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List.keys_kreplace | α : Type u
β : α → Type v
l l₁ l₂ : List (Sigma β)
inst✝ : DecidableEq α
a : α
b : β a
⊢ ∀ (a_1 b_1 : Sigma β), (b_1 ∈ if a = a_1.fst then some ⟨a, b⟩ else none) → a_1.fst = b_1.fst | rintro ⟨a₁, b₂⟩ ⟨a₂, b₂⟩ | case mk.mk
α : Type u
β : α → Type v
l l₁ l₂ : List (Sigma β)
inst✝ : DecidableEq α
a : α
b : β a
a₁ : α
b₂✝ : β a₁
a₂ : α
b₂ : β a₂
⊢ (⟨a₂, b₂⟩ ∈ if a = ⟨a₁, b₂✝⟩.fst then some ⟨a, b⟩ else none) → ⟨a₁, b₂✝⟩.fst = ⟨a₂, b₂⟩.fst | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/Data/List/Sigma.lean |
List.keys_kreplace | case mk.mk
α : Type u
β : α → Type v
l l₁ l₂ : List (Sigma β)
inst✝ : DecidableEq α
a : α
b : β a
a₁ : α
b₂✝ : β a₁
a₂ : α
b₂ : β a₂
⊢ (⟨a₂, b₂⟩ ∈ if a = ⟨a₁, b₂✝⟩.fst then some ⟨a, b⟩ else none) → ⟨a₁, b₂✝⟩.fst = ⟨a₂, b₂⟩.fst | dsimp | case mk.mk
α : Type u
β : α → Type v
l l₁ l₂ : List (Sigma β)
inst✝ : DecidableEq α
a : α
b : β a
a₁ : α
b₂✝ : β a₁
a₂ : α
b₂ : β a₂
⊢ (⟨a₂, b₂⟩ ∈ if a = a₁ then some ⟨a, b⟩ else none) → a₁ = a₂ | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/Data/List/Sigma.lean |
List.keys_kreplace | case mk.mk
α : Type u
β : α → Type v
l l₁ l₂ : List (Sigma β)
inst✝ : DecidableEq α
a : α
b : β a
a₁ : α
b₂✝ : β a₁
a₂ : α
b₂ : β a₂
⊢ (⟨a₂, b₂⟩ ∈ if a = a₁ then some ⟨a, b⟩ else none) → a₁ = a₂ | split_ifs with h <;> simp (config := { contextual := <a>Bool.true</a> }) [h] | no goals | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/Data/List/Sigma.lean |
ultrafilter_extend_extends | α : Type u
γ : Type u_1
inst✝¹ : TopologicalSpace γ
inst✝ : T2Space γ
f : α → γ
⊢ Ultrafilter.extend f ∘ pure = f | letI : <a>TopologicalSpace</a> α := ⊥ | α : Type u
γ : Type u_1
inst✝¹ : TopologicalSpace γ
inst✝ : T2Space γ
f : α → γ
this : TopologicalSpace α := ⊥
⊢ Ultrafilter.extend f ∘ pure = f | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/Topology/StoneCech.lean |
ultrafilter_extend_extends | α : Type u
γ : Type u_1
inst✝¹ : TopologicalSpace γ
inst✝ : T2Space γ
f : α → γ
this : TopologicalSpace α := ⊥
⊢ Ultrafilter.extend f ∘ pure = f | haveI : <a>DiscreteTopology</a> α := ⟨<a>rfl</a>⟩ | α : Type u
γ : Type u_1
inst✝¹ : TopologicalSpace γ
inst✝ : T2Space γ
f : α → γ
this✝ : TopologicalSpace α := ⊥
this : DiscreteTopology α
⊢ Ultrafilter.extend f ∘ pure = f | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/Topology/StoneCech.lean |
ultrafilter_extend_extends | α : Type u
γ : Type u_1
inst✝¹ : TopologicalSpace γ
inst✝ : T2Space γ
f : α → γ
this✝ : TopologicalSpace α := ⊥
this : DiscreteTopology α
⊢ Ultrafilter.extend f ∘ pure = f | exact <a>funext</a> (denseInducing_pure.extend_eq <a>continuous_of_discreteTopology</a>) | no goals | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/Topology/StoneCech.lean |
Monoid.PushoutI.induction_on | ι : Type u_1
G : ι → Type u_2
H : Type u_3
K : Type u_4
inst✝² : Monoid K
inst✝¹ : (i : ι) → Monoid (G i)
inst✝ : Monoid H
φ : (i : ι) → H →* G i
motive : PushoutI φ → Prop
x : PushoutI φ
of : ∀ (i : ι) (g : G i), motive ((Monoid.PushoutI.of i) g)
base : ∀ (h : H), motive ((Monoid.PushoutI.base φ) h)
mul : ∀ (x y : PushoutI φ), motive x → motive y → motive (x * y)
⊢ motive x | delta <a>Monoid.PushoutI</a> <a>Monoid.PushoutI.of</a> <a>Monoid.PushoutI.base</a> at * | ι : Type u_1
G : ι → Type u_2
H : Type u_3
K : Type u_4
inst✝² : Monoid K
inst✝¹ : (i : ι) → Monoid (G i)
inst✝ : Monoid H
φ : (i : ι) → H →* G i
motive : (con φ).Quotient → Prop
x : (con φ).Quotient
of : ∀ (i : ι) (g : G i), motive (((con φ).mk'.comp (inl.comp CoprodI.of)) g)
base : ∀ (h : H), motive (((con φ).mk'.comp inr) h)
mul : ∀ (x y : (con φ).Quotient), motive x → motive y → motive (x * y)
⊢ motive x | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/GroupTheory/PushoutI.lean |
Monoid.PushoutI.induction_on | ι : Type u_1
G : ι → Type u_2
H : Type u_3
K : Type u_4
inst✝² : Monoid K
inst✝¹ : (i : ι) → Monoid (G i)
inst✝ : Monoid H
φ : (i : ι) → H →* G i
motive : (con φ).Quotient → Prop
x : (con φ).Quotient
of : ∀ (i : ι) (g : G i), motive (((con φ).mk'.comp (inl.comp CoprodI.of)) g)
base : ∀ (h : H), motive (((con φ).mk'.comp inr) h)
mul : ∀ (x y : (con φ).Quotient), motive x → motive y → motive (x * y)
⊢ motive x | induction x using <a>Con.induction_on</a> with | H x => induction x using <a>Monoid.Coprod.induction_on</a> with | inl g => induction g using <a>Monoid.CoprodI.induction_on</a> with | h_of i g => exact of i g | h_mul x y ihx ihy => rw [<a>map_mul</a>] exact mul _ _ ihx ihy | h_one => simpa using base 1 | inr h => exact base h | mul x y ihx ihy => exact mul _ _ ihx ihy | no goals | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/GroupTheory/PushoutI.lean |
Monoid.PushoutI.induction_on | case H
ι : Type u_1
G : ι → Type u_2
H : Type u_3
K : Type u_4
inst✝² : Monoid K
inst✝¹ : (i : ι) → Monoid (G i)
inst✝ : Monoid H
φ : (i : ι) → H →* G i
motive : (con φ).Quotient → Prop
of : ∀ (i : ι) (g : G i), motive (((con φ).mk'.comp (inl.comp CoprodI.of)) g)
base : ∀ (h : H), motive (((con φ).mk'.comp inr) h)
mul : ∀ (x y : (con φ).Quotient), motive x → motive y → motive (x * y)
x : CoprodI G ∗ H
⊢ motive ↑x | induction x using <a>Monoid.Coprod.induction_on</a> with | inl g => induction g using <a>Monoid.CoprodI.induction_on</a> with | h_of i g => exact of i g | h_mul x y ihx ihy => rw [<a>map_mul</a>] exact mul _ _ ihx ihy | h_one => simpa using base 1 | inr h => exact base h | mul x y ihx ihy => exact mul _ _ ihx ihy | no goals | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/GroupTheory/PushoutI.lean |
Monoid.PushoutI.induction_on | case H.inl
ι : Type u_1
G : ι → Type u_2
H : Type u_3
K : Type u_4
inst✝² : Monoid K
inst✝¹ : (i : ι) → Monoid (G i)
inst✝ : Monoid H
φ : (i : ι) → H →* G i
motive : (con φ).Quotient → Prop
of : ∀ (i : ι) (g : G i), motive (((con φ).mk'.comp (inl.comp CoprodI.of)) g)
base : ∀ (h : H), motive (((con φ).mk'.comp inr) h)
mul : ∀ (x y : (con φ).Quotient), motive x → motive y → motive (x * y)
g : CoprodI G
⊢ motive ↑(inl g) | induction g using <a>Monoid.CoprodI.induction_on</a> with | h_of i g => exact of i g | h_mul x y ihx ihy => rw [<a>map_mul</a>] exact mul _ _ ihx ihy | h_one => simpa using base 1 | no goals | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/GroupTheory/PushoutI.lean |
Monoid.PushoutI.induction_on | case H.inl.h_of
ι : Type u_1
G : ι → Type u_2
H : Type u_3
K : Type u_4
inst✝² : Monoid K
inst✝¹ : (i : ι) → Monoid (G i)
inst✝ : Monoid H
φ : (i : ι) → H →* G i
motive : (con φ).Quotient → Prop
of : ∀ (i : ι) (g : G i), motive (((con φ).mk'.comp (inl.comp CoprodI.of)) g)
base : ∀ (h : H), motive (((con φ).mk'.comp inr) h)
mul : ∀ (x y : (con φ).Quotient), motive x → motive y → motive (x * y)
i : ι
g : G i
⊢ motive ↑(inl (CoprodI.of g)) | exact of i g | no goals | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/GroupTheory/PushoutI.lean |
Monoid.PushoutI.induction_on | case H.inl.h_mul
ι : Type u_1
G : ι → Type u_2
H : Type u_3
K : Type u_4
inst✝² : Monoid K
inst✝¹ : (i : ι) → Monoid (G i)
inst✝ : Monoid H
φ : (i : ι) → H →* G i
motive : (con φ).Quotient → Prop
of : ∀ (i : ι) (g : G i), motive (((con φ).mk'.comp (inl.comp CoprodI.of)) g)
base : ∀ (h : H), motive (((con φ).mk'.comp inr) h)
mul : ∀ (x y : (con φ).Quotient), motive x → motive y → motive (x * y)
x y : CoprodI G
ihx : motive ↑(inl x)
ihy : motive ↑(inl y)
⊢ motive ↑(inl (x * y)) | rw [<a>map_mul</a>] | case H.inl.h_mul
ι : Type u_1
G : ι → Type u_2
H : Type u_3
K : Type u_4
inst✝² : Monoid K
inst✝¹ : (i : ι) → Monoid (G i)
inst✝ : Monoid H
φ : (i : ι) → H →* G i
motive : (con φ).Quotient → Prop
of : ∀ (i : ι) (g : G i), motive (((con φ).mk'.comp (inl.comp CoprodI.of)) g)
base : ∀ (h : H), motive (((con φ).mk'.comp inr) h)
mul : ∀ (x y : (con φ).Quotient), motive x → motive y → motive (x * y)
x y : CoprodI G
ihx : motive ↑(inl x)
ihy : motive ↑(inl y)
⊢ motive ↑(inl x * inl y) | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/GroupTheory/PushoutI.lean |
Monoid.PushoutI.induction_on | case H.inl.h_mul
ι : Type u_1
G : ι → Type u_2
H : Type u_3
K : Type u_4
inst✝² : Monoid K
inst✝¹ : (i : ι) → Monoid (G i)
inst✝ : Monoid H
φ : (i : ι) → H →* G i
motive : (con φ).Quotient → Prop
of : ∀ (i : ι) (g : G i), motive (((con φ).mk'.comp (inl.comp CoprodI.of)) g)
base : ∀ (h : H), motive (((con φ).mk'.comp inr) h)
mul : ∀ (x y : (con φ).Quotient), motive x → motive y → motive (x * y)
x y : CoprodI G
ihx : motive ↑(inl x)
ihy : motive ↑(inl y)
⊢ motive ↑(inl x * inl y) | exact mul _ _ ihx ihy | no goals | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/GroupTheory/PushoutI.lean |
Monoid.PushoutI.induction_on | case H.inl.h_one
ι : Type u_1
G : ι → Type u_2
H : Type u_3
K : Type u_4
inst✝² : Monoid K
inst✝¹ : (i : ι) → Monoid (G i)
inst✝ : Monoid H
φ : (i : ι) → H →* G i
motive : (con φ).Quotient → Prop
of : ∀ (i : ι) (g : G i), motive (((con φ).mk'.comp (inl.comp CoprodI.of)) g)
base : ∀ (h : H), motive (((con φ).mk'.comp inr) h)
mul : ∀ (x y : (con φ).Quotient), motive x → motive y → motive (x * y)
⊢ motive ↑(inl 1) | simpa using base 1 | no goals | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/GroupTheory/PushoutI.lean |
Monoid.PushoutI.induction_on | case H.inr
ι : Type u_1
G : ι → Type u_2
H : Type u_3
K : Type u_4
inst✝² : Monoid K
inst✝¹ : (i : ι) → Monoid (G i)
inst✝ : Monoid H
φ : (i : ι) → H →* G i
motive : (con φ).Quotient → Prop
of : ∀ (i : ι) (g : G i), motive (((con φ).mk'.comp (inl.comp CoprodI.of)) g)
base : ∀ (h : H), motive (((con φ).mk'.comp inr) h)
mul : ∀ (x y : (con φ).Quotient), motive x → motive y → motive (x * y)
h : H
⊢ motive ↑(inr h) | exact base h | no goals | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/GroupTheory/PushoutI.lean |
Monoid.PushoutI.induction_on | case H.mul
ι : Type u_1
G : ι → Type u_2
H : Type u_3
K : Type u_4
inst✝² : Monoid K
inst✝¹ : (i : ι) → Monoid (G i)
inst✝ : Monoid H
φ : (i : ι) → H →* G i
motive : (con φ).Quotient → Prop
of : ∀ (i : ι) (g : G i), motive (((con φ).mk'.comp (inl.comp CoprodI.of)) g)
base : ∀ (h : H), motive (((con φ).mk'.comp inr) h)
mul : ∀ (x y : (con φ).Quotient), motive x → motive y → motive (x * y)
x y : CoprodI G ∗ H
ihx : motive ↑x
ihy : motive ↑y
⊢ motive ↑(x * y) | exact mul _ _ ihx ihy | no goals | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/GroupTheory/PushoutI.lean |
isProperMap_of_comp_of_t2 | X : Type u_1
Y : Type u_2
Z : Type u_3
W : Type u_4
ι : Type u_5
inst✝⁴ : TopologicalSpace X
inst✝³ : TopologicalSpace Y
inst✝² : TopologicalSpace Z
inst✝¹ : TopologicalSpace W
f : X → Y
g : Y → Z
inst✝ : T2Space Y
hf : Continuous f
hg : Continuous g
hgf : IsProperMap (g ∘ f)
⊢ IsProperMap f | rw [<a>isProperMap_iff_ultrafilter_of_t2</a>] | X : Type u_1
Y : Type u_2
Z : Type u_3
W : Type u_4
ι : Type u_5
inst✝⁴ : TopologicalSpace X
inst✝³ : TopologicalSpace Y
inst✝² : TopologicalSpace Z
inst✝¹ : TopologicalSpace W
f : X → Y
g : Y → Z
inst✝ : T2Space Y
hf : Continuous f
hg : Continuous g
hgf : IsProperMap (g ∘ f)
⊢ Continuous f ∧ ∀ ⦃𝒰 : Ultrafilter X⦄ ⦃y : Y⦄, Tendsto f (↑𝒰) (𝓝 y) → ∃ x, ↑𝒰 ≤ 𝓝 x | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/Topology/ProperMap.lean |
isProperMap_of_comp_of_t2 | X : Type u_1
Y : Type u_2
Z : Type u_3
W : Type u_4
ι : Type u_5
inst✝⁴ : TopologicalSpace X
inst✝³ : TopologicalSpace Y
inst✝² : TopologicalSpace Z
inst✝¹ : TopologicalSpace W
f : X → Y
g : Y → Z
inst✝ : T2Space Y
hf : Continuous f
hg : Continuous g
hgf : IsProperMap (g ∘ f)
⊢ Continuous f ∧ ∀ ⦃𝒰 : Ultrafilter X⦄ ⦃y : Y⦄, Tendsto f (↑𝒰) (𝓝 y) → ∃ x, ↑𝒰 ≤ 𝓝 x | refine ⟨hf, fun 𝒰 y h ↦ ?_⟩ | X : Type u_1
Y : Type u_2
Z : Type u_3
W : Type u_4
ι : Type u_5
inst✝⁴ : TopologicalSpace X
inst✝³ : TopologicalSpace Y
inst✝² : TopologicalSpace Z
inst✝¹ : TopologicalSpace W
f : X → Y
g : Y → Z
inst✝ : T2Space Y
hf : Continuous f
hg : Continuous g
hgf : IsProperMap (g ∘ f)
𝒰 : Ultrafilter X
y : Y
h : Tendsto f (↑𝒰) (𝓝 y)
⊢ ∃ x, ↑𝒰 ≤ 𝓝 x | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/Topology/ProperMap.lean |
isProperMap_of_comp_of_t2 | X : Type u_1
Y : Type u_2
Z : Type u_3
W : Type u_4
ι : Type u_5
inst✝⁴ : TopologicalSpace X
inst✝³ : TopologicalSpace Y
inst✝² : TopologicalSpace Z
inst✝¹ : TopologicalSpace W
f : X → Y
g : Y → Z
inst✝ : T2Space Y
hf : Continuous f
hg : Continuous g
hgf : IsProperMap (g ∘ f)
𝒰 : Ultrafilter X
y : Y
h : Tendsto f (↑𝒰) (𝓝 y)
⊢ ∃ x, ↑𝒰 ≤ 𝓝 x | rw [<a>isProperMap_iff_ultrafilter</a>] at hgf | X : Type u_1
Y : Type u_2
Z : Type u_3
W : Type u_4
ι : Type u_5
inst✝⁴ : TopologicalSpace X
inst✝³ : TopologicalSpace Y
inst✝² : TopologicalSpace Z
inst✝¹ : TopologicalSpace W
f : X → Y
g : Y → Z
inst✝ : T2Space Y
hf : Continuous f
hg : Continuous g
hgf : Continuous (g ∘ f) ∧ ∀ ⦃𝒰 : Ultrafilter X⦄ ⦃y : Z⦄, Tendsto (g ∘ f) (↑𝒰) (𝓝 y) → ∃ x, (g ∘ f) x = y ∧ ↑𝒰 ≤ 𝓝 x
𝒰 : Ultrafilter X
y : Y
h : Tendsto f (↑𝒰) (𝓝 y)
⊢ ∃ x, ↑𝒰 ≤ 𝓝 x | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/Topology/ProperMap.lean |
isProperMap_of_comp_of_t2 | X : Type u_1
Y : Type u_2
Z : Type u_3
W : Type u_4
ι : Type u_5
inst✝⁴ : TopologicalSpace X
inst✝³ : TopologicalSpace Y
inst✝² : TopologicalSpace Z
inst✝¹ : TopologicalSpace W
f : X → Y
g : Y → Z
inst✝ : T2Space Y
hf : Continuous f
hg : Continuous g
hgf : Continuous (g ∘ f) ∧ ∀ ⦃𝒰 : Ultrafilter X⦄ ⦃y : Z⦄, Tendsto (g ∘ f) (↑𝒰) (𝓝 y) → ∃ x, (g ∘ f) x = y ∧ ↑𝒰 ≤ 𝓝 x
𝒰 : Ultrafilter X
y : Y
h : Tendsto f (↑𝒰) (𝓝 y)
⊢ ∃ x, ↑𝒰 ≤ 𝓝 x | rcases hgf.2 ((hg.tendsto y).<a>Filter.Tendsto.comp</a> h) with ⟨x, -, hx⟩ | case intro.intro
X : Type u_1
Y : Type u_2
Z : Type u_3
W : Type u_4
ι : Type u_5
inst✝⁴ : TopologicalSpace X
inst✝³ : TopologicalSpace Y
inst✝² : TopologicalSpace Z
inst✝¹ : TopologicalSpace W
f : X → Y
g : Y → Z
inst✝ : T2Space Y
hf : Continuous f
hg : Continuous g
hgf : Continuous (g ∘ f) ∧ ∀ ⦃𝒰 : Ultrafilter X⦄ ⦃y : Z⦄, Tendsto (g ∘ f) (↑𝒰) (𝓝 y) → ∃ x, (g ∘ f) x = y ∧ ↑𝒰 ≤ 𝓝 x
𝒰 : Ultrafilter X
y : Y
h : Tendsto f (↑𝒰) (𝓝 y)
x : X
hx : ↑𝒰 ≤ 𝓝 x
⊢ ∃ x, ↑𝒰 ≤ 𝓝 x | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/Topology/ProperMap.lean |
isProperMap_of_comp_of_t2 | case intro.intro
X : Type u_1
Y : Type u_2
Z : Type u_3
W : Type u_4
ι : Type u_5
inst✝⁴ : TopologicalSpace X
inst✝³ : TopologicalSpace Y
inst✝² : TopologicalSpace Z
inst✝¹ : TopologicalSpace W
f : X → Y
g : Y → Z
inst✝ : T2Space Y
hf : Continuous f
hg : Continuous g
hgf : Continuous (g ∘ f) ∧ ∀ ⦃𝒰 : Ultrafilter X⦄ ⦃y : Z⦄, Tendsto (g ∘ f) (↑𝒰) (𝓝 y) → ∃ x, (g ∘ f) x = y ∧ ↑𝒰 ≤ 𝓝 x
𝒰 : Ultrafilter X
y : Y
h : Tendsto f (↑𝒰) (𝓝 y)
x : X
hx : ↑𝒰 ≤ 𝓝 x
⊢ ∃ x, ↑𝒰 ≤ 𝓝 x | exact ⟨x, hx⟩ | no goals | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/Topology/ProperMap.lean |
Nat.ofDigits_mod_eq_head! | n b : ℕ
l : List ℕ
⊢ ofDigits b l % b = l.head! % b | induction l <;> simp [<a>Nat.ofDigits</a>, <a>Int.ModEq</a>] | no goals | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/Data/Nat/Digits.lean |
untrop_sum_eq_sInf_image | R : Type u_1
S : Type u_2
inst✝ : ConditionallyCompleteLinearOrder R
s : Finset S
f : S → Tropical (WithTop R)
⊢ untrop (∑ i ∈ s, f i) = sInf (untrop ∘ f '' ↑s) | rcases s.eq_empty_or_nonempty with (rfl | h) | case inl
R : Type u_1
S : Type u_2
inst✝ : ConditionallyCompleteLinearOrder R
f : S → Tropical (WithTop R)
⊢ untrop (∑ i ∈ ∅, f i) = sInf (untrop ∘ f '' ↑∅)
case inr
R : Type u_1
S : Type u_2
inst✝ : ConditionallyCompleteLinearOrder R
s : Finset S
f : S → Tropical (WithTop R)
h : s.Nonempty
⊢ untrop (∑ i ∈ s, f i) = sInf (untrop ∘ f '' ↑s) | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/Algebra/Tropical/BigOperators.lean |
untrop_sum_eq_sInf_image | case inl
R : Type u_1
S : Type u_2
inst✝ : ConditionallyCompleteLinearOrder R
f : S → Tropical (WithTop R)
⊢ untrop (∑ i ∈ ∅, f i) = sInf (untrop ∘ f '' ↑∅) | simp only [<a>Set.image_empty</a>, <a>Finset.coe_empty</a>, <a>Finset.sum_empty</a>, <a>WithTop.sInf_empty</a>, <a>Tropical.untrop_zero</a>] | no goals | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/Algebra/Tropical/BigOperators.lean |
untrop_sum_eq_sInf_image | case inr
R : Type u_1
S : Type u_2
inst✝ : ConditionallyCompleteLinearOrder R
s : Finset S
f : S → Tropical (WithTop R)
h : s.Nonempty
⊢ untrop (∑ i ∈ s, f i) = sInf (untrop ∘ f '' ↑s) | rw [← <a>Finset.inf'_eq_csInf_image</a> _ h, <a>Finset.inf'_eq_inf</a>, <a>Finset.untrop_sum'</a>] | no goals | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/Algebra/Tropical/BigOperators.lean |
Ordinal.bsup_not_succ_of_ne_bsup | α : Type u_1
β : Type u_2
γ : Type u_3
r : α → α → Prop
s : β → β → Prop
t : γ → γ → Prop
o : Ordinal.{u}
f : (a : Ordinal.{u}) → a < o → Ordinal.{max u v}
hf : ∀ {i : Ordinal.{u}} (h : i < o), f i h ≠ o.bsup f
a : Ordinal.{max u v}
⊢ a < o.bsup f → succ a < o.bsup f | rw [← <a>Ordinal.sup_eq_bsup</a>] at * | α : Type u_1
β : Type u_2
γ : Type u_3
r : α → α → Prop
s : β → β → Prop
t : γ → γ → Prop
o : Ordinal.{u}
f : (a : Ordinal.{u}) → a < o → Ordinal.{max u v}
hf : ∀ {i : Ordinal.{u}} (h : i < o), f i h ≠ sup (o.familyOfBFamily f)
a : Ordinal.{max u v}
⊢ a < sup (o.familyOfBFamily f) → succ a < sup (o.familyOfBFamily f) | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/SetTheory/Ordinal/Arithmetic.lean |
Ordinal.bsup_not_succ_of_ne_bsup | α : Type u_1
β : Type u_2
γ : Type u_3
r : α → α → Prop
s : β → β → Prop
t : γ → γ → Prop
o : Ordinal.{u}
f : (a : Ordinal.{u}) → a < o → Ordinal.{max u v}
hf : ∀ {i : Ordinal.{u}} (h : i < o), f i h ≠ sup (o.familyOfBFamily f)
a : Ordinal.{max u v}
⊢ a < sup (o.familyOfBFamily f) → succ a < sup (o.familyOfBFamily f) | exact <a>Ordinal.sup_not_succ_of_ne_sup</a> fun i => hf _ | no goals | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/SetTheory/Ordinal/Arithmetic.lean |
Set.Finite.interior_sInter | X : Type u
Y : Type v
ι : Sort w
α : Type u_1
β : Type u_2
x : X
s s₁ s₂ t : Set X
p p₁ p₂ : X → Prop
inst✝ : TopologicalSpace X
S : Set (Set X)
hS : S.Finite
⊢ interior (⋂₀ S) = ⋂ s ∈ S, interior s | rw [<a>Set.sInter_eq_biInter</a>, hS.interior_biInter] | no goals | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/Topology/Basic.lean |
Path.delayReflRight_one | X : Type u
inst✝ : TopologicalSpace X
x y : X
γ : Path x y
⊢ delayReflRight 1 γ = γ | ext t | case a.h
X : Type u
inst✝ : TopologicalSpace X
x y : X
γ : Path x y
t : ↑I
⊢ (delayReflRight 1 γ) t = γ t | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/Topology/Homotopy/HSpaces.lean |
Path.delayReflRight_one | case a.h
X : Type u
inst✝ : TopologicalSpace X
x y : X
γ : Path x y
t : ↑I
⊢ (delayReflRight 1 γ) t = γ t | exact <a>congr_arg</a> γ (<a>unitInterval.qRight_one_right</a> t) | no goals | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/Topology/Homotopy/HSpaces.lean |
KummerDedekind.normalizedFactors_ideal_map_eq_normalizedFactors_min_poly_mk_map | R : Type u_1
S : Type u_2
inst✝⁶ : CommRing R
inst✝⁵ : CommRing S
inst✝⁴ : Algebra R S
x : S
I : Ideal R
inst✝³ : IsDomain R
inst✝² : IsIntegrallyClosed R
inst✝¹ : IsDedekindDomain S
inst✝ : NoZeroSMulDivisors R S
hI : I.IsMaximal
hI' : I ≠ ⊥
hx : comap (algebraMap R S) (conductor R x) ⊔ I = ⊤
hx' : _root_.IsIntegral R x
⊢ normalizedFactors (Ideal.map (algebraMap R S) I) =
Multiset.map (fun f => ↑((normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx').symm f))
(normalizedFactors (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))).attach | ext J | case a
R : Type u_1
S : Type u_2
inst✝⁶ : CommRing R
inst✝⁵ : CommRing S
inst✝⁴ : Algebra R S
x : S
I : Ideal R
inst✝³ : IsDomain R
inst✝² : IsIntegrallyClosed R
inst✝¹ : IsDedekindDomain S
inst✝ : NoZeroSMulDivisors R S
hI : I.IsMaximal
hI' : I ≠ ⊥
hx : comap (algebraMap R S) (conductor R x) ⊔ I = ⊤
hx' : _root_.IsIntegral R x
J : Ideal S
⊢ Multiset.count J (normalizedFactors (Ideal.map (algebraMap R S) I)) =
Multiset.count J
(Multiset.map (fun f => ↑((normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx').symm f))
(normalizedFactors (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))).attach) | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/NumberTheory/KummerDedekind.lean |
KummerDedekind.normalizedFactors_ideal_map_eq_normalizedFactors_min_poly_mk_map | case a
R : Type u_1
S : Type u_2
inst✝⁶ : CommRing R
inst✝⁵ : CommRing S
inst✝⁴ : Algebra R S
x : S
I : Ideal R
inst✝³ : IsDomain R
inst✝² : IsIntegrallyClosed R
inst✝¹ : IsDedekindDomain S
inst✝ : NoZeroSMulDivisors R S
hI : I.IsMaximal
hI' : I ≠ ⊥
hx : comap (algebraMap R S) (conductor R x) ⊔ I = ⊤
hx' : _root_.IsIntegral R x
J : Ideal S
⊢ Multiset.count J (normalizedFactors (Ideal.map (algebraMap R S) I)) =
Multiset.count J
(Multiset.map (fun f => ↑((normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx').symm f))
(normalizedFactors (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))).attach) | by_cases hJ : J ∈ <a>UniqueFactorizationMonoid.normalizedFactors</a> (I.map (<a>algebraMap</a> R S)) | case pos
R : Type u_1
S : Type u_2
inst✝⁶ : CommRing R
inst✝⁵ : CommRing S
inst✝⁴ : Algebra R S
x : S
I : Ideal R
inst✝³ : IsDomain R
inst✝² : IsIntegrallyClosed R
inst✝¹ : IsDedekindDomain S
inst✝ : NoZeroSMulDivisors R S
hI : I.IsMaximal
hI' : I ≠ ⊥
hx : comap (algebraMap R S) (conductor R x) ⊔ I = ⊤
hx' : _root_.IsIntegral R x
J : Ideal S
hJ : J ∈ normalizedFactors (Ideal.map (algebraMap R S) I)
⊢ Multiset.count J (normalizedFactors (Ideal.map (algebraMap R S) I)) =
Multiset.count J
(Multiset.map (fun f => ↑((normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx').symm f))
(normalizedFactors (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))).attach)
case neg
R : Type u_1
S : Type u_2
inst✝⁶ : CommRing R
inst✝⁵ : CommRing S
inst✝⁴ : Algebra R S
x : S
I : Ideal R
inst✝³ : IsDomain R
inst✝² : IsIntegrallyClosed R
inst✝¹ : IsDedekindDomain S
inst✝ : NoZeroSMulDivisors R S
hI : I.IsMaximal
hI' : I ≠ ⊥
hx : comap (algebraMap R S) (conductor R x) ⊔ I = ⊤
hx' : _root_.IsIntegral R x
J : Ideal S
hJ : J ∉ normalizedFactors (Ideal.map (algebraMap R S) I)
⊢ Multiset.count J (normalizedFactors (Ideal.map (algebraMap R S) I)) =
Multiset.count J
(Multiset.map (fun f => ↑((normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx').symm f))
(normalizedFactors (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))).attach) | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/NumberTheory/KummerDedekind.lean |
KummerDedekind.normalizedFactors_ideal_map_eq_normalizedFactors_min_poly_mk_map | case pos
R : Type u_1
S : Type u_2
inst✝⁶ : CommRing R
inst✝⁵ : CommRing S
inst✝⁴ : Algebra R S
x : S
I : Ideal R
inst✝³ : IsDomain R
inst✝² : IsIntegrallyClosed R
inst✝¹ : IsDedekindDomain S
inst✝ : NoZeroSMulDivisors R S
hI : I.IsMaximal
hI' : I ≠ ⊥
hx : comap (algebraMap R S) (conductor R x) ⊔ I = ⊤
hx' : _root_.IsIntegral R x
J : Ideal S
hJ : J ∈ normalizedFactors (Ideal.map (algebraMap R S) I)
⊢ Multiset.count J (normalizedFactors (Ideal.map (algebraMap R S) I)) =
Multiset.count J
(Multiset.map (fun f => ↑((normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx').symm f))
(normalizedFactors (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))).attach)
case neg
R : Type u_1
S : Type u_2
inst✝⁶ : CommRing R
inst✝⁵ : CommRing S
inst✝⁴ : Algebra R S
x : S
I : Ideal R
inst✝³ : IsDomain R
inst✝² : IsIntegrallyClosed R
inst✝¹ : IsDedekindDomain S
inst✝ : NoZeroSMulDivisors R S
hI : I.IsMaximal
hI' : I ≠ ⊥
hx : comap (algebraMap R S) (conductor R x) ⊔ I = ⊤
hx' : _root_.IsIntegral R x
J : Ideal S
hJ : J ∉ normalizedFactors (Ideal.map (algebraMap R S) I)
⊢ Multiset.count J (normalizedFactors (Ideal.map (algebraMap R S) I)) =
Multiset.count J
(Multiset.map (fun f => ↑((normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx').symm f))
(normalizedFactors (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))).attach) | swap | case neg
R : Type u_1
S : Type u_2
inst✝⁶ : CommRing R
inst✝⁵ : CommRing S
inst✝⁴ : Algebra R S
x : S
I : Ideal R
inst✝³ : IsDomain R
inst✝² : IsIntegrallyClosed R
inst✝¹ : IsDedekindDomain S
inst✝ : NoZeroSMulDivisors R S
hI : I.IsMaximal
hI' : I ≠ ⊥
hx : comap (algebraMap R S) (conductor R x) ⊔ I = ⊤
hx' : _root_.IsIntegral R x
J : Ideal S
hJ : J ∉ normalizedFactors (Ideal.map (algebraMap R S) I)
⊢ Multiset.count J (normalizedFactors (Ideal.map (algebraMap R S) I)) =
Multiset.count J
(Multiset.map (fun f => ↑((normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx').symm f))
(normalizedFactors (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))).attach)
case pos
R : Type u_1
S : Type u_2
inst✝⁶ : CommRing R
inst✝⁵ : CommRing S
inst✝⁴ : Algebra R S
x : S
I : Ideal R
inst✝³ : IsDomain R
inst✝² : IsIntegrallyClosed R
inst✝¹ : IsDedekindDomain S
inst✝ : NoZeroSMulDivisors R S
hI : I.IsMaximal
hI' : I ≠ ⊥
hx : comap (algebraMap R S) (conductor R x) ⊔ I = ⊤
hx' : _root_.IsIntegral R x
J : Ideal S
hJ : J ∈ normalizedFactors (Ideal.map (algebraMap R S) I)
⊢ Multiset.count J (normalizedFactors (Ideal.map (algebraMap R S) I)) =
Multiset.count J
(Multiset.map (fun f => ↑((normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx').symm f))
(normalizedFactors (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))).attach) | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/NumberTheory/KummerDedekind.lean |
KummerDedekind.normalizedFactors_ideal_map_eq_normalizedFactors_min_poly_mk_map | case pos
R : Type u_1
S : Type u_2
inst✝⁶ : CommRing R
inst✝⁵ : CommRing S
inst✝⁴ : Algebra R S
x : S
I : Ideal R
inst✝³ : IsDomain R
inst✝² : IsIntegrallyClosed R
inst✝¹ : IsDedekindDomain S
inst✝ : NoZeroSMulDivisors R S
hI : I.IsMaximal
hI' : I ≠ ⊥
hx : comap (algebraMap R S) (conductor R x) ⊔ I = ⊤
hx' : _root_.IsIntegral R x
J : Ideal S
hJ : J ∈ normalizedFactors (Ideal.map (algebraMap R S) I)
⊢ Multiset.count J (normalizedFactors (Ideal.map (algebraMap R S) I)) =
Multiset.count J
(Multiset.map (fun f => ↑((normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx').symm f))
(normalizedFactors (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))).attach) | have := <a>KummerDedekind.multiplicity_factors_map_eq_multiplicity</a> hI hI' hx hx' hJ | case pos
R : Type u_1
S : Type u_2
inst✝⁶ : CommRing R
inst✝⁵ : CommRing S
inst✝⁴ : Algebra R S
x : S
I : Ideal R
inst✝³ : IsDomain R
inst✝² : IsIntegrallyClosed R
inst✝¹ : IsDedekindDomain S
inst✝ : NoZeroSMulDivisors R S
hI : I.IsMaximal
hI' : I ≠ ⊥
hx : comap (algebraMap R S) (conductor R x) ⊔ I = ⊤
hx' : _root_.IsIntegral R x
J : Ideal S
hJ : J ∈ normalizedFactors (Ideal.map (algebraMap R S) I)
this :
multiplicity J (Ideal.map (algebraMap R S) I) =
multiplicity (↑((normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx') ⟨J, hJ⟩))
(Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))
⊢ Multiset.count J (normalizedFactors (Ideal.map (algebraMap R S) I)) =
Multiset.count J
(Multiset.map (fun f => ↑((normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx').symm f))
(normalizedFactors (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))).attach) | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/NumberTheory/KummerDedekind.lean |
KummerDedekind.normalizedFactors_ideal_map_eq_normalizedFactors_min_poly_mk_map | case pos
R : Type u_1
S : Type u_2
inst✝⁶ : CommRing R
inst✝⁵ : CommRing S
inst✝⁴ : Algebra R S
x : S
I : Ideal R
inst✝³ : IsDomain R
inst✝² : IsIntegrallyClosed R
inst✝¹ : IsDedekindDomain S
inst✝ : NoZeroSMulDivisors R S
hI : I.IsMaximal
hI' : I ≠ ⊥
hx : comap (algebraMap R S) (conductor R x) ⊔ I = ⊤
hx' : _root_.IsIntegral R x
J : Ideal S
hJ : J ∈ normalizedFactors (Ideal.map (algebraMap R S) I)
this :
multiplicity J (Ideal.map (algebraMap R S) I) =
multiplicity (↑((normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx') ⟨J, hJ⟩))
(Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))
⊢ Multiset.count J (normalizedFactors (Ideal.map (algebraMap R S) I)) =
Multiset.count J
(Multiset.map (fun f => ↑((normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx').symm f))
(normalizedFactors (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))).attach) | rw [<a>UniqueFactorizationMonoid.multiplicity_eq_count_normalizedFactors</a>, <a>UniqueFactorizationMonoid.multiplicity_eq_count_normalizedFactors</a>, <a>UniqueFactorizationMonoid.normalize_normalized_factor</a> _ hJ, <a>UniqueFactorizationMonoid.normalize_normalized_factor</a>, <a>PartENat.natCast_inj</a>] at this | case pos
R : Type u_1
S : Type u_2
inst✝⁶ : CommRing R
inst✝⁵ : CommRing S
inst✝⁴ : Algebra R S
x : S
I : Ideal R
inst✝³ : IsDomain R
inst✝² : IsIntegrallyClosed R
inst✝¹ : IsDedekindDomain S
inst✝ : NoZeroSMulDivisors R S
hI : I.IsMaximal
hI' : I ≠ ⊥
hx : comap (algebraMap R S) (conductor R x) ⊔ I = ⊤
hx' : _root_.IsIntegral R x
J : Ideal S
hJ : J ∈ normalizedFactors (Ideal.map (algebraMap R S) I)
this :
Multiset.count J (normalizedFactors (Ideal.map (algebraMap R S) I)) =
Multiset.count (↑((normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx') ⟨J, hJ⟩))
(normalizedFactors (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x)))
⊢ Multiset.count J (normalizedFactors (Ideal.map (algebraMap R S) I)) =
Multiset.count J
(Multiset.map (fun f => ↑((normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx').symm f))
(normalizedFactors (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))).attach)
case pos.a
R : Type u_1
S : Type u_2
inst✝⁶ : CommRing R
inst✝⁵ : CommRing S
inst✝⁴ : Algebra R S
x : S
I : Ideal R
inst✝³ : IsDomain R
inst✝² : IsIntegrallyClosed R
inst✝¹ : IsDedekindDomain S
inst✝ : NoZeroSMulDivisors R S
hI : I.IsMaximal
hI' : I ≠ ⊥
hx : comap (algebraMap R S) (conductor R x) ⊔ I = ⊤
hx' : _root_.IsIntegral R x
J : Ideal S
hJ : J ∈ normalizedFactors (Ideal.map (algebraMap R S) I)
this :
↑(Multiset.count J (normalizedFactors (Ideal.map (algebraMap R S) I))) =
↑(Multiset.count (normalize ↑((normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx') ⟨J, hJ⟩))
(normalizedFactors (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))))
⊢ ↑((normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx') ⟨J, hJ⟩) ∈ normalizedFactors ?m.128881
R : Type u_1
S : Type u_2
inst✝⁶ : CommRing R
inst✝⁵ : CommRing S
inst✝⁴ : Algebra R S
x : S
I : Ideal R
inst✝³ : IsDomain R
inst✝² : IsIntegrallyClosed R
inst✝¹ : IsDedekindDomain S
inst✝ : NoZeroSMulDivisors R S
hI : I.IsMaximal
hI' : I ≠ ⊥
hx : comap (algebraMap R S) (conductor R x) ⊔ I = ⊤
hx' : _root_.IsIntegral R x
J : Ideal S
hJ : J ∈ normalizedFactors (Ideal.map (algebraMap R S) I)
this :
↑(Multiset.count J (normalizedFactors (Ideal.map (algebraMap R S) I))) =
↑(Multiset.count (normalize ↑((normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx') ⟨J, hJ⟩))
(normalizedFactors (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))))
⊢ (R ⧸ I)[X]
case pos.ha
R : Type u_1
S : Type u_2
inst✝⁶ : CommRing R
inst✝⁵ : CommRing S
inst✝⁴ : Algebra R S
x : S
I : Ideal R
inst✝³ : IsDomain R
inst✝² : IsIntegrallyClosed R
inst✝¹ : IsDedekindDomain S
inst✝ : NoZeroSMulDivisors R S
hI : I.IsMaximal
hI' : I ≠ ⊥
hx : comap (algebraMap R S) (conductor R x) ⊔ I = ⊤
hx' : _root_.IsIntegral R x
J : Ideal S
hJ : J ∈ normalizedFactors (Ideal.map (algebraMap R S) I)
this :
↑(Multiset.count (normalize J) (normalizedFactors (Ideal.map (algebraMap R S) I))) =
multiplicity (↑((normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx') ⟨J, hJ⟩))
(Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))
⊢ Irreducible ↑((normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx') ⟨J, hJ⟩)
case pos.hb
R : Type u_1
S : Type u_2
inst✝⁶ : CommRing R
inst✝⁵ : CommRing S
inst✝⁴ : Algebra R S
x : S
I : Ideal R
inst✝³ : IsDomain R
inst✝² : IsIntegrallyClosed R
inst✝¹ : IsDedekindDomain S
inst✝ : NoZeroSMulDivisors R S
hI : I.IsMaximal
hI' : I ≠ ⊥
hx : comap (algebraMap R S) (conductor R x) ⊔ I = ⊤
hx' : _root_.IsIntegral R x
J : Ideal S
hJ : J ∈ normalizedFactors (Ideal.map (algebraMap R S) I)
this :
↑(Multiset.count (normalize J) (normalizedFactors (Ideal.map (algebraMap R S) I))) =
multiplicity (↑((normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx') ⟨J, hJ⟩))
(Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))
⊢ Polynomial.map (Ideal.Quotient.mk I) (minpoly R x) ≠ 0
case pos.ha
R : Type u_1
S : Type u_2
inst✝⁶ : CommRing R
inst✝⁵ : CommRing S
inst✝⁴ : Algebra R S
x : S
I : Ideal R
inst✝³ : IsDomain R
inst✝² : IsIntegrallyClosed R
inst✝¹ : IsDedekindDomain S
inst✝ : NoZeroSMulDivisors R S
hI : I.IsMaximal
hI' : I ≠ ⊥
hx : comap (algebraMap R S) (conductor R x) ⊔ I = ⊤
hx' : _root_.IsIntegral R x
J : Ideal S
hJ : J ∈ normalizedFactors (Ideal.map (algebraMap R S) I)
this :
multiplicity J (Ideal.map (algebraMap R S) I) =
multiplicity (↑((normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx') ⟨J, hJ⟩))
(Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))
⊢ Irreducible J
case pos.hb
R : Type u_1
S : Type u_2
inst✝⁶ : CommRing R
inst✝⁵ : CommRing S
inst✝⁴ : Algebra R S
x : S
I : Ideal R
inst✝³ : IsDomain R
inst✝² : IsIntegrallyClosed R
inst✝¹ : IsDedekindDomain S
inst✝ : NoZeroSMulDivisors R S
hI : I.IsMaximal
hI' : I ≠ ⊥
hx : comap (algebraMap R S) (conductor R x) ⊔ I = ⊤
hx' : _root_.IsIntegral R x
J : Ideal S
hJ : J ∈ normalizedFactors (Ideal.map (algebraMap R S) I)
this :
multiplicity J (Ideal.map (algebraMap R S) I) =
multiplicity (↑((normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx') ⟨J, hJ⟩))
(Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))
⊢ Ideal.map (algebraMap R S) I ≠ 0 | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/NumberTheory/KummerDedekind.lean |
KummerDedekind.normalizedFactors_ideal_map_eq_normalizedFactors_min_poly_mk_map | case neg
R : Type u_1
S : Type u_2
inst✝⁶ : CommRing R
inst✝⁵ : CommRing S
inst✝⁴ : Algebra R S
x : S
I : Ideal R
inst✝³ : IsDomain R
inst✝² : IsIntegrallyClosed R
inst✝¹ : IsDedekindDomain S
inst✝ : NoZeroSMulDivisors R S
hI : I.IsMaximal
hI' : I ≠ ⊥
hx : comap (algebraMap R S) (conductor R x) ⊔ I = ⊤
hx' : _root_.IsIntegral R x
J : Ideal S
hJ : J ∉ normalizedFactors (Ideal.map (algebraMap R S) I)
⊢ Multiset.count J (normalizedFactors (Ideal.map (algebraMap R S) I)) =
Multiset.count J
(Multiset.map (fun f => ↑((normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx').symm f))
(normalizedFactors (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))).attach) | rw [Multiset.count_eq_zero.mpr hJ, <a>eq_comm</a>, <a>Multiset.count_eq_zero</a>, <a>Multiset.mem_map</a>] | case neg
R : Type u_1
S : Type u_2
inst✝⁶ : CommRing R
inst✝⁵ : CommRing S
inst✝⁴ : Algebra R S
x : S
I : Ideal R
inst✝³ : IsDomain R
inst✝² : IsIntegrallyClosed R
inst✝¹ : IsDedekindDomain S
inst✝ : NoZeroSMulDivisors R S
hI : I.IsMaximal
hI' : I ≠ ⊥
hx : comap (algebraMap R S) (conductor R x) ⊔ I = ⊤
hx' : _root_.IsIntegral R x
J : Ideal S
hJ : J ∉ normalizedFactors (Ideal.map (algebraMap R S) I)
⊢ ¬∃ a ∈ (normalizedFactors (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))).attach,
↑((normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx').symm a) = J | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/NumberTheory/KummerDedekind.lean |
KummerDedekind.normalizedFactors_ideal_map_eq_normalizedFactors_min_poly_mk_map | case neg
R : Type u_1
S : Type u_2
inst✝⁶ : CommRing R
inst✝⁵ : CommRing S
inst✝⁴ : Algebra R S
x : S
I : Ideal R
inst✝³ : IsDomain R
inst✝² : IsIntegrallyClosed R
inst✝¹ : IsDedekindDomain S
inst✝ : NoZeroSMulDivisors R S
hI : I.IsMaximal
hI' : I ≠ ⊥
hx : comap (algebraMap R S) (conductor R x) ⊔ I = ⊤
hx' : _root_.IsIntegral R x
J : Ideal S
hJ : J ∉ normalizedFactors (Ideal.map (algebraMap R S) I)
⊢ ¬∃ a ∈ (normalizedFactors (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))).attach,
↑((normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx').symm a) = J | simp only [<a>Multiset.mem_attach</a>, <a>true_and_iff</a>, <a>not_exists</a>] | case neg
R : Type u_1
S : Type u_2
inst✝⁶ : CommRing R
inst✝⁵ : CommRing S
inst✝⁴ : Algebra R S
x : S
I : Ideal R
inst✝³ : IsDomain R
inst✝² : IsIntegrallyClosed R
inst✝¹ : IsDedekindDomain S
inst✝ : NoZeroSMulDivisors R S
hI : I.IsMaximal
hI' : I ≠ ⊥
hx : comap (algebraMap R S) (conductor R x) ⊔ I = ⊤
hx' : _root_.IsIntegral R x
J : Ideal S
hJ : J ∉ normalizedFactors (Ideal.map (algebraMap R S) I)
⊢ ∀ (x_1 : ↑{d | d ∈ normalizedFactors (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))}),
¬↑((normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx').symm x_1) = J | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/NumberTheory/KummerDedekind.lean |
KummerDedekind.normalizedFactors_ideal_map_eq_normalizedFactors_min_poly_mk_map | case neg
R : Type u_1
S : Type u_2
inst✝⁶ : CommRing R
inst✝⁵ : CommRing S
inst✝⁴ : Algebra R S
x : S
I : Ideal R
inst✝³ : IsDomain R
inst✝² : IsIntegrallyClosed R
inst✝¹ : IsDedekindDomain S
inst✝ : NoZeroSMulDivisors R S
hI : I.IsMaximal
hI' : I ≠ ⊥
hx : comap (algebraMap R S) (conductor R x) ⊔ I = ⊤
hx' : _root_.IsIntegral R x
J : Ideal S
hJ : J ∉ normalizedFactors (Ideal.map (algebraMap R S) I)
⊢ ∀ (x_1 : ↑{d | d ∈ normalizedFactors (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))}),
¬↑((normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx').symm x_1) = J | rintro J' rfl | case neg
R : Type u_1
S : Type u_2
inst✝⁶ : CommRing R
inst✝⁵ : CommRing S
inst✝⁴ : Algebra R S
x : S
I : Ideal R
inst✝³ : IsDomain R
inst✝² : IsIntegrallyClosed R
inst✝¹ : IsDedekindDomain S
inst✝ : NoZeroSMulDivisors R S
hI : I.IsMaximal
hI' : I ≠ ⊥
hx : comap (algebraMap R S) (conductor R x) ⊔ I = ⊤
hx' : _root_.IsIntegral R x
J' : ↑{d | d ∈ normalizedFactors (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))}
hJ :
↑((normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx').symm J') ∉
normalizedFactors (Ideal.map (algebraMap R S) I)
⊢ False | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/NumberTheory/KummerDedekind.lean |
KummerDedekind.normalizedFactors_ideal_map_eq_normalizedFactors_min_poly_mk_map | case neg
R : Type u_1
S : Type u_2
inst✝⁶ : CommRing R
inst✝⁵ : CommRing S
inst✝⁴ : Algebra R S
x : S
I : Ideal R
inst✝³ : IsDomain R
inst✝² : IsIntegrallyClosed R
inst✝¹ : IsDedekindDomain S
inst✝ : NoZeroSMulDivisors R S
hI : I.IsMaximal
hI' : I ≠ ⊥
hx : comap (algebraMap R S) (conductor R x) ⊔ I = ⊤
hx' : _root_.IsIntegral R x
J' : ↑{d | d ∈ normalizedFactors (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))}
hJ :
↑((normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx').symm J') ∉
normalizedFactors (Ideal.map (algebraMap R S) I)
⊢ False | exact hJ ((<a>KummerDedekind.normalizedFactorsMapEquivNormalizedFactorsMinPolyMk</a> hI hI' hx hx').<a>Equiv.symm</a> J').<a>Subtype.prop</a> | no goals | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/NumberTheory/KummerDedekind.lean |
KummerDedekind.normalizedFactors_ideal_map_eq_normalizedFactors_min_poly_mk_map | case pos
R : Type u_1
S : Type u_2
inst✝⁶ : CommRing R
inst✝⁵ : CommRing S
inst✝⁴ : Algebra R S
x : S
I : Ideal R
inst✝³ : IsDomain R
inst✝² : IsIntegrallyClosed R
inst✝¹ : IsDedekindDomain S
inst✝ : NoZeroSMulDivisors R S
hI : I.IsMaximal
hI' : I ≠ ⊥
hx : comap (algebraMap R S) (conductor R x) ⊔ I = ⊤
hx' : _root_.IsIntegral R x
J : Ideal S
hJ : J ∈ normalizedFactors (Ideal.map (algebraMap R S) I)
this :
Multiset.count J (normalizedFactors (Ideal.map (algebraMap R S) I)) =
Multiset.count (↑((normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx') ⟨J, hJ⟩))
(normalizedFactors (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x)))
⊢ Multiset.count J (normalizedFactors (Ideal.map (algebraMap R S) I)) =
Multiset.count J
(Multiset.map (fun f => ↑((normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx').symm f))
(normalizedFactors (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))).attach) | refine this.trans ?_ | case pos
R : Type u_1
S : Type u_2
inst✝⁶ : CommRing R
inst✝⁵ : CommRing S
inst✝⁴ : Algebra R S
x : S
I : Ideal R
inst✝³ : IsDomain R
inst✝² : IsIntegrallyClosed R
inst✝¹ : IsDedekindDomain S
inst✝ : NoZeroSMulDivisors R S
hI : I.IsMaximal
hI' : I ≠ ⊥
hx : comap (algebraMap R S) (conductor R x) ⊔ I = ⊤
hx' : _root_.IsIntegral R x
J : Ideal S
hJ : J ∈ normalizedFactors (Ideal.map (algebraMap R S) I)
this :
Multiset.count J (normalizedFactors (Ideal.map (algebraMap R S) I)) =
Multiset.count (↑((normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx') ⟨J, hJ⟩))
(normalizedFactors (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x)))
⊢ Multiset.count (↑((normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx') ⟨J, hJ⟩))
(normalizedFactors (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))) =
Multiset.count J
(Multiset.map (fun f => ↑((normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx').symm f))
(normalizedFactors (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))).attach) | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/NumberTheory/KummerDedekind.lean |
KummerDedekind.normalizedFactors_ideal_map_eq_normalizedFactors_min_poly_mk_map | case pos
R : Type u_1
S : Type u_2
inst✝⁶ : CommRing R
inst✝⁵ : CommRing S
inst✝⁴ : Algebra R S
x : S
I : Ideal R
inst✝³ : IsDomain R
inst✝² : IsIntegrallyClosed R
inst✝¹ : IsDedekindDomain S
inst✝ : NoZeroSMulDivisors R S
hI : I.IsMaximal
hI' : I ≠ ⊥
hx : comap (algebraMap R S) (conductor R x) ⊔ I = ⊤
hx' : _root_.IsIntegral R x
J : Ideal S
hJ : J ∈ normalizedFactors (Ideal.map (algebraMap R S) I)
this :
Multiset.count J (normalizedFactors (Ideal.map (algebraMap R S) I)) =
Multiset.count (↑((normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx') ⟨J, hJ⟩))
(normalizedFactors (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x)))
⊢ Multiset.count (↑((normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx') ⟨J, hJ⟩))
(normalizedFactors (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))) =
Multiset.count J
(Multiset.map (fun f => ↑((normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx').symm f))
(normalizedFactors (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))).attach) | generalize hJ' : (<a>KummerDedekind.normalizedFactorsMapEquivNormalizedFactorsMinPolyMk</a> hI hI' hx hx') ⟨J, hJ⟩ = J' | case pos
R : Type u_1
S : Type u_2
inst✝⁶ : CommRing R
inst✝⁵ : CommRing S
inst✝⁴ : Algebra R S
x : S
I : Ideal R
inst✝³ : IsDomain R
inst✝² : IsIntegrallyClosed R
inst✝¹ : IsDedekindDomain S
inst✝ : NoZeroSMulDivisors R S
hI : I.IsMaximal
hI' : I ≠ ⊥
hx : comap (algebraMap R S) (conductor R x) ⊔ I = ⊤
hx' : _root_.IsIntegral R x
J : Ideal S
hJ : J ∈ normalizedFactors (Ideal.map (algebraMap R S) I)
this :
Multiset.count J (normalizedFactors (Ideal.map (algebraMap R S) I)) =
Multiset.count (↑((normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx') ⟨J, hJ⟩))
(normalizedFactors (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x)))
J' : ↑{d | d ∈ normalizedFactors (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))}
hJ' : (normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx') ⟨J, hJ⟩ = J'
⊢ Multiset.count (↑J') (normalizedFactors (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))) =
Multiset.count J
(Multiset.map (fun f => ↑((normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx').symm f))
(normalizedFactors (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))).attach) | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/NumberTheory/KummerDedekind.lean |
KummerDedekind.normalizedFactors_ideal_map_eq_normalizedFactors_min_poly_mk_map | case pos
R : Type u_1
S : Type u_2
inst✝⁶ : CommRing R
inst✝⁵ : CommRing S
inst✝⁴ : Algebra R S
x : S
I : Ideal R
inst✝³ : IsDomain R
inst✝² : IsIntegrallyClosed R
inst✝¹ : IsDedekindDomain S
inst✝ : NoZeroSMulDivisors R S
hI : I.IsMaximal
hI' : I ≠ ⊥
hx : comap (algebraMap R S) (conductor R x) ⊔ I = ⊤
hx' : _root_.IsIntegral R x
J : Ideal S
hJ : J ∈ normalizedFactors (Ideal.map (algebraMap R S) I)
this :
Multiset.count J (normalizedFactors (Ideal.map (algebraMap R S) I)) =
Multiset.count (↑((normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx') ⟨J, hJ⟩))
(normalizedFactors (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x)))
J' : ↑{d | d ∈ normalizedFactors (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))}
hJ' : (normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx') ⟨J, hJ⟩ = J'
⊢ Multiset.count (↑J') (normalizedFactors (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))) =
Multiset.count J
(Multiset.map (fun f => ↑((normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx').symm f))
(normalizedFactors (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))).attach) | have : ((<a>KummerDedekind.normalizedFactorsMapEquivNormalizedFactorsMinPolyMk</a> hI hI' hx hx').<a>Equiv.symm</a> J' : <a>Ideal</a> S) = J := by rw [← hJ', <a>Equiv.symm_apply_apply</a> _ _, <a>Subtype.coe_mk</a>] | case pos
R : Type u_1
S : Type u_2
inst✝⁶ : CommRing R
inst✝⁵ : CommRing S
inst✝⁴ : Algebra R S
x : S
I : Ideal R
inst✝³ : IsDomain R
inst✝² : IsIntegrallyClosed R
inst✝¹ : IsDedekindDomain S
inst✝ : NoZeroSMulDivisors R S
hI : I.IsMaximal
hI' : I ≠ ⊥
hx : comap (algebraMap R S) (conductor R x) ⊔ I = ⊤
hx' : _root_.IsIntegral R x
J : Ideal S
hJ : J ∈ normalizedFactors (Ideal.map (algebraMap R S) I)
this✝ :
Multiset.count J (normalizedFactors (Ideal.map (algebraMap R S) I)) =
Multiset.count (↑((normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx') ⟨J, hJ⟩))
(normalizedFactors (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x)))
J' : ↑{d | d ∈ normalizedFactors (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))}
hJ' : (normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx') ⟨J, hJ⟩ = J'
this : ↑((normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx').symm J') = J
⊢ Multiset.count (↑J') (normalizedFactors (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))) =
Multiset.count J
(Multiset.map (fun f => ↑((normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx').symm f))
(normalizedFactors (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))).attach) | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/NumberTheory/KummerDedekind.lean |
KummerDedekind.normalizedFactors_ideal_map_eq_normalizedFactors_min_poly_mk_map | case pos
R : Type u_1
S : Type u_2
inst✝⁶ : CommRing R
inst✝⁵ : CommRing S
inst✝⁴ : Algebra R S
x : S
I : Ideal R
inst✝³ : IsDomain R
inst✝² : IsIntegrallyClosed R
inst✝¹ : IsDedekindDomain S
inst✝ : NoZeroSMulDivisors R S
hI : I.IsMaximal
hI' : I ≠ ⊥
hx : comap (algebraMap R S) (conductor R x) ⊔ I = ⊤
hx' : _root_.IsIntegral R x
J : Ideal S
hJ : J ∈ normalizedFactors (Ideal.map (algebraMap R S) I)
this✝ :
Multiset.count J (normalizedFactors (Ideal.map (algebraMap R S) I)) =
Multiset.count (↑((normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx') ⟨J, hJ⟩))
(normalizedFactors (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x)))
J' : ↑{d | d ∈ normalizedFactors (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))}
hJ' : (normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx') ⟨J, hJ⟩ = J'
this : ↑((normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx').symm J') = J
⊢ Multiset.count (↑J') (normalizedFactors (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))) =
Multiset.count J
(Multiset.map (fun f => ↑((normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx').symm f))
(normalizedFactors (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))).attach) | subst this | case pos
R : Type u_1
S : Type u_2
inst✝⁶ : CommRing R
inst✝⁵ : CommRing S
inst✝⁴ : Algebra R S
x : S
I : Ideal R
inst✝³ : IsDomain R
inst✝² : IsIntegrallyClosed R
inst✝¹ : IsDedekindDomain S
inst✝ : NoZeroSMulDivisors R S
hI : I.IsMaximal
hI' : I ≠ ⊥
hx : comap (algebraMap R S) (conductor R x) ⊔ I = ⊤
hx' : _root_.IsIntegral R x
J' : ↑{d | d ∈ normalizedFactors (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))}
hJ :
↑((normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx').symm J') ∈
normalizedFactors (Ideal.map (algebraMap R S) I)
this :
Multiset.count (↑((normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx').symm J'))
(normalizedFactors (Ideal.map (algebraMap R S) I)) =
Multiset.count
(↑((normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx')
⟨↑((normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx').symm J'), hJ⟩))
(normalizedFactors (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x)))
hJ' :
(normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx')
⟨↑((normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx').symm J'), hJ⟩ =
J'
⊢ Multiset.count (↑J') (normalizedFactors (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))) =
Multiset.count (↑((normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx').symm J'))
(Multiset.map (fun f => ↑((normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx').symm f))
(normalizedFactors (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))).attach) | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/NumberTheory/KummerDedekind.lean |
KummerDedekind.normalizedFactors_ideal_map_eq_normalizedFactors_min_poly_mk_map | case pos
R : Type u_1
S : Type u_2
inst✝⁶ : CommRing R
inst✝⁵ : CommRing S
inst✝⁴ : Algebra R S
x : S
I : Ideal R
inst✝³ : IsDomain R
inst✝² : IsIntegrallyClosed R
inst✝¹ : IsDedekindDomain S
inst✝ : NoZeroSMulDivisors R S
hI : I.IsMaximal
hI' : I ≠ ⊥
hx : comap (algebraMap R S) (conductor R x) ⊔ I = ⊤
hx' : _root_.IsIntegral R x
J' : ↑{d | d ∈ normalizedFactors (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))}
hJ :
↑((normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx').symm J') ∈
normalizedFactors (Ideal.map (algebraMap R S) I)
this :
Multiset.count (↑((normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx').symm J'))
(normalizedFactors (Ideal.map (algebraMap R S) I)) =
Multiset.count
(↑((normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx')
⟨↑((normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx').symm J'), hJ⟩))
(normalizedFactors (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x)))
hJ' :
(normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx')
⟨↑((normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx').symm J'), hJ⟩ =
J'
⊢ Multiset.count (↑J') (normalizedFactors (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))) =
Multiset.count (↑((normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx').symm J'))
(Multiset.map (fun f => ↑((normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx').symm f))
(normalizedFactors (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))).attach) | rw [<a>Multiset.count_map_eq_count'</a> fun f => ((<a>KummerDedekind.normalizedFactorsMapEquivNormalizedFactorsMinPolyMk</a> hI hI' hx hx').<a>Equiv.symm</a> f : <a>Ideal</a> S), <a>Multiset.count_attach</a>] | case pos.hf
R : Type u_1
S : Type u_2
inst✝⁶ : CommRing R
inst✝⁵ : CommRing S
inst✝⁴ : Algebra R S
x : S
I : Ideal R
inst✝³ : IsDomain R
inst✝² : IsIntegrallyClosed R
inst✝¹ : IsDedekindDomain S
inst✝ : NoZeroSMulDivisors R S
hI : I.IsMaximal
hI' : I ≠ ⊥
hx : comap (algebraMap R S) (conductor R x) ⊔ I = ⊤
hx' : _root_.IsIntegral R x
J' : ↑{d | d ∈ normalizedFactors (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))}
hJ :
↑((normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx').symm J') ∈
normalizedFactors (Ideal.map (algebraMap R S) I)
this :
Multiset.count (↑((normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx').symm J'))
(normalizedFactors (Ideal.map (algebraMap R S) I)) =
Multiset.count
(↑((normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx')
⟨↑((normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx').symm J'), hJ⟩))
(normalizedFactors (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x)))
hJ' :
(normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx')
⟨↑((normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx').symm J'), hJ⟩ =
J'
⊢ Function.Injective fun f => ↑((normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx').symm f) | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/NumberTheory/KummerDedekind.lean |
KummerDedekind.normalizedFactors_ideal_map_eq_normalizedFactors_min_poly_mk_map | R : Type u_1
S : Type u_2
inst✝⁶ : CommRing R
inst✝⁵ : CommRing S
inst✝⁴ : Algebra R S
x : S
I : Ideal R
inst✝³ : IsDomain R
inst✝² : IsIntegrallyClosed R
inst✝¹ : IsDedekindDomain S
inst✝ : NoZeroSMulDivisors R S
hI : I.IsMaximal
hI' : I ≠ ⊥
hx : comap (algebraMap R S) (conductor R x) ⊔ I = ⊤
hx' : _root_.IsIntegral R x
J : Ideal S
hJ : J ∈ normalizedFactors (Ideal.map (algebraMap R S) I)
this :
Multiset.count J (normalizedFactors (Ideal.map (algebraMap R S) I)) =
Multiset.count (↑((normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx') ⟨J, hJ⟩))
(normalizedFactors (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x)))
J' : ↑{d | d ∈ normalizedFactors (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))}
hJ' : (normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx') ⟨J, hJ⟩ = J'
⊢ ↑((normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx').symm J') = J | rw [← hJ', <a>Equiv.symm_apply_apply</a> _ _, <a>Subtype.coe_mk</a>] | no goals | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/NumberTheory/KummerDedekind.lean |
KummerDedekind.normalizedFactors_ideal_map_eq_normalizedFactors_min_poly_mk_map | case pos.hf
R : Type u_1
S : Type u_2
inst✝⁶ : CommRing R
inst✝⁵ : CommRing S
inst✝⁴ : Algebra R S
x : S
I : Ideal R
inst✝³ : IsDomain R
inst✝² : IsIntegrallyClosed R
inst✝¹ : IsDedekindDomain S
inst✝ : NoZeroSMulDivisors R S
hI : I.IsMaximal
hI' : I ≠ ⊥
hx : comap (algebraMap R S) (conductor R x) ⊔ I = ⊤
hx' : _root_.IsIntegral R x
J' : ↑{d | d ∈ normalizedFactors (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))}
hJ :
↑((normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx').symm J') ∈
normalizedFactors (Ideal.map (algebraMap R S) I)
this :
Multiset.count (↑((normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx').symm J'))
(normalizedFactors (Ideal.map (algebraMap R S) I)) =
Multiset.count
(↑((normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx')
⟨↑((normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx').symm J'), hJ⟩))
(normalizedFactors (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x)))
hJ' :
(normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx')
⟨↑((normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx').symm J'), hJ⟩ =
J'
⊢ Function.Injective fun f => ↑((normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx').symm f) | exact Subtype.coe_injective.comp (<a>Equiv.injective</a> _) | no goals | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/NumberTheory/KummerDedekind.lean |
KummerDedekind.normalizedFactors_ideal_map_eq_normalizedFactors_min_poly_mk_map | case pos.a
R : Type u_1
S : Type u_2
inst✝⁶ : CommRing R
inst✝⁵ : CommRing S
inst✝⁴ : Algebra R S
x : S
I : Ideal R
inst✝³ : IsDomain R
inst✝² : IsIntegrallyClosed R
inst✝¹ : IsDedekindDomain S
inst✝ : NoZeroSMulDivisors R S
hI : I.IsMaximal
hI' : I ≠ ⊥
hx : comap (algebraMap R S) (conductor R x) ⊔ I = ⊤
hx' : _root_.IsIntegral R x
J : Ideal S
hJ : J ∈ normalizedFactors (Ideal.map (algebraMap R S) I)
this :
↑(Multiset.count J (normalizedFactors (Ideal.map (algebraMap R S) I))) =
↑(Multiset.count (normalize ↑((normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx') ⟨J, hJ⟩))
(normalizedFactors (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))))
⊢ ↑((normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx') ⟨J, hJ⟩) ∈ normalizedFactors ?m.128881 | exact (<a>KummerDedekind.normalizedFactorsMapEquivNormalizedFactorsMinPolyMk</a> hI hI' hx hx' _).<a>Subtype.prop</a> | no goals | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/NumberTheory/KummerDedekind.lean |
KummerDedekind.normalizedFactors_ideal_map_eq_normalizedFactors_min_poly_mk_map | case pos.ha
R : Type u_1
S : Type u_2
inst✝⁶ : CommRing R
inst✝⁵ : CommRing S
inst✝⁴ : Algebra R S
x : S
I : Ideal R
inst✝³ : IsDomain R
inst✝² : IsIntegrallyClosed R
inst✝¹ : IsDedekindDomain S
inst✝ : NoZeroSMulDivisors R S
hI : I.IsMaximal
hI' : I ≠ ⊥
hx : comap (algebraMap R S) (conductor R x) ⊔ I = ⊤
hx' : _root_.IsIntegral R x
J : Ideal S
hJ : J ∈ normalizedFactors (Ideal.map (algebraMap R S) I)
this :
↑(Multiset.count (normalize J) (normalizedFactors (Ideal.map (algebraMap R S) I))) =
multiplicity (↑((normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx') ⟨J, hJ⟩))
(Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))
⊢ Irreducible ↑((normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx') ⟨J, hJ⟩) | exact <a>UniqueFactorizationMonoid.irreducible_of_normalized_factor</a> _ (<a>KummerDedekind.normalizedFactorsMapEquivNormalizedFactorsMinPolyMk</a> hI hI' hx hx' _).<a>Subtype.prop</a> | no goals | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/NumberTheory/KummerDedekind.lean |
KummerDedekind.normalizedFactors_ideal_map_eq_normalizedFactors_min_poly_mk_map | case pos.hb
R : Type u_1
S : Type u_2
inst✝⁶ : CommRing R
inst✝⁵ : CommRing S
inst✝⁴ : Algebra R S
x : S
I : Ideal R
inst✝³ : IsDomain R
inst✝² : IsIntegrallyClosed R
inst✝¹ : IsDedekindDomain S
inst✝ : NoZeroSMulDivisors R S
hI : I.IsMaximal
hI' : I ≠ ⊥
hx : comap (algebraMap R S) (conductor R x) ⊔ I = ⊤
hx' : _root_.IsIntegral R x
J : Ideal S
hJ : J ∈ normalizedFactors (Ideal.map (algebraMap R S) I)
this :
↑(Multiset.count (normalize J) (normalizedFactors (Ideal.map (algebraMap R S) I))) =
multiplicity (↑((normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx') ⟨J, hJ⟩))
(Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))
⊢ Polynomial.map (Ideal.Quotient.mk I) (minpoly R x) ≠ 0 | exact <a>Polynomial.map_monic_ne_zero</a> (<a>minpoly.monic</a> hx') | no goals | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/NumberTheory/KummerDedekind.lean |
KummerDedekind.normalizedFactors_ideal_map_eq_normalizedFactors_min_poly_mk_map | case pos.ha
R : Type u_1
S : Type u_2
inst✝⁶ : CommRing R
inst✝⁵ : CommRing S
inst✝⁴ : Algebra R S
x : S
I : Ideal R
inst✝³ : IsDomain R
inst✝² : IsIntegrallyClosed R
inst✝¹ : IsDedekindDomain S
inst✝ : NoZeroSMulDivisors R S
hI : I.IsMaximal
hI' : I ≠ ⊥
hx : comap (algebraMap R S) (conductor R x) ⊔ I = ⊤
hx' : _root_.IsIntegral R x
J : Ideal S
hJ : J ∈ normalizedFactors (Ideal.map (algebraMap R S) I)
this :
multiplicity J (Ideal.map (algebraMap R S) I) =
multiplicity (↑((normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx') ⟨J, hJ⟩))
(Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))
⊢ Irreducible J | exact <a>UniqueFactorizationMonoid.irreducible_of_normalized_factor</a> _ hJ | no goals | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/NumberTheory/KummerDedekind.lean |
KummerDedekind.normalizedFactors_ideal_map_eq_normalizedFactors_min_poly_mk_map | case pos.hb
R : Type u_1
S : Type u_2
inst✝⁶ : CommRing R
inst✝⁵ : CommRing S
inst✝⁴ : Algebra R S
x : S
I : Ideal R
inst✝³ : IsDomain R
inst✝² : IsIntegrallyClosed R
inst✝¹ : IsDedekindDomain S
inst✝ : NoZeroSMulDivisors R S
hI : I.IsMaximal
hI' : I ≠ ⊥
hx : comap (algebraMap R S) (conductor R x) ⊔ I = ⊤
hx' : _root_.IsIntegral R x
J : Ideal S
hJ : J ∈ normalizedFactors (Ideal.map (algebraMap R S) I)
this :
multiplicity J (Ideal.map (algebraMap R S) I) =
multiplicity (↑((normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx') ⟨J, hJ⟩))
(Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))
⊢ Ideal.map (algebraMap R S) I ≠ 0 | rwa [← <a>bot_eq_zero</a>, <a>Ne</a>, <a>Ideal.map_eq_bot_iff_of_injective</a> (<a>NoZeroSMulDivisors.algebraMap_injective</a> R S)] | no goals | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/NumberTheory/KummerDedekind.lean |
SetLike.coe_list_dProd | ι : Type u_1
R : Type u_2
α : Type u_3
S : Type u_4
inst✝³ : SetLike S R
inst✝² : Monoid R
inst✝¹ : AddMonoid ι
A : ι → S
inst✝ : GradedMonoid A
fι : α → ι
fA : (a : α) → ↥(A (fι a))
l : List α
⊢ ↑(l.dProd fι fA) = (List.map (fun a => ↑(fA a)) l).prod | match l with | [] => rw [<a>List.dProd_nil</a>, <a>SetLike.coe_gOne</a>, <a>List.map_nil</a>, <a>List.prod_nil</a>] | head::tail => rw [<a>List.dProd_cons</a>, <a>SetLike.coe_gMul</a>, <a>List.map_cons</a>, <a>List.prod_cons</a>, SetLike.coe_list_dProd _ _ _ tail] | no goals | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/Algebra/GradedMonoid.lean |
SetLike.coe_list_dProd | ι : Type u_1
R : Type u_2
α : Type u_3
S : Type u_4
inst✝³ : SetLike S R
inst✝² : Monoid R
inst✝¹ : AddMonoid ι
A : ι → S
inst✝ : GradedMonoid A
fι : α → ι
fA : (a : α) → ↥(A (fι a))
l : List α
⊢ ↑([].dProd fι fA) = (List.map (fun a => ↑(fA a)) []).prod | rw [<a>List.dProd_nil</a>, <a>SetLike.coe_gOne</a>, <a>List.map_nil</a>, <a>List.prod_nil</a>] | no goals | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/Algebra/GradedMonoid.lean |
SetLike.coe_list_dProd | ι : Type u_1
R : Type u_2
α : Type u_3
S : Type u_4
inst✝³ : SetLike S R
inst✝² : Monoid R
inst✝¹ : AddMonoid ι
A : ι → S
inst✝ : GradedMonoid A
fι : α → ι
fA : (a : α) → ↥(A (fι a))
l : List α
head : α
tail : List α
⊢ ↑((head :: tail).dProd fι fA) = (List.map (fun a => ↑(fA a)) (head :: tail)).prod | rw [<a>List.dProd_cons</a>, <a>SetLike.coe_gMul</a>, <a>List.map_cons</a>, <a>List.prod_cons</a>, SetLike.coe_list_dProd _ _ _ tail] | no goals | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/Algebra/GradedMonoid.lean |
Cubic.ne_zero | R : Type u_1
S : Type u_2
F : Type u_3
K : Type u_4
P Q : Cubic R
a b c d a' b' c' d' : R
inst✝ : Semiring R
h0 : P.a ≠ 0 ∨ P.b ≠ 0 ∨ P.c ≠ 0 ∨ P.d ≠ 0
⊢ P.toPoly ≠ 0 | contrapose! h0 | R : Type u_1
S : Type u_2
F : Type u_3
K : Type u_4
P Q : Cubic R
a b c d a' b' c' d' : R
inst✝ : Semiring R
h0 : P.toPoly = 0
⊢ P.a = 0 ∧ P.b = 0 ∧ P.c = 0 ∧ P.d = 0 | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/Algebra/CubicDiscriminant.lean |
Cubic.ne_zero | R : Type u_1
S : Type u_2
F : Type u_3
K : Type u_4
P Q : Cubic R
a b c d a' b' c' d' : R
inst✝ : Semiring R
h0 : P.toPoly = 0
⊢ P.a = 0 ∧ P.b = 0 ∧ P.c = 0 ∧ P.d = 0 | rw [(<a>Cubic.toPoly_eq_zero_iff</a> P).<a>Iff.mp</a> h0] | R : Type u_1
S : Type u_2
F : Type u_3
K : Type u_4
P Q : Cubic R
a b c d a' b' c' d' : R
inst✝ : Semiring R
h0 : P.toPoly = 0
⊢ Cubic.a 0 = 0 ∧ Cubic.b 0 = 0 ∧ Cubic.c 0 = 0 ∧ Cubic.d 0 = 0 | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/Algebra/CubicDiscriminant.lean |
Cubic.ne_zero | R : Type u_1
S : Type u_2
F : Type u_3
K : Type u_4
P Q : Cubic R
a b c d a' b' c' d' : R
inst✝ : Semiring R
h0 : P.toPoly = 0
⊢ Cubic.a 0 = 0 ∧ Cubic.b 0 = 0 ∧ Cubic.c 0 = 0 ∧ Cubic.d 0 = 0 | exact ⟨<a>rfl</a>, <a>rfl</a>, <a>rfl</a>, <a>rfl</a>⟩ | no goals | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/Algebra/CubicDiscriminant.lean |
Filter.Tendsto.continuousWithinAt_of_equicontinuousWithinAt | ι : Type u_1
κ : Type u_2
X : Type u_3
X' : Type u_4
Y : Type u_5
Z : Type u_6
α : Type u_7
α' : Type u_8
β : Type u_9
β' : Type u_10
γ : Type u_11
𝓕 : Type u_12
tX : TopologicalSpace X
tY : TopologicalSpace Y
tZ : TopologicalSpace Z
uα : UniformSpace α
uβ : UniformSpace β
uγ : UniformSpace γ
l : Filter ι
inst✝ : l.NeBot
F : ι → X → α
f : X → α
S : Set X
x₀ : X
h₁ : ∀ x ∈ S, Tendsto (fun x_1 => F x_1 x) l (𝓝 (f x))
h₂ : Tendsto (fun x => F x x₀) l (𝓝 (f x₀))
h₃ : EquicontinuousWithinAt F S x₀
⊢ ContinuousWithinAt f S x₀ | intro U hU | ι : Type u_1
κ : Type u_2
X : Type u_3
X' : Type u_4
Y : Type u_5
Z : Type u_6
α : Type u_7
α' : Type u_8
β : Type u_9
β' : Type u_10
γ : Type u_11
𝓕 : Type u_12
tX : TopologicalSpace X
tY : TopologicalSpace Y
tZ : TopologicalSpace Z
uα : UniformSpace α
uβ : UniformSpace β
uγ : UniformSpace γ
l : Filter ι
inst✝ : l.NeBot
F : ι → X → α
f : X → α
S : Set X
x₀ : X
h₁ : ∀ x ∈ S, Tendsto (fun x_1 => F x_1 x) l (𝓝 (f x))
h₂ : Tendsto (fun x => F x x₀) l (𝓝 (f x₀))
h₃ : EquicontinuousWithinAt F S x₀
U : Set α
hU : U ∈ 𝓝 (f x₀)
⊢ U ∈ map f (𝓝[S] x₀) | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/Topology/UniformSpace/Equicontinuity.lean |
Filter.Tendsto.continuousWithinAt_of_equicontinuousWithinAt | ι : Type u_1
κ : Type u_2
X : Type u_3
X' : Type u_4
Y : Type u_5
Z : Type u_6
α : Type u_7
α' : Type u_8
β : Type u_9
β' : Type u_10
γ : Type u_11
𝓕 : Type u_12
tX : TopologicalSpace X
tY : TopologicalSpace Y
tZ : TopologicalSpace Z
uα : UniformSpace α
uβ : UniformSpace β
uγ : UniformSpace γ
l : Filter ι
inst✝ : l.NeBot
F : ι → X → α
f : X → α
S : Set X
x₀ : X
h₁ : ∀ x ∈ S, Tendsto (fun x_1 => F x_1 x) l (𝓝 (f x))
h₂ : Tendsto (fun x => F x x₀) l (𝓝 (f x₀))
h₃ : EquicontinuousWithinAt F S x₀
U : Set α
hU : U ∈ 𝓝 (f x₀)
⊢ U ∈ map f (𝓝[S] x₀) | rw [<a>Filter.mem_map</a>] | ι : Type u_1
κ : Type u_2
X : Type u_3
X' : Type u_4
Y : Type u_5
Z : Type u_6
α : Type u_7
α' : Type u_8
β : Type u_9
β' : Type u_10
γ : Type u_11
𝓕 : Type u_12
tX : TopologicalSpace X
tY : TopologicalSpace Y
tZ : TopologicalSpace Z
uα : UniformSpace α
uβ : UniformSpace β
uγ : UniformSpace γ
l : Filter ι
inst✝ : l.NeBot
F : ι → X → α
f : X → α
S : Set X
x₀ : X
h₁ : ∀ x ∈ S, Tendsto (fun x_1 => F x_1 x) l (𝓝 (f x))
h₂ : Tendsto (fun x => F x x₀) l (𝓝 (f x₀))
h₃ : EquicontinuousWithinAt F S x₀
U : Set α
hU : U ∈ 𝓝 (f x₀)
⊢ f ⁻¹' U ∈ 𝓝[S] x₀ | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/Topology/UniformSpace/Equicontinuity.lean |
Filter.Tendsto.continuousWithinAt_of_equicontinuousWithinAt | ι : Type u_1
κ : Type u_2
X : Type u_3
X' : Type u_4
Y : Type u_5
Z : Type u_6
α : Type u_7
α' : Type u_8
β : Type u_9
β' : Type u_10
γ : Type u_11
𝓕 : Type u_12
tX : TopologicalSpace X
tY : TopologicalSpace Y
tZ : TopologicalSpace Z
uα : UniformSpace α
uβ : UniformSpace β
uγ : UniformSpace γ
l : Filter ι
inst✝ : l.NeBot
F : ι → X → α
f : X → α
S : Set X
x₀ : X
h₁ : ∀ x ∈ S, Tendsto (fun x_1 => F x_1 x) l (𝓝 (f x))
h₂ : Tendsto (fun x => F x x₀) l (𝓝 (f x₀))
h₃ : EquicontinuousWithinAt F S x₀
U : Set α
hU : U ∈ 𝓝 (f x₀)
⊢ f ⁻¹' U ∈ 𝓝[S] x₀ | rcases UniformSpace.mem_nhds_iff.mp hU with ⟨V, hV, hVU⟩ | case intro.intro
ι : Type u_1
κ : Type u_2
X : Type u_3
X' : Type u_4
Y : Type u_5
Z : Type u_6
α : Type u_7
α' : Type u_8
β : Type u_9
β' : Type u_10
γ : Type u_11
𝓕 : Type u_12
tX : TopologicalSpace X
tY : TopologicalSpace Y
tZ : TopologicalSpace Z
uα : UniformSpace α
uβ : UniformSpace β
uγ : UniformSpace γ
l : Filter ι
inst✝ : l.NeBot
F : ι → X → α
f : X → α
S : Set X
x₀ : X
h₁ : ∀ x ∈ S, Tendsto (fun x_1 => F x_1 x) l (𝓝 (f x))
h₂ : Tendsto (fun x => F x x₀) l (𝓝 (f x₀))
h₃ : EquicontinuousWithinAt F S x₀
U : Set α
hU : U ∈ 𝓝 (f x₀)
V : Set (α × α)
hV : V ∈ 𝓤 α
hVU : ball (f x₀) V ⊆ U
⊢ f ⁻¹' U ∈ 𝓝[S] x₀ | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/Topology/UniformSpace/Equicontinuity.lean |
Filter.Tendsto.continuousWithinAt_of_equicontinuousWithinAt | case intro.intro
ι : Type u_1
κ : Type u_2
X : Type u_3
X' : Type u_4
Y : Type u_5
Z : Type u_6
α : Type u_7
α' : Type u_8
β : Type u_9
β' : Type u_10
γ : Type u_11
𝓕 : Type u_12
tX : TopologicalSpace X
tY : TopologicalSpace Y
tZ : TopologicalSpace Z
uα : UniformSpace α
uβ : UniformSpace β
uγ : UniformSpace γ
l : Filter ι
inst✝ : l.NeBot
F : ι → X → α
f : X → α
S : Set X
x₀ : X
h₁ : ∀ x ∈ S, Tendsto (fun x_1 => F x_1 x) l (𝓝 (f x))
h₂ : Tendsto (fun x => F x x₀) l (𝓝 (f x₀))
h₃ : EquicontinuousWithinAt F S x₀
U : Set α
hU : U ∈ 𝓝 (f x₀)
V : Set (α × α)
hV : V ∈ 𝓤 α
hVU : ball (f x₀) V ⊆ U
⊢ f ⁻¹' U ∈ 𝓝[S] x₀ | rcases <a>mem_uniformity_isClosed</a> hV with ⟨W, hW, hWclosed, hWV⟩ | case intro.intro.intro.intro.intro
ι : Type u_1
κ : Type u_2
X : Type u_3
X' : Type u_4
Y : Type u_5
Z : Type u_6
α : Type u_7
α' : Type u_8
β : Type u_9
β' : Type u_10
γ : Type u_11
𝓕 : Type u_12
tX : TopologicalSpace X
tY : TopologicalSpace Y
tZ : TopologicalSpace Z
uα : UniformSpace α
uβ : UniformSpace β
uγ : UniformSpace γ
l : Filter ι
inst✝ : l.NeBot
F : ι → X → α
f : X → α
S : Set X
x₀ : X
h₁ : ∀ x ∈ S, Tendsto (fun x_1 => F x_1 x) l (𝓝 (f x))
h₂ : Tendsto (fun x => F x x₀) l (𝓝 (f x₀))
h₃ : EquicontinuousWithinAt F S x₀
U : Set α
hU : U ∈ 𝓝 (f x₀)
V : Set (α × α)
hV : V ∈ 𝓤 α
hVU : ball (f x₀) V ⊆ U
W : Set (α × α)
hW : W ∈ 𝓤 α
hWclosed : IsClosed W
hWV : W ⊆ V
⊢ f ⁻¹' U ∈ 𝓝[S] x₀ | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/Topology/UniformSpace/Equicontinuity.lean |
Filter.Tendsto.continuousWithinAt_of_equicontinuousWithinAt | case intro.intro.intro.intro.intro
ι : Type u_1
κ : Type u_2
X : Type u_3
X' : Type u_4
Y : Type u_5
Z : Type u_6
α : Type u_7
α' : Type u_8
β : Type u_9
β' : Type u_10
γ : Type u_11
𝓕 : Type u_12
tX : TopologicalSpace X
tY : TopologicalSpace Y
tZ : TopologicalSpace Z
uα : UniformSpace α
uβ : UniformSpace β
uγ : UniformSpace γ
l : Filter ι
inst✝ : l.NeBot
F : ι → X → α
f : X → α
S : Set X
x₀ : X
h₁ : ∀ x ∈ S, Tendsto (fun x_1 => F x_1 x) l (𝓝 (f x))
h₂ : Tendsto (fun x => F x x₀) l (𝓝 (f x₀))
h₃ : EquicontinuousWithinAt F S x₀
U : Set α
hU : U ∈ 𝓝 (f x₀)
V : Set (α × α)
hV : V ∈ 𝓤 α
hVU : ball (f x₀) V ⊆ U
W : Set (α × α)
hW : W ∈ 𝓤 α
hWclosed : IsClosed W
hWV : W ⊆ V
⊢ f ⁻¹' U ∈ 𝓝[S] x₀ | filter_upwards [h₃ W hW, <a>eventually_mem_nhdsWithin</a>] with x hx hxS using hVU <| <a>ball_mono</a> hWV (f x₀) <| hWclosed.mem_of_tendsto (h₂.prod_mk_nhds (h₁ x hxS)) <| <a>Filter.eventually_of_forall</a> hx | no goals | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/Topology/UniformSpace/Equicontinuity.lean |
Cardinal.ord_one | α : Type u
β : Type u_1
γ : Type u_2
r : α → α → Prop
s : β → β → Prop
t : γ → γ → Prop
⊢ ord 1 = 1 | simpa using <a>Cardinal.ord_nat</a> 1 | no goals | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/SetTheory/Ordinal/Basic.lean |
AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.pullbackConeOfLeftLift_fst | C : Type u
inst✝ : Category.{v, u} C
X Y Z : PresheafedSpace C
f : X ⟶ Z
hf : IsOpenImmersion f
g : Y ⟶ Z
s : PullbackCone f g
⊢ pullbackConeOfLeftLift f g s ≫ (pullbackConeOfLeft f g).fst = s.fst | refine <a>AlgebraicGeometry.PresheafedSpace.Hom.ext</a> _ _ ?_ <| <a>CategoryTheory.NatTrans.ext</a> _ _ <| <a>funext</a> fun x => ?_ | case refine_1
C : Type u
inst✝ : Category.{v, u} C
X Y Z : PresheafedSpace C
f : X ⟶ Z
hf : IsOpenImmersion f
g : Y ⟶ Z
s : PullbackCone f g
⊢ (pullbackConeOfLeftLift f g s ≫ (pullbackConeOfLeft f g).fst).base = s.fst.base
case refine_2
C : Type u
inst✝ : Category.{v, u} C
X Y Z : PresheafedSpace C
f : X ⟶ Z
hf : IsOpenImmersion f
g : Y ⟶ Z
s : PullbackCone f g
x : (Opens ↑↑X)ᵒᵖ
⊢ ((pullbackConeOfLeftLift f g s ≫ (pullbackConeOfLeft f g).fst).c ≫ whiskerRight (eqToHom ⋯) s.pt.presheaf).app x =
s.fst.c.app x | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/Geometry/RingedSpace/OpenImmersion.lean |
AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.pullbackConeOfLeftLift_fst | case refine_1
C : Type u
inst✝ : Category.{v, u} C
X Y Z : PresheafedSpace C
f : X ⟶ Z
hf : IsOpenImmersion f
g : Y ⟶ Z
s : PullbackCone f g
⊢ (pullbackConeOfLeftLift f g s ≫ (pullbackConeOfLeft f g).fst).base = s.fst.base | change <a>CategoryTheory.Limits.pullback.lift</a> _ _ _ ≫ <a>CategoryTheory.Limits.pullback.fst</a> = _ | case refine_1
C : Type u
inst✝ : Category.{v, u} C
X Y Z : PresheafedSpace C
f : X ⟶ Z
hf : IsOpenImmersion f
g : Y ⟶ Z
s : PullbackCone f g
⊢ pullback.lift s.fst.base s.snd.base ⋯ ≫ pullback.fst = s.fst.base | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/Geometry/RingedSpace/OpenImmersion.lean |
AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.pullbackConeOfLeftLift_fst | case refine_1
C : Type u
inst✝ : Category.{v, u} C
X Y Z : PresheafedSpace C
f : X ⟶ Z
hf : IsOpenImmersion f
g : Y ⟶ Z
s : PullbackCone f g
⊢ pullback.lift s.fst.base s.snd.base ⋯ ≫ pullback.fst = s.fst.base | simp | no goals | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/Geometry/RingedSpace/OpenImmersion.lean |
AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.pullbackConeOfLeftLift_fst | case refine_2
C : Type u
inst✝ : Category.{v, u} C
X Y Z : PresheafedSpace C
f : X ⟶ Z
hf : IsOpenImmersion f
g : Y ⟶ Z
s : PullbackCone f g
x : (Opens ↑↑X)ᵒᵖ
⊢ ((pullbackConeOfLeftLift f g s ≫ (pullbackConeOfLeft f g).fst).c ≫ whiskerRight (eqToHom ⋯) s.pt.presheaf).app x =
s.fst.c.app x | induction x using <a>Opposite.rec'</a> with | h x => ?_ | case refine_2.h
C : Type u
inst✝ : Category.{v, u} C
X Y Z : PresheafedSpace C
f : X ⟶ Z
hf : IsOpenImmersion f
g : Y ⟶ Z
s : PullbackCone f g
x : Opens ↑↑X
⊢ ((pullbackConeOfLeftLift f g s ≫ (pullbackConeOfLeft f g).fst).c ≫ whiskerRight (eqToHom ⋯) s.pt.presheaf).app
{ unop := x } =
s.fst.c.app { unop := x } | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/Geometry/RingedSpace/OpenImmersion.lean |
AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.pullbackConeOfLeftLift_fst | case refine_2.h
C : Type u
inst✝ : Category.{v, u} C
X Y Z : PresheafedSpace C
f : X ⟶ Z
hf : IsOpenImmersion f
g : Y ⟶ Z
s : PullbackCone f g
x : Opens ↑↑X
⊢ ((pullbackConeOfLeftLift f g s ≫ (pullbackConeOfLeft f g).fst).c ≫ whiskerRight (eqToHom ⋯) s.pt.presheaf).app
{ unop := x } =
s.fst.c.app { unop := x } | change ((_ ≫ _) ≫ _ ≫ _) ≫ _ = _ | case refine_2.h
C : Type u
inst✝ : Category.{v, u} C
X Y Z : PresheafedSpace C
f : X ⟶ Z
hf : IsOpenImmersion f
g : Y ⟶ Z
s : PullbackCone f g
x : Opens ↑↑X
⊢ ((invApp f { unop := x }.unop ≫ g.c.app { unop := ⋯.functor.obj { unop := x }.unop } ≫ Y.presheaf.map (eqToHom ⋯)) ≫
s.snd.c.app (⋯.functor.op.obj ((Opens.map (pullbackConeOfLeft f g).fst.base).op.obj { unop := x })) ≫
s.pt.presheaf.map (eqToHom ⋯)) ≫
(whiskerRight (eqToHom ⋯) s.pt.presheaf).app { unop := x } =
s.fst.c.app { unop := x } | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/Geometry/RingedSpace/OpenImmersion.lean |
AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.pullbackConeOfLeftLift_fst | case refine_2.h
C : Type u
inst✝ : Category.{v, u} C
X Y Z : PresheafedSpace C
f : X ⟶ Z
hf : IsOpenImmersion f
g : Y ⟶ Z
s : PullbackCone f g
x : Opens ↑↑X
⊢ ((invApp f { unop := x }.unop ≫ g.c.app { unop := ⋯.functor.obj { unop := x }.unop } ≫ Y.presheaf.map (eqToHom ⋯)) ≫
s.snd.c.app (⋯.functor.op.obj ((Opens.map (pullbackConeOfLeft f g).fst.base).op.obj { unop := x })) ≫
s.pt.presheaf.map (eqToHom ⋯)) ≫
(whiskerRight (eqToHom ⋯) s.pt.presheaf).app { unop := x } =
s.fst.c.app { unop := x } | simp_rw [<a>CategoryTheory.Category.assoc</a>] | case refine_2.h
C : Type u
inst✝ : Category.{v, u} C
X Y Z : PresheafedSpace C
f : X ⟶ Z
hf : IsOpenImmersion f
g : Y ⟶ Z
s : PullbackCone f g
x : Opens ↑↑X
⊢ invApp f x ≫
g.c.app { unop := ⋯.functor.obj x } ≫
Y.presheaf.map (eqToHom ⋯) ≫
s.snd.c.app (⋯.functor.op.obj ((Opens.map (pullbackConeOfLeft f g).fst.base).op.obj { unop := x })) ≫
s.pt.presheaf.map (eqToHom ⋯) ≫ (whiskerRight (eqToHom ⋯) s.pt.presheaf).app { unop := x } =
s.fst.c.app { unop := x } | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/Geometry/RingedSpace/OpenImmersion.lean |
AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.pullbackConeOfLeftLift_fst | case refine_2.h
C : Type u
inst✝ : Category.{v, u} C
X Y Z : PresheafedSpace C
f : X ⟶ Z
hf : IsOpenImmersion f
g : Y ⟶ Z
s : PullbackCone f g
x : Opens ↑↑X
⊢ invApp f x ≫
g.c.app { unop := ⋯.functor.obj x } ≫
Y.presheaf.map (eqToHom ⋯) ≫
s.snd.c.app (⋯.functor.op.obj ((Opens.map (pullbackConeOfLeft f g).fst.base).op.obj { unop := x })) ≫
s.pt.presheaf.map (eqToHom ⋯) ≫ (whiskerRight (eqToHom ⋯) s.pt.presheaf).app { unop := x } =
s.fst.c.app { unop := x } | erw [← s.pt.presheaf.map_comp] | case refine_2.h
C : Type u
inst✝ : Category.{v, u} C
X Y Z : PresheafedSpace C
f : X ⟶ Z
hf : IsOpenImmersion f
g : Y ⟶ Z
s : PullbackCone f g
x : Opens ↑↑X
⊢ invApp f x ≫
g.c.app { unop := ⋯.functor.obj x } ≫
Y.presheaf.map (eqToHom ⋯) ≫
s.snd.c.app (⋯.functor.op.obj ((Opens.map (pullbackConeOfLeft f g).fst.base).op.obj { unop := x })) ≫
s.pt.presheaf.map (eqToHom ⋯ ≫ (eqToHom ⋯).app { unop := x }) =
s.fst.c.app { unop := x } | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/Geometry/RingedSpace/OpenImmersion.lean |
AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.pullbackConeOfLeftLift_fst | case refine_2.h
C : Type u
inst✝ : Category.{v, u} C
X Y Z : PresheafedSpace C
f : X ⟶ Z
hf : IsOpenImmersion f
g : Y ⟶ Z
s : PullbackCone f g
x : Opens ↑↑X
⊢ invApp f x ≫
g.c.app { unop := ⋯.functor.obj x } ≫
Y.presheaf.map (eqToHom ⋯) ≫
s.snd.c.app (⋯.functor.op.obj ((Opens.map (pullbackConeOfLeft f g).fst.base).op.obj { unop := x })) ≫
s.pt.presheaf.map (eqToHom ⋯ ≫ (eqToHom ⋯).app { unop := x }) =
s.fst.c.app { unop := x } | erw [s.snd.c.naturality_assoc] | case refine_2.h
C : Type u
inst✝ : Category.{v, u} C
X Y Z : PresheafedSpace C
f : X ⟶ Z
hf : IsOpenImmersion f
g : Y ⟶ Z
s : PullbackCone f g
x : Opens ↑↑X
⊢ invApp f x ≫
g.c.app { unop := ⋯.functor.obj x } ≫
s.snd.c.app ((Opens.map g.base).op.obj { unop := ⋯.functor.obj x }) ≫
(s.snd.base _* s.pt.presheaf).map (eqToHom ⋯) ≫
s.pt.presheaf.map (eqToHom ⋯ ≫ (eqToHom ⋯).app { unop := x }) =
s.fst.c.app { unop := x } | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/Geometry/RingedSpace/OpenImmersion.lean |
AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.pullbackConeOfLeftLift_fst | case refine_2.h
C : Type u
inst✝ : Category.{v, u} C
X Y Z : PresheafedSpace C
f : X ⟶ Z
hf : IsOpenImmersion f
g : Y ⟶ Z
s : PullbackCone f g
x : Opens ↑↑X
⊢ invApp f x ≫
g.c.app { unop := ⋯.functor.obj x } ≫
s.snd.c.app ((Opens.map g.base).op.obj { unop := ⋯.functor.obj x }) ≫
(s.snd.base _* s.pt.presheaf).map (eqToHom ⋯) ≫
s.pt.presheaf.map (eqToHom ⋯ ≫ (eqToHom ⋯).app { unop := x }) =
s.fst.c.app { unop := x } | have := <a>AlgebraicGeometry.PresheafedSpace.congr_app</a> s.condition (<a>Opposite.op</a> (<a>AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.opensFunctor</a> f |>.<a>Prefunctor.obj</a> x)) | case refine_2.h
C : Type u
inst✝ : Category.{v, u} C
X Y Z : PresheafedSpace C
f : X ⟶ Z
hf : IsOpenImmersion f
g : Y ⟶ Z
s : PullbackCone f g
x : Opens ↑↑X
this :
(s.fst ≫ f).c.app { unop := (opensFunctor f).obj x } =
(s.snd ≫ g).c.app { unop := (opensFunctor f).obj x } ≫ s.pt.presheaf.map (eqToHom ⋯)
⊢ invApp f x ≫
g.c.app { unop := ⋯.functor.obj x } ≫
s.snd.c.app ((Opens.map g.base).op.obj { unop := ⋯.functor.obj x }) ≫
(s.snd.base _* s.pt.presheaf).map (eqToHom ⋯) ≫
s.pt.presheaf.map (eqToHom ⋯ ≫ (eqToHom ⋯).app { unop := x }) =
s.fst.c.app { unop := x } | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/Geometry/RingedSpace/OpenImmersion.lean |
AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.pullbackConeOfLeftLift_fst | case refine_2.h
C : Type u
inst✝ : Category.{v, u} C
X Y Z : PresheafedSpace C
f : X ⟶ Z
hf : IsOpenImmersion f
g : Y ⟶ Z
s : PullbackCone f g
x : Opens ↑↑X
this :
(s.fst ≫ f).c.app { unop := (opensFunctor f).obj x } =
(s.snd ≫ g).c.app { unop := (opensFunctor f).obj x } ≫ s.pt.presheaf.map (eqToHom ⋯)
⊢ invApp f x ≫
g.c.app { unop := ⋯.functor.obj x } ≫
s.snd.c.app ((Opens.map g.base).op.obj { unop := ⋯.functor.obj x }) ≫
(s.snd.base _* s.pt.presheaf).map (eqToHom ⋯) ≫
s.pt.presheaf.map (eqToHom ⋯ ≫ (eqToHom ⋯).app { unop := x }) =
s.fst.c.app { unop := x } | dsimp only [<a>AlgebraicGeometry.PresheafedSpace.comp_c_app</a>, <a>Opposite.unop_op</a>] at this | case refine_2.h
C : Type u
inst✝ : Category.{v, u} C
X Y Z : PresheafedSpace C
f : X ⟶ Z
hf : IsOpenImmersion f
g : Y ⟶ Z
s : PullbackCone f g
x : Opens ↑↑X
this :
f.c.app { unop := (opensFunctor f).obj x } ≫ s.fst.c.app { unop := (Opens.map f.base).obj ((opensFunctor f).obj x) } =
(g.c.app { unop := (opensFunctor f).obj x } ≫
s.snd.c.app { unop := (Opens.map g.base).obj ((opensFunctor f).obj x) }) ≫
s.pt.presheaf.map (eqToHom ⋯)
⊢ invApp f x ≫
g.c.app { unop := ⋯.functor.obj x } ≫
s.snd.c.app ((Opens.map g.base).op.obj { unop := ⋯.functor.obj x }) ≫
(s.snd.base _* s.pt.presheaf).map (eqToHom ⋯) ≫
s.pt.presheaf.map (eqToHom ⋯ ≫ (eqToHom ⋯).app { unop := x }) =
s.fst.c.app { unop := x } | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/Geometry/RingedSpace/OpenImmersion.lean |
AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.pullbackConeOfLeftLift_fst | case refine_2.h
C : Type u
inst✝ : Category.{v, u} C
X Y Z : PresheafedSpace C
f : X ⟶ Z
hf : IsOpenImmersion f
g : Y ⟶ Z
s : PullbackCone f g
x : Opens ↑↑X
this :
f.c.app { unop := (opensFunctor f).obj x } ≫ s.fst.c.app { unop := (Opens.map f.base).obj ((opensFunctor f).obj x) } =
(g.c.app { unop := (opensFunctor f).obj x } ≫
s.snd.c.app { unop := (Opens.map g.base).obj ((opensFunctor f).obj x) }) ≫
s.pt.presheaf.map (eqToHom ⋯)
⊢ invApp f x ≫
g.c.app { unop := ⋯.functor.obj x } ≫
s.snd.c.app ((Opens.map g.base).op.obj { unop := ⋯.functor.obj x }) ≫
(s.snd.base _* s.pt.presheaf).map (eqToHom ⋯) ≫
s.pt.presheaf.map (eqToHom ⋯ ≫ (eqToHom ⋯).app { unop := x }) =
s.fst.c.app { unop := x } | rw [← <a>CategoryTheory.IsIso.comp_inv_eq</a>] at this | case refine_2.h
C : Type u
inst✝ : Category.{v, u} C
X Y Z : PresheafedSpace C
f : X ⟶ Z
hf : IsOpenImmersion f
g : Y ⟶ Z
s : PullbackCone f g
x : Opens ↑↑X
this :
(f.c.app { unop := (opensFunctor f).obj x } ≫
s.fst.c.app { unop := (Opens.map f.base).obj ((opensFunctor f).obj x) }) ≫
inv (s.pt.presheaf.map (eqToHom ⋯)) =
g.c.app { unop := (opensFunctor f).obj x } ≫ s.snd.c.app { unop := (Opens.map g.base).obj ((opensFunctor f).obj x) }
⊢ invApp f x ≫
g.c.app { unop := ⋯.functor.obj x } ≫
s.snd.c.app ((Opens.map g.base).op.obj { unop := ⋯.functor.obj x }) ≫
(s.snd.base _* s.pt.presheaf).map (eqToHom ⋯) ≫
s.pt.presheaf.map (eqToHom ⋯ ≫ (eqToHom ⋯).app { unop := x }) =
s.fst.c.app { unop := x } | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/Geometry/RingedSpace/OpenImmersion.lean |
AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.pullbackConeOfLeftLift_fst | case refine_2.h
C : Type u
inst✝ : Category.{v, u} C
X Y Z : PresheafedSpace C
f : X ⟶ Z
hf : IsOpenImmersion f
g : Y ⟶ Z
s : PullbackCone f g
x : Opens ↑↑X
this :
(f.c.app { unop := (opensFunctor f).obj x } ≫
s.fst.c.app { unop := (Opens.map f.base).obj ((opensFunctor f).obj x) }) ≫
inv (s.pt.presheaf.map (eqToHom ⋯)) =
g.c.app { unop := (opensFunctor f).obj x } ≫ s.snd.c.app { unop := (Opens.map g.base).obj ((opensFunctor f).obj x) }
⊢ invApp f x ≫
g.c.app { unop := ⋯.functor.obj x } ≫
s.snd.c.app ((Opens.map g.base).op.obj { unop := ⋯.functor.obj x }) ≫
(s.snd.base _* s.pt.presheaf).map (eqToHom ⋯) ≫
s.pt.presheaf.map (eqToHom ⋯ ≫ (eqToHom ⋯).app { unop := x }) =
s.fst.c.app { unop := x } | replace this := reassoc_of% this | case refine_2.h
C : Type u
inst✝ : Category.{v, u} C
X Y Z : PresheafedSpace C
f : X ⟶ Z
hf : IsOpenImmersion f
g : Y ⟶ Z
s : PullbackCone f g
x : Opens ↑↑X
this :
∀ {Z_1 : C} (h : ((s.snd ≫ g).base _* s.pt.presheaf).obj { unop := (opensFunctor f).obj x } ⟶ Z_1),
f.c.app { unop := (opensFunctor f).obj x } ≫
s.fst.c.app { unop := (Opens.map f.base).obj ((opensFunctor f).obj x) } ≫
inv (s.pt.presheaf.map (eqToHom ⋯)) ≫ h =
g.c.app { unop := (opensFunctor f).obj x } ≫
s.snd.c.app { unop := (Opens.map g.base).obj ((opensFunctor f).obj x) } ≫ h
⊢ invApp f x ≫
g.c.app { unop := ⋯.functor.obj x } ≫
s.snd.c.app ((Opens.map g.base).op.obj { unop := ⋯.functor.obj x }) ≫
(s.snd.base _* s.pt.presheaf).map (eqToHom ⋯) ≫
s.pt.presheaf.map (eqToHom ⋯ ≫ (eqToHom ⋯).app { unop := x }) =
s.fst.c.app { unop := x } | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/Geometry/RingedSpace/OpenImmersion.lean |
AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.pullbackConeOfLeftLift_fst | case refine_2.h
C : Type u
inst✝ : Category.{v, u} C
X Y Z : PresheafedSpace C
f : X ⟶ Z
hf : IsOpenImmersion f
g : Y ⟶ Z
s : PullbackCone f g
x : Opens ↑↑X
this :
∀ {Z_1 : C} (h : ((s.snd ≫ g).base _* s.pt.presheaf).obj { unop := (opensFunctor f).obj x } ⟶ Z_1),
f.c.app { unop := (opensFunctor f).obj x } ≫
s.fst.c.app { unop := (Opens.map f.base).obj ((opensFunctor f).obj x) } ≫
inv (s.pt.presheaf.map (eqToHom ⋯)) ≫ h =
g.c.app { unop := (opensFunctor f).obj x } ≫
s.snd.c.app { unop := (Opens.map g.base).obj ((opensFunctor f).obj x) } ≫ h
⊢ invApp f x ≫
g.c.app { unop := ⋯.functor.obj x } ≫
s.snd.c.app ((Opens.map g.base).op.obj { unop := ⋯.functor.obj x }) ≫
(s.snd.base _* s.pt.presheaf).map (eqToHom ⋯) ≫
s.pt.presheaf.map (eqToHom ⋯ ≫ (eqToHom ⋯).app { unop := x }) =
s.fst.c.app { unop := x } | erw [← this, hf.invApp_app_assoc, s.fst.c.naturality_assoc] | case refine_2.h
C : Type u
inst✝ : Category.{v, u} C
X Y Z : PresheafedSpace C
f : X ⟶ Z
hf : IsOpenImmersion f
g : Y ⟶ Z
s : PullbackCone f g
x : Opens ↑↑X
this :
∀ {Z_1 : C} (h : ((s.snd ≫ g).base _* s.pt.presheaf).obj { unop := (opensFunctor f).obj x } ⟶ Z_1),
f.c.app { unop := (opensFunctor f).obj x } ≫
s.fst.c.app { unop := (Opens.map f.base).obj ((opensFunctor f).obj x) } ≫
inv (s.pt.presheaf.map (eqToHom ⋯)) ≫ h =
g.c.app { unop := (opensFunctor f).obj x } ≫
s.snd.c.app { unop := (Opens.map g.base).obj ((opensFunctor f).obj x) } ≫ h
⊢ s.fst.c.app { unop := x } ≫
(s.fst.base _* s.pt.presheaf).map (eqToHom ⋯) ≫
inv (s.pt.presheaf.map (eqToHom ⋯)) ≫
(s.snd.base _* s.pt.presheaf).map (eqToHom ⋯) ≫
s.pt.presheaf.map (eqToHom ⋯ ≫ (eqToHom ⋯).app { unop := x }) =
s.fst.c.app { unop := x } | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/Geometry/RingedSpace/OpenImmersion.lean |
AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.pullbackConeOfLeftLift_fst | case refine_2.h
C : Type u
inst✝ : Category.{v, u} C
X Y Z : PresheafedSpace C
f : X ⟶ Z
hf : IsOpenImmersion f
g : Y ⟶ Z
s : PullbackCone f g
x : Opens ↑↑X
this :
∀ {Z_1 : C} (h : ((s.snd ≫ g).base _* s.pt.presheaf).obj { unop := (opensFunctor f).obj x } ⟶ Z_1),
f.c.app { unop := (opensFunctor f).obj x } ≫
s.fst.c.app { unop := (Opens.map f.base).obj ((opensFunctor f).obj x) } ≫
inv (s.pt.presheaf.map (eqToHom ⋯)) ≫ h =
g.c.app { unop := (opensFunctor f).obj x } ≫
s.snd.c.app { unop := (Opens.map g.base).obj ((opensFunctor f).obj x) } ≫ h
⊢ s.fst.c.app { unop := x } ≫
(s.fst.base _* s.pt.presheaf).map (eqToHom ⋯) ≫
inv (s.pt.presheaf.map (eqToHom ⋯)) ≫
(s.snd.base _* s.pt.presheaf).map (eqToHom ⋯) ≫
s.pt.presheaf.map (eqToHom ⋯ ≫ (eqToHom ⋯).app { unop := x }) =
s.fst.c.app { unop := x } | simp [<a>CategoryTheory.eqToHom_map</a>] | no goals | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/Geometry/RingedSpace/OpenImmersion.lean |
MeasurableSet.tProd | α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
δ' : Type u_5
ι : Sort uι
s✝ t u : Set α
π : δ → Type u_6
inst✝ : (x : δ) → MeasurableSpace (π x)
l : List δ
s : (i : δ) → Set (π i)
hs : ∀ (i : δ), MeasurableSet (s i)
⊢ MeasurableSet (Set.tprod l s) | induction' l with i l ih | case nil
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
δ' : Type u_5
ι : Sort uι
s✝ t u : Set α
π : δ → Type u_6
inst✝ : (x : δ) → MeasurableSpace (π x)
s : (i : δ) → Set (π i)
hs : ∀ (i : δ), MeasurableSet (s i)
⊢ MeasurableSet (Set.tprod [] s)
case cons
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
δ' : Type u_5
ι : Sort uι
s✝ t u : Set α
π : δ → Type u_6
inst✝ : (x : δ) → MeasurableSpace (π x)
s : (i : δ) → Set (π i)
hs : ∀ (i : δ), MeasurableSet (s i)
i : δ
l : List δ
ih : MeasurableSet (Set.tprod l s)
⊢ MeasurableSet (Set.tprod (i :: l) s) | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/MeasureTheory/MeasurableSpace/Basic.lean |
MeasurableSet.tProd | case nil
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
δ' : Type u_5
ι : Sort uι
s✝ t u : Set α
π : δ → Type u_6
inst✝ : (x : δ) → MeasurableSpace (π x)
s : (i : δ) → Set (π i)
hs : ∀ (i : δ), MeasurableSet (s i)
⊢ MeasurableSet (Set.tprod [] s) | exact <a>MeasurableSet.univ</a> | no goals | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/MeasureTheory/MeasurableSpace/Basic.lean |
MeasurableSet.tProd | case cons
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
δ' : Type u_5
ι : Sort uι
s✝ t u : Set α
π : δ → Type u_6
inst✝ : (x : δ) → MeasurableSpace (π x)
s : (i : δ) → Set (π i)
hs : ∀ (i : δ), MeasurableSet (s i)
i : δ
l : List δ
ih : MeasurableSet (Set.tprod l s)
⊢ MeasurableSet (Set.tprod (i :: l) s) | exact (hs i).<a>MeasurableSet.prod</a> ih | no goals | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/MeasureTheory/MeasurableSpace/Basic.lean |
MeasureTheory.lintegral_singleton | α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
m : MeasurableSpace α
μ ν : Measure α
inst✝ : MeasurableSingletonClass α
f : α → ℝ≥0∞
a : α
⊢ ∫⁻ (x : α) in {a}, f x ∂μ = f a * μ {a} | simp only [<a>MeasureTheory.Measure.restrict_singleton</a>, <a>MeasureTheory.lintegral_smul_measure</a>, <a>MeasureTheory.lintegral_dirac</a>, <a>mul_comm</a>] | no goals | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/MeasureTheory/Integral/Lebesgue.lean |
LinearMap.isSymm_iff_eq_flip | R : Type u_1
R₁ : Type u_2
R₂ : Type u_3
R₃ : Type u_4
M : Type u_5
M₁ : Type u_6
M₂ : Type u_7
M₃ : Type u_8
Mₗ₁ : Type u_9
Mₗ₁' : Type u_10
Mₗ₂ : Type u_11
Mₗ₂' : Type u_12
K : Type u_13
K₁ : Type u_14
K₂ : Type u_15
V : Type u_16
V₁ : Type u_17
V₂ : Type u_18
n : Type u_19
inst✝² : CommSemiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
I : R →+* R
B✝ : M →ₛₗ[I] M →ₗ[R] R
B : LinearMap.BilinForm R M
⊢ IsSymm B ↔ B = flip B | constructor <;> intro h | case mp
R : Type u_1
R₁ : Type u_2
R₂ : Type u_3
R₃ : Type u_4
M : Type u_5
M₁ : Type u_6
M₂ : Type u_7
M₃ : Type u_8
Mₗ₁ : Type u_9
Mₗ₁' : Type u_10
Mₗ₂ : Type u_11
Mₗ₂' : Type u_12
K : Type u_13
K₁ : Type u_14
K₂ : Type u_15
V : Type u_16
V₁ : Type u_17
V₂ : Type u_18
n : Type u_19
inst✝² : CommSemiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
I : R →+* R
B✝ : M →ₛₗ[I] M →ₗ[R] R
B : LinearMap.BilinForm R M
h : IsSymm B
⊢ B = flip B
case mpr
R : Type u_1
R₁ : Type u_2
R₂ : Type u_3
R₃ : Type u_4
M : Type u_5
M₁ : Type u_6
M₂ : Type u_7
M₃ : Type u_8
Mₗ₁ : Type u_9
Mₗ₁' : Type u_10
Mₗ₂ : Type u_11
Mₗ₂' : Type u_12
K : Type u_13
K₁ : Type u_14
K₂ : Type u_15
V : Type u_16
V₁ : Type u_17
V₂ : Type u_18
n : Type u_19
inst✝² : CommSemiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
I : R →+* R
B✝ : M →ₛₗ[I] M →ₗ[R] R
B : LinearMap.BilinForm R M
h : B = flip B
⊢ IsSymm B | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/LinearAlgebra/SesquilinearForm.lean |
LinearMap.isSymm_iff_eq_flip | case mpr
R : Type u_1
R₁ : Type u_2
R₂ : Type u_3
R₃ : Type u_4
M : Type u_5
M₁ : Type u_6
M₂ : Type u_7
M₃ : Type u_8
Mₗ₁ : Type u_9
Mₗ₁' : Type u_10
Mₗ₂ : Type u_11
Mₗ₂' : Type u_12
K : Type u_13
K₁ : Type u_14
K₂ : Type u_15
V : Type u_16
V₁ : Type u_17
V₂ : Type u_18
n : Type u_19
inst✝² : CommSemiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
I : R →+* R
B✝ : M →ₛₗ[I] M →ₗ[R] R
B : LinearMap.BilinForm R M
h : B = flip B
⊢ IsSymm B | intro x y | case mpr
R : Type u_1
R₁ : Type u_2
R₂ : Type u_3
R₃ : Type u_4
M : Type u_5
M₁ : Type u_6
M₂ : Type u_7
M₃ : Type u_8
Mₗ₁ : Type u_9
Mₗ₁' : Type u_10
Mₗ₂ : Type u_11
Mₗ₂' : Type u_12
K : Type u_13
K₁ : Type u_14
K₂ : Type u_15
V : Type u_16
V₁ : Type u_17
V₂ : Type u_18
n : Type u_19
inst✝² : CommSemiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
I : R →+* R
B✝ : M →ₛₗ[I] M →ₗ[R] R
B : LinearMap.BilinForm R M
h : B = flip B
x y : M
⊢ (RingHom.id R) ((B x) y) = (B y) x | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/LinearAlgebra/SesquilinearForm.lean |
LinearMap.isSymm_iff_eq_flip | case mpr
R : Type u_1
R₁ : Type u_2
R₂ : Type u_3
R₃ : Type u_4
M : Type u_5
M₁ : Type u_6
M₂ : Type u_7
M₃ : Type u_8
Mₗ₁ : Type u_9
Mₗ₁' : Type u_10
Mₗ₂ : Type u_11
Mₗ₂' : Type u_12
K : Type u_13
K₁ : Type u_14
K₂ : Type u_15
V : Type u_16
V₁ : Type u_17
V₂ : Type u_18
n : Type u_19
inst✝² : CommSemiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
I : R →+* R
B✝ : M →ₛₗ[I] M →ₗ[R] R
B : LinearMap.BilinForm R M
h : B = flip B
x y : M
⊢ (RingHom.id R) ((B x) y) = (B y) x | conv_lhs => rw [h] | case mpr
R : Type u_1
R₁ : Type u_2
R₂ : Type u_3
R₃ : Type u_4
M : Type u_5
M₁ : Type u_6
M₂ : Type u_7
M₃ : Type u_8
Mₗ₁ : Type u_9
Mₗ₁' : Type u_10
Mₗ₂ : Type u_11
Mₗ₂' : Type u_12
K : Type u_13
K₁ : Type u_14
K₂ : Type u_15
V : Type u_16
V₁ : Type u_17
V₂ : Type u_18
n : Type u_19
inst✝² : CommSemiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
I : R →+* R
B✝ : M →ₛₗ[I] M →ₗ[R] R
B : LinearMap.BilinForm R M
h : B = flip B
x y : M
⊢ (RingHom.id R) (((flip B) x) y) = (B y) x | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/LinearAlgebra/SesquilinearForm.lean |
LinearMap.isSymm_iff_eq_flip | case mpr
R : Type u_1
R₁ : Type u_2
R₂ : Type u_3
R₃ : Type u_4
M : Type u_5
M₁ : Type u_6
M₂ : Type u_7
M₃ : Type u_8
Mₗ₁ : Type u_9
Mₗ₁' : Type u_10
Mₗ₂ : Type u_11
Mₗ₂' : Type u_12
K : Type u_13
K₁ : Type u_14
K₂ : Type u_15
V : Type u_16
V₁ : Type u_17
V₂ : Type u_18
n : Type u_19
inst✝² : CommSemiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
I : R →+* R
B✝ : M →ₛₗ[I] M →ₗ[R] R
B : LinearMap.BilinForm R M
h : B = flip B
x y : M
⊢ (RingHom.id R) (((flip B) x) y) = (B y) x | rfl | no goals | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/LinearAlgebra/SesquilinearForm.lean |
LinearMap.isSymm_iff_eq_flip | case mp
R : Type u_1
R₁ : Type u_2
R₂ : Type u_3
R₃ : Type u_4
M : Type u_5
M₁ : Type u_6
M₂ : Type u_7
M₃ : Type u_8
Mₗ₁ : Type u_9
Mₗ₁' : Type u_10
Mₗ₂ : Type u_11
Mₗ₂' : Type u_12
K : Type u_13
K₁ : Type u_14
K₂ : Type u_15
V : Type u_16
V₁ : Type u_17
V₂ : Type u_18
n : Type u_19
inst✝² : CommSemiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
I : R →+* R
B✝ : M →ₛₗ[I] M →ₗ[R] R
B : LinearMap.BilinForm R M
h : IsSymm B
⊢ B = flip B | ext | case mp.h.h
R : Type u_1
R₁ : Type u_2
R₂ : Type u_3
R₃ : Type u_4
M : Type u_5
M₁ : Type u_6
M₂ : Type u_7
M₃ : Type u_8
Mₗ₁ : Type u_9
Mₗ₁' : Type u_10
Mₗ₂ : Type u_11
Mₗ₂' : Type u_12
K : Type u_13
K₁ : Type u_14
K₂ : Type u_15
V : Type u_16
V₁ : Type u_17
V₂ : Type u_18
n : Type u_19
inst✝² : CommSemiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
I : R →+* R
B✝ : M →ₛₗ[I] M →ₗ[R] R
B : LinearMap.BilinForm R M
h : IsSymm B
x✝¹ x✝ : M
⊢ (B x✝¹) x✝ = ((flip B) x✝¹) x✝ | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/LinearAlgebra/SesquilinearForm.lean |
LinearMap.isSymm_iff_eq_flip | case mp.h.h
R : Type u_1
R₁ : Type u_2
R₂ : Type u_3
R₃ : Type u_4
M : Type u_5
M₁ : Type u_6
M₂ : Type u_7
M₃ : Type u_8
Mₗ₁ : Type u_9
Mₗ₁' : Type u_10
Mₗ₂ : Type u_11
Mₗ₂' : Type u_12
K : Type u_13
K₁ : Type u_14
K₂ : Type u_15
V : Type u_16
V₁ : Type u_17
V₂ : Type u_18
n : Type u_19
inst✝² : CommSemiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
I : R →+* R
B✝ : M →ₛₗ[I] M →ₗ[R] R
B : LinearMap.BilinForm R M
h : IsSymm B
x✝¹ x✝ : M
⊢ (B x✝¹) x✝ = ((flip B) x✝¹) x✝ | rw [← h, <a>LinearMap.flip_apply</a>, <a>RingHom.id_apply</a>] | no goals | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/LinearAlgebra/SesquilinearForm.lean |
Nat.shiftLeft'_sub | m✝ n✝ : ℕ
b : Bool
m n k : ℕ
h : k + 1 ≤ n + 1
⊢ shiftLeft' b m (n + 1 - (k + 1)) = shiftLeft' b m (n + 1) >>> (k + 1) | rw [<a>Nat.succ_sub_succ_eq_sub</a>, <a>Nat.shiftLeft'</a>, <a>Nat.add_comm</a>, <a>Nat.shiftRight_add</a>] | m✝ n✝ : ℕ
b : Bool
m n k : ℕ
h : k + 1 ≤ n + 1
⊢ shiftLeft' b m (n - k) = bit b (shiftLeft' b m n) >>> 1 >>> k | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/Data/Nat/Bits.lean |
Nat.shiftLeft'_sub | m✝ n✝ : ℕ
b : Bool
m n k : ℕ
h : k + 1 ≤ n + 1
⊢ shiftLeft' b m (n - k) = bit b (shiftLeft' b m n) >>> 1 >>> k | simp only [shiftLeft'_sub, <a>Nat.le_of_succ_le_succ</a> h, <a>Nat.shiftRight_succ</a>, <a>Nat.shiftRight_zero</a>] | m✝ n✝ : ℕ
b : Bool
m n k : ℕ
h : k + 1 ≤ n + 1
⊢ shiftLeft' b m n >>> k = (bit b (shiftLeft' b m n) / 2) >>> k | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/Data/Nat/Bits.lean |
Nat.shiftLeft'_sub | m✝ n✝ : ℕ
b : Bool
m n k : ℕ
h : k + 1 ≤ n + 1
⊢ shiftLeft' b m n >>> k = (bit b (shiftLeft' b m n) / 2) >>> k | simp [← <a>Nat.div2_val</a>, <a>Nat.div2_bit</a>] | no goals | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/Data/Nat/Bits.lean |
CategoryTheory.PreOneHypercover.sieve₁_eq_pullback_sieve₁' | C : Type u
inst✝² : Category.{v, u} C
A : Type u_1
inst✝¹ : Category.{?u.6702, u_1} A
S : C
E : PreOneHypercover S
i₁ i₂ : E.I₀
inst✝ : HasPullback (E.f i₁) (E.f i₂)
W : C
p₁ : W ⟶ E.X i₁
p₂ : W ⟶ E.X i₂
w : p₁ ≫ E.f i₁ = p₂ ≫ E.f i₂
⊢ E.sieve₁ p₁ p₂ = Sieve.pullback (pullback.lift p₁ p₂ w) (E.sieve₁' i₁ i₂) | ext Z g | case h
C : Type u
inst✝² : Category.{v, u} C
A : Type u_1
inst✝¹ : Category.{?u.6702, u_1} A
S : C
E : PreOneHypercover S
i₁ i₂ : E.I₀
inst✝ : HasPullback (E.f i₁) (E.f i₂)
W : C
p₁ : W ⟶ E.X i₁
p₂ : W ⟶ E.X i₂
w : p₁ ≫ E.f i₁ = p₂ ≫ E.f i₂
Z : C
g : Z ⟶ W
⊢ (E.sieve₁ p₁ p₂).arrows g ↔ (Sieve.pullback (pullback.lift p₁ p₂ w) (E.sieve₁' i₁ i₂)).arrows g | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/CategoryTheory/Sites/OneHypercover.lean |
CategoryTheory.PreOneHypercover.sieve₁_eq_pullback_sieve₁' | case h
C : Type u
inst✝² : Category.{v, u} C
A : Type u_1
inst✝¹ : Category.{?u.6702, u_1} A
S : C
E : PreOneHypercover S
i₁ i₂ : E.I₀
inst✝ : HasPullback (E.f i₁) (E.f i₂)
W : C
p₁ : W ⟶ E.X i₁
p₂ : W ⟶ E.X i₂
w : p₁ ≫ E.f i₁ = p₂ ≫ E.f i₂
Z : C
g : Z ⟶ W
⊢ (E.sieve₁ p₁ p₂).arrows g ↔ (Sieve.pullback (pullback.lift p₁ p₂ w) (E.sieve₁' i₁ i₂)).arrows g | constructor | case h.mp
C : Type u
inst✝² : Category.{v, u} C
A : Type u_1
inst✝¹ : Category.{?u.6702, u_1} A
S : C
E : PreOneHypercover S
i₁ i₂ : E.I₀
inst✝ : HasPullback (E.f i₁) (E.f i₂)
W : C
p₁ : W ⟶ E.X i₁
p₂ : W ⟶ E.X i₂
w : p₁ ≫ E.f i₁ = p₂ ≫ E.f i₂
Z : C
g : Z ⟶ W
⊢ (E.sieve₁ p₁ p₂).arrows g → (Sieve.pullback (pullback.lift p₁ p₂ w) (E.sieve₁' i₁ i₂)).arrows g
case h.mpr
C : Type u
inst✝² : Category.{v, u} C
A : Type u_1
inst✝¹ : Category.{?u.6702, u_1} A
S : C
E : PreOneHypercover S
i₁ i₂ : E.I₀
inst✝ : HasPullback (E.f i₁) (E.f i₂)
W : C
p₁ : W ⟶ E.X i₁
p₂ : W ⟶ E.X i₂
w : p₁ ≫ E.f i₁ = p₂ ≫ E.f i₂
Z : C
g : Z ⟶ W
⊢ (Sieve.pullback (pullback.lift p₁ p₂ w) (E.sieve₁' i₁ i₂)).arrows g → (E.sieve₁ p₁ p₂).arrows g | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/CategoryTheory/Sites/OneHypercover.lean |
CategoryTheory.PreOneHypercover.sieve₁_eq_pullback_sieve₁' | case h.mp
C : Type u
inst✝² : Category.{v, u} C
A : Type u_1
inst✝¹ : Category.{?u.6702, u_1} A
S : C
E : PreOneHypercover S
i₁ i₂ : E.I₀
inst✝ : HasPullback (E.f i₁) (E.f i₂)
W : C
p₁ : W ⟶ E.X i₁
p₂ : W ⟶ E.X i₂
w : p₁ ≫ E.f i₁ = p₂ ≫ E.f i₂
Z : C
g : Z ⟶ W
⊢ (E.sieve₁ p₁ p₂).arrows g → (Sieve.pullback (pullback.lift p₁ p₂ w) (E.sieve₁' i₁ i₂)).arrows g | rintro ⟨j, h, fac₁, fac₂⟩ | case h.mp.intro.intro.intro
C : Type u
inst✝² : Category.{v, u} C
A : Type u_1
inst✝¹ : Category.{?u.6702, u_1} A
S : C
E : PreOneHypercover S
i₁ i₂ : E.I₀
inst✝ : HasPullback (E.f i₁) (E.f i₂)
W : C
p₁ : W ⟶ E.X i₁
p₂ : W ⟶ E.X i₂
w : p₁ ≫ E.f i₁ = p₂ ≫ E.f i₂
Z : C
g : Z ⟶ W
j : E.I₁ i₁ i₂
h : Z ⟶ E.Y j
fac₁ : g ≫ p₁ = h ≫ E.p₁ j
fac₂ : g ≫ p₂ = h ≫ E.p₂ j
⊢ (Sieve.pullback (pullback.lift p₁ p₂ w) (E.sieve₁' i₁ i₂)).arrows g | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/CategoryTheory/Sites/OneHypercover.lean |
CategoryTheory.PreOneHypercover.sieve₁_eq_pullback_sieve₁' | case h.mp.intro.intro.intro
C : Type u
inst✝² : Category.{v, u} C
A : Type u_1
inst✝¹ : Category.{?u.6702, u_1} A
S : C
E : PreOneHypercover S
i₁ i₂ : E.I₀
inst✝ : HasPullback (E.f i₁) (E.f i₂)
W : C
p₁ : W ⟶ E.X i₁
p₂ : W ⟶ E.X i₂
w : p₁ ≫ E.f i₁ = p₂ ≫ E.f i₂
Z : C
g : Z ⟶ W
j : E.I₁ i₁ i₂
h : Z ⟶ E.Y j
fac₁ : g ≫ p₁ = h ≫ E.p₁ j
fac₂ : g ≫ p₂ = h ≫ E.p₂ j
⊢ (Sieve.pullback (pullback.lift p₁ p₂ w) (E.sieve₁' i₁ i₂)).arrows g | exact ⟨_, h, _, ⟨j⟩, by aesop_cat⟩ | no goals | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/CategoryTheory/Sites/OneHypercover.lean |
CategoryTheory.PreOneHypercover.sieve₁_eq_pullback_sieve₁' | C : Type u
inst✝² : Category.{v, u} C
A : Type u_1
inst✝¹ : Category.{?u.6702, u_1} A
S : C
E : PreOneHypercover S
i₁ i₂ : E.I₀
inst✝ : HasPullback (E.f i₁) (E.f i₂)
W : C
p₁ : W ⟶ E.X i₁
p₂ : W ⟶ E.X i₂
w : p₁ ≫ E.f i₁ = p₂ ≫ E.f i₂
Z : C
g : Z ⟶ W
j : E.I₁ i₁ i₂
h : Z ⟶ E.Y j
fac₁ : g ≫ p₁ = h ≫ E.p₁ j
fac₂ : g ≫ p₂ = h ≫ E.p₂ j
⊢ h ≫ E.toPullback j = g ≫ pullback.lift p₁ p₂ w | aesop_cat | no goals | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/CategoryTheory/Sites/OneHypercover.lean |
CategoryTheory.PreOneHypercover.sieve₁_eq_pullback_sieve₁' | case h.mpr
C : Type u
inst✝² : Category.{v, u} C
A : Type u_1
inst✝¹ : Category.{?u.6702, u_1} A
S : C
E : PreOneHypercover S
i₁ i₂ : E.I₀
inst✝ : HasPullback (E.f i₁) (E.f i₂)
W : C
p₁ : W ⟶ E.X i₁
p₂ : W ⟶ E.X i₂
w : p₁ ≫ E.f i₁ = p₂ ≫ E.f i₂
Z : C
g : Z ⟶ W
⊢ (Sieve.pullback (pullback.lift p₁ p₂ w) (E.sieve₁' i₁ i₂)).arrows g → (E.sieve₁ p₁ p₂).arrows g | rintro ⟨_, h, w, ⟨j⟩, fac⟩ | case h.mpr.intro.intro.intro.intro.mk
C : Type u
inst✝² : Category.{v, u} C
A : Type u_1
inst✝¹ : Category.{?u.6702, u_1} A
S : C
E : PreOneHypercover S
i₁ i₂ : E.I₀
inst✝ : HasPullback (E.f i₁) (E.f i₂)
W : C
p₁ : W ⟶ E.X i₁
p₂ : W ⟶ E.X i₂
w : p₁ ≫ E.f i₁ = p₂ ≫ E.f i₂
Z : C
g : Z ⟶ W
Y : C
j : E.I₁ i₁ i₂
h : Z ⟶ E.Y j
fac : h ≫ E.toPullback j = g ≫ pullback.lift p₁ p₂ w
⊢ (E.sieve₁ p₁ p₂).arrows g | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/CategoryTheory/Sites/OneHypercover.lean |
CategoryTheory.PreOneHypercover.sieve₁_eq_pullback_sieve₁' | case h.mpr.intro.intro.intro.intro.mk
C : Type u
inst✝² : Category.{v, u} C
A : Type u_1
inst✝¹ : Category.{?u.6702, u_1} A
S : C
E : PreOneHypercover S
i₁ i₂ : E.I₀
inst✝ : HasPullback (E.f i₁) (E.f i₂)
W : C
p₁ : W ⟶ E.X i₁
p₂ : W ⟶ E.X i₂
w : p₁ ≫ E.f i₁ = p₂ ≫ E.f i₂
Z : C
g : Z ⟶ W
Y : C
j : E.I₁ i₁ i₂
h : Z ⟶ E.Y j
fac : h ≫ E.toPullback j = g ≫ pullback.lift p₁ p₂ w
⊢ (E.sieve₁ p₁ p₂).arrows g | exact ⟨j, h, by simpa using fac.symm =≫ <a>CategoryTheory.Limits.pullback.fst</a>, by simpa using fac.symm =≫ <a>CategoryTheory.Limits.pullback.snd</a>⟩ | no goals | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/CategoryTheory/Sites/OneHypercover.lean |
CategoryTheory.PreOneHypercover.sieve₁_eq_pullback_sieve₁' | C : Type u
inst✝² : Category.{v, u} C
A : Type u_1
inst✝¹ : Category.{?u.6702, u_1} A
S : C
E : PreOneHypercover S
i₁ i₂ : E.I₀
inst✝ : HasPullback (E.f i₁) (E.f i₂)
W : C
p₁ : W ⟶ E.X i₁
p₂ : W ⟶ E.X i₂
w : p₁ ≫ E.f i₁ = p₂ ≫ E.f i₂
Z : C
g : Z ⟶ W
Y : C
j : E.I₁ i₁ i₂
h : Z ⟶ E.Y j
fac : h ≫ E.toPullback j = g ≫ pullback.lift p₁ p₂ w
⊢ g ≫ p₁ = h ≫ E.p₁ j | simpa using fac.symm =≫ <a>CategoryTheory.Limits.pullback.fst</a> | no goals | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/CategoryTheory/Sites/OneHypercover.lean |
CategoryTheory.PreOneHypercover.sieve₁_eq_pullback_sieve₁' | C : Type u
inst✝² : Category.{v, u} C
A : Type u_1
inst✝¹ : Category.{?u.6702, u_1} A
S : C
E : PreOneHypercover S
i₁ i₂ : E.I₀
inst✝ : HasPullback (E.f i₁) (E.f i₂)
W : C
p₁ : W ⟶ E.X i₁
p₂ : W ⟶ E.X i₂
w : p₁ ≫ E.f i₁ = p₂ ≫ E.f i₂
Z : C
g : Z ⟶ W
Y : C
j : E.I₁ i₁ i₂
h : Z ⟶ E.Y j
fac : h ≫ E.toPullback j = g ≫ pullback.lift p₁ p₂ w
⊢ g ≫ p₂ = h ≫ E.p₂ j | simpa using fac.symm =≫ <a>CategoryTheory.Limits.pullback.snd</a> | no goals | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/CategoryTheory/Sites/OneHypercover.lean |
Algebra.intTrace_eq_of_isLocalization | A : Type u_1
K : Type u_2
L : Type u_3
B : Type u_4
inst✝³⁵ : CommRing A
inst✝³⁴ : CommRing B
inst✝³³ : Algebra A B
inst✝³² : Field K
inst✝³¹ : Field L
inst✝³⁰ : Algebra A K
inst✝²⁹ : IsFractionRing A K
inst✝²⁸ : Algebra B L
inst✝²⁷ : Algebra K L
inst✝²⁶ : Algebra A L
inst✝²⁵ : IsScalarTower A B L
inst✝²⁴ : IsScalarTower A K L
inst✝²³ : IsIntegralClosure B A L
inst✝²² : FiniteDimensional K L
Aₘ : Type u_5
Bₘ : Type u_6
inst✝²¹ : CommRing Aₘ
inst✝²⁰ : CommRing Bₘ
inst✝¹⁹ : Algebra Aₘ Bₘ
inst✝¹⁸ : Algebra A Aₘ
inst✝¹⁷ : Algebra B Bₘ
inst✝¹⁶ : Algebra A Bₘ
inst✝¹⁵ : IsScalarTower A Aₘ Bₘ
inst✝¹⁴ : IsScalarTower A B Bₘ
M : Submonoid A
inst✝¹³ : IsLocalization M Aₘ
inst✝¹² : IsLocalization (algebraMapSubmonoid B M) Bₘ
inst✝¹¹ : IsDomain A
inst✝¹⁰ : IsIntegrallyClosed A
inst✝⁹ : IsDomain B
inst✝⁸ : IsIntegrallyClosed B
inst✝⁷ : Module.Finite A B
inst✝⁶ : NoZeroSMulDivisors A B
inst✝⁵ : IsDomain Aₘ
inst✝⁴ : IsIntegrallyClosed Aₘ
inst✝³ : IsDomain Bₘ
inst✝² : IsIntegrallyClosed Bₘ
inst✝¹ : NoZeroSMulDivisors Aₘ Bₘ
inst✝ : Module.Finite Aₘ Bₘ
x : B
⊢ (algebraMap A Aₘ) ((intTrace A B) x) = (intTrace Aₘ Bₘ) ((algebraMap B Bₘ) x) | by_cases hM : 0 ∈ M | case pos
A : Type u_1
K : Type u_2
L : Type u_3
B : Type u_4
inst✝³⁵ : CommRing A
inst✝³⁴ : CommRing B
inst✝³³ : Algebra A B
inst✝³² : Field K
inst✝³¹ : Field L
inst✝³⁰ : Algebra A K
inst✝²⁹ : IsFractionRing A K
inst✝²⁸ : Algebra B L
inst✝²⁷ : Algebra K L
inst✝²⁶ : Algebra A L
inst✝²⁵ : IsScalarTower A B L
inst✝²⁴ : IsScalarTower A K L
inst✝²³ : IsIntegralClosure B A L
inst✝²² : FiniteDimensional K L
Aₘ : Type u_5
Bₘ : Type u_6
inst✝²¹ : CommRing Aₘ
inst✝²⁰ : CommRing Bₘ
inst✝¹⁹ : Algebra Aₘ Bₘ
inst✝¹⁸ : Algebra A Aₘ
inst✝¹⁷ : Algebra B Bₘ
inst✝¹⁶ : Algebra A Bₘ
inst✝¹⁵ : IsScalarTower A Aₘ Bₘ
inst✝¹⁴ : IsScalarTower A B Bₘ
M : Submonoid A
inst✝¹³ : IsLocalization M Aₘ
inst✝¹² : IsLocalization (algebraMapSubmonoid B M) Bₘ
inst✝¹¹ : IsDomain A
inst✝¹⁰ : IsIntegrallyClosed A
inst✝⁹ : IsDomain B
inst✝⁸ : IsIntegrallyClosed B
inst✝⁷ : Module.Finite A B
inst✝⁶ : NoZeroSMulDivisors A B
inst✝⁵ : IsDomain Aₘ
inst✝⁴ : IsIntegrallyClosed Aₘ
inst✝³ : IsDomain Bₘ
inst✝² : IsIntegrallyClosed Bₘ
inst✝¹ : NoZeroSMulDivisors Aₘ Bₘ
inst✝ : Module.Finite Aₘ Bₘ
x : B
hM : 0 ∈ M
⊢ (algebraMap A Aₘ) ((intTrace A B) x) = (intTrace Aₘ Bₘ) ((algebraMap B Bₘ) x)
case neg
A : Type u_1
K : Type u_2
L : Type u_3
B : Type u_4
inst✝³⁵ : CommRing A
inst✝³⁴ : CommRing B
inst✝³³ : Algebra A B
inst✝³² : Field K
inst✝³¹ : Field L
inst✝³⁰ : Algebra A K
inst✝²⁹ : IsFractionRing A K
inst✝²⁸ : Algebra B L
inst✝²⁷ : Algebra K L
inst✝²⁶ : Algebra A L
inst✝²⁵ : IsScalarTower A B L
inst✝²⁴ : IsScalarTower A K L
inst✝²³ : IsIntegralClosure B A L
inst✝²² : FiniteDimensional K L
Aₘ : Type u_5
Bₘ : Type u_6
inst✝²¹ : CommRing Aₘ
inst✝²⁰ : CommRing Bₘ
inst✝¹⁹ : Algebra Aₘ Bₘ
inst✝¹⁸ : Algebra A Aₘ
inst✝¹⁷ : Algebra B Bₘ
inst✝¹⁶ : Algebra A Bₘ
inst✝¹⁵ : IsScalarTower A Aₘ Bₘ
inst✝¹⁴ : IsScalarTower A B Bₘ
M : Submonoid A
inst✝¹³ : IsLocalization M Aₘ
inst✝¹² : IsLocalization (algebraMapSubmonoid B M) Bₘ
inst✝¹¹ : IsDomain A
inst✝¹⁰ : IsIntegrallyClosed A
inst✝⁹ : IsDomain B
inst✝⁸ : IsIntegrallyClosed B
inst✝⁷ : Module.Finite A B
inst✝⁶ : NoZeroSMulDivisors A B
inst✝⁵ : IsDomain Aₘ
inst✝⁴ : IsIntegrallyClosed Aₘ
inst✝³ : IsDomain Bₘ
inst✝² : IsIntegrallyClosed Bₘ
inst✝¹ : NoZeroSMulDivisors Aₘ Bₘ
inst✝ : Module.Finite Aₘ Bₘ
x : B
hM : 0 ∉ M
⊢ (algebraMap A Aₘ) ((intTrace A B) x) = (intTrace Aₘ Bₘ) ((algebraMap B Bₘ) x) | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/RingTheory/IntegralRestrict.lean |