : N.FG
s : Finset M
hs : span R ↑s = N
this✝¹ : DecidableEq M
this✝ : DecidableEq R
this : ∀ x ∈ s, x ∈ N
n : M
hn✝ : n ∈ N
hn : ∃ l ∈ Finsupp.supported R R ↑s, (Finsupp.total M M R id) l = n
⊢ ∃ a, { toFun := fun f => ⟨∑ i ∈ s.attach, f i • ↑i, ⋯⟩, map_add' := ⋯, map_smul' := ⋯ } a = ⟨n, hn✝⟩ | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/RingTheory/Noetherian.lean |
isNoetherian_of_fg_of_noetherian | case refine_2.mk
R : Type u_1
M : Type u_2
inst✝² : Ring R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
N : Submodule R M
I : IsNoetherianRing R
hN : N.FG
s : Finset M
hs : span R ↑s = N
this✝¹ : DecidableEq M
this✝ : DecidableEq R
this : ∀ x ∈ s, x ∈ N
n : M
hn✝ : n ∈ N
hn : ∃ l ∈ Finsupp.supported R R ↑s, (Finsupp.total M M R id) l = n
⊢ ∃ a, { toFun := fun f => ⟨∑ i ∈ s.attach, f i • ↑i, ⋯⟩, map_add' := ⋯, map_smul' := ⋯ } a = ⟨n, hn✝⟩ | rcases hn with ⟨l, hl1, hl2⟩ | case refine_2.mk.intro.intro
R : Type u_1
M : Type u_2
inst✝² : Ring R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
N : Submodule R M
I : IsNoetherianRing R
hN : N.FG
s : Finset M
hs : span R ↑s = N
this✝¹ : DecidableEq M
this✝ : DecidableEq R
this : ∀ x ∈ s, x ∈ N
n : M
hn : n ∈ N
l : M →₀ R
hl1 : l ∈ Finsupp.supported R R ↑s
hl2 : (Finsupp.total M M R id) l = n
⊢ ∃ a, { toFun := fun f => ⟨∑ i ∈ s.attach, f i • ↑i, ⋯⟩, map_add' := ⋯, map_smul' := ⋯ } a = ⟨n, hn⟩ | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/RingTheory/Noetherian.lean |
isNoetherian_of_fg_of_noetherian | case refine_2.mk.intro.intro
R : Type u_1
M : Type u_2
inst✝² : Ring R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
N : Submodule R M
I : IsNoetherianRing R
hN : N.FG
s : Finset M
hs : span R ↑s = N
this✝¹ : DecidableEq M
this✝ : DecidableEq R
this : ∀ x ∈ s, x ∈ N
n : M
hn : n ∈ N
l : M →₀ R
hl1 : l ∈ Finsupp.supported R R ↑s
hl2 : (Finsupp.total M M R id) l = n
⊢ ∃ a, { toFun := fun f => ⟨∑ i ∈ s.attach, f i • ↑i, ⋯⟩, map_add' := ⋯, map_smul' := ⋯ } a = ⟨n, hn⟩ | refine ⟨fun x => l x, <a>Subtype.ext</a> ?_⟩ | case refine_2.mk.intro.intro
R : Type u_1
M : Type u_2
inst✝² : Ring R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
N : Submodule R M
I : IsNoetherianRing R
hN : N.FG
s : Finset M
hs : span R ↑s = N
this✝¹ : DecidableEq M
this✝ : DecidableEq R
this : ∀ x ∈ s, x ∈ N
n : M
hn : n ∈ N
l : M →₀ R
hl1 : l ∈ Finsupp.supported R R ↑s
hl2 : (Finsupp.total M M R id) l = n
⊢ ↑({ toFun := fun f => ⟨∑ i ∈ s.attach, f i • ↑i, ⋯⟩, map_add' := ⋯, map_smul' := ⋯ } fun x => l ↑x) = ↑⟨n, hn⟩ | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/RingTheory/Noetherian.lean |
isNoetherian_of_fg_of_noetherian | case refine_2.mk.intro.intro
R : Type u_1
M : Type u_2
inst✝² : Ring R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
N : Submodule R M
I : IsNoetherianRing R
hN : N.FG
s : Finset M
hs : span R ↑s = N
this✝¹ : DecidableEq M
this✝ : DecidableEq R
this : ∀ x ∈ s, x ∈ N
n : M
hn : n ∈ N
l : M →₀ R
hl1 : l ∈ Finsupp.supported R R ↑s
hl2 : (Finsupp.total M M R id) l = n
⊢ ↑({ toFun := fun f => ⟨∑ i ∈ s.attach, f i • ↑i, ⋯⟩, map_add' := ⋯, map_smul' := ⋯ } fun x => l ↑x) = ↑⟨n, hn⟩ | change (∑ i ∈ s.attach, l i • (i : M)) = n | case refine_2.mk.intro.intro
R : Type u_1
M : Type u_2
inst✝² : Ring R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
N : Submodule R M
I : IsNoetherianRing R
hN : N.FG
s : Finset M
hs : span R ↑s = N
this✝¹ : DecidableEq M
this✝ : DecidableEq R
this : ∀ x ∈ s, x ∈ N
n : M
hn : n ∈ N
l : M →₀ R
hl1 : l ∈ Finsupp.supported R R ↑s
hl2 : (Finsupp.total M M R id) l = n
⊢ ∑ i ∈ s.attach, l ↑i • ↑i = n | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/RingTheory/Noetherian.lean |
isNoetherian_of_fg_of_noetherian | case refine_2.mk.intro.intro
R : Type u_1
M : Type u_2
inst✝² : Ring R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
N : Submodule R M
I : IsNoetherianRing R
hN : N.FG
s : Finset M
hs : span R ↑s = N
this✝¹ : DecidableEq M
this✝ : DecidableEq R
this : ∀ x ∈ s, x ∈ N
n : M
hn : n ∈ N
l : M →₀ R
hl1 : l ∈ Finsupp.supported R R ↑s
hl2 : (Finsupp.total M M R id) l = n
⊢ ∑ i ∈ s.attach, l ↑i • ↑i = n | rw [s.sum_attach fun i ↦ l i • i, ← hl2, <a>Finsupp.total_apply</a>, <a>Finsupp.sum</a>, <a>eq_comm</a>] | case refine_2.mk.intro.intro
R : Type u_1
M : Type u_2
inst✝² : Ring R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
N : Submodule R M
I : IsNoetherianRing R
hN : N.FG
s : Finset M
hs : span R ↑s = N
this✝¹ : DecidableEq M
this✝ : DecidableEq R
this : ∀ x ∈ s, x ∈ N
n : M
hn : n ∈ N
l : M →₀ R
hl1 : l ∈ Finsupp.supported R R ↑s
hl2 : (Finsupp.total M M R id) l = n
⊢ ∑ a ∈ l.support, l a • id a = ∑ x ∈ s, l x • x | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/RingTheory/Noetherian.lean |
isNoetherian_of_fg_of_noetherian | case refine_2.mk.intro.intro
R : Type u_1
M : Type u_2
inst✝² : Ring R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
N : Submodule R M
I : IsNoetherianRing R
hN : N.FG
s : Finset M
hs : span R ↑s = N
this✝¹ : DecidableEq M
this✝ : DecidableEq R
this : ∀ x ∈ s, x ∈ N
n : M
hn : n ∈ N
l : M →₀ R
hl1 : l ∈ Finsupp.supported R R ↑s
hl2 : (Finsupp.total M M R id) l = n
⊢ ∑ a ∈ l.support, l a • id a = ∑ x ∈ s, l x • x | refine <a>Finset.sum_subset</a> hl1 fun x _ hx => ?_ | case refine_2.mk.intro.intro
R : Type u_1
M : Type u_2
inst✝² : Ring R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
N : Submodule R M
I : IsNoetherianRing R
hN : N.FG
s : Finset M
hs : span R ↑s = N
this✝¹ : DecidableEq M
this✝ : DecidableEq R
this : ∀ x ∈ s, x ∈ N
n : M
hn : n ∈ N
l : M →₀ R
hl1 : l ∈ Finsupp.supported R R ↑s
hl2 : (Finsupp.total M M R id) l = n
x : M
x✝ : x ∈ s
hx : x ∉ l.support
⊢ l x • id x = 0 | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/RingTheory/Noetherian.lean |
isNoetherian_of_fg_of_noetherian | case refine_2.mk.intro.intro
R : Type u_1
M : Type u_2
inst✝² : Ring R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
N : Submodule R M
I : IsNoetherianRing R
hN : N.FG
s : Finset M
hs : span R ↑s = N
this✝¹ : DecidableEq M
this✝ : DecidableEq R
this : ∀ x ∈ s, x ∈ N
n : M
hn : n ∈ N
l : M →₀ R
hl1 : l ∈ Finsupp.supported R R ↑s
hl2 : (Finsupp.total M M R id) l = n
x : M
x✝ : x ∈ s
hx : x ∉ l.support
⊢ l x • id x = 0 | rw [<a>Finsupp.not_mem_support_iff</a>.1 hx, <a>zero_smul</a>] | no goals | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/RingTheory/Noetherian.lean |
Ordnode.dual_rotateL | α : Type u_1
l : Ordnode α
x : α
r : Ordnode α
⊢ (l.rotateL x r).dual = r.dual.rotateR x l.dual | cases r <;> simp [<a>Ordnode.rotateL</a>, <a>Ordnode.rotateR</a>, <a>Ordnode.dual_node'</a>] | case node
α : Type u_1
l : Ordnode α
x : α
size✝ : ℕ
l✝ : Ordnode α
x✝ : α
r✝ : Ordnode α
⊢ (if l✝.size < ratio * r✝.size then l.node3L x l✝ x✝ r✝ else l.node4L x l✝ x✝ r✝).dual =
if l✝.size < ratio * r✝.size then r✝.dual.node3R x✝ l✝.dual x l.dual else r✝.dual.node4R x✝ l✝.dual x l.dual | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/Data/Ordmap/Ordset.lean |
Ordnode.dual_rotateL | case node
α : Type u_1
l : Ordnode α
x : α
size✝ : ℕ
l✝ : Ordnode α
x✝ : α
r✝ : Ordnode α
⊢ (if l✝.size < ratio * r✝.size then l.node3L x l✝ x✝ r✝ else l.node4L x l✝ x✝ r✝).dual =
if l✝.size < ratio * r✝.size then r✝.dual.node3R x✝ l✝.dual x l.dual else r✝.dual.node4R x✝ l✝.dual x l.dual | split_ifs <;> simp [<a>Ordnode.dual_node3L</a>, <a>Ordnode.dual_node4L</a>, <a>Ordnode.node3R</a>, <a>add_comm</a>] | no goals | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/Data/Ordmap/Ordset.lean |
LieSubalgebra.ofLe_eq_comap_incl | R : Type u
L : Type v
inst✝⁴ : CommRing R
inst✝³ : LieRing L
inst✝² : LieAlgebra R L
L₂ : Type w
inst✝¹ : LieRing L₂
inst✝ : LieAlgebra R L₂
f : L →ₗ⁅R⁆ L₂
K K' : LieSubalgebra R L
K₂ : LieSubalgebra R L₂
h : K ≤ K'
⊢ ofLe h = comap K'.incl K | ext | case h
R : Type u
L : Type v
inst✝⁴ : CommRing R
inst✝³ : LieRing L
inst✝² : LieAlgebra R L
L₂ : Type w
inst✝¹ : LieRing L₂
inst✝ : LieAlgebra R L₂
f : L →ₗ⁅R⁆ L₂
K K' : LieSubalgebra R L
K₂ : LieSubalgebra R L₂
h : K ≤ K'
x✝ : ↥K'
⊢ x✝ ∈ ofLe h ↔ x✝ ∈ comap K'.incl K | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/Algebra/Lie/Subalgebra.lean |
LieSubalgebra.ofLe_eq_comap_incl | case h
R : Type u
L : Type v
inst✝⁴ : CommRing R
inst✝³ : LieRing L
inst✝² : LieAlgebra R L
L₂ : Type w
inst✝¹ : LieRing L₂
inst✝ : LieAlgebra R L₂
f : L →ₗ⁅R⁆ L₂
K K' : LieSubalgebra R L
K₂ : LieSubalgebra R L₂
h : K ≤ K'
x✝ : ↥K'
⊢ x✝ ∈ ofLe h ↔ x✝ ∈ comap K'.incl K | rw [<a>LieSubalgebra.mem_ofLe</a>] | case h
R : Type u
L : Type v
inst✝⁴ : CommRing R
inst✝³ : LieRing L
inst✝² : LieAlgebra R L
L₂ : Type w
inst✝¹ : LieRing L₂
inst✝ : LieAlgebra R L₂
f : L →ₗ⁅R⁆ L₂
K K' : LieSubalgebra R L
K₂ : LieSubalgebra R L₂
h : K ≤ K'
x✝ : ↥K'
⊢ ↑x✝ ∈ K ↔ x✝ ∈ comap K'.incl K | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/Algebra/Lie/Subalgebra.lean |
LieSubalgebra.ofLe_eq_comap_incl | case h
R : Type u
L : Type v
inst✝⁴ : CommRing R
inst✝³ : LieRing L
inst✝² : LieAlgebra R L
L₂ : Type w
inst✝¹ : LieRing L₂
inst✝ : LieAlgebra R L₂
f : L →ₗ⁅R⁆ L₂
K K' : LieSubalgebra R L
K₂ : LieSubalgebra R L₂
h : K ≤ K'
x✝ : ↥K'
⊢ ↑x✝ ∈ K ↔ x✝ ∈ comap K'.incl K | rfl | no goals | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/Algebra/Lie/Subalgebra.lean |
parallelepiped_single | ι : Type u_1
ι' : Type u_2
E : Type u_3
F : Type u_4
inst✝⁶ : Fintype ι
inst✝⁵ : Fintype ι'
inst✝⁴ : AddCommGroup E
inst✝³ : Module ℝ E
inst✝² : AddCommGroup F
inst✝¹ : Module ℝ F
inst✝ : DecidableEq ι
a : ι → ℝ
⊢ (parallelepiped fun i => Pi.single i (a i)) = uIcc 0 a | ext x | case h
ι : Type u_1
ι' : Type u_2
E : Type u_3
F : Type u_4
inst✝⁶ : Fintype ι
inst✝⁵ : Fintype ι'
inst✝⁴ : AddCommGroup E
inst✝³ : Module ℝ E
inst✝² : AddCommGroup F
inst✝¹ : Module ℝ F
inst✝ : DecidableEq ι
a x : ι → ℝ
⊢ (x ∈ parallelepiped fun i => Pi.single i (a i)) ↔ x ∈ uIcc 0 a | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/MeasureTheory/Measure/Haar/OfBasis.lean |
parallelepiped_single | case h
ι : Type u_1
ι' : Type u_2
E : Type u_3
F : Type u_4
inst✝⁶ : Fintype ι
inst✝⁵ : Fintype ι'
inst✝⁴ : AddCommGroup E
inst✝³ : Module ℝ E
inst✝² : AddCommGroup F
inst✝¹ : Module ℝ F
inst✝ : DecidableEq ι
a x : ι → ℝ
⊢ (x ∈ parallelepiped fun i => Pi.single i (a i)) ↔ x ∈ uIcc 0 a | simp_rw [<a>Set.uIcc</a>, <a>mem_parallelepiped_iff</a>, <a>Set.mem_Icc</a>, <a>Pi.le_def</a>, ← <a>forall_and</a>, <a>Pi.inf_apply</a>, <a>Pi.sup_apply</a>, ← <a>Pi.single_smul'</a>, <a>Pi.one_apply</a>, <a>Pi.zero_apply</a>, ← <a>Pi.smul_apply'</a>, <a>Finset.univ_sum_single</a> (_ : ι → ℝ)] | case h
ι : Type u_1
ι' : Type u_2
E : Type u_3
F : Type u_4
inst✝⁶ : Fintype ι
inst✝⁵ : Fintype ι'
inst✝⁴ : AddCommGroup E
inst✝³ : Module ℝ E
inst✝² : AddCommGroup F
inst✝¹ : Module ℝ F
inst✝ : DecidableEq ι
a x : ι → ℝ
⊢ (∃ t, (∀ (x : ι), 0 ≤ t x ∧ t x ≤ 1) ∧ x = t • a) ↔ ∀ (x_1 : ι), 0 ⊓ a x_1 ≤ x x_1 ∧ x x_1 ≤ 0 ⊔ a x_1 | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/MeasureTheory/Measure/Haar/OfBasis.lean |
parallelepiped_single | case h
ι : Type u_1
ι' : Type u_2
E : Type u_3
F : Type u_4
inst✝⁶ : Fintype ι
inst✝⁵ : Fintype ι'
inst✝⁴ : AddCommGroup E
inst✝³ : Module ℝ E
inst✝² : AddCommGroup F
inst✝¹ : Module ℝ F
inst✝ : DecidableEq ι
a x : ι → ℝ
⊢ (∃ t, (∀ (x : ι), 0 ≤ t x ∧ t x ≤ 1) ∧ x = t • a) ↔ ∀ (x_1 : ι), 0 ⊓ a x_1 ≤ x x_1 ∧ x x_1 ≤ 0 ⊔ a x_1 | constructor | case h.mp
ι : Type u_1
ι' : Type u_2
E : Type u_3
F : Type u_4
inst✝⁶ : Fintype ι
inst✝⁵ : Fintype ι'
inst✝⁴ : AddCommGroup E
inst✝³ : Module ℝ E
inst✝² : AddCommGroup F
inst✝¹ : Module ℝ F
inst✝ : DecidableEq ι
a x : ι → ℝ
⊢ (∃ t, (∀ (x : ι), 0 ≤ t x ∧ t x ≤ 1) ∧ x = t • a) → ∀ (x_1 : ι), 0 ⊓ a x_1 ≤ x x_1 ∧ x x_1 ≤ 0 ⊔ a x_1
case h.mpr
ι : Type u_1
ι' : Type u_2
E : Type u_3
F : Type u_4
inst✝⁶ : Fintype ι
inst✝⁵ : Fintype ι'
inst✝⁴ : AddCommGroup E
inst✝³ : Module ℝ E
inst✝² : AddCommGroup F
inst✝¹ : Module ℝ F
inst✝ : DecidableEq ι
a x : ι → ℝ
⊢ (∀ (x_1 : ι), 0 ⊓ a x_1 ≤ x x_1 ∧ x x_1 ≤ 0 ⊔ a x_1) → ∃ t, (∀ (x : ι), 0 ≤ t x ∧ t x ≤ 1) ∧ x = t • a | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/MeasureTheory/Measure/Haar/OfBasis.lean |
parallelepiped_single | case h.mp
ι : Type u_1
ι' : Type u_2
E : Type u_3
F : Type u_4
inst✝⁶ : Fintype ι
inst✝⁵ : Fintype ι'
inst✝⁴ : AddCommGroup E
inst✝³ : Module ℝ E
inst✝² : AddCommGroup F
inst✝¹ : Module ℝ F
inst✝ : DecidableEq ι
a x : ι → ℝ
⊢ (∃ t, (∀ (x : ι), 0 ≤ t x ∧ t x ≤ 1) ∧ x = t • a) → ∀ (x_1 : ι), 0 ⊓ a x_1 ≤ x x_1 ∧ x x_1 ≤ 0 ⊔ a x_1 | rintro ⟨t, ht, rfl⟩ i | case h.mp.intro.intro
ι : Type u_1
ι' : Type u_2
E : Type u_3
F : Type u_4
inst✝⁶ : Fintype ι
inst✝⁵ : Fintype ι'
inst✝⁴ : AddCommGroup E
inst✝³ : Module ℝ E
inst✝² : AddCommGroup F
inst✝¹ : Module ℝ F
inst✝ : DecidableEq ι
a t : ι → ℝ
ht : ∀ (x : ι), 0 ≤ t x ∧ t x ≤ 1
i : ι
⊢ 0 ⊓ a i ≤ (t • a) i ∧ (t • a) i ≤ 0 ⊔ a i | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/MeasureTheory/Measure/Haar/OfBasis.lean |
parallelepiped_single | case h.mp.intro.intro
ι : Type u_1
ι' : Type u_2
E : Type u_3
F : Type u_4
inst✝⁶ : Fintype ι
inst✝⁵ : Fintype ι'
inst✝⁴ : AddCommGroup E
inst✝³ : Module ℝ E
inst✝² : AddCommGroup F
inst✝¹ : Module ℝ F
inst✝ : DecidableEq ι
a t : ι → ℝ
ht : ∀ (x : ι), 0 ≤ t x ∧ t x ≤ 1
i : ι
⊢ 0 ⊓ a i ≤ (t • a) i ∧ (t • a) i ≤ 0 ⊔ a i | specialize ht i | case h.mp.intro.intro
ι : Type u_1
ι' : Type u_2
E : Type u_3
F : Type u_4
inst✝⁶ : Fintype ι
inst✝⁵ : Fintype ι'
inst✝⁴ : AddCommGroup E
inst✝³ : Module ℝ E
inst✝² : AddCommGroup F
inst✝¹ : Module ℝ F
inst✝ : DecidableEq ι
a t : ι → ℝ
i : ι
ht : 0 ≤ t i ∧ t i ≤ 1
⊢ 0 ⊓ a i ≤ (t • a) i ∧ (t • a) i ≤ 0 ⊔ a i | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/MeasureTheory/Measure/Haar/OfBasis.lean |
parallelepiped_single | case h.mp.intro.intro
ι : Type u_1
ι' : Type u_2
E : Type u_3
F : Type u_4
inst✝⁶ : Fintype ι
inst✝⁵ : Fintype ι'
inst✝⁴ : AddCommGroup E
inst✝³ : Module ℝ E
inst✝² : AddCommGroup F
inst✝¹ : Module ℝ F
inst✝ : DecidableEq ι
a t : ι → ℝ
i : ι
ht : 0 ≤ t i ∧ t i ≤ 1
⊢ 0 ⊓ a i ≤ (t • a) i ∧ (t • a) i ≤ 0 ⊔ a i | simp_rw [<a>smul_eq_mul</a>, <a>Pi.mul_apply</a>] | case h.mp.intro.intro
ι : Type u_1
ι' : Type u_2
E : Type u_3
F : Type u_4
inst✝⁶ : Fintype ι
inst✝⁵ : Fintype ι'
inst✝⁴ : AddCommGroup E
inst✝³ : Module ℝ E
inst✝² : AddCommGroup F
inst✝¹ : Module ℝ F
inst✝ : DecidableEq ι
a t : ι → ℝ
i : ι
ht : 0 ≤ t i ∧ t i ≤ 1
⊢ 0 ⊓ a i ≤ t i * a i ∧ t i * a i ≤ 0 ⊔ a i | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/MeasureTheory/Measure/Haar/OfBasis.lean |
parallelepiped_single | case h.mp.intro.intro
ι : Type u_1
ι' : Type u_2
E : Type u_3
F : Type u_4
inst✝⁶ : Fintype ι
inst✝⁵ : Fintype ι'
inst✝⁴ : AddCommGroup E
inst✝³ : Module ℝ E
inst✝² : AddCommGroup F
inst✝¹ : Module ℝ F
inst✝ : DecidableEq ι
a t : ι → ℝ
i : ι
ht : 0 ≤ t i ∧ t i ≤ 1
⊢ 0 ⊓ a i ≤ t i * a i ∧ t i * a i ≤ 0 ⊔ a i | rcases <a>le_total</a> (a i) 0 with hai | hai | case h.mp.intro.intro.inl
ι : Type u_1
ι' : Type u_2
E : Type u_3
F : Type u_4
inst✝⁶ : Fintype ι
inst✝⁵ : Fintype ι'
inst✝⁴ : AddCommGroup E
inst✝³ : Module ℝ E
inst✝² : AddCommGroup F
inst✝¹ : Module ℝ F
inst✝ : DecidableEq ι
a t : ι → ℝ
i : ι
ht : 0 ≤ t i ∧ t i ≤ 1
hai : a i ≤ 0
⊢ 0 ⊓ a i ≤ t i * a i ∧ t i * a i ≤ 0 ⊔ a i
case h.mp.intro.intro.inr
ι : Type u_1
ι' : Type u_2
E : Type u_3
F : Type u_4
inst✝⁶ : Fintype ι
inst✝⁵ : Fintype ι'
inst✝⁴ : AddCommGroup E
inst✝³ : Module ℝ E
inst✝² : AddCommGroup F
inst✝¹ : Module ℝ F
inst✝ : DecidableEq ι
a t : ι → ℝ
i : ι
ht : 0 ≤ t i ∧ t i ≤ 1
hai : 0 ≤ a i
⊢ 0 ⊓ a i ≤ t i * a i ∧ t i * a i ≤ 0 ⊔ a i | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/MeasureTheory/Measure/Haar/OfBasis.lean |
parallelepiped_single | case h.mp.intro.intro.inl
ι : Type u_1
ι' : Type u_2
E : Type u_3
F : Type u_4
inst✝⁶ : Fintype ι
inst✝⁵ : Fintype ι'
inst✝⁴ : AddCommGroup E
inst✝³ : Module ℝ E
inst✝² : AddCommGroup F
inst✝¹ : Module ℝ F
inst✝ : DecidableEq ι
a t : ι → ℝ
i : ι
ht : 0 ≤ t i ∧ t i ≤ 1
hai : a i ≤ 0
⊢ 0 ⊓ a i ≤ t i * a i ∧ t i * a i ≤ 0 ⊔ a i | rw [sup_eq_left.mpr hai, inf_eq_right.mpr hai] | case h.mp.intro.intro.inl
ι : Type u_1
ι' : Type u_2
E : Type u_3
F : Type u_4
inst✝⁶ : Fintype ι
inst✝⁵ : Fintype ι'
inst✝⁴ : AddCommGroup E
inst✝³ : Module ℝ E
inst✝² : AddCommGroup F
inst✝¹ : Module ℝ F
inst✝ : DecidableEq ι
a t : ι → ℝ
i : ι
ht : 0 ≤ t i ∧ t i ≤ 1
hai : a i ≤ 0
⊢ a i ≤ t i * a i ∧ t i * a i ≤ 0 | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/MeasureTheory/Measure/Haar/OfBasis.lean |
parallelepiped_single | case h.mp.intro.intro.inl
ι : Type u_1
ι' : Type u_2
E : Type u_3
F : Type u_4
inst✝⁶ : Fintype ι
inst✝⁵ : Fintype ι'
inst✝⁴ : AddCommGroup E
inst✝³ : Module ℝ E
inst✝² : AddCommGroup F
inst✝¹ : Module ℝ F
inst✝ : DecidableEq ι
a t : ι → ℝ
i : ι
ht : 0 ≤ t i ∧ t i ≤ 1
hai : a i ≤ 0
⊢ a i ≤ t i * a i ∧ t i * a i ≤ 0 | exact ⟨<a>le_mul_of_le_one_left</a> hai ht.2, <a>mul_nonpos_of_nonneg_of_nonpos</a> ht.1 hai⟩ | no goals | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/MeasureTheory/Measure/Haar/OfBasis.lean |
parallelepiped_single | case h.mp.intro.intro.inr
ι : Type u_1
ι' : Type u_2
E : Type u_3
F : Type u_4
inst✝⁶ : Fintype ι
inst✝⁵ : Fintype ι'
inst✝⁴ : AddCommGroup E
inst✝³ : Module ℝ E
inst✝² : AddCommGroup F
inst✝¹ : Module ℝ F
inst✝ : DecidableEq ι
a t : ι → ℝ
i : ι
ht : 0 ≤ t i ∧ t i ≤ 1
hai : 0 ≤ a i
⊢ 0 ⊓ a i ≤ t i * a i ∧ t i * a i ≤ 0 ⊔ a i | rw [sup_eq_right.mpr hai, inf_eq_left.mpr hai] | case h.mp.intro.intro.inr
ι : Type u_1
ι' : Type u_2
E : Type u_3
F : Type u_4
inst✝⁶ : Fintype ι
inst✝⁵ : Fintype ι'
inst✝⁴ : AddCommGroup E
inst✝³ : Module ℝ E
inst✝² : AddCommGroup F
inst✝¹ : Module ℝ F
inst✝ : DecidableEq ι
a t : ι → ℝ
i : ι
ht : 0 ≤ t i ∧ t i ≤ 1
hai : 0 ≤ a i
⊢ 0 ≤ t i * a i ∧ t i * a i ≤ a i | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/MeasureTheory/Measure/Haar/OfBasis.lean |
parallelepiped_single | case h.mp.intro.intro.inr
ι : Type u_1
ι' : Type u_2
E : Type u_3
F : Type u_4
inst✝⁶ : Fintype ι
inst✝⁵ : Fintype ι'
inst✝⁴ : AddCommGroup E
inst✝³ : Module ℝ E
inst✝² : AddCommGroup F
inst✝¹ : Module ℝ F
inst✝ : DecidableEq ι
a t : ι → ℝ
i : ι
ht : 0 ≤ t i ∧ t i ≤ 1
hai : 0 ≤ a i
⊢ 0 ≤ t i * a i ∧ t i * a i ≤ a i | exact ⟨<a>mul_nonneg</a> ht.1 hai, <a>mul_le_of_le_one_left</a> hai ht.2⟩ | no goals | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/MeasureTheory/Measure/Haar/OfBasis.lean |
parallelepiped_single | case h.mpr
ι : Type u_1
ι' : Type u_2
E : Type u_3
F : Type u_4
inst✝⁶ : Fintype ι
inst✝⁵ : Fintype ι'
inst✝⁴ : AddCommGroup E
inst✝³ : Module ℝ E
inst✝² : AddCommGroup F
inst✝¹ : Module ℝ F
inst✝ : DecidableEq ι
a x : ι → ℝ
⊢ (∀ (x_1 : ι), 0 ⊓ a x_1 ≤ x x_1 ∧ x x_1 ≤ 0 ⊔ a x_1) → ∃ t, (∀ (x : ι), 0 ≤ t x ∧ t x ≤ 1) ∧ x = t • a | intro h | case h.mpr
ι : Type u_1
ι' : Type u_2
E : Type u_3
F : Type u_4
inst✝⁶ : Fintype ι
inst✝⁵ : Fintype ι'
inst✝⁴ : AddCommGroup E
inst✝³ : Module ℝ E
inst✝² : AddCommGroup F
inst✝¹ : Module ℝ F
inst✝ : DecidableEq ι
a x : ι → ℝ
h : ∀ (x_1 : ι), 0 ⊓ a x_1 ≤ x x_1 ∧ x x_1 ≤ 0 ⊔ a x_1
⊢ ∃ t, (∀ (x : ι), 0 ≤ t x ∧ t x ≤ 1) ∧ x = t • a | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/MeasureTheory/Measure/Haar/OfBasis.lean |
parallelepiped_single | case h.mpr
ι : Type u_1
ι' : Type u_2
E : Type u_3
F : Type u_4
inst✝⁶ : Fintype ι
inst✝⁵ : Fintype ι'
inst✝⁴ : AddCommGroup E
inst✝³ : Module ℝ E
inst✝² : AddCommGroup F
inst✝¹ : Module ℝ F
inst✝ : DecidableEq ι
a x : ι → ℝ
h : ∀ (x_1 : ι), 0 ⊓ a x_1 ≤ x x_1 ∧ x x_1 ≤ 0 ⊔ a x_1
⊢ ∃ t, (∀ (x : ι), 0 ≤ t x ∧ t x ≤ 1) ∧ x = t • a | refine ⟨fun i => x i / a i, fun i => ?_, <a>funext</a> fun i => ?_⟩ | case h.mpr.refine_1
ι : Type u_1
ι' : Type u_2
E : Type u_3
F : Type u_4
inst✝⁶ : Fintype ι
inst✝⁵ : Fintype ι'
inst✝⁴ : AddCommGroup E
inst✝³ : Module ℝ E
inst✝² : AddCommGroup F
inst✝¹ : Module ℝ F
inst✝ : DecidableEq ι
a x : ι → ℝ
h : ∀ (x_1 : ι), 0 ⊓ a x_1 ≤ x x_1 ∧ x x_1 ≤ 0 ⊔ a x_1
i : ι
⊢ 0 ≤ (fun i => x i / a i) i ∧ (fun i => x i / a i) i ≤ 1
case h.mpr.refine_2
ι : Type u_1
ι' : Type u_2
E : Type u_3
F : Type u_4
inst✝⁶ : Fintype ι
inst✝⁵ : Fintype ι'
inst✝⁴ : AddCommGroup E
inst✝³ : Module ℝ E
inst✝² : AddCommGroup F
inst✝¹ : Module ℝ F
inst✝ : DecidableEq ι
a x : ι → ℝ
h : ∀ (x_1 : ι), 0 ⊓ a x_1 ≤ x x_1 ∧ x x_1 ≤ 0 ⊔ a x_1
i : ι
⊢ x i = ((fun i => x i / a i) • a) i | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/MeasureTheory/Measure/Haar/OfBasis.lean |
parallelepiped_single | case h.mpr.refine_1
ι : Type u_1
ι' : Type u_2
E : Type u_3
F : Type u_4
inst✝⁶ : Fintype ι
inst✝⁵ : Fintype ι'
inst✝⁴ : AddCommGroup E
inst✝³ : Module ℝ E
inst✝² : AddCommGroup F
inst✝¹ : Module ℝ F
inst✝ : DecidableEq ι
a x : ι → ℝ
h : ∀ (x_1 : ι), 0 ⊓ a x_1 ≤ x x_1 ∧ x x_1 ≤ 0 ⊔ a x_1
i : ι
⊢ 0 ≤ (fun i => x i / a i) i ∧ (fun i => x i / a i) i ≤ 1 | specialize h i | case h.mpr.refine_1
ι : Type u_1
ι' : Type u_2
E : Type u_3
F : Type u_4
inst✝⁶ : Fintype ι
inst✝⁵ : Fintype ι'
inst✝⁴ : AddCommGroup E
inst✝³ : Module ℝ E
inst✝² : AddCommGroup F
inst✝¹ : Module ℝ F
inst✝ : DecidableEq ι
a x : ι → ℝ
i : ι
h : 0 ⊓ a i ≤ x i ∧ x i ≤ 0 ⊔ a i
⊢ 0 ≤ (fun i => x i / a i) i ∧ (fun i => x i / a i) i ≤ 1 | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/MeasureTheory/Measure/Haar/OfBasis.lean |
parallelepiped_single | case h.mpr.refine_1
ι : Type u_1
ι' : Type u_2
E : Type u_3
F : Type u_4
inst✝⁶ : Fintype ι
inst✝⁵ : Fintype ι'
inst✝⁴ : AddCommGroup E
inst✝³ : Module ℝ E
inst✝² : AddCommGroup F
inst✝¹ : Module ℝ F
inst✝ : DecidableEq ι
a x : ι → ℝ
i : ι
h : 0 ⊓ a i ≤ x i ∧ x i ≤ 0 ⊔ a i
⊢ 0 ≤ (fun i => x i / a i) i ∧ (fun i => x i / a i) i ≤ 1 | rcases <a>le_total</a> (a i) 0 with hai | hai | case h.mpr.refine_1.inl
ι : Type u_1
ι' : Type u_2
E : Type u_3
F : Type u_4
inst✝⁶ : Fintype ι
inst✝⁵ : Fintype ι'
inst✝⁴ : AddCommGroup E
inst✝³ : Module ℝ E
inst✝² : AddCommGroup F
inst✝¹ : Module ℝ F
inst✝ : DecidableEq ι
a x : ι → ℝ
i : ι
h : 0 ⊓ a i ≤ x i ∧ x i ≤ 0 ⊔ a i
hai : a i ≤ 0
⊢ 0 ≤ (fun i => x i / a i) i ∧ (fun i => x i / a i) i ≤ 1
case h.mpr.refine_1.inr
ι : Type u_1
ι' : Type u_2
E : Type u_3
F : Type u_4
inst✝⁶ : Fintype ι
inst✝⁵ : Fintype ι'
inst✝⁴ : AddCommGroup E
inst✝³ : Module ℝ E
inst✝² : AddCommGroup F
inst✝¹ : Module ℝ F
inst✝ : DecidableEq ι
a x : ι → ℝ
i : ι
h : 0 ⊓ a i ≤ x i ∧ x i ≤ 0 ⊔ a i
hai : 0 ≤ a i
⊢ 0 ≤ (fun i => x i / a i) i ∧ (fun i => x i / a i) i ≤ 1 | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/MeasureTheory/Measure/Haar/OfBasis.lean |
parallelepiped_single | case h.mpr.refine_1.inl
ι : Type u_1
ι' : Type u_2
E : Type u_3
F : Type u_4
inst✝⁶ : Fintype ι
inst✝⁵ : Fintype ι'
inst✝⁴ : AddCommGroup E
inst✝³ : Module ℝ E
inst✝² : AddCommGroup F
inst✝¹ : Module ℝ F
inst✝ : DecidableEq ι
a x : ι → ℝ
i : ι
h : 0 ⊓ a i ≤ x i ∧ x i ≤ 0 ⊔ a i
hai : a i ≤ 0
⊢ 0 ≤ (fun i => x i / a i) i ∧ (fun i => x i / a i) i ≤ 1 | rw [sup_eq_left.mpr hai, inf_eq_right.mpr hai] at h | case h.mpr.refine_1.inl
ι : Type u_1
ι' : Type u_2
E : Type u_3
F : Type u_4
inst✝⁶ : Fintype ι
inst✝⁵ : Fintype ι'
inst✝⁴ : AddCommGroup E
inst✝³ : Module ℝ E
inst✝² : AddCommGroup F
inst✝¹ : Module ℝ F
inst✝ : DecidableEq ι
a x : ι → ℝ
i : ι
h : a i ≤ x i ∧ x i ≤ 0
hai : a i ≤ 0
⊢ 0 ≤ (fun i => x i / a i) i ∧ (fun i => x i / a i) i ≤ 1 | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/MeasureTheory/Measure/Haar/OfBasis.lean |
parallelepiped_single | case h.mpr.refine_1.inl
ι : Type u_1
ι' : Type u_2
E : Type u_3
F : Type u_4
inst✝⁶ : Fintype ι
inst✝⁵ : Fintype ι'
inst✝⁴ : AddCommGroup E
inst✝³ : Module ℝ E
inst✝² : AddCommGroup F
inst✝¹ : Module ℝ F
inst✝ : DecidableEq ι
a x : ι → ℝ
i : ι
h : a i ≤ x i ∧ x i ≤ 0
hai : a i ≤ 0
⊢ 0 ≤ (fun i => x i / a i) i ∧ (fun i => x i / a i) i ≤ 1 | exact ⟨<a>div_nonneg_of_nonpos</a> h.2 hai, <a>div_le_one_of_ge</a> h.1 hai⟩ | no goals | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/MeasureTheory/Measure/Haar/OfBasis.lean |
parallelepiped_single | case h.mpr.refine_1.inr
ι : Type u_1
ι' : Type u_2
E : Type u_3
F : Type u_4
inst✝⁶ : Fintype ι
inst✝⁵ : Fintype ι'
inst✝⁴ : AddCommGroup E
inst✝³ : Module ℝ E
inst✝² : AddCommGroup F
inst✝¹ : Module ℝ F
inst✝ : DecidableEq ι
a x : ι → ℝ
i : ι
h : 0 ⊓ a i ≤ x i ∧ x i ≤ 0 ⊔ a i
hai : 0 ≤ a i
⊢ 0 ≤ (fun i => x i / a i) i ∧ (fun i => x i / a i) i ≤ 1 | rw [sup_eq_right.mpr hai, inf_eq_left.mpr hai] at h | case h.mpr.refine_1.inr
ι : Type u_1
ι' : Type u_2
E : Type u_3
F : Type u_4
inst✝⁶ : Fintype ι
inst✝⁵ : Fintype ι'
inst✝⁴ : AddCommGroup E
inst✝³ : Module ℝ E
inst✝² : AddCommGroup F
inst✝¹ : Module ℝ F
inst✝ : DecidableEq ι
a x : ι → ℝ
i : ι
h : 0 ≤ x i ∧ x i ≤ a i
hai : 0 ≤ a i
⊢ 0 ≤ (fun i => x i / a i) i ∧ (fun i => x i / a i) i ≤ 1 | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/MeasureTheory/Measure/Haar/OfBasis.lean |
parallelepiped_single | case h.mpr.refine_1.inr
ι : Type u_1
ι' : Type u_2
E : Type u_3
F : Type u_4
inst✝⁶ : Fintype ι
inst✝⁵ : Fintype ι'
inst✝⁴ : AddCommGroup E
inst✝³ : Module ℝ E
inst✝² : AddCommGroup F
inst✝¹ : Module ℝ F
inst✝ : DecidableEq ι
a x : ι → ℝ
i : ι
h : 0 ≤ x i ∧ x i ≤ a i
hai : 0 ≤ a i
⊢ 0 ≤ (fun i => x i / a i) i ∧ (fun i => x i / a i) i ≤ 1 | exact ⟨<a>div_nonneg</a> h.1 hai, <a>div_le_one_of_le</a> h.2 hai⟩ | no goals | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/MeasureTheory/Measure/Haar/OfBasis.lean |
parallelepiped_single | case h.mpr.refine_2
ι : Type u_1
ι' : Type u_2
E : Type u_3
F : Type u_4
inst✝⁶ : Fintype ι
inst✝⁵ : Fintype ι'
inst✝⁴ : AddCommGroup E
inst✝³ : Module ℝ E
inst✝² : AddCommGroup F
inst✝¹ : Module ℝ F
inst✝ : DecidableEq ι
a x : ι → ℝ
h : ∀ (x_1 : ι), 0 ⊓ a x_1 ≤ x x_1 ∧ x x_1 ≤ 0 ⊔ a x_1
i : ι
⊢ x i = ((fun i => x i / a i) • a) i | specialize h i | case h.mpr.refine_2
ι : Type u_1
ι' : Type u_2
E : Type u_3
F : Type u_4
inst✝⁶ : Fintype ι
inst✝⁵ : Fintype ι'
inst✝⁴ : AddCommGroup E
inst✝³ : Module ℝ E
inst✝² : AddCommGroup F
inst✝¹ : Module ℝ F
inst✝ : DecidableEq ι
a x : ι → ℝ
i : ι
h : 0 ⊓ a i ≤ x i ∧ x i ≤ 0 ⊔ a i
⊢ x i = ((fun i => x i / a i) • a) i | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/MeasureTheory/Measure/Haar/OfBasis.lean |
parallelepiped_single | case h.mpr.refine_2
ι : Type u_1
ι' : Type u_2
E : Type u_3
F : Type u_4
inst✝⁶ : Fintype ι
inst✝⁵ : Fintype ι'
inst✝⁴ : AddCommGroup E
inst✝³ : Module ℝ E
inst✝² : AddCommGroup F
inst✝¹ : Module ℝ F
inst✝ : DecidableEq ι
a x : ι → ℝ
i : ι
h : 0 ⊓ a i ≤ x i ∧ x i ≤ 0 ⊔ a i
⊢ x i = ((fun i => x i / a i) • a) i | simp only [<a>smul_eq_mul</a>, <a>Pi.mul_apply</a>] | case h.mpr.refine_2
ι : Type u_1
ι' : Type u_2
E : Type u_3
F : Type u_4
inst✝⁶ : Fintype ι
inst✝⁵ : Fintype ι'
inst✝⁴ : AddCommGroup E
inst✝³ : Module ℝ E
inst✝² : AddCommGroup F
inst✝¹ : Module ℝ F
inst✝ : DecidableEq ι
a x : ι → ℝ
i : ι
h : 0 ⊓ a i ≤ x i ∧ x i ≤ 0 ⊔ a i
⊢ x i = x i / a i * a i | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/MeasureTheory/Measure/Haar/OfBasis.lean |
parallelepiped_single | case h.mpr.refine_2
ι : Type u_1
ι' : Type u_2
E : Type u_3
F : Type u_4
inst✝⁶ : Fintype ι
inst✝⁵ : Fintype ι'
inst✝⁴ : AddCommGroup E
inst✝³ : Module ℝ E
inst✝² : AddCommGroup F
inst✝¹ : Module ℝ F
inst✝ : DecidableEq ι
a x : ι → ℝ
i : ι
h : 0 ⊓ a i ≤ x i ∧ x i ≤ 0 ⊔ a i
⊢ x i = x i / a i * a i | rcases <a>eq_or_ne</a> (a i) 0 with hai | hai | case h.mpr.refine_2.inl
ι : Type u_1
ι' : Type u_2
E : Type u_3
F : Type u_4
inst✝⁶ : Fintype ι
inst✝⁵ : Fintype ι'
inst✝⁴ : AddCommGroup E
inst✝³ : Module ℝ E
inst✝² : AddCommGroup F
inst✝¹ : Module ℝ F
inst✝ : DecidableEq ι
a x : ι → ℝ
i : ι
h : 0 ⊓ a i ≤ x i ∧ x i ≤ 0 ⊔ a i
hai : a i = 0
⊢ x i = x i / a i * a i
case h.mpr.refine_2.inr
ι : Type u_1
ι' : Type u_2
E : Type u_3
F : Type u_4
inst✝⁶ : Fintype ι
inst✝⁵ : Fintype ι'
inst✝⁴ : AddCommGroup E
inst✝³ : Module ℝ E
inst✝² : AddCommGroup F
inst✝¹ : Module ℝ F
inst✝ : DecidableEq ι
a x : ι → ℝ
i : ι
h : 0 ⊓ a i ≤ x i ∧ x i ≤ 0 ⊔ a i
hai : a i ≠ 0
⊢ x i = x i / a i * a i | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/MeasureTheory/Measure/Haar/OfBasis.lean |
parallelepiped_single | case h.mpr.refine_2.inl
ι : Type u_1
ι' : Type u_2
E : Type u_3
F : Type u_4
inst✝⁶ : Fintype ι
inst✝⁵ : Fintype ι'
inst✝⁴ : AddCommGroup E
inst✝³ : Module ℝ E
inst✝² : AddCommGroup F
inst✝¹ : Module ℝ F
inst✝ : DecidableEq ι
a x : ι → ℝ
i : ι
h : 0 ⊓ a i ≤ x i ∧ x i ≤ 0 ⊔ a i
hai : a i = 0
⊢ x i = x i / a i * a i | rw [hai, <a>inf_idem</a>, <a>sup_idem</a>, ← <a>le_antisymm_iff</a>] at h | case h.mpr.refine_2.inl
ι : Type u_1
ι' : Type u_2
E : Type u_3
F : Type u_4
inst✝⁶ : Fintype ι
inst✝⁵ : Fintype ι'
inst✝⁴ : AddCommGroup E
inst✝³ : Module ℝ E
inst✝² : AddCommGroup F
inst✝¹ : Module ℝ F
inst✝ : DecidableEq ι
a x : ι → ℝ
i : ι
h : 0 = x i
hai : a i = 0
⊢ x i = x i / a i * a i | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/MeasureTheory/Measure/Haar/OfBasis.lean |
parallelepiped_single | case h.mpr.refine_2.inl
ι : Type u_1
ι' : Type u_2
E : Type u_3
F : Type u_4
inst✝⁶ : Fintype ι
inst✝⁵ : Fintype ι'
inst✝⁴ : AddCommGroup E
inst✝³ : Module ℝ E
inst✝² : AddCommGroup F
inst✝¹ : Module ℝ F
inst✝ : DecidableEq ι
a x : ι → ℝ
i : ι
h : 0 = x i
hai : a i = 0
⊢ x i = x i / a i * a i | rw [hai, ← h, <a>zero_div</a>, <a>MulZeroClass.zero_mul</a>] | no goals | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/MeasureTheory/Measure/Haar/OfBasis.lean |
parallelepiped_single | case h.mpr.refine_2.inr
ι : Type u_1
ι' : Type u_2
E : Type u_3
F : Type u_4
inst✝⁶ : Fintype ι
inst✝⁵ : Fintype ι'
inst✝⁴ : AddCommGroup E
inst✝³ : Module ℝ E
inst✝² : AddCommGroup F
inst✝¹ : Module ℝ F
inst✝ : DecidableEq ι
a x : ι → ℝ
i : ι
h : 0 ⊓ a i ≤ x i ∧ x i ≤ 0 ⊔ a i
hai : a i ≠ 0
⊢ x i = x i / a i * a i | rw [<a>div_mul_cancel₀</a> _ hai] | no goals | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/MeasureTheory/Measure/Haar/OfBasis.lean |
Ordnode.dual_node3L | α : Type u_1
l : Ordnode α
x : α
m : Ordnode α
y : α
r : Ordnode α
⊢ (l.node3L x m y r).dual = r.dual.node3R y m.dual x l.dual | simp [<a>Ordnode.node3L</a>, <a>Ordnode.node3R</a>, <a>Ordnode.dual_node'</a>, <a>add_comm</a>] | no goals | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/Data/Ordmap/Ordset.lean |
LinearMap.trace_eq_contract_of_basis' | R : Type u_1
inst✝⁸ : CommRing R
M : Type u_2
inst✝⁷ : AddCommGroup M
inst✝⁶ : Module R M
N : Type u_3
P : Type u_4
inst✝⁵ : AddCommGroup N
inst✝⁴ : Module R N
inst✝³ : AddCommGroup P
inst✝² : Module R P
ι : Type u_5
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
b : Basis ι R M
⊢ trace R M = contractLeft R M ∘ₗ ↑(dualTensorHomEquivOfBasis b).symm | simp [<a>LinearEquiv.eq_comp_toLinearMap_symm</a>, <a>LinearMap.trace_eq_contract_of_basis</a> b] | no goals | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/LinearAlgebra/Trace.lean |
map_ratCast_smul | α : Type u_1
R✝ : Type u_2
M : Type u_3
M₂ : Type u_4
inst✝⁷ : AddCommGroup M
inst✝⁶ : AddCommGroup M₂
F : Type u_5
inst✝⁵ : FunLike F M M₂
inst✝⁴ : AddMonoidHomClass F M M₂
f : F
R : Type u_6
S : Type u_7
inst✝³ : DivisionRing R
inst✝² : DivisionRing S
inst✝¹ : Module R M
inst✝ : Module S M₂
c : ℚ
x : M
⊢ f (↑c • x) = ↑c • f x | rw [<a>Rat.cast_def</a>, <a>Rat.cast_def</a>, <a>div_eq_mul_inv</a>, <a>div_eq_mul_inv</a>, <a>MulAction.mul_smul</a>, <a>MulAction.mul_smul</a>, <a>map_intCast_smul</a> f R S, <a>map_inv_natCast_smul</a> f R S] | no goals | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/Algebra/Module/Basic.lean |
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The random split of LeanDojo Benchmark 4. Source data: https://zenodo.org/record/12740403/files/leandojo_benchmark_4.tar.gz
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