full_name
stringlengths 3
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stringlengths 7
9.32k
| tactic
stringlengths 3
5.35k
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stringlengths 7
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stringclasses 1
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stringclasses 1
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stringlengths 21
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|---|---|---|---|---|---|---|
tendsto_Ioo_atBot | α : Type u_1
β : Type u_2
γ : Type u_3
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : DenselyOrdered α
a b : α
s : Set α
l : Filter β
f✝ : α → β
f : β → ↑(Ioo a b)
⊢ Tendsto f l atBot ↔ Tendsto (fun x => ↑(f x)) l (𝓝[>] a) | rw [← <a>comap_coe_Ioo_nhdsWithin_Ioi</a>, <a>Filter.tendsto_comap_iff</a>] | α : Type u_1
β : Type u_2
γ : Type u_3
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : DenselyOrdered α
a b : α
s : Set α
l : Filter β
f✝ : α → β
f : β → ↑(Ioo a b)
⊢ Tendsto (Subtype.val ∘ f) l (𝓝[>] a) ↔ Tendsto (fun x => ↑(f x)) l (𝓝[>] a) | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/Topology/Order/DenselyOrdered.lean |
tendsto_Ioo_atBot | α : Type u_1
β : Type u_2
γ : Type u_3
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : DenselyOrdered α
a b : α
s : Set α
l : Filter β
f✝ : α → β
f : β → ↑(Ioo a b)
⊢ Tendsto (Subtype.val ∘ f) l (𝓝[>] a) ↔ Tendsto (fun x => ↑(f x)) l (𝓝[>] a) | rfl | no goals | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/Topology/Order/DenselyOrdered.lean |
LieModule.toEnd_lie | R : Type u
L : Type v
M : Type w
inst✝⁶ : CommRing R
inst✝⁵ : LieRing L
inst✝⁴ : LieAlgebra R L
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : LieRingModule L M
inst✝ : LieModule R L M
x y : L
z : M
⊢ (φ x) ⁅y, z⁆ = ⁅((ad R L) x) y, z⁆ + ⁅y, (φ x) z⁆ | simp | no goals | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/Algebra/Lie/OfAssociative.lean |
Set.iUnion_univ_pi | α : Type u_1
β : Type u_2
γ : Type u_3
ι✝ : Sort u_4
ι' : Sort u_5
ι₂ : Sort u_6
κ : ι✝ → Sort u_7
κ₁ : ι✝ → Sort u_8
κ₂ : ι✝ → Sort u_9
κ' : ι' → Sort u_10
π : α → Type u_11
ι : α → Type u_12
t : (a : α) → ι a → Set (π a)
⊢ (⋃ x, univ.pi fun a => t a (x a)) = univ.pi fun a => ⋃ j, t a j | ext | case h
α : Type u_1
β : Type u_2
γ : Type u_3
ι✝ : Sort u_4
ι' : Sort u_5
ι₂ : Sort u_6
κ : ι✝ → Sort u_7
κ₁ : ι✝ → Sort u_8
κ₂ : ι✝ → Sort u_9
κ' : ι' → Sort u_10
π : α → Type u_11
ι : α → Type u_12
t : (a : α) → ι a → Set (π a)
x✝ : (i : α) → π i
⊢ (x✝ ∈ ⋃ x, univ.pi fun a => t a (x a)) ↔ x✝ ∈ univ.pi fun a => ⋃ j, t a j | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/Data/Set/Lattice.lean |
Set.iUnion_univ_pi | case h
α : Type u_1
β : Type u_2
γ : Type u_3
ι✝ : Sort u_4
ι' : Sort u_5
ι₂ : Sort u_6
κ : ι✝ → Sort u_7
κ₁ : ι✝ → Sort u_8
κ₂ : ι✝ → Sort u_9
κ' : ι' → Sort u_10
π : α → Type u_11
ι : α → Type u_12
t : (a : α) → ι a → Set (π a)
x✝ : (i : α) → π i
⊢ (x✝ ∈ ⋃ x, univ.pi fun a => t a (x a)) ↔ x✝ ∈ univ.pi fun a => ⋃ j, t a j | simp [<a>Classical.skolem</a>] | no goals | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/Data/Set/Lattice.lean |
spectrum.units_smul_resolvent_self | R : Type u
A : Type v
inst✝² : CommSemiring R
inst✝¹ : Ring A
inst✝ : Algebra R A
r : Rˣ
a : A
⊢ r • resolvent a ↑r = resolvent (r⁻¹ • a) 1 | simpa only [<a>Units.smul_def</a>, <a>Algebra.id.smul_eq_mul</a>, <a>Units.inv_mul</a>] using @<a>spectrum.units_smul_resolvent</a> _ _ _ _ _ r r a | no goals | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/Algebra/Algebra/Spectrum.lean |
Int.gcd_a_modEq | m n a✝ b✝ c d : ℤ
a b : ℕ
⊢ ↑a * a.gcdA b ≡ ↑(a.gcd b) [ZMOD ↑b] | rw [← <a>add_zero</a> ((a : ℤ) * _), <a>Nat.gcd_eq_gcd_ab</a>] | m n a✝ b✝ c d : ℤ
a b : ℕ
⊢ ↑a * a.gcdA b + 0 ≡ ↑a * a.gcdA b + ↑b * a.gcdB b [ZMOD ↑b] | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/Data/Int/ModEq.lean |
Int.gcd_a_modEq | m n a✝ b✝ c d : ℤ
a b : ℕ
⊢ ↑a * a.gcdA b + 0 ≡ ↑a * a.gcdA b + ↑b * a.gcdB b [ZMOD ↑b] | exact (<a>dvd_mul_right</a> _ _).zero_modEq_int.add_left _ | no goals | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/Data/Int/ModEq.lean |
Function.Semiconj.bijOn_range | α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
π : α → Type u_5
fa : α → α
fb : β → β
f : α → β
g : β → γ
s t : Set α
h : Semiconj f fa fb
ha : Bijective fa
hf : Injective f
⊢ BijOn fb (range f) (range f) | rw [← <a>Set.image_univ</a>] | α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
π : α → Type u_5
fa : α → α
fb : β → β
f : α → β
g : β → γ
s t : Set α
h : Semiconj f fa fb
ha : Bijective fa
hf : Injective f
⊢ BijOn fb (f '' univ) (f '' univ) | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/Data/Set/Function.lean |
Function.Semiconj.bijOn_range | α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
π : α → Type u_5
fa : α → α
fb : β → β
f : α → β
g : β → γ
s t : Set α
h : Semiconj f fa fb
ha : Bijective fa
hf : Injective f
⊢ BijOn fb (f '' univ) (f '' univ) | exact h.bijOn_image (<a>Set.bijective_iff_bijOn_univ</a>.1 ha) hf.injOn | no goals | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/Data/Set/Function.lean |
Matroid.emptyOn_dual_eq | α : Type u_1
M : Matroid α
E B I X R J : Set α
⊢ (emptyOn α)✶ = emptyOn α | rw [← <a>Matroid.ground_eq_empty_iff</a>] | α : Type u_1
M : Matroid α
E B I X R J : Set α
⊢ (emptyOn α)✶.E = ∅ | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/Data/Matroid/Constructions.lean |
Matroid.emptyOn_dual_eq | α : Type u_1
M : Matroid α
E B I X R J : Set α
⊢ (emptyOn α)✶.E = ∅ | rfl | no goals | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/Data/Matroid/Constructions.lean |
Subgroup.fg_iff | M : Type u_1
N : Type u_2
inst✝³ : Monoid M
inst✝² : AddMonoid N
G : Type u_3
H : Type u_4
inst✝¹ : Group G
inst✝ : AddGroup H
P : Subgroup G
x✝ : ∃ S, closure S = P ∧ S.Finite
S : Set G
hS : closure S = P
hf : S.Finite
⊢ closure ↑hf.toFinset = P | simp [hS] | no goals | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/GroupTheory/Finiteness.lean |
IsFractional.sup | R : Type u_1
inst✝² : CommRing R
S : Submonoid R
P : Type u_2
inst✝¹ : CommRing P
inst✝ : Algebra R P
loc : IsLocalization S P
I J : Submodule R P
aI : R
haI : aI ∈ S
hI : ∀ b ∈ I, IsInteger R (aI • b)
aJ : R
haJ : aJ ∈ S
hJ : ∀ b ∈ J, IsInteger R (aJ • b)
b : P
hb : b ∈ I ⊔ J
⊢ IsInteger R ((aI * aJ) • b) | rcases mem_sup.mp hb with ⟨bI, hbI, bJ, hbJ, rfl⟩ | case intro.intro.intro.intro
R : Type u_1
inst✝² : CommRing R
S : Submonoid R
P : Type u_2
inst✝¹ : CommRing P
inst✝ : Algebra R P
loc : IsLocalization S P
I J : Submodule R P
aI : R
haI : aI ∈ S
hI : ∀ b ∈ I, IsInteger R (aI • b)
aJ : R
haJ : aJ ∈ S
hJ : ∀ b ∈ J, IsInteger R (aJ • b)
bI : P
hbI : bI ∈ I
bJ : P
hbJ : bJ ∈ J
hb : bI + bJ ∈ I ⊔ J
⊢ IsInteger R ((aI * aJ) • (bI + bJ)) | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/RingTheory/FractionalIdeal/Basic.lean |
IsFractional.sup | case intro.intro.intro.intro
R : Type u_1
inst✝² : CommRing R
S : Submonoid R
P : Type u_2
inst✝¹ : CommRing P
inst✝ : Algebra R P
loc : IsLocalization S P
I J : Submodule R P
aI : R
haI : aI ∈ S
hI : ∀ b ∈ I, IsInteger R (aI • b)
aJ : R
haJ : aJ ∈ S
hJ : ∀ b ∈ J, IsInteger R (aJ • b)
bI : P
hbI : bI ∈ I
bJ : P
hbJ : bJ ∈ J
hb : bI + bJ ∈ I ⊔ J
⊢ IsInteger R ((aI * aJ) • (bI + bJ)) | rw [<a>smul_add</a>] | case intro.intro.intro.intro
R : Type u_1
inst✝² : CommRing R
S : Submonoid R
P : Type u_2
inst✝¹ : CommRing P
inst✝ : Algebra R P
loc : IsLocalization S P
I J : Submodule R P
aI : R
haI : aI ∈ S
hI : ∀ b ∈ I, IsInteger R (aI • b)
aJ : R
haJ : aJ ∈ S
hJ : ∀ b ∈ J, IsInteger R (aJ • b)
bI : P
hbI : bI ∈ I
bJ : P
hbJ : bJ ∈ J
hb : bI + bJ ∈ I ⊔ J
⊢ IsInteger R ((aI * aJ) • bI + (aI * aJ) • bJ) | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/RingTheory/FractionalIdeal/Basic.lean |
IsFractional.sup | case intro.intro.intro.intro
R : Type u_1
inst✝² : CommRing R
S : Submonoid R
P : Type u_2
inst✝¹ : CommRing P
inst✝ : Algebra R P
loc : IsLocalization S P
I J : Submodule R P
aI : R
haI : aI ∈ S
hI : ∀ b ∈ I, IsInteger R (aI • b)
aJ : R
haJ : aJ ∈ S
hJ : ∀ b ∈ J, IsInteger R (aJ • b)
bI : P
hbI : bI ∈ I
bJ : P
hbJ : bJ ∈ J
hb : bI + bJ ∈ I ⊔ J
⊢ IsInteger R ((aI * aJ) • bI + (aI * aJ) • bJ) | apply <a>IsLocalization.isInteger_add</a> | case intro.intro.intro.intro.ha
R : Type u_1
inst✝² : CommRing R
S : Submonoid R
P : Type u_2
inst✝¹ : CommRing P
inst✝ : Algebra R P
loc : IsLocalization S P
I J : Submodule R P
aI : R
haI : aI ∈ S
hI : ∀ b ∈ I, IsInteger R (aI • b)
aJ : R
haJ : aJ ∈ S
hJ : ∀ b ∈ J, IsInteger R (aJ • b)
bI : P
hbI : bI ∈ I
bJ : P
hbJ : bJ ∈ J
hb : bI + bJ ∈ I ⊔ J
⊢ IsInteger R ((aI * aJ) • bI)
case intro.intro.intro.intro.hb
R : Type u_1
inst✝² : CommRing R
S : Submonoid R
P : Type u_2
inst✝¹ : CommRing P
inst✝ : Algebra R P
loc : IsLocalization S P
I J : Submodule R P
aI : R
haI : aI ∈ S
hI : ∀ b ∈ I, IsInteger R (aI • b)
aJ : R
haJ : aJ ∈ S
hJ : ∀ b ∈ J, IsInteger R (aJ • b)
bI : P
hbI : bI ∈ I
bJ : P
hbJ : bJ ∈ J
hb : bI + bJ ∈ I ⊔ J
⊢ IsInteger R ((aI * aJ) • bJ) | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/RingTheory/FractionalIdeal/Basic.lean |
IsFractional.sup | case intro.intro.intro.intro.ha
R : Type u_1
inst✝² : CommRing R
S : Submonoid R
P : Type u_2
inst✝¹ : CommRing P
inst✝ : Algebra R P
loc : IsLocalization S P
I J : Submodule R P
aI : R
haI : aI ∈ S
hI : ∀ b ∈ I, IsInteger R (aI • b)
aJ : R
haJ : aJ ∈ S
hJ : ∀ b ∈ J, IsInteger R (aJ • b)
bI : P
hbI : bI ∈ I
bJ : P
hbJ : bJ ∈ J
hb : bI + bJ ∈ I ⊔ J
⊢ IsInteger R ((aI * aJ) • bI) | rw [<a>MulAction.mul_smul</a>, <a>SMulCommClass.smul_comm</a>] | case intro.intro.intro.intro.ha
R : Type u_1
inst✝² : CommRing R
S : Submonoid R
P : Type u_2
inst✝¹ : CommRing P
inst✝ : Algebra R P
loc : IsLocalization S P
I J : Submodule R P
aI : R
haI : aI ∈ S
hI : ∀ b ∈ I, IsInteger R (aI • b)
aJ : R
haJ : aJ ∈ S
hJ : ∀ b ∈ J, IsInteger R (aJ • b)
bI : P
hbI : bI ∈ I
bJ : P
hbJ : bJ ∈ J
hb : bI + bJ ∈ I ⊔ J
⊢ IsInteger R (aJ • aI • bI) | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/RingTheory/FractionalIdeal/Basic.lean |
IsFractional.sup | case intro.intro.intro.intro.ha
R : Type u_1
inst✝² : CommRing R
S : Submonoid R
P : Type u_2
inst✝¹ : CommRing P
inst✝ : Algebra R P
loc : IsLocalization S P
I J : Submodule R P
aI : R
haI : aI ∈ S
hI : ∀ b ∈ I, IsInteger R (aI • b)
aJ : R
haJ : aJ ∈ S
hJ : ∀ b ∈ J, IsInteger R (aJ • b)
bI : P
hbI : bI ∈ I
bJ : P
hbJ : bJ ∈ J
hb : bI + bJ ∈ I ⊔ J
⊢ IsInteger R (aJ • aI • bI) | exact <a>IsLocalization.isInteger_smul</a> (hI bI hbI) | no goals | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/RingTheory/FractionalIdeal/Basic.lean |
IsFractional.sup | case intro.intro.intro.intro.hb
R : Type u_1
inst✝² : CommRing R
S : Submonoid R
P : Type u_2
inst✝¹ : CommRing P
inst✝ : Algebra R P
loc : IsLocalization S P
I J : Submodule R P
aI : R
haI : aI ∈ S
hI : ∀ b ∈ I, IsInteger R (aI • b)
aJ : R
haJ : aJ ∈ S
hJ : ∀ b ∈ J, IsInteger R (aJ • b)
bI : P
hbI : bI ∈ I
bJ : P
hbJ : bJ ∈ J
hb : bI + bJ ∈ I ⊔ J
⊢ IsInteger R ((aI * aJ) • bJ) | rw [<a>MulAction.mul_smul</a>] | case intro.intro.intro.intro.hb
R : Type u_1
inst✝² : CommRing R
S : Submonoid R
P : Type u_2
inst✝¹ : CommRing P
inst✝ : Algebra R P
loc : IsLocalization S P
I J : Submodule R P
aI : R
haI : aI ∈ S
hI : ∀ b ∈ I, IsInteger R (aI • b)
aJ : R
haJ : aJ ∈ S
hJ : ∀ b ∈ J, IsInteger R (aJ • b)
bI : P
hbI : bI ∈ I
bJ : P
hbJ : bJ ∈ J
hb : bI + bJ ∈ I ⊔ J
⊢ IsInteger R (aI • aJ • bJ) | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/RingTheory/FractionalIdeal/Basic.lean |
IsFractional.sup | case intro.intro.intro.intro.hb
R : Type u_1
inst✝² : CommRing R
S : Submonoid R
P : Type u_2
inst✝¹ : CommRing P
inst✝ : Algebra R P
loc : IsLocalization S P
I J : Submodule R P
aI : R
haI : aI ∈ S
hI : ∀ b ∈ I, IsInteger R (aI • b)
aJ : R
haJ : aJ ∈ S
hJ : ∀ b ∈ J, IsInteger R (aJ • b)
bI : P
hbI : bI ∈ I
bJ : P
hbJ : bJ ∈ J
hb : bI + bJ ∈ I ⊔ J
⊢ IsInteger R (aI • aJ • bJ) | exact <a>IsLocalization.isInteger_smul</a> (hJ bJ hbJ) | no goals | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/RingTheory/FractionalIdeal/Basic.lean |
TensorProduct.forall_vanishesTrivially_iff_forall_rTensor_injective | R : Type u
inst✝⁵ : CommRing R
M : Type u
inst✝⁴ : AddCommGroup M
inst✝³ : Module R M
N : Type u
inst✝² : AddCommGroup N
inst✝¹ : Module R N
ι : Type u
inst✝ : Fintype ι
m : ι → M
n : ι → N
⊢ (∀ {ι : Type u} [inst : Fintype ι] {m : ι → M} {n : ι → N}, ∑ i : ι, m i ⊗ₜ[R] n i = 0 → VanishesTrivially R m n) ↔
∀ (M' : Submodule R M), Injective ⇑(rTensor N M'.subtype) | constructor | case mp
R : Type u
inst✝⁵ : CommRing R
M : Type u
inst✝⁴ : AddCommGroup M
inst✝³ : Module R M
N : Type u
inst✝² : AddCommGroup N
inst✝¹ : Module R N
ι : Type u
inst✝ : Fintype ι
m : ι → M
n : ι → N
⊢ (∀ {ι : Type u} [inst : Fintype ι] {m : ι → M} {n : ι → N}, ∑ i : ι, m i ⊗ₜ[R] n i = 0 → VanishesTrivially R m n) →
∀ (M' : Submodule R M), Injective ⇑(rTensor N M'.subtype)
case mpr
R : Type u
inst✝⁵ : CommRing R
M : Type u
inst✝⁴ : AddCommGroup M
inst✝³ : Module R M
N : Type u
inst✝² : AddCommGroup N
inst✝¹ : Module R N
ι : Type u
inst✝ : Fintype ι
m : ι → M
n : ι → N
⊢ (∀ (M' : Submodule R M), Injective ⇑(rTensor N M'.subtype)) →
∀ {ι : Type u} [inst : Fintype ι] {m : ι → M} {n : ι → N}, ∑ i : ι, m i ⊗ₜ[R] n i = 0 → VanishesTrivially R m n | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/LinearAlgebra/TensorProduct/Vanishing.lean |
TensorProduct.forall_vanishesTrivially_iff_forall_rTensor_injective | case mp
R : Type u
inst✝⁵ : CommRing R
M : Type u
inst✝⁴ : AddCommGroup M
inst✝³ : Module R M
N : Type u
inst✝² : AddCommGroup N
inst✝¹ : Module R N
ι : Type u
inst✝ : Fintype ι
m : ι → M
n : ι → N
⊢ (∀ {ι : Type u} [inst : Fintype ι] {m : ι → M} {n : ι → N}, ∑ i : ι, m i ⊗ₜ[R] n i = 0 → VanishesTrivially R m n) →
∀ (M' : Submodule R M), Injective ⇑(rTensor N M'.subtype) | intro h | case mp
R : Type u
inst✝⁵ : CommRing R
M : Type u
inst✝⁴ : AddCommGroup M
inst✝³ : Module R M
N : Type u
inst✝² : AddCommGroup N
inst✝¹ : Module R N
ι : Type u
inst✝ : Fintype ι
m : ι → M
n : ι → N
h : ∀ {ι : Type u} [inst : Fintype ι] {m : ι → M} {n : ι → N}, ∑ i : ι, m i ⊗ₜ[R] n i = 0 → VanishesTrivially R m n
⊢ ∀ (M' : Submodule R M), Injective ⇑(rTensor N M'.subtype) | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/LinearAlgebra/TensorProduct/Vanishing.lean |
TensorProduct.forall_vanishesTrivially_iff_forall_rTensor_injective | case mp
R : Type u
inst✝⁵ : CommRing R
M : Type u
inst✝⁴ : AddCommGroup M
inst✝³ : Module R M
N : Type u
inst✝² : AddCommGroup N
inst✝¹ : Module R N
ι : Type u
inst✝ : Fintype ι
m : ι → M
n : ι → N
h : ∀ {ι : Type u} [inst : Fintype ι] {m : ι → M} {n : ι → N}, ∑ i : ι, m i ⊗ₜ[R] n i = 0 → VanishesTrivially R m n
⊢ ∀ (M' : Submodule R M), Injective ⇑(rTensor N M'.subtype) | exact <a>TensorProduct.rTensor_injective_of_forall_vanishesTrivially</a> R h | no goals | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/LinearAlgebra/TensorProduct/Vanishing.lean |
TensorProduct.forall_vanishesTrivially_iff_forall_rTensor_injective | case mpr
R : Type u
inst✝⁵ : CommRing R
M : Type u
inst✝⁴ : AddCommGroup M
inst✝³ : Module R M
N : Type u
inst✝² : AddCommGroup N
inst✝¹ : Module R N
ι : Type u
inst✝ : Fintype ι
m : ι → M
n : ι → N
⊢ (∀ (M' : Submodule R M), Injective ⇑(rTensor N M'.subtype)) →
∀ {ι : Type u} [inst : Fintype ι] {m : ι → M} {n : ι → N}, ∑ i : ι, m i ⊗ₜ[R] n i = 0 → VanishesTrivially R m n | intro h ι _ m n hmn | case mpr
R : Type u
inst✝⁶ : CommRing R
M : Type u
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
N : Type u
inst✝³ : AddCommGroup N
inst✝² : Module R N
ι✝ : Type u
inst✝¹ : Fintype ι✝
m✝ : ι✝ → M
n✝ : ι✝ → N
h : ∀ (M' : Submodule R M), Injective ⇑(rTensor N M'.subtype)
ι : Type u
inst✝ : Fintype ι
m : ι → M
n : ι → N
hmn : ∑ i : ι, m i ⊗ₜ[R] n i = 0
⊢ VanishesTrivially R m n | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/LinearAlgebra/TensorProduct/Vanishing.lean |
TensorProduct.forall_vanishesTrivially_iff_forall_rTensor_injective | case mpr
R : Type u
inst✝⁶ : CommRing R
M : Type u
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
N : Type u
inst✝³ : AddCommGroup N
inst✝² : Module R N
ι✝ : Type u
inst✝¹ : Fintype ι✝
m✝ : ι✝ → M
n✝ : ι✝ → N
h : ∀ (M' : Submodule R M), Injective ⇑(rTensor N M'.subtype)
ι : Type u
inst✝ : Fintype ι
m : ι → M
n : ι → N
hmn : ∑ i : ι, m i ⊗ₜ[R] n i = 0
⊢ VanishesTrivially R m n | exact <a>TensorProduct.vanishesTrivially_of_sum_tmul_eq_zero_of_rTensor_injective</a> R (h _) hmn | no goals | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/LinearAlgebra/TensorProduct/Vanishing.lean |
Finset.prod_powerset_cons | ι : Type u_1
κ : Type u_2
α : Type u_3
β : Type u_4
γ : Type u_5
s s₁ s₂ : Finset α
a : α
f✝ g : α → β
inst✝ : CommMonoid β
ha : a ∉ s
f : Finset α → β
⊢ ∏ t ∈ (cons a s ha).powerset, f t = (∏ t ∈ s.powerset, f t) * ∏ t ∈ s.powerset.attach, f (cons a ↑t ⋯) | classical simp_rw [<a>Finset.cons_eq_insert</a>] rw [<a>Finset.prod_powerset_insert</a> ha, <a>Finset.prod_attach</a> _ fun t ↦ f (<a>Insert.insert</a> a t)] | no goals | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/Algebra/BigOperators/Group/Finset.lean |
Finset.prod_powerset_cons | ι : Type u_1
κ : Type u_2
α : Type u_3
β : Type u_4
γ : Type u_5
s s₁ s₂ : Finset α
a : α
f✝ g : α → β
inst✝ : CommMonoid β
ha : a ∉ s
f : Finset α → β
⊢ ∏ t ∈ (cons a s ha).powerset, f t = (∏ t ∈ s.powerset, f t) * ∏ t ∈ s.powerset.attach, f (cons a ↑t ⋯) | simp_rw [<a>Finset.cons_eq_insert</a>] | ι : Type u_1
κ : Type u_2
α : Type u_3
β : Type u_4
γ : Type u_5
s s₁ s₂ : Finset α
a : α
f✝ g : α → β
inst✝ : CommMonoid β
ha : a ∉ s
f : Finset α → β
⊢ ∏ x ∈ (insert a s).powerset, f x = (∏ x ∈ s.powerset, f x) * ∏ x ∈ s.powerset.attach, f (insert a ↑x) | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/Algebra/BigOperators/Group/Finset.lean |
Finset.prod_powerset_cons | ι : Type u_1
κ : Type u_2
α : Type u_3
β : Type u_4
γ : Type u_5
s s₁ s₂ : Finset α
a : α
f✝ g : α → β
inst✝ : CommMonoid β
ha : a ∉ s
f : Finset α → β
⊢ ∏ x ∈ (insert a s).powerset, f x = (∏ x ∈ s.powerset, f x) * ∏ x ∈ s.powerset.attach, f (insert a ↑x) | rw [<a>Finset.prod_powerset_insert</a> ha, <a>Finset.prod_attach</a> _ fun t ↦ f (<a>Insert.insert</a> a t)] | no goals | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/Algebra/BigOperators/Group/Finset.lean |
MeasureTheory.Measure.compProd_add_left | α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
μ✝ : Measure α
κ✝ η : ↥(kernel α β)
μ ν : Measure α
inst✝² : SFinite μ
inst✝¹ : SFinite ν
κ : ↥(kernel α β)
inst✝ : IsSFiniteKernel κ
⊢ (μ + ν) ⊗ₘ κ = μ ⊗ₘ κ + ν ⊗ₘ κ | rw [<a>MeasureTheory.Measure.compProd</a>, <a>ProbabilityTheory.kernel.const_add</a>, <a>ProbabilityTheory.kernel.compProd_add_left</a>] | α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
μ✝ : Measure α
κ✝ η : ↥(kernel α β)
μ ν : Measure α
inst✝² : SFinite μ
inst✝¹ : SFinite ν
κ : ↥(kernel α β)
inst✝ : IsSFiniteKernel κ
⊢ (kernel.const Unit μ ⊗ₖ kernel.prodMkLeft Unit κ + kernel.const Unit ν ⊗ₖ kernel.prodMkLeft Unit κ) () =
μ ⊗ₘ κ + ν ⊗ₘ κ | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/Probability/Kernel/MeasureCompProd.lean |
MeasureTheory.Measure.compProd_add_left | α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
μ✝ : Measure α
κ✝ η : ↥(kernel α β)
μ ν : Measure α
inst✝² : SFinite μ
inst✝¹ : SFinite ν
κ : ↥(kernel α β)
inst✝ : IsSFiniteKernel κ
⊢ (kernel.const Unit μ ⊗ₖ kernel.prodMkLeft Unit κ + kernel.const Unit ν ⊗ₖ kernel.prodMkLeft Unit κ) () =
μ ⊗ₘ κ + ν ⊗ₘ κ | rfl | no goals | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/Probability/Kernel/MeasureCompProd.lean |
DirectSum.toAddMonoid.unique | ι : Type v
dec_ι : DecidableEq ι
β : ι → Type w
inst✝¹ : (i : ι) → AddCommMonoid (β i)
γ : Type u₁
inst✝ : AddCommMonoid γ
φ : (i : ι) → β i →+ γ
ψ : (⨁ (i : ι), β i) →+ γ
f : ⨁ (i : ι), β i
⊢ ψ f = (toAddMonoid fun i => ψ.comp (of β i)) f | congr | case e_a
ι : Type v
dec_ι : DecidableEq ι
β : ι → Type w
inst✝¹ : (i : ι) → AddCommMonoid (β i)
γ : Type u₁
inst✝ : AddCommMonoid γ
φ : (i : ι) → β i →+ γ
ψ : (⨁ (i : ι), β i) →+ γ
f : ⨁ (i : ι), β i
⊢ ψ = toAddMonoid fun i => ψ.comp (of β i) | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/Algebra/DirectSum/Basic.lean |
DirectSum.toAddMonoid.unique | case e_a
ι : Type v
dec_ι : DecidableEq ι
β : ι → Type w
inst✝¹ : (i : ι) → AddCommMonoid (β i)
γ : Type u₁
inst✝ : AddCommMonoid γ
φ : (i : ι) → β i →+ γ
ψ : (⨁ (i : ι), β i) →+ γ
f : ⨁ (i : ι), β i
⊢ ψ = toAddMonoid fun i => ψ.comp (of β i) | apply <a>DFinsupp.addHom_ext'</a> | case e_a.H
ι : Type v
dec_ι : DecidableEq ι
β : ι → Type w
inst✝¹ : (i : ι) → AddCommMonoid (β i)
γ : Type u₁
inst✝ : AddCommMonoid γ
φ : (i : ι) → β i →+ γ
ψ : (⨁ (i : ι), β i) →+ γ
f : ⨁ (i : ι), β i
⊢ ∀ (x : ι),
ψ.comp (DFinsupp.singleAddHom (fun i => β i) x) =
(toAddMonoid fun i => ψ.comp (of β i)).comp (DFinsupp.singleAddHom (fun i => β i) x) | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/Algebra/DirectSum/Basic.lean |
DirectSum.toAddMonoid.unique | case e_a.H
ι : Type v
dec_ι : DecidableEq ι
β : ι → Type w
inst✝¹ : (i : ι) → AddCommMonoid (β i)
γ : Type u₁
inst✝ : AddCommMonoid γ
φ : (i : ι) → β i →+ γ
ψ : (⨁ (i : ι), β i) →+ γ
f : ⨁ (i : ι), β i
⊢ ∀ (x : ι),
ψ.comp (DFinsupp.singleAddHom (fun i => β i) x) =
(toAddMonoid fun i => ψ.comp (of β i)).comp (DFinsupp.singleAddHom (fun i => β i) x) | simp [<a>DirectSum.toAddMonoid</a>, <a>DirectSum.of</a>] | no goals | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/Algebra/DirectSum/Basic.lean |
Fintype.card_eq_zero_iff | α : Type u_1
β : Type u_2
γ : Type u_3
inst✝¹ : Fintype α
inst✝ : Fintype β
⊢ card α = 0 ↔ IsEmpty α | rw [<a>Fintype.card</a>, <a>Finset.card_eq_zero</a>, <a>Finset.univ_eq_empty_iff</a>] | no goals | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/Data/Fintype/Card.lean |
mem_vectorSpan_pair | k : Type u_1
V : Type u_2
P : Type u_3
inst✝³ : Ring k
inst✝² : AddCommGroup V
inst✝¹ : Module k V
inst✝ : AffineSpace V P
ι : Type u_4
p₁ p₂ : P
v : V
⊢ v ∈ vectorSpan k {p₁, p₂} ↔ ∃ r, r • (p₁ -ᵥ p₂) = v | rw [<a>vectorSpan_pair</a>, <a>Submodule.mem_span_singleton</a>] | no goals | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/LinearAlgebra/AffineSpace/AffineSubspace.lean |
one_lt_tprod | ι : Type u_1
κ : Type u_2
α : Type u_3
inst✝³ : OrderedCommGroup α
inst✝² : TopologicalSpace α
inst✝¹ : TopologicalGroup α
inst✝ : OrderClosedTopology α
f g : ι → α
a₁ a₂ : α
i✝ : ι
hsum : Multipliable g
hg : ∀ (i : ι), 1 ≤ g i
i : ι
hi : 1 < g i
⊢ 1 < ∏' (i : ι), g i | rw [← <a>tprod_one</a>] | ι : Type u_1
κ : Type u_2
α : Type u_3
inst✝³ : OrderedCommGroup α
inst✝² : TopologicalSpace α
inst✝¹ : TopologicalGroup α
inst✝ : OrderClosedTopology α
f g : ι → α
a₁ a₂ : α
i✝ : ι
hsum : Multipliable g
hg : ∀ (i : ι), 1 ≤ g i
i : ι
hi : 1 < g i
⊢ ∏' (x : ?m.53617), 1 < ∏' (i : ι), g i
ι : Type u_1
κ : Type u_2
α : Type u_3
inst✝³ : OrderedCommGroup α
inst✝² : TopologicalSpace α
inst✝¹ : TopologicalGroup α
inst✝ : OrderClosedTopology α
f g : ι → α
a₁ a₂ : α
i✝ : ι
hsum : Multipliable g
hg : ∀ (i : ι), 1 ≤ g i
i : ι
hi : 1 < g i
⊢ Type ?u.53614 | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/Topology/Algebra/InfiniteSum/Order.lean |
one_lt_tprod | ι : Type u_1
κ : Type u_2
α : Type u_3
inst✝³ : OrderedCommGroup α
inst✝² : TopologicalSpace α
inst✝¹ : TopologicalGroup α
inst✝ : OrderClosedTopology α
f g : ι → α
a₁ a₂ : α
i✝ : ι
hsum : Multipliable g
hg : ∀ (i : ι), 1 ≤ g i
i : ι
hi : 1 < g i
⊢ ∏' (x : ?m.53617), 1 < ∏' (i : ι), g i
ι : Type u_1
κ : Type u_2
α : Type u_3
inst✝³ : OrderedCommGroup α
inst✝² : TopologicalSpace α
inst✝¹ : TopologicalGroup α
inst✝ : OrderClosedTopology α
f g : ι → α
a₁ a₂ : α
i✝ : ι
hsum : Multipliable g
hg : ∀ (i : ι), 1 ≤ g i
i : ι
hi : 1 < g i
⊢ Type ?u.53614 | exact <a>tprod_lt_tprod</a> hg hi <a>multipliable_one</a> hsum | no goals | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/Topology/Algebra/InfiniteSum/Order.lean |
AlgebraicGeometry.IsAffineOpen.isQuasiSeparated | X✝ Y : Scheme
f : X✝ ⟶ Y
X : Scheme
U : Opens ↑↑X.toPresheafedSpace
hU : IsAffineOpen U
⊢ IsQuasiSeparated ↑U | rw [<a>isQuasiSeparated_iff_quasiSeparatedSpace</a>] | X✝ Y : Scheme
f : X✝ ⟶ Y
X : Scheme
U : Opens ↑↑X.toPresheafedSpace
hU : IsAffineOpen U
⊢ QuasiSeparatedSpace ↑↑U
case hs
X✝ Y : Scheme
f : X✝ ⟶ Y
X : Scheme
U : Opens ↑↑X.toPresheafedSpace
hU : IsAffineOpen U
⊢ IsOpen ↑U | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/AlgebraicGeometry/Morphisms/QuasiSeparated.lean |
AlgebraicGeometry.IsAffineOpen.isQuasiSeparated | X✝ Y : Scheme
f : X✝ ⟶ Y
X : Scheme
U : Opens ↑↑X.toPresheafedSpace
hU : IsAffineOpen U
⊢ QuasiSeparatedSpace ↑↑U
case hs
X✝ Y : Scheme
f : X✝ ⟶ Y
X : Scheme
U : Opens ↑↑X.toPresheafedSpace
hU : IsAffineOpen U
⊢ IsOpen ↑U | exacts [@<a>AlgebraicGeometry.quasiSeparatedSpace_of_isAffine</a> _ hU, U.isOpen] | no goals | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/AlgebraicGeometry/Morphisms/QuasiSeparated.lean |
FdRep.average_char_eq_finrank_invariants | k : Type u
inst✝³ : Field k
G : Type u
inst✝² : Group G
inst✝¹ : Fintype G
inst✝ : Invertible ↑(Fintype.card G)
V : FdRep k G
⊢ ⅟↑(Fintype.card G) • ∑ g : G, V.character g = ↑(finrank k ↥(invariants V.ρ)) | erw [← (<a>Representation.isProj_averageMap</a> V.ρ).<a>LinearMap.IsProj.trace</a>] | k : Type u
inst✝³ : Field k
G : Type u
inst✝² : Group G
inst✝¹ : Fintype G
inst✝ : Invertible ↑(Fintype.card G)
V : FdRep k G
⊢ ⅟↑(Fintype.card G) • ∑ g : G, V.character g = (trace k (CoeSort.coe V)) (averageMap V.ρ) | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/RepresentationTheory/Character.lean |
FdRep.average_char_eq_finrank_invariants | k : Type u
inst✝³ : Field k
G : Type u
inst✝² : Group G
inst✝¹ : Fintype G
inst✝ : Invertible ↑(Fintype.card G)
V : FdRep k G
⊢ ⅟↑(Fintype.card G) • ∑ g : G, V.character g = (trace k (CoeSort.coe V)) (averageMap V.ρ) | simp [<a>FdRep.character</a>, <a>GroupAlgebra.average</a>, <a>map_sum</a>] | no goals | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/RepresentationTheory/Character.lean |
symmDiff_hnot_self | ι : Type u_1
α : Type u_2
β : Type u_3
π : ι → Type u_4
inst✝ : CoheytingAlgebra α
a : α
⊢ a ∆ (￢a) = ⊤ | rw [<a>symmDiff_comm</a>, <a>hnot_symmDiff_self</a>] | no goals | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/Order/SymmDiff.lean |
le_self_pow | α : Type u_1
M : Type u_2
R : Type u_3
inst✝ : OrderedSemiring R
a b x y : R
n m : ℕ
ha : 1 ≤ a
h : m ≠ 0
⊢ a ≤ a ^ m | simpa only [<a>pow_one</a>] using <a>pow_le_pow_right</a> ha <| <a>Nat.pos_iff_ne_zero</a>.2 h | no goals | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/Algebra/Order/Ring/Basic.lean |
Computation.results_bind | α : Type u
β : Type v
γ : Type w
s : Computation α
f : α → Computation β
a : α
b : β
m n : ℕ
h1 : s.Results a m
h2 : (f a).Results b n
⊢ (s.bind f).Results b (n + m) | have := h1.mem | α : Type u
β : Type v
γ : Type w
s : Computation α
f : α → Computation β
a : α
b : β
m n : ℕ
h1 : s.Results a m
h2 : (f a).Results b n
this : a ∈ s
⊢ (s.bind f).Results b (n + m) | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/Data/Seq/Computation.lean |
Computation.results_bind | α : Type u
β : Type v
γ : Type w
s : Computation α
f : α → Computation β
a : α
b : β
m n : ℕ
h1 : s.Results a m
h2 : (f a).Results b n
this : a ∈ s
⊢ (s.bind f).Results b (n + m) | revert m | α : Type u
β : Type v
γ : Type w
s : Computation α
f : α → Computation β
a : α
b : β
n : ℕ
h2 : (f a).Results b n
this : a ∈ s
⊢ ∀ {m : ℕ}, s.Results a m → (s.bind f).Results b (n + m) | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/Data/Seq/Computation.lean |
Computation.results_bind | α : Type u
β : Type v
γ : Type w
s : Computation α
f : α → Computation β
a : α
b : β
n : ℕ
h2 : (f a).Results b n
this : a ∈ s
⊢ ∀ {m : ℕ}, s.Results a m → (s.bind f).Results b (n + m) | apply <a>Computation.memRecOn</a> this _ fun s IH => _ | α : Type u
β : Type v
γ : Type w
s : Computation α
f : α → Computation β
a : α
b : β
n : ℕ
h2 : (f a).Results b n
this : a ∈ s
⊢ ∀ {m : ℕ}, (pure a).Results a m → ((pure a).bind f).Results b (n + m)
α : Type u
β : Type v
γ : Type w
s : Computation α
f : α → Computation β
a : α
b : β
n : ℕ
h2 : (f a).Results b n
this : a ∈ s
⊢ ∀ (s : Computation α),
(∀ {m : ℕ}, s.Results a m → (s.bind f).Results b (n + m)) →
∀ {m : ℕ}, s.think.Results a m → (s.think.bind f).Results b (n + m) | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/Data/Seq/Computation.lean |
Computation.results_bind | α : Type u
β : Type v
γ : Type w
s : Computation α
f : α → Computation β
a : α
b : β
n : ℕ
h2 : (f a).Results b n
this : a ∈ s
⊢ ∀ {m : ℕ}, (pure a).Results a m → ((pure a).bind f).Results b (n + m) | intro _ h1 | α : Type u
β : Type v
γ : Type w
s : Computation α
f : α → Computation β
a : α
b : β
n : ℕ
h2 : (f a).Results b n
this : a ∈ s
m✝ : ℕ
h1 : (pure a).Results a m✝
⊢ ((pure a).bind f).Results b (n + m✝) | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/Data/Seq/Computation.lean |
Computation.results_bind | α : Type u
β : Type v
γ : Type w
s : Computation α
f : α → Computation β
a : α
b : β
n : ℕ
h2 : (f a).Results b n
this : a ∈ s
m✝ : ℕ
h1 : (pure a).Results a m✝
⊢ ((pure a).bind f).Results b (n + m✝) | rw [<a>Computation.ret_bind</a>] | α : Type u
β : Type v
γ : Type w
s : Computation α
f : α → Computation β
a : α
b : β
n : ℕ
h2 : (f a).Results b n
this : a ∈ s
m✝ : ℕ
h1 : (pure a).Results a m✝
⊢ (f a).Results b (n + m✝) | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/Data/Seq/Computation.lean |
Computation.results_bind | α : Type u
β : Type v
γ : Type w
s : Computation α
f : α → Computation β
a : α
b : β
n : ℕ
h2 : (f a).Results b n
this : a ∈ s
m✝ : ℕ
h1 : (pure a).Results a m✝
⊢ (f a).Results b (n + m✝) | rw [h1.len_unique (<a>Computation.results_pure</a> _)] | α : Type u
β : Type v
γ : Type w
s : Computation α
f : α → Computation β
a : α
b : β
n : ℕ
h2 : (f a).Results b n
this : a ∈ s
m✝ : ℕ
h1 : (pure a).Results a m✝
⊢ (f a).Results b (n + 0) | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/Data/Seq/Computation.lean |
Computation.results_bind | α : Type u
β : Type v
γ : Type w
s : Computation α
f : α → Computation β
a : α
b : β
n : ℕ
h2 : (f a).Results b n
this : a ∈ s
m✝ : ℕ
h1 : (pure a).Results a m✝
⊢ (f a).Results b (n + 0) | exact h2 | no goals | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/Data/Seq/Computation.lean |
Computation.results_bind | α : Type u
β : Type v
γ : Type w
s : Computation α
f : α → Computation β
a : α
b : β
n : ℕ
h2 : (f a).Results b n
this : a ∈ s
⊢ ∀ (s : Computation α),
(∀ {m : ℕ}, s.Results a m → (s.bind f).Results b (n + m)) →
∀ {m : ℕ}, s.think.Results a m → (s.think.bind f).Results b (n + m) | intro _ h3 _ h1 | α : Type u
β : Type v
γ : Type w
s : Computation α
f : α → Computation β
a : α
b : β
n : ℕ
h2 : (f a).Results b n
this : a ∈ s
s✝ : Computation α
h3 : ∀ {m : ℕ}, s✝.Results a m → (s✝.bind f).Results b (n + m)
m✝ : ℕ
h1 : s✝.think.Results a m✝
⊢ (s✝.think.bind f).Results b (n + m✝) | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/Data/Seq/Computation.lean |
Computation.results_bind | α : Type u
β : Type v
γ : Type w
s : Computation α
f : α → Computation β
a : α
b : β
n : ℕ
h2 : (f a).Results b n
this : a ∈ s
s✝ : Computation α
h3 : ∀ {m : ℕ}, s✝.Results a m → (s✝.bind f).Results b (n + m)
m✝ : ℕ
h1 : s✝.think.Results a m✝
⊢ (s✝.think.bind f).Results b (n + m✝) | rw [<a>Computation.think_bind</a>] | α : Type u
β : Type v
γ : Type w
s : Computation α
f : α → Computation β
a : α
b : β
n : ℕ
h2 : (f a).Results b n
this : a ∈ s
s✝ : Computation α
h3 : ∀ {m : ℕ}, s✝.Results a m → (s✝.bind f).Results b (n + m)
m✝ : ℕ
h1 : s✝.think.Results a m✝
⊢ (s✝.bind f).think.Results b (n + m✝) | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/Data/Seq/Computation.lean |
Computation.results_bind | α : Type u
β : Type v
γ : Type w
s : Computation α
f : α → Computation β
a : α
b : β
n : ℕ
h2 : (f a).Results b n
this : a ∈ s
s✝ : Computation α
h3 : ∀ {m : ℕ}, s✝.Results a m → (s✝.bind f).Results b (n + m)
m✝ : ℕ
h1 : s✝.think.Results a m✝
⊢ (s✝.bind f).think.Results b (n + m✝) | cases' <a>Computation.of_results_think</a> h1 with m' h | case intro
α : Type u
β : Type v
γ : Type w
s : Computation α
f : α → Computation β
a : α
b : β
n : ℕ
h2 : (f a).Results b n
this : a ∈ s
s✝ : Computation α
h3 : ∀ {m : ℕ}, s✝.Results a m → (s✝.bind f).Results b (n + m)
m✝ : ℕ
h1 : s✝.think.Results a m✝
m' : ℕ
h : s✝.Results a m' ∧ m✝ = m' + 1
⊢ (s✝.bind f).think.Results b (n + m✝) | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/Data/Seq/Computation.lean |
Computation.results_bind | case intro
α : Type u
β : Type v
γ : Type w
s : Computation α
f : α → Computation β
a : α
b : β
n : ℕ
h2 : (f a).Results b n
this : a ∈ s
s✝ : Computation α
h3 : ∀ {m : ℕ}, s✝.Results a m → (s✝.bind f).Results b (n + m)
m✝ : ℕ
h1 : s✝.think.Results a m✝
m' : ℕ
h : s✝.Results a m' ∧ m✝ = m' + 1
⊢ (s✝.bind f).think.Results b (n + m✝) | cases' h with h1 e | case intro.intro
α : Type u
β : Type v
γ : Type w
s : Computation α
f : α → Computation β
a : α
b : β
n : ℕ
h2 : (f a).Results b n
this : a ∈ s
s✝ : Computation α
h3 : ∀ {m : ℕ}, s✝.Results a m → (s✝.bind f).Results b (n + m)
m✝ : ℕ
h1✝ : s✝.think.Results a m✝
m' : ℕ
h1 : s✝.Results a m'
e : m✝ = m' + 1
⊢ (s✝.bind f).think.Results b (n + m✝) | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/Data/Seq/Computation.lean |
Computation.results_bind | case intro.intro
α : Type u
β : Type v
γ : Type w
s : Computation α
f : α → Computation β
a : α
b : β
n : ℕ
h2 : (f a).Results b n
this : a ∈ s
s✝ : Computation α
h3 : ∀ {m : ℕ}, s✝.Results a m → (s✝.bind f).Results b (n + m)
m✝ : ℕ
h1✝ : s✝.think.Results a m✝
m' : ℕ
h1 : s✝.Results a m'
e : m✝ = m' + 1
⊢ (s✝.bind f).think.Results b (n + m✝) | rw [e] | case intro.intro
α : Type u
β : Type v
γ : Type w
s : Computation α
f : α → Computation β
a : α
b : β
n : ℕ
h2 : (f a).Results b n
this : a ∈ s
s✝ : Computation α
h3 : ∀ {m : ℕ}, s✝.Results a m → (s✝.bind f).Results b (n + m)
m✝ : ℕ
h1✝ : s✝.think.Results a m✝
m' : ℕ
h1 : s✝.Results a m'
e : m✝ = m' + 1
⊢ (s✝.bind f).think.Results b (n + (m' + 1)) | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/Data/Seq/Computation.lean |
Computation.results_bind | case intro.intro
α : Type u
β : Type v
γ : Type w
s : Computation α
f : α → Computation β
a : α
b : β
n : ℕ
h2 : (f a).Results b n
this : a ∈ s
s✝ : Computation α
h3 : ∀ {m : ℕ}, s✝.Results a m → (s✝.bind f).Results b (n + m)
m✝ : ℕ
h1✝ : s✝.think.Results a m✝
m' : ℕ
h1 : s✝.Results a m'
e : m✝ = m' + 1
⊢ (s✝.bind f).think.Results b (n + (m' + 1)) | exact <a>Computation.results_think</a> (h3 h1) | no goals | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/Data/Seq/Computation.lean |
Substring.Valid.foldr | α : Type u_1
f : Char → α → α
init : α
x✝ : Substring
h✝ : x✝.Valid
w✝² w✝¹ w✝ : List Char
h : ValidFor w✝² w✝¹ w✝ x✝
⊢ Substring.foldr f init x✝ = List.foldr f init x✝.toString.data | simp [h.foldr, h.toString] | no goals | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | .lake/packages/batteries/Batteries/Data/String/Lemmas.lean |
contDiff_succ_iff_hasFDerivAt | 𝕜 : Type u
inst✝⁸ : NontriviallyNormedField 𝕜
E : Type uE
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace 𝕜 E
F : Type uF
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
G : Type uG
inst✝³ : NormedAddCommGroup G
inst✝² : NormedSpace 𝕜 G
X : Type uX
inst✝¹ : NormedAddCommGroup X
inst✝ : NormedSpace 𝕜 X
s s₁ t u : Set E
f f₁ : E → F
g : F → G
x x₀ : E
c : F
m n✝ : ℕ∞
p : E → FormalMultilinearSeries 𝕜 E F
n : ℕ
⊢ ContDiff 𝕜 (↑(n + 1)) f ↔ ∃ f', ContDiff 𝕜 (↑n) f' ∧ ∀ (x : E), HasFDerivAt f (f' x) x | simp only [← <a>contDiffOn_univ</a>, ← <a>hasFDerivWithinAt_univ</a>, <a>contDiffOn_succ_iff_hasFDerivWithin</a> <a>uniqueDiffOn_univ</a>, <a>Set.mem_univ</a>, <a>forall_true_left</a>] | no goals | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/Analysis/Calculus/ContDiff/Defs.lean |
Real.exp_neg_one_gt_d9 | ⊢ 0.36787944116 < rexp (-1) | rw [<a>Real.exp_neg</a>, <a>lt_inv</a> _ (<a>Real.exp_pos</a> _)] | ⊢ rexp 1 < 0.36787944116⁻¹
⊢ 0 < 0.36787944116 | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/Data/Complex/ExponentialBounds.lean |
Real.exp_neg_one_gt_d9 | ⊢ rexp 1 < 0.36787944116⁻¹ | refine <a>lt_of_le_of_lt</a> (<a>sub_le_iff_le_add</a>.1 (<a>abs_sub_le_iff</a>.1 <a>Real.exp_one_near_10</a>).1) ?_ | ⊢ 1 / 10 ^ 10 + 2244083 / 825552 < 0.36787944116⁻¹ | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/Data/Complex/ExponentialBounds.lean |
Real.exp_neg_one_gt_d9 | ⊢ 1 / 10 ^ 10 + 2244083 / 825552 < 0.36787944116⁻¹ | norm_num | no goals | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/Data/Complex/ExponentialBounds.lean |
Real.exp_neg_one_gt_d9 | ⊢ 0 < 0.36787944116 | norm_num | no goals | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/Data/Complex/ExponentialBounds.lean |
LinearMap.adjoint_inner_left | 𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
inst✝⁹ : RCLike 𝕜
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : NormedAddCommGroup G
inst✝⁵ : InnerProductSpace 𝕜 E
inst✝⁴ : InnerProductSpace 𝕜 F
inst✝³ : InnerProductSpace 𝕜 G
inst✝² : FiniteDimensional 𝕜 E
inst✝¹ : FiniteDimensional 𝕜 F
inst✝ : FiniteDimensional 𝕜 G
A : E →ₗ[𝕜] F
x : E
y : F
⊢ ⟪(adjoint A) y, x⟫_𝕜 = ⟪y, A x⟫_𝕜 | haveI := <a>FiniteDimensional.complete</a> 𝕜 E | 𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
inst✝⁹ : RCLike 𝕜
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : NormedAddCommGroup G
inst✝⁵ : InnerProductSpace 𝕜 E
inst✝⁴ : InnerProductSpace 𝕜 F
inst✝³ : InnerProductSpace 𝕜 G
inst✝² : FiniteDimensional 𝕜 E
inst✝¹ : FiniteDimensional 𝕜 F
inst✝ : FiniteDimensional 𝕜 G
A : E →ₗ[𝕜] F
x : E
y : F
this : CompleteSpace E
⊢ ⟪(adjoint A) y, x⟫_𝕜 = ⟪y, A x⟫_𝕜 | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/Analysis/InnerProductSpace/Adjoint.lean |
LinearMap.adjoint_inner_left | 𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
inst✝⁹ : RCLike 𝕜
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : NormedAddCommGroup G
inst✝⁵ : InnerProductSpace 𝕜 E
inst✝⁴ : InnerProductSpace 𝕜 F
inst✝³ : InnerProductSpace 𝕜 G
inst✝² : FiniteDimensional 𝕜 E
inst✝¹ : FiniteDimensional 𝕜 F
inst✝ : FiniteDimensional 𝕜 G
A : E →ₗ[𝕜] F
x : E
y : F
this : CompleteSpace E
⊢ ⟪(adjoint A) y, x⟫_𝕜 = ⟪y, A x⟫_𝕜 | haveI := <a>FiniteDimensional.complete</a> 𝕜 F | 𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
inst✝⁹ : RCLike 𝕜
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : NormedAddCommGroup G
inst✝⁵ : InnerProductSpace 𝕜 E
inst✝⁴ : InnerProductSpace 𝕜 F
inst✝³ : InnerProductSpace 𝕜 G
inst✝² : FiniteDimensional 𝕜 E
inst✝¹ : FiniteDimensional 𝕜 F
inst✝ : FiniteDimensional 𝕜 G
A : E →ₗ[𝕜] F
x : E
y : F
this✝ : CompleteSpace E
this : CompleteSpace F
⊢ ⟪(adjoint A) y, x⟫_𝕜 = ⟪y, A x⟫_𝕜 | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/Analysis/InnerProductSpace/Adjoint.lean |
LinearMap.adjoint_inner_left | 𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
inst✝⁹ : RCLike 𝕜
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : NormedAddCommGroup G
inst✝⁵ : InnerProductSpace 𝕜 E
inst✝⁴ : InnerProductSpace 𝕜 F
inst✝³ : InnerProductSpace 𝕜 G
inst✝² : FiniteDimensional 𝕜 E
inst✝¹ : FiniteDimensional 𝕜 F
inst✝ : FiniteDimensional 𝕜 G
A : E →ₗ[𝕜] F
x : E
y : F
this✝ : CompleteSpace E
this : CompleteSpace F
⊢ ⟪(adjoint A) y, x⟫_𝕜 = ⟪y, A x⟫_𝕜 | rw [← <a>LinearMap.coe_toContinuousLinearMap</a> A, <a>LinearMap.adjoint_eq_toCLM_adjoint</a>] | 𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
inst✝⁹ : RCLike 𝕜
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : NormedAddCommGroup G
inst✝⁵ : InnerProductSpace 𝕜 E
inst✝⁴ : InnerProductSpace 𝕜 F
inst✝³ : InnerProductSpace 𝕜 G
inst✝² : FiniteDimensional 𝕜 E
inst✝¹ : FiniteDimensional 𝕜 F
inst✝ : FiniteDimensional 𝕜 G
A : E →ₗ[𝕜] F
x : E
y : F
this✝ : CompleteSpace E
this : CompleteSpace F
⊢ ⟪↑(ContinuousLinearMap.adjoint (toContinuousLinearMap ↑(toContinuousLinearMap A))) y, x⟫_𝕜 =
⟪y, ↑(toContinuousLinearMap A) x⟫_𝕜 | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/Analysis/InnerProductSpace/Adjoint.lean |
LinearMap.adjoint_inner_left | 𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
inst✝⁹ : RCLike 𝕜
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : NormedAddCommGroup G
inst✝⁵ : InnerProductSpace 𝕜 E
inst✝⁴ : InnerProductSpace 𝕜 F
inst✝³ : InnerProductSpace 𝕜 G
inst✝² : FiniteDimensional 𝕜 E
inst✝¹ : FiniteDimensional 𝕜 F
inst✝ : FiniteDimensional 𝕜 G
A : E →ₗ[𝕜] F
x : E
y : F
this✝ : CompleteSpace E
this : CompleteSpace F
⊢ ⟪↑(ContinuousLinearMap.adjoint (toContinuousLinearMap ↑(toContinuousLinearMap A))) y, x⟫_𝕜 =
⟪y, ↑(toContinuousLinearMap A) x⟫_𝕜 | exact <a>ContinuousLinearMap.adjoint_inner_left</a> _ x y | no goals | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/Analysis/InnerProductSpace/Adjoint.lean |
homology'.π'_map | A : Type u
inst✝¹ : Category.{v, u} A
inst✝ : Abelian A
X Y Z : A
f : X ⟶ Y
g : Y ⟶ Z
w : f ≫ g = 0
X' Y' Z' : A
f' : X' ⟶ Y'
g' : Y' ⟶ Z'
w' : f' ≫ g' = 0
α : Arrow.mk f ⟶ Arrow.mk f'
β : Arrow.mk g ⟶ Arrow.mk g'
h : α.right = β.left
⊢ g ≫ β.right = α.right ≫ g' | simp [h, β.w.symm] | no goals | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/CategoryTheory/Abelian/Homology.lean |
homology'.π'_map | A : Type u
inst✝¹ : Category.{v, u} A
inst✝ : Abelian A
X Y Z : A
f : X ⟶ Y
g : Y ⟶ Z
w : f ≫ g = 0
X' Y' Z' : A
f' : X' ⟶ Y'
g' : Y' ⟶ Z'
w' : f' ≫ g' = 0
α : Arrow.mk f ⟶ Arrow.mk f'
β : Arrow.mk g ⟶ Arrow.mk g'
h : α.right = β.left
⊢ π' f g w ≫ map w w' α β h = kernel.map g g' α.right β.right ⋯ ≫ π' f' g' w' | apply_fun fun e => (<a>CategoryTheory.Limits.kernelSubobjectIso</a> _).<a>CategoryTheory.Iso.hom</a> ≫ e | A : Type u
inst✝¹ : Category.{v, u} A
inst✝ : Abelian A
X Y Z : A
f : X ⟶ Y
g : Y ⟶ Z
w : f ≫ g = 0
X' Y' Z' : A
f' : X' ⟶ Y'
g' : Y' ⟶ Z'
w' : f' ≫ g' = 0
α : Arrow.mk f ⟶ Arrow.mk f'
β : Arrow.mk g ⟶ Arrow.mk g'
h : α.right = β.left
⊢ (fun e => (kernelSubobjectIso g).hom ≫ e) (π' f g w ≫ map w w' α β h) =
(fun e => (kernelSubobjectIso g).hom ≫ e) (kernel.map g g' α.right β.right ⋯ ≫ π' f' g' w')
case inj
A : Type u
inst✝¹ : Category.{v, u} A
inst✝ : Abelian A
X Y Z : A
f : X ⟶ Y
g : Y ⟶ Z
w : f ≫ g = 0
X' Y' Z' : A
f' : X' ⟶ Y'
g' : Y' ⟶ Z'
w' : f' ≫ g' = 0
α : Arrow.mk f ⟶ Arrow.mk f'
β : Arrow.mk g ⟶ Arrow.mk g'
h : α.right = β.left
⊢ Function.Injective fun e => (kernelSubobjectIso g).hom ≫ e | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/CategoryTheory/Abelian/Homology.lean |
homology'.π'_map | A : Type u
inst✝¹ : Category.{v, u} A
inst✝ : Abelian A
X Y Z : A
f : X ⟶ Y
g : Y ⟶ Z
w : f ≫ g = 0
X' Y' Z' : A
f' : X' ⟶ Y'
g' : Y' ⟶ Z'
w' : f' ≫ g' = 0
α : Arrow.mk f ⟶ Arrow.mk f'
β : Arrow.mk g ⟶ Arrow.mk g'
h : α.right = β.left
⊢ (fun e => (kernelSubobjectIso g).hom ≫ e) (π' f g w ≫ map w w' α β h) =
(fun e => (kernelSubobjectIso g).hom ≫ e) (kernel.map g g' α.right β.right ⋯ ≫ π' f' g' w')
case inj
A : Type u
inst✝¹ : Category.{v, u} A
inst✝ : Abelian A
X Y Z : A
f : X ⟶ Y
g : Y ⟶ Z
w : f ≫ g = 0
X' Y' Z' : A
f' : X' ⟶ Y'
g' : Y' ⟶ Z'
w' : f' ≫ g' = 0
α : Arrow.mk f ⟶ Arrow.mk f'
β : Arrow.mk g ⟶ Arrow.mk g'
h : α.right = β.left
⊢ Function.Injective fun e => (kernelSubobjectIso g).hom ≫ e | swap | case inj
A : Type u
inst✝¹ : Category.{v, u} A
inst✝ : Abelian A
X Y Z : A
f : X ⟶ Y
g : Y ⟶ Z
w : f ≫ g = 0
X' Y' Z' : A
f' : X' ⟶ Y'
g' : Y' ⟶ Z'
w' : f' ≫ g' = 0
α : Arrow.mk f ⟶ Arrow.mk f'
β : Arrow.mk g ⟶ Arrow.mk g'
h : α.right = β.left
⊢ Function.Injective fun e => (kernelSubobjectIso g).hom ≫ e
A : Type u
inst✝¹ : Category.{v, u} A
inst✝ : Abelian A
X Y Z : A
f : X ⟶ Y
g : Y ⟶ Z
w : f ≫ g = 0
X' Y' Z' : A
f' : X' ⟶ Y'
g' : Y' ⟶ Z'
w' : f' ≫ g' = 0
α : Arrow.mk f ⟶ Arrow.mk f'
β : Arrow.mk g ⟶ Arrow.mk g'
h : α.right = β.left
⊢ (fun e => (kernelSubobjectIso g).hom ≫ e) (π' f g w ≫ map w w' α β h) =
(fun e => (kernelSubobjectIso g).hom ≫ e) (kernel.map g g' α.right β.right ⋯ ≫ π' f' g' w') | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/CategoryTheory/Abelian/Homology.lean |
homology'.π'_map | A : Type u
inst✝¹ : Category.{v, u} A
inst✝ : Abelian A
X Y Z : A
f : X ⟶ Y
g : Y ⟶ Z
w : f ≫ g = 0
X' Y' Z' : A
f' : X' ⟶ Y'
g' : Y' ⟶ Z'
w' : f' ≫ g' = 0
α : Arrow.mk f ⟶ Arrow.mk f'
β : Arrow.mk g ⟶ Arrow.mk g'
h : α.right = β.left
⊢ (fun e => (kernelSubobjectIso g).hom ≫ e) (π' f g w ≫ map w w' α β h) =
(fun e => (kernelSubobjectIso g).hom ≫ e) (kernel.map g g' α.right β.right ⋯ ≫ π' f' g' w') | dsimp [<a>homology'.map</a>] | A : Type u
inst✝¹ : Category.{v, u} A
inst✝ : Abelian A
X Y Z : A
f : X ⟶ Y
g : Y ⟶ Z
w : f ≫ g = 0
X' Y' Z' : A
f' : X' ⟶ Y'
g' : Y' ⟶ Z'
w' : f' ≫ g' = 0
α : Arrow.mk f ⟶ Arrow.mk f'
β : Arrow.mk g ⟶ Arrow.mk g'
h : α.right = β.left
⊢ (kernelSubobjectIso g).hom ≫
π' f g w ≫ cokernel.desc (imageToKernel f g w) (kernelSubobjectMap β ≫ cokernel.π (imageToKernel f' g' w')) ⋯ =
(kernelSubobjectIso g).hom ≫ kernel.map g g' α.right β.right ⋯ ≫ π' f' g' w' | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/CategoryTheory/Abelian/Homology.lean |
homology'.π'_map | A : Type u
inst✝¹ : Category.{v, u} A
inst✝ : Abelian A
X Y Z : A
f : X ⟶ Y
g : Y ⟶ Z
w : f ≫ g = 0
X' Y' Z' : A
f' : X' ⟶ Y'
g' : Y' ⟶ Z'
w' : f' ≫ g' = 0
α : Arrow.mk f ⟶ Arrow.mk f'
β : Arrow.mk g ⟶ Arrow.mk g'
h : α.right = β.left
⊢ (kernelSubobjectIso g).hom ≫
π' f g w ≫ cokernel.desc (imageToKernel f g w) (kernelSubobjectMap β ≫ cokernel.π (imageToKernel f' g' w')) ⋯ =
(kernelSubobjectIso g).hom ≫ kernel.map g g' α.right β.right ⋯ ≫ π' f' g' w' | simp only [<a>homology'.π'_eq_π_assoc</a>] | A : Type u
inst✝¹ : Category.{v, u} A
inst✝ : Abelian A
X Y Z : A
f : X ⟶ Y
g : Y ⟶ Z
w : f ≫ g = 0
X' Y' Z' : A
f' : X' ⟶ Y'
g' : Y' ⟶ Z'
w' : f' ≫ g' = 0
α : Arrow.mk f ⟶ Arrow.mk f'
β : Arrow.mk g ⟶ Arrow.mk g'
h : α.right = β.left
⊢ π f g w ≫ cokernel.desc (imageToKernel f g w) (kernelSubobjectMap β ≫ cokernel.π (imageToKernel f' g' w')) ⋯ =
(kernelSubobjectIso g).hom ≫ kernel.map g g' α.right β.right ⋯ ≫ π' f' g' w' | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/CategoryTheory/Abelian/Homology.lean |
homology'.π'_map | A : Type u
inst✝¹ : Category.{v, u} A
inst✝ : Abelian A
X Y Z : A
f : X ⟶ Y
g : Y ⟶ Z
w : f ≫ g = 0
X' Y' Z' : A
f' : X' ⟶ Y'
g' : Y' ⟶ Z'
w' : f' ≫ g' = 0
α : Arrow.mk f ⟶ Arrow.mk f'
β : Arrow.mk g ⟶ Arrow.mk g'
h : α.right = β.left
⊢ π f g w ≫ cokernel.desc (imageToKernel f g w) (kernelSubobjectMap β ≫ cokernel.π (imageToKernel f' g' w')) ⋯ =
(kernelSubobjectIso g).hom ≫ kernel.map g g' α.right β.right ⋯ ≫ π' f' g' w' | dsimp [<a>homology'.π</a>] | A : Type u
inst✝¹ : Category.{v, u} A
inst✝ : Abelian A
X Y Z : A
f : X ⟶ Y
g : Y ⟶ Z
w : f ≫ g = 0
X' Y' Z' : A
f' : X' ⟶ Y'
g' : Y' ⟶ Z'
w' : f' ≫ g' = 0
α : Arrow.mk f ⟶ Arrow.mk f'
β : Arrow.mk g ⟶ Arrow.mk g'
h : α.right = β.left
⊢ cokernel.π (imageToKernel f g w) ≫
cokernel.desc (imageToKernel f g w) (kernelSubobjectMap β ≫ cokernel.π (imageToKernel f' g' w')) ⋯ =
(kernelSubobjectIso g).hom ≫ kernel.map g g' α.right β.right ⋯ ≫ π' f' g' w' | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/CategoryTheory/Abelian/Homology.lean |
homology'.π'_map | A : Type u
inst✝¹ : Category.{v, u} A
inst✝ : Abelian A
X Y Z : A
f : X ⟶ Y
g : Y ⟶ Z
w : f ≫ g = 0
X' Y' Z' : A
f' : X' ⟶ Y'
g' : Y' ⟶ Z'
w' : f' ≫ g' = 0
α : Arrow.mk f ⟶ Arrow.mk f'
β : Arrow.mk g ⟶ Arrow.mk g'
h : α.right = β.left
⊢ cokernel.π (imageToKernel f g w) ≫
cokernel.desc (imageToKernel f g w) (kernelSubobjectMap β ≫ cokernel.π (imageToKernel f' g' w')) ⋯ =
(kernelSubobjectIso g).hom ≫ kernel.map g g' α.right β.right ⋯ ≫ π' f' g' w' | simp only [<a>CategoryTheory.Limits.cokernel.π_desc</a>] | A : Type u
inst✝¹ : Category.{v, u} A
inst✝ : Abelian A
X Y Z : A
f : X ⟶ Y
g : Y ⟶ Z
w : f ≫ g = 0
X' Y' Z' : A
f' : X' ⟶ Y'
g' : Y' ⟶ Z'
w' : f' ≫ g' = 0
α : Arrow.mk f ⟶ Arrow.mk f'
β : Arrow.mk g ⟶ Arrow.mk g'
h : α.right = β.left
⊢ kernelSubobjectMap β ≫ cokernel.π (imageToKernel f' g' w') =
(kernelSubobjectIso g).hom ≫ kernel.map g g' α.right β.right ⋯ ≫ π' f' g' w' | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/CategoryTheory/Abelian/Homology.lean |
homology'.π'_map | A : Type u
inst✝¹ : Category.{v, u} A
inst✝ : Abelian A
X Y Z : A
f : X ⟶ Y
g : Y ⟶ Z
w : f ≫ g = 0
X' Y' Z' : A
f' : X' ⟶ Y'
g' : Y' ⟶ Z'
w' : f' ≫ g' = 0
α : Arrow.mk f ⟶ Arrow.mk f'
β : Arrow.mk g ⟶ Arrow.mk g'
h : α.right = β.left
⊢ kernelSubobjectMap β ≫ cokernel.π (imageToKernel f' g' w') =
(kernelSubobjectIso g).hom ≫ kernel.map g g' α.right β.right ⋯ ≫ π' f' g' w' | rw [← <a>CategoryTheory.Iso.inv_comp_eq</a>, ← <a>CategoryTheory.Category.assoc</a>] | A : Type u
inst✝¹ : Category.{v, u} A
inst✝ : Abelian A
X Y Z : A
f : X ⟶ Y
g : Y ⟶ Z
w : f ≫ g = 0
X' Y' Z' : A
f' : X' ⟶ Y'
g' : Y' ⟶ Z'
w' : f' ≫ g' = 0
α : Arrow.mk f ⟶ Arrow.mk f'
β : Arrow.mk g ⟶ Arrow.mk g'
h : α.right = β.left
⊢ ((kernelSubobjectIso g).inv ≫ kernelSubobjectMap β) ≫ cokernel.π (imageToKernel f' g' w') =
kernel.map g g' α.right β.right ⋯ ≫ π' f' g' w' | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/CategoryTheory/Abelian/Homology.lean |
homology'.π'_map | A : Type u
inst✝¹ : Category.{v, u} A
inst✝ : Abelian A
X Y Z : A
f : X ⟶ Y
g : Y ⟶ Z
w : f ≫ g = 0
X' Y' Z' : A
f' : X' ⟶ Y'
g' : Y' ⟶ Z'
w' : f' ≫ g' = 0
α : Arrow.mk f ⟶ Arrow.mk f'
β : Arrow.mk g ⟶ Arrow.mk g'
h : α.right = β.left
⊢ ((kernelSubobjectIso g).inv ≫ kernelSubobjectMap β) ≫ cokernel.π (imageToKernel f' g' w') =
kernel.map g g' α.right β.right ⋯ ≫ π' f' g' w' | have : (<a>CategoryTheory.Limits.kernelSubobjectIso</a> g).<a>CategoryTheory.Iso.inv</a> ≫ <a>CategoryTheory.Limits.kernelSubobjectMap</a> β = <a>CategoryTheory.Limits.kernel.map</a> _ _ β.left β.right β.w.symm ≫ (<a>CategoryTheory.Limits.kernelSubobjectIso</a> _).<a>CategoryTheory.Iso.inv</a> := by rw [<a>CategoryTheory.Iso.inv_comp_eq</a>, ← <a>CategoryTheory.Category.assoc</a>, <a>CategoryTheory.Iso.eq_comp_inv</a>] ext dsimp simp | A : Type u
inst✝¹ : Category.{v, u} A
inst✝ : Abelian A
X Y Z : A
f : X ⟶ Y
g : Y ⟶ Z
w : f ≫ g = 0
X' Y' Z' : A
f' : X' ⟶ Y'
g' : Y' ⟶ Z'
w' : f' ≫ g' = 0
α : Arrow.mk f ⟶ Arrow.mk f'
β : Arrow.mk g ⟶ Arrow.mk g'
h : α.right = β.left
this :
(kernelSubobjectIso g).inv ≫ kernelSubobjectMap β =
kernel.map (Arrow.mk g).hom (Arrow.mk g').hom β.left β.right ⋯ ≫ (kernelSubobjectIso (Arrow.mk g').hom).inv
⊢ ((kernelSubobjectIso g).inv ≫ kernelSubobjectMap β) ≫ cokernel.π (imageToKernel f' g' w') =
kernel.map g g' α.right β.right ⋯ ≫ π' f' g' w' | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/CategoryTheory/Abelian/Homology.lean |
homology'.π'_map | A : Type u
inst✝¹ : Category.{v, u} A
inst✝ : Abelian A
X Y Z : A
f : X ⟶ Y
g : Y ⟶ Z
w : f ≫ g = 0
X' Y' Z' : A
f' : X' ⟶ Y'
g' : Y' ⟶ Z'
w' : f' ≫ g' = 0
α : Arrow.mk f ⟶ Arrow.mk f'
β : Arrow.mk g ⟶ Arrow.mk g'
h : α.right = β.left
this :
(kernelSubobjectIso g).inv ≫ kernelSubobjectMap β =
kernel.map (Arrow.mk g).hom (Arrow.mk g').hom β.left β.right ⋯ ≫ (kernelSubobjectIso (Arrow.mk g').hom).inv
⊢ ((kernelSubobjectIso g).inv ≫ kernelSubobjectMap β) ≫ cokernel.π (imageToKernel f' g' w') =
kernel.map g g' α.right β.right ⋯ ≫ π' f' g' w' | rw [this] | A : Type u
inst✝¹ : Category.{v, u} A
inst✝ : Abelian A
X Y Z : A
f : X ⟶ Y
g : Y ⟶ Z
w : f ≫ g = 0
X' Y' Z' : A
f' : X' ⟶ Y'
g' : Y' ⟶ Z'
w' : f' ≫ g' = 0
α : Arrow.mk f ⟶ Arrow.mk f'
β : Arrow.mk g ⟶ Arrow.mk g'
h : α.right = β.left
this :
(kernelSubobjectIso g).inv ≫ kernelSubobjectMap β =
kernel.map (Arrow.mk g).hom (Arrow.mk g').hom β.left β.right ⋯ ≫ (kernelSubobjectIso (Arrow.mk g').hom).inv
⊢ (kernel.map (Arrow.mk g).hom (Arrow.mk g').hom β.left β.right ⋯ ≫ (kernelSubobjectIso (Arrow.mk g').hom).inv) ≫
cokernel.π (imageToKernel f' g' w') =
kernel.map g g' α.right β.right ⋯ ≫ π' f' g' w' | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/CategoryTheory/Abelian/Homology.lean |
homology'.π'_map | A : Type u
inst✝¹ : Category.{v, u} A
inst✝ : Abelian A
X Y Z : A
f : X ⟶ Y
g : Y ⟶ Z
w : f ≫ g = 0
X' Y' Z' : A
f' : X' ⟶ Y'
g' : Y' ⟶ Z'
w' : f' ≫ g' = 0
α : Arrow.mk f ⟶ Arrow.mk f'
β : Arrow.mk g ⟶ Arrow.mk g'
h : α.right = β.left
this :
(kernelSubobjectIso g).inv ≫ kernelSubobjectMap β =
kernel.map (Arrow.mk g).hom (Arrow.mk g').hom β.left β.right ⋯ ≫ (kernelSubobjectIso (Arrow.mk g').hom).inv
⊢ (kernel.map (Arrow.mk g).hom (Arrow.mk g').hom β.left β.right ⋯ ≫ (kernelSubobjectIso (Arrow.mk g').hom).inv) ≫
cokernel.π (imageToKernel f' g' w') =
kernel.map g g' α.right β.right ⋯ ≫ π' f' g' w' | simp only [<a>CategoryTheory.Category.assoc</a>] | A : Type u
inst✝¹ : Category.{v, u} A
inst✝ : Abelian A
X Y Z : A
f : X ⟶ Y
g : Y ⟶ Z
w : f ≫ g = 0
X' Y' Z' : A
f' : X' ⟶ Y'
g' : Y' ⟶ Z'
w' : f' ≫ g' = 0
α : Arrow.mk f ⟶ Arrow.mk f'
β : Arrow.mk g ⟶ Arrow.mk g'
h : α.right = β.left
this :
(kernelSubobjectIso g).inv ≫ kernelSubobjectMap β =
kernel.map (Arrow.mk g).hom (Arrow.mk g').hom β.left β.right ⋯ ≫ (kernelSubobjectIso (Arrow.mk g').hom).inv
⊢ kernel.map (Arrow.mk g).hom (Arrow.mk g').hom β.left β.right ⋯ ≫
(kernelSubobjectIso (Arrow.mk g').hom).inv ≫ cokernel.π (imageToKernel f' g' w') =
kernel.map g g' α.right β.right ⋯ ≫ π' f' g' w' | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/CategoryTheory/Abelian/Homology.lean |
homology'.π'_map | A : Type u
inst✝¹ : Category.{v, u} A
inst✝ : Abelian A
X Y Z : A
f : X ⟶ Y
g : Y ⟶ Z
w : f ≫ g = 0
X' Y' Z' : A
f' : X' ⟶ Y'
g' : Y' ⟶ Z'
w' : f' ≫ g' = 0
α : Arrow.mk f ⟶ Arrow.mk f'
β : Arrow.mk g ⟶ Arrow.mk g'
h : α.right = β.left
this :
(kernelSubobjectIso g).inv ≫ kernelSubobjectMap β =
kernel.map (Arrow.mk g).hom (Arrow.mk g').hom β.left β.right ⋯ ≫ (kernelSubobjectIso (Arrow.mk g').hom).inv
⊢ kernel.map (Arrow.mk g).hom (Arrow.mk g').hom β.left β.right ⋯ ≫
(kernelSubobjectIso (Arrow.mk g').hom).inv ≫ cokernel.π (imageToKernel f' g' w') =
kernel.map g g' α.right β.right ⋯ ≫ π' f' g' w' | dsimp [<a>homology'.π'</a>, <a>homology'IsoCokernelLift</a>] | A : Type u
inst✝¹ : Category.{v, u} A
inst✝ : Abelian A
X Y Z : A
f : X ⟶ Y
g : Y ⟶ Z
w : f ≫ g = 0
X' Y' Z' : A
f' : X' ⟶ Y'
g' : Y' ⟶ Z'
w' : f' ≫ g' = 0
α : Arrow.mk f ⟶ Arrow.mk f'
β : Arrow.mk g ⟶ Arrow.mk g'
h : α.right = β.left
this :
(kernelSubobjectIso g).inv ≫ kernelSubobjectMap β =
kernel.map (Arrow.mk g).hom (Arrow.mk g').hom β.left β.right ⋯ ≫ (kernelSubobjectIso (Arrow.mk g').hom).inv
⊢ kernel.map g g' β.left β.right ⋯ ≫ (kernelSubobjectIso g').inv ≫ cokernel.π (imageToKernel f' g' w') =
kernel.map g g' α.right β.right ⋯ ≫
cokernel.π (kernel.lift g' f' w') ≫
((cokernelIsoOfEq ⋯).inv ≫
cokernel.desc (factorThruImage f' ≫ imageToKernel' f' g' w') (cokernel.π (imageToKernel' f' g' w')) ⋯) ≫
(homology'IsoCokernelImageToKernel' f' g' w').inv | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/CategoryTheory/Abelian/Homology.lean |
homology'.π'_map | A : Type u
inst✝¹ : Category.{v, u} A
inst✝ : Abelian A
X Y Z : A
f : X ⟶ Y
g : Y ⟶ Z
w : f ≫ g = 0
X' Y' Z' : A
f' : X' ⟶ Y'
g' : Y' ⟶ Z'
w' : f' ≫ g' = 0
α : Arrow.mk f ⟶ Arrow.mk f'
β : Arrow.mk g ⟶ Arrow.mk g'
h : α.right = β.left
this :
(kernelSubobjectIso g).inv ≫ kernelSubobjectMap β =
kernel.map (Arrow.mk g).hom (Arrow.mk g').hom β.left β.right ⋯ ≫ (kernelSubobjectIso (Arrow.mk g').hom).inv
⊢ kernel.map g g' β.left β.right ⋯ ≫ (kernelSubobjectIso g').inv ≫ cokernel.π (imageToKernel f' g' w') =
kernel.map g g' α.right β.right ⋯ ≫
cokernel.π (kernel.lift g' f' w') ≫
((cokernelIsoOfEq ⋯).inv ≫
cokernel.desc (factorThruImage f' ≫ imageToKernel' f' g' w') (cokernel.π (imageToKernel' f' g' w')) ⋯) ≫
(homology'IsoCokernelImageToKernel' f' g' w').inv | simp only [<a>CategoryTheory.Limits.cokernelIsoOfEq_inv_comp_desc</a>, <a>CategoryTheory.Limits.cokernel.π_desc_assoc</a>] | A : Type u
inst✝¹ : Category.{v, u} A
inst✝ : Abelian A
X Y Z : A
f : X ⟶ Y
g : Y ⟶ Z
w : f ≫ g = 0
X' Y' Z' : A
f' : X' ⟶ Y'
g' : Y' ⟶ Z'
w' : f' ≫ g' = 0
α : Arrow.mk f ⟶ Arrow.mk f'
β : Arrow.mk g ⟶ Arrow.mk g'
h : α.right = β.left
this :
(kernelSubobjectIso g).inv ≫ kernelSubobjectMap β =
kernel.map (Arrow.mk g).hom (Arrow.mk g').hom β.left β.right ⋯ ≫ (kernelSubobjectIso (Arrow.mk g').hom).inv
⊢ kernel.map g g' β.left β.right ⋯ ≫ (kernelSubobjectIso g').inv ≫ cokernel.π (imageToKernel f' g' w') =
kernel.map g g' α.right β.right ⋯ ≫
cokernel.π (imageToKernel' f' g' w') ≫ (homology'IsoCokernelImageToKernel' f' g' w').inv | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/CategoryTheory/Abelian/Homology.lean |
homology'.π'_map | A : Type u
inst✝¹ : Category.{v, u} A
inst✝ : Abelian A
X Y Z : A
f : X ⟶ Y
g : Y ⟶ Z
w : f ≫ g = 0
X' Y' Z' : A
f' : X' ⟶ Y'
g' : Y' ⟶ Z'
w' : f' ≫ g' = 0
α : Arrow.mk f ⟶ Arrow.mk f'
β : Arrow.mk g ⟶ Arrow.mk g'
h : α.right = β.left
this :
(kernelSubobjectIso g).inv ≫ kernelSubobjectMap β =
kernel.map (Arrow.mk g).hom (Arrow.mk g').hom β.left β.right ⋯ ≫ (kernelSubobjectIso (Arrow.mk g').hom).inv
⊢ kernel.map g g' β.left β.right ⋯ ≫ (kernelSubobjectIso g').inv ≫ cokernel.π (imageToKernel f' g' w') =
kernel.map g g' α.right β.right ⋯ ≫
cokernel.π (imageToKernel' f' g' w') ≫ (homology'IsoCokernelImageToKernel' f' g' w').inv | congr 1 | case e_a
A : Type u
inst✝¹ : Category.{v, u} A
inst✝ : Abelian A
X Y Z : A
f : X ⟶ Y
g : Y ⟶ Z
w : f ≫ g = 0
X' Y' Z' : A
f' : X' ⟶ Y'
g' : Y' ⟶ Z'
w' : f' ≫ g' = 0
α : Arrow.mk f ⟶ Arrow.mk f'
β : Arrow.mk g ⟶ Arrow.mk g'
h : α.right = β.left
this :
(kernelSubobjectIso g).inv ≫ kernelSubobjectMap β =
kernel.map (Arrow.mk g).hom (Arrow.mk g').hom β.left β.right ⋯ ≫ (kernelSubobjectIso (Arrow.mk g').hom).inv
⊢ kernel.map g g' β.left β.right ⋯ = kernel.map g g' α.right β.right ⋯
case e_a
A : Type u
inst✝¹ : Category.{v, u} A
inst✝ : Abelian A
X Y Z : A
f : X ⟶ Y
g : Y ⟶ Z
w : f ≫ g = 0
X' Y' Z' : A
f' : X' ⟶ Y'
g' : Y' ⟶ Z'
w' : f' ≫ g' = 0
α : Arrow.mk f ⟶ Arrow.mk f'
β : Arrow.mk g ⟶ Arrow.mk g'
h : α.right = β.left
this :
(kernelSubobjectIso g).inv ≫ kernelSubobjectMap β =
kernel.map (Arrow.mk g).hom (Arrow.mk g').hom β.left β.right ⋯ ≫ (kernelSubobjectIso (Arrow.mk g').hom).inv
⊢ (kernelSubobjectIso g').inv ≫ cokernel.π (imageToKernel f' g' w') =
cokernel.π (imageToKernel' f' g' w') ≫ (homology'IsoCokernelImageToKernel' f' g' w').inv | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/CategoryTheory/Abelian/Homology.lean |
homology'.π'_map | case inj
A : Type u
inst✝¹ : Category.{v, u} A
inst✝ : Abelian A
X Y Z : A
f : X ⟶ Y
g : Y ⟶ Z
w : f ≫ g = 0
X' Y' Z' : A
f' : X' ⟶ Y'
g' : Y' ⟶ Z'
w' : f' ≫ g' = 0
α : Arrow.mk f ⟶ Arrow.mk f'
β : Arrow.mk g ⟶ Arrow.mk g'
h : α.right = β.left
⊢ Function.Injective fun e => (kernelSubobjectIso g).hom ≫ e | intro i j hh | case inj
A : Type u
inst✝¹ : Category.{v, u} A
inst✝ : Abelian A
X Y Z : A
f : X ⟶ Y
g : Y ⟶ Z
w : f ≫ g = 0
X' Y' Z' : A
f' : X' ⟶ Y'
g' : Y' ⟶ Z'
w' : f' ≫ g' = 0
α : Arrow.mk f ⟶ Arrow.mk f'
β : Arrow.mk g ⟶ Arrow.mk g'
h : α.right = β.left
i j : kernel g ⟶ homology' f' g' w'
hh : (fun e => (kernelSubobjectIso g).hom ≫ e) i = (fun e => (kernelSubobjectIso g).hom ≫ e) j
⊢ i = j | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/CategoryTheory/Abelian/Homology.lean |
homology'.π'_map | case inj
A : Type u
inst✝¹ : Category.{v, u} A
inst✝ : Abelian A
X Y Z : A
f : X ⟶ Y
g : Y ⟶ Z
w : f ≫ g = 0
X' Y' Z' : A
f' : X' ⟶ Y'
g' : Y' ⟶ Z'
w' : f' ≫ g' = 0
α : Arrow.mk f ⟶ Arrow.mk f'
β : Arrow.mk g ⟶ Arrow.mk g'
h : α.right = β.left
i j : kernel g ⟶ homology' f' g' w'
hh : (fun e => (kernelSubobjectIso g).hom ≫ e) i = (fun e => (kernelSubobjectIso g).hom ≫ e) j
⊢ i = j | apply_fun fun e => (<a>CategoryTheory.Limits.kernelSubobjectIso</a> _).<a>CategoryTheory.Iso.inv</a> ≫ e at hh | case inj
A : Type u
inst✝¹ : Category.{v, u} A
inst✝ : Abelian A
X Y Z : A
f : X ⟶ Y
g : Y ⟶ Z
w : f ≫ g = 0
X' Y' Z' : A
f' : X' ⟶ Y'
g' : Y' ⟶ Z'
w' : f' ≫ g' = 0
α : Arrow.mk f ⟶ Arrow.mk f'
β : Arrow.mk g ⟶ Arrow.mk g'
h : α.right = β.left
i j : kernel g ⟶ homology' f' g' w'
hh :
(kernelSubobjectIso g).inv ≫ (fun e => (kernelSubobjectIso g).hom ≫ e) i =
(kernelSubobjectIso g).inv ≫ (fun e => (kernelSubobjectIso g).hom ≫ e) j
⊢ i = j | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/CategoryTheory/Abelian/Homology.lean |
homology'.π'_map | case inj
A : Type u
inst✝¹ : Category.{v, u} A
inst✝ : Abelian A
X Y Z : A
f : X ⟶ Y
g : Y ⟶ Z
w : f ≫ g = 0
X' Y' Z' : A
f' : X' ⟶ Y'
g' : Y' ⟶ Z'
w' : f' ≫ g' = 0
α : Arrow.mk f ⟶ Arrow.mk f'
β : Arrow.mk g ⟶ Arrow.mk g'
h : α.right = β.left
i j : kernel g ⟶ homology' f' g' w'
hh :
(kernelSubobjectIso g).inv ≫ (fun e => (kernelSubobjectIso g).hom ≫ e) i =
(kernelSubobjectIso g).inv ≫ (fun e => (kernelSubobjectIso g).hom ≫ e) j
⊢ i = j | simpa using hh | no goals | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/CategoryTheory/Abelian/Homology.lean |
homology'.π'_map | A : Type u
inst✝¹ : Category.{v, u} A
inst✝ : Abelian A
X Y Z : A
f : X ⟶ Y
g : Y ⟶ Z
w : f ≫ g = 0
X' Y' Z' : A
f' : X' ⟶ Y'
g' : Y' ⟶ Z'
w' : f' ≫ g' = 0
α : Arrow.mk f ⟶ Arrow.mk f'
β : Arrow.mk g ⟶ Arrow.mk g'
h : α.right = β.left
⊢ (kernelSubobjectIso g).inv ≫ kernelSubobjectMap β =
kernel.map (Arrow.mk g).hom (Arrow.mk g').hom β.left β.right ⋯ ≫ (kernelSubobjectIso (Arrow.mk g').hom).inv | rw [<a>CategoryTheory.Iso.inv_comp_eq</a>, ← <a>CategoryTheory.Category.assoc</a>, <a>CategoryTheory.Iso.eq_comp_inv</a>] | A : Type u
inst✝¹ : Category.{v, u} A
inst✝ : Abelian A
X Y Z : A
f : X ⟶ Y
g : Y ⟶ Z
w : f ≫ g = 0
X' Y' Z' : A
f' : X' ⟶ Y'
g' : Y' ⟶ Z'
w' : f' ≫ g' = 0
α : Arrow.mk f ⟶ Arrow.mk f'
β : Arrow.mk g ⟶ Arrow.mk g'
h : α.right = β.left
⊢ kernelSubobjectMap β ≫ (kernelSubobjectIso (Arrow.mk g').hom).hom =
(kernelSubobjectIso g).hom ≫ kernel.map (Arrow.mk g).hom (Arrow.mk g').hom β.left β.right ⋯ | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/CategoryTheory/Abelian/Homology.lean |
homology'.π'_map | A : Type u
inst✝¹ : Category.{v, u} A
inst✝ : Abelian A
X Y Z : A
f : X ⟶ Y
g : Y ⟶ Z
w : f ≫ g = 0
X' Y' Z' : A
f' : X' ⟶ Y'
g' : Y' ⟶ Z'
w' : f' ≫ g' = 0
α : Arrow.mk f ⟶ Arrow.mk f'
β : Arrow.mk g ⟶ Arrow.mk g'
h : α.right = β.left
⊢ kernelSubobjectMap β ≫ (kernelSubobjectIso (Arrow.mk g').hom).hom =
(kernelSubobjectIso g).hom ≫ kernel.map (Arrow.mk g).hom (Arrow.mk g').hom β.left β.right ⋯ | ext | case h
A : Type u
inst✝¹ : Category.{v, u} A
inst✝ : Abelian A
X Y Z : A
f : X ⟶ Y
g : Y ⟶ Z
w : f ≫ g = 0
X' Y' Z' : A
f' : X' ⟶ Y'
g' : Y' ⟶ Z'
w' : f' ≫ g' = 0
α : Arrow.mk f ⟶ Arrow.mk f'
β : Arrow.mk g ⟶ Arrow.mk g'
h : α.right = β.left
⊢ (kernelSubobjectMap β ≫ (kernelSubobjectIso (Arrow.mk g').hom).hom) ≫ equalizer.ι (Arrow.mk g').hom 0 =
((kernelSubobjectIso g).hom ≫ kernel.map (Arrow.mk g).hom (Arrow.mk g').hom β.left β.right ⋯) ≫
equalizer.ι (Arrow.mk g').hom 0 | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/CategoryTheory/Abelian/Homology.lean |
homology'.π'_map | case h
A : Type u
inst✝¹ : Category.{v, u} A
inst✝ : Abelian A
X Y Z : A
f : X ⟶ Y
g : Y ⟶ Z
w : f ≫ g = 0
X' Y' Z' : A
f' : X' ⟶ Y'
g' : Y' ⟶ Z'
w' : f' ≫ g' = 0
α : Arrow.mk f ⟶ Arrow.mk f'
β : Arrow.mk g ⟶ Arrow.mk g'
h : α.right = β.left
⊢ (kernelSubobjectMap β ≫ (kernelSubobjectIso (Arrow.mk g').hom).hom) ≫ equalizer.ι (Arrow.mk g').hom 0 =
((kernelSubobjectIso g).hom ≫ kernel.map (Arrow.mk g).hom (Arrow.mk g').hom β.left β.right ⋯) ≫
equalizer.ι (Arrow.mk g').hom 0 | dsimp | case h
A : Type u
inst✝¹ : Category.{v, u} A
inst✝ : Abelian A
X Y Z : A
f : X ⟶ Y
g : Y ⟶ Z
w : f ≫ g = 0
X' Y' Z' : A
f' : X' ⟶ Y'
g' : Y' ⟶ Z'
w' : f' ≫ g' = 0
α : Arrow.mk f ⟶ Arrow.mk f'
β : Arrow.mk g ⟶ Arrow.mk g'
h : α.right = β.left
⊢ (kernelSubobjectMap β ≫ (kernelSubobjectIso g').hom) ≫ kernel.ι g' =
((kernelSubobjectIso g).hom ≫ kernel.map g g' β.left β.right ⋯) ≫ kernel.ι g' | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/CategoryTheory/Abelian/Homology.lean |
homology'.π'_map | case h
A : Type u
inst✝¹ : Category.{v, u} A
inst✝ : Abelian A
X Y Z : A
f : X ⟶ Y
g : Y ⟶ Z
w : f ≫ g = 0
X' Y' Z' : A
f' : X' ⟶ Y'
g' : Y' ⟶ Z'
w' : f' ≫ g' = 0
α : Arrow.mk f ⟶ Arrow.mk f'
β : Arrow.mk g ⟶ Arrow.mk g'
h : α.right = β.left
⊢ (kernelSubobjectMap β ≫ (kernelSubobjectIso g').hom) ≫ kernel.ι g' =
((kernelSubobjectIso g).hom ≫ kernel.map g g' β.left β.right ⋯) ≫ kernel.ι g' | simp | no goals | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/CategoryTheory/Abelian/Homology.lean |
homology'.π'_map | case e_a
A : Type u
inst✝¹ : Category.{v, u} A
inst✝ : Abelian A
X Y Z : A
f : X ⟶ Y
g : Y ⟶ Z
w : f ≫ g = 0
X' Y' Z' : A
f' : X' ⟶ Y'
g' : Y' ⟶ Z'
w' : f' ≫ g' = 0
α : Arrow.mk f ⟶ Arrow.mk f'
β : Arrow.mk g ⟶ Arrow.mk g'
h : α.right = β.left
this :
(kernelSubobjectIso g).inv ≫ kernelSubobjectMap β =
kernel.map (Arrow.mk g).hom (Arrow.mk g').hom β.left β.right ⋯ ≫ (kernelSubobjectIso (Arrow.mk g').hom).inv
⊢ kernel.map g g' β.left β.right ⋯ = kernel.map g g' α.right β.right ⋯ | congr | case e_a.e_p
A : Type u
inst✝¹ : Category.{v, u} A
inst✝ : Abelian A
X Y Z : A
f : X ⟶ Y
g : Y ⟶ Z
w : f ≫ g = 0
X' Y' Z' : A
f' : X' ⟶ Y'
g' : Y' ⟶ Z'
w' : f' ≫ g' = 0
α : Arrow.mk f ⟶ Arrow.mk f'
β : Arrow.mk g ⟶ Arrow.mk g'
h : α.right = β.left
this :
(kernelSubobjectIso g).inv ≫ kernelSubobjectMap β =
kernel.map (Arrow.mk g).hom (Arrow.mk g').hom β.left β.right ⋯ ≫ (kernelSubobjectIso (Arrow.mk g').hom).inv
⊢ β.left = α.right | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/CategoryTheory/Abelian/Homology.lean |
homology'.π'_map | case e_a.e_p
A : Type u
inst✝¹ : Category.{v, u} A
inst✝ : Abelian A
X Y Z : A
f : X ⟶ Y
g : Y ⟶ Z
w : f ≫ g = 0
X' Y' Z' : A
f' : X' ⟶ Y'
g' : Y' ⟶ Z'
w' : f' ≫ g' = 0
α : Arrow.mk f ⟶ Arrow.mk f'
β : Arrow.mk g ⟶ Arrow.mk g'
h : α.right = β.left
this :
(kernelSubobjectIso g).inv ≫ kernelSubobjectMap β =
kernel.map (Arrow.mk g).hom (Arrow.mk g').hom β.left β.right ⋯ ≫ (kernelSubobjectIso (Arrow.mk g').hom).inv
⊢ β.left = α.right | exact h.symm | no goals | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/CategoryTheory/Abelian/Homology.lean |
homology'.π'_map | case e_a
A : Type u
inst✝¹ : Category.{v, u} A
inst✝ : Abelian A
X Y Z : A
f : X ⟶ Y
g : Y ⟶ Z
w : f ≫ g = 0
X' Y' Z' : A
f' : X' ⟶ Y'
g' : Y' ⟶ Z'
w' : f' ≫ g' = 0
α : Arrow.mk f ⟶ Arrow.mk f'
β : Arrow.mk g ⟶ Arrow.mk g'
h : α.right = β.left
this :
(kernelSubobjectIso g).inv ≫ kernelSubobjectMap β =
kernel.map (Arrow.mk g).hom (Arrow.mk g').hom β.left β.right ⋯ ≫ (kernelSubobjectIso (Arrow.mk g').hom).inv
⊢ (kernelSubobjectIso g').inv ≫ cokernel.π (imageToKernel f' g' w') =
cokernel.π (imageToKernel' f' g' w') ≫ (homology'IsoCokernelImageToKernel' f' g' w').inv | rw [<a>CategoryTheory.Iso.inv_comp_eq</a>, ← <a>CategoryTheory.Category.assoc</a>, <a>CategoryTheory.Iso.eq_comp_inv</a>] | case e_a
A : Type u
inst✝¹ : Category.{v, u} A
inst✝ : Abelian A
X Y Z : A
f : X ⟶ Y
g : Y ⟶ Z
w : f ≫ g = 0
X' Y' Z' : A
f' : X' ⟶ Y'
g' : Y' ⟶ Z'
w' : f' ≫ g' = 0
α : Arrow.mk f ⟶ Arrow.mk f'
β : Arrow.mk g ⟶ Arrow.mk g'
h : α.right = β.left
this :
(kernelSubobjectIso g).inv ≫ kernelSubobjectMap β =
kernel.map (Arrow.mk g).hom (Arrow.mk g').hom β.left β.right ⋯ ≫ (kernelSubobjectIso (Arrow.mk g').hom).inv
⊢ cokernel.π (imageToKernel f' g' w') ≫ (homology'IsoCokernelImageToKernel' f' g' w').hom =
(kernelSubobjectIso g').hom ≫ cokernel.π (imageToKernel' f' g' w') | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/CategoryTheory/Abelian/Homology.lean |
homology'.π'_map | case e_a
A : Type u
inst✝¹ : Category.{v, u} A
inst✝ : Abelian A
X Y Z : A
f : X ⟶ Y
g : Y ⟶ Z
w : f ≫ g = 0
X' Y' Z' : A
f' : X' ⟶ Y'
g' : Y' ⟶ Z'
w' : f' ≫ g' = 0
α : Arrow.mk f ⟶ Arrow.mk f'
β : Arrow.mk g ⟶ Arrow.mk g'
h : α.right = β.left
this :
(kernelSubobjectIso g).inv ≫ kernelSubobjectMap β =
kernel.map (Arrow.mk g).hom (Arrow.mk g').hom β.left β.right ⋯ ≫ (kernelSubobjectIso (Arrow.mk g').hom).inv
⊢ cokernel.π (imageToKernel f' g' w') ≫ (homology'IsoCokernelImageToKernel' f' g' w').hom =
(kernelSubobjectIso g').hom ≫ cokernel.π (imageToKernel' f' g' w') | dsimp [<a>homology'IsoCokernelImageToKernel'</a>] | case e_a
A : Type u
inst✝¹ : Category.{v, u} A
inst✝ : Abelian A
X Y Z : A
f : X ⟶ Y
g : Y ⟶ Z
w : f ≫ g = 0
X' Y' Z' : A
f' : X' ⟶ Y'
g' : Y' ⟶ Z'
w' : f' ≫ g' = 0
α : Arrow.mk f ⟶ Arrow.mk f'
β : Arrow.mk g ⟶ Arrow.mk g'
h : α.right = β.left
this :
(kernelSubobjectIso g).inv ≫ kernelSubobjectMap β =
kernel.map (Arrow.mk g).hom (Arrow.mk g').hom β.left β.right ⋯ ≫ (kernelSubobjectIso (Arrow.mk g').hom).inv
⊢ cokernel.π (imageToKernel f' g' w') ≫
cokernel.map (imageToKernel f' g' w') (imageToKernel' f' g' w') (imageSubobjectIso f').hom
(kernelSubobjectIso g').hom ⋯ =
(kernelSubobjectIso g').hom ≫ cokernel.π (imageToKernel' f' g' w') | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/CategoryTheory/Abelian/Homology.lean |
homology'.π'_map | case e_a
A : Type u
inst✝¹ : Category.{v, u} A
inst✝ : Abelian A
X Y Z : A
f : X ⟶ Y
g : Y ⟶ Z
w : f ≫ g = 0
X' Y' Z' : A
f' : X' ⟶ Y'
g' : Y' ⟶ Z'
w' : f' ≫ g' = 0
α : Arrow.mk f ⟶ Arrow.mk f'
β : Arrow.mk g ⟶ Arrow.mk g'
h : α.right = β.left
this :
(kernelSubobjectIso g).inv ≫ kernelSubobjectMap β =
kernel.map (Arrow.mk g).hom (Arrow.mk g').hom β.left β.right ⋯ ≫ (kernelSubobjectIso (Arrow.mk g').hom).inv
⊢ cokernel.π (imageToKernel f' g' w') ≫
cokernel.map (imageToKernel f' g' w') (imageToKernel' f' g' w') (imageSubobjectIso f').hom
(kernelSubobjectIso g').hom ⋯ =
(kernelSubobjectIso g').hom ≫ cokernel.π (imageToKernel' f' g' w') | simp | no goals | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/CategoryTheory/Abelian/Homology.lean |
List.IsRotated.refl | α : Type u
l✝ l' l : List α
⊢ l.rotate 0 = l | simp | no goals | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/Data/List/Rotate.lean |
Finset.Ioc_eq_empty_iff | ι : Type u_1
α : Type u_2
inst✝¹ : Preorder α
inst✝ : LocallyFiniteOrder α
a a₁ a₂ b b₁ b₂ c x : α
⊢ Ioc a b = ∅ ↔ ¬a < b | rw [← <a>Finset.coe_eq_empty</a>, <a>Finset.coe_Ioc</a>, <a>Set.Ioc_eq_empty_iff</a>] | no goals | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/Order/Interval/Finset/Basic.lean |
Matrix.sum_cramer_apply | m : Type u
n : Type v
α : Type w
inst✝⁴ : DecidableEq n
inst✝³ : Fintype n
inst✝² : DecidableEq m
inst✝¹ : Fintype m
inst✝ : CommRing α
A : Matrix n n α
b : n → α
β : Type u_1
s : Finset β
f : n → β → α
i : n
⊢ (∑ x ∈ s, A.cramer fun j => f j x) i = A.cramer (fun j => ∑ x ∈ s, f j x) i | rw [<a>Matrix.sum_cramer</a>, <a>Matrix.cramer_apply</a>, <a>Matrix.cramer_apply</a>] | m : Type u
n : Type v
α : Type w
inst✝⁴ : DecidableEq n
inst✝³ : Fintype n
inst✝² : DecidableEq m
inst✝¹ : Fintype m
inst✝ : CommRing α
A : Matrix n n α
b : n → α
β : Type u_1
s : Finset β
f : n → β → α
i : n
⊢ (A.updateColumn i (∑ x ∈ s, fun j => f j x)).det = (A.updateColumn i fun j => ∑ x ∈ s, f j x).det | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/LinearAlgebra/Matrix/Adjugate.lean |
Matrix.sum_cramer_apply | m : Type u
n : Type v
α : Type w
inst✝⁴ : DecidableEq n
inst✝³ : Fintype n
inst✝² : DecidableEq m
inst✝¹ : Fintype m
inst✝ : CommRing α
A : Matrix n n α
b : n → α
β : Type u_1
s : Finset β
f : n → β → α
i : n
⊢ (A.updateColumn i (∑ x ∈ s, fun j => f j x)).det = (A.updateColumn i fun j => ∑ x ∈ s, f j x).det | simp only [<a>Matrix.updateColumn</a>] | m : Type u
n : Type v
α : Type w
inst✝⁴ : DecidableEq n
inst✝³ : Fintype n
inst✝² : DecidableEq m
inst✝¹ : Fintype m
inst✝ : CommRing α
A : Matrix n n α
b : n → α
β : Type u_1
s : Finset β
f : n → β → α
i : n
⊢ (of fun i_1 => Function.update (A i_1) i ((∑ x ∈ s, fun j => f j x) i_1)).det =
(of fun i_1 => Function.update (A i_1) i (∑ x ∈ s, f i_1 x)).det | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/LinearAlgebra/Matrix/Adjugate.lean |
Matrix.sum_cramer_apply | m : Type u
n : Type v
α : Type w
inst✝⁴ : DecidableEq n
inst✝³ : Fintype n
inst✝² : DecidableEq m
inst✝¹ : Fintype m
inst✝ : CommRing α
A : Matrix n n α
b : n → α
β : Type u_1
s : Finset β
f : n → β → α
i : n
⊢ (of fun i_1 => Function.update (A i_1) i ((∑ x ∈ s, fun j => f j x) i_1)).det =
(of fun i_1 => Function.update (A i_1) i (∑ x ∈ s, f i_1 x)).det | congr with j | case e_M.h.e_6.h.h.h
m : Type u
n : Type v
α : Type w
inst✝⁴ : DecidableEq n
inst✝³ : Fintype n
inst✝² : DecidableEq m
inst✝¹ : Fintype m
inst✝ : CommRing α
A : Matrix n n α
b : n → α
β : Type u_1
s : Finset β
f : n → β → α
i j x✝ : n
⊢ Function.update (A j) i ((∑ x ∈ s, fun j => f j x) j) x✝ = Function.update (A j) i (∑ x ∈ s, f j x) x✝ | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/LinearAlgebra/Matrix/Adjugate.lean |
Matrix.sum_cramer_apply | case e_M.h.e_6.h.h.h
m : Type u
n : Type v
α : Type w
inst✝⁴ : DecidableEq n
inst✝³ : Fintype n
inst✝² : DecidableEq m
inst✝¹ : Fintype m
inst✝ : CommRing α
A : Matrix n n α
b : n → α
β : Type u_1
s : Finset β
f : n → β → α
i j x✝ : n
⊢ Function.update (A j) i ((∑ x ∈ s, fun j => f j x) j) x✝ = Function.update (A j) i (∑ x ∈ s, f j x) x✝ | congr | case e_M.h.e_6.h.h.h.e_v
m : Type u
n : Type v
α : Type w
inst✝⁴ : DecidableEq n
inst✝³ : Fintype n
inst✝² : DecidableEq m
inst✝¹ : Fintype m
inst✝ : CommRing α
A : Matrix n n α
b : n → α
β : Type u_1
s : Finset β
f : n → β → α
i j x✝ : n
⊢ (∑ x ∈ s, fun j => f j x) j = ∑ x ∈ s, f j x | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/LinearAlgebra/Matrix/Adjugate.lean |
Matrix.sum_cramer_apply | case e_M.h.e_6.h.h.h.e_v
m : Type u
n : Type v
α : Type w
inst✝⁴ : DecidableEq n
inst✝³ : Fintype n
inst✝² : DecidableEq m
inst✝¹ : Fintype m
inst✝ : CommRing α
A : Matrix n n α
b : n → α
β : Type u_1
s : Finset β
f : n → β → α
i j x✝ : n
⊢ (∑ x ∈ s, fun j => f j x) j = ∑ x ∈ s, f j x | apply <a>Finset.sum_apply</a> | no goals | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/LinearAlgebra/Matrix/Adjugate.lean |
Polynomial.rootSet_prod | R : Type u
S : Type v
k : Type y
A : Type z
a b : R
n : ℕ
inst✝³ : Field R
p q : R[X]
inst✝² : CommRing S
inst✝¹ : IsDomain S
inst✝ : Algebra R S
ι : Type u_1
f : ι → R[X]
s : Finset ι
h : s.prod f ≠ 0
⊢ (s.prod f).rootSet S = ⋃ i ∈ s, (f i).rootSet S | classical simp only [<a>Polynomial.rootSet</a>, <a>Polynomial.aroots</a>, ← <a>Finset.mem_coe</a>] rw [<a>Polynomial.map_prod</a>, <a>Polynomial.roots_prod</a>, <a>Finset.bind_toFinset</a>, s.val_toFinset, <a>Finset.coe_biUnion</a>] rwa [← <a>Polynomial.map_prod</a>, <a>Ne</a>, <a>Polynomial.map_eq_zero</a>] | no goals | https://github.com/leanprover-community/mathlib4 | 29dcec074de168ac2bf835a77ef68bbe069194c5 | Mathlib/Algebra/Polynomial/FieldDivision.lean |