full_name
stringlengths 3
121
| state
stringlengths 7
9.32k
| tactic
stringlengths 3
5.35k
| target_state
stringlengths 7
19k
| url
stringclasses 1
value | commit
stringclasses 1
value | file_path
stringlengths 21
79
|
|---|---|---|---|---|---|---|
Set.Finite.exists_maximal_wrt
|
case neg.inl
α : Type u
β : Type v
ι : Sort w
γ : Type x
s✝¹ t : Set α
inst✝ : PartialOrder β
f : α → β
s✝ s : Set α
h✝¹ : s.Finite
ih✝ : s.Nonempty → ∃ a ∈ s, ∀ a' ∈ s, f a ≤ f a' → f a = f a'
h✝ : s.Nonempty
b : α
hb : b ∈ s
ih : ∀ a' ∈ s, f b ≤ f a' → f b = f a'
c : α
hbc : f b ≤ f c
his : c ∉ s
hs : (insert c s).Nonempty
h : ¬f b ≤ f c
hc : c ∈ insert c s
⊢ f b = f c
|
exact (h hbc).<a>False.elim</a>
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Data/Set/Finite.lean
|
Set.Finite.exists_maximal_wrt
|
case neg.inr
α : Type u
β : Type v
ι : Sort w
γ : Type x
s✝¹ t : Set α
inst✝ : PartialOrder β
f : α → β
s✝ : Set α
a : α
s : Set α
his : a ∉ s
h✝¹ : s.Finite
ih✝ : s.Nonempty → ∃ a ∈ s, ∀ a' ∈ s, f a ≤ f a' → f a = f a'
hs : (insert a s).Nonempty
h✝ : s.Nonempty
b : α
hb : b ∈ s
ih : ∀ a' ∈ s, f b ≤ f a' → f b = f a'
h : ¬f b ≤ f a
c : α
hc : c ∈ insert a s
hbc : f b ≤ f c
hcs : c ∈ s
⊢ f b = f c
|
exact ih c hcs hbc
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Data/Set/Finite.lean
|
MeasureTheory.SignedMeasure.totalVariation_neg
|
α : Type u_1
β : Type u_2
inst✝ : MeasurableSpace α
s : SignedMeasure α
⊢ (-s).totalVariation = s.totalVariation
|
simp [<a>MeasureTheory.SignedMeasure.totalVariation</a>, <a>MeasureTheory.SignedMeasure.toJordanDecomposition_neg</a>, <a>add_comm</a>]
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/MeasureTheory/Decomposition/Jordan.lean
|
LinearMap.BilinForm.orthogonal_span_singleton_eq_toLin_ker
|
R : Type u_1
M : Type u_2
inst✝⁸ : CommSemiring R
inst✝⁷ : AddCommMonoid M
inst✝⁶ : Module R M
R₁ : Type u_3
M₁ : Type u_4
inst✝⁵ : CommRing R₁
inst✝⁴ : AddCommGroup M₁
inst✝³ : Module R₁ M₁
V : Type u_5
K : Type u_6
inst✝² : Field K
inst✝¹ : AddCommGroup V
inst✝ : Module K V
B✝ : BilinForm R M
B₁ : BilinForm R₁ M₁
N L : Submodule R M
B : BilinForm K V
x : V
⊢ B.orthogonal (Submodule.span K {x}) = ker (B.toLinHomAux₁ x)
|
ext y
|
case h
R : Type u_1
M : Type u_2
inst✝⁸ : CommSemiring R
inst✝⁷ : AddCommMonoid M
inst✝⁶ : Module R M
R₁ : Type u_3
M₁ : Type u_4
inst✝⁵ : CommRing R₁
inst✝⁴ : AddCommGroup M₁
inst✝³ : Module R₁ M₁
V : Type u_5
K : Type u_6
inst✝² : Field K
inst✝¹ : AddCommGroup V
inst✝ : Module K V
B✝ : BilinForm R M
B₁ : BilinForm R₁ M₁
N L : Submodule R M
B : BilinForm K V
x y : V
⊢ y ∈ B.orthogonal (Submodule.span K {x}) ↔ y ∈ ker (B.toLinHomAux₁ x)
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/LinearAlgebra/BilinearForm/Orthogonal.lean
|
LinearMap.BilinForm.orthogonal_span_singleton_eq_toLin_ker
|
case h
R : Type u_1
M : Type u_2
inst✝⁸ : CommSemiring R
inst✝⁷ : AddCommMonoid M
inst✝⁶ : Module R M
R₁ : Type u_3
M₁ : Type u_4
inst✝⁵ : CommRing R₁
inst✝⁴ : AddCommGroup M₁
inst✝³ : Module R₁ M₁
V : Type u_5
K : Type u_6
inst✝² : Field K
inst✝¹ : AddCommGroup V
inst✝ : Module K V
B✝ : BilinForm R M
B₁ : BilinForm R₁ M₁
N L : Submodule R M
B : BilinForm K V
x y : V
⊢ y ∈ B.orthogonal (Submodule.span K {x}) ↔ y ∈ ker (B.toLinHomAux₁ x)
|
simp_rw [<a>LinearMap.BilinForm.mem_orthogonal_iff</a>, <a>LinearMap.mem_ker</a>, <a>Submodule.mem_span_singleton</a>]
|
case h
R : Type u_1
M : Type u_2
inst✝⁸ : CommSemiring R
inst✝⁷ : AddCommMonoid M
inst✝⁶ : Module R M
R₁ : Type u_3
M₁ : Type u_4
inst✝⁵ : CommRing R₁
inst✝⁴ : AddCommGroup M₁
inst✝³ : Module R₁ M₁
V : Type u_5
K : Type u_6
inst✝² : Field K
inst✝¹ : AddCommGroup V
inst✝ : Module K V
B✝ : BilinForm R M
B₁ : BilinForm R₁ M₁
N L : Submodule R M
B : BilinForm K V
x y : V
⊢ (∀ (n : V), (∃ a, a • x = n) → B.IsOrtho n y) ↔ (B.toLinHomAux₁ x) y = 0
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/LinearAlgebra/BilinearForm/Orthogonal.lean
|
LinearMap.BilinForm.orthogonal_span_singleton_eq_toLin_ker
|
case h
R : Type u_1
M : Type u_2
inst✝⁸ : CommSemiring R
inst✝⁷ : AddCommMonoid M
inst✝⁶ : Module R M
R₁ : Type u_3
M₁ : Type u_4
inst✝⁵ : CommRing R₁
inst✝⁴ : AddCommGroup M₁
inst✝³ : Module R₁ M₁
V : Type u_5
K : Type u_6
inst✝² : Field K
inst✝¹ : AddCommGroup V
inst✝ : Module K V
B✝ : BilinForm R M
B₁ : BilinForm R₁ M₁
N L : Submodule R M
B : BilinForm K V
x y : V
⊢ (∀ (n : V), (∃ a, a • x = n) → B.IsOrtho n y) ↔ (B.toLinHomAux₁ x) y = 0
|
constructor
|
case h.mp
R : Type u_1
M : Type u_2
inst✝⁸ : CommSemiring R
inst✝⁷ : AddCommMonoid M
inst✝⁶ : Module R M
R₁ : Type u_3
M₁ : Type u_4
inst✝⁵ : CommRing R₁
inst✝⁴ : AddCommGroup M₁
inst✝³ : Module R₁ M₁
V : Type u_5
K : Type u_6
inst✝² : Field K
inst✝¹ : AddCommGroup V
inst✝ : Module K V
B✝ : BilinForm R M
B₁ : BilinForm R₁ M₁
N L : Submodule R M
B : BilinForm K V
x y : V
⊢ (∀ (n : V), (∃ a, a • x = n) → B.IsOrtho n y) → (B.toLinHomAux₁ x) y = 0
case h.mpr
R : Type u_1
M : Type u_2
inst✝⁸ : CommSemiring R
inst✝⁷ : AddCommMonoid M
inst✝⁶ : Module R M
R₁ : Type u_3
M₁ : Type u_4
inst✝⁵ : CommRing R₁
inst✝⁴ : AddCommGroup M₁
inst✝³ : Module R₁ M₁
V : Type u_5
K : Type u_6
inst✝² : Field K
inst✝¹ : AddCommGroup V
inst✝ : Module K V
B✝ : BilinForm R M
B₁ : BilinForm R₁ M₁
N L : Submodule R M
B : BilinForm K V
x y : V
⊢ (B.toLinHomAux₁ x) y = 0 → ∀ (n : V), (∃ a, a • x = n) → B.IsOrtho n y
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/LinearAlgebra/BilinearForm/Orthogonal.lean
|
LinearMap.BilinForm.orthogonal_span_singleton_eq_toLin_ker
|
case h.mp
R : Type u_1
M : Type u_2
inst✝⁸ : CommSemiring R
inst✝⁷ : AddCommMonoid M
inst✝⁶ : Module R M
R₁ : Type u_3
M₁ : Type u_4
inst✝⁵ : CommRing R₁
inst✝⁴ : AddCommGroup M₁
inst✝³ : Module R₁ M₁
V : Type u_5
K : Type u_6
inst✝² : Field K
inst✝¹ : AddCommGroup V
inst✝ : Module K V
B✝ : BilinForm R M
B₁ : BilinForm R₁ M₁
N L : Submodule R M
B : BilinForm K V
x y : V
⊢ (∀ (n : V), (∃ a, a • x = n) → B.IsOrtho n y) → (B.toLinHomAux₁ x) y = 0
|
exact fun h => h x ⟨1, <a>one_smul</a> _ _⟩
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/LinearAlgebra/BilinearForm/Orthogonal.lean
|
LinearMap.BilinForm.orthogonal_span_singleton_eq_toLin_ker
|
case h.mpr
R : Type u_1
M : Type u_2
inst✝⁸ : CommSemiring R
inst✝⁷ : AddCommMonoid M
inst✝⁶ : Module R M
R₁ : Type u_3
M₁ : Type u_4
inst✝⁵ : CommRing R₁
inst✝⁴ : AddCommGroup M₁
inst✝³ : Module R₁ M₁
V : Type u_5
K : Type u_6
inst✝² : Field K
inst✝¹ : AddCommGroup V
inst✝ : Module K V
B✝ : BilinForm R M
B₁ : BilinForm R₁ M₁
N L : Submodule R M
B : BilinForm K V
x y : V
⊢ (B.toLinHomAux₁ x) y = 0 → ∀ (n : V), (∃ a, a • x = n) → B.IsOrtho n y
|
rintro h _ ⟨z, rfl⟩
|
case h.mpr.intro
R : Type u_1
M : Type u_2
inst✝⁸ : CommSemiring R
inst✝⁷ : AddCommMonoid M
inst✝⁶ : Module R M
R₁ : Type u_3
M₁ : Type u_4
inst✝⁵ : CommRing R₁
inst✝⁴ : AddCommGroup M₁
inst✝³ : Module R₁ M₁
V : Type u_5
K : Type u_6
inst✝² : Field K
inst✝¹ : AddCommGroup V
inst✝ : Module K V
B✝ : BilinForm R M
B₁ : BilinForm R₁ M₁
N L : Submodule R M
B : BilinForm K V
x y : V
h : (B.toLinHomAux₁ x) y = 0
z : K
⊢ B.IsOrtho (z • x) y
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/LinearAlgebra/BilinearForm/Orthogonal.lean
|
LinearMap.BilinForm.orthogonal_span_singleton_eq_toLin_ker
|
case h.mpr.intro
R : Type u_1
M : Type u_2
inst✝⁸ : CommSemiring R
inst✝⁷ : AddCommMonoid M
inst✝⁶ : Module R M
R₁ : Type u_3
M₁ : Type u_4
inst✝⁵ : CommRing R₁
inst✝⁴ : AddCommGroup M₁
inst✝³ : Module R₁ M₁
V : Type u_5
K : Type u_6
inst✝² : Field K
inst✝¹ : AddCommGroup V
inst✝ : Module K V
B✝ : BilinForm R M
B₁ : BilinForm R₁ M₁
N L : Submodule R M
B : BilinForm K V
x y : V
h : (B.toLinHomAux₁ x) y = 0
z : K
⊢ B.IsOrtho (z • x) y
|
rw [<a>LinearMap.BilinForm.IsOrtho</a>, <a>LinearMap.BilinForm.smul_left</a>, <a>mul_eq_zero</a>]
|
case h.mpr.intro
R : Type u_1
M : Type u_2
inst✝⁸ : CommSemiring R
inst✝⁷ : AddCommMonoid M
inst✝⁶ : Module R M
R₁ : Type u_3
M₁ : Type u_4
inst✝⁵ : CommRing R₁
inst✝⁴ : AddCommGroup M₁
inst✝³ : Module R₁ M₁
V : Type u_5
K : Type u_6
inst✝² : Field K
inst✝¹ : AddCommGroup V
inst✝ : Module K V
B✝ : BilinForm R M
B₁ : BilinForm R₁ M₁
N L : Submodule R M
B : BilinForm K V
x y : V
h : (B.toLinHomAux₁ x) y = 0
z : K
⊢ z = 0 ∨ (B x) y = 0
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/LinearAlgebra/BilinearForm/Orthogonal.lean
|
LinearMap.BilinForm.orthogonal_span_singleton_eq_toLin_ker
|
case h.mpr.intro
R : Type u_1
M : Type u_2
inst✝⁸ : CommSemiring R
inst✝⁷ : AddCommMonoid M
inst✝⁶ : Module R M
R₁ : Type u_3
M₁ : Type u_4
inst✝⁵ : CommRing R₁
inst✝⁴ : AddCommGroup M₁
inst✝³ : Module R₁ M₁
V : Type u_5
K : Type u_6
inst✝² : Field K
inst✝¹ : AddCommGroup V
inst✝ : Module K V
B✝ : BilinForm R M
B₁ : BilinForm R₁ M₁
N L : Submodule R M
B : BilinForm K V
x y : V
h : (B.toLinHomAux₁ x) y = 0
z : K
⊢ z = 0 ∨ (B x) y = 0
|
exact <a>Or.intro_right</a> _ h
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/LinearAlgebra/BilinearForm/Orthogonal.lean
|
intervalIntegral.integral_nonneg_of_ae_restrict
|
ι : Type u_1
𝕜 : Type u_2
E : Type u_3
F : Type u_4
A : Type u_5
inst✝² : NormedAddCommGroup E
inst✝¹ : CompleteSpace E
inst✝ : NormedSpace ℝ E
f g : ℝ → ℝ
a b : ℝ
μ : Measure ℝ
hab : a ≤ b
hf : 0 ≤ᶠ[ae (μ.restrict (Icc a b))] f
⊢ 0 ≤ ∫ (u : ℝ) in a..b, f u ∂μ
|
let H := <a>MeasureTheory.ae_restrict_of_ae_restrict_of_subset</a> <a>Set.Ioc_subset_Icc_self</a> hf
|
ι : Type u_1
𝕜 : Type u_2
E : Type u_3
F : Type u_4
A : Type u_5
inst✝² : NormedAddCommGroup E
inst✝¹ : CompleteSpace E
inst✝ : NormedSpace ℝ E
f g : ℝ → ℝ
a b : ℝ
μ : Measure ℝ
hab : a ≤ b
hf : 0 ≤ᶠ[ae (μ.restrict (Icc a b))] f
H : ∀ᵐ (x : ℝ) ∂μ.restrict (Ioc a b), 0 x ≤ f x := ae_restrict_of_ae_restrict_of_subset Ioc_subset_Icc_self hf
⊢ 0 ≤ ∫ (u : ℝ) in a..b, f u ∂μ
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/MeasureTheory/Integral/IntervalIntegral.lean
|
intervalIntegral.integral_nonneg_of_ae_restrict
|
ι : Type u_1
𝕜 : Type u_2
E : Type u_3
F : Type u_4
A : Type u_5
inst✝² : NormedAddCommGroup E
inst✝¹ : CompleteSpace E
inst✝ : NormedSpace ℝ E
f g : ℝ → ℝ
a b : ℝ
μ : Measure ℝ
hab : a ≤ b
hf : 0 ≤ᶠ[ae (μ.restrict (Icc a b))] f
H : ∀ᵐ (x : ℝ) ∂μ.restrict (Ioc a b), 0 x ≤ f x := ae_restrict_of_ae_restrict_of_subset Ioc_subset_Icc_self hf
⊢ 0 ≤ ∫ (u : ℝ) in a..b, f u ∂μ
|
simpa only [<a>intervalIntegral.integral_of_le</a> hab] using <a>MeasureTheory.setIntegral_nonneg_of_ae_restrict</a> H
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/MeasureTheory/Integral/IntervalIntegral.lean
|
Finset.mul_prod_Ioc_eq_prod_Icc
|
α : Type u_1
M : Type u_2
inst✝² : PartialOrder α
inst✝¹ : CommMonoid M
f : α → M
a b : α
inst✝ : LocallyFiniteOrder α
h : a ≤ b
⊢ f a * ∏ x ∈ Ioc a b, f x = ∏ x ∈ Icc a b, f x
|
rw [<a>Finset.Icc_eq_cons_Ioc</a> h, <a>Finset.prod_cons</a>]
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Algebra/BigOperators/Intervals.lean
|
isSplittingField_iff_intermediateField
|
F : Type u_1
inst✝² : Field F
E : Type u_2
inst✝¹ : Field E
inst✝ : Algebra F E
S : Set E
α : E
p : F[X]
⊢ IsSplittingField F E p ↔ Splits (algebraMap F E) p ∧ adjoin F (p.rootSet E) = ⊤
|
rw [← toSubalgebra_injective.eq_iff, <a>IntermediateField.adjoin_algebraic_toSubalgebra</a> fun _ ↦ <a>isAlgebraic_of_mem_rootSet</a>]
|
F : Type u_1
inst✝² : Field F
E : Type u_2
inst✝¹ : Field E
inst✝ : Algebra F E
S : Set E
α : E
p : F[X]
⊢ IsSplittingField F E p ↔ Splits (algebraMap F E) p ∧ Algebra.adjoin F (p.rootSet E) = ⊤.toSubalgebra
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/FieldTheory/Adjoin.lean
|
isSplittingField_iff_intermediateField
|
F : Type u_1
inst✝² : Field F
E : Type u_2
inst✝¹ : Field E
inst✝ : Algebra F E
S : Set E
α : E
p : F[X]
⊢ IsSplittingField F E p ↔ Splits (algebraMap F E) p ∧ Algebra.adjoin F (p.rootSet E) = ⊤.toSubalgebra
|
exact ⟨fun ⟨spl, adj⟩ ↦ ⟨spl, adj⟩, fun ⟨spl, adj⟩ ↦ ⟨spl, adj⟩⟩
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/FieldTheory/Adjoin.lean
|
ZMod.natAbs_min_of_le_div_two
|
n✝ a n : ℕ
x y : ℤ
he : ↑x = ↑y
hl : x.natAbs ≤ n / 2
⊢ x.natAbs ≤ y.natAbs
|
rw [<a>ZMod.intCast_eq_intCast_iff_dvd_sub</a>] at he
|
n✝ a n : ℕ
x y : ℤ
he : ↑n ∣ y - x
hl : x.natAbs ≤ n / 2
⊢ x.natAbs ≤ y.natAbs
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Data/ZMod/Basic.lean
|
ZMod.natAbs_min_of_le_div_two
|
n✝ a n : ℕ
x y : ℤ
he : ↑n ∣ y - x
hl : x.natAbs ≤ n / 2
⊢ x.natAbs ≤ y.natAbs
|
obtain ⟨m, he⟩ := he
|
case intro
n✝ a n : ℕ
x y : ℤ
hl : x.natAbs ≤ n / 2
m : ℤ
he : y - x = ↑n * m
⊢ x.natAbs ≤ y.natAbs
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Data/ZMod/Basic.lean
|
ZMod.natAbs_min_of_le_div_two
|
case intro
n✝ a n : ℕ
x y : ℤ
hl : x.natAbs ≤ n / 2
m : ℤ
he : y - x = ↑n * m
⊢ x.natAbs ≤ y.natAbs
|
rw [<a>sub_eq_iff_eq_add</a>] at he
|
case intro
n✝ a n : ℕ
x y : ℤ
hl : x.natAbs ≤ n / 2
m : ℤ
he : y = ↑n * m + x
⊢ x.natAbs ≤ y.natAbs
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Data/ZMod/Basic.lean
|
ZMod.natAbs_min_of_le_div_two
|
case intro
n✝ a n : ℕ
x y : ℤ
hl : x.natAbs ≤ n / 2
m : ℤ
he : y = ↑n * m + x
⊢ x.natAbs ≤ y.natAbs
|
subst he
|
case intro
n✝ a n : ℕ
x : ℤ
hl : x.natAbs ≤ n / 2
m : ℤ
⊢ x.natAbs ≤ (↑n * m + x).natAbs
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Data/ZMod/Basic.lean
|
ZMod.natAbs_min_of_le_div_two
|
case intro
n✝ a n : ℕ
x : ℤ
hl : x.natAbs ≤ n / 2
m : ℤ
⊢ x.natAbs ≤ (↑n * m + x).natAbs
|
obtain rfl | hm := <a>eq_or_ne</a> m 0
|
case intro.inl
n✝ a n : ℕ
x : ℤ
hl : x.natAbs ≤ n / 2
⊢ x.natAbs ≤ (↑n * 0 + x).natAbs
case intro.inr
n✝ a n : ℕ
x : ℤ
hl : x.natAbs ≤ n / 2
m : ℤ
hm : m ≠ 0
⊢ x.natAbs ≤ (↑n * m + x).natAbs
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Data/ZMod/Basic.lean
|
ZMod.natAbs_min_of_le_div_two
|
case intro.inr
n✝ a n : ℕ
x : ℤ
hl : x.natAbs ≤ n / 2
m : ℤ
hm : m ≠ 0
⊢ x.natAbs ≤ (↑n * m + x).natAbs
|
apply hl.trans
|
case intro.inr
n✝ a n : ℕ
x : ℤ
hl : x.natAbs ≤ n / 2
m : ℤ
hm : m ≠ 0
⊢ n / 2 ≤ (↑n * m + x).natAbs
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Data/ZMod/Basic.lean
|
ZMod.natAbs_min_of_le_div_two
|
case intro.inr
n✝ a n : ℕ
x : ℤ
hl : x.natAbs ≤ n / 2
m : ℤ
hm : m ≠ 0
⊢ n / 2 ≤ (↑n * m + x).natAbs
|
rw [← <a>add_le_add_iff_right</a> x.natAbs]
|
case intro.inr
n✝ a n : ℕ
x : ℤ
hl : x.natAbs ≤ n / 2
m : ℤ
hm : m ≠ 0
⊢ n / 2 + x.natAbs ≤ (↑n * m + x).natAbs + x.natAbs
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Data/ZMod/Basic.lean
|
ZMod.natAbs_min_of_le_div_two
|
case intro.inr
n✝ a n : ℕ
x : ℤ
hl : x.natAbs ≤ n / 2
m : ℤ
hm : m ≠ 0
⊢ n / 2 + x.natAbs ≤ (↑n * m + x).natAbs + x.natAbs
|
refine <a>le_trans</a> (<a>le_trans</a> ((<a>add_le_add_iff_left</a> _).2 hl) ?_) (<a>Int.natAbs_sub_le</a> _ _)
|
case intro.inr
n✝ a n : ℕ
x : ℤ
hl : x.natAbs ≤ n / 2
m : ℤ
hm : m ≠ 0
⊢ n / 2 + n / 2 ≤ (↑n * m + x - x).natAbs
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Data/ZMod/Basic.lean
|
ZMod.natAbs_min_of_le_div_two
|
case intro.inr
n✝ a n : ℕ
x : ℤ
hl : x.natAbs ≤ n / 2
m : ℤ
hm : m ≠ 0
⊢ n / 2 + n / 2 ≤ (↑n * m + x - x).natAbs
|
rw [<a>add_sub_cancel_right</a>, <a>Int.natAbs_mul</a>, <a>Int.natAbs_ofNat</a>]
|
case intro.inr
n✝ a n : ℕ
x : ℤ
hl : x.natAbs ≤ n / 2
m : ℤ
hm : m ≠ 0
⊢ n / 2 + n / 2 ≤ n * m.natAbs
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Data/ZMod/Basic.lean
|
ZMod.natAbs_min_of_le_div_two
|
case intro.inr
n✝ a n : ℕ
x : ℤ
hl : x.natAbs ≤ n / 2
m : ℤ
hm : m ≠ 0
⊢ n / 2 + n / 2 ≤ n * m.natAbs
|
refine <a>le_trans</a> ?_ (<a>Nat.le_mul_of_pos_right</a> _ <| <a>Int.natAbs_pos</a>.2 hm)
|
case intro.inr
n✝ a n : ℕ
x : ℤ
hl : x.natAbs ≤ n / 2
m : ℤ
hm : m ≠ 0
⊢ n / 2 + n / 2 ≤ n
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Data/ZMod/Basic.lean
|
ZMod.natAbs_min_of_le_div_two
|
case intro.inr
n✝ a n : ℕ
x : ℤ
hl : x.natAbs ≤ n / 2
m : ℤ
hm : m ≠ 0
⊢ n / 2 + n / 2 ≤ n
|
rw [← <a>mul_two</a>]
|
case intro.inr
n✝ a n : ℕ
x : ℤ
hl : x.natAbs ≤ n / 2
m : ℤ
hm : m ≠ 0
⊢ n / 2 * 2 ≤ n
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Data/ZMod/Basic.lean
|
ZMod.natAbs_min_of_le_div_two
|
case intro.inr
n✝ a n : ℕ
x : ℤ
hl : x.natAbs ≤ n / 2
m : ℤ
hm : m ≠ 0
⊢ n / 2 * 2 ≤ n
|
apply <a>Nat.div_mul_le_self</a>
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Data/ZMod/Basic.lean
|
ZMod.natAbs_min_of_le_div_two
|
case intro.inl
n✝ a n : ℕ
x : ℤ
hl : x.natAbs ≤ n / 2
⊢ x.natAbs ≤ (↑n * 0 + x).natAbs
|
rw [<a>MulZeroClass.mul_zero</a>, <a>zero_add</a>]
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Data/ZMod/Basic.lean
|
Ideal.add_eq_one_iff
|
R : Type u
ι : Type u_1
inst✝ : CommSemiring R
I J K L : Ideal R
⊢ I + J = 1 ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1
|
rw [<a>Ideal.one_eq_top</a>, <a>Ideal.eq_top_iff_one</a>, <a>Ideal.add_eq_sup</a>, <a>Submodule.mem_sup</a>]
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/RingTheory/Ideal/Operations.lean
|
Set.iUnion_accumulate
|
α : Type u_1
β : Type u_2
γ : Type u_3
s : α → Set β
t : α → Set γ
inst✝ : Preorder α
⊢ ⋃ x, Accumulate s x = ⋃ x, s x
|
apply <a>Set.Subset.antisymm</a>
|
case h₁
α : Type u_1
β : Type u_2
γ : Type u_3
s : α → Set β
t : α → Set γ
inst✝ : Preorder α
⊢ ⋃ x, Accumulate s x ⊆ ⋃ x, s x
case h₂
α : Type u_1
β : Type u_2
γ : Type u_3
s : α → Set β
t : α → Set γ
inst✝ : Preorder α
⊢ ⋃ x, s x ⊆ ⋃ x, Accumulate s x
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Data/Set/Accumulate.lean
|
Set.iUnion_accumulate
|
case h₁
α : Type u_1
β : Type u_2
γ : Type u_3
s : α → Set β
t : α → Set γ
inst✝ : Preorder α
⊢ ⋃ x, Accumulate s x ⊆ ⋃ x, s x
|
simp only [<a>Set.subset_def</a>, <a>Set.mem_iUnion</a>, <a>exists_imp</a>, <a>Set.mem_accumulate</a>]
|
case h₁
α : Type u_1
β : Type u_2
γ : Type u_3
s : α → Set β
t : α → Set γ
inst✝ : Preorder α
⊢ ∀ (x : β) (x_1 x_2 : α), x_2 ≤ x_1 ∧ x ∈ s x_2 → ∃ i, x ∈ s i
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Data/Set/Accumulate.lean
|
Set.iUnion_accumulate
|
case h₁
α : Type u_1
β : Type u_2
γ : Type u_3
s : α → Set β
t : α → Set γ
inst✝ : Preorder α
⊢ ∀ (x : β) (x_1 x_2 : α), x_2 ≤ x_1 ∧ x ∈ s x_2 → ∃ i, x ∈ s i
|
intro z x x' ⟨_, hz⟩
|
case h₁
α : Type u_1
β : Type u_2
γ : Type u_3
s : α → Set β
t : α → Set γ
inst✝ : Preorder α
z : β
x x' : α
left✝ : x' ≤ x
hz : z ∈ s x'
⊢ ∃ i, z ∈ s i
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Data/Set/Accumulate.lean
|
Set.iUnion_accumulate
|
case h₁
α : Type u_1
β : Type u_2
γ : Type u_3
s : α → Set β
t : α → Set γ
inst✝ : Preorder α
z : β
x x' : α
left✝ : x' ≤ x
hz : z ∈ s x'
⊢ ∃ i, z ∈ s i
|
exact ⟨x', hz⟩
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Data/Set/Accumulate.lean
|
Set.iUnion_accumulate
|
case h₂
α : Type u_1
β : Type u_2
γ : Type u_3
s : α → Set β
t : α → Set γ
inst✝ : Preorder α
⊢ ⋃ x, s x ⊆ ⋃ x, Accumulate s x
|
exact <a>Set.iUnion_mono</a> fun i => <a>Set.subset_accumulate</a>
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Data/Set/Accumulate.lean
|
MeasureTheory.L2.integral_inner_eq_sq_snorm
|
α : Type u_1
E : Type u_2
F : Type u_3
𝕜 : Type u_4
inst✝⁴ : RCLike 𝕜
inst✝³ : MeasurableSpace α
μ : Measure α
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace 𝕜 E
inst✝ : NormedAddCommGroup F
f : ↥(Lp E 2 μ)
⊢ ∫ (a : α), ⟪↑↑f a, ↑↑f a⟫_𝕜 ∂μ = ↑(∫⁻ (a : α), ↑‖↑↑f a‖₊ ^ 2 ∂μ).toReal
|
simp_rw [<a>inner_self_eq_norm_sq_to_K</a>]
|
α : Type u_1
E : Type u_2
F : Type u_3
𝕜 : Type u_4
inst✝⁴ : RCLike 𝕜
inst✝³ : MeasurableSpace α
μ : Measure α
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace 𝕜 E
inst✝ : NormedAddCommGroup F
f : ↥(Lp E 2 μ)
⊢ ∫ (a : α), ↑‖↑↑f a‖ ^ 2 ∂μ = ↑(∫⁻ (a : α), ↑‖↑↑f a‖₊ ^ 2 ∂μ).toReal
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/MeasureTheory/Function/L2Space.lean
|
MeasureTheory.L2.integral_inner_eq_sq_snorm
|
α : Type u_1
E : Type u_2
F : Type u_3
𝕜 : Type u_4
inst✝⁴ : RCLike 𝕜
inst✝³ : MeasurableSpace α
μ : Measure α
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace 𝕜 E
inst✝ : NormedAddCommGroup F
f : ↥(Lp E 2 μ)
⊢ ∫ (a : α), ↑‖↑↑f a‖ ^ 2 ∂μ = ↑(∫⁻ (a : α), ↑‖↑↑f a‖₊ ^ 2 ∂μ).toReal
|
norm_cast
|
α : Type u_1
E : Type u_2
F : Type u_3
𝕜 : Type u_4
inst✝⁴ : RCLike 𝕜
inst✝³ : MeasurableSpace α
μ : Measure α
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace 𝕜 E
inst✝ : NormedAddCommGroup F
f : ↥(Lp E 2 μ)
⊢ ∫ (x : α), ‖↑↑f x‖ ^ 2 ∂μ = (∫⁻ (a : α), ↑(‖↑↑f a‖₊ ^ 2) ∂μ).toReal
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/MeasureTheory/Function/L2Space.lean
|
MeasureTheory.L2.integral_inner_eq_sq_snorm
|
α : Type u_1
E : Type u_2
F : Type u_3
𝕜 : Type u_4
inst✝⁴ : RCLike 𝕜
inst✝³ : MeasurableSpace α
μ : Measure α
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace 𝕜 E
inst✝ : NormedAddCommGroup F
f : ↥(Lp E 2 μ)
⊢ ∫ (x : α), ‖↑↑f x‖ ^ 2 ∂μ = (∫⁻ (a : α), ↑(‖↑↑f a‖₊ ^ 2) ∂μ).toReal
|
rw [<a>MeasureTheory.integral_eq_lintegral_of_nonneg_ae</a>]
|
α : Type u_1
E : Type u_2
F : Type u_3
𝕜 : Type u_4
inst✝⁴ : RCLike 𝕜
inst✝³ : MeasurableSpace α
μ : Measure α
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace 𝕜 E
inst✝ : NormedAddCommGroup F
f : ↥(Lp E 2 μ)
⊢ (∫⁻ (a : α), ENNReal.ofReal (‖↑↑f a‖ ^ 2) ∂μ).toReal = (∫⁻ (a : α), ↑(‖↑↑f a‖₊ ^ 2) ∂μ).toReal
case hf
α : Type u_1
E : Type u_2
F : Type u_3
𝕜 : Type u_4
inst✝⁴ : RCLike 𝕜
inst✝³ : MeasurableSpace α
μ : Measure α
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace 𝕜 E
inst✝ : NormedAddCommGroup F
f : ↥(Lp E 2 μ)
⊢ 0 ≤ᶠ[ae μ] fun x => ‖↑↑f x‖ ^ 2
case hfm
α : Type u_1
E : Type u_2
F : Type u_3
𝕜 : Type u_4
inst✝⁴ : RCLike 𝕜
inst✝³ : MeasurableSpace α
μ : Measure α
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace 𝕜 E
inst✝ : NormedAddCommGroup F
f : ↥(Lp E 2 μ)
⊢ AEStronglyMeasurable (fun x => ‖↑↑f x‖ ^ 2) μ
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/MeasureTheory/Function/L2Space.lean
|
MeasureTheory.L2.integral_inner_eq_sq_snorm
|
α : Type u_1
E : Type u_2
F : Type u_3
𝕜 : Type u_4
inst✝⁴ : RCLike 𝕜
inst✝³ : MeasurableSpace α
μ : Measure α
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace 𝕜 E
inst✝ : NormedAddCommGroup F
f : ↥(Lp E 2 μ)
⊢ (∫⁻ (a : α), ENNReal.ofReal (‖↑↑f a‖ ^ 2) ∂μ).toReal = (∫⁻ (a : α), ↑(‖↑↑f a‖₊ ^ 2) ∂μ).toReal
case hf
α : Type u_1
E : Type u_2
F : Type u_3
𝕜 : Type u_4
inst✝⁴ : RCLike 𝕜
inst✝³ : MeasurableSpace α
μ : Measure α
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace 𝕜 E
inst✝ : NormedAddCommGroup F
f : ↥(Lp E 2 μ)
⊢ 0 ≤ᶠ[ae μ] fun x => ‖↑↑f x‖ ^ 2
case hfm
α : Type u_1
E : Type u_2
F : Type u_3
𝕜 : Type u_4
inst✝⁴ : RCLike 𝕜
inst✝³ : MeasurableSpace α
μ : Measure α
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace 𝕜 E
inst✝ : NormedAddCommGroup F
f : ↥(Lp E 2 μ)
⊢ AEStronglyMeasurable (fun x => ‖↑↑f x‖ ^ 2) μ
|
rotate_left
|
case hf
α : Type u_1
E : Type u_2
F : Type u_3
𝕜 : Type u_4
inst✝⁴ : RCLike 𝕜
inst✝³ : MeasurableSpace α
μ : Measure α
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace 𝕜 E
inst✝ : NormedAddCommGroup F
f : ↥(Lp E 2 μ)
⊢ 0 ≤ᶠ[ae μ] fun x => ‖↑↑f x‖ ^ 2
case hfm
α : Type u_1
E : Type u_2
F : Type u_3
𝕜 : Type u_4
inst✝⁴ : RCLike 𝕜
inst✝³ : MeasurableSpace α
μ : Measure α
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace 𝕜 E
inst✝ : NormedAddCommGroup F
f : ↥(Lp E 2 μ)
⊢ AEStronglyMeasurable (fun x => ‖↑↑f x‖ ^ 2) μ
α : Type u_1
E : Type u_2
F : Type u_3
𝕜 : Type u_4
inst✝⁴ : RCLike 𝕜
inst✝³ : MeasurableSpace α
μ : Measure α
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace 𝕜 E
inst✝ : NormedAddCommGroup F
f : ↥(Lp E 2 μ)
⊢ (∫⁻ (a : α), ENNReal.ofReal (‖↑↑f a‖ ^ 2) ∂μ).toReal = (∫⁻ (a : α), ↑(‖↑↑f a‖₊ ^ 2) ∂μ).toReal
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/MeasureTheory/Function/L2Space.lean
|
MeasureTheory.L2.integral_inner_eq_sq_snorm
|
α : Type u_1
E : Type u_2
F : Type u_3
𝕜 : Type u_4
inst✝⁴ : RCLike 𝕜
inst✝³ : MeasurableSpace α
μ : Measure α
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace 𝕜 E
inst✝ : NormedAddCommGroup F
f : ↥(Lp E 2 μ)
⊢ (∫⁻ (a : α), ENNReal.ofReal (‖↑↑f a‖ ^ 2) ∂μ).toReal = (∫⁻ (a : α), ↑(‖↑↑f a‖₊ ^ 2) ∂μ).toReal
|
congr
|
case e_a.e_f
α : Type u_1
E : Type u_2
F : Type u_3
𝕜 : Type u_4
inst✝⁴ : RCLike 𝕜
inst✝³ : MeasurableSpace α
μ : Measure α
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace 𝕜 E
inst✝ : NormedAddCommGroup F
f : ↥(Lp E 2 μ)
⊢ (fun a => ENNReal.ofReal (‖↑↑f a‖ ^ 2)) = fun a => ↑(‖↑↑f a‖₊ ^ 2)
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/MeasureTheory/Function/L2Space.lean
|
MeasureTheory.L2.integral_inner_eq_sq_snorm
|
case e_a.e_f
α : Type u_1
E : Type u_2
F : Type u_3
𝕜 : Type u_4
inst✝⁴ : RCLike 𝕜
inst✝³ : MeasurableSpace α
μ : Measure α
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace 𝕜 E
inst✝ : NormedAddCommGroup F
f : ↥(Lp E 2 μ)
⊢ (fun a => ENNReal.ofReal (‖↑↑f a‖ ^ 2)) = fun a => ↑(‖↑↑f a‖₊ ^ 2)
|
ext1 x
|
case e_a.e_f.h
α : Type u_1
E : Type u_2
F : Type u_3
𝕜 : Type u_4
inst✝⁴ : RCLike 𝕜
inst✝³ : MeasurableSpace α
μ : Measure α
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace 𝕜 E
inst✝ : NormedAddCommGroup F
f : ↥(Lp E 2 μ)
x : α
⊢ ENNReal.ofReal (‖↑↑f x‖ ^ 2) = ↑(‖↑↑f x‖₊ ^ 2)
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/MeasureTheory/Function/L2Space.lean
|
MeasureTheory.L2.integral_inner_eq_sq_snorm
|
case e_a.e_f.h
α : Type u_1
E : Type u_2
F : Type u_3
𝕜 : Type u_4
inst✝⁴ : RCLike 𝕜
inst✝³ : MeasurableSpace α
μ : Measure α
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace 𝕜 E
inst✝ : NormedAddCommGroup F
f : ↥(Lp E 2 μ)
x : α
⊢ ENNReal.ofReal (‖↑↑f x‖ ^ 2) = ↑(‖↑↑f x‖₊ ^ 2)
|
have h_two : (2 : ℝ) = ((2 : ℕ) : ℝ) := by simp
|
case e_a.e_f.h
α : Type u_1
E : Type u_2
F : Type u_3
𝕜 : Type u_4
inst✝⁴ : RCLike 𝕜
inst✝³ : MeasurableSpace α
μ : Measure α
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace 𝕜 E
inst✝ : NormedAddCommGroup F
f : ↥(Lp E 2 μ)
x : α
h_two : 2 = ↑2
⊢ ENNReal.ofReal (‖↑↑f x‖ ^ 2) = ↑(‖↑↑f x‖₊ ^ 2)
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/MeasureTheory/Function/L2Space.lean
|
MeasureTheory.L2.integral_inner_eq_sq_snorm
|
case e_a.e_f.h
α : Type u_1
E : Type u_2
F : Type u_3
𝕜 : Type u_4
inst✝⁴ : RCLike 𝕜
inst✝³ : MeasurableSpace α
μ : Measure α
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace 𝕜 E
inst✝ : NormedAddCommGroup F
f : ↥(Lp E 2 μ)
x : α
h_two : 2 = ↑2
⊢ ENNReal.ofReal (‖↑↑f x‖ ^ 2) = ↑(‖↑↑f x‖₊ ^ 2)
|
rw [← <a>Real.rpow_natCast</a> _ 2, ← h_two, ← <a>ENNReal.ofReal_rpow_of_nonneg</a> (<a>norm_nonneg</a> _) <a>zero_le_two</a>, <a>ofReal_norm_eq_coe_nnnorm</a>]
|
case e_a.e_f.h
α : Type u_1
E : Type u_2
F : Type u_3
𝕜 : Type u_4
inst✝⁴ : RCLike 𝕜
inst✝³ : MeasurableSpace α
μ : Measure α
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace 𝕜 E
inst✝ : NormedAddCommGroup F
f : ↥(Lp E 2 μ)
x : α
h_two : 2 = ↑2
⊢ ↑‖↑↑f x‖₊ ^ 2 = ↑(‖↑↑f x‖₊ ^ 2)
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/MeasureTheory/Function/L2Space.lean
|
MeasureTheory.L2.integral_inner_eq_sq_snorm
|
case e_a.e_f.h
α : Type u_1
E : Type u_2
F : Type u_3
𝕜 : Type u_4
inst✝⁴ : RCLike 𝕜
inst✝³ : MeasurableSpace α
μ : Measure α
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace 𝕜 E
inst✝ : NormedAddCommGroup F
f : ↥(Lp E 2 μ)
x : α
h_two : 2 = ↑2
⊢ ↑‖↑↑f x‖₊ ^ 2 = ↑(‖↑↑f x‖₊ ^ 2)
|
norm_cast
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/MeasureTheory/Function/L2Space.lean
|
MeasureTheory.L2.integral_inner_eq_sq_snorm
|
case hf
α : Type u_1
E : Type u_2
F : Type u_3
𝕜 : Type u_4
inst✝⁴ : RCLike 𝕜
inst✝³ : MeasurableSpace α
μ : Measure α
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace 𝕜 E
inst✝ : NormedAddCommGroup F
f : ↥(Lp E 2 μ)
⊢ 0 ≤ᶠ[ae μ] fun x => ‖↑↑f x‖ ^ 2
|
exact <a>Filter.eventually_of_forall</a> fun x => <a>sq_nonneg</a> _
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/MeasureTheory/Function/L2Space.lean
|
MeasureTheory.L2.integral_inner_eq_sq_snorm
|
case hfm
α : Type u_1
E : Type u_2
F : Type u_3
𝕜 : Type u_4
inst✝⁴ : RCLike 𝕜
inst✝³ : MeasurableSpace α
μ : Measure α
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace 𝕜 E
inst✝ : NormedAddCommGroup F
f : ↥(Lp E 2 μ)
⊢ AEStronglyMeasurable (fun x => ‖↑↑f x‖ ^ 2) μ
|
exact ((<a>MeasureTheory.Lp.aestronglyMeasurable</a> f).norm.aemeasurable.pow_const _).<a>AEMeasurable.aestronglyMeasurable</a>
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/MeasureTheory/Function/L2Space.lean
|
MeasureTheory.L2.integral_inner_eq_sq_snorm
|
α : Type u_1
E : Type u_2
F : Type u_3
𝕜 : Type u_4
inst✝⁴ : RCLike 𝕜
inst✝³ : MeasurableSpace α
μ : Measure α
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace 𝕜 E
inst✝ : NormedAddCommGroup F
f : ↥(Lp E 2 μ)
x : α
⊢ 2 = ↑2
|
simp
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/MeasureTheory/Function/L2Space.lean
|
unitInterval.one_minus_nonneg
|
x : ↑I
⊢ 0 ≤ 1 - ↑x
|
simpa using x.2.2
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Topology/UnitInterval.lean
|
Polynomial.as_sum_range_C_mul_X_pow
|
R : Type u
S : Type v
a b c d : R
n m : ℕ
inst✝ : Semiring R
p✝ q r p : R[X]
⊢ ∑ i ∈ range (p.natDegree + 1), (monomial i) (p.coeff i) = ∑ i ∈ range (p.natDegree + 1), C (p.coeff i) * X ^ i
|
simp only [<a>Polynomial.C_mul_X_pow_eq_monomial</a>]
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Algebra/Polynomial/Degree/Definitions.lean
|
LinearMap.equivariantProjection_apply
|
k : Type u
inst✝¹¹ : CommRing k
G : Type u
inst✝¹⁰ : Group G
V : Type v
inst✝⁹ : AddCommGroup V
inst✝⁸ : Module k V
inst✝⁷ : Module (MonoidAlgebra k G) V
inst✝⁶ : IsScalarTower k (MonoidAlgebra k G) V
W : Type w
inst✝⁵ : AddCommGroup W
inst✝⁴ : Module k W
inst✝³ : Module (MonoidAlgebra k G) W
inst✝² : IsScalarTower k (MonoidAlgebra k G) W
π : W →ₗ[k] V
i : V →ₗ[MonoidAlgebra k G] W
h : ∀ (v : V), π (i v) = v
inst✝¹ : Fintype G
inst✝ : Invertible ↑(Fintype.card G)
v : W
⊢ (equivariantProjection G π) v = ⅟↑(Fintype.card G) • ∑ g : G, (π.conjugate g) v
|
simp only [<a>LinearMap.equivariantProjection</a>, <a>LinearMap.smul_apply</a>, <a>LinearMap.sumOfConjugatesEquivariant_apply</a>]
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/RepresentationTheory/Maschke.lean
|
MeasureTheory.Lp.simpleFunc.add_toSimpleFunc
|
α : Type u_1
β : Type u_2
ι : Type u_3
E : Type u_4
F : Type u_5
𝕜 : Type u_6
inst✝² : MeasurableSpace α
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedAddCommGroup F
p : ℝ≥0∞
μ : Measure α
f g : ↥(simpleFunc E p μ)
⊢ ↑(toSimpleFunc (f + g)) =ᶠ[ae μ] ↑(toSimpleFunc f) + ↑(toSimpleFunc g)
|
filter_upwards [<a>MeasureTheory.Lp.simpleFunc.toSimpleFunc_eq_toFun</a> (f + g), <a>MeasureTheory.Lp.simpleFunc.toSimpleFunc_eq_toFun</a> f, <a>MeasureTheory.Lp.simpleFunc.toSimpleFunc_eq_toFun</a> g, <a>MeasureTheory.Lp.coeFn_add</a> (f : <a>MeasureTheory.Lp</a> E p μ) g] with _
|
case h
α : Type u_1
β : Type u_2
ι : Type u_3
E : Type u_4
F : Type u_5
𝕜 : Type u_6
inst✝² : MeasurableSpace α
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedAddCommGroup F
p : ℝ≥0∞
μ : Measure α
f g : ↥(simpleFunc E p μ)
a✝ : α
⊢ ↑(toSimpleFunc (f + g)) a✝ = ↑↑↑(f + g) a✝ →
↑(toSimpleFunc f) a✝ = ↑↑↑f a✝ →
↑(toSimpleFunc g) a✝ = ↑↑↑g a✝ →
↑↑(↑f + ↑g) a✝ = (↑↑↑f + ↑↑↑g) a✝ → ↑(toSimpleFunc (f + g)) a✝ = (↑(toSimpleFunc f) + ↑(toSimpleFunc g)) a✝
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/MeasureTheory/Function/SimpleFuncDenseLp.lean
|
MeasureTheory.Lp.simpleFunc.add_toSimpleFunc
|
case h
α : Type u_1
β : Type u_2
ι : Type u_3
E : Type u_4
F : Type u_5
𝕜 : Type u_6
inst✝² : MeasurableSpace α
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedAddCommGroup F
p : ℝ≥0∞
μ : Measure α
f g : ↥(simpleFunc E p μ)
a✝ : α
⊢ ↑(toSimpleFunc (f + g)) a✝ = ↑↑↑(f + g) a✝ →
↑(toSimpleFunc f) a✝ = ↑↑↑f a✝ →
↑(toSimpleFunc g) a✝ = ↑↑↑g a✝ →
↑↑(↑f + ↑g) a✝ = (↑↑↑f + ↑↑↑g) a✝ → ↑(toSimpleFunc (f + g)) a✝ = (↑(toSimpleFunc f) + ↑(toSimpleFunc g)) a✝
|
simp only [<a>AddSubgroup.coe_add</a>, <a>Pi.add_apply</a>]
|
case h
α : Type u_1
β : Type u_2
ι : Type u_3
E : Type u_4
F : Type u_5
𝕜 : Type u_6
inst✝² : MeasurableSpace α
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedAddCommGroup F
p : ℝ≥0∞
μ : Measure α
f g : ↥(simpleFunc E p μ)
a✝ : α
⊢ ↑(toSimpleFunc (f + g)) a✝ = ↑(↑↑f + ↑↑g) a✝ →
↑(toSimpleFunc f) a✝ = ↑↑↑f a✝ →
↑(toSimpleFunc g) a✝ = ↑↑↑g a✝ →
↑(↑↑f + ↑↑g) a✝ = ↑↑↑f a✝ + ↑↑↑g a✝ → ↑(toSimpleFunc (f + g)) a✝ = ↑(toSimpleFunc f) a✝ + ↑(toSimpleFunc g) a✝
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/MeasureTheory/Function/SimpleFuncDenseLp.lean
|
MeasureTheory.Lp.simpleFunc.add_toSimpleFunc
|
case h
α : Type u_1
β : Type u_2
ι : Type u_3
E : Type u_4
F : Type u_5
𝕜 : Type u_6
inst✝² : MeasurableSpace α
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedAddCommGroup F
p : ℝ≥0∞
μ : Measure α
f g : ↥(simpleFunc E p μ)
a✝ : α
⊢ ↑(toSimpleFunc (f + g)) a✝ = ↑(↑↑f + ↑↑g) a✝ →
↑(toSimpleFunc f) a✝ = ↑↑↑f a✝ →
↑(toSimpleFunc g) a✝ = ↑↑↑g a✝ →
↑(↑↑f + ↑↑g) a✝ = ↑↑↑f a✝ + ↑↑↑g a✝ → ↑(toSimpleFunc (f + g)) a✝ = ↑(toSimpleFunc f) a✝ + ↑(toSimpleFunc g) a✝
|
iterate 4 intro h; rw [h]
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/MeasureTheory/Function/SimpleFuncDenseLp.lean
|
MeasureTheory.Lp.simpleFunc.add_toSimpleFunc
|
case h
α : Type u_1
β : Type u_2
ι : Type u_3
E : Type u_4
F : Type u_5
𝕜 : Type u_6
inst✝² : MeasurableSpace α
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedAddCommGroup F
p : ℝ≥0∞
μ : Measure α
f g : ↥(simpleFunc E p μ)
a✝ : α
h✝¹ : ↑(toSimpleFunc (f + g)) a✝ = ↑(↑↑f + ↑↑g) a✝
h✝ : ↑(toSimpleFunc f) a✝ = ↑↑↑f a✝
h : ↑(toSimpleFunc g) a✝ = ↑↑↑g a✝
⊢ ↑(↑↑f + ↑↑g) a✝ = ↑↑↑f a✝ + ↑↑↑g a✝ → ↑(↑↑f + ↑↑g) a✝ = ↑↑↑f a✝ + ↑↑↑g a✝
|
intro h
|
case h
α : Type u_1
β : Type u_2
ι : Type u_3
E : Type u_4
F : Type u_5
𝕜 : Type u_6
inst✝² : MeasurableSpace α
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedAddCommGroup F
p : ℝ≥0∞
μ : Measure α
f g : ↥(simpleFunc E p μ)
a✝ : α
h✝² : ↑(toSimpleFunc (f + g)) a✝ = ↑(↑↑f + ↑↑g) a✝
h✝¹ : ↑(toSimpleFunc f) a✝ = ↑↑↑f a✝
h✝ : ↑(toSimpleFunc g) a✝ = ↑↑↑g a✝
h : ↑(↑↑f + ↑↑g) a✝ = ↑↑↑f a✝ + ↑↑↑g a✝
⊢ ↑(↑↑f + ↑↑g) a✝ = ↑↑↑f a✝ + ↑↑↑g a✝
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/MeasureTheory/Function/SimpleFuncDenseLp.lean
|
MeasureTheory.Lp.simpleFunc.add_toSimpleFunc
|
case h
α : Type u_1
β : Type u_2
ι : Type u_3
E : Type u_4
F : Type u_5
𝕜 : Type u_6
inst✝² : MeasurableSpace α
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedAddCommGroup F
p : ℝ≥0∞
μ : Measure α
f g : ↥(simpleFunc E p μ)
a✝ : α
h✝² : ↑(toSimpleFunc (f + g)) a✝ = ↑(↑↑f + ↑↑g) a✝
h✝¹ : ↑(toSimpleFunc f) a✝ = ↑↑↑f a✝
h✝ : ↑(toSimpleFunc g) a✝ = ↑↑↑g a✝
h : ↑(↑↑f + ↑↑g) a✝ = ↑↑↑f a✝ + ↑↑↑g a✝
⊢ ↑(↑↑f + ↑↑g) a✝ = ↑↑↑f a✝ + ↑↑↑g a✝
|
rw [h]
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/MeasureTheory/Function/SimpleFuncDenseLp.lean
|
EuclideanGeometry.reflection_eq_iff_orthogonalProjection_eq
|
V : Type u_1
P : Type u_2
inst✝⁷ : NormedAddCommGroup V
inst✝⁶ : InnerProductSpace ℝ V
inst✝⁵ : MetricSpace P
inst✝⁴ : NormedAddTorsor V P
s₁ s₂ : AffineSubspace ℝ P
inst✝³ : Nonempty ↥s₁
inst✝² : Nonempty ↥s₂
inst✝¹ : HasOrthogonalProjection s₁.direction
inst✝ : HasOrthogonalProjection s₂.direction
p : P
⊢ (reflection s₁) p = (reflection s₂) p ↔ ↑((orthogonalProjection s₁) p) = ↑((orthogonalProjection s₂) p)
|
rw [<a>EuclideanGeometry.reflection_apply</a>, <a>EuclideanGeometry.reflection_apply</a>]
|
V : Type u_1
P : Type u_2
inst✝⁷ : NormedAddCommGroup V
inst✝⁶ : InnerProductSpace ℝ V
inst✝⁵ : MetricSpace P
inst✝⁴ : NormedAddTorsor V P
s₁ s₂ : AffineSubspace ℝ P
inst✝³ : Nonempty ↥s₁
inst✝² : Nonempty ↥s₂
inst✝¹ : HasOrthogonalProjection s₁.direction
inst✝ : HasOrthogonalProjection s₂.direction
p : P
⊢ ↑((orthogonalProjection s₁) p) -ᵥ p +ᵥ ↑((orthogonalProjection s₁) p) =
↑((orthogonalProjection s₂) p) -ᵥ p +ᵥ ↑((orthogonalProjection s₂) p) ↔
↑((orthogonalProjection s₁) p) = ↑((orthogonalProjection s₂) p)
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Geometry/Euclidean/Basic.lean
|
EuclideanGeometry.reflection_eq_iff_orthogonalProjection_eq
|
V : Type u_1
P : Type u_2
inst✝⁷ : NormedAddCommGroup V
inst✝⁶ : InnerProductSpace ℝ V
inst✝⁵ : MetricSpace P
inst✝⁴ : NormedAddTorsor V P
s₁ s₂ : AffineSubspace ℝ P
inst✝³ : Nonempty ↥s₁
inst✝² : Nonempty ↥s₂
inst✝¹ : HasOrthogonalProjection s₁.direction
inst✝ : HasOrthogonalProjection s₂.direction
p : P
⊢ ↑((orthogonalProjection s₁) p) -ᵥ p +ᵥ ↑((orthogonalProjection s₁) p) =
↑((orthogonalProjection s₂) p) -ᵥ p +ᵥ ↑((orthogonalProjection s₂) p) ↔
↑((orthogonalProjection s₁) p) = ↑((orthogonalProjection s₂) p)
|
constructor
|
case mp
V : Type u_1
P : Type u_2
inst✝⁷ : NormedAddCommGroup V
inst✝⁶ : InnerProductSpace ℝ V
inst✝⁵ : MetricSpace P
inst✝⁴ : NormedAddTorsor V P
s₁ s₂ : AffineSubspace ℝ P
inst✝³ : Nonempty ↥s₁
inst✝² : Nonempty ↥s₂
inst✝¹ : HasOrthogonalProjection s₁.direction
inst✝ : HasOrthogonalProjection s₂.direction
p : P
⊢ ↑((orthogonalProjection s₁) p) -ᵥ p +ᵥ ↑((orthogonalProjection s₁) p) =
↑((orthogonalProjection s₂) p) -ᵥ p +ᵥ ↑((orthogonalProjection s₂) p) →
↑((orthogonalProjection s₁) p) = ↑((orthogonalProjection s₂) p)
case mpr
V : Type u_1
P : Type u_2
inst✝⁷ : NormedAddCommGroup V
inst✝⁶ : InnerProductSpace ℝ V
inst✝⁵ : MetricSpace P
inst✝⁴ : NormedAddTorsor V P
s₁ s₂ : AffineSubspace ℝ P
inst✝³ : Nonempty ↥s₁
inst✝² : Nonempty ↥s₂
inst✝¹ : HasOrthogonalProjection s₁.direction
inst✝ : HasOrthogonalProjection s₂.direction
p : P
⊢ ↑((orthogonalProjection s₁) p) = ↑((orthogonalProjection s₂) p) →
↑((orthogonalProjection s₁) p) -ᵥ p +ᵥ ↑((orthogonalProjection s₁) p) =
↑((orthogonalProjection s₂) p) -ᵥ p +ᵥ ↑((orthogonalProjection s₂) p)
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Geometry/Euclidean/Basic.lean
|
EuclideanGeometry.reflection_eq_iff_orthogonalProjection_eq
|
case mp
V : Type u_1
P : Type u_2
inst✝⁷ : NormedAddCommGroup V
inst✝⁶ : InnerProductSpace ℝ V
inst✝⁵ : MetricSpace P
inst✝⁴ : NormedAddTorsor V P
s₁ s₂ : AffineSubspace ℝ P
inst✝³ : Nonempty ↥s₁
inst✝² : Nonempty ↥s₂
inst✝¹ : HasOrthogonalProjection s₁.direction
inst✝ : HasOrthogonalProjection s₂.direction
p : P
⊢ ↑((orthogonalProjection s₁) p) -ᵥ p +ᵥ ↑((orthogonalProjection s₁) p) =
↑((orthogonalProjection s₂) p) -ᵥ p +ᵥ ↑((orthogonalProjection s₂) p) →
↑((orthogonalProjection s₁) p) = ↑((orthogonalProjection s₂) p)
|
intro h
|
case mp
V : Type u_1
P : Type u_2
inst✝⁷ : NormedAddCommGroup V
inst✝⁶ : InnerProductSpace ℝ V
inst✝⁵ : MetricSpace P
inst✝⁴ : NormedAddTorsor V P
s₁ s₂ : AffineSubspace ℝ P
inst✝³ : Nonempty ↥s₁
inst✝² : Nonempty ↥s₂
inst✝¹ : HasOrthogonalProjection s₁.direction
inst✝ : HasOrthogonalProjection s₂.direction
p : P
h :
↑((orthogonalProjection s₁) p) -ᵥ p +ᵥ ↑((orthogonalProjection s₁) p) =
↑((orthogonalProjection s₂) p) -ᵥ p +ᵥ ↑((orthogonalProjection s₂) p)
⊢ ↑((orthogonalProjection s₁) p) = ↑((orthogonalProjection s₂) p)
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Geometry/Euclidean/Basic.lean
|
EuclideanGeometry.reflection_eq_iff_orthogonalProjection_eq
|
case mp
V : Type u_1
P : Type u_2
inst✝⁷ : NormedAddCommGroup V
inst✝⁶ : InnerProductSpace ℝ V
inst✝⁵ : MetricSpace P
inst✝⁴ : NormedAddTorsor V P
s₁ s₂ : AffineSubspace ℝ P
inst✝³ : Nonempty ↥s₁
inst✝² : Nonempty ↥s₂
inst✝¹ : HasOrthogonalProjection s₁.direction
inst✝ : HasOrthogonalProjection s₂.direction
p : P
h :
↑((orthogonalProjection s₁) p) -ᵥ p +ᵥ ↑((orthogonalProjection s₁) p) =
↑((orthogonalProjection s₂) p) -ᵥ p +ᵥ ↑((orthogonalProjection s₂) p)
⊢ ↑((orthogonalProjection s₁) p) = ↑((orthogonalProjection s₂) p)
|
rw [← @<a>vsub_eq_zero_iff_eq</a> V, <a>vsub_vadd_eq_vsub_sub</a>, <a>vadd_vsub_assoc</a>, <a>add_comm</a>, <a>add_sub_assoc</a>, <a>vsub_sub_vsub_cancel_right</a>, ← <a>two_smul</a> ℝ ((<a>EuclideanGeometry.orthogonalProjection</a> s₁ p : P) -ᵥ <a>EuclideanGeometry.orthogonalProjection</a> s₂ p), <a>smul_eq_zero</a>] at h
|
case mp
V : Type u_1
P : Type u_2
inst✝⁷ : NormedAddCommGroup V
inst✝⁶ : InnerProductSpace ℝ V
inst✝⁵ : MetricSpace P
inst✝⁴ : NormedAddTorsor V P
s₁ s₂ : AffineSubspace ℝ P
inst✝³ : Nonempty ↥s₁
inst✝² : Nonempty ↥s₂
inst✝¹ : HasOrthogonalProjection s₁.direction
inst✝ : HasOrthogonalProjection s₂.direction
p : P
h : 2 = 0 ∨ ↑((orthogonalProjection s₁) p) -ᵥ ↑((orthogonalProjection s₂) p) = 0
⊢ ↑((orthogonalProjection s₁) p) = ↑((orthogonalProjection s₂) p)
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Geometry/Euclidean/Basic.lean
|
EuclideanGeometry.reflection_eq_iff_orthogonalProjection_eq
|
case mp
V : Type u_1
P : Type u_2
inst✝⁷ : NormedAddCommGroup V
inst✝⁶ : InnerProductSpace ℝ V
inst✝⁵ : MetricSpace P
inst✝⁴ : NormedAddTorsor V P
s₁ s₂ : AffineSubspace ℝ P
inst✝³ : Nonempty ↥s₁
inst✝² : Nonempty ↥s₂
inst✝¹ : HasOrthogonalProjection s₁.direction
inst✝ : HasOrthogonalProjection s₂.direction
p : P
h : 2 = 0 ∨ ↑((orthogonalProjection s₁) p) -ᵥ ↑((orthogonalProjection s₂) p) = 0
⊢ ↑((orthogonalProjection s₁) p) = ↑((orthogonalProjection s₂) p)
|
norm_num at h
|
case mp
V : Type u_1
P : Type u_2
inst✝⁷ : NormedAddCommGroup V
inst✝⁶ : InnerProductSpace ℝ V
inst✝⁵ : MetricSpace P
inst✝⁴ : NormedAddTorsor V P
s₁ s₂ : AffineSubspace ℝ P
inst✝³ : Nonempty ↥s₁
inst✝² : Nonempty ↥s₂
inst✝¹ : HasOrthogonalProjection s₁.direction
inst✝ : HasOrthogonalProjection s₂.direction
p : P
h : ↑((orthogonalProjection s₁) p) = ↑((orthogonalProjection s₂) p)
⊢ ↑((orthogonalProjection s₁) p) = ↑((orthogonalProjection s₂) p)
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Geometry/Euclidean/Basic.lean
|
EuclideanGeometry.reflection_eq_iff_orthogonalProjection_eq
|
case mp
V : Type u_1
P : Type u_2
inst✝⁷ : NormedAddCommGroup V
inst✝⁶ : InnerProductSpace ℝ V
inst✝⁵ : MetricSpace P
inst✝⁴ : NormedAddTorsor V P
s₁ s₂ : AffineSubspace ℝ P
inst✝³ : Nonempty ↥s₁
inst✝² : Nonempty ↥s₂
inst✝¹ : HasOrthogonalProjection s₁.direction
inst✝ : HasOrthogonalProjection s₂.direction
p : P
h : ↑((orthogonalProjection s₁) p) = ↑((orthogonalProjection s₂) p)
⊢ ↑((orthogonalProjection s₁) p) = ↑((orthogonalProjection s₂) p)
|
exact h
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Geometry/Euclidean/Basic.lean
|
EuclideanGeometry.reflection_eq_iff_orthogonalProjection_eq
|
case mpr
V : Type u_1
P : Type u_2
inst✝⁷ : NormedAddCommGroup V
inst✝⁶ : InnerProductSpace ℝ V
inst✝⁵ : MetricSpace P
inst✝⁴ : NormedAddTorsor V P
s₁ s₂ : AffineSubspace ℝ P
inst✝³ : Nonempty ↥s₁
inst✝² : Nonempty ↥s₂
inst✝¹ : HasOrthogonalProjection s₁.direction
inst✝ : HasOrthogonalProjection s₂.direction
p : P
⊢ ↑((orthogonalProjection s₁) p) = ↑((orthogonalProjection s₂) p) →
↑((orthogonalProjection s₁) p) -ᵥ p +ᵥ ↑((orthogonalProjection s₁) p) =
↑((orthogonalProjection s₂) p) -ᵥ p +ᵥ ↑((orthogonalProjection s₂) p)
|
intro h
|
case mpr
V : Type u_1
P : Type u_2
inst✝⁷ : NormedAddCommGroup V
inst✝⁶ : InnerProductSpace ℝ V
inst✝⁵ : MetricSpace P
inst✝⁴ : NormedAddTorsor V P
s₁ s₂ : AffineSubspace ℝ P
inst✝³ : Nonempty ↥s₁
inst✝² : Nonempty ↥s₂
inst✝¹ : HasOrthogonalProjection s₁.direction
inst✝ : HasOrthogonalProjection s₂.direction
p : P
h : ↑((orthogonalProjection s₁) p) = ↑((orthogonalProjection s₂) p)
⊢ ↑((orthogonalProjection s₁) p) -ᵥ p +ᵥ ↑((orthogonalProjection s₁) p) =
↑((orthogonalProjection s₂) p) -ᵥ p +ᵥ ↑((orthogonalProjection s₂) p)
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Geometry/Euclidean/Basic.lean
|
EuclideanGeometry.reflection_eq_iff_orthogonalProjection_eq
|
case mpr
V : Type u_1
P : Type u_2
inst✝⁷ : NormedAddCommGroup V
inst✝⁶ : InnerProductSpace ℝ V
inst✝⁵ : MetricSpace P
inst✝⁴ : NormedAddTorsor V P
s₁ s₂ : AffineSubspace ℝ P
inst✝³ : Nonempty ↥s₁
inst✝² : Nonempty ↥s₂
inst✝¹ : HasOrthogonalProjection s₁.direction
inst✝ : HasOrthogonalProjection s₂.direction
p : P
h : ↑((orthogonalProjection s₁) p) = ↑((orthogonalProjection s₂) p)
⊢ ↑((orthogonalProjection s₁) p) -ᵥ p +ᵥ ↑((orthogonalProjection s₁) p) =
↑((orthogonalProjection s₂) p) -ᵥ p +ᵥ ↑((orthogonalProjection s₂) p)
|
rw [h]
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Geometry/Euclidean/Basic.lean
|
Polynomial.rootMultiplicity_expand_pow
|
R : Type u
inst✝¹ : CommRing R
p n : ℕ
inst✝ : ExpChar R p
f : R[X]
r : R
⊢ rootMultiplicity r ((expand R (p ^ n)) f) = p ^ n * rootMultiplicity (r ^ p ^ n) f
|
obtain rfl | h0 := <a>eq_or_ne</a> f 0
|
case inl
R : Type u
inst✝¹ : CommRing R
p n : ℕ
inst✝ : ExpChar R p
r : R
⊢ rootMultiplicity r ((expand R (p ^ n)) 0) = p ^ n * rootMultiplicity (r ^ p ^ n) 0
case inr
R : Type u
inst✝¹ : CommRing R
p n : ℕ
inst✝ : ExpChar R p
f : R[X]
r : R
h0 : f ≠ 0
⊢ rootMultiplicity r ((expand R (p ^ n)) f) = p ^ n * rootMultiplicity (r ^ p ^ n) f
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Algebra/Polynomial/Expand.lean
|
Polynomial.rootMultiplicity_expand_pow
|
case inr
R : Type u
inst✝¹ : CommRing R
p n : ℕ
inst✝ : ExpChar R p
f : R[X]
r : R
h0 : f ≠ 0
⊢ rootMultiplicity r ((expand R (p ^ n)) f) = p ^ n * rootMultiplicity (r ^ p ^ n) f
|
obtain ⟨g, hg, ndvd⟩ := f.exists_eq_pow_rootMultiplicity_mul_and_not_dvd h0 (r ^ p ^ n)
|
case inr.intro.intro
R : Type u
inst✝¹ : CommRing R
p n : ℕ
inst✝ : ExpChar R p
f : R[X]
r : R
h0 : f ≠ 0
g : R[X]
hg : f = (X - C (r ^ p ^ n)) ^ rootMultiplicity (r ^ p ^ n) f * g
ndvd : ¬X - C (r ^ p ^ n) ∣ g
⊢ rootMultiplicity r ((expand R (p ^ n)) f) = p ^ n * rootMultiplicity (r ^ p ^ n) f
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Algebra/Polynomial/Expand.lean
|
Polynomial.rootMultiplicity_expand_pow
|
case inr.intro.intro
R : Type u
inst✝¹ : CommRing R
p n : ℕ
inst✝ : ExpChar R p
f : R[X]
r : R
h0 : f ≠ 0
g : R[X]
hg : f = (X - C (r ^ p ^ n)) ^ rootMultiplicity (r ^ p ^ n) f * g
ndvd : ¬X - C (r ^ p ^ n) ∣ g
⊢ rootMultiplicity r ((expand R (p ^ n)) f) = p ^ n * rootMultiplicity (r ^ p ^ n) f
|
rw [<a>Polynomial.dvd_iff_isRoot</a>, ← <a>Polynomial.eval_X</a> (x := r), ← <a>Polynomial.eval_pow</a>, ← <a>Polynomial.isRoot_comp</a>, ← <a>Polynomial.expand_eq_comp_X_pow</a>] at ndvd
|
case inr.intro.intro
R : Type u
inst✝¹ : CommRing R
p n : ℕ
inst✝ : ExpChar R p
f : R[X]
r : R
h0 : f ≠ 0
g : R[X]
hg : f = (X - C (r ^ p ^ n)) ^ rootMultiplicity (r ^ p ^ n) f * g
ndvd : ¬((expand R (p ^ n)) g).IsRoot r
⊢ rootMultiplicity r ((expand R (p ^ n)) f) = p ^ n * rootMultiplicity (r ^ p ^ n) f
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Algebra/Polynomial/Expand.lean
|
Polynomial.rootMultiplicity_expand_pow
|
case inr.intro.intro
R : Type u
inst✝¹ : CommRing R
p n : ℕ
inst✝ : ExpChar R p
f : R[X]
r : R
h0 : f ≠ 0
g : R[X]
hg : f = (X - C (r ^ p ^ n)) ^ rootMultiplicity (r ^ p ^ n) f * g
ndvd : ¬((expand R (p ^ n)) g).IsRoot r
⊢ rootMultiplicity r ((expand R (p ^ n)) f) = p ^ n * rootMultiplicity (r ^ p ^ n) f
|
conv_lhs => rw [hg, <a>map_mul</a>, <a>map_pow</a>, <a>map_sub</a>, <a>Polynomial.expand_X</a>, <a>Polynomial.expand_C</a>, <a>map_pow</a>, ← <a>sub_pow_expChar_pow</a>, ← <a>pow_mul</a>, <a>mul_comm</a>, <a>Polynomial.rootMultiplicity_mul_X_sub_C_pow</a> (<a>Polynomial.expand_ne_zero</a> (<a>expChar_pow_pos</a> R p n) |>.<a>Iff.mpr</a> <| <a>right_ne_zero_of_mul</a> <| hg ▸ h0), <a>Polynomial.rootMultiplicity_eq_zero</a> ndvd, <a>zero_add</a>]
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Algebra/Polynomial/Expand.lean
|
Polynomial.rootMultiplicity_expand_pow
|
case inl
R : Type u
inst✝¹ : CommRing R
p n : ℕ
inst✝ : ExpChar R p
r : R
⊢ rootMultiplicity r ((expand R (p ^ n)) 0) = p ^ n * rootMultiplicity (r ^ p ^ n) 0
|
simp
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Algebra/Polynomial/Expand.lean
|
convexJoin_comm
|
ι : Sort u_1
𝕜 : Type u_2
E : Type u_3
inst✝² : OrderedSemiring 𝕜
inst✝¹ : AddCommMonoid E
inst✝ : Module 𝕜 E
s✝ t✝ s₁ s₂ t₁ t₂ u : Set E
x y : E
s t : Set E
⊢ ⋃ i₂ ∈ t, ⋃ i₁ ∈ s, segment 𝕜 i₁ i₂ = convexJoin 𝕜 t s
|
simp_rw [<a>convexJoin</a>, <a>segment_symm</a>]
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Analysis/Convex/Join.lean
|
CochainComplex.mapBifunctorShift₁Iso_trans_mapBifunctorShift₂Iso
|
C₁ : Type u_1
C₂ : Type u_2
D : Type u_3
inst✝⁸ : Category.{u_5, u_1} C₁
inst✝⁷ : Category.{u_6, u_2} C₂
inst✝⁶ : Category.{u_4, u_3} D
inst✝⁵ : Preadditive C₁
inst✝⁴ : Preadditive C₂
inst✝³ : Preadditive D
K₁ : CochainComplex C₁ ℤ
K₂ : CochainComplex C₂ ℤ
F : C₁ ⥤ C₂ ⥤ D
inst✝² : F.Additive
inst✝¹ : ∀ (X₁ : C₁), (F.obj X₁).Additive
x y : ℤ
inst✝ : K₁.HasMapBifunctor K₂ F
⊢ K₁.mapBifunctorShift₁Iso ((CategoryTheory.shiftFunctor (CochainComplex C₂ ℤ) y).obj K₂) F x ≪≫
(CategoryTheory.shiftFunctor (CochainComplex D ℤ) x).mapIso (K₁.mapBifunctorShift₂Iso K₂ F y) =
(x * y).negOnePow •
((CategoryTheory.shiftFunctor (CochainComplex C₁ ℤ) x).obj K₁).mapBifunctorShift₂Iso K₂ F y ≪≫
(CategoryTheory.shiftFunctor (CochainComplex D ℤ) y).mapIso (K₁.mapBifunctorShift₁Iso K₂ F x) ≪≫
(shiftFunctorComm (CochainComplex D ℤ) x y).app (K₁.mapBifunctor K₂ F)
|
ext1
|
case w
C₁ : Type u_1
C₂ : Type u_2
D : Type u_3
inst✝⁸ : Category.{u_5, u_1} C₁
inst✝⁷ : Category.{u_6, u_2} C₂
inst✝⁶ : Category.{u_4, u_3} D
inst✝⁵ : Preadditive C₁
inst✝⁴ : Preadditive C₂
inst✝³ : Preadditive D
K₁ : CochainComplex C₁ ℤ
K₂ : CochainComplex C₂ ℤ
F : C₁ ⥤ C₂ ⥤ D
inst✝² : F.Additive
inst✝¹ : ∀ (X₁ : C₁), (F.obj X₁).Additive
x y : ℤ
inst✝ : K₁.HasMapBifunctor K₂ F
⊢ (K₁.mapBifunctorShift₁Iso ((CategoryTheory.shiftFunctor (CochainComplex C₂ ℤ) y).obj K₂) F x ≪≫
(CategoryTheory.shiftFunctor (CochainComplex D ℤ) x).mapIso (K₁.mapBifunctorShift₂Iso K₂ F y)).hom =
((x * y).negOnePow •
((CategoryTheory.shiftFunctor (CochainComplex C₁ ℤ) x).obj K₁).mapBifunctorShift₂Iso K₂ F y ≪≫
(CategoryTheory.shiftFunctor (CochainComplex D ℤ) y).mapIso (K₁.mapBifunctorShift₁Iso K₂ F x) ≪≫
(shiftFunctorComm (CochainComplex D ℤ) x y).app (K₁.mapBifunctor K₂ F)).hom
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Algebra/Homology/BifunctorShift.lean
|
CochainComplex.mapBifunctorShift₁Iso_trans_mapBifunctorShift₂Iso
|
case w
C₁ : Type u_1
C₂ : Type u_2
D : Type u_3
inst✝⁸ : Category.{u_5, u_1} C₁
inst✝⁷ : Category.{u_6, u_2} C₂
inst✝⁶ : Category.{u_4, u_3} D
inst✝⁵ : Preadditive C₁
inst✝⁴ : Preadditive C₂
inst✝³ : Preadditive D
K₁ : CochainComplex C₁ ℤ
K₂ : CochainComplex C₂ ℤ
F : C₁ ⥤ C₂ ⥤ D
inst✝² : F.Additive
inst✝¹ : ∀ (X₁ : C₁), (F.obj X₁).Additive
x y : ℤ
inst✝ : K₁.HasMapBifunctor K₂ F
⊢ (K₁.mapBifunctorShift₁Iso ((CategoryTheory.shiftFunctor (CochainComplex C₂ ℤ) y).obj K₂) F x ≪≫
(CategoryTheory.shiftFunctor (CochainComplex D ℤ) x).mapIso (K₁.mapBifunctorShift₂Iso K₂ F y)).hom =
((x * y).negOnePow •
((CategoryTheory.shiftFunctor (CochainComplex C₁ ℤ) x).obj K₁).mapBifunctorShift₂Iso K₂ F y ≪≫
(CategoryTheory.shiftFunctor (CochainComplex D ℤ) y).mapIso (K₁.mapBifunctorShift₁Iso K₂ F x) ≪≫
(shiftFunctorComm (CochainComplex D ℤ) x y).app (K₁.mapBifunctor K₂ F)).hom
|
dsimp [<a>CochainComplex.mapBifunctorShift₁Iso</a>, <a>CochainComplex.mapBifunctorShift₂Iso</a>]
|
case w
C₁ : Type u_1
C₂ : Type u_2
D : Type u_3
inst✝⁸ : Category.{u_5, u_1} C₁
inst✝⁷ : Category.{u_6, u_2} C₂
inst✝⁶ : Category.{u_4, u_3} D
inst✝⁵ : Preadditive C₁
inst✝⁴ : Preadditive C₂
inst✝³ : Preadditive D
K₁ : CochainComplex C₁ ℤ
K₂ : CochainComplex C₂ ℤ
F : C₁ ⥤ C₂ ⥤ D
inst✝² : F.Additive
inst✝¹ : ∀ (X₁ : C₁), (F.obj X₁).Additive
x y : ℤ
inst✝ : K₁.HasMapBifunctor K₂ F
⊢ (HomologicalComplex₂.total.map
(K₁.mapBifunctorHomologicalComplexShift₁Iso ((CategoryTheory.shiftFunctor (CochainComplex C₂ ℤ) y).obj K₂) F
x).hom
(ComplexShape.up ℤ) ≫
((((F.mapBifunctorHomologicalComplex (ComplexShape.up ℤ) (ComplexShape.up ℤ)).obj K₁).obj
((CategoryTheory.shiftFunctor (CochainComplex C₂ ℤ) y).obj K₂)).totalShift₁Iso
x).hom) ≫
(CategoryTheory.shiftFunctor (CochainComplex D ℤ) x).map
(HomologicalComplex₂.total.map (K₁.mapBifunctorHomologicalComplexShift₂Iso K₂ F y).hom (ComplexShape.up ℤ) ≫
((((F.mapBifunctorHomologicalComplex (ComplexShape.up ℤ) (ComplexShape.up ℤ)).obj K₁).obj K₂).totalShift₂Iso
y).hom) =
(x * y).negOnePow •
(HomologicalComplex₂.total.map
(((CategoryTheory.shiftFunctor (CochainComplex C₁ ℤ) x).obj K₁).mapBifunctorHomologicalComplexShift₂Iso K₂ F
y).hom
(ComplexShape.up ℤ) ≫
((((F.mapBifunctorHomologicalComplex (ComplexShape.up ℤ) (ComplexShape.up ℤ)).obj
((CategoryTheory.shiftFunctor (CochainComplex C₁ ℤ) x).obj K₁)).obj
K₂).totalShift₂Iso
y).hom) ≫
(CategoryTheory.shiftFunctor (CochainComplex D ℤ) y).map
(HomologicalComplex₂.total.map (K₁.mapBifunctorHomologicalComplexShift₁Iso K₂ F x).hom (ComplexShape.up ℤ) ≫
((((F.mapBifunctorHomologicalComplex (ComplexShape.up ℤ) (ComplexShape.up ℤ)).obj K₁).obj
K₂).totalShift₁Iso
x).hom) ≫
(shiftFunctorComm (CochainComplex D ℤ) x y).hom.app (K₁.mapBifunctor K₂ F)
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Algebra/Homology/BifunctorShift.lean
|
CochainComplex.mapBifunctorShift₁Iso_trans_mapBifunctorShift₂Iso
|
case w
C₁ : Type u_1
C₂ : Type u_2
D : Type u_3
inst✝⁸ : Category.{u_5, u_1} C₁
inst✝⁷ : Category.{u_6, u_2} C₂
inst✝⁶ : Category.{u_4, u_3} D
inst✝⁵ : Preadditive C₁
inst✝⁴ : Preadditive C₂
inst✝³ : Preadditive D
K₁ : CochainComplex C₁ ℤ
K₂ : CochainComplex C₂ ℤ
F : C₁ ⥤ C₂ ⥤ D
inst✝² : F.Additive
inst✝¹ : ∀ (X₁ : C₁), (F.obj X₁).Additive
x y : ℤ
inst✝ : K₁.HasMapBifunctor K₂ F
⊢ (HomologicalComplex₂.total.map
(K₁.mapBifunctorHomologicalComplexShift₁Iso ((CategoryTheory.shiftFunctor (CochainComplex C₂ ℤ) y).obj K₂) F
x).hom
(ComplexShape.up ℤ) ≫
((((F.mapBifunctorHomologicalComplex (ComplexShape.up ℤ) (ComplexShape.up ℤ)).obj K₁).obj
((CategoryTheory.shiftFunctor (CochainComplex C₂ ℤ) y).obj K₂)).totalShift₁Iso
x).hom) ≫
(CategoryTheory.shiftFunctor (CochainComplex D ℤ) x).map
(HomologicalComplex₂.total.map (K₁.mapBifunctorHomologicalComplexShift₂Iso K₂ F y).hom (ComplexShape.up ℤ) ≫
((((F.mapBifunctorHomologicalComplex (ComplexShape.up ℤ) (ComplexShape.up ℤ)).obj K₁).obj K₂).totalShift₂Iso
y).hom) =
(x * y).negOnePow •
(HomologicalComplex₂.total.map
(((CategoryTheory.shiftFunctor (CochainComplex C₁ ℤ) x).obj K₁).mapBifunctorHomologicalComplexShift₂Iso K₂ F
y).hom
(ComplexShape.up ℤ) ≫
((((F.mapBifunctorHomologicalComplex (ComplexShape.up ℤ) (ComplexShape.up ℤ)).obj
((CategoryTheory.shiftFunctor (CochainComplex C₁ ℤ) x).obj K₁)).obj
K₂).totalShift₂Iso
y).hom) ≫
(CategoryTheory.shiftFunctor (CochainComplex D ℤ) y).map
(HomologicalComplex₂.total.map (K₁.mapBifunctorHomologicalComplexShift₁Iso K₂ F x).hom (ComplexShape.up ℤ) ≫
((((F.mapBifunctorHomologicalComplex (ComplexShape.up ℤ) (ComplexShape.up ℤ)).obj K₁).obj
K₂).totalShift₁Iso
x).hom) ≫
(shiftFunctorComm (CochainComplex D ℤ) x y).hom.app (K₁.mapBifunctor K₂ F)
|
rw [<a>CategoryTheory.Functor.map_comp</a>, <a>CategoryTheory.Functor.map_comp</a>, <a>CategoryTheory.Category.assoc</a>, <a>CategoryTheory.Category.assoc</a>, <a>CategoryTheory.Category.assoc</a>, ← <a>HomologicalComplex₂.totalShift₁Iso_hom_naturality_assoc</a>, <a>HomologicalComplex₂.totalShift₁Iso_hom_totalShift₂Iso_hom</a>, ← <a>HomologicalComplex₂.totalShift₂Iso_hom_naturality_assoc</a>, <a>CategoryTheory.Linear.comp_units_smul</a>, <a>CategoryTheory.Linear.comp_units_smul</a>, <a>smul_left_cancel_iff</a>, ← <a>HomologicalComplex₂.total.map_comp_assoc</a>, ← <a>HomologicalComplex₂.total.map_comp_assoc</a>, ← <a>HomologicalComplex₂.total.map_comp_assoc</a>]
|
case w
C₁ : Type u_1
C₂ : Type u_2
D : Type u_3
inst✝⁸ : Category.{u_5, u_1} C₁
inst✝⁷ : Category.{u_6, u_2} C₂
inst✝⁶ : Category.{u_4, u_3} D
inst✝⁵ : Preadditive C₁
inst✝⁴ : Preadditive C₂
inst✝³ : Preadditive D
K₁ : CochainComplex C₁ ℤ
K₂ : CochainComplex C₂ ℤ
F : C₁ ⥤ C₂ ⥤ D
inst✝² : F.Additive
inst✝¹ : ∀ (X₁ : C₁), (F.obj X₁).Additive
x y : ℤ
inst✝ : K₁.HasMapBifunctor K₂ F
⊢ HomologicalComplex₂.total.map
(((K₁.mapBifunctorHomologicalComplexShift₁Iso ((CategoryTheory.shiftFunctor (CochainComplex C₂ ℤ) y).obj K₂) F
x).hom ≫
(HomologicalComplex₂.shiftFunctor₁ D x).map (K₁.mapBifunctorHomologicalComplexShift₂Iso K₂ F y).hom) ≫
(HomologicalComplex₂.shiftFunctor₁₂CommIso D x y).hom.app
(((F.mapBifunctorHomologicalComplex (ComplexShape.up ℤ) (ComplexShape.up ℤ)).obj K₁).obj K₂))
(ComplexShape.up ℤ) ≫
(((HomologicalComplex₂.shiftFunctor₁ D x).obj
(((F.mapBifunctorHomologicalComplex (ComplexShape.up ℤ) (ComplexShape.up ℤ)).obj K₁).obj
K₂)).totalShift₂Iso
y).hom ≫
(CategoryTheory.shiftFunctor (HomologicalComplex D (ComplexShape.up ℤ)) y).map
((((F.mapBifunctorHomologicalComplex (ComplexShape.up ℤ) (ComplexShape.up ℤ)).obj K₁).obj K₂).totalShift₁Iso
x).hom ≫
(shiftFunctorComm (CochainComplex D ℤ) x y).hom.app
((((F.mapBifunctorHomologicalComplex (ComplexShape.up ℤ) (ComplexShape.up ℤ)).obj K₁).obj K₂).total
(ComplexShape.up ℤ)) =
HomologicalComplex₂.total.map
((((CategoryTheory.shiftFunctor (CochainComplex C₁ ℤ) x).obj K₁).mapBifunctorHomologicalComplexShift₂Iso K₂ F
y).hom ≫
(HomologicalComplex₂.shiftFunctor₂ D y).map (K₁.mapBifunctorHomologicalComplexShift₁Iso K₂ F x).hom)
(ComplexShape.up ℤ) ≫
(((HomologicalComplex₂.shiftFunctor₁ D x).obj
(((F.mapBifunctorHomologicalComplex (ComplexShape.up ℤ) (ComplexShape.up ℤ)).obj K₁).obj
K₂)).totalShift₂Iso
y).hom ≫
(CategoryTheory.shiftFunctor (CochainComplex D ℤ) y).map
((((F.mapBifunctorHomologicalComplex (ComplexShape.up ℤ) (ComplexShape.up ℤ)).obj K₁).obj K₂).totalShift₁Iso
x).hom ≫
(shiftFunctorComm (CochainComplex D ℤ) x y).hom.app (K₁.mapBifunctor K₂ F)
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Algebra/Homology/BifunctorShift.lean
|
CochainComplex.mapBifunctorShift₁Iso_trans_mapBifunctorShift₂Iso
|
case w
C₁ : Type u_1
C₂ : Type u_2
D : Type u_3
inst✝⁸ : Category.{u_5, u_1} C₁
inst✝⁷ : Category.{u_6, u_2} C₂
inst✝⁶ : Category.{u_4, u_3} D
inst✝⁵ : Preadditive C₁
inst✝⁴ : Preadditive C₂
inst✝³ : Preadditive D
K₁ : CochainComplex C₁ ℤ
K₂ : CochainComplex C₂ ℤ
F : C₁ ⥤ C₂ ⥤ D
inst✝² : F.Additive
inst✝¹ : ∀ (X₁ : C₁), (F.obj X₁).Additive
x y : ℤ
inst✝ : K₁.HasMapBifunctor K₂ F
⊢ HomologicalComplex₂.total.map
(((K₁.mapBifunctorHomologicalComplexShift₁Iso ((CategoryTheory.shiftFunctor (CochainComplex C₂ ℤ) y).obj K₂) F
x).hom ≫
(HomologicalComplex₂.shiftFunctor₁ D x).map (K₁.mapBifunctorHomologicalComplexShift₂Iso K₂ F y).hom) ≫
(HomologicalComplex₂.shiftFunctor₁₂CommIso D x y).hom.app
(((F.mapBifunctorHomologicalComplex (ComplexShape.up ℤ) (ComplexShape.up ℤ)).obj K₁).obj K₂))
(ComplexShape.up ℤ) ≫
(((HomologicalComplex₂.shiftFunctor₁ D x).obj
(((F.mapBifunctorHomologicalComplex (ComplexShape.up ℤ) (ComplexShape.up ℤ)).obj K₁).obj
K₂)).totalShift₂Iso
y).hom ≫
(CategoryTheory.shiftFunctor (HomologicalComplex D (ComplexShape.up ℤ)) y).map
((((F.mapBifunctorHomologicalComplex (ComplexShape.up ℤ) (ComplexShape.up ℤ)).obj K₁).obj K₂).totalShift₁Iso
x).hom ≫
(shiftFunctorComm (CochainComplex D ℤ) x y).hom.app
((((F.mapBifunctorHomologicalComplex (ComplexShape.up ℤ) (ComplexShape.up ℤ)).obj K₁).obj K₂).total
(ComplexShape.up ℤ)) =
HomologicalComplex₂.total.map
((((CategoryTheory.shiftFunctor (CochainComplex C₁ ℤ) x).obj K₁).mapBifunctorHomologicalComplexShift₂Iso K₂ F
y).hom ≫
(HomologicalComplex₂.shiftFunctor₂ D y).map (K₁.mapBifunctorHomologicalComplexShift₁Iso K₂ F x).hom)
(ComplexShape.up ℤ) ≫
(((HomologicalComplex₂.shiftFunctor₁ D x).obj
(((F.mapBifunctorHomologicalComplex (ComplexShape.up ℤ) (ComplexShape.up ℤ)).obj K₁).obj
K₂)).totalShift₂Iso
y).hom ≫
(CategoryTheory.shiftFunctor (CochainComplex D ℤ) y).map
((((F.mapBifunctorHomologicalComplex (ComplexShape.up ℤ) (ComplexShape.up ℤ)).obj K₁).obj K₂).totalShift₁Iso
x).hom ≫
(shiftFunctorComm (CochainComplex D ℤ) x y).hom.app (K₁.mapBifunctor K₂ F)
|
congr 2
|
case w.e_a.e_φ
C₁ : Type u_1
C₂ : Type u_2
D : Type u_3
inst✝⁸ : Category.{u_5, u_1} C₁
inst✝⁷ : Category.{u_6, u_2} C₂
inst✝⁶ : Category.{u_4, u_3} D
inst✝⁵ : Preadditive C₁
inst✝⁴ : Preadditive C₂
inst✝³ : Preadditive D
K₁ : CochainComplex C₁ ℤ
K₂ : CochainComplex C₂ ℤ
F : C₁ ⥤ C₂ ⥤ D
inst✝² : F.Additive
inst✝¹ : ∀ (X₁ : C₁), (F.obj X₁).Additive
x y : ℤ
inst✝ : K₁.HasMapBifunctor K₂ F
⊢ ((K₁.mapBifunctorHomologicalComplexShift₁Iso ((CategoryTheory.shiftFunctor (CochainComplex C₂ ℤ) y).obj K₂) F x).hom ≫
(HomologicalComplex₂.shiftFunctor₁ D x).map (K₁.mapBifunctorHomologicalComplexShift₂Iso K₂ F y).hom) ≫
(HomologicalComplex₂.shiftFunctor₁₂CommIso D x y).hom.app
(((F.mapBifunctorHomologicalComplex (ComplexShape.up ℤ) (ComplexShape.up ℤ)).obj K₁).obj K₂) =
(((CategoryTheory.shiftFunctor (CochainComplex C₁ ℤ) x).obj K₁).mapBifunctorHomologicalComplexShift₂Iso K₂ F
y).hom ≫
(HomologicalComplex₂.shiftFunctor₂ D y).map (K₁.mapBifunctorHomologicalComplexShift₁Iso K₂ F x).hom
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Algebra/Homology/BifunctorShift.lean
|
CochainComplex.mapBifunctorShift₁Iso_trans_mapBifunctorShift₂Iso
|
case w.e_a.e_φ
C₁ : Type u_1
C₂ : Type u_2
D : Type u_3
inst✝⁸ : Category.{u_5, u_1} C₁
inst✝⁷ : Category.{u_6, u_2} C₂
inst✝⁶ : Category.{u_4, u_3} D
inst✝⁵ : Preadditive C₁
inst✝⁴ : Preadditive C₂
inst✝³ : Preadditive D
K₁ : CochainComplex C₁ ℤ
K₂ : CochainComplex C₂ ℤ
F : C₁ ⥤ C₂ ⥤ D
inst✝² : F.Additive
inst✝¹ : ∀ (X₁ : C₁), (F.obj X₁).Additive
x y : ℤ
inst✝ : K₁.HasMapBifunctor K₂ F
⊢ ((K₁.mapBifunctorHomologicalComplexShift₁Iso ((CategoryTheory.shiftFunctor (CochainComplex C₂ ℤ) y).obj K₂) F x).hom ≫
(HomologicalComplex₂.shiftFunctor₁ D x).map (K₁.mapBifunctorHomologicalComplexShift₂Iso K₂ F y).hom) ≫
(HomologicalComplex₂.shiftFunctor₁₂CommIso D x y).hom.app
(((F.mapBifunctorHomologicalComplex (ComplexShape.up ℤ) (ComplexShape.up ℤ)).obj K₁).obj K₂) =
(((CategoryTheory.shiftFunctor (CochainComplex C₁ ℤ) x).obj K₁).mapBifunctorHomologicalComplexShift₂Iso K₂ F
y).hom ≫
(HomologicalComplex₂.shiftFunctor₂ D y).map (K₁.mapBifunctorHomologicalComplexShift₁Iso K₂ F x).hom
|
ext a b
|
case w.e_a.e_φ.h.h
C₁ : Type u_1
C₂ : Type u_2
D : Type u_3
inst✝⁸ : Category.{u_5, u_1} C₁
inst✝⁷ : Category.{u_6, u_2} C₂
inst✝⁶ : Category.{u_4, u_3} D
inst✝⁵ : Preadditive C₁
inst✝⁴ : Preadditive C₂
inst✝³ : Preadditive D
K₁ : CochainComplex C₁ ℤ
K₂ : CochainComplex C₂ ℤ
F : C₁ ⥤ C₂ ⥤ D
inst✝² : F.Additive
inst✝¹ : ∀ (X₁ : C₁), (F.obj X₁).Additive
x y : ℤ
inst✝ : K₁.HasMapBifunctor K₂ F
a b : ℤ
⊢ ((((K₁.mapBifunctorHomologicalComplexShift₁Iso ((CategoryTheory.shiftFunctor (CochainComplex C₂ ℤ) y).obj K₂) F
x).hom ≫
(HomologicalComplex₂.shiftFunctor₁ D x).map (K₁.mapBifunctorHomologicalComplexShift₂Iso K₂ F y).hom) ≫
(HomologicalComplex₂.shiftFunctor₁₂CommIso D x y).hom.app
(((F.mapBifunctorHomologicalComplex (ComplexShape.up ℤ) (ComplexShape.up ℤ)).obj K₁).obj K₂)).f
a).f
b =
(((((CategoryTheory.shiftFunctor (CochainComplex C₁ ℤ) x).obj K₁).mapBifunctorHomologicalComplexShift₂Iso K₂ F
y).hom ≫
(HomologicalComplex₂.shiftFunctor₂ D y).map (K₁.mapBifunctorHomologicalComplexShift₁Iso K₂ F x).hom).f
a).f
b
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Algebra/Homology/BifunctorShift.lean
|
CochainComplex.mapBifunctorShift₁Iso_trans_mapBifunctorShift₂Iso
|
case w.e_a.e_φ.h.h
C₁ : Type u_1
C₂ : Type u_2
D : Type u_3
inst✝⁸ : Category.{u_5, u_1} C₁
inst✝⁷ : Category.{u_6, u_2} C₂
inst✝⁶ : Category.{u_4, u_3} D
inst✝⁵ : Preadditive C₁
inst✝⁴ : Preadditive C₂
inst✝³ : Preadditive D
K₁ : CochainComplex C₁ ℤ
K₂ : CochainComplex C₂ ℤ
F : C₁ ⥤ C₂ ⥤ D
inst✝² : F.Additive
inst✝¹ : ∀ (X₁ : C₁), (F.obj X₁).Additive
x y : ℤ
inst✝ : K₁.HasMapBifunctor K₂ F
a b : ℤ
⊢ ((((K₁.mapBifunctorHomologicalComplexShift₁Iso ((CategoryTheory.shiftFunctor (CochainComplex C₂ ℤ) y).obj K₂) F
x).hom ≫
(HomologicalComplex₂.shiftFunctor₁ D x).map (K₁.mapBifunctorHomologicalComplexShift₂Iso K₂ F y).hom) ≫
(HomologicalComplex₂.shiftFunctor₁₂CommIso D x y).hom.app
(((F.mapBifunctorHomologicalComplex (ComplexShape.up ℤ) (ComplexShape.up ℤ)).obj K₁).obj K₂)).f
a).f
b =
(((((CategoryTheory.shiftFunctor (CochainComplex C₁ ℤ) x).obj K₁).mapBifunctorHomologicalComplexShift₂Iso K₂ F
y).hom ≫
(HomologicalComplex₂.shiftFunctor₂ D y).map (K₁.mapBifunctorHomologicalComplexShift₁Iso K₂ F x).hom).f
a).f
b
|
dsimp [<a>HomologicalComplex₂.shiftFunctor₁₂CommIso</a>]
|
case w.e_a.e_φ.h.h
C₁ : Type u_1
C₂ : Type u_2
D : Type u_3
inst✝⁸ : Category.{u_5, u_1} C₁
inst✝⁷ : Category.{u_6, u_2} C₂
inst✝⁶ : Category.{u_4, u_3} D
inst✝⁵ : Preadditive C₁
inst✝⁴ : Preadditive C₂
inst✝³ : Preadditive D
K₁ : CochainComplex C₁ ℤ
K₂ : CochainComplex C₂ ℤ
F : C₁ ⥤ C₂ ⥤ D
inst✝² : F.Additive
inst✝¹ : ∀ (X₁ : C₁), (F.obj X₁).Additive
x y : ℤ
inst✝ : K₁.HasMapBifunctor K₂ F
a b : ℤ
⊢ (𝟙 ((F.obj (K₁.X (a + x))).obj (K₂.X (b + y))) ≫ 𝟙 ((F.obj (K₁.X (a + x))).obj (K₂.X (b + y)))) ≫
𝟙 ((F.obj (K₁.X (a + x))).obj (K₂.X (b + y))) =
𝟙 ((F.obj (K₁.X (a + x))).obj (K₂.X (b + y))) ≫ 𝟙 ((F.obj (K₁.X (a + x))).obj (K₂.X (b + y)))
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Algebra/Homology/BifunctorShift.lean
|
CochainComplex.mapBifunctorShift₁Iso_trans_mapBifunctorShift₂Iso
|
case w.e_a.e_φ.h.h
C₁ : Type u_1
C₂ : Type u_2
D : Type u_3
inst✝⁸ : Category.{u_5, u_1} C₁
inst✝⁷ : Category.{u_6, u_2} C₂
inst✝⁶ : Category.{u_4, u_3} D
inst✝⁵ : Preadditive C₁
inst✝⁴ : Preadditive C₂
inst✝³ : Preadditive D
K₁ : CochainComplex C₁ ℤ
K₂ : CochainComplex C₂ ℤ
F : C₁ ⥤ C₂ ⥤ D
inst✝² : F.Additive
inst✝¹ : ∀ (X₁ : C₁), (F.obj X₁).Additive
x y : ℤ
inst✝ : K₁.HasMapBifunctor K₂ F
a b : ℤ
⊢ (𝟙 ((F.obj (K₁.X (a + x))).obj (K₂.X (b + y))) ≫ 𝟙 ((F.obj (K₁.X (a + x))).obj (K₂.X (b + y)))) ≫
𝟙 ((F.obj (K₁.X (a + x))).obj (K₂.X (b + y))) =
𝟙 ((F.obj (K₁.X (a + x))).obj (K₂.X (b + y))) ≫ 𝟙 ((F.obj (K₁.X (a + x))).obj (K₂.X (b + y)))
|
simp only [<a>CategoryTheory.Category.id_comp</a>]
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Algebra/Homology/BifunctorShift.lean
|
DFinsupp.toMultiset_lt_toMultiset
|
α : Type u_1
β : α → Type u_2
inst✝ : DecidableEq α
f g : Π₀ (_a : α), ℕ
⊢ toMultiset f < toMultiset g ↔ f < g
|
simp_rw [← <a>Multiset.toDFinsupp_lt_toDFinsupp</a>, <a>DFinsupp.toMultiset_toDFinsupp</a>]
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Data/DFinsupp/Multiset.lean
|
Cardinal.toENat_strictMonoOn
|
⊢ StrictMonoOn (⇑toENat) (Iic ℵ₀)
|
simp only [← <a>Cardinal.range_ofENat</a>, <a>StrictMonoOn</a>, <a>Set.forall_mem_range</a>, <a>Cardinal.toENat_ofENat</a>, <a>Cardinal.ofENat_lt_ofENat</a>]
|
⊢ ∀ (i i_1 : ℕ∞), i < i_1 → i < i_1
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/SetTheory/Cardinal/ENat.lean
|
Cardinal.toENat_strictMonoOn
|
⊢ ∀ (i i_1 : ℕ∞), i < i_1 → i < i_1
|
exact fun _ _ ↦ <a>id</a>
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/SetTheory/Cardinal/ENat.lean
|
MeasureTheory.lintegral_iSup_directed
|
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
m : MeasurableSpace α
μ ν : Measure α
inst✝ : Countable β
f : β → α → ℝ≥0∞
hf : ∀ (b : β), AEMeasurable (f b) μ
h_directed : Directed (fun x x_1 => x ≤ x_1) f
⊢ ∫⁻ (a : α), ⨆ b, f b a ∂μ = ⨆ b, ∫⁻ (a : α), f b a ∂μ
|
simp_rw [← <a>iSup_apply</a>]
|
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
m : MeasurableSpace α
μ ν : Measure α
inst✝ : Countable β
f : β → α → ℝ≥0∞
hf : ∀ (b : β), AEMeasurable (f b) μ
h_directed : Directed (fun x x_1 => x ≤ x_1) f
⊢ ∫⁻ (a : α), (⨆ i, f i) a ∂μ = ⨆ b, ∫⁻ (a : α), f b a ∂μ
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/MeasureTheory/Integral/Lebesgue.lean
|
MeasureTheory.lintegral_iSup_directed
|
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
m : MeasurableSpace α
μ ν : Measure α
inst✝ : Countable β
f : β → α → ℝ≥0∞
hf : ∀ (b : β), AEMeasurable (f b) μ
h_directed : Directed (fun x x_1 => x ≤ x_1) f
⊢ ∫⁻ (a : α), (⨆ i, f i) a ∂μ = ⨆ b, ∫⁻ (a : α), f b a ∂μ
|
let p : α → (β → <a>ENNReal</a>) → Prop := fun x f' => <a>Directed</a> <a>LE.le</a> f'
|
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
m : MeasurableSpace α
μ ν : Measure α
inst✝ : Countable β
f : β → α → ℝ≥0∞
hf : ∀ (b : β), AEMeasurable (f b) μ
h_directed : Directed (fun x x_1 => x ≤ x_1) f
p : α → (β → ℝ≥0∞) → Prop := fun x f' => Directed LE.le f'
⊢ ∫⁻ (a : α), (⨆ i, f i) a ∂μ = ⨆ b, ∫⁻ (a : α), f b a ∂μ
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/MeasureTheory/Integral/Lebesgue.lean
|
MeasureTheory.lintegral_iSup_directed
|
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
m : MeasurableSpace α
μ ν : Measure α
inst✝ : Countable β
f : β → α → ℝ≥0∞
hf : ∀ (b : β), AEMeasurable (f b) μ
h_directed : Directed (fun x x_1 => x ≤ x_1) f
p : α → (β → ℝ≥0∞) → Prop := fun x f' => Directed LE.le f'
⊢ ∫⁻ (a : α), (⨆ i, f i) a ∂μ = ⨆ b, ∫⁻ (a : α), f b a ∂μ
|
have hp : ∀ᵐ x ∂μ, p x fun i => f i x := by filter_upwards [] with x i j obtain ⟨z, hz₁, hz₂⟩ := h_directed i j exact ⟨z, hz₁ x, hz₂ x⟩
|
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
m : MeasurableSpace α
μ ν : Measure α
inst✝ : Countable β
f : β → α → ℝ≥0∞
hf : ∀ (b : β), AEMeasurable (f b) μ
h_directed : Directed (fun x x_1 => x ≤ x_1) f
p : α → (β → ℝ≥0∞) → Prop := fun x f' => Directed LE.le f'
hp : ∀ᵐ (x : α) ∂μ, p x fun i => f i x
⊢ ∫⁻ (a : α), (⨆ i, f i) a ∂μ = ⨆ b, ∫⁻ (a : α), f b a ∂μ
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/MeasureTheory/Integral/Lebesgue.lean
|
MeasureTheory.lintegral_iSup_directed
|
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
m : MeasurableSpace α
μ ν : Measure α
inst✝ : Countable β
f : β → α → ℝ≥0∞
hf : ∀ (b : β), AEMeasurable (f b) μ
h_directed : Directed (fun x x_1 => x ≤ x_1) f
p : α → (β → ℝ≥0∞) → Prop := fun x f' => Directed LE.le f'
hp : ∀ᵐ (x : α) ∂μ, p x fun i => f i x
h_ae_seq_directed : Directed LE.le (aeSeq hf p)
⊢ ∫⁻ (a : α), (⨆ i, f i) a ∂μ = ⨆ b, ∫⁻ (a : α), f b a ∂μ
|
convert <a>MeasureTheory.lintegral_iSup_directed_of_measurable</a> (<a>aeSeq.measurable</a> hf p) h_ae_seq_directed using 1
|
case h.e'_2
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
m : MeasurableSpace α
μ ν : Measure α
inst✝ : Countable β
f : β → α → ℝ≥0∞
hf : ∀ (b : β), AEMeasurable (f b) μ
h_directed : Directed (fun x x_1 => x ≤ x_1) f
p : α → (β → ℝ≥0∞) → Prop := fun x f' => Directed LE.le f'
hp : ∀ᵐ (x : α) ∂μ, p x fun i => f i x
h_ae_seq_directed : Directed LE.le (aeSeq hf p)
⊢ ∫⁻ (a : α), (⨆ i, f i) a ∂μ = ∫⁻ (a : α), ⨆ b, aeSeq hf p b a ∂?m.323592
case h.e'_3
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
m : MeasurableSpace α
μ ν : Measure α
inst✝ : Countable β
f : β → α → ℝ≥0∞
hf : ∀ (b : β), AEMeasurable (f b) μ
h_directed : Directed (fun x x_1 => x ≤ x_1) f
p : α → (β → ℝ≥0∞) → Prop := fun x f' => Directed LE.le f'
hp : ∀ᵐ (x : α) ∂μ, p x fun i => f i x
h_ae_seq_directed : Directed LE.le (aeSeq hf p)
⊢ ⨆ b, ∫⁻ (a : α), f b a ∂μ = ⨆ b, ∫⁻ (a : α), aeSeq hf p b a ∂?m.323592
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
m : MeasurableSpace α
μ ν : Measure α
inst✝ : Countable β
f : β → α → ℝ≥0∞
hf : ∀ (b : β), AEMeasurable (f b) μ
h_directed : Directed (fun x x_1 => x ≤ x_1) f
p : α → (β → ℝ≥0∞) → Prop := fun x f' => Directed LE.le f'
hp : ∀ᵐ (x : α) ∂μ, p x fun i => f i x
h_ae_seq_directed : Directed LE.le (aeSeq hf p)
⊢ Measure α
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/MeasureTheory/Integral/Lebesgue.lean
|
MeasureTheory.lintegral_iSup_directed
|
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
m : MeasurableSpace α
μ ν : Measure α
inst✝ : Countable β
f : β → α → ℝ≥0∞
hf : ∀ (b : β), AEMeasurable (f b) μ
h_directed : Directed (fun x x_1 => x ≤ x_1) f
p : α → (β → ℝ≥0∞) → Prop := fun x f' => Directed LE.le f'
⊢ ∀ᵐ (x : α) ∂μ, p x fun i => f i x
|
filter_upwards [] with x i j
|
case h
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
m : MeasurableSpace α
μ ν : Measure α
inst✝ : Countable β
f : β → α → ℝ≥0∞
hf : ∀ (b : β), AEMeasurable (f b) μ
h_directed : Directed (fun x x_1 => x ≤ x_1) f
p : α → (β → ℝ≥0∞) → Prop := fun x f' => Directed LE.le f'
x : α
i j : β
⊢ ∃ z, (fun i => f i x) i ≤ (fun i => f i x) z ∧ (fun i => f i x) j ≤ (fun i => f i x) z
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/MeasureTheory/Integral/Lebesgue.lean
|
MeasureTheory.lintegral_iSup_directed
|
case h
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
m : MeasurableSpace α
μ ν : Measure α
inst✝ : Countable β
f : β → α → ℝ≥0∞
hf : ∀ (b : β), AEMeasurable (f b) μ
h_directed : Directed (fun x x_1 => x ≤ x_1) f
p : α → (β → ℝ≥0∞) → Prop := fun x f' => Directed LE.le f'
x : α
i j : β
⊢ ∃ z, (fun i => f i x) i ≤ (fun i => f i x) z ∧ (fun i => f i x) j ≤ (fun i => f i x) z
|
obtain ⟨z, hz₁, hz₂⟩ := h_directed i j
|
case h.intro.intro
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
m : MeasurableSpace α
μ ν : Measure α
inst✝ : Countable β
f : β → α → ℝ≥0∞
hf : ∀ (b : β), AEMeasurable (f b) μ
h_directed : Directed (fun x x_1 => x ≤ x_1) f
p : α → (β → ℝ≥0∞) → Prop := fun x f' => Directed LE.le f'
x : α
i j z : β
hz₁ : f i ≤ f z
hz₂ : f j ≤ f z
⊢ ∃ z, (fun i => f i x) i ≤ (fun i => f i x) z ∧ (fun i => f i x) j ≤ (fun i => f i x) z
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/MeasureTheory/Integral/Lebesgue.lean
|
MeasureTheory.lintegral_iSup_directed
|
case h.intro.intro
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
m : MeasurableSpace α
μ ν : Measure α
inst✝ : Countable β
f : β → α → ℝ≥0∞
hf : ∀ (b : β), AEMeasurable (f b) μ
h_directed : Directed (fun x x_1 => x ≤ x_1) f
p : α → (β → ℝ≥0∞) → Prop := fun x f' => Directed LE.le f'
x : α
i j z : β
hz₁ : f i ≤ f z
hz₂ : f j ≤ f z
⊢ ∃ z, (fun i => f i x) i ≤ (fun i => f i x) z ∧ (fun i => f i x) j ≤ (fun i => f i x) z
|
exact ⟨z, hz₁ x, hz₂ x⟩
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/MeasureTheory/Integral/Lebesgue.lean
|
MeasureTheory.lintegral_iSup_directed
|
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
m : MeasurableSpace α
μ ν : Measure α
inst✝ : Countable β
f : β → α → ℝ≥0∞
hf : ∀ (b : β), AEMeasurable (f b) μ
h_directed : Directed (fun x x_1 => x ≤ x_1) f
p : α → (β → ℝ≥0∞) → Prop := fun x f' => Directed LE.le f'
hp : ∀ᵐ (x : α) ∂μ, p x fun i => f i x
⊢ Directed LE.le (aeSeq hf p)
|
intro b₁ b₂
|
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
m : MeasurableSpace α
μ ν : Measure α
inst✝ : Countable β
f : β → α → ℝ≥0∞
hf : ∀ (b : β), AEMeasurable (f b) μ
h_directed : Directed (fun x x_1 => x ≤ x_1) f
p : α → (β → ℝ≥0∞) → Prop := fun x f' => Directed LE.le f'
hp : ∀ᵐ (x : α) ∂μ, p x fun i => f i x
b₁ b₂ : β
⊢ ∃ z, aeSeq hf p b₁ ≤ aeSeq hf p z ∧ aeSeq hf p b₂ ≤ aeSeq hf p z
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/MeasureTheory/Integral/Lebesgue.lean
|
MeasureTheory.lintegral_iSup_directed
|
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
m : MeasurableSpace α
μ ν : Measure α
inst✝ : Countable β
f : β → α → ℝ≥0∞
hf : ∀ (b : β), AEMeasurable (f b) μ
h_directed : Directed (fun x x_1 => x ≤ x_1) f
p : α → (β → ℝ≥0∞) → Prop := fun x f' => Directed LE.le f'
hp : ∀ᵐ (x : α) ∂μ, p x fun i => f i x
b₁ b₂ : β
⊢ ∃ z, aeSeq hf p b₁ ≤ aeSeq hf p z ∧ aeSeq hf p b₂ ≤ aeSeq hf p z
|
obtain ⟨z, hz₁, hz₂⟩ := h_directed b₁ b₂
|
case intro.intro
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
m : MeasurableSpace α
μ ν : Measure α
inst✝ : Countable β
f : β → α → ℝ≥0∞
hf : ∀ (b : β), AEMeasurable (f b) μ
h_directed : Directed (fun x x_1 => x ≤ x_1) f
p : α → (β → ℝ≥0∞) → Prop := fun x f' => Directed LE.le f'
hp : ∀ᵐ (x : α) ∂μ, p x fun i => f i x
b₁ b₂ z : β
hz₁ : f b₁ ≤ f z
hz₂ : f b₂ ≤ f z
⊢ ∃ z, aeSeq hf p b₁ ≤ aeSeq hf p z ∧ aeSeq hf p b₂ ≤ aeSeq hf p z
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/MeasureTheory/Integral/Lebesgue.lean
|
MeasureTheory.lintegral_iSup_directed
|
case intro.intro.refine_2
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
m : MeasurableSpace α
μ ν : Measure α
inst✝ : Countable β
f : β → α → ℝ≥0∞
hf : ∀ (b : β), AEMeasurable (f b) μ
h_directed : Directed (fun x x_1 => x ≤ x_1) f
p : α → (β → ℝ≥0∞) → Prop := fun x f' => Directed LE.le f'
hp : ∀ᵐ (x : α) ∂μ, p x fun i => f i x
b₁ b₂ z : β
hz₁ : f b₁ ≤ f z
hz₂ : f b₂ ≤ f z
⊢ aeSeq hf p b₂ ≤ aeSeq hf p z
|
intro x
|
case intro.intro.refine_2
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
m : MeasurableSpace α
μ ν : Measure α
inst✝ : Countable β
f : β → α → ℝ≥0∞
hf : ∀ (b : β), AEMeasurable (f b) μ
h_directed : Directed (fun x x_1 => x ≤ x_1) f
p : α → (β → ℝ≥0∞) → Prop := fun x f' => Directed LE.le f'
hp : ∀ᵐ (x : α) ∂μ, p x fun i => f i x
b₁ b₂ z : β
hz₁ : f b₁ ≤ f z
hz₂ : f b₂ ≤ f z
x : α
⊢ aeSeq hf p b₂ x ≤ aeSeq hf p z x
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/MeasureTheory/Integral/Lebesgue.lean
|
MeasureTheory.lintegral_iSup_directed
|
case intro.intro.refine_2
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
m : MeasurableSpace α
μ ν : Measure α
inst✝ : Countable β
f : β → α → ℝ≥0∞
hf : ∀ (b : β), AEMeasurable (f b) μ
h_directed : Directed (fun x x_1 => x ≤ x_1) f
p : α → (β → ℝ≥0∞) → Prop := fun x f' => Directed LE.le f'
hp : ∀ᵐ (x : α) ∂μ, p x fun i => f i x
b₁ b₂ z : β
hz₁ : f b₁ ≤ f z
hz₂ : f b₂ ≤ f z
x : α
⊢ aeSeq hf p b₂ x ≤ aeSeq hf p z x
|
by_cases hx : x ∈ <a>aeSeqSet</a> hf p
|
case pos
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
m : MeasurableSpace α
μ ν : Measure α
inst✝ : Countable β
f : β → α → ℝ≥0∞
hf : ∀ (b : β), AEMeasurable (f b) μ
h_directed : Directed (fun x x_1 => x ≤ x_1) f
p : α → (β → ℝ≥0∞) → Prop := fun x f' => Directed LE.le f'
hp : ∀ᵐ (x : α) ∂μ, p x fun i => f i x
b₁ b₂ z : β
hz₁ : f b₁ ≤ f z
hz₂ : f b₂ ≤ f z
x : α
hx : x ∈ aeSeqSet hf p
⊢ aeSeq hf p b₂ x ≤ aeSeq hf p z x
case neg
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
m : MeasurableSpace α
μ ν : Measure α
inst✝ : Countable β
f : β → α → ℝ≥0∞
hf : ∀ (b : β), AEMeasurable (f b) μ
h_directed : Directed (fun x x_1 => x ≤ x_1) f
p : α → (β → ℝ≥0∞) → Prop := fun x f' => Directed LE.le f'
hp : ∀ᵐ (x : α) ∂μ, p x fun i => f i x
b₁ b₂ z : β
hz₁ : f b₁ ≤ f z
hz₂ : f b₂ ≤ f z
x : α
hx : x ∉ aeSeqSet hf p
⊢ aeSeq hf p b₂ x ≤ aeSeq hf p z x
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/MeasureTheory/Integral/Lebesgue.lean
|
MeasureTheory.lintegral_iSup_directed
|
case pos
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
m : MeasurableSpace α
μ ν : Measure α
inst✝ : Countable β
f : β → α → ℝ≥0∞
hf : ∀ (b : β), AEMeasurable (f b) μ
h_directed : Directed (fun x x_1 => x ≤ x_1) f
p : α → (β → ℝ≥0∞) → Prop := fun x f' => Directed LE.le f'
hp : ∀ᵐ (x : α) ∂μ, p x fun i => f i x
b₁ b₂ z : β
hz₁ : f b₁ ≤ f z
hz₂ : f b₂ ≤ f z
x : α
hx : x ∈ aeSeqSet hf p
⊢ aeSeq hf p b₂ x ≤ aeSeq hf p z x
|
repeat rw [<a>aeSeq.aeSeq_eq_fun_of_mem_aeSeqSet</a> hf hx]
|
case pos
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
m : MeasurableSpace α
μ ν : Measure α
inst✝ : Countable β
f : β → α → ℝ≥0∞
hf : ∀ (b : β), AEMeasurable (f b) μ
h_directed : Directed (fun x x_1 => x ≤ x_1) f
p : α → (β → ℝ≥0∞) → Prop := fun x f' => Directed LE.le f'
hp : ∀ᵐ (x : α) ∂μ, p x fun i => f i x
b₁ b₂ z : β
hz₁ : f b₁ ≤ f z
hz₂ : f b₂ ≤ f z
x : α
hx : x ∈ aeSeqSet hf p
⊢ f b₂ x ≤ f z x
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/MeasureTheory/Integral/Lebesgue.lean
|
MeasureTheory.lintegral_iSup_directed
|
case pos
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
m : MeasurableSpace α
μ ν : Measure α
inst✝ : Countable β
f : β → α → ℝ≥0∞
hf : ∀ (b : β), AEMeasurable (f b) μ
h_directed : Directed (fun x x_1 => x ≤ x_1) f
p : α → (β → ℝ≥0∞) → Prop := fun x f' => Directed LE.le f'
hp : ∀ᵐ (x : α) ∂μ, p x fun i => f i x
b₁ b₂ z : β
hz₁ : f b₁ ≤ f z
hz₂ : f b₂ ≤ f z
x : α
hx : x ∈ aeSeqSet hf p
⊢ f b₂ x ≤ f z x
|
apply_rules [hz₁, hz₂]
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/MeasureTheory/Integral/Lebesgue.lean
|
MeasureTheory.lintegral_iSup_directed
|
case pos
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
m : MeasurableSpace α
μ ν : Measure α
inst✝ : Countable β
f : β → α → ℝ≥0∞
hf : ∀ (b : β), AEMeasurable (f b) μ
h_directed : Directed (fun x x_1 => x ≤ x_1) f
p : α → (β → ℝ≥0∞) → Prop := fun x f' => Directed LE.le f'
hp : ∀ᵐ (x : α) ∂μ, p x fun i => f i x
b₁ b₂ z : β
hz₁ : f b₁ ≤ f z
hz₂ : f b₂ ≤ f z
x : α
hx : x ∈ aeSeqSet hf p
⊢ f b₂ x ≤ aeSeq hf p z x
|
rw [<a>aeSeq.aeSeq_eq_fun_of_mem_aeSeqSet</a> hf hx]
|
case pos
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
m : MeasurableSpace α
μ ν : Measure α
inst✝ : Countable β
f : β → α → ℝ≥0∞
hf : ∀ (b : β), AEMeasurable (f b) μ
h_directed : Directed (fun x x_1 => x ≤ x_1) f
p : α → (β → ℝ≥0∞) → Prop := fun x f' => Directed LE.le f'
hp : ∀ᵐ (x : α) ∂μ, p x fun i => f i x
b₁ b₂ z : β
hz₁ : f b₁ ≤ f z
hz₂ : f b₂ ≤ f z
x : α
hx : x ∈ aeSeqSet hf p
⊢ f b₂ x ≤ f z x
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/MeasureTheory/Integral/Lebesgue.lean
|
MeasureTheory.lintegral_iSup_directed
|
case neg
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
m : MeasurableSpace α
μ ν : Measure α
inst✝ : Countable β
f : β → α → ℝ≥0∞
hf : ∀ (b : β), AEMeasurable (f b) μ
h_directed : Directed (fun x x_1 => x ≤ x_1) f
p : α → (β → ℝ≥0∞) → Prop := fun x f' => Directed LE.le f'
hp : ∀ᵐ (x : α) ∂μ, p x fun i => f i x
b₁ b₂ z : β
hz₁ : f b₁ ≤ f z
hz₂ : f b₂ ≤ f z
x : α
hx : x ∉ aeSeqSet hf p
⊢ aeSeq hf p b₂ x ≤ aeSeq hf p z x
|
simp only [<a>aeSeq</a>, hx, <a>if_false</a>]
|
case neg
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
m : MeasurableSpace α
μ ν : Measure α
inst✝ : Countable β
f : β → α → ℝ≥0∞
hf : ∀ (b : β), AEMeasurable (f b) μ
h_directed : Directed (fun x x_1 => x ≤ x_1) f
p : α → (β → ℝ≥0∞) → Prop := fun x f' => Directed LE.le f'
hp : ∀ᵐ (x : α) ∂μ, p x fun i => f i x
b₁ b₂ z : β
hz₁ : f b₁ ≤ f z
hz₂ : f b₂ ≤ f z
x : α
hx : x ∉ aeSeqSet hf p
⊢ ⋯.some ≤ ⋯.some
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/MeasureTheory/Integral/Lebesgue.lean
|
MeasureTheory.lintegral_iSup_directed
|
case neg
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
m : MeasurableSpace α
μ ν : Measure α
inst✝ : Countable β
f : β → α → ℝ≥0∞
hf : ∀ (b : β), AEMeasurable (f b) μ
h_directed : Directed (fun x x_1 => x ≤ x_1) f
p : α → (β → ℝ≥0∞) → Prop := fun x f' => Directed LE.le f'
hp : ∀ᵐ (x : α) ∂μ, p x fun i => f i x
b₁ b₂ z : β
hz₁ : f b₁ ≤ f z
hz₂ : f b₂ ≤ f z
x : α
hx : x ∉ aeSeqSet hf p
⊢ ⋯.some ≤ ⋯.some
|
exact <a>le_rfl</a>
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/MeasureTheory/Integral/Lebesgue.lean
|
MeasureTheory.lintegral_iSup_directed
|
case h.e'_2
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
m : MeasurableSpace α
μ ν : Measure α
inst✝ : Countable β
f : β → α → ℝ≥0∞
hf : ∀ (b : β), AEMeasurable (f b) μ
h_directed : Directed (fun x x_1 => x ≤ x_1) f
p : α → (β → ℝ≥0∞) → Prop := fun x f' => Directed LE.le f'
hp : ∀ᵐ (x : α) ∂μ, p x fun i => f i x
h_ae_seq_directed : Directed LE.le (aeSeq hf p)
⊢ ∫⁻ (a : α), (⨆ i, f i) a ∂μ = ∫⁻ (a : α), ⨆ b, aeSeq hf p b a ∂?m.323592
|
simp_rw [← <a>iSup_apply</a>]
|
case h.e'_2
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
m : MeasurableSpace α
μ ν : Measure α
inst✝ : Countable β
f : β → α → ℝ≥0∞
hf : ∀ (b : β), AEMeasurable (f b) μ
h_directed : Directed (fun x x_1 => x ≤ x_1) f
p : α → (β → ℝ≥0∞) → Prop := fun x f' => Directed LE.le f'
hp : ∀ᵐ (x : α) ∂μ, p x fun i => f i x
h_ae_seq_directed : Directed LE.le (aeSeq hf p)
⊢ ∫⁻ (a : α), (⨆ i, f i) a ∂μ = ∫⁻ (a : α), (⨆ i, aeSeq hf p i) a ∂?m.323592
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/MeasureTheory/Integral/Lebesgue.lean
|
MeasureTheory.lintegral_iSup_directed
|
case h.e'_2
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
m : MeasurableSpace α
μ ν : Measure α
inst✝ : Countable β
f : β → α → ℝ≥0∞
hf : ∀ (b : β), AEMeasurable (f b) μ
h_directed : Directed (fun x x_1 => x ≤ x_1) f
p : α → (β → ℝ≥0∞) → Prop := fun x f' => Directed LE.le f'
hp : ∀ᵐ (x : α) ∂μ, p x fun i => f i x
h_ae_seq_directed : Directed LE.le (aeSeq hf p)
⊢ ∫⁻ (a : α), (⨆ i, f i) a ∂μ = ∫⁻ (a : α), (⨆ i, aeSeq hf p i) a ∂?m.323592
|
rw [<a>MeasureTheory.lintegral_congr_ae</a> (<a>aeSeq.iSup</a> hf hp).<a>Filter.EventuallyEq.symm</a>]
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/MeasureTheory/Integral/Lebesgue.lean
|
MeasureTheory.lintegral_iSup_directed
|
case h.e'_3
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
m : MeasurableSpace α
μ ν : Measure α
inst✝ : Countable β
f : β → α → ℝ≥0∞
hf : ∀ (b : β), AEMeasurable (f b) μ
h_directed : Directed (fun x x_1 => x ≤ x_1) f
p : α → (β → ℝ≥0∞) → Prop := fun x f' => Directed LE.le f'
hp : ∀ᵐ (x : α) ∂μ, p x fun i => f i x
h_ae_seq_directed : Directed LE.le (aeSeq hf p)
⊢ ⨆ b, ∫⁻ (a : α), f b a ∂μ = ⨆ b, ∫⁻ (a : α), aeSeq hf p b a ∂μ
|
congr 1
|
case h.e'_3.e_s
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
m : MeasurableSpace α
μ ν : Measure α
inst✝ : Countable β
f : β → α → ℝ≥0∞
hf : ∀ (b : β), AEMeasurable (f b) μ
h_directed : Directed (fun x x_1 => x ≤ x_1) f
p : α → (β → ℝ≥0∞) → Prop := fun x f' => Directed LE.le f'
hp : ∀ᵐ (x : α) ∂μ, p x fun i => f i x
h_ae_seq_directed : Directed LE.le (aeSeq hf p)
⊢ (fun b => ∫⁻ (a : α), f b a ∂μ) = fun b => ∫⁻ (a : α), aeSeq hf p b a ∂μ
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/MeasureTheory/Integral/Lebesgue.lean
|
MeasureTheory.lintegral_iSup_directed
|
case h.e'_3.e_s
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
m : MeasurableSpace α
μ ν : Measure α
inst✝ : Countable β
f : β → α → ℝ≥0∞
hf : ∀ (b : β), AEMeasurable (f b) μ
h_directed : Directed (fun x x_1 => x ≤ x_1) f
p : α → (β → ℝ≥0∞) → Prop := fun x f' => Directed LE.le f'
hp : ∀ᵐ (x : α) ∂μ, p x fun i => f i x
h_ae_seq_directed : Directed LE.le (aeSeq hf p)
⊢ (fun b => ∫⁻ (a : α), f b a ∂μ) = fun b => ∫⁻ (a : α), aeSeq hf p b a ∂μ
|
ext1 b
|
case h.e'_3.e_s.h
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
m : MeasurableSpace α
μ ν : Measure α
inst✝ : Countable β
f : β → α → ℝ≥0∞
hf : ∀ (b : β), AEMeasurable (f b) μ
h_directed : Directed (fun x x_1 => x ≤ x_1) f
p : α → (β → ℝ≥0∞) → Prop := fun x f' => Directed LE.le f'
hp : ∀ᵐ (x : α) ∂μ, p x fun i => f i x
h_ae_seq_directed : Directed LE.le (aeSeq hf p)
b : β
⊢ ∫⁻ (a : α), f b a ∂μ = ∫⁻ (a : α), aeSeq hf p b a ∂μ
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/MeasureTheory/Integral/Lebesgue.lean
|
MeasureTheory.lintegral_iSup_directed
|
case h.e'_3.e_s.h
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
m : MeasurableSpace α
μ ν : Measure α
inst✝ : Countable β
f : β → α → ℝ≥0∞
hf : ∀ (b : β), AEMeasurable (f b) μ
h_directed : Directed (fun x x_1 => x ≤ x_1) f
p : α → (β → ℝ≥0∞) → Prop := fun x f' => Directed LE.le f'
hp : ∀ᵐ (x : α) ∂μ, p x fun i => f i x
h_ae_seq_directed : Directed LE.le (aeSeq hf p)
b : β
⊢ ∫⁻ (a : α), f b a ∂μ = ∫⁻ (a : α), aeSeq hf p b a ∂μ
|
rw [<a>MeasureTheory.lintegral_congr_ae</a>]
|
case h.e'_3.e_s.h
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
m : MeasurableSpace α
μ ν : Measure α
inst✝ : Countable β
f : β → α → ℝ≥0∞
hf : ∀ (b : β), AEMeasurable (f b) μ
h_directed : Directed (fun x x_1 => x ≤ x_1) f
p : α → (β → ℝ≥0∞) → Prop := fun x f' => Directed LE.le f'
hp : ∀ᵐ (x : α) ∂μ, p x fun i => f i x
h_ae_seq_directed : Directed LE.le (aeSeq hf p)
b : β
⊢ f b =ᶠ[ae μ] aeSeq hf p b
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/MeasureTheory/Integral/Lebesgue.lean
|
MeasureTheory.lintegral_iSup_directed
|
case h.e'_3.e_s.h
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
m : MeasurableSpace α
μ ν : Measure α
inst✝ : Countable β
f : β → α → ℝ≥0∞
hf : ∀ (b : β), AEMeasurable (f b) μ
h_directed : Directed (fun x x_1 => x ≤ x_1) f
p : α → (β → ℝ≥0∞) → Prop := fun x f' => Directed LE.le f'
hp : ∀ᵐ (x : α) ∂μ, p x fun i => f i x
h_ae_seq_directed : Directed LE.le (aeSeq hf p)
b : β
⊢ f b =ᶠ[ae μ] aeSeq hf p b
|
apply <a>Filter.EventuallyEq.symm</a>
|
case h.e'_3.e_s.h.H
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
m : MeasurableSpace α
μ ν : Measure α
inst✝ : Countable β
f : β → α → ℝ≥0∞
hf : ∀ (b : β), AEMeasurable (f b) μ
h_directed : Directed (fun x x_1 => x ≤ x_1) f
p : α → (β → ℝ≥0∞) → Prop := fun x f' => Directed LE.le f'
hp : ∀ᵐ (x : α) ∂μ, p x fun i => f i x
h_ae_seq_directed : Directed LE.le (aeSeq hf p)
b : β
⊢ aeSeq hf p b =ᶠ[ae μ] f b
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/MeasureTheory/Integral/Lebesgue.lean
|
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