full_name
stringlengths 3
121
| state
stringlengths 7
9.32k
| tactic
stringlengths 3
5.35k
| target_state
stringlengths 7
19k
| url
stringclasses 1
value | commit
stringclasses 1
value | file_path
stringlengths 21
79
|
|---|---|---|---|---|---|---|
SimpleGraph.mem_incidence_iff_neighbor
|
ι : Sort u_1
V : Type u
G : SimpleGraph V
a b c u v✝ w✝ : V
e : Sym2 V
v w : V
⊢ s(v, w) ∈ G.incidenceSet v ↔ w ∈ G.neighborSet v
|
simp only [<a>SimpleGraph.mem_incidenceSet</a>, <a>SimpleGraph.mem_neighborSet</a>]
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Combinatorics/SimpleGraph/Basic.lean
|
isAdjointPair_toLinearMap₂'
|
R : Type u_1
R₁ : Type u_2
R₂ : Type u_3
M : Type u_4
M₁ : Type u_5
M₂ : Type u_6
M₁' : Type u_7
M₂' : Type u_8
n : Type u_9
m : Type u_10
n' : Type u_11
m' : Type u_12
ι : Type u_13
inst✝⁸ : CommRing R
inst✝⁷ : AddCommMonoid M₁
inst✝⁶ : Module R M₁
inst✝⁵ : AddCommMonoid M₂
inst✝⁴ : Module R M₂
inst✝³ : Fintype n
inst✝² : Fintype n'
b₁ : Basis n R M₁
b₂ : Basis n' R M₂
J J₂ : Matrix n n R
J' : Matrix n' n' R
A : Matrix n' n R
A' : Matrix n n' R
A₁ A₂ : Matrix n n R
inst✝¹ : DecidableEq n
inst✝ : DecidableEq n'
⊢ (toLinearMap₂' J).IsAdjointPair (toLinearMap₂' J') (toLin' A) (toLin' A') ↔ J.IsAdjointPair J' A A'
|
rw [<a>LinearMap.isAdjointPair_iff_comp_eq_compl₂</a>]
|
R : Type u_1
R₁ : Type u_2
R₂ : Type u_3
M : Type u_4
M₁ : Type u_5
M₂ : Type u_6
M₁' : Type u_7
M₂' : Type u_8
n : Type u_9
m : Type u_10
n' : Type u_11
m' : Type u_12
ι : Type u_13
inst✝⁸ : CommRing R
inst✝⁷ : AddCommMonoid M₁
inst✝⁶ : Module R M₁
inst✝⁵ : AddCommMonoid M₂
inst✝⁴ : Module R M₂
inst✝³ : Fintype n
inst✝² : Fintype n'
b₁ : Basis n R M₁
b₂ : Basis n' R M₂
J J₂ : Matrix n n R
J' : Matrix n' n' R
A : Matrix n' n R
A' : Matrix n n' R
A₁ A₂ : Matrix n n R
inst✝¹ : DecidableEq n
inst✝ : DecidableEq n'
⊢ toLinearMap₂' J' ∘ₗ toLin' A = (toLinearMap₂' J).compl₂ (toLin' A') ↔ J.IsAdjointPair J' A A'
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/LinearAlgebra/Matrix/SesquilinearForm.lean
|
isAdjointPair_toLinearMap₂'
|
R : Type u_1
R₁ : Type u_2
R₂ : Type u_3
M : Type u_4
M₁ : Type u_5
M₂ : Type u_6
M₁' : Type u_7
M₂' : Type u_8
n : Type u_9
m : Type u_10
n' : Type u_11
m' : Type u_12
ι : Type u_13
inst✝⁸ : CommRing R
inst✝⁷ : AddCommMonoid M₁
inst✝⁶ : Module R M₁
inst✝⁵ : AddCommMonoid M₂
inst✝⁴ : Module R M₂
inst✝³ : Fintype n
inst✝² : Fintype n'
b₁ : Basis n R M₁
b₂ : Basis n' R M₂
J J₂ : Matrix n n R
J' : Matrix n' n' R
A : Matrix n' n R
A' : Matrix n n' R
A₁ A₂ : Matrix n n R
inst✝¹ : DecidableEq n
inst✝ : DecidableEq n'
h : ∀ (B B' : (n → R) →ₗ[R] (n' → R) →ₗ[R] R), B = B' ↔ toMatrix₂' B = toMatrix₂' B'
⊢ toLinearMap₂' J' ∘ₗ toLin' A = (toLinearMap₂' J).compl₂ (toLin' A') ↔ J.IsAdjointPair J' A A'
|
simp_rw [h, <a>LinearMap.toMatrix₂'_comp</a>, <a>LinearMap.toMatrix₂'_compl₂</a>, <a>LinearMap.toMatrix'_toLin'</a>, <a>LinearMap.toMatrix'_toLinearMap₂'</a>]
|
R : Type u_1
R₁ : Type u_2
R₂ : Type u_3
M : Type u_4
M₁ : Type u_5
M₂ : Type u_6
M₁' : Type u_7
M₂' : Type u_8
n : Type u_9
m : Type u_10
n' : Type u_11
m' : Type u_12
ι : Type u_13
inst✝⁸ : CommRing R
inst✝⁷ : AddCommMonoid M₁
inst✝⁶ : Module R M₁
inst✝⁵ : AddCommMonoid M₂
inst✝⁴ : Module R M₂
inst✝³ : Fintype n
inst✝² : Fintype n'
b₁ : Basis n R M₁
b₂ : Basis n' R M₂
J J₂ : Matrix n n R
J' : Matrix n' n' R
A : Matrix n' n R
A' : Matrix n n' R
A₁ A₂ : Matrix n n R
inst✝¹ : DecidableEq n
inst✝ : DecidableEq n'
h : ∀ (B B' : (n → R) →ₗ[R] (n' → R) →ₗ[R] R), B = B' ↔ toMatrix₂' B = toMatrix₂' B'
⊢ Aᵀ * J' = J * A' ↔ J.IsAdjointPair J' A A'
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/LinearAlgebra/Matrix/SesquilinearForm.lean
|
isAdjointPair_toLinearMap₂'
|
R : Type u_1
R₁ : Type u_2
R₂ : Type u_3
M : Type u_4
M₁ : Type u_5
M₂ : Type u_6
M₁' : Type u_7
M₂' : Type u_8
n : Type u_9
m : Type u_10
n' : Type u_11
m' : Type u_12
ι : Type u_13
inst✝⁸ : CommRing R
inst✝⁷ : AddCommMonoid M₁
inst✝⁶ : Module R M₁
inst✝⁵ : AddCommMonoid M₂
inst✝⁴ : Module R M₂
inst✝³ : Fintype n
inst✝² : Fintype n'
b₁ : Basis n R M₁
b₂ : Basis n' R M₂
J J₂ : Matrix n n R
J' : Matrix n' n' R
A : Matrix n' n R
A' : Matrix n n' R
A₁ A₂ : Matrix n n R
inst✝¹ : DecidableEq n
inst✝ : DecidableEq n'
h : ∀ (B B' : (n → R) →ₗ[R] (n' → R) →ₗ[R] R), B = B' ↔ toMatrix₂' B = toMatrix₂' B'
⊢ Aᵀ * J' = J * A' ↔ J.IsAdjointPair J' A A'
|
rfl
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/LinearAlgebra/Matrix/SesquilinearForm.lean
|
isAdjointPair_toLinearMap₂'
|
R : Type u_1
R₁ : Type u_2
R₂ : Type u_3
M : Type u_4
M₁ : Type u_5
M₂ : Type u_6
M₁' : Type u_7
M₂' : Type u_8
n : Type u_9
m : Type u_10
n' : Type u_11
m' : Type u_12
ι : Type u_13
inst✝⁸ : CommRing R
inst✝⁷ : AddCommMonoid M₁
inst✝⁶ : Module R M₁
inst✝⁵ : AddCommMonoid M₂
inst✝⁴ : Module R M₂
inst✝³ : Fintype n
inst✝² : Fintype n'
b₁ : Basis n R M₁
b₂ : Basis n' R M₂
J J₂ : Matrix n n R
J' : Matrix n' n' R
A : Matrix n' n R
A' : Matrix n n' R
A₁ A₂ : Matrix n n R
inst✝¹ : DecidableEq n
inst✝ : DecidableEq n'
⊢ ∀ (B B' : (n → R) →ₗ[R] (n' → R) →ₗ[R] R), B = B' ↔ toMatrix₂' B = toMatrix₂' B'
|
intro B B'
|
R : Type u_1
R₁ : Type u_2
R₂ : Type u_3
M : Type u_4
M₁ : Type u_5
M₂ : Type u_6
M₁' : Type u_7
M₂' : Type u_8
n : Type u_9
m : Type u_10
n' : Type u_11
m' : Type u_12
ι : Type u_13
inst✝⁸ : CommRing R
inst✝⁷ : AddCommMonoid M₁
inst✝⁶ : Module R M₁
inst✝⁵ : AddCommMonoid M₂
inst✝⁴ : Module R M₂
inst✝³ : Fintype n
inst✝² : Fintype n'
b₁ : Basis n R M₁
b₂ : Basis n' R M₂
J J₂ : Matrix n n R
J' : Matrix n' n' R
A : Matrix n' n R
A' : Matrix n n' R
A₁ A₂ : Matrix n n R
inst✝¹ : DecidableEq n
inst✝ : DecidableEq n'
B B' : (n → R) →ₗ[R] (n' → R) →ₗ[R] R
⊢ B = B' ↔ toMatrix₂' B = toMatrix₂' B'
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/LinearAlgebra/Matrix/SesquilinearForm.lean
|
isAdjointPair_toLinearMap₂'
|
R : Type u_1
R₁ : Type u_2
R₂ : Type u_3
M : Type u_4
M₁ : Type u_5
M₂ : Type u_6
M₁' : Type u_7
M₂' : Type u_8
n : Type u_9
m : Type u_10
n' : Type u_11
m' : Type u_12
ι : Type u_13
inst✝⁸ : CommRing R
inst✝⁷ : AddCommMonoid M₁
inst✝⁶ : Module R M₁
inst✝⁵ : AddCommMonoid M₂
inst✝⁴ : Module R M₂
inst✝³ : Fintype n
inst✝² : Fintype n'
b₁ : Basis n R M₁
b₂ : Basis n' R M₂
J J₂ : Matrix n n R
J' : Matrix n' n' R
A : Matrix n' n R
A' : Matrix n n' R
A₁ A₂ : Matrix n n R
inst✝¹ : DecidableEq n
inst✝ : DecidableEq n'
B B' : (n → R) →ₗ[R] (n' → R) →ₗ[R] R
⊢ B = B' ↔ toMatrix₂' B = toMatrix₂' B'
|
constructor <;> intro h
|
case mp
R : Type u_1
R₁ : Type u_2
R₂ : Type u_3
M : Type u_4
M₁ : Type u_5
M₂ : Type u_6
M₁' : Type u_7
M₂' : Type u_8
n : Type u_9
m : Type u_10
n' : Type u_11
m' : Type u_12
ι : Type u_13
inst✝⁸ : CommRing R
inst✝⁷ : AddCommMonoid M₁
inst✝⁶ : Module R M₁
inst✝⁵ : AddCommMonoid M₂
inst✝⁴ : Module R M₂
inst✝³ : Fintype n
inst✝² : Fintype n'
b₁ : Basis n R M₁
b₂ : Basis n' R M₂
J J₂ : Matrix n n R
J' : Matrix n' n' R
A : Matrix n' n R
A' : Matrix n n' R
A₁ A₂ : Matrix n n R
inst✝¹ : DecidableEq n
inst✝ : DecidableEq n'
B B' : (n → R) →ₗ[R] (n' → R) →ₗ[R] R
h : B = B'
⊢ toMatrix₂' B = toMatrix₂' B'
case mpr
R : Type u_1
R₁ : Type u_2
R₂ : Type u_3
M : Type u_4
M₁ : Type u_5
M₂ : Type u_6
M₁' : Type u_7
M₂' : Type u_8
n : Type u_9
m : Type u_10
n' : Type u_11
m' : Type u_12
ι : Type u_13
inst✝⁸ : CommRing R
inst✝⁷ : AddCommMonoid M₁
inst✝⁶ : Module R M₁
inst✝⁵ : AddCommMonoid M₂
inst✝⁴ : Module R M₂
inst✝³ : Fintype n
inst✝² : Fintype n'
b₁ : Basis n R M₁
b₂ : Basis n' R M₂
J J₂ : Matrix n n R
J' : Matrix n' n' R
A : Matrix n' n R
A' : Matrix n n' R
A₁ A₂ : Matrix n n R
inst✝¹ : DecidableEq n
inst✝ : DecidableEq n'
B B' : (n → R) →ₗ[R] (n' → R) →ₗ[R] R
h : toMatrix₂' B = toMatrix₂' B'
⊢ B = B'
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/LinearAlgebra/Matrix/SesquilinearForm.lean
|
isAdjointPair_toLinearMap₂'
|
case mp
R : Type u_1
R₁ : Type u_2
R₂ : Type u_3
M : Type u_4
M₁ : Type u_5
M₂ : Type u_6
M₁' : Type u_7
M₂' : Type u_8
n : Type u_9
m : Type u_10
n' : Type u_11
m' : Type u_12
ι : Type u_13
inst✝⁸ : CommRing R
inst✝⁷ : AddCommMonoid M₁
inst✝⁶ : Module R M₁
inst✝⁵ : AddCommMonoid M₂
inst✝⁴ : Module R M₂
inst✝³ : Fintype n
inst✝² : Fintype n'
b₁ : Basis n R M₁
b₂ : Basis n' R M₂
J J₂ : Matrix n n R
J' : Matrix n' n' R
A : Matrix n' n R
A' : Matrix n n' R
A₁ A₂ : Matrix n n R
inst✝¹ : DecidableEq n
inst✝ : DecidableEq n'
B B' : (n → R) →ₗ[R] (n' → R) →ₗ[R] R
h : B = B'
⊢ toMatrix₂' B = toMatrix₂' B'
|
rw [h]
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/LinearAlgebra/Matrix/SesquilinearForm.lean
|
isAdjointPair_toLinearMap₂'
|
case mpr
R : Type u_1
R₁ : Type u_2
R₂ : Type u_3
M : Type u_4
M₁ : Type u_5
M₂ : Type u_6
M₁' : Type u_7
M₂' : Type u_8
n : Type u_9
m : Type u_10
n' : Type u_11
m' : Type u_12
ι : Type u_13
inst✝⁸ : CommRing R
inst✝⁷ : AddCommMonoid M₁
inst✝⁶ : Module R M₁
inst✝⁵ : AddCommMonoid M₂
inst✝⁴ : Module R M₂
inst✝³ : Fintype n
inst✝² : Fintype n'
b₁ : Basis n R M₁
b₂ : Basis n' R M₂
J J₂ : Matrix n n R
J' : Matrix n' n' R
A : Matrix n' n R
A' : Matrix n n' R
A₁ A₂ : Matrix n n R
inst✝¹ : DecidableEq n
inst✝ : DecidableEq n'
B B' : (n → R) →ₗ[R] (n' → R) →ₗ[R] R
h : toMatrix₂' B = toMatrix₂' B'
⊢ B = B'
|
exact LinearMap.toMatrix₂'.injective h
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/LinearAlgebra/Matrix/SesquilinearForm.lean
|
Real.sin_sq_pi_over_two_pow
|
x : ℝ
n : ℕ
⊢ sin (π / 2 ^ (n + 1)) ^ 2 = 1 - (sqrtTwoAddSeries 0 n / 2) ^ 2
|
rw [<a>Real.sin_sq</a>, <a>Real.cos_pi_over_two_pow</a>]
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Analysis/SpecialFunctions/Trigonometric/Basic.lean
|
Trivialization.frontier_preimage
|
ι : Type u_1
B : Type u_2
F : Type u_3
E : B → Type u_4
Z : Type u_5
inst✝⁴ : TopologicalSpace B
inst✝³ : TopologicalSpace F
proj : Z → B
inst✝² : TopologicalSpace Z
inst✝¹ : TopologicalSpace (TotalSpace F E)
e✝ : Trivialization F proj
x : Z
e' : Trivialization F TotalSpace.proj
x' : TotalSpace F E
b : B
y : E b
B' : Type u_6
inst✝ : TopologicalSpace B'
e : Trivialization F proj
s : Set B
⊢ e.source ∩ frontier (proj ⁻¹' s) = proj ⁻¹' (e.baseSet ∩ frontier s)
|
rw [← (e.isImage_preimage_prod s).frontier.preimage_eq, <a>frontier_prod_univ_eq</a>, (e.isImage_preimage_prod _).<a>PartialHomeomorph.IsImage.preimage_eq</a>, e.source_eq, <a>Set.preimage_inter</a>]
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Topology/FiberBundle/Trivialization.lean
|
IsFractionRing.integerNormalization_eq_zero_iff
|
R : Type u_1
inst✝⁹ : CommRing R
M : Submonoid R
S : Type u_2
inst✝⁸ : CommRing S
inst✝⁷ : Algebra R S
P : Type u_3
inst✝⁶ : CommRing P
A : Type u_4
K : Type u_5
C : Type u_6
inst✝⁵ : CommRing A
inst✝⁴ : IsDomain A
inst✝³ : Field K
inst✝² : Algebra A K
inst✝¹ : IsFractionRing A K
inst✝ : CommRing C
p : K[X]
⊢ integerNormalization (nonZeroDivisors A) p = 0 ↔ p = 0
|
refine Polynomial.ext_iff.trans (Polynomial.ext_iff.trans ?_).<a>Iff.symm</a>
|
R : Type u_1
inst✝⁹ : CommRing R
M : Submonoid R
S : Type u_2
inst✝⁸ : CommRing S
inst✝⁷ : Algebra R S
P : Type u_3
inst✝⁶ : CommRing P
A : Type u_4
K : Type u_5
C : Type u_6
inst✝⁵ : CommRing A
inst✝⁴ : IsDomain A
inst✝³ : Field K
inst✝² : Algebra A K
inst✝¹ : IsFractionRing A K
inst✝ : CommRing C
p : K[X]
⊢ (∀ (n : ℕ), p.coeff n = coeff 0 n) ↔ ∀ (n : ℕ), (integerNormalization (nonZeroDivisors A) p).coeff n = coeff 0 n
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/RingTheory/Localization/Integral.lean
|
IsFractionRing.integerNormalization_eq_zero_iff
|
R : Type u_1
inst✝⁹ : CommRing R
M : Submonoid R
S : Type u_2
inst✝⁸ : CommRing S
inst✝⁷ : Algebra R S
P : Type u_3
inst✝⁶ : CommRing P
A : Type u_4
K : Type u_5
C : Type u_6
inst✝⁵ : CommRing A
inst✝⁴ : IsDomain A
inst✝³ : Field K
inst✝² : Algebra A K
inst✝¹ : IsFractionRing A K
inst✝ : CommRing C
p : K[X]
⊢ (∀ (n : ℕ), p.coeff n = coeff 0 n) ↔ ∀ (n : ℕ), (integerNormalization (nonZeroDivisors A) p).coeff n = coeff 0 n
|
obtain ⟨⟨b, nonzero⟩, hb⟩ := <a>IsLocalization.integerNormalization_spec</a> (<a>nonZeroDivisors</a> A) p
|
case intro.mk
R : Type u_1
inst✝⁹ : CommRing R
M : Submonoid R
S : Type u_2
inst✝⁸ : CommRing S
inst✝⁷ : Algebra R S
P : Type u_3
inst✝⁶ : CommRing P
A : Type u_4
K : Type u_5
C : Type u_6
inst✝⁵ : CommRing A
inst✝⁴ : IsDomain A
inst✝³ : Field K
inst✝² : Algebra A K
inst✝¹ : IsFractionRing A K
inst✝ : CommRing C
p : K[X]
b : A
nonzero : b ∈ nonZeroDivisors A
hb : ∀ (i : ℕ), (algebraMap A K) ((integerNormalization (nonZeroDivisors A) p).coeff i) = ↑⟨b, nonzero⟩ • p.coeff i
⊢ (∀ (n : ℕ), p.coeff n = coeff 0 n) ↔ ∀ (n : ℕ), (integerNormalization (nonZeroDivisors A) p).coeff n = coeff 0 n
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/RingTheory/Localization/Integral.lean
|
IsFractionRing.integerNormalization_eq_zero_iff
|
case intro.mk
R : Type u_1
inst✝⁹ : CommRing R
M : Submonoid R
S : Type u_2
inst✝⁸ : CommRing S
inst✝⁷ : Algebra R S
P : Type u_3
inst✝⁶ : CommRing P
A : Type u_4
K : Type u_5
C : Type u_6
inst✝⁵ : CommRing A
inst✝⁴ : IsDomain A
inst✝³ : Field K
inst✝² : Algebra A K
inst✝¹ : IsFractionRing A K
inst✝ : CommRing C
p : K[X]
b : A
nonzero : b ∈ nonZeroDivisors A
hb : ∀ (i : ℕ), (algebraMap A K) ((integerNormalization (nonZeroDivisors A) p).coeff i) = ↑⟨b, nonzero⟩ • p.coeff i
⊢ (∀ (n : ℕ), p.coeff n = coeff 0 n) ↔ ∀ (n : ℕ), (integerNormalization (nonZeroDivisors A) p).coeff n = coeff 0 n
|
constructor <;> intro h i
|
case intro.mk.mp
R : Type u_1
inst✝⁹ : CommRing R
M : Submonoid R
S : Type u_2
inst✝⁸ : CommRing S
inst✝⁷ : Algebra R S
P : Type u_3
inst✝⁶ : CommRing P
A : Type u_4
K : Type u_5
C : Type u_6
inst✝⁵ : CommRing A
inst✝⁴ : IsDomain A
inst✝³ : Field K
inst✝² : Algebra A K
inst✝¹ : IsFractionRing A K
inst✝ : CommRing C
p : K[X]
b : A
nonzero : b ∈ nonZeroDivisors A
hb : ∀ (i : ℕ), (algebraMap A K) ((integerNormalization (nonZeroDivisors A) p).coeff i) = ↑⟨b, nonzero⟩ • p.coeff i
h : ∀ (n : ℕ), p.coeff n = coeff 0 n
i : ℕ
⊢ (integerNormalization (nonZeroDivisors A) p).coeff i = coeff 0 i
case intro.mk.mpr
R : Type u_1
inst✝⁹ : CommRing R
M : Submonoid R
S : Type u_2
inst✝⁸ : CommRing S
inst✝⁷ : Algebra R S
P : Type u_3
inst✝⁶ : CommRing P
A : Type u_4
K : Type u_5
C : Type u_6
inst✝⁵ : CommRing A
inst✝⁴ : IsDomain A
inst✝³ : Field K
inst✝² : Algebra A K
inst✝¹ : IsFractionRing A K
inst✝ : CommRing C
p : K[X]
b : A
nonzero : b ∈ nonZeroDivisors A
hb : ∀ (i : ℕ), (algebraMap A K) ((integerNormalization (nonZeroDivisors A) p).coeff i) = ↑⟨b, nonzero⟩ • p.coeff i
h : ∀ (n : ℕ), (integerNormalization (nonZeroDivisors A) p).coeff n = coeff 0 n
i : ℕ
⊢ p.coeff i = coeff 0 i
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/RingTheory/Localization/Integral.lean
|
IsFractionRing.integerNormalization_eq_zero_iff
|
case intro.mk.mp
R : Type u_1
inst✝⁹ : CommRing R
M : Submonoid R
S : Type u_2
inst✝⁸ : CommRing S
inst✝⁷ : Algebra R S
P : Type u_3
inst✝⁶ : CommRing P
A : Type u_4
K : Type u_5
C : Type u_6
inst✝⁵ : CommRing A
inst✝⁴ : IsDomain A
inst✝³ : Field K
inst✝² : Algebra A K
inst✝¹ : IsFractionRing A K
inst✝ : CommRing C
p : K[X]
b : A
nonzero : b ∈ nonZeroDivisors A
hb : ∀ (i : ℕ), (algebraMap A K) ((integerNormalization (nonZeroDivisors A) p).coeff i) = ↑⟨b, nonzero⟩ • p.coeff i
h : ∀ (n : ℕ), p.coeff n = coeff 0 n
i : ℕ
⊢ (integerNormalization (nonZeroDivisors A) p).coeff i = coeff 0 i
|
rw [<a>Polynomial.coeff_zero</a>, ← <a>IsFractionRing.to_map_eq_zero_iff</a> (K := K), hb i, h i, <a>Polynomial.coeff_zero</a>, <a>smul_zero</a>]
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/RingTheory/Localization/Integral.lean
|
IsFractionRing.integerNormalization_eq_zero_iff
|
case intro.mk.mpr
R : Type u_1
inst✝⁹ : CommRing R
M : Submonoid R
S : Type u_2
inst✝⁸ : CommRing S
inst✝⁷ : Algebra R S
P : Type u_3
inst✝⁶ : CommRing P
A : Type u_4
K : Type u_5
C : Type u_6
inst✝⁵ : CommRing A
inst✝⁴ : IsDomain A
inst✝³ : Field K
inst✝² : Algebra A K
inst✝¹ : IsFractionRing A K
inst✝ : CommRing C
p : K[X]
b : A
nonzero : b ∈ nonZeroDivisors A
hb : ∀ (i : ℕ), (algebraMap A K) ((integerNormalization (nonZeroDivisors A) p).coeff i) = ↑⟨b, nonzero⟩ • p.coeff i
h : ∀ (n : ℕ), (integerNormalization (nonZeroDivisors A) p).coeff n = coeff 0 n
i : ℕ
⊢ p.coeff i = coeff 0 i
|
have hi := h i
|
case intro.mk.mpr
R : Type u_1
inst✝⁹ : CommRing R
M : Submonoid R
S : Type u_2
inst✝⁸ : CommRing S
inst✝⁷ : Algebra R S
P : Type u_3
inst✝⁶ : CommRing P
A : Type u_4
K : Type u_5
C : Type u_6
inst✝⁵ : CommRing A
inst✝⁴ : IsDomain A
inst✝³ : Field K
inst✝² : Algebra A K
inst✝¹ : IsFractionRing A K
inst✝ : CommRing C
p : K[X]
b : A
nonzero : b ∈ nonZeroDivisors A
hb : ∀ (i : ℕ), (algebraMap A K) ((integerNormalization (nonZeroDivisors A) p).coeff i) = ↑⟨b, nonzero⟩ • p.coeff i
h : ∀ (n : ℕ), (integerNormalization (nonZeroDivisors A) p).coeff n = coeff 0 n
i : ℕ
hi : (integerNormalization (nonZeroDivisors A) p).coeff i = coeff 0 i
⊢ p.coeff i = coeff 0 i
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/RingTheory/Localization/Integral.lean
|
IsFractionRing.integerNormalization_eq_zero_iff
|
case intro.mk.mpr
R : Type u_1
inst✝⁹ : CommRing R
M : Submonoid R
S : Type u_2
inst✝⁸ : CommRing S
inst✝⁷ : Algebra R S
P : Type u_3
inst✝⁶ : CommRing P
A : Type u_4
K : Type u_5
C : Type u_6
inst✝⁵ : CommRing A
inst✝⁴ : IsDomain A
inst✝³ : Field K
inst✝² : Algebra A K
inst✝¹ : IsFractionRing A K
inst✝ : CommRing C
p : K[X]
b : A
nonzero : b ∈ nonZeroDivisors A
hb : ∀ (i : ℕ), (algebraMap A K) ((integerNormalization (nonZeroDivisors A) p).coeff i) = ↑⟨b, nonzero⟩ • p.coeff i
h : ∀ (n : ℕ), (integerNormalization (nonZeroDivisors A) p).coeff n = coeff 0 n
i : ℕ
hi : (integerNormalization (nonZeroDivisors A) p).coeff i = coeff 0 i
⊢ p.coeff i = coeff 0 i
|
rw [<a>Polynomial.coeff_zero</a>, ← @<a>IsFractionRing.to_map_eq_zero_iff</a> A _ K, hb i, <a>Algebra.smul_def</a>] at hi
|
case intro.mk.mpr
R : Type u_1
inst✝⁹ : CommRing R
M : Submonoid R
S : Type u_2
inst✝⁸ : CommRing S
inst✝⁷ : Algebra R S
P : Type u_3
inst✝⁶ : CommRing P
A : Type u_4
K : Type u_5
C : Type u_6
inst✝⁵ : CommRing A
inst✝⁴ : IsDomain A
inst✝³ : Field K
inst✝² : Algebra A K
inst✝¹ : IsFractionRing A K
inst✝ : CommRing C
p : K[X]
b : A
nonzero : b ∈ nonZeroDivisors A
hb : ∀ (i : ℕ), (algebraMap A K) ((integerNormalization (nonZeroDivisors A) p).coeff i) = ↑⟨b, nonzero⟩ • p.coeff i
h : ∀ (n : ℕ), (integerNormalization (nonZeroDivisors A) p).coeff n = coeff 0 n
i : ℕ
hi : (algebraMap A K) ↑⟨b, nonzero⟩ * p.coeff i = 0
⊢ p.coeff i = coeff 0 i
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/RingTheory/Localization/Integral.lean
|
IsFractionRing.integerNormalization_eq_zero_iff
|
case intro.mk.mpr
R : Type u_1
inst✝⁹ : CommRing R
M : Submonoid R
S : Type u_2
inst✝⁸ : CommRing S
inst✝⁷ : Algebra R S
P : Type u_3
inst✝⁶ : CommRing P
A : Type u_4
K : Type u_5
C : Type u_6
inst✝⁵ : CommRing A
inst✝⁴ : IsDomain A
inst✝³ : Field K
inst✝² : Algebra A K
inst✝¹ : IsFractionRing A K
inst✝ : CommRing C
p : K[X]
b : A
nonzero : b ∈ nonZeroDivisors A
hb : ∀ (i : ℕ), (algebraMap A K) ((integerNormalization (nonZeroDivisors A) p).coeff i) = ↑⟨b, nonzero⟩ • p.coeff i
h : ∀ (n : ℕ), (integerNormalization (nonZeroDivisors A) p).coeff n = coeff 0 n
i : ℕ
hi : (algebraMap A K) ↑⟨b, nonzero⟩ * p.coeff i = 0
⊢ p.coeff i = coeff 0 i
|
apply <a>Or.resolve_left</a> (<a>NoZeroDivisors.eq_zero_or_eq_zero_of_mul_eq_zero</a> hi)
|
case intro.mk.mpr
R : Type u_1
inst✝⁹ : CommRing R
M : Submonoid R
S : Type u_2
inst✝⁸ : CommRing S
inst✝⁷ : Algebra R S
P : Type u_3
inst✝⁶ : CommRing P
A : Type u_4
K : Type u_5
C : Type u_6
inst✝⁵ : CommRing A
inst✝⁴ : IsDomain A
inst✝³ : Field K
inst✝² : Algebra A K
inst✝¹ : IsFractionRing A K
inst✝ : CommRing C
p : K[X]
b : A
nonzero : b ∈ nonZeroDivisors A
hb : ∀ (i : ℕ), (algebraMap A K) ((integerNormalization (nonZeroDivisors A) p).coeff i) = ↑⟨b, nonzero⟩ • p.coeff i
h : ∀ (n : ℕ), (integerNormalization (nonZeroDivisors A) p).coeff n = coeff 0 n
i : ℕ
hi : (algebraMap A K) ↑⟨b, nonzero⟩ * p.coeff i = 0
⊢ ¬(algebraMap A K) ↑⟨b, nonzero⟩ = 0
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/RingTheory/Localization/Integral.lean
|
IsFractionRing.integerNormalization_eq_zero_iff
|
case intro.mk.mpr
R : Type u_1
inst✝⁹ : CommRing R
M : Submonoid R
S : Type u_2
inst✝⁸ : CommRing S
inst✝⁷ : Algebra R S
P : Type u_3
inst✝⁶ : CommRing P
A : Type u_4
K : Type u_5
C : Type u_6
inst✝⁵ : CommRing A
inst✝⁴ : IsDomain A
inst✝³ : Field K
inst✝² : Algebra A K
inst✝¹ : IsFractionRing A K
inst✝ : CommRing C
p : K[X]
b : A
nonzero : b ∈ nonZeroDivisors A
hb : ∀ (i : ℕ), (algebraMap A K) ((integerNormalization (nonZeroDivisors A) p).coeff i) = ↑⟨b, nonzero⟩ • p.coeff i
h : ∀ (n : ℕ), (integerNormalization (nonZeroDivisors A) p).coeff n = coeff 0 n
i : ℕ
hi : (algebraMap A K) ↑⟨b, nonzero⟩ * p.coeff i = 0
⊢ ¬(algebraMap A K) ↑⟨b, nonzero⟩ = 0
|
intro h
|
case intro.mk.mpr
R : Type u_1
inst✝⁹ : CommRing R
M : Submonoid R
S : Type u_2
inst✝⁸ : CommRing S
inst✝⁷ : Algebra R S
P : Type u_3
inst✝⁶ : CommRing P
A : Type u_4
K : Type u_5
C : Type u_6
inst✝⁵ : CommRing A
inst✝⁴ : IsDomain A
inst✝³ : Field K
inst✝² : Algebra A K
inst✝¹ : IsFractionRing A K
inst✝ : CommRing C
p : K[X]
b : A
nonzero : b ∈ nonZeroDivisors A
hb : ∀ (i : ℕ), (algebraMap A K) ((integerNormalization (nonZeroDivisors A) p).coeff i) = ↑⟨b, nonzero⟩ • p.coeff i
h✝ : ∀ (n : ℕ), (integerNormalization (nonZeroDivisors A) p).coeff n = coeff 0 n
i : ℕ
hi : (algebraMap A K) ↑⟨b, nonzero⟩ * p.coeff i = 0
h : (algebraMap A K) ↑⟨b, nonzero⟩ = 0
⊢ False
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/RingTheory/Localization/Integral.lean
|
IsFractionRing.integerNormalization_eq_zero_iff
|
case intro.mk.mpr
R : Type u_1
inst✝⁹ : CommRing R
M : Submonoid R
S : Type u_2
inst✝⁸ : CommRing S
inst✝⁷ : Algebra R S
P : Type u_3
inst✝⁶ : CommRing P
A : Type u_4
K : Type u_5
C : Type u_6
inst✝⁵ : CommRing A
inst✝⁴ : IsDomain A
inst✝³ : Field K
inst✝² : Algebra A K
inst✝¹ : IsFractionRing A K
inst✝ : CommRing C
p : K[X]
b : A
nonzero : b ∈ nonZeroDivisors A
hb : ∀ (i : ℕ), (algebraMap A K) ((integerNormalization (nonZeroDivisors A) p).coeff i) = ↑⟨b, nonzero⟩ • p.coeff i
h✝ : ∀ (n : ℕ), (integerNormalization (nonZeroDivisors A) p).coeff n = coeff 0 n
i : ℕ
hi : (algebraMap A K) ↑⟨b, nonzero⟩ * p.coeff i = 0
h : (algebraMap A K) ↑⟨b, nonzero⟩ = 0
⊢ False
|
apply mem_nonZeroDivisors_iff_ne_zero.mp nonzero
|
case intro.mk.mpr
R : Type u_1
inst✝⁹ : CommRing R
M : Submonoid R
S : Type u_2
inst✝⁸ : CommRing S
inst✝⁷ : Algebra R S
P : Type u_3
inst✝⁶ : CommRing P
A : Type u_4
K : Type u_5
C : Type u_6
inst✝⁵ : CommRing A
inst✝⁴ : IsDomain A
inst✝³ : Field K
inst✝² : Algebra A K
inst✝¹ : IsFractionRing A K
inst✝ : CommRing C
p : K[X]
b : A
nonzero : b ∈ nonZeroDivisors A
hb : ∀ (i : ℕ), (algebraMap A K) ((integerNormalization (nonZeroDivisors A) p).coeff i) = ↑⟨b, nonzero⟩ • p.coeff i
h✝ : ∀ (n : ℕ), (integerNormalization (nonZeroDivisors A) p).coeff n = coeff 0 n
i : ℕ
hi : (algebraMap A K) ↑⟨b, nonzero⟩ * p.coeff i = 0
h : (algebraMap A K) ↑⟨b, nonzero⟩ = 0
⊢ b = 0
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/RingTheory/Localization/Integral.lean
|
IsFractionRing.integerNormalization_eq_zero_iff
|
case intro.mk.mpr
R : Type u_1
inst✝⁹ : CommRing R
M : Submonoid R
S : Type u_2
inst✝⁸ : CommRing S
inst✝⁷ : Algebra R S
P : Type u_3
inst✝⁶ : CommRing P
A : Type u_4
K : Type u_5
C : Type u_6
inst✝⁵ : CommRing A
inst✝⁴ : IsDomain A
inst✝³ : Field K
inst✝² : Algebra A K
inst✝¹ : IsFractionRing A K
inst✝ : CommRing C
p : K[X]
b : A
nonzero : b ∈ nonZeroDivisors A
hb : ∀ (i : ℕ), (algebraMap A K) ((integerNormalization (nonZeroDivisors A) p).coeff i) = ↑⟨b, nonzero⟩ • p.coeff i
h✝ : ∀ (n : ℕ), (integerNormalization (nonZeroDivisors A) p).coeff n = coeff 0 n
i : ℕ
hi : (algebraMap A K) ↑⟨b, nonzero⟩ * p.coeff i = 0
h : (algebraMap A K) ↑⟨b, nonzero⟩ = 0
⊢ b = 0
|
exact to_map_eq_zero_iff.mp h
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/RingTheory/Localization/Integral.lean
|
Int.Ico_filter_modEq_eq
|
a b r : ℤ
hr : 0 < r
v : ℤ
⊢ filter (fun x => x ≡ v [ZMOD r]) (Ico a b) =
map { toFun := fun x => x + v, inj' := ⋯ } (filter (fun x => r ∣ x) (Ico (a - v) (b - v)))
|
ext x
|
case a
a b r : ℤ
hr : 0 < r
v x : ℤ
⊢ x ∈ filter (fun x => x ≡ v [ZMOD r]) (Ico a b) ↔
x ∈ map { toFun := fun x => x + v, inj' := ⋯ } (filter (fun x => r ∣ x) (Ico (a - v) (b - v)))
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Data/Int/CardIntervalMod.lean
|
Int.Ico_filter_modEq_eq
|
case a
a b r : ℤ
hr : 0 < r
v x : ℤ
⊢ x ∈ filter (fun x => x ≡ v [ZMOD r]) (Ico a b) ↔
x ∈ map { toFun := fun x => x + v, inj' := ⋯ } (filter (fun x => r ∣ x) (Ico (a - v) (b - v)))
|
simp_rw [<a>Finset.mem_map</a>, <a>Finset.mem_filter</a>, <a>Finset.mem_Ico</a>, <a>Function.Embedding.coeFn_mk</a>, ← <a>eq_sub_iff_add_eq</a>, <a>exists_eq_right</a>, <a>Int.modEq_comm</a>, <a>Int.modEq_iff_dvd</a>, <a>sub_lt_sub_iff_right</a>, <a>sub_le_sub_iff_right</a>]
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Data/Int/CardIntervalMod.lean
|
Polynomial.natDegree_le_of_dvd
|
R : Type u
S : Type v
T : Type w
a b : R
n : ℕ
inst✝¹ : Semiring R
inst✝ : NoZeroDivisors R
p✝ q✝ p q : R[X]
h1 : p ∣ q
h2 : q ≠ 0
⊢ p.natDegree ≤ q.natDegree
|
rcases h1 with ⟨q, rfl⟩
|
case intro
R : Type u
S : Type v
T : Type w
a b : R
n : ℕ
inst✝¹ : Semiring R
inst✝ : NoZeroDivisors R
p✝ q✝ p q : R[X]
h2 : p * q ≠ 0
⊢ p.natDegree ≤ (p * q).natDegree
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Algebra/Polynomial/RingDivision.lean
|
Polynomial.natDegree_le_of_dvd
|
case intro
R : Type u
S : Type v
T : Type w
a b : R
n : ℕ
inst✝¹ : Semiring R
inst✝ : NoZeroDivisors R
p✝ q✝ p q : R[X]
h2 : p * q ≠ 0
⊢ p.natDegree ≤ (p * q).natDegree
|
rw [<a>mul_ne_zero_iff</a>] at h2
|
case intro
R : Type u
S : Type v
T : Type w
a b : R
n : ℕ
inst✝¹ : Semiring R
inst✝ : NoZeroDivisors R
p✝ q✝ p q : R[X]
h2 : p ≠ 0 ∧ q ≠ 0
⊢ p.natDegree ≤ (p * q).natDegree
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Algebra/Polynomial/RingDivision.lean
|
Polynomial.natDegree_le_of_dvd
|
case intro
R : Type u
S : Type v
T : Type w
a b : R
n : ℕ
inst✝¹ : Semiring R
inst✝ : NoZeroDivisors R
p✝ q✝ p q : R[X]
h2 : p ≠ 0 ∧ q ≠ 0
⊢ p.natDegree ≤ (p * q).natDegree
|
rw [<a>Polynomial.natDegree_mul</a> h2.1 h2.2]
|
case intro
R : Type u
S : Type v
T : Type w
a b : R
n : ℕ
inst✝¹ : Semiring R
inst✝ : NoZeroDivisors R
p✝ q✝ p q : R[X]
h2 : p ≠ 0 ∧ q ≠ 0
⊢ p.natDegree ≤ p.natDegree + q.natDegree
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Algebra/Polynomial/RingDivision.lean
|
Polynomial.natDegree_le_of_dvd
|
case intro
R : Type u
S : Type v
T : Type w
a b : R
n : ℕ
inst✝¹ : Semiring R
inst✝ : NoZeroDivisors R
p✝ q✝ p q : R[X]
h2 : p ≠ 0 ∧ q ≠ 0
⊢ p.natDegree ≤ p.natDegree + q.natDegree
|
exact <a>Nat.le_add_right</a> _ _
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Algebra/Polynomial/RingDivision.lean
|
lt_iff_lt_of_cmp_eq_cmp
|
α : Type u_1
β✝ : Type u_2
inst✝¹ : LinearOrder α
x y : α
β : Type u_3
inst✝ : LinearOrder β
x' y' : β
h : cmp x y = cmp x' y'
⊢ x < y ↔ x' < y'
|
rw [← <a>cmp_eq_lt_iff</a>, ← <a>cmp_eq_lt_iff</a>, h]
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Order/Compare.lean
|
Filter.pi_inf_principal_univ_pi_neBot
|
ι : Type u_1
α : ι → Type u_2
f f₁ f₂ : (i : ι) → Filter (α i)
s : (i : ι) → Set (α i)
p : (i : ι) → α i → Prop
⊢ (pi f ⊓ 𝓟 (univ.pi s)).NeBot ↔ ∀ (i : ι), (f i ⊓ 𝓟 (s i)).NeBot
|
simp [<a>Filter.neBot_iff</a>]
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Order/Filter/Pi.lean
|
separableClosure.comap_eq_of_algHom
|
F : Type u
E : Type v
inst✝⁴ : Field F
inst✝³ : Field E
inst✝² : Algebra F E
K : Type w
inst✝¹ : Field K
inst✝ : Algebra F K
i : E →ₐ[F] K
⊢ comap i (separableClosure F K) = separableClosure F E
|
ext x
|
case h
F : Type u
E : Type v
inst✝⁴ : Field F
inst✝³ : Field E
inst✝² : Algebra F E
K : Type w
inst✝¹ : Field K
inst✝ : Algebra F K
i : E →ₐ[F] K
x : E
⊢ x ∈ comap i (separableClosure F K) ↔ x ∈ separableClosure F E
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/FieldTheory/SeparableClosure.lean
|
separableClosure.comap_eq_of_algHom
|
case h
F : Type u
E : Type v
inst✝⁴ : Field F
inst✝³ : Field E
inst✝² : Algebra F E
K : Type w
inst✝¹ : Field K
inst✝ : Algebra F K
i : E →ₐ[F] K
x : E
⊢ x ∈ comap i (separableClosure F K) ↔ x ∈ separableClosure F E
|
exact <a>map_mem_separableClosure_iff</a> i
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/FieldTheory/SeparableClosure.lean
|
MeasureTheory.integral_Ioi_deriv_mul_eq_sub
|
A : Type u_1
inst✝² : NormedRing A
inst✝¹ : NormedAlgebra ℝ A
a b : ℝ
a' b' : A
u v u' v' : ℝ → A
inst✝ : CompleteSpace A
hu : ∀ x ∈ Ioi a, HasDerivAt u (u' x) x
hv : ∀ x ∈ Ioi a, HasDerivAt v (v' x) x
huv : IntegrableOn (u' * v + u * v') (Ioi a) volume
h_zero : Tendsto (u * v) (𝓝[>] a) (𝓝 a')
h_infty : Tendsto (u * v) atTop (𝓝 b')
⊢ ∫ (x : ℝ) in Ioi a, u' x * v x + u x * v' x = b' - a'
|
rw [← <a>Set.Ici_diff_left</a>] at h_zero
|
A : Type u_1
inst✝² : NormedRing A
inst✝¹ : NormedAlgebra ℝ A
a b : ℝ
a' b' : A
u v u' v' : ℝ → A
inst✝ : CompleteSpace A
hu : ∀ x ∈ Ioi a, HasDerivAt u (u' x) x
hv : ∀ x ∈ Ioi a, HasDerivAt v (v' x) x
huv : IntegrableOn (u' * v + u * v') (Ioi a) volume
h_zero : Tendsto (u * v) (𝓝[Ici a \ {a}] a) (𝓝 a')
h_infty : Tendsto (u * v) atTop (𝓝 b')
⊢ ∫ (x : ℝ) in Ioi a, u' x * v x + u x * v' x = b' - a'
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/MeasureTheory/Integral/IntegralEqImproper.lean
|
MeasureTheory.integral_Ioi_deriv_mul_eq_sub
|
A : Type u_1
inst✝² : NormedRing A
inst✝¹ : NormedAlgebra ℝ A
a b : ℝ
a' b' : A
u v u' v' : ℝ → A
inst✝ : CompleteSpace A
hu : ∀ x ∈ Ioi a, HasDerivAt u (u' x) x
hv : ∀ x ∈ Ioi a, HasDerivAt v (v' x) x
huv : IntegrableOn (u' * v + u * v') (Ioi a) volume
h_zero : Tendsto (u * v) (𝓝[Ici a \ {a}] a) (𝓝 a')
h_infty : Tendsto (u * v) atTop (𝓝 b')
⊢ ∫ (x : ℝ) in Ioi a, u' x * v x + u x * v' x = b' - a'
|
let f := <a>Function.update</a> (u * v) a a'
|
A : Type u_1
inst✝² : NormedRing A
inst✝¹ : NormedAlgebra ℝ A
a b : ℝ
a' b' : A
u v u' v' : ℝ → A
inst✝ : CompleteSpace A
hu : ∀ x ∈ Ioi a, HasDerivAt u (u' x) x
hv : ∀ x ∈ Ioi a, HasDerivAt v (v' x) x
huv : IntegrableOn (u' * v + u * v') (Ioi a) volume
h_zero : Tendsto (u * v) (𝓝[Ici a \ {a}] a) (𝓝 a')
h_infty : Tendsto (u * v) atTop (𝓝 b')
f : ℝ → A := Function.update (u * v) a a'
⊢ ∫ (x : ℝ) in Ioi a, u' x * v x + u x * v' x = b' - a'
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/MeasureTheory/Integral/IntegralEqImproper.lean
|
MeasureTheory.integral_Ioi_deriv_mul_eq_sub
|
A : Type u_1
inst✝² : NormedRing A
inst✝¹ : NormedAlgebra ℝ A
a b : ℝ
a' b' : A
u v u' v' : ℝ → A
inst✝ : CompleteSpace A
hu : ∀ x ∈ Ioi a, HasDerivAt u (u' x) x
hv : ∀ x ∈ Ioi a, HasDerivAt v (v' x) x
huv : IntegrableOn (u' * v + u * v') (Ioi a) volume
h_zero : Tendsto (u * v) (𝓝[Ici a \ {a}] a) (𝓝 a')
h_infty : Tendsto (u * v) atTop (𝓝 b')
f : ℝ → A := Function.update (u * v) a a'
⊢ ∫ (x : ℝ) in Ioi a, u' x * v x + u x * v' x = b' - a'
|
have hderiv : ∀ x ∈ <a>Set.Ioi</a> a, <a>HasDerivAt</a> f (u' x * v x + u x * v' x) x := by intro x (hx : a < x) apply ((hu x hx).<a>HasDerivAt.mul</a> (hv x hx)).<a>HasDerivAt.congr_of_eventuallyEq</a> filter_upwards [<a>eventually_ne_nhds</a> hx.ne.symm] with y hy exact <a>Function.update_noteq</a> hy a' (u * v)
|
A : Type u_1
inst✝² : NormedRing A
inst✝¹ : NormedAlgebra ℝ A
a b : ℝ
a' b' : A
u v u' v' : ℝ → A
inst✝ : CompleteSpace A
hu : ∀ x ∈ Ioi a, HasDerivAt u (u' x) x
hv : ∀ x ∈ Ioi a, HasDerivAt v (v' x) x
huv : IntegrableOn (u' * v + u * v') (Ioi a) volume
h_zero : Tendsto (u * v) (𝓝[Ici a \ {a}] a) (𝓝 a')
h_infty : Tendsto (u * v) atTop (𝓝 b')
f : ℝ → A := Function.update (u * v) a a'
hderiv : ∀ x ∈ Ioi a, HasDerivAt f (u' x * v x + u x * v' x) x
⊢ ∫ (x : ℝ) in Ioi a, u' x * v x + u x * v' x = b' - a'
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/MeasureTheory/Integral/IntegralEqImproper.lean
|
MeasureTheory.integral_Ioi_deriv_mul_eq_sub
|
A : Type u_1
inst✝² : NormedRing A
inst✝¹ : NormedAlgebra ℝ A
a b : ℝ
a' b' : A
u v u' v' : ℝ → A
inst✝ : CompleteSpace A
hu : ∀ x ∈ Ioi a, HasDerivAt u (u' x) x
hv : ∀ x ∈ Ioi a, HasDerivAt v (v' x) x
huv : IntegrableOn (u' * v + u * v') (Ioi a) volume
h_zero : Tendsto (u * v) (𝓝[Ici a \ {a}] a) (𝓝 a')
h_infty : Tendsto (u * v) atTop (𝓝 b')
f : ℝ → A := Function.update (u * v) a a'
hderiv : ∀ x ∈ Ioi a, HasDerivAt f (u' x * v x + u x * v' x) x
⊢ ∫ (x : ℝ) in Ioi a, u' x * v x + u x * v' x = b' - a'
|
have htendsto : <a>Filter.Tendsto</a> f <a>Filter.atTop</a> (𝓝 b') := by apply h_infty.congr' filter_upwards [<a>Filter.eventually_ne_atTop</a> a] with x hx exact (<a>Function.update_noteq</a> hx a' (u * v)).<a>Eq.symm</a>
|
A : Type u_1
inst✝² : NormedRing A
inst✝¹ : NormedAlgebra ℝ A
a b : ℝ
a' b' : A
u v u' v' : ℝ → A
inst✝ : CompleteSpace A
hu : ∀ x ∈ Ioi a, HasDerivAt u (u' x) x
hv : ∀ x ∈ Ioi a, HasDerivAt v (v' x) x
huv : IntegrableOn (u' * v + u * v') (Ioi a) volume
h_zero : Tendsto (u * v) (𝓝[Ici a \ {a}] a) (𝓝 a')
h_infty : Tendsto (u * v) atTop (𝓝 b')
f : ℝ → A := Function.update (u * v) a a'
hderiv : ∀ x ∈ Ioi a, HasDerivAt f (u' x * v x + u x * v' x) x
htendsto : Tendsto f atTop (𝓝 b')
⊢ ∫ (x : ℝ) in Ioi a, u' x * v x + u x * v' x = b' - a'
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/MeasureTheory/Integral/IntegralEqImproper.lean
|
MeasureTheory.integral_Ioi_deriv_mul_eq_sub
|
A : Type u_1
inst✝² : NormedRing A
inst✝¹ : NormedAlgebra ℝ A
a b : ℝ
a' b' : A
u v u' v' : ℝ → A
inst✝ : CompleteSpace A
hu : ∀ x ∈ Ioi a, HasDerivAt u (u' x) x
hv : ∀ x ∈ Ioi a, HasDerivAt v (v' x) x
huv : IntegrableOn (u' * v + u * v') (Ioi a) volume
h_zero : Tendsto (u * v) (𝓝[Ici a \ {a}] a) (𝓝 a')
h_infty : Tendsto (u * v) atTop (𝓝 b')
f : ℝ → A := Function.update (u * v) a a'
hderiv : ∀ x ∈ Ioi a, HasDerivAt f (u' x * v x + u x * v' x) x
htendsto : Tendsto f atTop (𝓝 b')
⊢ ∫ (x : ℝ) in Ioi a, u' x * v x + u x * v' x = b' - a'
|
simpa using <a>MeasureTheory.integral_Ioi_of_hasDerivAt_of_tendsto</a> (continuousWithinAt_update_same.mpr h_zero) hderiv huv htendsto
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/MeasureTheory/Integral/IntegralEqImproper.lean
|
MeasureTheory.integral_Ioi_deriv_mul_eq_sub
|
A : Type u_1
inst✝² : NormedRing A
inst✝¹ : NormedAlgebra ℝ A
a b : ℝ
a' b' : A
u v u' v' : ℝ → A
inst✝ : CompleteSpace A
hu : ∀ x ∈ Ioi a, HasDerivAt u (u' x) x
hv : ∀ x ∈ Ioi a, HasDerivAt v (v' x) x
huv : IntegrableOn (u' * v + u * v') (Ioi a) volume
h_zero : Tendsto (u * v) (𝓝[Ici a \ {a}] a) (𝓝 a')
h_infty : Tendsto (u * v) atTop (𝓝 b')
f : ℝ → A := Function.update (u * v) a a'
⊢ ∀ x ∈ Ioi a, HasDerivAt f (u' x * v x + u x * v' x) x
|
intro x (hx : a < x)
|
A : Type u_1
inst✝² : NormedRing A
inst✝¹ : NormedAlgebra ℝ A
a b : ℝ
a' b' : A
u v u' v' : ℝ → A
inst✝ : CompleteSpace A
hu : ∀ x ∈ Ioi a, HasDerivAt u (u' x) x
hv : ∀ x ∈ Ioi a, HasDerivAt v (v' x) x
huv : IntegrableOn (u' * v + u * v') (Ioi a) volume
h_zero : Tendsto (u * v) (𝓝[Ici a \ {a}] a) (𝓝 a')
h_infty : Tendsto (u * v) atTop (𝓝 b')
f : ℝ → A := Function.update (u * v) a a'
x : ℝ
hx : a < x
⊢ HasDerivAt f (u' x * v x + u x * v' x) x
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/MeasureTheory/Integral/IntegralEqImproper.lean
|
MeasureTheory.integral_Ioi_deriv_mul_eq_sub
|
A : Type u_1
inst✝² : NormedRing A
inst✝¹ : NormedAlgebra ℝ A
a b : ℝ
a' b' : A
u v u' v' : ℝ → A
inst✝ : CompleteSpace A
hu : ∀ x ∈ Ioi a, HasDerivAt u (u' x) x
hv : ∀ x ∈ Ioi a, HasDerivAt v (v' x) x
huv : IntegrableOn (u' * v + u * v') (Ioi a) volume
h_zero : Tendsto (u * v) (𝓝[Ici a \ {a}] a) (𝓝 a')
h_infty : Tendsto (u * v) atTop (𝓝 b')
f : ℝ → A := Function.update (u * v) a a'
x : ℝ
hx : a < x
⊢ HasDerivAt f (u' x * v x + u x * v' x) x
|
apply ((hu x hx).<a>HasDerivAt.mul</a> (hv x hx)).<a>HasDerivAt.congr_of_eventuallyEq</a>
|
A : Type u_1
inst✝² : NormedRing A
inst✝¹ : NormedAlgebra ℝ A
a b : ℝ
a' b' : A
u v u' v' : ℝ → A
inst✝ : CompleteSpace A
hu : ∀ x ∈ Ioi a, HasDerivAt u (u' x) x
hv : ∀ x ∈ Ioi a, HasDerivAt v (v' x) x
huv : IntegrableOn (u' * v + u * v') (Ioi a) volume
h_zero : Tendsto (u * v) (𝓝[Ici a \ {a}] a) (𝓝 a')
h_infty : Tendsto (u * v) atTop (𝓝 b')
f : ℝ → A := Function.update (u * v) a a'
x : ℝ
hx : a < x
⊢ f =ᶠ[𝓝 x] fun y => u y * v y
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/MeasureTheory/Integral/IntegralEqImproper.lean
|
MeasureTheory.integral_Ioi_deriv_mul_eq_sub
|
A : Type u_1
inst✝² : NormedRing A
inst✝¹ : NormedAlgebra ℝ A
a b : ℝ
a' b' : A
u v u' v' : ℝ → A
inst✝ : CompleteSpace A
hu : ∀ x ∈ Ioi a, HasDerivAt u (u' x) x
hv : ∀ x ∈ Ioi a, HasDerivAt v (v' x) x
huv : IntegrableOn (u' * v + u * v') (Ioi a) volume
h_zero : Tendsto (u * v) (𝓝[Ici a \ {a}] a) (𝓝 a')
h_infty : Tendsto (u * v) atTop (𝓝 b')
f : ℝ → A := Function.update (u * v) a a'
x : ℝ
hx : a < x
⊢ f =ᶠ[𝓝 x] fun y => u y * v y
|
filter_upwards [<a>eventually_ne_nhds</a> hx.ne.symm] with y hy
|
case h
A : Type u_1
inst✝² : NormedRing A
inst✝¹ : NormedAlgebra ℝ A
a b : ℝ
a' b' : A
u v u' v' : ℝ → A
inst✝ : CompleteSpace A
hu : ∀ x ∈ Ioi a, HasDerivAt u (u' x) x
hv : ∀ x ∈ Ioi a, HasDerivAt v (v' x) x
huv : IntegrableOn (u' * v + u * v') (Ioi a) volume
h_zero : Tendsto (u * v) (𝓝[Ici a \ {a}] a) (𝓝 a')
h_infty : Tendsto (u * v) atTop (𝓝 b')
f : ℝ → A := Function.update (u * v) a a'
x : ℝ
hx : a < x
y : ℝ
hy : y ≠ a
⊢ f y = u y * v y
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/MeasureTheory/Integral/IntegralEqImproper.lean
|
MeasureTheory.integral_Ioi_deriv_mul_eq_sub
|
case h
A : Type u_1
inst✝² : NormedRing A
inst✝¹ : NormedAlgebra ℝ A
a b : ℝ
a' b' : A
u v u' v' : ℝ → A
inst✝ : CompleteSpace A
hu : ∀ x ∈ Ioi a, HasDerivAt u (u' x) x
hv : ∀ x ∈ Ioi a, HasDerivAt v (v' x) x
huv : IntegrableOn (u' * v + u * v') (Ioi a) volume
h_zero : Tendsto (u * v) (𝓝[Ici a \ {a}] a) (𝓝 a')
h_infty : Tendsto (u * v) atTop (𝓝 b')
f : ℝ → A := Function.update (u * v) a a'
x : ℝ
hx : a < x
y : ℝ
hy : y ≠ a
⊢ f y = u y * v y
|
exact <a>Function.update_noteq</a> hy a' (u * v)
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/MeasureTheory/Integral/IntegralEqImproper.lean
|
MeasureTheory.integral_Ioi_deriv_mul_eq_sub
|
A : Type u_1
inst✝² : NormedRing A
inst✝¹ : NormedAlgebra ℝ A
a b : ℝ
a' b' : A
u v u' v' : ℝ → A
inst✝ : CompleteSpace A
hu : ∀ x ∈ Ioi a, HasDerivAt u (u' x) x
hv : ∀ x ∈ Ioi a, HasDerivAt v (v' x) x
huv : IntegrableOn (u' * v + u * v') (Ioi a) volume
h_zero : Tendsto (u * v) (𝓝[Ici a \ {a}] a) (𝓝 a')
h_infty : Tendsto (u * v) atTop (𝓝 b')
f : ℝ → A := Function.update (u * v) a a'
hderiv : ∀ x ∈ Ioi a, HasDerivAt f (u' x * v x + u x * v' x) x
⊢ Tendsto f atTop (𝓝 b')
|
apply h_infty.congr'
|
A : Type u_1
inst✝² : NormedRing A
inst✝¹ : NormedAlgebra ℝ A
a b : ℝ
a' b' : A
u v u' v' : ℝ → A
inst✝ : CompleteSpace A
hu : ∀ x ∈ Ioi a, HasDerivAt u (u' x) x
hv : ∀ x ∈ Ioi a, HasDerivAt v (v' x) x
huv : IntegrableOn (u' * v + u * v') (Ioi a) volume
h_zero : Tendsto (u * v) (𝓝[Ici a \ {a}] a) (𝓝 a')
h_infty : Tendsto (u * v) atTop (𝓝 b')
f : ℝ → A := Function.update (u * v) a a'
hderiv : ∀ x ∈ Ioi a, HasDerivAt f (u' x * v x + u x * v' x) x
⊢ u * v =ᶠ[atTop] f
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/MeasureTheory/Integral/IntegralEqImproper.lean
|
MeasureTheory.integral_Ioi_deriv_mul_eq_sub
|
A : Type u_1
inst✝² : NormedRing A
inst✝¹ : NormedAlgebra ℝ A
a b : ℝ
a' b' : A
u v u' v' : ℝ → A
inst✝ : CompleteSpace A
hu : ∀ x ∈ Ioi a, HasDerivAt u (u' x) x
hv : ∀ x ∈ Ioi a, HasDerivAt v (v' x) x
huv : IntegrableOn (u' * v + u * v') (Ioi a) volume
h_zero : Tendsto (u * v) (𝓝[Ici a \ {a}] a) (𝓝 a')
h_infty : Tendsto (u * v) atTop (𝓝 b')
f : ℝ → A := Function.update (u * v) a a'
hderiv : ∀ x ∈ Ioi a, HasDerivAt f (u' x * v x + u x * v' x) x
⊢ u * v =ᶠ[atTop] f
|
filter_upwards [<a>Filter.eventually_ne_atTop</a> a] with x hx
|
case h
A : Type u_1
inst✝² : NormedRing A
inst✝¹ : NormedAlgebra ℝ A
a b : ℝ
a' b' : A
u v u' v' : ℝ → A
inst✝ : CompleteSpace A
hu : ∀ x ∈ Ioi a, HasDerivAt u (u' x) x
hv : ∀ x ∈ Ioi a, HasDerivAt v (v' x) x
huv : IntegrableOn (u' * v + u * v') (Ioi a) volume
h_zero : Tendsto (u * v) (𝓝[Ici a \ {a}] a) (𝓝 a')
h_infty : Tendsto (u * v) atTop (𝓝 b')
f : ℝ → A := Function.update (u * v) a a'
hderiv : ∀ x ∈ Ioi a, HasDerivAt f (u' x * v x + u x * v' x) x
x : ℝ
hx : x ≠ a
⊢ (u * v) x = f x
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/MeasureTheory/Integral/IntegralEqImproper.lean
|
MeasureTheory.integral_Ioi_deriv_mul_eq_sub
|
case h
A : Type u_1
inst✝² : NormedRing A
inst✝¹ : NormedAlgebra ℝ A
a b : ℝ
a' b' : A
u v u' v' : ℝ → A
inst✝ : CompleteSpace A
hu : ∀ x ∈ Ioi a, HasDerivAt u (u' x) x
hv : ∀ x ∈ Ioi a, HasDerivAt v (v' x) x
huv : IntegrableOn (u' * v + u * v') (Ioi a) volume
h_zero : Tendsto (u * v) (𝓝[Ici a \ {a}] a) (𝓝 a')
h_infty : Tendsto (u * v) atTop (𝓝 b')
f : ℝ → A := Function.update (u * v) a a'
hderiv : ∀ x ∈ Ioi a, HasDerivAt f (u' x * v x + u x * v' x) x
x : ℝ
hx : x ≠ a
⊢ (u * v) x = f x
|
exact (<a>Function.update_noteq</a> hx a' (u * v)).<a>Eq.symm</a>
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/MeasureTheory/Integral/IntegralEqImproper.lean
|
Memℓp.infty_mul
|
α : Type u_1
E : α → Type u_2
p q : ℝ≥0∞
inst✝¹ : (i : α) → NormedAddCommGroup (E i)
I : Type u_3
B : I → Type u_4
inst✝ : (i : I) → NonUnitalNormedRing (B i)
f g : (i : I) → B i
hf : Memℓp f ⊤
hg : Memℓp g ⊤
⊢ Memℓp (f * g) ⊤
|
rw [<a>memℓp_infty_iff</a>]
|
α : Type u_1
E : α → Type u_2
p q : ℝ≥0∞
inst✝¹ : (i : α) → NormedAddCommGroup (E i)
I : Type u_3
B : I → Type u_4
inst✝ : (i : I) → NonUnitalNormedRing (B i)
f g : (i : I) → B i
hf : Memℓp f ⊤
hg : Memℓp g ⊤
⊢ BddAbove (Set.range fun i => ‖(f * g) i‖)
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Analysis/NormedSpace/lpSpace.lean
|
Memℓp.infty_mul
|
α : Type u_1
E : α → Type u_2
p q : ℝ≥0∞
inst✝¹ : (i : α) → NormedAddCommGroup (E i)
I : Type u_3
B : I → Type u_4
inst✝ : (i : I) → NonUnitalNormedRing (B i)
f g : (i : I) → B i
hf : Memℓp f ⊤
hg : Memℓp g ⊤
⊢ BddAbove (Set.range fun i => ‖(f * g) i‖)
|
obtain ⟨⟨Cf, hCf⟩, ⟨Cg, hCg⟩⟩ := hf.bddAbove, hg.bddAbove
|
case intro.intro
α : Type u_1
E : α → Type u_2
p q : ℝ≥0∞
inst✝¹ : (i : α) → NormedAddCommGroup (E i)
I : Type u_3
B : I → Type u_4
inst✝ : (i : I) → NonUnitalNormedRing (B i)
f g : (i : I) → B i
hf : Memℓp f ⊤
hg : Memℓp g ⊤
Cf : ℝ
hCf : Cf ∈ upperBounds (Set.range fun i => ‖f i‖)
Cg : ℝ
hCg : Cg ∈ upperBounds (Set.range fun i => ‖g i‖)
⊢ BddAbove (Set.range fun i => ‖(f * g) i‖)
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Analysis/NormedSpace/lpSpace.lean
|
Memℓp.infty_mul
|
case intro.intro
α : Type u_1
E : α → Type u_2
p q : ℝ≥0∞
inst✝¹ : (i : α) → NormedAddCommGroup (E i)
I : Type u_3
B : I → Type u_4
inst✝ : (i : I) → NonUnitalNormedRing (B i)
f g : (i : I) → B i
hf : Memℓp f ⊤
hg : Memℓp g ⊤
Cf : ℝ
hCf : Cf ∈ upperBounds (Set.range fun i => ‖f i‖)
Cg : ℝ
hCg : Cg ∈ upperBounds (Set.range fun i => ‖g i‖)
⊢ BddAbove (Set.range fun i => ‖(f * g) i‖)
|
refine ⟨Cf * Cg, ?_⟩
|
case intro.intro
α : Type u_1
E : α → Type u_2
p q : ℝ≥0∞
inst✝¹ : (i : α) → NormedAddCommGroup (E i)
I : Type u_3
B : I → Type u_4
inst✝ : (i : I) → NonUnitalNormedRing (B i)
f g : (i : I) → B i
hf : Memℓp f ⊤
hg : Memℓp g ⊤
Cf : ℝ
hCf : Cf ∈ upperBounds (Set.range fun i => ‖f i‖)
Cg : ℝ
hCg : Cg ∈ upperBounds (Set.range fun i => ‖g i‖)
⊢ Cf * Cg ∈ upperBounds (Set.range fun i => ‖(f * g) i‖)
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Analysis/NormedSpace/lpSpace.lean
|
Memℓp.infty_mul
|
case intro.intro
α : Type u_1
E : α → Type u_2
p q : ℝ≥0∞
inst✝¹ : (i : α) → NormedAddCommGroup (E i)
I : Type u_3
B : I → Type u_4
inst✝ : (i : I) → NonUnitalNormedRing (B i)
f g : (i : I) → B i
hf : Memℓp f ⊤
hg : Memℓp g ⊤
Cf : ℝ
hCf : Cf ∈ upperBounds (Set.range fun i => ‖f i‖)
Cg : ℝ
hCg : Cg ∈ upperBounds (Set.range fun i => ‖g i‖)
⊢ Cf * Cg ∈ upperBounds (Set.range fun i => ‖(f * g) i‖)
|
rintro _ ⟨i, rfl⟩
|
case intro.intro.intro
α : Type u_1
E : α → Type u_2
p q : ℝ≥0∞
inst✝¹ : (i : α) → NormedAddCommGroup (E i)
I : Type u_3
B : I → Type u_4
inst✝ : (i : I) → NonUnitalNormedRing (B i)
f g : (i : I) → B i
hf : Memℓp f ⊤
hg : Memℓp g ⊤
Cf : ℝ
hCf : Cf ∈ upperBounds (Set.range fun i => ‖f i‖)
Cg : ℝ
hCg : Cg ∈ upperBounds (Set.range fun i => ‖g i‖)
i : I
⊢ (fun i => ‖(f * g) i‖) i ≤ Cf * Cg
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Analysis/NormedSpace/lpSpace.lean
|
Memℓp.infty_mul
|
case intro.intro.intro
α : Type u_1
E : α → Type u_2
p q : ℝ≥0∞
inst✝¹ : (i : α) → NormedAddCommGroup (E i)
I : Type u_3
B : I → Type u_4
inst✝ : (i : I) → NonUnitalNormedRing (B i)
f g : (i : I) → B i
hf : Memℓp f ⊤
hg : Memℓp g ⊤
Cf : ℝ
hCf : Cf ∈ upperBounds (Set.range fun i => ‖f i‖)
Cg : ℝ
hCg : Cg ∈ upperBounds (Set.range fun i => ‖g i‖)
i : I
⊢ (fun i => ‖(f * g) i‖) i ≤ Cf * Cg
|
calc ‖(f * g) i‖ ≤ ‖f i‖ * ‖g i‖ := <a>norm_mul_le</a> (f i) (g i) _ ≤ Cf * Cg := <a>mul_le_mul</a> (hCf ⟨i, <a>rfl</a>⟩) (hCg ⟨i, <a>rfl</a>⟩) (<a>norm_nonneg</a> _) ((<a>norm_nonneg</a> _).<a>LE.le.trans</a> (hCf ⟨i, <a>rfl</a>⟩))
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Analysis/NormedSpace/lpSpace.lean
|
UniformSpace.compactSpace_iff_seqCompactSpace
|
X : Type u_1
Y : Type u_2
inst✝¹ : UniformSpace X
s : Set X
inst✝ : (𝓤 X).IsCountablyGenerated
⊢ CompactSpace X ↔ SeqCompactSpace X
|
simp only [← <a>isCompact_univ_iff</a>, <a>seqCompactSpace_iff</a>, <a>UniformSpace.isCompact_iff_isSeqCompact</a>]
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Topology/Sequences.lean
|
MeasureTheory.IsStoppingTime.measurableSet_lt_of_countable_range'
|
Ω : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace Ω
inst✝ : LinearOrder ι
f : Filtration ι m
τ π : Ω → ι
hτ : IsStoppingTime f τ
h_countable : (Set.range τ).Countable
i : ι
⊢ MeasurableSet {ω | τ ω < i}
|
have : {ω | τ ω < i} = {ω | τ ω ≤ i} \ {ω | τ ω = i} := by ext1 ω simp only [<a>lt_iff_le_and_ne</a>, <a>Set.mem_setOf_eq</a>, <a>Set.mem_diff</a>]
|
Ω : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace Ω
inst✝ : LinearOrder ι
f : Filtration ι m
τ π : Ω → ι
hτ : IsStoppingTime f τ
h_countable : (Set.range τ).Countable
i : ι
this : {ω | τ ω < i} = {ω | τ ω ≤ i} \ {ω | τ ω = i}
⊢ MeasurableSet {ω | τ ω < i}
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Probability/Process/Stopping.lean
|
MeasureTheory.IsStoppingTime.measurableSet_lt_of_countable_range'
|
Ω : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace Ω
inst✝ : LinearOrder ι
f : Filtration ι m
τ π : Ω → ι
hτ : IsStoppingTime f τ
h_countable : (Set.range τ).Countable
i : ι
this : {ω | τ ω < i} = {ω | τ ω ≤ i} \ {ω | τ ω = i}
⊢ MeasurableSet {ω | τ ω < i}
|
rw [this]
|
Ω : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace Ω
inst✝ : LinearOrder ι
f : Filtration ι m
τ π : Ω → ι
hτ : IsStoppingTime f τ
h_countable : (Set.range τ).Countable
i : ι
this : {ω | τ ω < i} = {ω | τ ω ≤ i} \ {ω | τ ω = i}
⊢ MeasurableSet ({ω | τ ω ≤ i} \ {ω | τ ω = i})
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Probability/Process/Stopping.lean
|
MeasureTheory.IsStoppingTime.measurableSet_lt_of_countable_range'
|
Ω : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace Ω
inst✝ : LinearOrder ι
f : Filtration ι m
τ π : Ω → ι
hτ : IsStoppingTime f τ
h_countable : (Set.range τ).Countable
i : ι
this : {ω | τ ω < i} = {ω | τ ω ≤ i} \ {ω | τ ω = i}
⊢ MeasurableSet ({ω | τ ω ≤ i} \ {ω | τ ω = i})
|
exact (hτ.measurableSet_le' i).<a>MeasurableSet.diff</a> (hτ.measurableSet_eq_of_countable_range' h_countable i)
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Probability/Process/Stopping.lean
|
MeasureTheory.IsStoppingTime.measurableSet_lt_of_countable_range'
|
Ω : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace Ω
inst✝ : LinearOrder ι
f : Filtration ι m
τ π : Ω → ι
hτ : IsStoppingTime f τ
h_countable : (Set.range τ).Countable
i : ι
⊢ {ω | τ ω < i} = {ω | τ ω ≤ i} \ {ω | τ ω = i}
|
ext1 ω
|
case h
Ω : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace Ω
inst✝ : LinearOrder ι
f : Filtration ι m
τ π : Ω → ι
hτ : IsStoppingTime f τ
h_countable : (Set.range τ).Countable
i : ι
ω : Ω
⊢ ω ∈ {ω | τ ω < i} ↔ ω ∈ {ω | τ ω ≤ i} \ {ω | τ ω = i}
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Probability/Process/Stopping.lean
|
MeasureTheory.IsStoppingTime.measurableSet_lt_of_countable_range'
|
case h
Ω : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace Ω
inst✝ : LinearOrder ι
f : Filtration ι m
τ π : Ω → ι
hτ : IsStoppingTime f τ
h_countable : (Set.range τ).Countable
i : ι
ω : Ω
⊢ ω ∈ {ω | τ ω < i} ↔ ω ∈ {ω | τ ω ≤ i} \ {ω | τ ω = i}
|
simp only [<a>lt_iff_le_and_ne</a>, <a>Set.mem_setOf_eq</a>, <a>Set.mem_diff</a>]
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Probability/Process/Stopping.lean
|
Fin.map_valEmbedding_Ioi
|
n : ℕ
a b : Fin n
⊢ map valEmbedding (Ioi a) = Ioc (↑a) (n - 1)
|
clear b
|
n : ℕ
a : Fin n
⊢ map valEmbedding (Ioi a) = Ioc (↑a) (n - 1)
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Order/Interval/Finset/Fin.lean
|
Fin.map_valEmbedding_Ioi
|
n : ℕ
a : Fin n
⊢ map valEmbedding (Ioi a) = Ioc (↑a) (n - 1)
|
ext x
|
case a
n : ℕ
a : Fin n
x : ℕ
⊢ x ∈ map valEmbedding (Ioi a) ↔ x ∈ Ioc (↑a) (n - 1)
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Order/Interval/Finset/Fin.lean
|
Fin.map_valEmbedding_Ioi
|
case a
n : ℕ
a : Fin n
x : ℕ
⊢ x ∈ map valEmbedding (Ioi a) ↔ x ∈ Ioc (↑a) (n - 1)
|
simp only [<a>exists_prop</a>, <a>Function.Embedding.coe_subtype</a>, <a>Finset.mem_Ioi</a>, <a>Finset.mem_map</a>, <a>Finset.mem_Ioc</a>]
|
case a
n : ℕ
a : Fin n
x : ℕ
⊢ (∃ a_1, a < a_1 ∧ valEmbedding a_1 = x) ↔ ↑a < x ∧ x ≤ n - 1
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Order/Interval/Finset/Fin.lean
|
Fin.map_valEmbedding_Ioi
|
case a
n : ℕ
a : Fin n
x : ℕ
⊢ (∃ a_1, a < a_1 ∧ valEmbedding a_1 = x) ↔ ↑a < x ∧ x ≤ n - 1
|
constructor
|
case a.mp
n : ℕ
a : Fin n
x : ℕ
⊢ (∃ a_1, a < a_1 ∧ valEmbedding a_1 = x) → ↑a < x ∧ x ≤ n - 1
case a.mpr
n : ℕ
a : Fin n
x : ℕ
⊢ ↑a < x ∧ x ≤ n - 1 → ∃ a_2, a < a_2 ∧ valEmbedding a_2 = x
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Order/Interval/Finset/Fin.lean
|
Fin.map_valEmbedding_Ioi
|
case a.mpr
n : ℕ
a : Fin n
x : ℕ
⊢ ↑a < x ∧ x ≤ n - 1 → ∃ a_2, a < a_2 ∧ valEmbedding a_2 = x
|
cases n
|
case a.mpr.zero
x : ℕ
a : Fin 0
⊢ ↑a < x ∧ x ≤ 0 - 1 → ∃ a_2, a < a_2 ∧ valEmbedding a_2 = x
case a.mpr.succ
x n✝ : ℕ
a : Fin (n✝ + 1)
⊢ ↑a < x ∧ x ≤ n✝ + 1 - 1 → ∃ a_2, a < a_2 ∧ valEmbedding a_2 = x
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Order/Interval/Finset/Fin.lean
|
Fin.map_valEmbedding_Ioi
|
case a.mp
n : ℕ
a : Fin n
x : ℕ
⊢ (∃ a_1, a < a_1 ∧ valEmbedding a_1 = x) → ↑a < x ∧ x ≤ n - 1
|
rintro ⟨x, hx, rfl⟩
|
case a.mp.intro.intro
n : ℕ
a x : Fin n
hx : a < x
⊢ ↑a < valEmbedding x ∧ valEmbedding x ≤ n - 1
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Order/Interval/Finset/Fin.lean
|
Fin.map_valEmbedding_Ioi
|
case a.mp.intro.intro
n : ℕ
a x : Fin n
hx : a < x
⊢ ↑a < valEmbedding x ∧ valEmbedding x ≤ n - 1
|
exact ⟨hx, <a>Nat.le_sub_of_add_le</a> <| x.2⟩
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Order/Interval/Finset/Fin.lean
|
Fin.map_valEmbedding_Ioi
|
case a.mpr.zero
x : ℕ
a : Fin 0
⊢ ↑a < x ∧ x ≤ 0 - 1 → ∃ a_2, a < a_2 ∧ valEmbedding a_2 = x
|
exact <a>Fin.elim0</a> a
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Order/Interval/Finset/Fin.lean
|
Fin.map_valEmbedding_Ioi
|
case a.mpr.succ
x n✝ : ℕ
a : Fin (n✝ + 1)
⊢ ↑a < x ∧ x ≤ n✝ + 1 - 1 → ∃ a_2, a < a_2 ∧ valEmbedding a_2 = x
|
exact fun hx => ⟨⟨x, <a>Nat.lt_succ_iff</a>.2 hx.2⟩, hx.1, <a>rfl</a>⟩
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Order/Interval/Finset/Fin.lean
|
LinearPMap.ext_iff
|
R : Type u_1
inst✝⁶ : Ring R
E : Type u_2
inst✝⁵ : AddCommGroup E
inst✝⁴ : Module R E
F : Type u_3
inst✝³ : AddCommGroup F
inst✝² : Module R F
G : Type u_4
inst✝¹ : AddCommGroup G
inst✝ : Module R G
f g : E →ₗ.[R] F
EQ : f = g
x y : ↥f.domain
h : ↑x = ↑y
⊢ ↑f x = ↑f y
|
congr
|
case e_a
R : Type u_1
inst✝⁶ : Ring R
E : Type u_2
inst✝⁵ : AddCommGroup E
inst✝⁴ : Module R E
F : Type u_3
inst✝³ : AddCommGroup F
inst✝² : Module R F
G : Type u_4
inst✝¹ : AddCommGroup G
inst✝ : Module R G
f g : E →ₗ.[R] F
EQ : f = g
x y : ↥f.domain
h : ↑x = ↑y
⊢ x = y
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/LinearAlgebra/LinearPMap.lean
|
LinearPMap.ext_iff
|
case e_a
R : Type u_1
inst✝⁶ : Ring R
E : Type u_2
inst✝⁵ : AddCommGroup E
inst✝⁴ : Module R E
F : Type u_3
inst✝³ : AddCommGroup F
inst✝² : Module R F
G : Type u_4
inst✝¹ : AddCommGroup G
inst✝ : Module R G
f g : E →ₗ.[R] F
EQ : f = g
x y : ↥f.domain
h : ↑x = ↑y
⊢ x = y
|
exact mod_cast h
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/LinearAlgebra/LinearPMap.lean
|
CategoryTheory.Paths.lift_unique
|
V : Type u₁
inst✝¹ : Quiver V
C : Type u_1
inst✝ : Category.{u_2, u_1} C
φ : V ⥤q C
Φ : Paths V ⥤ C
hΦ : of ⋙q Φ.toPrefunctor = φ
⊢ Φ = lift φ
|
subst_vars
|
V : Type u₁
inst✝¹ : Quiver V
C : Type u_1
inst✝ : Category.{u_2, u_1} C
Φ : Paths V ⥤ C
⊢ Φ = lift (of ⋙q Φ.toPrefunctor)
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/CategoryTheory/PathCategory.lean
|
CategoryTheory.Paths.lift_unique
|
V : Type u₁
inst✝¹ : Quiver V
C : Type u_1
inst✝ : Category.{u_2, u_1} C
Φ : Paths V ⥤ C
⊢ Φ = lift (of ⋙q Φ.toPrefunctor)
|
fapply <a>CategoryTheory.Functor.ext</a>
|
case h_obj
V : Type u₁
inst✝¹ : Quiver V
C : Type u_1
inst✝ : Category.{u_2, u_1} C
Φ : Paths V ⥤ C
⊢ ∀ (X : Paths V), Φ.obj X = (lift (of ⋙q Φ.toPrefunctor)).obj X
case h_map
V : Type u₁
inst✝¹ : Quiver V
C : Type u_1
inst✝ : Category.{u_2, u_1} C
Φ : Paths V ⥤ C
⊢ autoParam (∀ (X Y : Paths V) (f : X ⟶ Y), Φ.map f = eqToHom ⋯ ≫ (lift (of ⋙q Φ.toPrefunctor)).map f ≫ eqToHom ⋯)
_auto✝
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/CategoryTheory/PathCategory.lean
|
CategoryTheory.Paths.lift_unique
|
case h_obj
V : Type u₁
inst✝¹ : Quiver V
C : Type u_1
inst✝ : Category.{u_2, u_1} C
Φ : Paths V ⥤ C
⊢ ∀ (X : Paths V), Φ.obj X = (lift (of ⋙q Φ.toPrefunctor)).obj X
|
rintro X
|
case h_obj
V : Type u₁
inst✝¹ : Quiver V
C : Type u_1
inst✝ : Category.{u_2, u_1} C
Φ : Paths V ⥤ C
X : Paths V
⊢ Φ.obj X = (lift (of ⋙q Φ.toPrefunctor)).obj X
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/CategoryTheory/PathCategory.lean
|
CategoryTheory.Paths.lift_unique
|
case h_obj
V : Type u₁
inst✝¹ : Quiver V
C : Type u_1
inst✝ : Category.{u_2, u_1} C
Φ : Paths V ⥤ C
X : Paths V
⊢ Φ.obj X = (lift (of ⋙q Φ.toPrefunctor)).obj X
|
rfl
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/CategoryTheory/PathCategory.lean
|
CategoryTheory.Paths.lift_unique
|
case h_map
V : Type u₁
inst✝¹ : Quiver V
C : Type u_1
inst✝ : Category.{u_2, u_1} C
Φ : Paths V ⥤ C
⊢ autoParam (∀ (X Y : Paths V) (f : X ⟶ Y), Φ.map f = eqToHom ⋯ ≫ (lift (of ⋙q Φ.toPrefunctor)).map f ≫ eqToHom ⋯)
_auto✝
|
rintro X Y f
|
case h_map
V : Type u₁
inst✝¹ : Quiver V
C : Type u_1
inst✝ : Category.{u_2, u_1} C
Φ : Paths V ⥤ C
X Y : Paths V
f : X ⟶ Y
⊢ Φ.map f = eqToHom ⋯ ≫ (lift (of ⋙q Φ.toPrefunctor)).map f ≫ eqToHom ⋯
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/CategoryTheory/PathCategory.lean
|
CategoryTheory.Paths.lift_unique
|
case h_map
V : Type u₁
inst✝¹ : Quiver V
C : Type u_1
inst✝ : Category.{u_2, u_1} C
Φ : Paths V ⥤ C
X Y : Paths V
f : X ⟶ Y
⊢ Φ.map f = eqToHom ⋯ ≫ (lift (of ⋙q Φ.toPrefunctor)).map f ≫ eqToHom ⋯
|
dsimp [<a>CategoryTheory.Paths.lift</a>]
|
case h_map
V : Type u₁
inst✝¹ : Quiver V
C : Type u_1
inst✝ : Category.{u_2, u_1} C
Φ : Paths V ⥤ C
X Y : Paths V
f : X ⟶ Y
⊢ Φ.map f = 𝟙 (Φ.obj X) ≫ Quiver.Path.rec (𝟙 (Φ.obj X)) (fun {b c} x f ihp => ihp ≫ Φ.map f.toPath) f ≫ 𝟙 (Φ.obj Y)
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/CategoryTheory/PathCategory.lean
|
CategoryTheory.Paths.lift_unique
|
case h_map
V : Type u₁
inst✝¹ : Quiver V
C : Type u_1
inst✝ : Category.{u_2, u_1} C
Φ : Paths V ⥤ C
X Y : Paths V
f : X ⟶ Y
⊢ Φ.map f = 𝟙 (Φ.obj X) ≫ Quiver.Path.rec (𝟙 (Φ.obj X)) (fun {b c} x f ihp => ihp ≫ Φ.map f.toPath) f ≫ 𝟙 (Φ.obj Y)
|
induction' f with _ _ p f' ih
|
case h_map.nil
V : Type u₁
inst✝¹ : Quiver V
C : Type u_1
inst✝ : Category.{u_2, u_1} C
Φ : Paths V ⥤ C
X Y : Paths V
⊢ Φ.map Quiver.Path.nil =
𝟙 (Φ.obj X) ≫
Quiver.Path.rec (𝟙 (Φ.obj X)) (fun {b c} x f ihp => ihp ≫ Φ.map f.toPath) Quiver.Path.nil ≫ 𝟙 (Φ.obj X)
case h_map.cons
V : Type u₁
inst✝¹ : Quiver V
C : Type u_1
inst✝ : Category.{u_2, u_1} C
Φ : Paths V ⥤ C
X Y b✝ c✝ : Paths V
p : Quiver.Path X b✝
f' : b✝ ⟶ c✝
ih : Φ.map p = 𝟙 (Φ.obj X) ≫ Quiver.Path.rec (𝟙 (Φ.obj X)) (fun {b c} x f ihp => ihp ≫ Φ.map f.toPath) p ≫ 𝟙 (Φ.obj b✝)
⊢ Φ.map (p.cons f') =
𝟙 (Φ.obj X) ≫ Quiver.Path.rec (𝟙 (Φ.obj X)) (fun {b c} x f ihp => ihp ≫ Φ.map f.toPath) (p.cons f') ≫ 𝟙 (Φ.obj c✝)
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/CategoryTheory/PathCategory.lean
|
CategoryTheory.Paths.lift_unique
|
case h_map.nil
V : Type u₁
inst✝¹ : Quiver V
C : Type u_1
inst✝ : Category.{u_2, u_1} C
Φ : Paths V ⥤ C
X Y : Paths V
⊢ Φ.map Quiver.Path.nil =
𝟙 (Φ.obj X) ≫
Quiver.Path.rec (𝟙 (Φ.obj X)) (fun {b c} x f ihp => ihp ≫ Φ.map f.toPath) Quiver.Path.nil ≫ 𝟙 (Φ.obj X)
|
simp only [<a>CategoryTheory.Category.comp_id</a>]
|
case h_map.nil
V : Type u₁
inst✝¹ : Quiver V
C : Type u_1
inst✝ : Category.{u_2, u_1} C
Φ : Paths V ⥤ C
X Y : Paths V
⊢ Φ.map Quiver.Path.nil = 𝟙 (Φ.obj X)
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/CategoryTheory/PathCategory.lean
|
CategoryTheory.Paths.lift_unique
|
case h_map.nil
V : Type u₁
inst✝¹ : Quiver V
C : Type u_1
inst✝ : Category.{u_2, u_1} C
Φ : Paths V ⥤ C
X Y : Paths V
⊢ Φ.map Quiver.Path.nil = 𝟙 (Φ.obj X)
|
apply <a>CategoryTheory.Functor.map_id</a>
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/CategoryTheory/PathCategory.lean
|
CategoryTheory.Paths.lift_unique
|
case h_map.cons
V : Type u₁
inst✝¹ : Quiver V
C : Type u_1
inst✝ : Category.{u_2, u_1} C
Φ : Paths V ⥤ C
X Y b✝ c✝ : Paths V
p : Quiver.Path X b✝
f' : b✝ ⟶ c✝
ih : Φ.map p = 𝟙 (Φ.obj X) ≫ Quiver.Path.rec (𝟙 (Φ.obj X)) (fun {b c} x f ihp => ihp ≫ Φ.map f.toPath) p ≫ 𝟙 (Φ.obj b✝)
⊢ Φ.map (p.cons f') =
𝟙 (Φ.obj X) ≫ Quiver.Path.rec (𝟙 (Φ.obj X)) (fun {b c} x f ihp => ihp ≫ Φ.map f.toPath) (p.cons f') ≫ 𝟙 (Φ.obj c✝)
|
simp only [<a>CategoryTheory.Category.comp_id</a>, <a>CategoryTheory.Category.id_comp</a>] at ih ⊢
|
case h_map.cons
V : Type u₁
inst✝¹ : Quiver V
C : Type u_1
inst✝ : Category.{u_2, u_1} C
Φ : Paths V ⥤ C
X Y b✝ c✝ : Paths V
p : Quiver.Path X b✝
f' : b✝ ⟶ c✝
ih : Φ.map p = Quiver.Path.rec (𝟙 (Φ.obj X)) (fun {b c} x f ihp => ihp ≫ Φ.map f.toPath) p
⊢ Φ.map (p.cons f') = Quiver.Path.rec (𝟙 (Φ.obj X)) (fun {b c} x f ihp => ihp ≫ Φ.map f.toPath) p ≫ Φ.map f'.toPath
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/CategoryTheory/PathCategory.lean
|
CategoryTheory.Paths.lift_unique
|
case h_map.cons
V : Type u₁
inst✝¹ : Quiver V
C : Type u_1
inst✝ : Category.{u_2, u_1} C
Φ : Paths V ⥤ C
X Y b✝ c✝ : Paths V
p : Quiver.Path X b✝
f' : b✝ ⟶ c✝
ih : Φ.map p = Quiver.Path.rec (𝟙 (Φ.obj X)) (fun {b c} x f ihp => ihp ≫ Φ.map f.toPath) p
⊢ Φ.map (p.cons f') = Quiver.Path.rec (𝟙 (Φ.obj X)) (fun {b c} x f ihp => ihp ≫ Φ.map f.toPath) p ≫ Φ.map f'.toPath
|
have : Φ.map (<a>Quiver.Path.cons</a> p f') = Φ.map p ≫ Φ.map (<a>Quiver.Hom.toPath</a> f') := by convert <a>CategoryTheory.Functor.map_comp</a> Φ p (<a>Quiver.Hom.toPath</a> f')
|
case h_map.cons
V : Type u₁
inst✝¹ : Quiver V
C : Type u_1
inst✝ : Category.{u_2, u_1} C
Φ : Paths V ⥤ C
X Y b✝ c✝ : Paths V
p : Quiver.Path X b✝
f' : b✝ ⟶ c✝
ih : Φ.map p = Quiver.Path.rec (𝟙 (Φ.obj X)) (fun {b c} x f ihp => ihp ≫ Φ.map f.toPath) p
this : Φ.map (p.cons f') = Φ.map p ≫ Φ.map f'.toPath
⊢ Φ.map (p.cons f') = Quiver.Path.rec (𝟙 (Φ.obj X)) (fun {b c} x f ihp => ihp ≫ Φ.map f.toPath) p ≫ Φ.map f'.toPath
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/CategoryTheory/PathCategory.lean
|
CategoryTheory.Paths.lift_unique
|
case h_map.cons
V : Type u₁
inst✝¹ : Quiver V
C : Type u_1
inst✝ : Category.{u_2, u_1} C
Φ : Paths V ⥤ C
X Y b✝ c✝ : Paths V
p : Quiver.Path X b✝
f' : b✝ ⟶ c✝
ih : Φ.map p = Quiver.Path.rec (𝟙 (Φ.obj X)) (fun {b c} x f ihp => ihp ≫ Φ.map f.toPath) p
this : Φ.map (p.cons f') = Φ.map p ≫ Φ.map f'.toPath
⊢ Φ.map (p.cons f') = Quiver.Path.rec (𝟙 (Φ.obj X)) (fun {b c} x f ihp => ihp ≫ Φ.map f.toPath) p ≫ Φ.map f'.toPath
|
rw [this, ih]
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/CategoryTheory/PathCategory.lean
|
CategoryTheory.Paths.lift_unique
|
V : Type u₁
inst✝¹ : Quiver V
C : Type u_1
inst✝ : Category.{u_2, u_1} C
Φ : Paths V ⥤ C
X Y b✝ c✝ : Paths V
p : Quiver.Path X b✝
f' : b✝ ⟶ c✝
ih : Φ.map p = Quiver.Path.rec (𝟙 (Φ.obj X)) (fun {b c} x f ihp => ihp ≫ Φ.map f.toPath) p
⊢ Φ.map (p.cons f') = Φ.map p ≫ Φ.map f'.toPath
|
convert <a>CategoryTheory.Functor.map_comp</a> Φ p (<a>Quiver.Hom.toPath</a> f')
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/CategoryTheory/PathCategory.lean
|
Bimod.pentagon_bimod
|
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
inst✝³ : MonoidalCategory C
A B : Mon_ C
M✝ : Bimod A B
inst✝² : HasCoequalizers C
inst✝¹ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
inst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)
V W X Y Z : Mon_ C
M : Bimod V W
N : Bimod W X
P : Bimod X Y
Q : Bimod Y Z
⊢ whiskerRight (M.associatorBimod N P).hom Q ≫
(M.associatorBimod (N.tensorBimod P) Q).hom ≫ M.whiskerLeft (N.associatorBimod P Q).hom =
((M.tensorBimod N).associatorBimod P Q).hom ≫ (M.associatorBimod N (P.tensorBimod Q)).hom
|
dsimp [<a>Bimod.associatorBimod</a>]
|
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
inst✝³ : MonoidalCategory C
A B : Mon_ C
M✝ : Bimod A B
inst✝² : HasCoequalizers C
inst✝¹ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
inst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)
V W X Y Z : Mon_ C
M : Bimod V W
N : Bimod W X
P : Bimod X Y
Q : Bimod Y Z
⊢ whiskerRight
(isoOfIso
{ hom := AssociatorBimod.hom M N P, inv := AssociatorBimod.inv M N P, hom_inv_id := ⋯, inv_hom_id := ⋯ } ⋯
⋯).hom
Q ≫
(isoOfIso
{ hom := AssociatorBimod.hom M (N.tensorBimod P) Q, inv := AssociatorBimod.inv M (N.tensorBimod P) Q,
hom_inv_id := ⋯, inv_hom_id := ⋯ }
⋯ ⋯).hom ≫
M.whiskerLeft
(isoOfIso
{ hom := AssociatorBimod.hom N P Q, inv := AssociatorBimod.inv N P Q, hom_inv_id := ⋯, inv_hom_id := ⋯ } ⋯
⋯).hom =
(isoOfIso
{ hom := AssociatorBimod.hom (M.tensorBimod N) P Q, inv := AssociatorBimod.inv (M.tensorBimod N) P Q,
hom_inv_id := ⋯, inv_hom_id := ⋯ }
⋯ ⋯).hom ≫
(isoOfIso
{ hom := AssociatorBimod.hom M N (P.tensorBimod Q), inv := AssociatorBimod.inv M N (P.tensorBimod Q),
hom_inv_id := ⋯, inv_hom_id := ⋯ }
⋯ ⋯).hom
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/CategoryTheory/Monoidal/Bimod.lean
|
Bimod.pentagon_bimod
|
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
inst✝³ : MonoidalCategory C
A B : Mon_ C
M✝ : Bimod A B
inst✝² : HasCoequalizers C
inst✝¹ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
inst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)
V W X Y Z : Mon_ C
M : Bimod V W
N : Bimod W X
P : Bimod X Y
Q : Bimod Y Z
⊢ whiskerRight
(isoOfIso
{ hom := AssociatorBimod.hom M N P, inv := AssociatorBimod.inv M N P, hom_inv_id := ⋯, inv_hom_id := ⋯ } ⋯
⋯).hom
Q ≫
(isoOfIso
{ hom := AssociatorBimod.hom M (N.tensorBimod P) Q, inv := AssociatorBimod.inv M (N.tensorBimod P) Q,
hom_inv_id := ⋯, inv_hom_id := ⋯ }
⋯ ⋯).hom ≫
M.whiskerLeft
(isoOfIso
{ hom := AssociatorBimod.hom N P Q, inv := AssociatorBimod.inv N P Q, hom_inv_id := ⋯, inv_hom_id := ⋯ } ⋯
⋯).hom =
(isoOfIso
{ hom := AssociatorBimod.hom (M.tensorBimod N) P Q, inv := AssociatorBimod.inv (M.tensorBimod N) P Q,
hom_inv_id := ⋯, inv_hom_id := ⋯ }
⋯ ⋯).hom ≫
(isoOfIso
{ hom := AssociatorBimod.hom M N (P.tensorBimod Q), inv := AssociatorBimod.inv M N (P.tensorBimod Q),
hom_inv_id := ⋯, inv_hom_id := ⋯ }
⋯ ⋯).hom
|
ext
|
case h
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
inst✝³ : MonoidalCategory C
A B : Mon_ C
M✝ : Bimod A B
inst✝² : HasCoequalizers C
inst✝¹ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
inst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)
V W X Y Z : Mon_ C
M : Bimod V W
N : Bimod W X
P : Bimod X Y
Q : Bimod Y Z
⊢ (whiskerRight
(isoOfIso
{ hom := AssociatorBimod.hom M N P, inv := AssociatorBimod.inv M N P, hom_inv_id := ⋯, inv_hom_id := ⋯ } ⋯
⋯).hom
Q ≫
(isoOfIso
{ hom := AssociatorBimod.hom M (N.tensorBimod P) Q, inv := AssociatorBimod.inv M (N.tensorBimod P) Q,
hom_inv_id := ⋯, inv_hom_id := ⋯ }
⋯ ⋯).hom ≫
M.whiskerLeft
(isoOfIso
{ hom := AssociatorBimod.hom N P Q, inv := AssociatorBimod.inv N P Q, hom_inv_id := ⋯, inv_hom_id := ⋯ }
⋯ ⋯).hom).hom =
((isoOfIso
{ hom := AssociatorBimod.hom (M.tensorBimod N) P Q, inv := AssociatorBimod.inv (M.tensorBimod N) P Q,
hom_inv_id := ⋯, inv_hom_id := ⋯ }
⋯ ⋯).hom ≫
(isoOfIso
{ hom := AssociatorBimod.hom M N (P.tensorBimod Q), inv := AssociatorBimod.inv M N (P.tensorBimod Q),
hom_inv_id := ⋯, inv_hom_id := ⋯ }
⋯ ⋯).hom).hom
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/CategoryTheory/Monoidal/Bimod.lean
|
Bimod.pentagon_bimod
|
case h
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
inst✝³ : MonoidalCategory C
A B : Mon_ C
M✝ : Bimod A B
inst✝² : HasCoequalizers C
inst✝¹ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
inst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)
V W X Y Z : Mon_ C
M : Bimod V W
N : Bimod W X
P : Bimod X Y
Q : Bimod Y Z
⊢ (whiskerRight
(isoOfIso
{ hom := AssociatorBimod.hom M N P, inv := AssociatorBimod.inv M N P, hom_inv_id := ⋯, inv_hom_id := ⋯ } ⋯
⋯).hom
Q ≫
(isoOfIso
{ hom := AssociatorBimod.hom M (N.tensorBimod P) Q, inv := AssociatorBimod.inv M (N.tensorBimod P) Q,
hom_inv_id := ⋯, inv_hom_id := ⋯ }
⋯ ⋯).hom ≫
M.whiskerLeft
(isoOfIso
{ hom := AssociatorBimod.hom N P Q, inv := AssociatorBimod.inv N P Q, hom_inv_id := ⋯, inv_hom_id := ⋯ }
⋯ ⋯).hom).hom =
((isoOfIso
{ hom := AssociatorBimod.hom (M.tensorBimod N) P Q, inv := AssociatorBimod.inv (M.tensorBimod N) P Q,
hom_inv_id := ⋯, inv_hom_id := ⋯ }
⋯ ⋯).hom ≫
(isoOfIso
{ hom := AssociatorBimod.hom M N (P.tensorBimod Q), inv := AssociatorBimod.inv M N (P.tensorBimod Q),
hom_inv_id := ⋯, inv_hom_id := ⋯ }
⋯ ⋯).hom).hom
|
apply <a>CategoryTheory.Limits.coequalizer.hom_ext</a>
|
case h.h
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
inst✝³ : MonoidalCategory C
A B : Mon_ C
M✝ : Bimod A B
inst✝² : HasCoequalizers C
inst✝¹ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
inst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)
V W X Y Z : Mon_ C
M : Bimod V W
N : Bimod W X
P : Bimod X Y
Q : Bimod Y Z
⊢ coequalizer.π (((M.tensorBimod N).tensorBimod P).actRight ▷ Q.X)
((α_ ((M.tensorBimod N).tensorBimod P).X Y.X Q.X).hom ≫ ((M.tensorBimod N).tensorBimod P).X ◁ Q.actLeft) ≫
(whiskerRight
(isoOfIso
{ hom := AssociatorBimod.hom M N P, inv := AssociatorBimod.inv M N P, hom_inv_id := ⋯, inv_hom_id := ⋯ }
⋯ ⋯).hom
Q ≫
(isoOfIso
{ hom := AssociatorBimod.hom M (N.tensorBimod P) Q, inv := AssociatorBimod.inv M (N.tensorBimod P) Q,
hom_inv_id := ⋯, inv_hom_id := ⋯ }
⋯ ⋯).hom ≫
M.whiskerLeft
(isoOfIso
{ hom := AssociatorBimod.hom N P Q, inv := AssociatorBimod.inv N P Q, hom_inv_id := ⋯,
inv_hom_id := ⋯ }
⋯ ⋯).hom).hom =
coequalizer.π (((M.tensorBimod N).tensorBimod P).actRight ▷ Q.X)
((α_ ((M.tensorBimod N).tensorBimod P).X Y.X Q.X).hom ≫ ((M.tensorBimod N).tensorBimod P).X ◁ Q.actLeft) ≫
((isoOfIso
{ hom := AssociatorBimod.hom (M.tensorBimod N) P Q, inv := AssociatorBimod.inv (M.tensorBimod N) P Q,
hom_inv_id := ⋯, inv_hom_id := ⋯ }
⋯ ⋯).hom ≫
(isoOfIso
{ hom := AssociatorBimod.hom M N (P.tensorBimod Q), inv := AssociatorBimod.inv M N (P.tensorBimod Q),
hom_inv_id := ⋯, inv_hom_id := ⋯ }
⋯ ⋯).hom).hom
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/CategoryTheory/Monoidal/Bimod.lean
|
Bimod.pentagon_bimod
|
case h.h
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
inst✝³ : MonoidalCategory C
A B : Mon_ C
M✝ : Bimod A B
inst✝² : HasCoequalizers C
inst✝¹ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
inst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)
V W X Y Z : Mon_ C
M : Bimod V W
N : Bimod W X
P : Bimod X Y
Q : Bimod Y Z
⊢ coequalizer.π (((M.tensorBimod N).tensorBimod P).actRight ▷ Q.X)
((α_ ((M.tensorBimod N).tensorBimod P).X Y.X Q.X).hom ≫ ((M.tensorBimod N).tensorBimod P).X ◁ Q.actLeft) ≫
(whiskerRight
(isoOfIso
{ hom := AssociatorBimod.hom M N P, inv := AssociatorBimod.inv M N P, hom_inv_id := ⋯, inv_hom_id := ⋯ }
⋯ ⋯).hom
Q ≫
(isoOfIso
{ hom := AssociatorBimod.hom M (N.tensorBimod P) Q, inv := AssociatorBimod.inv M (N.tensorBimod P) Q,
hom_inv_id := ⋯, inv_hom_id := ⋯ }
⋯ ⋯).hom ≫
M.whiskerLeft
(isoOfIso
{ hom := AssociatorBimod.hom N P Q, inv := AssociatorBimod.inv N P Q, hom_inv_id := ⋯,
inv_hom_id := ⋯ }
⋯ ⋯).hom).hom =
coequalizer.π (((M.tensorBimod N).tensorBimod P).actRight ▷ Q.X)
((α_ ((M.tensorBimod N).tensorBimod P).X Y.X Q.X).hom ≫ ((M.tensorBimod N).tensorBimod P).X ◁ Q.actLeft) ≫
((isoOfIso
{ hom := AssociatorBimod.hom (M.tensorBimod N) P Q, inv := AssociatorBimod.inv (M.tensorBimod N) P Q,
hom_inv_id := ⋯, inv_hom_id := ⋯ }
⋯ ⋯).hom ≫
(isoOfIso
{ hom := AssociatorBimod.hom M N (P.tensorBimod Q), inv := AssociatorBimod.inv M N (P.tensorBimod Q),
hom_inv_id := ⋯, inv_hom_id := ⋯ }
⋯ ⋯).hom).hom
|
dsimp
|
case h.h
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
inst✝³ : MonoidalCategory C
A B : Mon_ C
M✝ : Bimod A B
inst✝² : HasCoequalizers C
inst✝¹ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
inst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)
V W X Y Z : Mon_ C
M : Bimod V W
N : Bimod W X
P : Bimod X Y
Q : Bimod Y Z
⊢ coequalizer.π (TensorBimod.actRight (M.tensorBimod N) P ▷ Q.X)
((α_ (TensorBimod.X (M.tensorBimod N) P) Y.X Q.X).hom ≫ TensorBimod.X (M.tensorBimod N) P ◁ Q.actLeft) ≫
colimMap
(parallelPairHom (TensorBimod.actRight (M.tensorBimod N) P ▷ Q.X)
((α_ (TensorBimod.X (M.tensorBimod N) P) Y.X Q.X).hom ≫ TensorBimod.X (M.tensorBimod N) P ◁ Q.actLeft)
(TensorBimod.actRight M (N.tensorBimod P) ▷ Q.X)
((α_ (TensorBimod.X M (N.tensorBimod P)) Y.X Q.X).hom ≫ TensorBimod.X M (N.tensorBimod P) ◁ Q.actLeft)
(AssociatorBimod.hom M N P ▷ Y.X ▷ Q.X) (AssociatorBimod.hom M N P ▷ Q.X) ⋯ ⋯) ≫
AssociatorBimod.hom M (N.tensorBimod P) Q ≫
colimMap
(parallelPairHom (M.actRight ▷ TensorBimod.X (N.tensorBimod P) Q)
((α_ M.X W.X (TensorBimod.X (N.tensorBimod P) Q)).hom ≫ M.X ◁ TensorBimod.actLeft (N.tensorBimod P) Q)
(M.actRight ▷ TensorBimod.X N (P.tensorBimod Q))
((α_ M.X W.X (TensorBimod.X N (P.tensorBimod Q))).hom ≫ M.X ◁ TensorBimod.actLeft N (P.tensorBimod Q))
((M.X ⊗ W.X) ◁ AssociatorBimod.hom N P Q) (M.X ◁ AssociatorBimod.hom N P Q) ⋯ ⋯) =
coequalizer.π (TensorBimod.actRight (M.tensorBimod N) P ▷ Q.X)
((α_ (TensorBimod.X (M.tensorBimod N) P) Y.X Q.X).hom ≫ TensorBimod.X (M.tensorBimod N) P ◁ Q.actLeft) ≫
AssociatorBimod.hom (M.tensorBimod N) P Q ≫ AssociatorBimod.hom M N (P.tensorBimod Q)
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/CategoryTheory/Monoidal/Bimod.lean
|
Bimod.pentagon_bimod
|
case h.h
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
inst✝³ : MonoidalCategory C
A B : Mon_ C
M✝ : Bimod A B
inst✝² : HasCoequalizers C
inst✝¹ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
inst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)
V W X Y Z : Mon_ C
M : Bimod V W
N : Bimod W X
P : Bimod X Y
Q : Bimod Y Z
⊢ coequalizer.π (TensorBimod.actRight (M.tensorBimod N) P ▷ Q.X)
((α_ (TensorBimod.X (M.tensorBimod N) P) Y.X Q.X).hom ≫ TensorBimod.X (M.tensorBimod N) P ◁ Q.actLeft) ≫
colimMap
(parallelPairHom (TensorBimod.actRight (M.tensorBimod N) P ▷ Q.X)
((α_ (TensorBimod.X (M.tensorBimod N) P) Y.X Q.X).hom ≫ TensorBimod.X (M.tensorBimod N) P ◁ Q.actLeft)
(TensorBimod.actRight M (N.tensorBimod P) ▷ Q.X)
((α_ (TensorBimod.X M (N.tensorBimod P)) Y.X Q.X).hom ≫ TensorBimod.X M (N.tensorBimod P) ◁ Q.actLeft)
(AssociatorBimod.hom M N P ▷ Y.X ▷ Q.X) (AssociatorBimod.hom M N P ▷ Q.X) ⋯ ⋯) ≫
AssociatorBimod.hom M (N.tensorBimod P) Q ≫
colimMap
(parallelPairHom (M.actRight ▷ TensorBimod.X (N.tensorBimod P) Q)
((α_ M.X W.X (TensorBimod.X (N.tensorBimod P) Q)).hom ≫ M.X ◁ TensorBimod.actLeft (N.tensorBimod P) Q)
(M.actRight ▷ TensorBimod.X N (P.tensorBimod Q))
((α_ M.X W.X (TensorBimod.X N (P.tensorBimod Q))).hom ≫ M.X ◁ TensorBimod.actLeft N (P.tensorBimod Q))
((M.X ⊗ W.X) ◁ AssociatorBimod.hom N P Q) (M.X ◁ AssociatorBimod.hom N P Q) ⋯ ⋯) =
coequalizer.π (TensorBimod.actRight (M.tensorBimod N) P ▷ Q.X)
((α_ (TensorBimod.X (M.tensorBimod N) P) Y.X Q.X).hom ≫ TensorBimod.X (M.tensorBimod N) P ◁ Q.actLeft) ≫
AssociatorBimod.hom (M.tensorBimod N) P Q ≫ AssociatorBimod.hom M N (P.tensorBimod Q)
|
dsimp only [<a>Bimod.AssociatorBimod.hom</a>]
|
case h.h
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
inst✝³ : MonoidalCategory C
A B : Mon_ C
M✝ : Bimod A B
inst✝² : HasCoequalizers C
inst✝¹ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
inst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)
V W X Y Z : Mon_ C
M : Bimod V W
N : Bimod W X
P : Bimod X Y
Q : Bimod Y Z
⊢ coequalizer.π (TensorBimod.actRight (M.tensorBimod N) P ▷ Q.X)
((α_ (TensorBimod.X (M.tensorBimod N) P) Y.X Q.X).hom ≫ TensorBimod.X (M.tensorBimod N) P ◁ Q.actLeft) ≫
colimMap
(parallelPairHom (TensorBimod.actRight (M.tensorBimod N) P ▷ Q.X)
((α_ (TensorBimod.X (M.tensorBimod N) P) Y.X Q.X).hom ≫ TensorBimod.X (M.tensorBimod N) P ◁ Q.actLeft)
(TensorBimod.actRight M (N.tensorBimod P) ▷ Q.X)
((α_ (TensorBimod.X M (N.tensorBimod P)) Y.X Q.X).hom ≫ TensorBimod.X M (N.tensorBimod P) ◁ Q.actLeft)
(coequalizer.desc (AssociatorBimod.homAux M N P) ⋯ ▷ Y.X ▷ Q.X)
(coequalizer.desc (AssociatorBimod.homAux M N P) ⋯ ▷ Q.X) ⋯ ⋯) ≫
coequalizer.desc (AssociatorBimod.homAux M (N.tensorBimod P) Q) ⋯ ≫
colimMap
(parallelPairHom (M.actRight ▷ TensorBimod.X (N.tensorBimod P) Q)
((α_ M.X W.X (TensorBimod.X (N.tensorBimod P) Q)).hom ≫ M.X ◁ TensorBimod.actLeft (N.tensorBimod P) Q)
(M.actRight ▷ TensorBimod.X N (P.tensorBimod Q))
((α_ M.X W.X (TensorBimod.X N (P.tensorBimod Q))).hom ≫ M.X ◁ TensorBimod.actLeft N (P.tensorBimod Q))
((M.X ⊗ W.X) ◁ coequalizer.desc (AssociatorBimod.homAux N P Q) ⋯)
(M.X ◁ coequalizer.desc (AssociatorBimod.homAux N P Q) ⋯) ⋯ ⋯) =
coequalizer.π (TensorBimod.actRight (M.tensorBimod N) P ▷ Q.X)
((α_ (TensorBimod.X (M.tensorBimod N) P) Y.X Q.X).hom ≫ TensorBimod.X (M.tensorBimod N) P ◁ Q.actLeft) ≫
coequalizer.desc (AssociatorBimod.homAux (M.tensorBimod N) P Q) ⋯ ≫
coequalizer.desc (AssociatorBimod.homAux M N (P.tensorBimod Q)) ⋯
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/CategoryTheory/Monoidal/Bimod.lean
|
Bimod.pentagon_bimod
|
case h.h
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
inst✝³ : MonoidalCategory C
A B : Mon_ C
M✝ : Bimod A B
inst✝² : HasCoequalizers C
inst✝¹ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
inst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)
V W X Y Z : Mon_ C
M : Bimod V W
N : Bimod W X
P : Bimod X Y
Q : Bimod Y Z
⊢ coequalizer.π (TensorBimod.actRight (M.tensorBimod N) P ▷ Q.X)
((α_ (TensorBimod.X (M.tensorBimod N) P) Y.X Q.X).hom ≫ TensorBimod.X (M.tensorBimod N) P ◁ Q.actLeft) ≫
colimMap
(parallelPairHom (TensorBimod.actRight (M.tensorBimod N) P ▷ Q.X)
((α_ (TensorBimod.X (M.tensorBimod N) P) Y.X Q.X).hom ≫ TensorBimod.X (M.tensorBimod N) P ◁ Q.actLeft)
(TensorBimod.actRight M (N.tensorBimod P) ▷ Q.X)
((α_ (TensorBimod.X M (N.tensorBimod P)) Y.X Q.X).hom ≫ TensorBimod.X M (N.tensorBimod P) ◁ Q.actLeft)
(coequalizer.desc (AssociatorBimod.homAux M N P) ⋯ ▷ Y.X ▷ Q.X)
(coequalizer.desc (AssociatorBimod.homAux M N P) ⋯ ▷ Q.X) ⋯ ⋯) ≫
coequalizer.desc (AssociatorBimod.homAux M (N.tensorBimod P) Q) ⋯ ≫
colimMap
(parallelPairHom (M.actRight ▷ TensorBimod.X (N.tensorBimod P) Q)
((α_ M.X W.X (TensorBimod.X (N.tensorBimod P) Q)).hom ≫ M.X ◁ TensorBimod.actLeft (N.tensorBimod P) Q)
(M.actRight ▷ TensorBimod.X N (P.tensorBimod Q))
((α_ M.X W.X (TensorBimod.X N (P.tensorBimod Q))).hom ≫ M.X ◁ TensorBimod.actLeft N (P.tensorBimod Q))
((M.X ⊗ W.X) ◁ coequalizer.desc (AssociatorBimod.homAux N P Q) ⋯)
(M.X ◁ coequalizer.desc (AssociatorBimod.homAux N P Q) ⋯) ⋯ ⋯) =
coequalizer.π (TensorBimod.actRight (M.tensorBimod N) P ▷ Q.X)
((α_ (TensorBimod.X (M.tensorBimod N) P) Y.X Q.X).hom ≫ TensorBimod.X (M.tensorBimod N) P ◁ Q.actLeft) ≫
coequalizer.desc (AssociatorBimod.homAux (M.tensorBimod N) P Q) ⋯ ≫
coequalizer.desc (AssociatorBimod.homAux M N (P.tensorBimod Q)) ⋯
|
slice_lhs 1 2 => rw [<a>CategoryTheory.Limits.ι_colimMap</a>, <a>CategoryTheory.Limits.parallelPairHom_app_one</a>]
|
case h.h
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
inst✝³ : MonoidalCategory C
A B : Mon_ C
M✝ : Bimod A B
inst✝² : HasCoequalizers C
inst✝¹ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
inst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)
V W X Y Z : Mon_ C
M : Bimod V W
N : Bimod W X
P : Bimod X Y
Q : Bimod Y Z
⊢ ((coequalizer.desc (AssociatorBimod.homAux M N P) ⋯ ▷ Q.X ≫
colimit.ι
(parallelPair (TensorBimod.actRight M (N.tensorBimod P) ▷ Q.X)
((α_ (TensorBimod.X M (N.tensorBimod P)) Y.X Q.X).hom ≫ TensorBimod.X M (N.tensorBimod P) ◁ Q.actLeft))
WalkingParallelPair.one) ≫
coequalizer.desc (AssociatorBimod.homAux M (N.tensorBimod P) Q) ⋯) ≫
colimMap
(parallelPairHom (M.actRight ▷ TensorBimod.X (N.tensorBimod P) Q)
((α_ M.X W.X (TensorBimod.X (N.tensorBimod P) Q)).hom ≫ M.X ◁ TensorBimod.actLeft (N.tensorBimod P) Q)
(M.actRight ▷ TensorBimod.X N (P.tensorBimod Q))
((α_ M.X W.X (TensorBimod.X N (P.tensorBimod Q))).hom ≫ M.X ◁ TensorBimod.actLeft N (P.tensorBimod Q))
((M.X ⊗ W.X) ◁ coequalizer.desc (AssociatorBimod.homAux N P Q) ⋯)
(M.X ◁ coequalizer.desc (AssociatorBimod.homAux N P Q) ⋯) ⋯ ⋯) =
coequalizer.π (TensorBimod.actRight (M.tensorBimod N) P ▷ Q.X)
((α_ (TensorBimod.X (M.tensorBimod N) P) Y.X Q.X).hom ≫ TensorBimod.X (M.tensorBimod N) P ◁ Q.actLeft) ≫
coequalizer.desc (AssociatorBimod.homAux (M.tensorBimod N) P Q) ⋯ ≫
coequalizer.desc (AssociatorBimod.homAux M N (P.tensorBimod Q)) ⋯
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/CategoryTheory/Monoidal/Bimod.lean
|
Bimod.pentagon_bimod
|
case h.h
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
inst✝³ : MonoidalCategory C
A B : Mon_ C
M✝ : Bimod A B
inst✝² : HasCoequalizers C
inst✝¹ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
inst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)
V W X Y Z : Mon_ C
M : Bimod V W
N : Bimod W X
P : Bimod X Y
Q : Bimod Y Z
⊢ ((coequalizer.desc (AssociatorBimod.homAux M N P) ⋯ ▷ Q.X ≫
colimit.ι
(parallelPair (TensorBimod.actRight M (N.tensorBimod P) ▷ Q.X)
((α_ (TensorBimod.X M (N.tensorBimod P)) Y.X Q.X).hom ≫ TensorBimod.X M (N.tensorBimod P) ◁ Q.actLeft))
WalkingParallelPair.one) ≫
coequalizer.desc (AssociatorBimod.homAux M (N.tensorBimod P) Q) ⋯) ≫
colimMap
(parallelPairHom (M.actRight ▷ TensorBimod.X (N.tensorBimod P) Q)
((α_ M.X W.X (TensorBimod.X (N.tensorBimod P) Q)).hom ≫ M.X ◁ TensorBimod.actLeft (N.tensorBimod P) Q)
(M.actRight ▷ TensorBimod.X N (P.tensorBimod Q))
((α_ M.X W.X (TensorBimod.X N (P.tensorBimod Q))).hom ≫ M.X ◁ TensorBimod.actLeft N (P.tensorBimod Q))
((M.X ⊗ W.X) ◁ coequalizer.desc (AssociatorBimod.homAux N P Q) ⋯)
(M.X ◁ coequalizer.desc (AssociatorBimod.homAux N P Q) ⋯) ⋯ ⋯) =
coequalizer.π (TensorBimod.actRight (M.tensorBimod N) P ▷ Q.X)
((α_ (TensorBimod.X (M.tensorBimod N) P) Y.X Q.X).hom ≫ TensorBimod.X (M.tensorBimod N) P ◁ Q.actLeft) ≫
coequalizer.desc (AssociatorBimod.homAux (M.tensorBimod N) P Q) ⋯ ≫
coequalizer.desc (AssociatorBimod.homAux M N (P.tensorBimod Q)) ⋯
|
slice_lhs 2 3 => rw [<a>CategoryTheory.Limits.coequalizer.π_desc</a>]
|
case h.h
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
inst✝³ : MonoidalCategory C
A B : Mon_ C
M✝ : Bimod A B
inst✝² : HasCoequalizers C
inst✝¹ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
inst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)
V W X Y Z : Mon_ C
M : Bimod V W
N : Bimod W X
P : Bimod X Y
Q : Bimod Y Z
⊢ coequalizer.desc (AssociatorBimod.homAux M N P) ⋯ ▷ Q.X ≫
AssociatorBimod.homAux M (N.tensorBimod P) Q ≫
colimMap
(parallelPairHom (M.actRight ▷ TensorBimod.X (N.tensorBimod P) Q)
((α_ M.X W.X (TensorBimod.X (N.tensorBimod P) Q)).hom ≫ M.X ◁ TensorBimod.actLeft (N.tensorBimod P) Q)
(M.actRight ▷ TensorBimod.X N (P.tensorBimod Q))
((α_ M.X W.X (TensorBimod.X N (P.tensorBimod Q))).hom ≫ M.X ◁ TensorBimod.actLeft N (P.tensorBimod Q))
((M.X ⊗ W.X) ◁ coequalizer.desc (AssociatorBimod.homAux N P Q) ⋯)
(M.X ◁ coequalizer.desc (AssociatorBimod.homAux N P Q) ⋯) ⋯ ⋯) =
coequalizer.π (TensorBimod.actRight (M.tensorBimod N) P ▷ Q.X)
((α_ (TensorBimod.X (M.tensorBimod N) P) Y.X Q.X).hom ≫ TensorBimod.X (M.tensorBimod N) P ◁ Q.actLeft) ≫
coequalizer.desc (AssociatorBimod.homAux (M.tensorBimod N) P Q) ⋯ ≫
coequalizer.desc (AssociatorBimod.homAux M N (P.tensorBimod Q)) ⋯
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/CategoryTheory/Monoidal/Bimod.lean
|
Bimod.pentagon_bimod
|
case h.h
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
inst✝³ : MonoidalCategory C
A B : Mon_ C
M✝ : Bimod A B
inst✝² : HasCoequalizers C
inst✝¹ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
inst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)
V W X Y Z : Mon_ C
M : Bimod V W
N : Bimod W X
P : Bimod X Y
Q : Bimod Y Z
⊢ coequalizer.desc (AssociatorBimod.homAux M N P) ⋯ ▷ Q.X ≫
AssociatorBimod.homAux M (N.tensorBimod P) Q ≫
colimMap
(parallelPairHom (M.actRight ▷ TensorBimod.X (N.tensorBimod P) Q)
((α_ M.X W.X (TensorBimod.X (N.tensorBimod P) Q)).hom ≫ M.X ◁ TensorBimod.actLeft (N.tensorBimod P) Q)
(M.actRight ▷ TensorBimod.X N (P.tensorBimod Q))
((α_ M.X W.X (TensorBimod.X N (P.tensorBimod Q))).hom ≫ M.X ◁ TensorBimod.actLeft N (P.tensorBimod Q))
((M.X ⊗ W.X) ◁ coequalizer.desc (AssociatorBimod.homAux N P Q) ⋯)
(M.X ◁ coequalizer.desc (AssociatorBimod.homAux N P Q) ⋯) ⋯ ⋯) =
coequalizer.π (TensorBimod.actRight (M.tensorBimod N) P ▷ Q.X)
((α_ (TensorBimod.X (M.tensorBimod N) P) Y.X Q.X).hom ≫ TensorBimod.X (M.tensorBimod N) P ◁ Q.actLeft) ≫
coequalizer.desc (AssociatorBimod.homAux (M.tensorBimod N) P Q) ⋯ ≫
coequalizer.desc (AssociatorBimod.homAux M N (P.tensorBimod Q)) ⋯
|
slice_rhs 1 2 => rw [<a>CategoryTheory.Limits.coequalizer.π_desc</a>]
|
case h.h
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
inst✝³ : MonoidalCategory C
A B : Mon_ C
M✝ : Bimod A B
inst✝² : HasCoequalizers C
inst✝¹ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
inst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)
V W X Y Z : Mon_ C
M : Bimod V W
N : Bimod W X
P : Bimod X Y
Q : Bimod Y Z
⊢ coequalizer.desc (AssociatorBimod.homAux M N P) ⋯ ▷ Q.X ≫
AssociatorBimod.homAux M (N.tensorBimod P) Q ≫
colimMap
(parallelPairHom (M.actRight ▷ TensorBimod.X (N.tensorBimod P) Q)
((α_ M.X W.X (TensorBimod.X (N.tensorBimod P) Q)).hom ≫ M.X ◁ TensorBimod.actLeft (N.tensorBimod P) Q)
(M.actRight ▷ TensorBimod.X N (P.tensorBimod Q))
((α_ M.X W.X (TensorBimod.X N (P.tensorBimod Q))).hom ≫ M.X ◁ TensorBimod.actLeft N (P.tensorBimod Q))
((M.X ⊗ W.X) ◁ coequalizer.desc (AssociatorBimod.homAux N P Q) ⋯)
(M.X ◁ coequalizer.desc (AssociatorBimod.homAux N P Q) ⋯) ⋯ ⋯) =
AssociatorBimod.homAux (M.tensorBimod N) P Q ≫ coequalizer.desc (AssociatorBimod.homAux M N (P.tensorBimod Q)) ⋯
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/CategoryTheory/Monoidal/Bimod.lean
|
Bimod.pentagon_bimod
|
case h.h
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
inst✝³ : MonoidalCategory C
A B : Mon_ C
M✝ : Bimod A B
inst✝² : HasCoequalizers C
inst✝¹ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
inst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)
V W X Y Z : Mon_ C
M : Bimod V W
N : Bimod W X
P : Bimod X Y
Q : Bimod Y Z
⊢ coequalizer.desc (AssociatorBimod.homAux M N P) ⋯ ▷ Q.X ≫
AssociatorBimod.homAux M (N.tensorBimod P) Q ≫
colimMap
(parallelPairHom (M.actRight ▷ TensorBimod.X (N.tensorBimod P) Q)
((α_ M.X W.X (TensorBimod.X (N.tensorBimod P) Q)).hom ≫ M.X ◁ TensorBimod.actLeft (N.tensorBimod P) Q)
(M.actRight ▷ TensorBimod.X N (P.tensorBimod Q))
((α_ M.X W.X (TensorBimod.X N (P.tensorBimod Q))).hom ≫ M.X ◁ TensorBimod.actLeft N (P.tensorBimod Q))
((M.X ⊗ W.X) ◁ coequalizer.desc (AssociatorBimod.homAux N P Q) ⋯)
(M.X ◁ coequalizer.desc (AssociatorBimod.homAux N P Q) ⋯) ⋯ ⋯) =
AssociatorBimod.homAux (M.tensorBimod N) P Q ≫ coequalizer.desc (AssociatorBimod.homAux M N (P.tensorBimod Q)) ⋯
|
dsimp [<a>Bimod.AssociatorBimod.homAux</a>]
|
case h.h
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
inst✝³ : MonoidalCategory C
A B : Mon_ C
M✝ : Bimod A B
inst✝² : HasCoequalizers C
inst✝¹ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
inst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)
V W X Y Z : Mon_ C
M : Bimod V W
N : Bimod W X
P : Bimod X Y
Q : Bimod Y Z
⊢ coequalizer.desc
((PreservesCoequalizer.iso (tensorRight P.X) (M.actRight ▷ N.X)
((α_ M.X W.X N.X).hom ≫ M.X ◁ N.actLeft)).inv ≫
coequalizer.desc
((α_ M.X N.X P.X).hom ≫
M.X ◁ coequalizer.π (N.actRight ▷ P.X) ((α_ N.X X.X P.X).hom ≫ N.X ◁ P.actLeft) ≫
coequalizer.π (M.actRight ▷ TensorBimod.X N P)
((α_ M.X W.X (TensorBimod.X N P)).hom ≫ M.X ◁ TensorBimod.actLeft N P))
⋯)
⋯ ▷
Q.X ≫
((PreservesCoequalizer.iso (tensorRight Q.X) (M.actRight ▷ TensorBimod.X N P)
((α_ M.X W.X (TensorBimod.X N P)).hom ≫ M.X ◁ TensorBimod.actLeft N P)).inv ≫
coequalizer.desc
((α_ M.X (TensorBimod.X N P) Q.X).hom ≫
M.X ◁
coequalizer.π (TensorBimod.actRight N P ▷ Q.X)
((α_ (TensorBimod.X N P) Y.X Q.X).hom ≫ TensorBimod.X N P ◁ Q.actLeft) ≫
coequalizer.π (M.actRight ▷ TensorBimod.X (N.tensorBimod P) Q)
((α_ M.X W.X (TensorBimod.X (N.tensorBimod P) Q)).hom ≫
M.X ◁ TensorBimod.actLeft (N.tensorBimod P) Q))
⋯) ≫
colimMap
(parallelPairHom (M.actRight ▷ TensorBimod.X (N.tensorBimod P) Q)
((α_ M.X W.X (TensorBimod.X (N.tensorBimod P) Q)).hom ≫ M.X ◁ TensorBimod.actLeft (N.tensorBimod P) Q)
(M.actRight ▷ TensorBimod.X N (P.tensorBimod Q))
((α_ M.X W.X (TensorBimod.X N (P.tensorBimod Q))).hom ≫ M.X ◁ TensorBimod.actLeft N (P.tensorBimod Q))
((M.X ⊗ W.X) ◁
coequalizer.desc
((PreservesCoequalizer.iso (tensorRight Q.X) (N.actRight ▷ P.X)
((α_ N.X X.X P.X).hom ≫ N.X ◁ P.actLeft)).inv ≫
coequalizer.desc
((α_ N.X P.X Q.X).hom ≫
N.X ◁ coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X Y.X Q.X).hom ≫ P.X ◁ Q.actLeft) ≫
coequalizer.π (N.actRight ▷ TensorBimod.X P Q)
((α_ N.X X.X (TensorBimod.X P Q)).hom ≫ N.X ◁ TensorBimod.actLeft P Q))
⋯)
⋯)
(M.X ◁
coequalizer.desc
((PreservesCoequalizer.iso (tensorRight Q.X) (N.actRight ▷ P.X)
((α_ N.X X.X P.X).hom ≫ N.X ◁ P.actLeft)).inv ≫
coequalizer.desc
((α_ N.X P.X Q.X).hom ≫
N.X ◁ coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X Y.X Q.X).hom ≫ P.X ◁ Q.actLeft) ≫
coequalizer.π (N.actRight ▷ TensorBimod.X P Q)
((α_ N.X X.X (TensorBimod.X P Q)).hom ≫ N.X ◁ TensorBimod.actLeft P Q))
⋯)
⋯)
⋯ ⋯) =
((PreservesCoequalizer.iso (tensorRight Q.X) (TensorBimod.actRight M N ▷ P.X)
((α_ (TensorBimod.X M N) X.X P.X).hom ≫ TensorBimod.X M N ◁ P.actLeft)).inv ≫
coequalizer.desc
((α_ (TensorBimod.X M N) P.X Q.X).hom ≫
TensorBimod.X M N ◁ coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X Y.X Q.X).hom ≫ P.X ◁ Q.actLeft) ≫
coequalizer.π (TensorBimod.actRight M N ▷ TensorBimod.X P Q)
((α_ (TensorBimod.X M N) X.X (TensorBimod.X P Q)).hom ≫ TensorBimod.X M N ◁ TensorBimod.actLeft P Q))
⋯) ≫
coequalizer.desc
((PreservesCoequalizer.iso (tensorRight (TensorBimod.X P Q)) (M.actRight ▷ N.X)
((α_ M.X W.X N.X).hom ≫ M.X ◁ N.actLeft)).inv ≫
coequalizer.desc
((α_ M.X N.X (TensorBimod.X P Q)).hom ≫
M.X ◁
coequalizer.π (N.actRight ▷ TensorBimod.X P Q)
((α_ N.X X.X (TensorBimod.X P Q)).hom ≫ N.X ◁ TensorBimod.actLeft P Q) ≫
coequalizer.π (M.actRight ▷ TensorBimod.X N (P.tensorBimod Q))
((α_ M.X W.X (TensorBimod.X N (P.tensorBimod Q))).hom ≫
M.X ◁ TensorBimod.actLeft N (P.tensorBimod Q)))
⋯)
⋯
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/CategoryTheory/Monoidal/Bimod.lean
|
Bimod.pentagon_bimod
|
case h.h
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
inst✝³ : MonoidalCategory C
A B : Mon_ C
M✝ : Bimod A B
inst✝² : HasCoequalizers C
inst✝¹ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
inst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)
V W X Y Z : Mon_ C
M : Bimod V W
N : Bimod W X
P : Bimod X Y
Q : Bimod Y Z
⊢ coequalizer.desc
((PreservesCoequalizer.iso (tensorRight P.X) (M.actRight ▷ N.X)
((α_ M.X W.X N.X).hom ≫ M.X ◁ N.actLeft)).inv ≫
coequalizer.desc
((α_ M.X N.X P.X).hom ≫
M.X ◁ coequalizer.π (N.actRight ▷ P.X) ((α_ N.X X.X P.X).hom ≫ N.X ◁ P.actLeft) ≫
coequalizer.π (M.actRight ▷ TensorBimod.X N P)
((α_ M.X W.X (TensorBimod.X N P)).hom ≫ M.X ◁ TensorBimod.actLeft N P))
⋯)
⋯ ▷
Q.X ≫
((PreservesCoequalizer.iso (tensorRight Q.X) (M.actRight ▷ TensorBimod.X N P)
((α_ M.X W.X (TensorBimod.X N P)).hom ≫ M.X ◁ TensorBimod.actLeft N P)).inv ≫
coequalizer.desc
((α_ M.X (TensorBimod.X N P) Q.X).hom ≫
M.X ◁
coequalizer.π (TensorBimod.actRight N P ▷ Q.X)
((α_ (TensorBimod.X N P) Y.X Q.X).hom ≫ TensorBimod.X N P ◁ Q.actLeft) ≫
coequalizer.π (M.actRight ▷ TensorBimod.X (N.tensorBimod P) Q)
((α_ M.X W.X (TensorBimod.X (N.tensorBimod P) Q)).hom ≫
M.X ◁ TensorBimod.actLeft (N.tensorBimod P) Q))
⋯) ≫
colimMap
(parallelPairHom (M.actRight ▷ TensorBimod.X (N.tensorBimod P) Q)
((α_ M.X W.X (TensorBimod.X (N.tensorBimod P) Q)).hom ≫ M.X ◁ TensorBimod.actLeft (N.tensorBimod P) Q)
(M.actRight ▷ TensorBimod.X N (P.tensorBimod Q))
((α_ M.X W.X (TensorBimod.X N (P.tensorBimod Q))).hom ≫ M.X ◁ TensorBimod.actLeft N (P.tensorBimod Q))
((M.X ⊗ W.X) ◁
coequalizer.desc
((PreservesCoequalizer.iso (tensorRight Q.X) (N.actRight ▷ P.X)
((α_ N.X X.X P.X).hom ≫ N.X ◁ P.actLeft)).inv ≫
coequalizer.desc
((α_ N.X P.X Q.X).hom ≫
N.X ◁ coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X Y.X Q.X).hom ≫ P.X ◁ Q.actLeft) ≫
coequalizer.π (N.actRight ▷ TensorBimod.X P Q)
((α_ N.X X.X (TensorBimod.X P Q)).hom ≫ N.X ◁ TensorBimod.actLeft P Q))
⋯)
⋯)
(M.X ◁
coequalizer.desc
((PreservesCoequalizer.iso (tensorRight Q.X) (N.actRight ▷ P.X)
((α_ N.X X.X P.X).hom ≫ N.X ◁ P.actLeft)).inv ≫
coequalizer.desc
((α_ N.X P.X Q.X).hom ≫
N.X ◁ coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X Y.X Q.X).hom ≫ P.X ◁ Q.actLeft) ≫
coequalizer.π (N.actRight ▷ TensorBimod.X P Q)
((α_ N.X X.X (TensorBimod.X P Q)).hom ≫ N.X ◁ TensorBimod.actLeft P Q))
⋯)
⋯)
⋯ ⋯) =
((PreservesCoequalizer.iso (tensorRight Q.X) (TensorBimod.actRight M N ▷ P.X)
((α_ (TensorBimod.X M N) X.X P.X).hom ≫ TensorBimod.X M N ◁ P.actLeft)).inv ≫
coequalizer.desc
((α_ (TensorBimod.X M N) P.X Q.X).hom ≫
TensorBimod.X M N ◁ coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X Y.X Q.X).hom ≫ P.X ◁ Q.actLeft) ≫
coequalizer.π (TensorBimod.actRight M N ▷ TensorBimod.X P Q)
((α_ (TensorBimod.X M N) X.X (TensorBimod.X P Q)).hom ≫ TensorBimod.X M N ◁ TensorBimod.actLeft P Q))
⋯) ≫
coequalizer.desc
((PreservesCoequalizer.iso (tensorRight (TensorBimod.X P Q)) (M.actRight ▷ N.X)
((α_ M.X W.X N.X).hom ≫ M.X ◁ N.actLeft)).inv ≫
coequalizer.desc
((α_ M.X N.X (TensorBimod.X P Q)).hom ≫
M.X ◁
coequalizer.π (N.actRight ▷ TensorBimod.X P Q)
((α_ N.X X.X (TensorBimod.X P Q)).hom ≫ N.X ◁ TensorBimod.actLeft P Q) ≫
coequalizer.π (M.actRight ▷ TensorBimod.X N (P.tensorBimod Q))
((α_ M.X W.X (TensorBimod.X N (P.tensorBimod Q))).hom ≫
M.X ◁ TensorBimod.actLeft N (P.tensorBimod Q)))
⋯)
⋯
|
refine (<a>CategoryTheory.cancel_epi</a> ((<a>CategoryTheory.MonoidalCategory.tensorRight</a> _).<a>Prefunctor.map</a> (<a>CategoryTheory.Limits.coequalizer.π</a> _ _))).1 ?_
|
case h.h
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
inst✝³ : MonoidalCategory C
A B : Mon_ C
M✝ : Bimod A B
inst✝² : HasCoequalizers C
inst✝¹ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
inst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)
V W X Y Z : Mon_ C
M : Bimod V W
N : Bimod W X
P : Bimod X Y
Q : Bimod Y Z
⊢ (tensorRight Q.X).map
(coequalizer.π ((M.tensorBimod N).actRight ▷ P.X)
((α_ (M.tensorBimod N).X X.X P.X).hom ≫ (M.tensorBimod N).X ◁ P.actLeft)) ≫
coequalizer.desc
((PreservesCoequalizer.iso (tensorRight P.X) (M.actRight ▷ N.X)
((α_ M.X W.X N.X).hom ≫ M.X ◁ N.actLeft)).inv ≫
coequalizer.desc
((α_ M.X N.X P.X).hom ≫
M.X ◁ coequalizer.π (N.actRight ▷ P.X) ((α_ N.X X.X P.X).hom ≫ N.X ◁ P.actLeft) ≫
coequalizer.π (M.actRight ▷ TensorBimod.X N P)
((α_ M.X W.X (TensorBimod.X N P)).hom ≫ M.X ◁ TensorBimod.actLeft N P))
⋯)
⋯ ▷
Q.X ≫
((PreservesCoequalizer.iso (tensorRight Q.X) (M.actRight ▷ TensorBimod.X N P)
((α_ M.X W.X (TensorBimod.X N P)).hom ≫ M.X ◁ TensorBimod.actLeft N P)).inv ≫
coequalizer.desc
((α_ M.X (TensorBimod.X N P) Q.X).hom ≫
M.X ◁
coequalizer.π (TensorBimod.actRight N P ▷ Q.X)
((α_ (TensorBimod.X N P) Y.X Q.X).hom ≫ TensorBimod.X N P ◁ Q.actLeft) ≫
coequalizer.π (M.actRight ▷ TensorBimod.X (N.tensorBimod P) Q)
((α_ M.X W.X (TensorBimod.X (N.tensorBimod P) Q)).hom ≫
M.X ◁ TensorBimod.actLeft (N.tensorBimod P) Q))
⋯) ≫
colimMap
(parallelPairHom (M.actRight ▷ TensorBimod.X (N.tensorBimod P) Q)
((α_ M.X W.X (TensorBimod.X (N.tensorBimod P) Q)).hom ≫ M.X ◁ TensorBimod.actLeft (N.tensorBimod P) Q)
(M.actRight ▷ TensorBimod.X N (P.tensorBimod Q))
((α_ M.X W.X (TensorBimod.X N (P.tensorBimod Q))).hom ≫ M.X ◁ TensorBimod.actLeft N (P.tensorBimod Q))
((M.X ⊗ W.X) ◁
coequalizer.desc
((PreservesCoequalizer.iso (tensorRight Q.X) (N.actRight ▷ P.X)
((α_ N.X X.X P.X).hom ≫ N.X ◁ P.actLeft)).inv ≫
coequalizer.desc
((α_ N.X P.X Q.X).hom ≫
N.X ◁ coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X Y.X Q.X).hom ≫ P.X ◁ Q.actLeft) ≫
coequalizer.π (N.actRight ▷ TensorBimod.X P Q)
((α_ N.X X.X (TensorBimod.X P Q)).hom ≫ N.X ◁ TensorBimod.actLeft P Q))
⋯)
⋯)
(M.X ◁
coequalizer.desc
((PreservesCoequalizer.iso (tensorRight Q.X) (N.actRight ▷ P.X)
((α_ N.X X.X P.X).hom ≫ N.X ◁ P.actLeft)).inv ≫
coequalizer.desc
((α_ N.X P.X Q.X).hom ≫
N.X ◁ coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X Y.X Q.X).hom ≫ P.X ◁ Q.actLeft) ≫
coequalizer.π (N.actRight ▷ TensorBimod.X P Q)
((α_ N.X X.X (TensorBimod.X P Q)).hom ≫ N.X ◁ TensorBimod.actLeft P Q))
⋯)
⋯)
⋯ ⋯) =
(tensorRight Q.X).map
(coequalizer.π ((M.tensorBimod N).actRight ▷ P.X)
((α_ (M.tensorBimod N).X X.X P.X).hom ≫ (M.tensorBimod N).X ◁ P.actLeft)) ≫
((PreservesCoequalizer.iso (tensorRight Q.X) (TensorBimod.actRight M N ▷ P.X)
((α_ (TensorBimod.X M N) X.X P.X).hom ≫ TensorBimod.X M N ◁ P.actLeft)).inv ≫
coequalizer.desc
((α_ (TensorBimod.X M N) P.X Q.X).hom ≫
TensorBimod.X M N ◁ coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X Y.X Q.X).hom ≫ P.X ◁ Q.actLeft) ≫
coequalizer.π (TensorBimod.actRight M N ▷ TensorBimod.X P Q)
((α_ (TensorBimod.X M N) X.X (TensorBimod.X P Q)).hom ≫ TensorBimod.X M N ◁ TensorBimod.actLeft P Q))
⋯) ≫
coequalizer.desc
((PreservesCoequalizer.iso (tensorRight (TensorBimod.X P Q)) (M.actRight ▷ N.X)
((α_ M.X W.X N.X).hom ≫ M.X ◁ N.actLeft)).inv ≫
coequalizer.desc
((α_ M.X N.X (TensorBimod.X P Q)).hom ≫
M.X ◁
coequalizer.π (N.actRight ▷ TensorBimod.X P Q)
((α_ N.X X.X (TensorBimod.X P Q)).hom ≫ N.X ◁ TensorBimod.actLeft P Q) ≫
coequalizer.π (M.actRight ▷ TensorBimod.X N (P.tensorBimod Q))
((α_ M.X W.X (TensorBimod.X N (P.tensorBimod Q))).hom ≫
M.X ◁ TensorBimod.actLeft N (P.tensorBimod Q)))
⋯)
⋯
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/CategoryTheory/Monoidal/Bimod.lean
|
Bimod.pentagon_bimod
|
case h.h
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
inst✝³ : MonoidalCategory C
A B : Mon_ C
M✝ : Bimod A B
inst✝² : HasCoequalizers C
inst✝¹ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
inst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)
V W X Y Z : Mon_ C
M : Bimod V W
N : Bimod W X
P : Bimod X Y
Q : Bimod Y Z
⊢ (tensorRight Q.X).map
(coequalizer.π ((M.tensorBimod N).actRight ▷ P.X)
((α_ (M.tensorBimod N).X X.X P.X).hom ≫ (M.tensorBimod N).X ◁ P.actLeft)) ≫
coequalizer.desc
((PreservesCoequalizer.iso (tensorRight P.X) (M.actRight ▷ N.X)
((α_ M.X W.X N.X).hom ≫ M.X ◁ N.actLeft)).inv ≫
coequalizer.desc
((α_ M.X N.X P.X).hom ≫
M.X ◁ coequalizer.π (N.actRight ▷ P.X) ((α_ N.X X.X P.X).hom ≫ N.X ◁ P.actLeft) ≫
coequalizer.π (M.actRight ▷ TensorBimod.X N P)
((α_ M.X W.X (TensorBimod.X N P)).hom ≫ M.X ◁ TensorBimod.actLeft N P))
⋯)
⋯ ▷
Q.X ≫
((PreservesCoequalizer.iso (tensorRight Q.X) (M.actRight ▷ TensorBimod.X N P)
((α_ M.X W.X (TensorBimod.X N P)).hom ≫ M.X ◁ TensorBimod.actLeft N P)).inv ≫
coequalizer.desc
((α_ M.X (TensorBimod.X N P) Q.X).hom ≫
M.X ◁
coequalizer.π (TensorBimod.actRight N P ▷ Q.X)
((α_ (TensorBimod.X N P) Y.X Q.X).hom ≫ TensorBimod.X N P ◁ Q.actLeft) ≫
coequalizer.π (M.actRight ▷ TensorBimod.X (N.tensorBimod P) Q)
((α_ M.X W.X (TensorBimod.X (N.tensorBimod P) Q)).hom ≫
M.X ◁ TensorBimod.actLeft (N.tensorBimod P) Q))
⋯) ≫
colimMap
(parallelPairHom (M.actRight ▷ TensorBimod.X (N.tensorBimod P) Q)
((α_ M.X W.X (TensorBimod.X (N.tensorBimod P) Q)).hom ≫ M.X ◁ TensorBimod.actLeft (N.tensorBimod P) Q)
(M.actRight ▷ TensorBimod.X N (P.tensorBimod Q))
((α_ M.X W.X (TensorBimod.X N (P.tensorBimod Q))).hom ≫ M.X ◁ TensorBimod.actLeft N (P.tensorBimod Q))
((M.X ⊗ W.X) ◁
coequalizer.desc
((PreservesCoequalizer.iso (tensorRight Q.X) (N.actRight ▷ P.X)
((α_ N.X X.X P.X).hom ≫ N.X ◁ P.actLeft)).inv ≫
coequalizer.desc
((α_ N.X P.X Q.X).hom ≫
N.X ◁ coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X Y.X Q.X).hom ≫ P.X ◁ Q.actLeft) ≫
coequalizer.π (N.actRight ▷ TensorBimod.X P Q)
((α_ N.X X.X (TensorBimod.X P Q)).hom ≫ N.X ◁ TensorBimod.actLeft P Q))
⋯)
⋯)
(M.X ◁
coequalizer.desc
((PreservesCoequalizer.iso (tensorRight Q.X) (N.actRight ▷ P.X)
((α_ N.X X.X P.X).hom ≫ N.X ◁ P.actLeft)).inv ≫
coequalizer.desc
((α_ N.X P.X Q.X).hom ≫
N.X ◁ coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X Y.X Q.X).hom ≫ P.X ◁ Q.actLeft) ≫
coequalizer.π (N.actRight ▷ TensorBimod.X P Q)
((α_ N.X X.X (TensorBimod.X P Q)).hom ≫ N.X ◁ TensorBimod.actLeft P Q))
⋯)
⋯)
⋯ ⋯) =
(tensorRight Q.X).map
(coequalizer.π ((M.tensorBimod N).actRight ▷ P.X)
((α_ (M.tensorBimod N).X X.X P.X).hom ≫ (M.tensorBimod N).X ◁ P.actLeft)) ≫
((PreservesCoequalizer.iso (tensorRight Q.X) (TensorBimod.actRight M N ▷ P.X)
((α_ (TensorBimod.X M N) X.X P.X).hom ≫ TensorBimod.X M N ◁ P.actLeft)).inv ≫
coequalizer.desc
((α_ (TensorBimod.X M N) P.X Q.X).hom ≫
TensorBimod.X M N ◁ coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X Y.X Q.X).hom ≫ P.X ◁ Q.actLeft) ≫
coequalizer.π (TensorBimod.actRight M N ▷ TensorBimod.X P Q)
((α_ (TensorBimod.X M N) X.X (TensorBimod.X P Q)).hom ≫ TensorBimod.X M N ◁ TensorBimod.actLeft P Q))
⋯) ≫
coequalizer.desc
((PreservesCoequalizer.iso (tensorRight (TensorBimod.X P Q)) (M.actRight ▷ N.X)
((α_ M.X W.X N.X).hom ≫ M.X ◁ N.actLeft)).inv ≫
coequalizer.desc
((α_ M.X N.X (TensorBimod.X P Q)).hom ≫
M.X ◁
coequalizer.π (N.actRight ▷ TensorBimod.X P Q)
((α_ N.X X.X (TensorBimod.X P Q)).hom ≫ N.X ◁ TensorBimod.actLeft P Q) ≫
coequalizer.π (M.actRight ▷ TensorBimod.X N (P.tensorBimod Q))
((α_ M.X W.X (TensorBimod.X N (P.tensorBimod Q))).hom ≫
M.X ◁ TensorBimod.actLeft N (P.tensorBimod Q)))
⋯)
⋯
|
dsimp
|
case h.h
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
inst✝³ : MonoidalCategory C
A B : Mon_ C
M✝ : Bimod A B
inst✝² : HasCoequalizers C
inst✝¹ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
inst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)
V W X Y Z : Mon_ C
M : Bimod V W
N : Bimod W X
P : Bimod X Y
Q : Bimod Y Z
⊢ coequalizer.π (TensorBimod.actRight M N ▷ P.X)
((α_ (TensorBimod.X M N) X.X P.X).hom ≫ TensorBimod.X M N ◁ P.actLeft) ▷
Q.X ≫
coequalizer.desc
((PreservesCoequalizer.iso (tensorRight P.X) (M.actRight ▷ N.X)
((α_ M.X W.X N.X).hom ≫ M.X ◁ N.actLeft)).inv ≫
coequalizer.desc
((α_ M.X N.X P.X).hom ≫
M.X ◁ coequalizer.π (N.actRight ▷ P.X) ((α_ N.X X.X P.X).hom ≫ N.X ◁ P.actLeft) ≫
coequalizer.π (M.actRight ▷ TensorBimod.X N P)
((α_ M.X W.X (TensorBimod.X N P)).hom ≫ M.X ◁ TensorBimod.actLeft N P))
⋯)
⋯ ▷
Q.X ≫
((PreservesCoequalizer.iso (tensorRight Q.X) (M.actRight ▷ TensorBimod.X N P)
((α_ M.X W.X (TensorBimod.X N P)).hom ≫ M.X ◁ TensorBimod.actLeft N P)).inv ≫
coequalizer.desc
((α_ M.X (TensorBimod.X N P) Q.X).hom ≫
M.X ◁
coequalizer.π (TensorBimod.actRight N P ▷ Q.X)
((α_ (TensorBimod.X N P) Y.X Q.X).hom ≫ TensorBimod.X N P ◁ Q.actLeft) ≫
coequalizer.π (M.actRight ▷ TensorBimod.X (N.tensorBimod P) Q)
((α_ M.X W.X (TensorBimod.X (N.tensorBimod P) Q)).hom ≫
M.X ◁ TensorBimod.actLeft (N.tensorBimod P) Q))
⋯) ≫
colimMap
(parallelPairHom (M.actRight ▷ TensorBimod.X (N.tensorBimod P) Q)
((α_ M.X W.X (TensorBimod.X (N.tensorBimod P) Q)).hom ≫ M.X ◁ TensorBimod.actLeft (N.tensorBimod P) Q)
(M.actRight ▷ TensorBimod.X N (P.tensorBimod Q))
((α_ M.X W.X (TensorBimod.X N (P.tensorBimod Q))).hom ≫ M.X ◁ TensorBimod.actLeft N (P.tensorBimod Q))
((M.X ⊗ W.X) ◁
coequalizer.desc
((PreservesCoequalizer.iso (tensorRight Q.X) (N.actRight ▷ P.X)
((α_ N.X X.X P.X).hom ≫ N.X ◁ P.actLeft)).inv ≫
coequalizer.desc
((α_ N.X P.X Q.X).hom ≫
N.X ◁ coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X Y.X Q.X).hom ≫ P.X ◁ Q.actLeft) ≫
coequalizer.π (N.actRight ▷ TensorBimod.X P Q)
((α_ N.X X.X (TensorBimod.X P Q)).hom ≫ N.X ◁ TensorBimod.actLeft P Q))
⋯)
⋯)
(M.X ◁
coequalizer.desc
((PreservesCoequalizer.iso (tensorRight Q.X) (N.actRight ▷ P.X)
((α_ N.X X.X P.X).hom ≫ N.X ◁ P.actLeft)).inv ≫
coequalizer.desc
((α_ N.X P.X Q.X).hom ≫
N.X ◁ coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X Y.X Q.X).hom ≫ P.X ◁ Q.actLeft) ≫
coequalizer.π (N.actRight ▷ TensorBimod.X P Q)
((α_ N.X X.X (TensorBimod.X P Q)).hom ≫ N.X ◁ TensorBimod.actLeft P Q))
⋯)
⋯)
⋯ ⋯) =
coequalizer.π (TensorBimod.actRight M N ▷ P.X)
((α_ (TensorBimod.X M N) X.X P.X).hom ≫ TensorBimod.X M N ◁ P.actLeft) ▷
Q.X ≫
((PreservesCoequalizer.iso (tensorRight Q.X) (TensorBimod.actRight M N ▷ P.X)
((α_ (TensorBimod.X M N) X.X P.X).hom ≫ TensorBimod.X M N ◁ P.actLeft)).inv ≫
coequalizer.desc
((α_ (TensorBimod.X M N) P.X Q.X).hom ≫
TensorBimod.X M N ◁ coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X Y.X Q.X).hom ≫ P.X ◁ Q.actLeft) ≫
coequalizer.π (TensorBimod.actRight M N ▷ TensorBimod.X P Q)
((α_ (TensorBimod.X M N) X.X (TensorBimod.X P Q)).hom ≫ TensorBimod.X M N ◁ TensorBimod.actLeft P Q))
⋯) ≫
coequalizer.desc
((PreservesCoequalizer.iso (tensorRight (TensorBimod.X P Q)) (M.actRight ▷ N.X)
((α_ M.X W.X N.X).hom ≫ M.X ◁ N.actLeft)).inv ≫
coequalizer.desc
((α_ M.X N.X (TensorBimod.X P Q)).hom ≫
M.X ◁
coequalizer.π (N.actRight ▷ TensorBimod.X P Q)
((α_ N.X X.X (TensorBimod.X P Q)).hom ≫ N.X ◁ TensorBimod.actLeft P Q) ≫
coequalizer.π (M.actRight ▷ TensorBimod.X N (P.tensorBimod Q))
((α_ M.X W.X (TensorBimod.X N (P.tensorBimod Q))).hom ≫
M.X ◁ TensorBimod.actLeft N (P.tensorBimod Q)))
⋯)
⋯
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/CategoryTheory/Monoidal/Bimod.lean
|
Bimod.pentagon_bimod
|
case h.h
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
inst✝³ : MonoidalCategory C
A B : Mon_ C
M✝ : Bimod A B
inst✝² : HasCoequalizers C
inst✝¹ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
inst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)
V W X Y Z : Mon_ C
M : Bimod V W
N : Bimod W X
P : Bimod X Y
Q : Bimod Y Z
⊢ coequalizer.π (TensorBimod.actRight M N ▷ P.X)
((α_ (TensorBimod.X M N) X.X P.X).hom ≫ TensorBimod.X M N ◁ P.actLeft) ▷
Q.X ≫
coequalizer.desc
((PreservesCoequalizer.iso (tensorRight P.X) (M.actRight ▷ N.X)
((α_ M.X W.X N.X).hom ≫ M.X ◁ N.actLeft)).inv ≫
coequalizer.desc
((α_ M.X N.X P.X).hom ≫
M.X ◁ coequalizer.π (N.actRight ▷ P.X) ((α_ N.X X.X P.X).hom ≫ N.X ◁ P.actLeft) ≫
coequalizer.π (M.actRight ▷ TensorBimod.X N P)
((α_ M.X W.X (TensorBimod.X N P)).hom ≫ M.X ◁ TensorBimod.actLeft N P))
⋯)
⋯ ▷
Q.X ≫
((PreservesCoequalizer.iso (tensorRight Q.X) (M.actRight ▷ TensorBimod.X N P)
((α_ M.X W.X (TensorBimod.X N P)).hom ≫ M.X ◁ TensorBimod.actLeft N P)).inv ≫
coequalizer.desc
((α_ M.X (TensorBimod.X N P) Q.X).hom ≫
M.X ◁
coequalizer.π (TensorBimod.actRight N P ▷ Q.X)
((α_ (TensorBimod.X N P) Y.X Q.X).hom ≫ TensorBimod.X N P ◁ Q.actLeft) ≫
coequalizer.π (M.actRight ▷ TensorBimod.X (N.tensorBimod P) Q)
((α_ M.X W.X (TensorBimod.X (N.tensorBimod P) Q)).hom ≫
M.X ◁ TensorBimod.actLeft (N.tensorBimod P) Q))
⋯) ≫
colimMap
(parallelPairHom (M.actRight ▷ TensorBimod.X (N.tensorBimod P) Q)
((α_ M.X W.X (TensorBimod.X (N.tensorBimod P) Q)).hom ≫ M.X ◁ TensorBimod.actLeft (N.tensorBimod P) Q)
(M.actRight ▷ TensorBimod.X N (P.tensorBimod Q))
((α_ M.X W.X (TensorBimod.X N (P.tensorBimod Q))).hom ≫ M.X ◁ TensorBimod.actLeft N (P.tensorBimod Q))
((M.X ⊗ W.X) ◁
coequalizer.desc
((PreservesCoequalizer.iso (tensorRight Q.X) (N.actRight ▷ P.X)
((α_ N.X X.X P.X).hom ≫ N.X ◁ P.actLeft)).inv ≫
coequalizer.desc
((α_ N.X P.X Q.X).hom ≫
N.X ◁ coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X Y.X Q.X).hom ≫ P.X ◁ Q.actLeft) ≫
coequalizer.π (N.actRight ▷ TensorBimod.X P Q)
((α_ N.X X.X (TensorBimod.X P Q)).hom ≫ N.X ◁ TensorBimod.actLeft P Q))
⋯)
⋯)
(M.X ◁
coequalizer.desc
((PreservesCoequalizer.iso (tensorRight Q.X) (N.actRight ▷ P.X)
((α_ N.X X.X P.X).hom ≫ N.X ◁ P.actLeft)).inv ≫
coequalizer.desc
((α_ N.X P.X Q.X).hom ≫
N.X ◁ coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X Y.X Q.X).hom ≫ P.X ◁ Q.actLeft) ≫
coequalizer.π (N.actRight ▷ TensorBimod.X P Q)
((α_ N.X X.X (TensorBimod.X P Q)).hom ≫ N.X ◁ TensorBimod.actLeft P Q))
⋯)
⋯)
⋯ ⋯) =
coequalizer.π (TensorBimod.actRight M N ▷ P.X)
((α_ (TensorBimod.X M N) X.X P.X).hom ≫ TensorBimod.X M N ◁ P.actLeft) ▷
Q.X ≫
((PreservesCoequalizer.iso (tensorRight Q.X) (TensorBimod.actRight M N ▷ P.X)
((α_ (TensorBimod.X M N) X.X P.X).hom ≫ TensorBimod.X M N ◁ P.actLeft)).inv ≫
coequalizer.desc
((α_ (TensorBimod.X M N) P.X Q.X).hom ≫
TensorBimod.X M N ◁ coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X Y.X Q.X).hom ≫ P.X ◁ Q.actLeft) ≫
coequalizer.π (TensorBimod.actRight M N ▷ TensorBimod.X P Q)
((α_ (TensorBimod.X M N) X.X (TensorBimod.X P Q)).hom ≫ TensorBimod.X M N ◁ TensorBimod.actLeft P Q))
⋯) ≫
coequalizer.desc
((PreservesCoequalizer.iso (tensorRight (TensorBimod.X P Q)) (M.actRight ▷ N.X)
((α_ M.X W.X N.X).hom ≫ M.X ◁ N.actLeft)).inv ≫
coequalizer.desc
((α_ M.X N.X (TensorBimod.X P Q)).hom ≫
M.X ◁
coequalizer.π (N.actRight ▷ TensorBimod.X P Q)
((α_ N.X X.X (TensorBimod.X P Q)).hom ≫ N.X ◁ TensorBimod.actLeft P Q) ≫
coequalizer.π (M.actRight ▷ TensorBimod.X N (P.tensorBimod Q))
((α_ M.X W.X (TensorBimod.X N (P.tensorBimod Q))).hom ≫
M.X ◁ TensorBimod.actLeft N (P.tensorBimod Q)))
⋯)
⋯
|
slice_lhs 1 2 => rw [← <a>CategoryTheory.MonoidalCategory.comp_whiskerRight</a>, <a>CategoryTheory.Limits.coequalizer.π_desc</a>]
|
case h.h
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
inst✝³ : MonoidalCategory C
A B : Mon_ C
M✝ : Bimod A B
inst✝² : HasCoequalizers C
inst✝¹ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
inst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)
V W X Y Z : Mon_ C
M : Bimod V W
N : Bimod W X
P : Bimod X Y
Q : Bimod Y Z
⊢ ((((PreservesCoequalizer.iso (tensorRight P.X) (M.actRight ▷ N.X) ((α_ M.X W.X N.X).hom ≫ M.X ◁ N.actLeft)).inv ≫
coequalizer.desc
((α_ M.X N.X P.X).hom ≫
M.X ◁ coequalizer.π (N.actRight ▷ P.X) ((α_ N.X X.X P.X).hom ≫ N.X ◁ P.actLeft) ≫
coequalizer.π (M.actRight ▷ TensorBimod.X N P)
((α_ M.X W.X (TensorBimod.X N P)).hom ≫ M.X ◁ TensorBimod.actLeft N P))
⋯) ▷
Q.X ≫
(PreservesCoequalizer.iso (tensorRight Q.X) (M.actRight ▷ TensorBimod.X N P)
((α_ M.X W.X (TensorBimod.X N P)).hom ≫ M.X ◁ TensorBimod.actLeft N P)).inv) ≫
coequalizer.desc
((α_ M.X (TensorBimod.X N P) Q.X).hom ≫
M.X ◁
coequalizer.π (TensorBimod.actRight N P ▷ Q.X)
((α_ (TensorBimod.X N P) Y.X Q.X).hom ≫ TensorBimod.X N P ◁ Q.actLeft) ≫
coequalizer.π (M.actRight ▷ TensorBimod.X (N.tensorBimod P) Q)
((α_ M.X W.X (TensorBimod.X (N.tensorBimod P) Q)).hom ≫ M.X ◁ TensorBimod.actLeft (N.tensorBimod P) Q))
⋯) ≫
colimMap
(parallelPairHom (M.actRight ▷ TensorBimod.X (N.tensorBimod P) Q)
((α_ M.X W.X (TensorBimod.X (N.tensorBimod P) Q)).hom ≫ M.X ◁ TensorBimod.actLeft (N.tensorBimod P) Q)
(M.actRight ▷ TensorBimod.X N (P.tensorBimod Q))
((α_ M.X W.X (TensorBimod.X N (P.tensorBimod Q))).hom ≫ M.X ◁ TensorBimod.actLeft N (P.tensorBimod Q))
((M.X ⊗ W.X) ◁
coequalizer.desc
((PreservesCoequalizer.iso (tensorRight Q.X) (N.actRight ▷ P.X)
((α_ N.X X.X P.X).hom ≫ N.X ◁ P.actLeft)).inv ≫
coequalizer.desc
((α_ N.X P.X Q.X).hom ≫
N.X ◁ coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X Y.X Q.X).hom ≫ P.X ◁ Q.actLeft) ≫
coequalizer.π (N.actRight ▷ TensorBimod.X P Q)
((α_ N.X X.X (TensorBimod.X P Q)).hom ≫ N.X ◁ TensorBimod.actLeft P Q))
⋯)
⋯)
(M.X ◁
coequalizer.desc
((PreservesCoequalizer.iso (tensorRight Q.X) (N.actRight ▷ P.X)
((α_ N.X X.X P.X).hom ≫ N.X ◁ P.actLeft)).inv ≫
coequalizer.desc
((α_ N.X P.X Q.X).hom ≫
N.X ◁ coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X Y.X Q.X).hom ≫ P.X ◁ Q.actLeft) ≫
coequalizer.π (N.actRight ▷ TensorBimod.X P Q)
((α_ N.X X.X (TensorBimod.X P Q)).hom ≫ N.X ◁ TensorBimod.actLeft P Q))
⋯)
⋯)
⋯ ⋯) =
coequalizer.π (TensorBimod.actRight M N ▷ P.X)
((α_ (TensorBimod.X M N) X.X P.X).hom ≫ TensorBimod.X M N ◁ P.actLeft) ▷
Q.X ≫
((PreservesCoequalizer.iso (tensorRight Q.X) (TensorBimod.actRight M N ▷ P.X)
((α_ (TensorBimod.X M N) X.X P.X).hom ≫ TensorBimod.X M N ◁ P.actLeft)).inv ≫
coequalizer.desc
((α_ (TensorBimod.X M N) P.X Q.X).hom ≫
TensorBimod.X M N ◁ coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X Y.X Q.X).hom ≫ P.X ◁ Q.actLeft) ≫
coequalizer.π (TensorBimod.actRight M N ▷ TensorBimod.X P Q)
((α_ (TensorBimod.X M N) X.X (TensorBimod.X P Q)).hom ≫ TensorBimod.X M N ◁ TensorBimod.actLeft P Q))
⋯) ≫
coequalizer.desc
((PreservesCoequalizer.iso (tensorRight (TensorBimod.X P Q)) (M.actRight ▷ N.X)
((α_ M.X W.X N.X).hom ≫ M.X ◁ N.actLeft)).inv ≫
coequalizer.desc
((α_ M.X N.X (TensorBimod.X P Q)).hom ≫
M.X ◁
coequalizer.π (N.actRight ▷ TensorBimod.X P Q)
((α_ N.X X.X (TensorBimod.X P Q)).hom ≫ N.X ◁ TensorBimod.actLeft P Q) ≫
coequalizer.π (M.actRight ▷ TensorBimod.X N (P.tensorBimod Q))
((α_ M.X W.X (TensorBimod.X N (P.tensorBimod Q))).hom ≫
M.X ◁ TensorBimod.actLeft N (P.tensorBimod Q)))
⋯)
⋯
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/CategoryTheory/Monoidal/Bimod.lean
|
Bimod.pentagon_bimod
|
case h.h
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
inst✝³ : MonoidalCategory C
A B : Mon_ C
M✝ : Bimod A B
inst✝² : HasCoequalizers C
inst✝¹ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
inst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)
V W X Y Z : Mon_ C
M : Bimod V W
N : Bimod W X
P : Bimod X Y
Q : Bimod Y Z
⊢ ((((PreservesCoequalizer.iso (tensorRight P.X) (M.actRight ▷ N.X) ((α_ M.X W.X N.X).hom ≫ M.X ◁ N.actLeft)).inv ≫
coequalizer.desc
((α_ M.X N.X P.X).hom ≫
M.X ◁ coequalizer.π (N.actRight ▷ P.X) ((α_ N.X X.X P.X).hom ≫ N.X ◁ P.actLeft) ≫
coequalizer.π (M.actRight ▷ TensorBimod.X N P)
((α_ M.X W.X (TensorBimod.X N P)).hom ≫ M.X ◁ TensorBimod.actLeft N P))
⋯) ▷
Q.X ≫
(PreservesCoequalizer.iso (tensorRight Q.X) (M.actRight ▷ TensorBimod.X N P)
((α_ M.X W.X (TensorBimod.X N P)).hom ≫ M.X ◁ TensorBimod.actLeft N P)).inv) ≫
coequalizer.desc
((α_ M.X (TensorBimod.X N P) Q.X).hom ≫
M.X ◁
coequalizer.π (TensorBimod.actRight N P ▷ Q.X)
((α_ (TensorBimod.X N P) Y.X Q.X).hom ≫ TensorBimod.X N P ◁ Q.actLeft) ≫
coequalizer.π (M.actRight ▷ TensorBimod.X (N.tensorBimod P) Q)
((α_ M.X W.X (TensorBimod.X (N.tensorBimod P) Q)).hom ≫ M.X ◁ TensorBimod.actLeft (N.tensorBimod P) Q))
⋯) ≫
colimMap
(parallelPairHom (M.actRight ▷ TensorBimod.X (N.tensorBimod P) Q)
((α_ M.X W.X (TensorBimod.X (N.tensorBimod P) Q)).hom ≫ M.X ◁ TensorBimod.actLeft (N.tensorBimod P) Q)
(M.actRight ▷ TensorBimod.X N (P.tensorBimod Q))
((α_ M.X W.X (TensorBimod.X N (P.tensorBimod Q))).hom ≫ M.X ◁ TensorBimod.actLeft N (P.tensorBimod Q))
((M.X ⊗ W.X) ◁
coequalizer.desc
((PreservesCoequalizer.iso (tensorRight Q.X) (N.actRight ▷ P.X)
((α_ N.X X.X P.X).hom ≫ N.X ◁ P.actLeft)).inv ≫
coequalizer.desc
((α_ N.X P.X Q.X).hom ≫
N.X ◁ coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X Y.X Q.X).hom ≫ P.X ◁ Q.actLeft) ≫
coequalizer.π (N.actRight ▷ TensorBimod.X P Q)
((α_ N.X X.X (TensorBimod.X P Q)).hom ≫ N.X ◁ TensorBimod.actLeft P Q))
⋯)
⋯)
(M.X ◁
coequalizer.desc
((PreservesCoequalizer.iso (tensorRight Q.X) (N.actRight ▷ P.X)
((α_ N.X X.X P.X).hom ≫ N.X ◁ P.actLeft)).inv ≫
coequalizer.desc
((α_ N.X P.X Q.X).hom ≫
N.X ◁ coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X Y.X Q.X).hom ≫ P.X ◁ Q.actLeft) ≫
coequalizer.π (N.actRight ▷ TensorBimod.X P Q)
((α_ N.X X.X (TensorBimod.X P Q)).hom ≫ N.X ◁ TensorBimod.actLeft P Q))
⋯)
⋯)
⋯ ⋯) =
coequalizer.π (TensorBimod.actRight M N ▷ P.X)
((α_ (TensorBimod.X M N) X.X P.X).hom ≫ TensorBimod.X M N ◁ P.actLeft) ▷
Q.X ≫
((PreservesCoequalizer.iso (tensorRight Q.X) (TensorBimod.actRight M N ▷ P.X)
((α_ (TensorBimod.X M N) X.X P.X).hom ≫ TensorBimod.X M N ◁ P.actLeft)).inv ≫
coequalizer.desc
((α_ (TensorBimod.X M N) P.X Q.X).hom ≫
TensorBimod.X M N ◁ coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X Y.X Q.X).hom ≫ P.X ◁ Q.actLeft) ≫
coequalizer.π (TensorBimod.actRight M N ▷ TensorBimod.X P Q)
((α_ (TensorBimod.X M N) X.X (TensorBimod.X P Q)).hom ≫ TensorBimod.X M N ◁ TensorBimod.actLeft P Q))
⋯) ≫
coequalizer.desc
((PreservesCoequalizer.iso (tensorRight (TensorBimod.X P Q)) (M.actRight ▷ N.X)
((α_ M.X W.X N.X).hom ≫ M.X ◁ N.actLeft)).inv ≫
coequalizer.desc
((α_ M.X N.X (TensorBimod.X P Q)).hom ≫
M.X ◁
coequalizer.π (N.actRight ▷ TensorBimod.X P Q)
((α_ N.X X.X (TensorBimod.X P Q)).hom ≫ N.X ◁ TensorBimod.actLeft P Q) ≫
coequalizer.π (M.actRight ▷ TensorBimod.X N (P.tensorBimod Q))
((α_ M.X W.X (TensorBimod.X N (P.tensorBimod Q))).hom ≫
M.X ◁ TensorBimod.actLeft N (P.tensorBimod Q)))
⋯)
⋯
|
slice_rhs 1 3 => rw [<a>π_tensor_id_preserves_coequalizer_inv_desc</a>]
|
case h.h
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
inst✝³ : MonoidalCategory C
A B : Mon_ C
M✝ : Bimod A B
inst✝² : HasCoequalizers C
inst✝¹ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
inst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)
V W X Y Z : Mon_ C
M : Bimod V W
N : Bimod W X
P : Bimod X Y
Q : Bimod Y Z
⊢ ((((PreservesCoequalizer.iso (tensorRight P.X) (M.actRight ▷ N.X) ((α_ M.X W.X N.X).hom ≫ M.X ◁ N.actLeft)).inv ≫
coequalizer.desc
((α_ M.X N.X P.X).hom ≫
M.X ◁ coequalizer.π (N.actRight ▷ P.X) ((α_ N.X X.X P.X).hom ≫ N.X ◁ P.actLeft) ≫
coequalizer.π (M.actRight ▷ TensorBimod.X N P)
((α_ M.X W.X (TensorBimod.X N P)).hom ≫ M.X ◁ TensorBimod.actLeft N P))
⋯) ▷
Q.X ≫
(PreservesCoequalizer.iso (tensorRight Q.X) (M.actRight ▷ TensorBimod.X N P)
((α_ M.X W.X (TensorBimod.X N P)).hom ≫ M.X ◁ TensorBimod.actLeft N P)).inv) ≫
coequalizer.desc
((α_ M.X (TensorBimod.X N P) Q.X).hom ≫
M.X ◁
coequalizer.π (TensorBimod.actRight N P ▷ Q.X)
((α_ (TensorBimod.X N P) Y.X Q.X).hom ≫ TensorBimod.X N P ◁ Q.actLeft) ≫
coequalizer.π (M.actRight ▷ TensorBimod.X (N.tensorBimod P) Q)
((α_ M.X W.X (TensorBimod.X (N.tensorBimod P) Q)).hom ≫ M.X ◁ TensorBimod.actLeft (N.tensorBimod P) Q))
⋯) ≫
colimMap
(parallelPairHom (M.actRight ▷ TensorBimod.X (N.tensorBimod P) Q)
((α_ M.X W.X (TensorBimod.X (N.tensorBimod P) Q)).hom ≫ M.X ◁ TensorBimod.actLeft (N.tensorBimod P) Q)
(M.actRight ▷ TensorBimod.X N (P.tensorBimod Q))
((α_ M.X W.X (TensorBimod.X N (P.tensorBimod Q))).hom ≫ M.X ◁ TensorBimod.actLeft N (P.tensorBimod Q))
((M.X ⊗ W.X) ◁
coequalizer.desc
((PreservesCoequalizer.iso (tensorRight Q.X) (N.actRight ▷ P.X)
((α_ N.X X.X P.X).hom ≫ N.X ◁ P.actLeft)).inv ≫
coequalizer.desc
((α_ N.X P.X Q.X).hom ≫
N.X ◁ coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X Y.X Q.X).hom ≫ P.X ◁ Q.actLeft) ≫
coequalizer.π (N.actRight ▷ TensorBimod.X P Q)
((α_ N.X X.X (TensorBimod.X P Q)).hom ≫ N.X ◁ TensorBimod.actLeft P Q))
⋯)
⋯)
(M.X ◁
coequalizer.desc
((PreservesCoequalizer.iso (tensorRight Q.X) (N.actRight ▷ P.X)
((α_ N.X X.X P.X).hom ≫ N.X ◁ P.actLeft)).inv ≫
coequalizer.desc
((α_ N.X P.X Q.X).hom ≫
N.X ◁ coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X Y.X Q.X).hom ≫ P.X ◁ Q.actLeft) ≫
coequalizer.π (N.actRight ▷ TensorBimod.X P Q)
((α_ N.X X.X (TensorBimod.X P Q)).hom ≫ N.X ◁ TensorBimod.actLeft P Q))
⋯)
⋯)
⋯ ⋯) =
((α_ (TensorBimod.X M N) P.X Q.X).hom ≫
TensorBimod.X M N ◁ coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X Y.X Q.X).hom ≫ P.X ◁ Q.actLeft) ≫
coequalizer.π (TensorBimod.actRight M N ▷ TensorBimod.X P Q)
((α_ (TensorBimod.X M N) X.X (TensorBimod.X P Q)).hom ≫ TensorBimod.X M N ◁ TensorBimod.actLeft P Q)) ≫
coequalizer.desc
((PreservesCoequalizer.iso (tensorRight (TensorBimod.X P Q)) (M.actRight ▷ N.X)
((α_ M.X W.X N.X).hom ≫ M.X ◁ N.actLeft)).inv ≫
coequalizer.desc
((α_ M.X N.X (TensorBimod.X P Q)).hom ≫
M.X ◁
coequalizer.π (N.actRight ▷ TensorBimod.X P Q)
((α_ N.X X.X (TensorBimod.X P Q)).hom ≫ N.X ◁ TensorBimod.actLeft P Q) ≫
coequalizer.π (M.actRight ▷ TensorBimod.X N (P.tensorBimod Q))
((α_ M.X W.X (TensorBimod.X N (P.tensorBimod Q))).hom ≫
M.X ◁ TensorBimod.actLeft N (P.tensorBimod Q)))
⋯)
⋯
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/CategoryTheory/Monoidal/Bimod.lean
|
Bimod.pentagon_bimod
|
case h.h
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
inst✝³ : MonoidalCategory C
A B : Mon_ C
M✝ : Bimod A B
inst✝² : HasCoequalizers C
inst✝¹ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
inst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)
V W X Y Z : Mon_ C
M : Bimod V W
N : Bimod W X
P : Bimod X Y
Q : Bimod Y Z
⊢ ((((PreservesCoequalizer.iso (tensorRight P.X) (M.actRight ▷ N.X) ((α_ M.X W.X N.X).hom ≫ M.X ◁ N.actLeft)).inv ≫
coequalizer.desc
((α_ M.X N.X P.X).hom ≫
M.X ◁ coequalizer.π (N.actRight ▷ P.X) ((α_ N.X X.X P.X).hom ≫ N.X ◁ P.actLeft) ≫
coequalizer.π (M.actRight ▷ TensorBimod.X N P)
((α_ M.X W.X (TensorBimod.X N P)).hom ≫ M.X ◁ TensorBimod.actLeft N P))
⋯) ▷
Q.X ≫
(PreservesCoequalizer.iso (tensorRight Q.X) (M.actRight ▷ TensorBimod.X N P)
((α_ M.X W.X (TensorBimod.X N P)).hom ≫ M.X ◁ TensorBimod.actLeft N P)).inv) ≫
coequalizer.desc
((α_ M.X (TensorBimod.X N P) Q.X).hom ≫
M.X ◁
coequalizer.π (TensorBimod.actRight N P ▷ Q.X)
((α_ (TensorBimod.X N P) Y.X Q.X).hom ≫ TensorBimod.X N P ◁ Q.actLeft) ≫
coequalizer.π (M.actRight ▷ TensorBimod.X (N.tensorBimod P) Q)
((α_ M.X W.X (TensorBimod.X (N.tensorBimod P) Q)).hom ≫ M.X ◁ TensorBimod.actLeft (N.tensorBimod P) Q))
⋯) ≫
colimMap
(parallelPairHom (M.actRight ▷ TensorBimod.X (N.tensorBimod P) Q)
((α_ M.X W.X (TensorBimod.X (N.tensorBimod P) Q)).hom ≫ M.X ◁ TensorBimod.actLeft (N.tensorBimod P) Q)
(M.actRight ▷ TensorBimod.X N (P.tensorBimod Q))
((α_ M.X W.X (TensorBimod.X N (P.tensorBimod Q))).hom ≫ M.X ◁ TensorBimod.actLeft N (P.tensorBimod Q))
((M.X ⊗ W.X) ◁
coequalizer.desc
((PreservesCoequalizer.iso (tensorRight Q.X) (N.actRight ▷ P.X)
((α_ N.X X.X P.X).hom ≫ N.X ◁ P.actLeft)).inv ≫
coequalizer.desc
((α_ N.X P.X Q.X).hom ≫
N.X ◁ coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X Y.X Q.X).hom ≫ P.X ◁ Q.actLeft) ≫
coequalizer.π (N.actRight ▷ TensorBimod.X P Q)
((α_ N.X X.X (TensorBimod.X P Q)).hom ≫ N.X ◁ TensorBimod.actLeft P Q))
⋯)
⋯)
(M.X ◁
coequalizer.desc
((PreservesCoequalizer.iso (tensorRight Q.X) (N.actRight ▷ P.X)
((α_ N.X X.X P.X).hom ≫ N.X ◁ P.actLeft)).inv ≫
coequalizer.desc
((α_ N.X P.X Q.X).hom ≫
N.X ◁ coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X Y.X Q.X).hom ≫ P.X ◁ Q.actLeft) ≫
coequalizer.π (N.actRight ▷ TensorBimod.X P Q)
((α_ N.X X.X (TensorBimod.X P Q)).hom ≫ N.X ◁ TensorBimod.actLeft P Q))
⋯)
⋯)
⋯ ⋯) =
((α_ (TensorBimod.X M N) P.X Q.X).hom ≫
TensorBimod.X M N ◁ coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X Y.X Q.X).hom ≫ P.X ◁ Q.actLeft) ≫
coequalizer.π (TensorBimod.actRight M N ▷ TensorBimod.X P Q)
((α_ (TensorBimod.X M N) X.X (TensorBimod.X P Q)).hom ≫ TensorBimod.X M N ◁ TensorBimod.actLeft P Q)) ≫
coequalizer.desc
((PreservesCoequalizer.iso (tensorRight (TensorBimod.X P Q)) (M.actRight ▷ N.X)
((α_ M.X W.X N.X).hom ≫ M.X ◁ N.actLeft)).inv ≫
coequalizer.desc
((α_ M.X N.X (TensorBimod.X P Q)).hom ≫
M.X ◁
coequalizer.π (N.actRight ▷ TensorBimod.X P Q)
((α_ N.X X.X (TensorBimod.X P Q)).hom ≫ N.X ◁ TensorBimod.actLeft P Q) ≫
coequalizer.π (M.actRight ▷ TensorBimod.X N (P.tensorBimod Q))
((α_ M.X W.X (TensorBimod.X N (P.tensorBimod Q))).hom ≫
M.X ◁ TensorBimod.actLeft N (P.tensorBimod Q)))
⋯)
⋯
|
slice_rhs 3 4 => rw [<a>CategoryTheory.Limits.coequalizer.π_desc</a>]
|
case h.h
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
inst✝³ : MonoidalCategory C
A B : Mon_ C
M✝ : Bimod A B
inst✝² : HasCoequalizers C
inst✝¹ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
inst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)
V W X Y Z : Mon_ C
M : Bimod V W
N : Bimod W X
P : Bimod X Y
Q : Bimod Y Z
⊢ ((((PreservesCoequalizer.iso (tensorRight P.X) (M.actRight ▷ N.X) ((α_ M.X W.X N.X).hom ≫ M.X ◁ N.actLeft)).inv ≫
coequalizer.desc
((α_ M.X N.X P.X).hom ≫
M.X ◁ coequalizer.π (N.actRight ▷ P.X) ((α_ N.X X.X P.X).hom ≫ N.X ◁ P.actLeft) ≫
coequalizer.π (M.actRight ▷ TensorBimod.X N P)
((α_ M.X W.X (TensorBimod.X N P)).hom ≫ M.X ◁ TensorBimod.actLeft N P))
⋯) ▷
Q.X ≫
(PreservesCoequalizer.iso (tensorRight Q.X) (M.actRight ▷ TensorBimod.X N P)
((α_ M.X W.X (TensorBimod.X N P)).hom ≫ M.X ◁ TensorBimod.actLeft N P)).inv) ≫
coequalizer.desc
((α_ M.X (TensorBimod.X N P) Q.X).hom ≫
M.X ◁
coequalizer.π (TensorBimod.actRight N P ▷ Q.X)
((α_ (TensorBimod.X N P) Y.X Q.X).hom ≫ TensorBimod.X N P ◁ Q.actLeft) ≫
coequalizer.π (M.actRight ▷ TensorBimod.X (N.tensorBimod P) Q)
((α_ M.X W.X (TensorBimod.X (N.tensorBimod P) Q)).hom ≫ M.X ◁ TensorBimod.actLeft (N.tensorBimod P) Q))
⋯) ≫
colimMap
(parallelPairHom (M.actRight ▷ TensorBimod.X (N.tensorBimod P) Q)
((α_ M.X W.X (TensorBimod.X (N.tensorBimod P) Q)).hom ≫ M.X ◁ TensorBimod.actLeft (N.tensorBimod P) Q)
(M.actRight ▷ TensorBimod.X N (P.tensorBimod Q))
((α_ M.X W.X (TensorBimod.X N (P.tensorBimod Q))).hom ≫ M.X ◁ TensorBimod.actLeft N (P.tensorBimod Q))
((M.X ⊗ W.X) ◁
coequalizer.desc
((PreservesCoequalizer.iso (tensorRight Q.X) (N.actRight ▷ P.X)
((α_ N.X X.X P.X).hom ≫ N.X ◁ P.actLeft)).inv ≫
coequalizer.desc
((α_ N.X P.X Q.X).hom ≫
N.X ◁ coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X Y.X Q.X).hom ≫ P.X ◁ Q.actLeft) ≫
coequalizer.π (N.actRight ▷ TensorBimod.X P Q)
((α_ N.X X.X (TensorBimod.X P Q)).hom ≫ N.X ◁ TensorBimod.actLeft P Q))
⋯)
⋯)
(M.X ◁
coequalizer.desc
((PreservesCoequalizer.iso (tensorRight Q.X) (N.actRight ▷ P.X)
((α_ N.X X.X P.X).hom ≫ N.X ◁ P.actLeft)).inv ≫
coequalizer.desc
((α_ N.X P.X Q.X).hom ≫
N.X ◁ coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X Y.X Q.X).hom ≫ P.X ◁ Q.actLeft) ≫
coequalizer.π (N.actRight ▷ TensorBimod.X P Q)
((α_ N.X X.X (TensorBimod.X P Q)).hom ≫ N.X ◁ TensorBimod.actLeft P Q))
⋯)
⋯)
⋯ ⋯) =
(α_ (TensorBimod.X M N) P.X Q.X).hom ≫
TensorBimod.X M N ◁ coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X Y.X Q.X).hom ≫ P.X ◁ Q.actLeft) ≫
(PreservesCoequalizer.iso (tensorRight (TensorBimod.X P Q)) (M.actRight ▷ N.X)
((α_ M.X W.X N.X).hom ≫ M.X ◁ N.actLeft)).inv ≫
coequalizer.desc
((α_ M.X N.X (TensorBimod.X P Q)).hom ≫
M.X ◁
coequalizer.π (N.actRight ▷ TensorBimod.X P Q)
((α_ N.X X.X (TensorBimod.X P Q)).hom ≫ N.X ◁ TensorBimod.actLeft P Q) ≫
coequalizer.π (M.actRight ▷ TensorBimod.X N (P.tensorBimod Q))
((α_ M.X W.X (TensorBimod.X N (P.tensorBimod Q))).hom ≫
M.X ◁ TensorBimod.actLeft N (P.tensorBimod Q)))
⋯
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/CategoryTheory/Monoidal/Bimod.lean
|
Bimod.pentagon_bimod
|
case h.h
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
inst✝³ : MonoidalCategory C
A B : Mon_ C
M✝ : Bimod A B
inst✝² : HasCoequalizers C
inst✝¹ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
inst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)
V W X Y Z : Mon_ C
M : Bimod V W
N : Bimod W X
P : Bimod X Y
Q : Bimod Y Z
⊢ ((((PreservesCoequalizer.iso (tensorRight P.X) (M.actRight ▷ N.X) ((α_ M.X W.X N.X).hom ≫ M.X ◁ N.actLeft)).inv ≫
coequalizer.desc
((α_ M.X N.X P.X).hom ≫
M.X ◁ coequalizer.π (N.actRight ▷ P.X) ((α_ N.X X.X P.X).hom ≫ N.X ◁ P.actLeft) ≫
coequalizer.π (M.actRight ▷ TensorBimod.X N P)
((α_ M.X W.X (TensorBimod.X N P)).hom ≫ M.X ◁ TensorBimod.actLeft N P))
⋯) ▷
Q.X ≫
(PreservesCoequalizer.iso (tensorRight Q.X) (M.actRight ▷ TensorBimod.X N P)
((α_ M.X W.X (TensorBimod.X N P)).hom ≫ M.X ◁ TensorBimod.actLeft N P)).inv) ≫
coequalizer.desc
((α_ M.X (TensorBimod.X N P) Q.X).hom ≫
M.X ◁
coequalizer.π (TensorBimod.actRight N P ▷ Q.X)
((α_ (TensorBimod.X N P) Y.X Q.X).hom ≫ TensorBimod.X N P ◁ Q.actLeft) ≫
coequalizer.π (M.actRight ▷ TensorBimod.X (N.tensorBimod P) Q)
((α_ M.X W.X (TensorBimod.X (N.tensorBimod P) Q)).hom ≫ M.X ◁ TensorBimod.actLeft (N.tensorBimod P) Q))
⋯) ≫
colimMap
(parallelPairHom (M.actRight ▷ TensorBimod.X (N.tensorBimod P) Q)
((α_ M.X W.X (TensorBimod.X (N.tensorBimod P) Q)).hom ≫ M.X ◁ TensorBimod.actLeft (N.tensorBimod P) Q)
(M.actRight ▷ TensorBimod.X N (P.tensorBimod Q))
((α_ M.X W.X (TensorBimod.X N (P.tensorBimod Q))).hom ≫ M.X ◁ TensorBimod.actLeft N (P.tensorBimod Q))
((M.X ⊗ W.X) ◁
coequalizer.desc
((PreservesCoequalizer.iso (tensorRight Q.X) (N.actRight ▷ P.X)
((α_ N.X X.X P.X).hom ≫ N.X ◁ P.actLeft)).inv ≫
coequalizer.desc
((α_ N.X P.X Q.X).hom ≫
N.X ◁ coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X Y.X Q.X).hom ≫ P.X ◁ Q.actLeft) ≫
coequalizer.π (N.actRight ▷ TensorBimod.X P Q)
((α_ N.X X.X (TensorBimod.X P Q)).hom ≫ N.X ◁ TensorBimod.actLeft P Q))
⋯)
⋯)
(M.X ◁
coequalizer.desc
((PreservesCoequalizer.iso (tensorRight Q.X) (N.actRight ▷ P.X)
((α_ N.X X.X P.X).hom ≫ N.X ◁ P.actLeft)).inv ≫
coequalizer.desc
((α_ N.X P.X Q.X).hom ≫
N.X ◁ coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X Y.X Q.X).hom ≫ P.X ◁ Q.actLeft) ≫
coequalizer.π (N.actRight ▷ TensorBimod.X P Q)
((α_ N.X X.X (TensorBimod.X P Q)).hom ≫ N.X ◁ TensorBimod.actLeft P Q))
⋯)
⋯)
⋯ ⋯) =
(α_ (TensorBimod.X M N) P.X Q.X).hom ≫
TensorBimod.X M N ◁ coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X Y.X Q.X).hom ≫ P.X ◁ Q.actLeft) ≫
(PreservesCoequalizer.iso (tensorRight (TensorBimod.X P Q)) (M.actRight ▷ N.X)
((α_ M.X W.X N.X).hom ≫ M.X ◁ N.actLeft)).inv ≫
coequalizer.desc
((α_ M.X N.X (TensorBimod.X P Q)).hom ≫
M.X ◁
coequalizer.π (N.actRight ▷ TensorBimod.X P Q)
((α_ N.X X.X (TensorBimod.X P Q)).hom ≫ N.X ◁ TensorBimod.actLeft P Q) ≫
coequalizer.π (M.actRight ▷ TensorBimod.X N (P.tensorBimod Q))
((α_ M.X W.X (TensorBimod.X N (P.tensorBimod Q))).hom ≫
M.X ◁ TensorBimod.actLeft N (P.tensorBimod Q)))
⋯
|
refine (<a>CategoryTheory.cancel_epi</a> ((<a>CategoryTheory.MonoidalCategory.tensorRight</a> _ ⋙ <a>CategoryTheory.MonoidalCategory.tensorRight</a> _).<a>Prefunctor.map</a> (<a>CategoryTheory.Limits.coequalizer.π</a> _ _))).1 ?_
|
case h.h
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
inst✝³ : MonoidalCategory C
A B : Mon_ C
M✝ : Bimod A B
inst✝² : HasCoequalizers C
inst✝¹ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
inst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)
V W X Y Z : Mon_ C
M : Bimod V W
N : Bimod W X
P : Bimod X Y
Q : Bimod Y Z
⊢ (tensorRight P.X ⋙ tensorRight Q.X).map (coequalizer.π (M.actRight ▷ N.X) ((α_ M.X W.X N.X).hom ≫ M.X ◁ N.actLeft)) ≫
((((PreservesCoequalizer.iso (tensorRight P.X) (M.actRight ▷ N.X) ((α_ M.X W.X N.X).hom ≫ M.X ◁ N.actLeft)).inv ≫
coequalizer.desc
((α_ M.X N.X P.X).hom ≫
M.X ◁ coequalizer.π (N.actRight ▷ P.X) ((α_ N.X X.X P.X).hom ≫ N.X ◁ P.actLeft) ≫
coequalizer.π (M.actRight ▷ TensorBimod.X N P)
((α_ M.X W.X (TensorBimod.X N P)).hom ≫ M.X ◁ TensorBimod.actLeft N P))
⋯) ▷
Q.X ≫
(PreservesCoequalizer.iso (tensorRight Q.X) (M.actRight ▷ TensorBimod.X N P)
((α_ M.X W.X (TensorBimod.X N P)).hom ≫ M.X ◁ TensorBimod.actLeft N P)).inv) ≫
coequalizer.desc
((α_ M.X (TensorBimod.X N P) Q.X).hom ≫
M.X ◁
coequalizer.π (TensorBimod.actRight N P ▷ Q.X)
((α_ (TensorBimod.X N P) Y.X Q.X).hom ≫ TensorBimod.X N P ◁ Q.actLeft) ≫
coequalizer.π (M.actRight ▷ TensorBimod.X (N.tensorBimod P) Q)
((α_ M.X W.X (TensorBimod.X (N.tensorBimod P) Q)).hom ≫
M.X ◁ TensorBimod.actLeft (N.tensorBimod P) Q))
⋯) ≫
colimMap
(parallelPairHom (M.actRight ▷ TensorBimod.X (N.tensorBimod P) Q)
((α_ M.X W.X (TensorBimod.X (N.tensorBimod P) Q)).hom ≫ M.X ◁ TensorBimod.actLeft (N.tensorBimod P) Q)
(M.actRight ▷ TensorBimod.X N (P.tensorBimod Q))
((α_ M.X W.X (TensorBimod.X N (P.tensorBimod Q))).hom ≫ M.X ◁ TensorBimod.actLeft N (P.tensorBimod Q))
((M.X ⊗ W.X) ◁
coequalizer.desc
((PreservesCoequalizer.iso (tensorRight Q.X) (N.actRight ▷ P.X)
((α_ N.X X.X P.X).hom ≫ N.X ◁ P.actLeft)).inv ≫
coequalizer.desc
((α_ N.X P.X Q.X).hom ≫
N.X ◁ coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X Y.X Q.X).hom ≫ P.X ◁ Q.actLeft) ≫
coequalizer.π (N.actRight ▷ TensorBimod.X P Q)
((α_ N.X X.X (TensorBimod.X P Q)).hom ≫ N.X ◁ TensorBimod.actLeft P Q))
⋯)
⋯)
(M.X ◁
coequalizer.desc
((PreservesCoequalizer.iso (tensorRight Q.X) (N.actRight ▷ P.X)
((α_ N.X X.X P.X).hom ≫ N.X ◁ P.actLeft)).inv ≫
coequalizer.desc
((α_ N.X P.X Q.X).hom ≫
N.X ◁ coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X Y.X Q.X).hom ≫ P.X ◁ Q.actLeft) ≫
coequalizer.π (N.actRight ▷ TensorBimod.X P Q)
((α_ N.X X.X (TensorBimod.X P Q)).hom ≫ N.X ◁ TensorBimod.actLeft P Q))
⋯)
⋯)
⋯ ⋯) =
(tensorRight P.X ⋙ tensorRight Q.X).map
(coequalizer.π (M.actRight ▷ N.X) ((α_ M.X W.X N.X).hom ≫ M.X ◁ N.actLeft)) ≫
(α_ (TensorBimod.X M N) P.X Q.X).hom ≫
TensorBimod.X M N ◁ coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X Y.X Q.X).hom ≫ P.X ◁ Q.actLeft) ≫
(PreservesCoequalizer.iso (tensorRight (TensorBimod.X P Q)) (M.actRight ▷ N.X)
((α_ M.X W.X N.X).hom ≫ M.X ◁ N.actLeft)).inv ≫
coequalizer.desc
((α_ M.X N.X (TensorBimod.X P Q)).hom ≫
M.X ◁
coequalizer.π (N.actRight ▷ TensorBimod.X P Q)
((α_ N.X X.X (TensorBimod.X P Q)).hom ≫ N.X ◁ TensorBimod.actLeft P Q) ≫
coequalizer.π (M.actRight ▷ TensorBimod.X N (P.tensorBimod Q))
((α_ M.X W.X (TensorBimod.X N (P.tensorBimod Q))).hom ≫
M.X ◁ TensorBimod.actLeft N (P.tensorBimod Q)))
⋯
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/CategoryTheory/Monoidal/Bimod.lean
|
Bimod.pentagon_bimod
|
case h.h
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
inst✝³ : MonoidalCategory C
A B : Mon_ C
M✝ : Bimod A B
inst✝² : HasCoequalizers C
inst✝¹ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
inst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)
V W X Y Z : Mon_ C
M : Bimod V W
N : Bimod W X
P : Bimod X Y
Q : Bimod Y Z
⊢ (tensorRight P.X ⋙ tensorRight Q.X).map (coequalizer.π (M.actRight ▷ N.X) ((α_ M.X W.X N.X).hom ≫ M.X ◁ N.actLeft)) ≫
((((PreservesCoequalizer.iso (tensorRight P.X) (M.actRight ▷ N.X) ((α_ M.X W.X N.X).hom ≫ M.X ◁ N.actLeft)).inv ≫
coequalizer.desc
((α_ M.X N.X P.X).hom ≫
M.X ◁ coequalizer.π (N.actRight ▷ P.X) ((α_ N.X X.X P.X).hom ≫ N.X ◁ P.actLeft) ≫
coequalizer.π (M.actRight ▷ TensorBimod.X N P)
((α_ M.X W.X (TensorBimod.X N P)).hom ≫ M.X ◁ TensorBimod.actLeft N P))
⋯) ▷
Q.X ≫
(PreservesCoequalizer.iso (tensorRight Q.X) (M.actRight ▷ TensorBimod.X N P)
((α_ M.X W.X (TensorBimod.X N P)).hom ≫ M.X ◁ TensorBimod.actLeft N P)).inv) ≫
coequalizer.desc
((α_ M.X (TensorBimod.X N P) Q.X).hom ≫
M.X ◁
coequalizer.π (TensorBimod.actRight N P ▷ Q.X)
((α_ (TensorBimod.X N P) Y.X Q.X).hom ≫ TensorBimod.X N P ◁ Q.actLeft) ≫
coequalizer.π (M.actRight ▷ TensorBimod.X (N.tensorBimod P) Q)
((α_ M.X W.X (TensorBimod.X (N.tensorBimod P) Q)).hom ≫
M.X ◁ TensorBimod.actLeft (N.tensorBimod P) Q))
⋯) ≫
colimMap
(parallelPairHom (M.actRight ▷ TensorBimod.X (N.tensorBimod P) Q)
((α_ M.X W.X (TensorBimod.X (N.tensorBimod P) Q)).hom ≫ M.X ◁ TensorBimod.actLeft (N.tensorBimod P) Q)
(M.actRight ▷ TensorBimod.X N (P.tensorBimod Q))
((α_ M.X W.X (TensorBimod.X N (P.tensorBimod Q))).hom ≫ M.X ◁ TensorBimod.actLeft N (P.tensorBimod Q))
((M.X ⊗ W.X) ◁
coequalizer.desc
((PreservesCoequalizer.iso (tensorRight Q.X) (N.actRight ▷ P.X)
((α_ N.X X.X P.X).hom ≫ N.X ◁ P.actLeft)).inv ≫
coequalizer.desc
((α_ N.X P.X Q.X).hom ≫
N.X ◁ coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X Y.X Q.X).hom ≫ P.X ◁ Q.actLeft) ≫
coequalizer.π (N.actRight ▷ TensorBimod.X P Q)
((α_ N.X X.X (TensorBimod.X P Q)).hom ≫ N.X ◁ TensorBimod.actLeft P Q))
⋯)
⋯)
(M.X ◁
coequalizer.desc
((PreservesCoequalizer.iso (tensorRight Q.X) (N.actRight ▷ P.X)
((α_ N.X X.X P.X).hom ≫ N.X ◁ P.actLeft)).inv ≫
coequalizer.desc
((α_ N.X P.X Q.X).hom ≫
N.X ◁ coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X Y.X Q.X).hom ≫ P.X ◁ Q.actLeft) ≫
coequalizer.π (N.actRight ▷ TensorBimod.X P Q)
((α_ N.X X.X (TensorBimod.X P Q)).hom ≫ N.X ◁ TensorBimod.actLeft P Q))
⋯)
⋯)
⋯ ⋯) =
(tensorRight P.X ⋙ tensorRight Q.X).map
(coequalizer.π (M.actRight ▷ N.X) ((α_ M.X W.X N.X).hom ≫ M.X ◁ N.actLeft)) ≫
(α_ (TensorBimod.X M N) P.X Q.X).hom ≫
TensorBimod.X M N ◁ coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X Y.X Q.X).hom ≫ P.X ◁ Q.actLeft) ≫
(PreservesCoequalizer.iso (tensorRight (TensorBimod.X P Q)) (M.actRight ▷ N.X)
((α_ M.X W.X N.X).hom ≫ M.X ◁ N.actLeft)).inv ≫
coequalizer.desc
((α_ M.X N.X (TensorBimod.X P Q)).hom ≫
M.X ◁
coequalizer.π (N.actRight ▷ TensorBimod.X P Q)
((α_ N.X X.X (TensorBimod.X P Q)).hom ≫ N.X ◁ TensorBimod.actLeft P Q) ≫
coequalizer.π (M.actRight ▷ TensorBimod.X N (P.tensorBimod Q))
((α_ M.X W.X (TensorBimod.X N (P.tensorBimod Q))).hom ≫
M.X ◁ TensorBimod.actLeft N (P.tensorBimod Q)))
⋯
|
dsimp
|
case h.h
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
inst✝³ : MonoidalCategory C
A B : Mon_ C
M✝ : Bimod A B
inst✝² : HasCoequalizers C
inst✝¹ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
inst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)
V W X Y Z : Mon_ C
M : Bimod V W
N : Bimod W X
P : Bimod X Y
Q : Bimod Y Z
⊢ coequalizer.π (M.actRight ▷ N.X) ((α_ M.X W.X N.X).hom ≫ M.X ◁ N.actLeft) ▷ P.X ▷ Q.X ≫
((((PreservesCoequalizer.iso (tensorRight P.X) (M.actRight ▷ N.X) ((α_ M.X W.X N.X).hom ≫ M.X ◁ N.actLeft)).inv ≫
coequalizer.desc
((α_ M.X N.X P.X).hom ≫
M.X ◁ coequalizer.π (N.actRight ▷ P.X) ((α_ N.X X.X P.X).hom ≫ N.X ◁ P.actLeft) ≫
coequalizer.π (M.actRight ▷ TensorBimod.X N P)
((α_ M.X W.X (TensorBimod.X N P)).hom ≫ M.X ◁ TensorBimod.actLeft N P))
⋯) ▷
Q.X ≫
(PreservesCoequalizer.iso (tensorRight Q.X) (M.actRight ▷ TensorBimod.X N P)
((α_ M.X W.X (TensorBimod.X N P)).hom ≫ M.X ◁ TensorBimod.actLeft N P)).inv) ≫
coequalizer.desc
((α_ M.X (TensorBimod.X N P) Q.X).hom ≫
M.X ◁
coequalizer.π (TensorBimod.actRight N P ▷ Q.X)
((α_ (TensorBimod.X N P) Y.X Q.X).hom ≫ TensorBimod.X N P ◁ Q.actLeft) ≫
coequalizer.π (M.actRight ▷ TensorBimod.X (N.tensorBimod P) Q)
((α_ M.X W.X (TensorBimod.X (N.tensorBimod P) Q)).hom ≫
M.X ◁ TensorBimod.actLeft (N.tensorBimod P) Q))
⋯) ≫
colimMap
(parallelPairHom (M.actRight ▷ TensorBimod.X (N.tensorBimod P) Q)
((α_ M.X W.X (TensorBimod.X (N.tensorBimod P) Q)).hom ≫ M.X ◁ TensorBimod.actLeft (N.tensorBimod P) Q)
(M.actRight ▷ TensorBimod.X N (P.tensorBimod Q))
((α_ M.X W.X (TensorBimod.X N (P.tensorBimod Q))).hom ≫ M.X ◁ TensorBimod.actLeft N (P.tensorBimod Q))
((M.X ⊗ W.X) ◁
coequalizer.desc
((PreservesCoequalizer.iso (tensorRight Q.X) (N.actRight ▷ P.X)
((α_ N.X X.X P.X).hom ≫ N.X ◁ P.actLeft)).inv ≫
coequalizer.desc
((α_ N.X P.X Q.X).hom ≫
N.X ◁ coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X Y.X Q.X).hom ≫ P.X ◁ Q.actLeft) ≫
coequalizer.π (N.actRight ▷ TensorBimod.X P Q)
((α_ N.X X.X (TensorBimod.X P Q)).hom ≫ N.X ◁ TensorBimod.actLeft P Q))
⋯)
⋯)
(M.X ◁
coequalizer.desc
((PreservesCoequalizer.iso (tensorRight Q.X) (N.actRight ▷ P.X)
((α_ N.X X.X P.X).hom ≫ N.X ◁ P.actLeft)).inv ≫
coequalizer.desc
((α_ N.X P.X Q.X).hom ≫
N.X ◁ coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X Y.X Q.X).hom ≫ P.X ◁ Q.actLeft) ≫
coequalizer.π (N.actRight ▷ TensorBimod.X P Q)
((α_ N.X X.X (TensorBimod.X P Q)).hom ≫ N.X ◁ TensorBimod.actLeft P Q))
⋯)
⋯)
⋯ ⋯) =
coequalizer.π (M.actRight ▷ N.X) ((α_ M.X W.X N.X).hom ≫ M.X ◁ N.actLeft) ▷ P.X ▷ Q.X ≫
(α_ (TensorBimod.X M N) P.X Q.X).hom ≫
TensorBimod.X M N ◁ coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X Y.X Q.X).hom ≫ P.X ◁ Q.actLeft) ≫
(PreservesCoequalizer.iso (tensorRight (TensorBimod.X P Q)) (M.actRight ▷ N.X)
((α_ M.X W.X N.X).hom ≫ M.X ◁ N.actLeft)).inv ≫
coequalizer.desc
((α_ M.X N.X (TensorBimod.X P Q)).hom ≫
M.X ◁
coequalizer.π (N.actRight ▷ TensorBimod.X P Q)
((α_ N.X X.X (TensorBimod.X P Q)).hom ≫ N.X ◁ TensorBimod.actLeft P Q) ≫
coequalizer.π (M.actRight ▷ TensorBimod.X N (P.tensorBimod Q))
((α_ M.X W.X (TensorBimod.X N (P.tensorBimod Q))).hom ≫
M.X ◁ TensorBimod.actLeft N (P.tensorBimod Q)))
⋯
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/CategoryTheory/Monoidal/Bimod.lean
|
Bimod.pentagon_bimod
|
case h.h
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
inst✝³ : MonoidalCategory C
A B : Mon_ C
M✝ : Bimod A B
inst✝² : HasCoequalizers C
inst✝¹ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
inst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)
V W X Y Z : Mon_ C
M : Bimod V W
N : Bimod W X
P : Bimod X Y
Q : Bimod Y Z
⊢ coequalizer.π (M.actRight ▷ N.X) ((α_ M.X W.X N.X).hom ≫ M.X ◁ N.actLeft) ▷ P.X ▷ Q.X ≫
((((PreservesCoequalizer.iso (tensorRight P.X) (M.actRight ▷ N.X) ((α_ M.X W.X N.X).hom ≫ M.X ◁ N.actLeft)).inv ≫
coequalizer.desc
((α_ M.X N.X P.X).hom ≫
M.X ◁ coequalizer.π (N.actRight ▷ P.X) ((α_ N.X X.X P.X).hom ≫ N.X ◁ P.actLeft) ≫
coequalizer.π (M.actRight ▷ TensorBimod.X N P)
((α_ M.X W.X (TensorBimod.X N P)).hom ≫ M.X ◁ TensorBimod.actLeft N P))
⋯) ▷
Q.X ≫
(PreservesCoequalizer.iso (tensorRight Q.X) (M.actRight ▷ TensorBimod.X N P)
((α_ M.X W.X (TensorBimod.X N P)).hom ≫ M.X ◁ TensorBimod.actLeft N P)).inv) ≫
coequalizer.desc
((α_ M.X (TensorBimod.X N P) Q.X).hom ≫
M.X ◁
coequalizer.π (TensorBimod.actRight N P ▷ Q.X)
((α_ (TensorBimod.X N P) Y.X Q.X).hom ≫ TensorBimod.X N P ◁ Q.actLeft) ≫
coequalizer.π (M.actRight ▷ TensorBimod.X (N.tensorBimod P) Q)
((α_ M.X W.X (TensorBimod.X (N.tensorBimod P) Q)).hom ≫
M.X ◁ TensorBimod.actLeft (N.tensorBimod P) Q))
⋯) ≫
colimMap
(parallelPairHom (M.actRight ▷ TensorBimod.X (N.tensorBimod P) Q)
((α_ M.X W.X (TensorBimod.X (N.tensorBimod P) Q)).hom ≫ M.X ◁ TensorBimod.actLeft (N.tensorBimod P) Q)
(M.actRight ▷ TensorBimod.X N (P.tensorBimod Q))
((α_ M.X W.X (TensorBimod.X N (P.tensorBimod Q))).hom ≫ M.X ◁ TensorBimod.actLeft N (P.tensorBimod Q))
((M.X ⊗ W.X) ◁
coequalizer.desc
((PreservesCoequalizer.iso (tensorRight Q.X) (N.actRight ▷ P.X)
((α_ N.X X.X P.X).hom ≫ N.X ◁ P.actLeft)).inv ≫
coequalizer.desc
((α_ N.X P.X Q.X).hom ≫
N.X ◁ coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X Y.X Q.X).hom ≫ P.X ◁ Q.actLeft) ≫
coequalizer.π (N.actRight ▷ TensorBimod.X P Q)
((α_ N.X X.X (TensorBimod.X P Q)).hom ≫ N.X ◁ TensorBimod.actLeft P Q))
⋯)
⋯)
(M.X ◁
coequalizer.desc
((PreservesCoequalizer.iso (tensorRight Q.X) (N.actRight ▷ P.X)
((α_ N.X X.X P.X).hom ≫ N.X ◁ P.actLeft)).inv ≫
coequalizer.desc
((α_ N.X P.X Q.X).hom ≫
N.X ◁ coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X Y.X Q.X).hom ≫ P.X ◁ Q.actLeft) ≫
coequalizer.π (N.actRight ▷ TensorBimod.X P Q)
((α_ N.X X.X (TensorBimod.X P Q)).hom ≫ N.X ◁ TensorBimod.actLeft P Q))
⋯)
⋯)
⋯ ⋯) =
coequalizer.π (M.actRight ▷ N.X) ((α_ M.X W.X N.X).hom ≫ M.X ◁ N.actLeft) ▷ P.X ▷ Q.X ≫
(α_ (TensorBimod.X M N) P.X Q.X).hom ≫
TensorBimod.X M N ◁ coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X Y.X Q.X).hom ≫ P.X ◁ Q.actLeft) ≫
(PreservesCoequalizer.iso (tensorRight (TensorBimod.X P Q)) (M.actRight ▷ N.X)
((α_ M.X W.X N.X).hom ≫ M.X ◁ N.actLeft)).inv ≫
coequalizer.desc
((α_ M.X N.X (TensorBimod.X P Q)).hom ≫
M.X ◁
coequalizer.π (N.actRight ▷ TensorBimod.X P Q)
((α_ N.X X.X (TensorBimod.X P Q)).hom ≫ N.X ◁ TensorBimod.actLeft P Q) ≫
coequalizer.π (M.actRight ▷ TensorBimod.X N (P.tensorBimod Q))
((α_ M.X W.X (TensorBimod.X N (P.tensorBimod Q))).hom ≫
M.X ◁ TensorBimod.actLeft N (P.tensorBimod Q)))
⋯
|
slice_lhs 1 2 => rw [← <a>CategoryTheory.MonoidalCategory.comp_whiskerRight</a>, <a>π_tensor_id_preserves_coequalizer_inv_desc</a>, <a>CategoryTheory.MonoidalCategory.comp_whiskerRight</a>, <a>CategoryTheory.MonoidalCategory.comp_whiskerRight</a>]
|
case h.h
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
inst✝³ : MonoidalCategory C
A B : Mon_ C
M✝ : Bimod A B
inst✝² : HasCoequalizers C
inst✝¹ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
inst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)
V W X Y Z : Mon_ C
M : Bimod V W
N : Bimod W X
P : Bimod X Y
Q : Bimod Y Z
⊢ ((((α_ M.X N.X P.X).hom ▷ Q.X ≫
(M.X ◁ coequalizer.π (N.actRight ▷ P.X) ((α_ N.X X.X P.X).hom ≫ N.X ◁ P.actLeft)) ▷ Q.X ≫
coequalizer.π (M.actRight ▷ TensorBimod.X N P)
((α_ M.X W.X (TensorBimod.X N P)).hom ≫ M.X ◁ TensorBimod.actLeft N P) ▷
Q.X) ≫
(PreservesCoequalizer.iso (tensorRight Q.X) (M.actRight ▷ TensorBimod.X N P)
((α_ M.X W.X (TensorBimod.X N P)).hom ≫ M.X ◁ TensorBimod.actLeft N P)).inv) ≫
coequalizer.desc
((α_ M.X (TensorBimod.X N P) Q.X).hom ≫
M.X ◁
coequalizer.π (TensorBimod.actRight N P ▷ Q.X)
((α_ (TensorBimod.X N P) Y.X Q.X).hom ≫ TensorBimod.X N P ◁ Q.actLeft) ≫
coequalizer.π (M.actRight ▷ TensorBimod.X (N.tensorBimod P) Q)
((α_ M.X W.X (TensorBimod.X (N.tensorBimod P) Q)).hom ≫ M.X ◁ TensorBimod.actLeft (N.tensorBimod P) Q))
⋯) ≫
colimMap
(parallelPairHom (M.actRight ▷ TensorBimod.X (N.tensorBimod P) Q)
((α_ M.X W.X (TensorBimod.X (N.tensorBimod P) Q)).hom ≫ M.X ◁ TensorBimod.actLeft (N.tensorBimod P) Q)
(M.actRight ▷ TensorBimod.X N (P.tensorBimod Q))
((α_ M.X W.X (TensorBimod.X N (P.tensorBimod Q))).hom ≫ M.X ◁ TensorBimod.actLeft N (P.tensorBimod Q))
((M.X ⊗ W.X) ◁
coequalizer.desc
((PreservesCoequalizer.iso (tensorRight Q.X) (N.actRight ▷ P.X)
((α_ N.X X.X P.X).hom ≫ N.X ◁ P.actLeft)).inv ≫
coequalizer.desc
((α_ N.X P.X Q.X).hom ≫
N.X ◁ coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X Y.X Q.X).hom ≫ P.X ◁ Q.actLeft) ≫
coequalizer.π (N.actRight ▷ TensorBimod.X P Q)
((α_ N.X X.X (TensorBimod.X P Q)).hom ≫ N.X ◁ TensorBimod.actLeft P Q))
⋯)
⋯)
(M.X ◁
coequalizer.desc
((PreservesCoequalizer.iso (tensorRight Q.X) (N.actRight ▷ P.X)
((α_ N.X X.X P.X).hom ≫ N.X ◁ P.actLeft)).inv ≫
coequalizer.desc
((α_ N.X P.X Q.X).hom ≫
N.X ◁ coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X Y.X Q.X).hom ≫ P.X ◁ Q.actLeft) ≫
coequalizer.π (N.actRight ▷ TensorBimod.X P Q)
((α_ N.X X.X (TensorBimod.X P Q)).hom ≫ N.X ◁ TensorBimod.actLeft P Q))
⋯)
⋯)
⋯ ⋯) =
coequalizer.π (M.actRight ▷ N.X) ((α_ M.X W.X N.X).hom ≫ M.X ◁ N.actLeft) ▷ P.X ▷ Q.X ≫
(α_ (TensorBimod.X M N) P.X Q.X).hom ≫
TensorBimod.X M N ◁ coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X Y.X Q.X).hom ≫ P.X ◁ Q.actLeft) ≫
(PreservesCoequalizer.iso (tensorRight (TensorBimod.X P Q)) (M.actRight ▷ N.X)
((α_ M.X W.X N.X).hom ≫ M.X ◁ N.actLeft)).inv ≫
coequalizer.desc
((α_ M.X N.X (TensorBimod.X P Q)).hom ≫
M.X ◁
coequalizer.π (N.actRight ▷ TensorBimod.X P Q)
((α_ N.X X.X (TensorBimod.X P Q)).hom ≫ N.X ◁ TensorBimod.actLeft P Q) ≫
coequalizer.π (M.actRight ▷ TensorBimod.X N (P.tensorBimod Q))
((α_ M.X W.X (TensorBimod.X N (P.tensorBimod Q))).hom ≫
M.X ◁ TensorBimod.actLeft N (P.tensorBimod Q)))
⋯
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/CategoryTheory/Monoidal/Bimod.lean
|
Bimod.pentagon_bimod
|
case h.h
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
inst✝³ : MonoidalCategory C
A B : Mon_ C
M✝ : Bimod A B
inst✝² : HasCoequalizers C
inst✝¹ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
inst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)
V W X Y Z : Mon_ C
M : Bimod V W
N : Bimod W X
P : Bimod X Y
Q : Bimod Y Z
⊢ ((((α_ M.X N.X P.X).hom ▷ Q.X ≫
(M.X ◁ coequalizer.π (N.actRight ▷ P.X) ((α_ N.X X.X P.X).hom ≫ N.X ◁ P.actLeft)) ▷ Q.X ≫
coequalizer.π (M.actRight ▷ TensorBimod.X N P)
((α_ M.X W.X (TensorBimod.X N P)).hom ≫ M.X ◁ TensorBimod.actLeft N P) ▷
Q.X) ≫
(PreservesCoequalizer.iso (tensorRight Q.X) (M.actRight ▷ TensorBimod.X N P)
((α_ M.X W.X (TensorBimod.X N P)).hom ≫ M.X ◁ TensorBimod.actLeft N P)).inv) ≫
coequalizer.desc
((α_ M.X (TensorBimod.X N P) Q.X).hom ≫
M.X ◁
coequalizer.π (TensorBimod.actRight N P ▷ Q.X)
((α_ (TensorBimod.X N P) Y.X Q.X).hom ≫ TensorBimod.X N P ◁ Q.actLeft) ≫
coequalizer.π (M.actRight ▷ TensorBimod.X (N.tensorBimod P) Q)
((α_ M.X W.X (TensorBimod.X (N.tensorBimod P) Q)).hom ≫ M.X ◁ TensorBimod.actLeft (N.tensorBimod P) Q))
⋯) ≫
colimMap
(parallelPairHom (M.actRight ▷ TensorBimod.X (N.tensorBimod P) Q)
((α_ M.X W.X (TensorBimod.X (N.tensorBimod P) Q)).hom ≫ M.X ◁ TensorBimod.actLeft (N.tensorBimod P) Q)
(M.actRight ▷ TensorBimod.X N (P.tensorBimod Q))
((α_ M.X W.X (TensorBimod.X N (P.tensorBimod Q))).hom ≫ M.X ◁ TensorBimod.actLeft N (P.tensorBimod Q))
((M.X ⊗ W.X) ◁
coequalizer.desc
((PreservesCoequalizer.iso (tensorRight Q.X) (N.actRight ▷ P.X)
((α_ N.X X.X P.X).hom ≫ N.X ◁ P.actLeft)).inv ≫
coequalizer.desc
((α_ N.X P.X Q.X).hom ≫
N.X ◁ coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X Y.X Q.X).hom ≫ P.X ◁ Q.actLeft) ≫
coequalizer.π (N.actRight ▷ TensorBimod.X P Q)
((α_ N.X X.X (TensorBimod.X P Q)).hom ≫ N.X ◁ TensorBimod.actLeft P Q))
⋯)
⋯)
(M.X ◁
coequalizer.desc
((PreservesCoequalizer.iso (tensorRight Q.X) (N.actRight ▷ P.X)
((α_ N.X X.X P.X).hom ≫ N.X ◁ P.actLeft)).inv ≫
coequalizer.desc
((α_ N.X P.X Q.X).hom ≫
N.X ◁ coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X Y.X Q.X).hom ≫ P.X ◁ Q.actLeft) ≫
coequalizer.π (N.actRight ▷ TensorBimod.X P Q)
((α_ N.X X.X (TensorBimod.X P Q)).hom ≫ N.X ◁ TensorBimod.actLeft P Q))
⋯)
⋯)
⋯ ⋯) =
coequalizer.π (M.actRight ▷ N.X) ((α_ M.X W.X N.X).hom ≫ M.X ◁ N.actLeft) ▷ P.X ▷ Q.X ≫
(α_ (TensorBimod.X M N) P.X Q.X).hom ≫
TensorBimod.X M N ◁ coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X Y.X Q.X).hom ≫ P.X ◁ Q.actLeft) ≫
(PreservesCoequalizer.iso (tensorRight (TensorBimod.X P Q)) (M.actRight ▷ N.X)
((α_ M.X W.X N.X).hom ≫ M.X ◁ N.actLeft)).inv ≫
coequalizer.desc
((α_ M.X N.X (TensorBimod.X P Q)).hom ≫
M.X ◁
coequalizer.π (N.actRight ▷ TensorBimod.X P Q)
((α_ N.X X.X (TensorBimod.X P Q)).hom ≫ N.X ◁ TensorBimod.actLeft P Q) ≫
coequalizer.π (M.actRight ▷ TensorBimod.X N (P.tensorBimod Q))
((α_ M.X W.X (TensorBimod.X N (P.tensorBimod Q))).hom ≫
M.X ◁ TensorBimod.actLeft N (P.tensorBimod Q)))
⋯
|
slice_lhs 3 5 => rw [<a>π_tensor_id_preserves_coequalizer_inv_desc</a>]
|
case h.h
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
inst✝³ : MonoidalCategory C
A B : Mon_ C
M✝ : Bimod A B
inst✝² : HasCoequalizers C
inst✝¹ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
inst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)
V W X Y Z : Mon_ C
M : Bimod V W
N : Bimod W X
P : Bimod X Y
Q : Bimod Y Z
⊢ (α_ M.X N.X P.X).hom ▷ Q.X ≫
(M.X ◁ coequalizer.π (N.actRight ▷ P.X) ((α_ N.X X.X P.X).hom ≫ N.X ◁ P.actLeft)) ▷ Q.X ≫
((α_ M.X (TensorBimod.X N P) Q.X).hom ≫
M.X ◁
coequalizer.π (TensorBimod.actRight N P ▷ Q.X)
((α_ (TensorBimod.X N P) Y.X Q.X).hom ≫ TensorBimod.X N P ◁ Q.actLeft) ≫
coequalizer.π (M.actRight ▷ TensorBimod.X (N.tensorBimod P) Q)
((α_ M.X W.X (TensorBimod.X (N.tensorBimod P) Q)).hom ≫
M.X ◁ TensorBimod.actLeft (N.tensorBimod P) Q)) ≫
colimMap
(parallelPairHom (M.actRight ▷ TensorBimod.X (N.tensorBimod P) Q)
((α_ M.X W.X (TensorBimod.X (N.tensorBimod P) Q)).hom ≫ M.X ◁ TensorBimod.actLeft (N.tensorBimod P) Q)
(M.actRight ▷ TensorBimod.X N (P.tensorBimod Q))
((α_ M.X W.X (TensorBimod.X N (P.tensorBimod Q))).hom ≫ M.X ◁ TensorBimod.actLeft N (P.tensorBimod Q))
((M.X ⊗ W.X) ◁
coequalizer.desc
((PreservesCoequalizer.iso (tensorRight Q.X) (N.actRight ▷ P.X)
((α_ N.X X.X P.X).hom ≫ N.X ◁ P.actLeft)).inv ≫
coequalizer.desc
((α_ N.X P.X Q.X).hom ≫
N.X ◁ coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X Y.X Q.X).hom ≫ P.X ◁ Q.actLeft) ≫
coequalizer.π (N.actRight ▷ TensorBimod.X P Q)
((α_ N.X X.X (TensorBimod.X P Q)).hom ≫ N.X ◁ TensorBimod.actLeft P Q))
⋯)
⋯)
(M.X ◁
coequalizer.desc
((PreservesCoequalizer.iso (tensorRight Q.X) (N.actRight ▷ P.X)
((α_ N.X X.X P.X).hom ≫ N.X ◁ P.actLeft)).inv ≫
coequalizer.desc
((α_ N.X P.X Q.X).hom ≫
N.X ◁ coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X Y.X Q.X).hom ≫ P.X ◁ Q.actLeft) ≫
coequalizer.π (N.actRight ▷ TensorBimod.X P Q)
((α_ N.X X.X (TensorBimod.X P Q)).hom ≫ N.X ◁ TensorBimod.actLeft P Q))
⋯)
⋯)
⋯ ⋯) =
coequalizer.π (M.actRight ▷ N.X) ((α_ M.X W.X N.X).hom ≫ M.X ◁ N.actLeft) ▷ P.X ▷ Q.X ≫
(α_ (TensorBimod.X M N) P.X Q.X).hom ≫
TensorBimod.X M N ◁ coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X Y.X Q.X).hom ≫ P.X ◁ Q.actLeft) ≫
(PreservesCoequalizer.iso (tensorRight (TensorBimod.X P Q)) (M.actRight ▷ N.X)
((α_ M.X W.X N.X).hom ≫ M.X ◁ N.actLeft)).inv ≫
coequalizer.desc
((α_ M.X N.X (TensorBimod.X P Q)).hom ≫
M.X ◁
coequalizer.π (N.actRight ▷ TensorBimod.X P Q)
((α_ N.X X.X (TensorBimod.X P Q)).hom ≫ N.X ◁ TensorBimod.actLeft P Q) ≫
coequalizer.π (M.actRight ▷ TensorBimod.X N (P.tensorBimod Q))
((α_ M.X W.X (TensorBimod.X N (P.tensorBimod Q))).hom ≫
M.X ◁ TensorBimod.actLeft N (P.tensorBimod Q)))
⋯
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/CategoryTheory/Monoidal/Bimod.lean
|
Bimod.pentagon_bimod
|
case h.h
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
inst✝³ : MonoidalCategory C
A B : Mon_ C
M✝ : Bimod A B
inst✝² : HasCoequalizers C
inst✝¹ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
inst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)
V W X Y Z : Mon_ C
M : Bimod V W
N : Bimod W X
P : Bimod X Y
Q : Bimod Y Z
⊢ (α_ M.X N.X P.X).hom ▷ Q.X ≫
(M.X ◁ coequalizer.π (N.actRight ▷ P.X) ((α_ N.X X.X P.X).hom ≫ N.X ◁ P.actLeft)) ▷ Q.X ≫
((α_ M.X (TensorBimod.X N P) Q.X).hom ≫
M.X ◁
coequalizer.π (TensorBimod.actRight N P ▷ Q.X)
((α_ (TensorBimod.X N P) Y.X Q.X).hom ≫ TensorBimod.X N P ◁ Q.actLeft) ≫
coequalizer.π (M.actRight ▷ TensorBimod.X (N.tensorBimod P) Q)
((α_ M.X W.X (TensorBimod.X (N.tensorBimod P) Q)).hom ≫
M.X ◁ TensorBimod.actLeft (N.tensorBimod P) Q)) ≫
colimMap
(parallelPairHom (M.actRight ▷ TensorBimod.X (N.tensorBimod P) Q)
((α_ M.X W.X (TensorBimod.X (N.tensorBimod P) Q)).hom ≫ M.X ◁ TensorBimod.actLeft (N.tensorBimod P) Q)
(M.actRight ▷ TensorBimod.X N (P.tensorBimod Q))
((α_ M.X W.X (TensorBimod.X N (P.tensorBimod Q))).hom ≫ M.X ◁ TensorBimod.actLeft N (P.tensorBimod Q))
((M.X ⊗ W.X) ◁
coequalizer.desc
((PreservesCoequalizer.iso (tensorRight Q.X) (N.actRight ▷ P.X)
((α_ N.X X.X P.X).hom ≫ N.X ◁ P.actLeft)).inv ≫
coequalizer.desc
((α_ N.X P.X Q.X).hom ≫
N.X ◁ coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X Y.X Q.X).hom ≫ P.X ◁ Q.actLeft) ≫
coequalizer.π (N.actRight ▷ TensorBimod.X P Q)
((α_ N.X X.X (TensorBimod.X P Q)).hom ≫ N.X ◁ TensorBimod.actLeft P Q))
⋯)
⋯)
(M.X ◁
coequalizer.desc
((PreservesCoequalizer.iso (tensorRight Q.X) (N.actRight ▷ P.X)
((α_ N.X X.X P.X).hom ≫ N.X ◁ P.actLeft)).inv ≫
coequalizer.desc
((α_ N.X P.X Q.X).hom ≫
N.X ◁ coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X Y.X Q.X).hom ≫ P.X ◁ Q.actLeft) ≫
coequalizer.π (N.actRight ▷ TensorBimod.X P Q)
((α_ N.X X.X (TensorBimod.X P Q)).hom ≫ N.X ◁ TensorBimod.actLeft P Q))
⋯)
⋯)
⋯ ⋯) =
coequalizer.π (M.actRight ▷ N.X) ((α_ M.X W.X N.X).hom ≫ M.X ◁ N.actLeft) ▷ P.X ▷ Q.X ≫
(α_ (TensorBimod.X M N) P.X Q.X).hom ≫
TensorBimod.X M N ◁ coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X Y.X Q.X).hom ≫ P.X ◁ Q.actLeft) ≫
(PreservesCoequalizer.iso (tensorRight (TensorBimod.X P Q)) (M.actRight ▷ N.X)
((α_ M.X W.X N.X).hom ≫ M.X ◁ N.actLeft)).inv ≫
coequalizer.desc
((α_ M.X N.X (TensorBimod.X P Q)).hom ≫
M.X ◁
coequalizer.π (N.actRight ▷ TensorBimod.X P Q)
((α_ N.X X.X (TensorBimod.X P Q)).hom ≫ N.X ◁ TensorBimod.actLeft P Q) ≫
coequalizer.π (M.actRight ▷ TensorBimod.X N (P.tensorBimod Q))
((α_ M.X W.X (TensorBimod.X N (P.tensorBimod Q))).hom ≫
M.X ◁ TensorBimod.actLeft N (P.tensorBimod Q)))
⋯
|
dsimp only [<a>Bimod.TensorBimod.X</a>]
|
case h.h
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
inst✝³ : MonoidalCategory C
A B : Mon_ C
M✝ : Bimod A B
inst✝² : HasCoequalizers C
inst✝¹ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
inst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)
V W X Y Z : Mon_ C
M : Bimod V W
N : Bimod W X
P : Bimod X Y
Q : Bimod Y Z
⊢ (α_ M.X N.X P.X).hom ▷ Q.X ≫
(M.X ◁ coequalizer.π (N.actRight ▷ P.X) ((α_ N.X X.X P.X).hom ≫ N.X ◁ P.actLeft)) ▷ Q.X ≫
((α_ M.X (coequalizer (N.actRight ▷ P.X) ((α_ N.X X.X P.X).hom ≫ N.X ◁ P.actLeft)) Q.X).hom ≫
M.X ◁
coequalizer.π (TensorBimod.actRight N P ▷ Q.X)
((α_ (coequalizer (N.actRight ▷ P.X) ((α_ N.X X.X P.X).hom ≫ N.X ◁ P.actLeft)) Y.X Q.X).hom ≫
coequalizer (N.actRight ▷ P.X) ((α_ N.X X.X P.X).hom ≫ N.X ◁ P.actLeft) ◁ Q.actLeft) ≫
coequalizer.π
(M.actRight ▷
coequalizer ((N.tensorBimod P).actRight ▷ Q.X)
((α_ (N.tensorBimod P).X Y.X Q.X).hom ≫ (N.tensorBimod P).X ◁ Q.actLeft))
((α_ M.X W.X
(coequalizer ((N.tensorBimod P).actRight ▷ Q.X)
((α_ (N.tensorBimod P).X Y.X Q.X).hom ≫ (N.tensorBimod P).X ◁ Q.actLeft))).hom ≫
M.X ◁ TensorBimod.actLeft (N.tensorBimod P) Q)) ≫
colimMap
(parallelPairHom
(M.actRight ▷
coequalizer ((N.tensorBimod P).actRight ▷ Q.X)
((α_ (N.tensorBimod P).X Y.X Q.X).hom ≫ (N.tensorBimod P).X ◁ Q.actLeft))
((α_ M.X W.X
(coequalizer ((N.tensorBimod P).actRight ▷ Q.X)
((α_ (N.tensorBimod P).X Y.X Q.X).hom ≫ (N.tensorBimod P).X ◁ Q.actLeft))).hom ≫
M.X ◁ TensorBimod.actLeft (N.tensorBimod P) Q)
(M.actRight ▷
coequalizer (N.actRight ▷ (P.tensorBimod Q).X)
((α_ N.X X.X (P.tensorBimod Q).X).hom ≫ N.X ◁ (P.tensorBimod Q).actLeft))
((α_ M.X W.X
(coequalizer (N.actRight ▷ (P.tensorBimod Q).X)
((α_ N.X X.X (P.tensorBimod Q).X).hom ≫ N.X ◁ (P.tensorBimod Q).actLeft))).hom ≫
M.X ◁ TensorBimod.actLeft N (P.tensorBimod Q))
((M.X ⊗ W.X) ◁
coequalizer.desc
((PreservesCoequalizer.iso (tensorRight Q.X) (N.actRight ▷ P.X)
((α_ N.X X.X P.X).hom ≫ N.X ◁ P.actLeft)).inv ≫
coequalizer.desc
((α_ N.X P.X Q.X).hom ≫
N.X ◁ coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X Y.X Q.X).hom ≫ P.X ◁ Q.actLeft) ≫
coequalizer.π
(N.actRight ▷ coequalizer (P.actRight ▷ Q.X) ((α_ P.X Y.X Q.X).hom ≫ P.X ◁ Q.actLeft))
((α_ N.X X.X
(coequalizer (P.actRight ▷ Q.X) ((α_ P.X Y.X Q.X).hom ≫ P.X ◁ Q.actLeft))).hom ≫
N.X ◁ TensorBimod.actLeft P Q))
⋯)
⋯)
(M.X ◁
coequalizer.desc
((PreservesCoequalizer.iso (tensorRight Q.X) (N.actRight ▷ P.X)
((α_ N.X X.X P.X).hom ≫ N.X ◁ P.actLeft)).inv ≫
coequalizer.desc
((α_ N.X P.X Q.X).hom ≫
N.X ◁ coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X Y.X Q.X).hom ≫ P.X ◁ Q.actLeft) ≫
coequalizer.π
(N.actRight ▷ coequalizer (P.actRight ▷ Q.X) ((α_ P.X Y.X Q.X).hom ≫ P.X ◁ Q.actLeft))
((α_ N.X X.X
(coequalizer (P.actRight ▷ Q.X) ((α_ P.X Y.X Q.X).hom ≫ P.X ◁ Q.actLeft))).hom ≫
N.X ◁ TensorBimod.actLeft P Q))
⋯)
⋯)
⋯ ⋯) =
coequalizer.π (M.actRight ▷ N.X) ((α_ M.X W.X N.X).hom ≫ M.X ◁ N.actLeft) ▷ P.X ▷ Q.X ≫
(α_ (coequalizer (M.actRight ▷ N.X) ((α_ M.X W.X N.X).hom ≫ M.X ◁ N.actLeft)) P.X Q.X).hom ≫
coequalizer (M.actRight ▷ N.X) ((α_ M.X W.X N.X).hom ≫ M.X ◁ N.actLeft) ◁
coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X Y.X Q.X).hom ≫ P.X ◁ Q.actLeft) ≫
(PreservesCoequalizer.iso
(tensorRight (coequalizer (P.actRight ▷ Q.X) ((α_ P.X Y.X Q.X).hom ≫ P.X ◁ Q.actLeft)))
(M.actRight ▷ N.X) ((α_ M.X W.X N.X).hom ≫ M.X ◁ N.actLeft)).inv ≫
coequalizer.desc
((α_ M.X N.X (coequalizer (P.actRight ▷ Q.X) ((α_ P.X Y.X Q.X).hom ≫ P.X ◁ Q.actLeft))).hom ≫
M.X ◁
coequalizer.π (N.actRight ▷ coequalizer (P.actRight ▷ Q.X) ((α_ P.X Y.X Q.X).hom ≫ P.X ◁ Q.actLeft))
((α_ N.X X.X (coequalizer (P.actRight ▷ Q.X) ((α_ P.X Y.X Q.X).hom ≫ P.X ◁ Q.actLeft))).hom ≫
N.X ◁ TensorBimod.actLeft P Q) ≫
coequalizer.π
(M.actRight ▷
coequalizer (N.actRight ▷ (P.tensorBimod Q).X)
((α_ N.X X.X (P.tensorBimod Q).X).hom ≫ N.X ◁ (P.tensorBimod Q).actLeft))
((α_ M.X W.X
(coequalizer (N.actRight ▷ (P.tensorBimod Q).X)
((α_ N.X X.X (P.tensorBimod Q).X).hom ≫ N.X ◁ (P.tensorBimod Q).actLeft))).hom ≫
M.X ◁ TensorBimod.actLeft N (P.tensorBimod Q)))
⋯
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/CategoryTheory/Monoidal/Bimod.lean
|
Bimod.pentagon_bimod
|
case h.h
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
inst✝³ : MonoidalCategory C
A B : Mon_ C
M✝ : Bimod A B
inst✝² : HasCoequalizers C
inst✝¹ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
inst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)
V W X Y Z : Mon_ C
M : Bimod V W
N : Bimod W X
P : Bimod X Y
Q : Bimod Y Z
⊢ (α_ M.X N.X P.X).hom ▷ Q.X ≫
(M.X ◁ coequalizer.π (N.actRight ▷ P.X) ((α_ N.X X.X P.X).hom ≫ N.X ◁ P.actLeft)) ▷ Q.X ≫
((α_ M.X (coequalizer (N.actRight ▷ P.X) ((α_ N.X X.X P.X).hom ≫ N.X ◁ P.actLeft)) Q.X).hom ≫
M.X ◁
coequalizer.π (TensorBimod.actRight N P ▷ Q.X)
((α_ (coequalizer (N.actRight ▷ P.X) ((α_ N.X X.X P.X).hom ≫ N.X ◁ P.actLeft)) Y.X Q.X).hom ≫
coequalizer (N.actRight ▷ P.X) ((α_ N.X X.X P.X).hom ≫ N.X ◁ P.actLeft) ◁ Q.actLeft) ≫
coequalizer.π
(M.actRight ▷
coequalizer ((N.tensorBimod P).actRight ▷ Q.X)
((α_ (N.tensorBimod P).X Y.X Q.X).hom ≫ (N.tensorBimod P).X ◁ Q.actLeft))
((α_ M.X W.X
(coequalizer ((N.tensorBimod P).actRight ▷ Q.X)
((α_ (N.tensorBimod P).X Y.X Q.X).hom ≫ (N.tensorBimod P).X ◁ Q.actLeft))).hom ≫
M.X ◁ TensorBimod.actLeft (N.tensorBimod P) Q)) ≫
colimMap
(parallelPairHom
(M.actRight ▷
coequalizer ((N.tensorBimod P).actRight ▷ Q.X)
((α_ (N.tensorBimod P).X Y.X Q.X).hom ≫ (N.tensorBimod P).X ◁ Q.actLeft))
((α_ M.X W.X
(coequalizer ((N.tensorBimod P).actRight ▷ Q.X)
((α_ (N.tensorBimod P).X Y.X Q.X).hom ≫ (N.tensorBimod P).X ◁ Q.actLeft))).hom ≫
M.X ◁ TensorBimod.actLeft (N.tensorBimod P) Q)
(M.actRight ▷
coequalizer (N.actRight ▷ (P.tensorBimod Q).X)
((α_ N.X X.X (P.tensorBimod Q).X).hom ≫ N.X ◁ (P.tensorBimod Q).actLeft))
((α_ M.X W.X
(coequalizer (N.actRight ▷ (P.tensorBimod Q).X)
((α_ N.X X.X (P.tensorBimod Q).X).hom ≫ N.X ◁ (P.tensorBimod Q).actLeft))).hom ≫
M.X ◁ TensorBimod.actLeft N (P.tensorBimod Q))
((M.X ⊗ W.X) ◁
coequalizer.desc
((PreservesCoequalizer.iso (tensorRight Q.X) (N.actRight ▷ P.X)
((α_ N.X X.X P.X).hom ≫ N.X ◁ P.actLeft)).inv ≫
coequalizer.desc
((α_ N.X P.X Q.X).hom ≫
N.X ◁ coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X Y.X Q.X).hom ≫ P.X ◁ Q.actLeft) ≫
coequalizer.π
(N.actRight ▷ coequalizer (P.actRight ▷ Q.X) ((α_ P.X Y.X Q.X).hom ≫ P.X ◁ Q.actLeft))
((α_ N.X X.X
(coequalizer (P.actRight ▷ Q.X) ((α_ P.X Y.X Q.X).hom ≫ P.X ◁ Q.actLeft))).hom ≫
N.X ◁ TensorBimod.actLeft P Q))
⋯)
⋯)
(M.X ◁
coequalizer.desc
((PreservesCoequalizer.iso (tensorRight Q.X) (N.actRight ▷ P.X)
((α_ N.X X.X P.X).hom ≫ N.X ◁ P.actLeft)).inv ≫
coequalizer.desc
((α_ N.X P.X Q.X).hom ≫
N.X ◁ coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X Y.X Q.X).hom ≫ P.X ◁ Q.actLeft) ≫
coequalizer.π
(N.actRight ▷ coequalizer (P.actRight ▷ Q.X) ((α_ P.X Y.X Q.X).hom ≫ P.X ◁ Q.actLeft))
((α_ N.X X.X
(coequalizer (P.actRight ▷ Q.X) ((α_ P.X Y.X Q.X).hom ≫ P.X ◁ Q.actLeft))).hom ≫
N.X ◁ TensorBimod.actLeft P Q))
⋯)
⋯)
⋯ ⋯) =
coequalizer.π (M.actRight ▷ N.X) ((α_ M.X W.X N.X).hom ≫ M.X ◁ N.actLeft) ▷ P.X ▷ Q.X ≫
(α_ (coequalizer (M.actRight ▷ N.X) ((α_ M.X W.X N.X).hom ≫ M.X ◁ N.actLeft)) P.X Q.X).hom ≫
coequalizer (M.actRight ▷ N.X) ((α_ M.X W.X N.X).hom ≫ M.X ◁ N.actLeft) ◁
coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X Y.X Q.X).hom ≫ P.X ◁ Q.actLeft) ≫
(PreservesCoequalizer.iso
(tensorRight (coequalizer (P.actRight ▷ Q.X) ((α_ P.X Y.X Q.X).hom ≫ P.X ◁ Q.actLeft)))
(M.actRight ▷ N.X) ((α_ M.X W.X N.X).hom ≫ M.X ◁ N.actLeft)).inv ≫
coequalizer.desc
((α_ M.X N.X (coequalizer (P.actRight ▷ Q.X) ((α_ P.X Y.X Q.X).hom ≫ P.X ◁ Q.actLeft))).hom ≫
M.X ◁
coequalizer.π (N.actRight ▷ coequalizer (P.actRight ▷ Q.X) ((α_ P.X Y.X Q.X).hom ≫ P.X ◁ Q.actLeft))
((α_ N.X X.X (coequalizer (P.actRight ▷ Q.X) ((α_ P.X Y.X Q.X).hom ≫ P.X ◁ Q.actLeft))).hom ≫
N.X ◁ TensorBimod.actLeft P Q) ≫
coequalizer.π
(M.actRight ▷
coequalizer (N.actRight ▷ (P.tensorBimod Q).X)
((α_ N.X X.X (P.tensorBimod Q).X).hom ≫ N.X ◁ (P.tensorBimod Q).actLeft))
((α_ M.X W.X
(coequalizer (N.actRight ▷ (P.tensorBimod Q).X)
((α_ N.X X.X (P.tensorBimod Q).X).hom ≫ N.X ◁ (P.tensorBimod Q).actLeft))).hom ≫
M.X ◁ TensorBimod.actLeft N (P.tensorBimod Q)))
⋯
|
slice_lhs 2 3 => rw [<a>CategoryTheory.MonoidalCategory.associator_naturality_middle</a>]
|
case h.h
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
inst✝³ : MonoidalCategory C
A B : Mon_ C
M✝ : Bimod A B
inst✝² : HasCoequalizers C
inst✝¹ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
inst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)
V W X Y Z : Mon_ C
M : Bimod V W
N : Bimod W X
P : Bimod X Y
Q : Bimod Y Z
⊢ (α_ M.X N.X P.X).hom ▷ Q.X ≫
((((α_ M.X (N.X ⊗ P.X) Q.X).hom ≫
M.X ◁ coequalizer.π (N.actRight ▷ P.X) ((α_ N.X X.X P.X).hom ≫ N.X ◁ P.actLeft) ▷ Q.X) ≫
M.X ◁
coequalizer.π (TensorBimod.actRight N P ▷ Q.X)
((α_ (coequalizer (N.actRight ▷ P.X) ((α_ N.X X.X P.X).hom ≫ N.X ◁ P.actLeft)) Y.X Q.X).hom ≫
coequalizer (N.actRight ▷ P.X) ((α_ N.X X.X P.X).hom ≫ N.X ◁ P.actLeft) ◁ Q.actLeft)) ≫
coequalizer.π
(M.actRight ▷
coequalizer ((N.tensorBimod P).actRight ▷ Q.X)
((α_ (N.tensorBimod P).X Y.X Q.X).hom ≫ (N.tensorBimod P).X ◁ Q.actLeft))
((α_ M.X W.X
(coequalizer ((N.tensorBimod P).actRight ▷ Q.X)
((α_ (N.tensorBimod P).X Y.X Q.X).hom ≫ (N.tensorBimod P).X ◁ Q.actLeft))).hom ≫
M.X ◁ TensorBimod.actLeft (N.tensorBimod P) Q)) ≫
colimMap
(parallelPairHom
(M.actRight ▷
coequalizer ((N.tensorBimod P).actRight ▷ Q.X)
((α_ (N.tensorBimod P).X Y.X Q.X).hom ≫ (N.tensorBimod P).X ◁ Q.actLeft))
((α_ M.X W.X
(coequalizer ((N.tensorBimod P).actRight ▷ Q.X)
((α_ (N.tensorBimod P).X Y.X Q.X).hom ≫ (N.tensorBimod P).X ◁ Q.actLeft))).hom ≫
M.X ◁ TensorBimod.actLeft (N.tensorBimod P) Q)
(M.actRight ▷
coequalizer (N.actRight ▷ (P.tensorBimod Q).X)
((α_ N.X X.X (P.tensorBimod Q).X).hom ≫ N.X ◁ (P.tensorBimod Q).actLeft))
((α_ M.X W.X
(coequalizer (N.actRight ▷ (P.tensorBimod Q).X)
((α_ N.X X.X (P.tensorBimod Q).X).hom ≫ N.X ◁ (P.tensorBimod Q).actLeft))).hom ≫
M.X ◁ TensorBimod.actLeft N (P.tensorBimod Q))
((M.X ⊗ W.X) ◁
coequalizer.desc
((PreservesCoequalizer.iso (tensorRight Q.X) (N.actRight ▷ P.X)
((α_ N.X X.X P.X).hom ≫ N.X ◁ P.actLeft)).inv ≫
coequalizer.desc
((α_ N.X P.X Q.X).hom ≫
N.X ◁ coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X Y.X Q.X).hom ≫ P.X ◁ Q.actLeft) ≫
coequalizer.π
(N.actRight ▷ coequalizer (P.actRight ▷ Q.X) ((α_ P.X Y.X Q.X).hom ≫ P.X ◁ Q.actLeft))
((α_ N.X X.X (coequalizer (P.actRight ▷ Q.X) ((α_ P.X Y.X Q.X).hom ≫ P.X ◁ Q.actLeft))).hom ≫
N.X ◁ TensorBimod.actLeft P Q))
⋯)
⋯)
(M.X ◁
coequalizer.desc
((PreservesCoequalizer.iso (tensorRight Q.X) (N.actRight ▷ P.X)
((α_ N.X X.X P.X).hom ≫ N.X ◁ P.actLeft)).inv ≫
coequalizer.desc
((α_ N.X P.X Q.X).hom ≫
N.X ◁ coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X Y.X Q.X).hom ≫ P.X ◁ Q.actLeft) ≫
coequalizer.π
(N.actRight ▷ coequalizer (P.actRight ▷ Q.X) ((α_ P.X Y.X Q.X).hom ≫ P.X ◁ Q.actLeft))
((α_ N.X X.X (coequalizer (P.actRight ▷ Q.X) ((α_ P.X Y.X Q.X).hom ≫ P.X ◁ Q.actLeft))).hom ≫
N.X ◁ TensorBimod.actLeft P Q))
⋯)
⋯)
⋯ ⋯) =
coequalizer.π (M.actRight ▷ N.X) ((α_ M.X W.X N.X).hom ≫ M.X ◁ N.actLeft) ▷ P.X ▷ Q.X ≫
(α_ (coequalizer (M.actRight ▷ N.X) ((α_ M.X W.X N.X).hom ≫ M.X ◁ N.actLeft)) P.X Q.X).hom ≫
coequalizer (M.actRight ▷ N.X) ((α_ M.X W.X N.X).hom ≫ M.X ◁ N.actLeft) ◁
coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X Y.X Q.X).hom ≫ P.X ◁ Q.actLeft) ≫
(PreservesCoequalizer.iso
(tensorRight (coequalizer (P.actRight ▷ Q.X) ((α_ P.X Y.X Q.X).hom ≫ P.X ◁ Q.actLeft)))
(M.actRight ▷ N.X) ((α_ M.X W.X N.X).hom ≫ M.X ◁ N.actLeft)).inv ≫
coequalizer.desc
((α_ M.X N.X (coequalizer (P.actRight ▷ Q.X) ((α_ P.X Y.X Q.X).hom ≫ P.X ◁ Q.actLeft))).hom ≫
M.X ◁
coequalizer.π (N.actRight ▷ coequalizer (P.actRight ▷ Q.X) ((α_ P.X Y.X Q.X).hom ≫ P.X ◁ Q.actLeft))
((α_ N.X X.X (coequalizer (P.actRight ▷ Q.X) ((α_ P.X Y.X Q.X).hom ≫ P.X ◁ Q.actLeft))).hom ≫
N.X ◁ TensorBimod.actLeft P Q) ≫
coequalizer.π
(M.actRight ▷
coequalizer (N.actRight ▷ (P.tensorBimod Q).X)
((α_ N.X X.X (P.tensorBimod Q).X).hom ≫ N.X ◁ (P.tensorBimod Q).actLeft))
((α_ M.X W.X
(coequalizer (N.actRight ▷ (P.tensorBimod Q).X)
((α_ N.X X.X (P.tensorBimod Q).X).hom ≫ N.X ◁ (P.tensorBimod Q).actLeft))).hom ≫
M.X ◁ TensorBimod.actLeft N (P.tensorBimod Q)))
⋯
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/CategoryTheory/Monoidal/Bimod.lean
|
Bimod.pentagon_bimod
|
case h.h
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
inst✝³ : MonoidalCategory C
A B : Mon_ C
M✝ : Bimod A B
inst✝² : HasCoequalizers C
inst✝¹ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
inst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)
V W X Y Z : Mon_ C
M : Bimod V W
N : Bimod W X
P : Bimod X Y
Q : Bimod Y Z
⊢ (α_ M.X N.X P.X).hom ▷ Q.X ≫
((((α_ M.X (N.X ⊗ P.X) Q.X).hom ≫
M.X ◁ coequalizer.π (N.actRight ▷ P.X) ((α_ N.X X.X P.X).hom ≫ N.X ◁ P.actLeft) ▷ Q.X) ≫
M.X ◁
coequalizer.π (TensorBimod.actRight N P ▷ Q.X)
((α_ (coequalizer (N.actRight ▷ P.X) ((α_ N.X X.X P.X).hom ≫ N.X ◁ P.actLeft)) Y.X Q.X).hom ≫
coequalizer (N.actRight ▷ P.X) ((α_ N.X X.X P.X).hom ≫ N.X ◁ P.actLeft) ◁ Q.actLeft)) ≫
coequalizer.π
(M.actRight ▷
coequalizer ((N.tensorBimod P).actRight ▷ Q.X)
((α_ (N.tensorBimod P).X Y.X Q.X).hom ≫ (N.tensorBimod P).X ◁ Q.actLeft))
((α_ M.X W.X
(coequalizer ((N.tensorBimod P).actRight ▷ Q.X)
((α_ (N.tensorBimod P).X Y.X Q.X).hom ≫ (N.tensorBimod P).X ◁ Q.actLeft))).hom ≫
M.X ◁ TensorBimod.actLeft (N.tensorBimod P) Q)) ≫
colimMap
(parallelPairHom
(M.actRight ▷
coequalizer ((N.tensorBimod P).actRight ▷ Q.X)
((α_ (N.tensorBimod P).X Y.X Q.X).hom ≫ (N.tensorBimod P).X ◁ Q.actLeft))
((α_ M.X W.X
(coequalizer ((N.tensorBimod P).actRight ▷ Q.X)
((α_ (N.tensorBimod P).X Y.X Q.X).hom ≫ (N.tensorBimod P).X ◁ Q.actLeft))).hom ≫
M.X ◁ TensorBimod.actLeft (N.tensorBimod P) Q)
(M.actRight ▷
coequalizer (N.actRight ▷ (P.tensorBimod Q).X)
((α_ N.X X.X (P.tensorBimod Q).X).hom ≫ N.X ◁ (P.tensorBimod Q).actLeft))
((α_ M.X W.X
(coequalizer (N.actRight ▷ (P.tensorBimod Q).X)
((α_ N.X X.X (P.tensorBimod Q).X).hom ≫ N.X ◁ (P.tensorBimod Q).actLeft))).hom ≫
M.X ◁ TensorBimod.actLeft N (P.tensorBimod Q))
((M.X ⊗ W.X) ◁
coequalizer.desc
((PreservesCoequalizer.iso (tensorRight Q.X) (N.actRight ▷ P.X)
((α_ N.X X.X P.X).hom ≫ N.X ◁ P.actLeft)).inv ≫
coequalizer.desc
((α_ N.X P.X Q.X).hom ≫
N.X ◁ coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X Y.X Q.X).hom ≫ P.X ◁ Q.actLeft) ≫
coequalizer.π
(N.actRight ▷ coequalizer (P.actRight ▷ Q.X) ((α_ P.X Y.X Q.X).hom ≫ P.X ◁ Q.actLeft))
((α_ N.X X.X (coequalizer (P.actRight ▷ Q.X) ((α_ P.X Y.X Q.X).hom ≫ P.X ◁ Q.actLeft))).hom ≫
N.X ◁ TensorBimod.actLeft P Q))
⋯)
⋯)
(M.X ◁
coequalizer.desc
((PreservesCoequalizer.iso (tensorRight Q.X) (N.actRight ▷ P.X)
((α_ N.X X.X P.X).hom ≫ N.X ◁ P.actLeft)).inv ≫
coequalizer.desc
((α_ N.X P.X Q.X).hom ≫
N.X ◁ coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X Y.X Q.X).hom ≫ P.X ◁ Q.actLeft) ≫
coequalizer.π
(N.actRight ▷ coequalizer (P.actRight ▷ Q.X) ((α_ P.X Y.X Q.X).hom ≫ P.X ◁ Q.actLeft))
((α_ N.X X.X (coequalizer (P.actRight ▷ Q.X) ((α_ P.X Y.X Q.X).hom ≫ P.X ◁ Q.actLeft))).hom ≫
N.X ◁ TensorBimod.actLeft P Q))
⋯)
⋯)
⋯ ⋯) =
coequalizer.π (M.actRight ▷ N.X) ((α_ M.X W.X N.X).hom ≫ M.X ◁ N.actLeft) ▷ P.X ▷ Q.X ≫
(α_ (coequalizer (M.actRight ▷ N.X) ((α_ M.X W.X N.X).hom ≫ M.X ◁ N.actLeft)) P.X Q.X).hom ≫
coequalizer (M.actRight ▷ N.X) ((α_ M.X W.X N.X).hom ≫ M.X ◁ N.actLeft) ◁
coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X Y.X Q.X).hom ≫ P.X ◁ Q.actLeft) ≫
(PreservesCoequalizer.iso
(tensorRight (coequalizer (P.actRight ▷ Q.X) ((α_ P.X Y.X Q.X).hom ≫ P.X ◁ Q.actLeft)))
(M.actRight ▷ N.X) ((α_ M.X W.X N.X).hom ≫ M.X ◁ N.actLeft)).inv ≫
coequalizer.desc
((α_ M.X N.X (coequalizer (P.actRight ▷ Q.X) ((α_ P.X Y.X Q.X).hom ≫ P.X ◁ Q.actLeft))).hom ≫
M.X ◁
coequalizer.π (N.actRight ▷ coequalizer (P.actRight ▷ Q.X) ((α_ P.X Y.X Q.X).hom ≫ P.X ◁ Q.actLeft))
((α_ N.X X.X (coequalizer (P.actRight ▷ Q.X) ((α_ P.X Y.X Q.X).hom ≫ P.X ◁ Q.actLeft))).hom ≫
N.X ◁ TensorBimod.actLeft P Q) ≫
coequalizer.π
(M.actRight ▷
coequalizer (N.actRight ▷ (P.tensorBimod Q).X)
((α_ N.X X.X (P.tensorBimod Q).X).hom ≫ N.X ◁ (P.tensorBimod Q).actLeft))
((α_ M.X W.X
(coequalizer (N.actRight ▷ (P.tensorBimod Q).X)
((α_ N.X X.X (P.tensorBimod Q).X).hom ≫ N.X ◁ (P.tensorBimod Q).actLeft))).hom ≫
M.X ◁ TensorBimod.actLeft N (P.tensorBimod Q)))
⋯
|
slice_lhs 5 6 => rw [<a>CategoryTheory.Limits.ι_colimMap</a>, <a>CategoryTheory.Limits.parallelPairHom_app_one</a>]
|
case h.h
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
inst✝³ : MonoidalCategory C
A B : Mon_ C
M✝ : Bimod A B
inst✝² : HasCoequalizers C
inst✝¹ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
inst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)
V W X Y Z : Mon_ C
M : Bimod V W
N : Bimod W X
P : Bimod X Y
Q : Bimod Y Z
⊢ (α_ M.X N.X P.X).hom ▷ Q.X ≫
(α_ M.X (N.X ⊗ P.X) Q.X).hom ≫
M.X ◁ coequalizer.π (N.actRight ▷ P.X) ((α_ N.X X.X P.X).hom ≫ N.X ◁ P.actLeft) ▷ Q.X ≫
M.X ◁
coequalizer.π (TensorBimod.actRight N P ▷ Q.X)
((α_ (coequalizer (N.actRight ▷ P.X) ((α_ N.X X.X P.X).hom ≫ N.X ◁ P.actLeft)) Y.X Q.X).hom ≫
coequalizer (N.actRight ▷ P.X) ((α_ N.X X.X P.X).hom ≫ N.X ◁ P.actLeft) ◁ Q.actLeft) ≫
M.X ◁
coequalizer.desc
((PreservesCoequalizer.iso (tensorRight Q.X) (N.actRight ▷ P.X)
((α_ N.X X.X P.X).hom ≫ N.X ◁ P.actLeft)).inv ≫
coequalizer.desc
((α_ N.X P.X Q.X).hom ≫
N.X ◁ coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X Y.X Q.X).hom ≫ P.X ◁ Q.actLeft) ≫
coequalizer.π
(N.actRight ▷ coequalizer (P.actRight ▷ Q.X) ((α_ P.X Y.X Q.X).hom ≫ P.X ◁ Q.actLeft))
((α_ N.X X.X
(coequalizer (P.actRight ▷ Q.X) ((α_ P.X Y.X Q.X).hom ≫ P.X ◁ Q.actLeft))).hom ≫
N.X ◁ TensorBimod.actLeft P Q))
⋯)
⋯ ≫
colimit.ι
(parallelPair
(M.actRight ▷
coequalizer (N.actRight ▷ (P.tensorBimod Q).X)
((α_ N.X X.X (P.tensorBimod Q).X).hom ≫ N.X ◁ (P.tensorBimod Q).actLeft))
((α_ M.X W.X
(coequalizer (N.actRight ▷ (P.tensorBimod Q).X)
((α_ N.X X.X (P.tensorBimod Q).X).hom ≫ N.X ◁ (P.tensorBimod Q).actLeft))).hom ≫
M.X ◁ TensorBimod.actLeft N (P.tensorBimod Q)))
WalkingParallelPair.one =
coequalizer.π (M.actRight ▷ N.X) ((α_ M.X W.X N.X).hom ≫ M.X ◁ N.actLeft) ▷ P.X ▷ Q.X ≫
(α_ (coequalizer (M.actRight ▷ N.X) ((α_ M.X W.X N.X).hom ≫ M.X ◁ N.actLeft)) P.X Q.X).hom ≫
coequalizer (M.actRight ▷ N.X) ((α_ M.X W.X N.X).hom ≫ M.X ◁ N.actLeft) ◁
coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X Y.X Q.X).hom ≫ P.X ◁ Q.actLeft) ≫
(PreservesCoequalizer.iso
(tensorRight (coequalizer (P.actRight ▷ Q.X) ((α_ P.X Y.X Q.X).hom ≫ P.X ◁ Q.actLeft)))
(M.actRight ▷ N.X) ((α_ M.X W.X N.X).hom ≫ M.X ◁ N.actLeft)).inv ≫
coequalizer.desc
((α_ M.X N.X (coequalizer (P.actRight ▷ Q.X) ((α_ P.X Y.X Q.X).hom ≫ P.X ◁ Q.actLeft))).hom ≫
M.X ◁
coequalizer.π (N.actRight ▷ coequalizer (P.actRight ▷ Q.X) ((α_ P.X Y.X Q.X).hom ≫ P.X ◁ Q.actLeft))
((α_ N.X X.X (coequalizer (P.actRight ▷ Q.X) ((α_ P.X Y.X Q.X).hom ≫ P.X ◁ Q.actLeft))).hom ≫
N.X ◁ TensorBimod.actLeft P Q) ≫
coequalizer.π
(M.actRight ▷
coequalizer (N.actRight ▷ (P.tensorBimod Q).X)
((α_ N.X X.X (P.tensorBimod Q).X).hom ≫ N.X ◁ (P.tensorBimod Q).actLeft))
((α_ M.X W.X
(coequalizer (N.actRight ▷ (P.tensorBimod Q).X)
((α_ N.X X.X (P.tensorBimod Q).X).hom ≫ N.X ◁ (P.tensorBimod Q).actLeft))).hom ≫
M.X ◁ TensorBimod.actLeft N (P.tensorBimod Q)))
⋯
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/CategoryTheory/Monoidal/Bimod.lean
|
Bimod.pentagon_bimod
|
case h.h
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
inst✝³ : MonoidalCategory C
A B : Mon_ C
M✝ : Bimod A B
inst✝² : HasCoequalizers C
inst✝¹ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
inst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)
V W X Y Z : Mon_ C
M : Bimod V W
N : Bimod W X
P : Bimod X Y
Q : Bimod Y Z
⊢ (α_ M.X N.X P.X).hom ▷ Q.X ≫
(α_ M.X (N.X ⊗ P.X) Q.X).hom ≫
M.X ◁ coequalizer.π (N.actRight ▷ P.X) ((α_ N.X X.X P.X).hom ≫ N.X ◁ P.actLeft) ▷ Q.X ≫
M.X ◁
coequalizer.π (TensorBimod.actRight N P ▷ Q.X)
((α_ (coequalizer (N.actRight ▷ P.X) ((α_ N.X X.X P.X).hom ≫ N.X ◁ P.actLeft)) Y.X Q.X).hom ≫
coequalizer (N.actRight ▷ P.X) ((α_ N.X X.X P.X).hom ≫ N.X ◁ P.actLeft) ◁ Q.actLeft) ≫
M.X ◁
coequalizer.desc
((PreservesCoequalizer.iso (tensorRight Q.X) (N.actRight ▷ P.X)
((α_ N.X X.X P.X).hom ≫ N.X ◁ P.actLeft)).inv ≫
coequalizer.desc
((α_ N.X P.X Q.X).hom ≫
N.X ◁ coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X Y.X Q.X).hom ≫ P.X ◁ Q.actLeft) ≫
coequalizer.π
(N.actRight ▷ coequalizer (P.actRight ▷ Q.X) ((α_ P.X Y.X Q.X).hom ≫ P.X ◁ Q.actLeft))
((α_ N.X X.X
(coequalizer (P.actRight ▷ Q.X) ((α_ P.X Y.X Q.X).hom ≫ P.X ◁ Q.actLeft))).hom ≫
N.X ◁ TensorBimod.actLeft P Q))
⋯)
⋯ ≫
colimit.ι
(parallelPair
(M.actRight ▷
coequalizer (N.actRight ▷ (P.tensorBimod Q).X)
((α_ N.X X.X (P.tensorBimod Q).X).hom ≫ N.X ◁ (P.tensorBimod Q).actLeft))
((α_ M.X W.X
(coequalizer (N.actRight ▷ (P.tensorBimod Q).X)
((α_ N.X X.X (P.tensorBimod Q).X).hom ≫ N.X ◁ (P.tensorBimod Q).actLeft))).hom ≫
M.X ◁ TensorBimod.actLeft N (P.tensorBimod Q)))
WalkingParallelPair.one =
coequalizer.π (M.actRight ▷ N.X) ((α_ M.X W.X N.X).hom ≫ M.X ◁ N.actLeft) ▷ P.X ▷ Q.X ≫
(α_ (coequalizer (M.actRight ▷ N.X) ((α_ M.X W.X N.X).hom ≫ M.X ◁ N.actLeft)) P.X Q.X).hom ≫
coequalizer (M.actRight ▷ N.X) ((α_ M.X W.X N.X).hom ≫ M.X ◁ N.actLeft) ◁
coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X Y.X Q.X).hom ≫ P.X ◁ Q.actLeft) ≫
(PreservesCoequalizer.iso
(tensorRight (coequalizer (P.actRight ▷ Q.X) ((α_ P.X Y.X Q.X).hom ≫ P.X ◁ Q.actLeft)))
(M.actRight ▷ N.X) ((α_ M.X W.X N.X).hom ≫ M.X ◁ N.actLeft)).inv ≫
coequalizer.desc
((α_ M.X N.X (coequalizer (P.actRight ▷ Q.X) ((α_ P.X Y.X Q.X).hom ≫ P.X ◁ Q.actLeft))).hom ≫
M.X ◁
coequalizer.π (N.actRight ▷ coequalizer (P.actRight ▷ Q.X) ((α_ P.X Y.X Q.X).hom ≫ P.X ◁ Q.actLeft))
((α_ N.X X.X (coequalizer (P.actRight ▷ Q.X) ((α_ P.X Y.X Q.X).hom ≫ P.X ◁ Q.actLeft))).hom ≫
N.X ◁ TensorBimod.actLeft P Q) ≫
coequalizer.π
(M.actRight ▷
coequalizer (N.actRight ▷ (P.tensorBimod Q).X)
((α_ N.X X.X (P.tensorBimod Q).X).hom ≫ N.X ◁ (P.tensorBimod Q).actLeft))
((α_ M.X W.X
(coequalizer (N.actRight ▷ (P.tensorBimod Q).X)
((α_ N.X X.X (P.tensorBimod Q).X).hom ≫ N.X ◁ (P.tensorBimod Q).actLeft))).hom ≫
M.X ◁ TensorBimod.actLeft N (P.tensorBimod Q)))
⋯
|
slice_lhs 4 5 => rw [← <a>CategoryTheory.MonoidalCategory.whiskerLeft_comp</a>, <a>CategoryTheory.Limits.coequalizer.π_desc</a>]
|
case h.h
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
inst✝³ : MonoidalCategory C
A B : Mon_ C
M✝ : Bimod A B
inst✝² : HasCoequalizers C
inst✝¹ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
inst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)
V W X Y Z : Mon_ C
M : Bimod V W
N : Bimod W X
P : Bimod X Y
Q : Bimod Y Z
⊢ (α_ M.X N.X P.X).hom ▷ Q.X ≫
(α_ M.X (N.X ⊗ P.X) Q.X).hom ≫
M.X ◁ coequalizer.π (N.actRight ▷ P.X) ((α_ N.X X.X P.X).hom ≫ N.X ◁ P.actLeft) ▷ Q.X ≫
M.X ◁
((PreservesCoequalizer.iso (tensorRight Q.X) (N.actRight ▷ P.X)
((α_ N.X X.X P.X).hom ≫ N.X ◁ P.actLeft)).inv ≫
coequalizer.desc
((α_ N.X P.X Q.X).hom ≫
N.X ◁ coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X Y.X Q.X).hom ≫ P.X ◁ Q.actLeft) ≫
coequalizer.π
(N.actRight ▷ coequalizer (P.actRight ▷ Q.X) ((α_ P.X Y.X Q.X).hom ≫ P.X ◁ Q.actLeft))
((α_ N.X X.X (coequalizer (P.actRight ▷ Q.X) ((α_ P.X Y.X Q.X).hom ≫ P.X ◁ Q.actLeft))).hom ≫
N.X ◁ TensorBimod.actLeft P Q))
⋯) ≫
colimit.ι
(parallelPair
(M.actRight ▷
coequalizer (N.actRight ▷ (P.tensorBimod Q).X)
((α_ N.X X.X (P.tensorBimod Q).X).hom ≫ N.X ◁ (P.tensorBimod Q).actLeft))
((α_ M.X W.X
(coequalizer (N.actRight ▷ (P.tensorBimod Q).X)
((α_ N.X X.X (P.tensorBimod Q).X).hom ≫ N.X ◁ (P.tensorBimod Q).actLeft))).hom ≫
M.X ◁ TensorBimod.actLeft N (P.tensorBimod Q)))
WalkingParallelPair.one =
coequalizer.π (M.actRight ▷ N.X) ((α_ M.X W.X N.X).hom ≫ M.X ◁ N.actLeft) ▷ P.X ▷ Q.X ≫
(α_ (coequalizer (M.actRight ▷ N.X) ((α_ M.X W.X N.X).hom ≫ M.X ◁ N.actLeft)) P.X Q.X).hom ≫
coequalizer (M.actRight ▷ N.X) ((α_ M.X W.X N.X).hom ≫ M.X ◁ N.actLeft) ◁
coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X Y.X Q.X).hom ≫ P.X ◁ Q.actLeft) ≫
(PreservesCoequalizer.iso
(tensorRight (coequalizer (P.actRight ▷ Q.X) ((α_ P.X Y.X Q.X).hom ≫ P.X ◁ Q.actLeft)))
(M.actRight ▷ N.X) ((α_ M.X W.X N.X).hom ≫ M.X ◁ N.actLeft)).inv ≫
coequalizer.desc
((α_ M.X N.X (coequalizer (P.actRight ▷ Q.X) ((α_ P.X Y.X Q.X).hom ≫ P.X ◁ Q.actLeft))).hom ≫
M.X ◁
coequalizer.π (N.actRight ▷ coequalizer (P.actRight ▷ Q.X) ((α_ P.X Y.X Q.X).hom ≫ P.X ◁ Q.actLeft))
((α_ N.X X.X (coequalizer (P.actRight ▷ Q.X) ((α_ P.X Y.X Q.X).hom ≫ P.X ◁ Q.actLeft))).hom ≫
N.X ◁ TensorBimod.actLeft P Q) ≫
coequalizer.π
(M.actRight ▷
coequalizer (N.actRight ▷ (P.tensorBimod Q).X)
((α_ N.X X.X (P.tensorBimod Q).X).hom ≫ N.X ◁ (P.tensorBimod Q).actLeft))
((α_ M.X W.X
(coequalizer (N.actRight ▷ (P.tensorBimod Q).X)
((α_ N.X X.X (P.tensorBimod Q).X).hom ≫ N.X ◁ (P.tensorBimod Q).actLeft))).hom ≫
M.X ◁ TensorBimod.actLeft N (P.tensorBimod Q)))
⋯
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/CategoryTheory/Monoidal/Bimod.lean
|
Bimod.pentagon_bimod
|
case h.h
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
inst✝³ : MonoidalCategory C
A B : Mon_ C
M✝ : Bimod A B
inst✝² : HasCoequalizers C
inst✝¹ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
inst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)
V W X Y Z : Mon_ C
M : Bimod V W
N : Bimod W X
P : Bimod X Y
Q : Bimod Y Z
⊢ (α_ M.X N.X P.X).hom ▷ Q.X ≫
(α_ M.X (N.X ⊗ P.X) Q.X).hom ≫
M.X ◁ coequalizer.π (N.actRight ▷ P.X) ((α_ N.X X.X P.X).hom ≫ N.X ◁ P.actLeft) ▷ Q.X ≫
M.X ◁
((PreservesCoequalizer.iso (tensorRight Q.X) (N.actRight ▷ P.X)
((α_ N.X X.X P.X).hom ≫ N.X ◁ P.actLeft)).inv ≫
coequalizer.desc
((α_ N.X P.X Q.X).hom ≫
N.X ◁ coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X Y.X Q.X).hom ≫ P.X ◁ Q.actLeft) ≫
coequalizer.π
(N.actRight ▷ coequalizer (P.actRight ▷ Q.X) ((α_ P.X Y.X Q.X).hom ≫ P.X ◁ Q.actLeft))
((α_ N.X X.X (coequalizer (P.actRight ▷ Q.X) ((α_ P.X Y.X Q.X).hom ≫ P.X ◁ Q.actLeft))).hom ≫
N.X ◁ TensorBimod.actLeft P Q))
⋯) ≫
colimit.ι
(parallelPair
(M.actRight ▷
coequalizer (N.actRight ▷ (P.tensorBimod Q).X)
((α_ N.X X.X (P.tensorBimod Q).X).hom ≫ N.X ◁ (P.tensorBimod Q).actLeft))
((α_ M.X W.X
(coequalizer (N.actRight ▷ (P.tensorBimod Q).X)
((α_ N.X X.X (P.tensorBimod Q).X).hom ≫ N.X ◁ (P.tensorBimod Q).actLeft))).hom ≫
M.X ◁ TensorBimod.actLeft N (P.tensorBimod Q)))
WalkingParallelPair.one =
coequalizer.π (M.actRight ▷ N.X) ((α_ M.X W.X N.X).hom ≫ M.X ◁ N.actLeft) ▷ P.X ▷ Q.X ≫
(α_ (coequalizer (M.actRight ▷ N.X) ((α_ M.X W.X N.X).hom ≫ M.X ◁ N.actLeft)) P.X Q.X).hom ≫
coequalizer (M.actRight ▷ N.X) ((α_ M.X W.X N.X).hom ≫ M.X ◁ N.actLeft) ◁
coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X Y.X Q.X).hom ≫ P.X ◁ Q.actLeft) ≫
(PreservesCoequalizer.iso
(tensorRight (coequalizer (P.actRight ▷ Q.X) ((α_ P.X Y.X Q.X).hom ≫ P.X ◁ Q.actLeft)))
(M.actRight ▷ N.X) ((α_ M.X W.X N.X).hom ≫ M.X ◁ N.actLeft)).inv ≫
coequalizer.desc
((α_ M.X N.X (coequalizer (P.actRight ▷ Q.X) ((α_ P.X Y.X Q.X).hom ≫ P.X ◁ Q.actLeft))).hom ≫
M.X ◁
coequalizer.π (N.actRight ▷ coequalizer (P.actRight ▷ Q.X) ((α_ P.X Y.X Q.X).hom ≫ P.X ◁ Q.actLeft))
((α_ N.X X.X (coequalizer (P.actRight ▷ Q.X) ((α_ P.X Y.X Q.X).hom ≫ P.X ◁ Q.actLeft))).hom ≫
N.X ◁ TensorBimod.actLeft P Q) ≫
coequalizer.π
(M.actRight ▷
coequalizer (N.actRight ▷ (P.tensorBimod Q).X)
((α_ N.X X.X (P.tensorBimod Q).X).hom ≫ N.X ◁ (P.tensorBimod Q).actLeft))
((α_ M.X W.X
(coequalizer (N.actRight ▷ (P.tensorBimod Q).X)
((α_ N.X X.X (P.tensorBimod Q).X).hom ≫ N.X ◁ (P.tensorBimod Q).actLeft))).hom ≫
M.X ◁ TensorBimod.actLeft N (P.tensorBimod Q)))
⋯
|
slice_lhs 3 4 => rw [← <a>CategoryTheory.MonoidalCategory.whiskerLeft_comp</a>, <a>π_tensor_id_preserves_coequalizer_inv_desc</a>, <a>CategoryTheory.MonoidalCategory.whiskerLeft_comp</a>, <a>CategoryTheory.MonoidalCategory.whiskerLeft_comp</a>]
|
case h.h
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
inst✝³ : MonoidalCategory C
A B : Mon_ C
M✝ : Bimod A B
inst✝² : HasCoequalizers C
inst✝¹ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
inst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)
V W X Y Z : Mon_ C
M : Bimod V W
N : Bimod W X
P : Bimod X Y
Q : Bimod Y Z
⊢ (α_ M.X N.X P.X).hom ▷ Q.X ≫
(α_ M.X (N.X ⊗ P.X) Q.X).hom ≫
(M.X ◁ (α_ N.X P.X Q.X).hom ≫
M.X ◁ N.X ◁ coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X Y.X Q.X).hom ≫ P.X ◁ Q.actLeft) ≫
M.X ◁
coequalizer.π (N.actRight ▷ coequalizer (P.actRight ▷ Q.X) ((α_ P.X Y.X Q.X).hom ≫ P.X ◁ Q.actLeft))
((α_ N.X X.X (coequalizer (P.actRight ▷ Q.X) ((α_ P.X Y.X Q.X).hom ≫ P.X ◁ Q.actLeft))).hom ≫
N.X ◁ TensorBimod.actLeft P Q)) ≫
colimit.ι
(parallelPair
(M.actRight ▷
coequalizer (N.actRight ▷ (P.tensorBimod Q).X)
((α_ N.X X.X (P.tensorBimod Q).X).hom ≫ N.X ◁ (P.tensorBimod Q).actLeft))
((α_ M.X W.X
(coequalizer (N.actRight ▷ (P.tensorBimod Q).X)
((α_ N.X X.X (P.tensorBimod Q).X).hom ≫ N.X ◁ (P.tensorBimod Q).actLeft))).hom ≫
M.X ◁ TensorBimod.actLeft N (P.tensorBimod Q)))
WalkingParallelPair.one =
coequalizer.π (M.actRight ▷ N.X) ((α_ M.X W.X N.X).hom ≫ M.X ◁ N.actLeft) ▷ P.X ▷ Q.X ≫
(α_ (coequalizer (M.actRight ▷ N.X) ((α_ M.X W.X N.X).hom ≫ M.X ◁ N.actLeft)) P.X Q.X).hom ≫
coequalizer (M.actRight ▷ N.X) ((α_ M.X W.X N.X).hom ≫ M.X ◁ N.actLeft) ◁
coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X Y.X Q.X).hom ≫ P.X ◁ Q.actLeft) ≫
(PreservesCoequalizer.iso
(tensorRight (coequalizer (P.actRight ▷ Q.X) ((α_ P.X Y.X Q.X).hom ≫ P.X ◁ Q.actLeft)))
(M.actRight ▷ N.X) ((α_ M.X W.X N.X).hom ≫ M.X ◁ N.actLeft)).inv ≫
coequalizer.desc
((α_ M.X N.X (coequalizer (P.actRight ▷ Q.X) ((α_ P.X Y.X Q.X).hom ≫ P.X ◁ Q.actLeft))).hom ≫
M.X ◁
coequalizer.π (N.actRight ▷ coequalizer (P.actRight ▷ Q.X) ((α_ P.X Y.X Q.X).hom ≫ P.X ◁ Q.actLeft))
((α_ N.X X.X (coequalizer (P.actRight ▷ Q.X) ((α_ P.X Y.X Q.X).hom ≫ P.X ◁ Q.actLeft))).hom ≫
N.X ◁ TensorBimod.actLeft P Q) ≫
coequalizer.π
(M.actRight ▷
coequalizer (N.actRight ▷ (P.tensorBimod Q).X)
((α_ N.X X.X (P.tensorBimod Q).X).hom ≫ N.X ◁ (P.tensorBimod Q).actLeft))
((α_ M.X W.X
(coequalizer (N.actRight ▷ (P.tensorBimod Q).X)
((α_ N.X X.X (P.tensorBimod Q).X).hom ≫ N.X ◁ (P.tensorBimod Q).actLeft))).hom ≫
M.X ◁ TensorBimod.actLeft N (P.tensorBimod Q)))
⋯
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/CategoryTheory/Monoidal/Bimod.lean
|
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