mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Algebra.Category.CommAlgCat.Monoidal

Goal state before:
pp: R : Type u
inst✝ : CommRing R
A B C D X✝ Y✝ Z✝ : CommAlgCat R
f✝ : Y✝ ⟶ Z✝
⊢ X✝ ◁ f✝ ≫ (isoMk (Algebra.TensorProduct.comm R ↑X✝ ↑Z✝)).hom =
    (isoMk (Algebra.TensorProduct.comm R ↑X✝ ↑Y✝)).hom ≫ f✝ ▷ X✝

Tactic:
;

Goal state after:
R : Type u
inst✝ : CommRing R
A B C D X✝ Y✝ Z✝ : CommAlgCat R
f✝ : Y✝ ⟶ Z✝
⊢ X✝ ◁ f✝ ≫ (isoMk (Algebra.TensorProduct.comm R ↑X✝ ↑Z✝)).hom =
    (isoMk (Algebra.TensorProduct.comm R ↑X✝ ↑Y✝)).hom ≫ f✝ ▷ X✝
Module: Mathlib.Algebra.Category.CommAlgCat.Monoidal

Goal state before:
pp: R : Type u
inst✝ : CommRing R
A B C D X✝ Y✝ Z✝ : CommAlgCat R
f✝ : Y✝ ⟶ Z✝
⊢ Hom.hom (X✝ ◁ f✝ ≫ (isoMk (Algebra.TensorProduct.comm R ↑X✝ ↑Z✝)).hom) =
    Hom.hom ((isoMk (Algebra.TensorProduct.comm R ↑X✝ ↑Y✝)).hom ≫ f✝ ▷ X✝)

Tactic:
;

Goal state after:
R : Type u
inst✝ : CommRing R
A B C D X✝ Y✝ Z✝ : CommAlgCat R
f✝ : Y✝ ⟶ Z✝
⊢ Hom.hom (X✝ ◁ f✝ ≫ (isoMk (Algebra.TensorProduct.comm R ↑X✝ ↑Z✝)).hom) =
    Hom.hom ((isoMk (Algebra.TensorProduct.comm R ↑X✝ ↑Y✝)).hom ≫ f✝ ▷ X✝)
Module: Mathlib.Algebra.Category.CommAlgCat.Monoidal

Goal state before:
pp: R : Type u
inst✝ : CommRing R
A B C D X✝ Y✝ Z✝ : CommAlgCat R
f✝ : Y✝ ⟶ Z✝
⊢ (Hom.hom (isoMk (Algebra.TensorProduct.comm R ↑X✝ ↑Z✝)).hom).comp
      (Algebra.TensorProduct.map (AlgHom.id R ↑X✝) (Hom.hom f✝)) =
    (Algebra.TensorProduct.map (Hom.hom f✝) (AlgHom.id R ↑X✝)).comp
      (Hom.hom (isoMk (Algebra.TensorProduct.comm R ↑X✝ ↑Y✝)).hom)

Tactic:
;

Goal state after:
R : Type u
inst✝ : CommRing R
A B C D X✝ Y✝ Z✝ : CommAlgCat R
f✝ : Y✝ ⟶ Z✝
⊢ (Hom.hom (isoMk (Algebra.TensorProduct.comm R ↑X✝ ↑Z✝)).hom).comp
      (Algebra.TensorProduct.map (AlgHom.id R ↑X✝) (Hom.hom f✝)) =
    (Algebra.TensorProduct.map (Hom.hom f✝) (AlgHom.id R ↑X✝)).comp
      (Hom.hom (isoMk (Algebra.TensorProduct.comm R ↑X✝ ↑Y✝)).hom)
Module: Mathlib.Algebra.Category.CommAlgCat.Monoidal

Goal state before:
pp: R : Type u
inst✝ : CommRing R
A B C D X✝ Y✝ : CommAlgCat R
f✝ : X✝ ⟶ Y✝
Z✝ : CommAlgCat R
⊢ f✝ ▷ Z✝ ≫ (isoMk (Algebra.TensorProduct.comm R ↑Y✝ ↑Z✝)).hom =
    (isoMk (Algebra.TensorProduct.comm R ↑X✝ ↑Z✝)).hom ≫ Z✝ ◁ f✝

Tactic:
;

Goal state after:
R : Type u
inst✝ : CommRing R
A B C D X✝ Y✝ : CommAlgCat R
f✝ : X✝ ⟶ Y✝
Z✝ : CommAlgCat R
⊢ f✝ ▷ Z✝ ≫ (isoMk (Algebra.TensorProduct.comm R ↑Y✝ ↑Z✝)).hom =
    (isoMk (Algebra.TensorProduct.comm R ↑X✝ ↑Z✝)).hom ≫ Z✝ ◁ f✝
Module: Mathlib.Algebra.Category.CommAlgCat.Monoidal

Goal state before:
pp: R : Type u
inst✝ : CommRing R
A B C D X✝ Y✝ : CommAlgCat R
f✝ : X✝ ⟶ Y✝
Z✝ : CommAlgCat R
⊢ Hom.hom (f✝ ▷ Z✝ ≫ (isoMk (Algebra.TensorProduct.comm R ↑Y✝ ↑Z✝)).hom) =
    Hom.hom ((isoMk (Algebra.TensorProduct.comm R ↑X✝ ↑Z✝)).hom ≫ Z✝ ◁ f✝)

Tactic:
;

Goal state after:
R : Type u
inst✝ : CommRing R
A B C D X✝ Y✝ : CommAlgCat R
f✝ : X✝ ⟶ Y✝
Z✝ : CommAlgCat R
⊢ Hom.hom (f✝ ▷ Z✝ ≫ (isoMk (Algebra.TensorProduct.comm R ↑Y✝ ↑Z✝)).hom) =
    Hom.hom ((isoMk (Algebra.TensorProduct.comm R ↑X✝ ↑Z✝)).hom ≫ Z✝ ◁ f✝)
Module: Mathlib.Algebra.Category.CommAlgCat.Monoidal

Goal state before:
pp: R : Type u
inst✝ : CommRing R
A B C D X✝ Y✝ : CommAlgCat R
f✝ : X✝ ⟶ Y✝
Z✝ : CommAlgCat R
⊢ (Hom.hom (isoMk (Algebra.TensorProduct.comm R ↑Y✝ ↑Z✝)).hom).comp
      (Algebra.TensorProduct.map (Hom.hom f✝) (AlgHom.id R ↑Z✝)) =
    (Algebra.TensorProduct.map (AlgHom.id R ↑Z✝) (Hom.hom f✝)).comp
      (Hom.hom (isoMk (Algebra.TensorProduct.comm R ↑X✝ ↑Z✝)).hom)

Tactic:
;

Goal state after:
R : Type u
inst✝ : CommRing R
A B C D X✝ Y✝ : CommAlgCat R
f✝ : X✝ ⟶ Y✝
Z✝ : CommAlgCat R
⊢ (Hom.hom (isoMk (Algebra.TensorProduct.comm R ↑Y✝ ↑Z✝)).hom).comp
      (Algebra.TensorProduct.map (Hom.hom f✝) (AlgHom.id R ↑Z✝)) =
    (Algebra.TensorProduct.map (AlgHom.id R ↑Z✝) (Hom.hom f✝)).comp
      (Hom.hom (isoMk (Algebra.TensorProduct.comm R ↑X✝ ↑Z✝)).hom)
Module: Mathlib.Algebra.Category.CommAlgCat.Monoidal

Goal state before:
pp: R : Type u
inst✝ : CommRing R
A B C D S T U : CommAlgCat R
⊢ Hom.hom ((α_ S T U).hom ≫ (isoMk (Algebra.TensorProduct.comm R ↑S ↑(T ⊗ U))).hom ≫ (α_ T U S).hom) =
    Hom.hom
      ((isoMk (Algebra.TensorProduct.comm R ↑S ↑T)).hom ▷ U ≫
        (α_ T S U).hom ≫ T ◁ (isoMk (Algebra.TensorProduct.comm R ↑S ↑U)).hom)

Tactic:
;

Goal state after:
R : Type u
inst✝ : CommRing R
A B C D S T U : CommAlgCat R
⊢ Hom.hom ((α_ S T U).hom ≫ (isoMk (Algebra.TensorProduct.comm R ↑S ↑(T ⊗ U))).hom ≫ (α_ T U S).hom) =
    Hom.hom
      ((isoMk (Algebra.TensorProduct.comm R ↑S ↑T)).hom ▷ U ≫
        (α_ T S U).hom ≫ T ◁ (isoMk (Algebra.TensorProduct.comm R ↑S ↑U)).hom)
Module: Mathlib.Algebra.Category.CommAlgCat.Monoidal

Goal state before:
pp: R : Type u
inst✝ : CommRing R
A B C D S T U : CommAlgCat R
⊢ ((↑(assoc R R R ↑T ↑U ↑S)).comp (Hom.hom (isoMk (Algebra.TensorProduct.comm R (↑S) (↑T ⊗[R] ↑U))).hom)).comp
      ↑(assoc R R R ↑S ↑T ↑U) =
    ((Algebra.TensorProduct.map (AlgHom.id R ↑T) (Hom.hom (isoMk (Algebra.TensorProduct.comm R ↑S ↑U)).hom)).comp
          ↑(assoc R R R ↑T ↑S ↑U)).comp
      (Algebra.TensorProduct.map (Hom.hom (isoMk (Algebra.TensorProduct.comm R ↑S ↑T)).hom) (AlgHom.id R ↑U))

Tactic:
;

Goal state after:
R : Type u
inst✝ : CommRing R
A B C D S T U : CommAlgCat R
⊢ ((↑(assoc R R R ↑T ↑U ↑S)).comp (Hom.hom (isoMk (Algebra.TensorProduct.comm R (↑S) (↑T ⊗[R] ↑U))).hom)).comp
      ↑(assoc R R R ↑S ↑T ↑U) =
    ((Algebra.TensorProduct.map (AlgHom.id R ↑T) (Hom.hom (isoMk (Algebra.TensorProduct.comm R ↑S ↑U)).hom)).comp
          ↑(assoc R R R ↑T ↑S ↑U)).comp
      (Algebra.TensorProduct.map (Hom.hom (isoMk (Algebra.TensorProduct.comm R ↑S ↑T)).hom) (AlgHom.id R ↑U))
Module: Mathlib.RingTheory.Coalgebra.Convolution

Goal state before:
pp: R : Type u_1
S : Type u_2
A : Type u_3
B : Type u_4
C : Type u_5
ι : Type u_6
inst✝⁷ : CommSemiring R
inst✝⁶ : NonUnitalSemiring A
inst✝⁵ : Module R A
inst✝⁴ : SMulCommClass R A A
inst✝³ : IsScalarTower R A A
inst✝² : AddCommMonoid C
inst✝¹ : Module R C
inst✝ : Coalgebra R C
f g h : WithConv (C →ₗ[R] A)
x✝ : C
⊢ (mul' R A ∘ₗ ((f * g).ofConv ⊗ₘ h.ofConv) ∘ₗ δ) x✝ =
    ((mul' R A ∘ₗ rTensor A (mul' R A)) ∘ₗ
        (((f.ofConv ⊗ₘ g.ofConv) ⊗ₘ h.ofConv) ∘ₗ ↑(TensorProduct.assoc R C C C).symm) ∘ₗ lTensor C δ ∘ₗ δ)
      x✝

Tactic:
;

Goal state after:
R : Type u_1
S : Type u_2
A : Type u_3
B : Type u_4
C : Type u_5
ι : Type u_6
inst✝⁷ : CommSemiring R
inst✝⁶ : NonUnitalSemiring A
inst✝⁵ : Module R A
inst✝⁴ : SMulCommClass R A A
inst✝³ : IsScalarTower R A A
inst✝² : AddCommMonoid C
inst✝¹ : Module R C
inst✝ : Coalgebra R C
f g h : WithConv (C →ₗ[R] A)
x✝ : C
⊢ (mul' R A ∘ₗ ((f * g).ofConv ⊗ₘ h.ofConv) ∘ₗ δ) x✝ =
    ((mul' R A ∘ₗ rTensor A (mul' R A)) ∘ₗ
        (((f.ofConv ⊗ₘ g.ofConv) ⊗ₘ h.ofConv) ∘ₗ ↑(TensorProduct.assoc R C C C).symm) ∘ₗ lTensor C δ ∘ₗ δ)
      x✝
Module: Mathlib.Algebra.Category.CommAlgCat.Monoidal

Goal state before:
pp: R : Type u
inst✝ : CommRing R
A B C D S T U : CommAlgCat R
⊢ Hom.hom ((α_ S T U).inv ≫ (isoMk (Algebra.TensorProduct.comm R ↑(S ⊗ T) ↑U)).hom ≫ (α_ U S T).inv) =
    Hom.hom
      (S ◁ (isoMk (Algebra.TensorProduct.comm R ↑T ↑U)).hom ≫
        (α_ S U T).inv ≫ (isoMk (Algebra.TensorProduct.comm R ↑S ↑U)).hom ▷ T)

Tactic:
;

Goal state after:
R : Type u
inst✝ : CommRing R
A B C D S T U : CommAlgCat R
⊢ Hom.hom ((α_ S T U).inv ≫ (isoMk (Algebra.TensorProduct.comm R ↑(S ⊗ T) ↑U)).hom ≫ (α_ U S T).inv) =
    Hom.hom
      (S ◁ (isoMk (Algebra.TensorProduct.comm R ↑T ↑U)).hom ≫
        (α_ S U T).inv ≫ (isoMk (Algebra.TensorProduct.comm R ↑S ↑U)).hom ▷ T)
Module: Mathlib.CategoryTheory.Monoidal.Cartesian.Basic

Goal state before:
pp: C✝ : Type u
inst✝⁴ : Category.{v, u} C✝
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : CartesianMonoidalCategory C
P : ObjectProperty C
inst✝¹ : P.IsClosedUnderLimitsOfShape (Discrete PEmpty.{1})
inst✝ : P.IsClosedUnderLimitsOfShape (Discrete WalkingPair)
x✝¹ : P.FullSubcategory
x✝ : x✝¹ ⟶ 𝟙_ P.FullSubcategory
⊢ x✝.hom = (ObjectProperty.homMk (toUnit x✝¹.obj)).hom

Tactic:
;

Goal state after:
C✝ : Type u
inst✝⁴ : Category.{v, u} C✝
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : CartesianMonoidalCategory C
P : ObjectProperty C
inst✝¹ : P.IsClosedUnderLimitsOfShape (Discrete PEmpty.{1})
inst✝ : P.IsClosedUnderLimitsOfShape (Discrete WalkingPair)
x✝¹ : P.FullSubcategory
x✝ : x✝¹ ⟶ 𝟙_ P.FullSubcategory
⊢ x✝.hom = (ObjectProperty.homMk (toUnit x✝¹.obj)).hom
Module: Mathlib.Algebra.Category.CommAlgCat.Monoidal

Goal state before:
pp: R : Type u
inst✝ : CommRing R
A B C D S T U : CommAlgCat R
⊢ ((↑(assoc R R R ↑U ↑S ↑T).symm).comp (Hom.hom (isoMk (Algebra.TensorProduct.comm R (↑S ⊗[R] ↑T) ↑U)).hom)).comp
      ↑(assoc R R R ↑S ↑T ↑U).symm =
    ((Algebra.TensorProduct.map (Hom.hom (isoMk (Algebra.TensorProduct.comm R ↑S ↑U)).hom) (AlgHom.id R ↑T)).comp
          ↑(assoc R R R ↑S ↑U ↑T).symm).comp
      (Algebra.TensorProduct.map (AlgHom.id R ↑S) (Hom.hom (isoMk (Algebra.TensorProduct.comm R ↑T ↑U)).hom))

Tactic:
;

Goal state after:
R : Type u
inst✝ : CommRing R
A B C D S T U : CommAlgCat R
⊢ ((↑(assoc R R R ↑U ↑S ↑T).symm).comp (Hom.hom (isoMk (Algebra.TensorProduct.comm R (↑S ⊗[R] ↑T) ↑U)).hom)).comp
      ↑(assoc R R R ↑S ↑T ↑U).symm =
    ((Algebra.TensorProduct.map (Hom.hom (isoMk (Algebra.TensorProduct.comm R ↑S ↑U)).hom) (AlgHom.id R ↑T)).comp
          ↑(assoc R R R ↑S ↑U ↑T).symm).comp
      (Algebra.TensorProduct.map (AlgHom.id R ↑S) (Hom.hom (isoMk (Algebra.TensorProduct.comm R ↑T ↑U)).hom))
Module: Mathlib.CategoryTheory.Monoidal.Cartesian.Basic

Goal state before:
pp: C✝ : Type u
inst✝⁴ : Category.{v, u} C✝
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : CartesianMonoidalCategory C
P : ObjectProperty C
inst✝¹ : P.IsClosedUnderLimitsOfShape (Discrete PEmpty.{1})
inst✝ : P.IsClosedUnderLimitsOfShape (Discrete WalkingPair)
X Y : P.FullSubcategory
⊢ (ObjectProperty.homMk (fst X.obj Y.obj)).hom =
    (X ◁ (IsTerminal.ofUniqueHom (fun X ↦ ObjectProperty.homMk (toUnit X.obj)) ⋯).from Y ≫ (ρ_ X).hom).hom

Tactic:
;

Goal state after:
C✝ : Type u
inst✝⁴ : Category.{v, u} C✝
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : CartesianMonoidalCategory C
P : ObjectProperty C
inst✝¹ : P.IsClosedUnderLimitsOfShape (Discrete PEmpty.{1})
inst✝ : P.IsClosedUnderLimitsOfShape (Discrete WalkingPair)
X Y : P.FullSubcategory
⊢ (ObjectProperty.homMk (fst X.obj Y.obj)).hom =
    (X ◁ (IsTerminal.ofUniqueHom (fun X ↦ ObjectProperty.homMk (toUnit X.obj)) ⋯).from Y ≫ (ρ_ X).hom).hom
Module: Mathlib.RingTheory.Coalgebra.Convolution

Goal state before:
pp: R : Type u_1
S : Type u_2
A : Type u_3
B : Type u_4
C : Type u_5
ι : Type u_6
inst✝⁷ : CommSemiring R
inst✝⁶ : NonUnitalSemiring A
inst✝⁵ : Module R A
inst✝⁴ : SMulCommClass R A A
inst✝³ : IsScalarTower R A A
inst✝² : AddCommMonoid C
inst✝¹ : Module R C
inst✝ : Coalgebra R C
f g h : WithConv (C →ₗ[R] A)
x✝ : C
⊢ ((mul' R A ∘ₗ lTensor A (mul' R A)) ∘ₗ (f.ofConv ⊗ₘ (g.ofConv ⊗ₘ h.ofConv)) ∘ₗ lTensor C δ ∘ₗ δ) x✝ =
    (mul' R A ∘ₗ (f.ofConv ⊗ₘ mul' R A ∘ₗ (g.ofConv ⊗ₘ h.ofConv) ∘ₗ δ) ∘ₗ δ) x✝

Tactic:
;

Goal state after:
R : Type u_1
S : Type u_2
A : Type u_3
B : Type u_4
C : Type u_5
ι : Type u_6
inst✝⁷ : CommSemiring R
inst✝⁶ : NonUnitalSemiring A
inst✝⁵ : Module R A
inst✝⁴ : SMulCommClass R A A
inst✝³ : IsScalarTower R A A
inst✝² : AddCommMonoid C
inst✝¹ : Module R C
inst✝ : Coalgebra R C
f g h : WithConv (C →ₗ[R] A)
x✝ : C
⊢ ((mul' R A ∘ₗ lTensor A (mul' R A)) ∘ₗ (f.ofConv ⊗ₘ (g.ofConv ⊗ₘ h.ofConv)) ∘ₗ lTensor C δ ∘ₗ δ) x✝ =
    (mul' R A ∘ₗ (f.ofConv ⊗ₘ mul' R A ∘ₗ (g.ofConv ⊗ₘ h.ofConv) ∘ₗ δ) ∘ₗ δ) x✝
Module: Mathlib.CategoryTheory.Monoidal.Cartesian.Basic

Goal state before:
pp: C✝ : Type u
inst✝⁴ : Category.{v, u} C✝
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : CartesianMonoidalCategory C
P : ObjectProperty C
inst✝¹ : P.IsClosedUnderLimitsOfShape (Discrete PEmpty.{1})
inst✝ : P.IsClosedUnderLimitsOfShape (Discrete WalkingPair)
X Y : P.FullSubcategory
⊢ (ObjectProperty.homMk (snd X.obj Y.obj)).hom =
    ((IsTerminal.ofUniqueHom (fun X ↦ ObjectProperty.homMk (toUnit X.obj)) ⋯).from X ▷ Y ≫ (λ_ Y).hom).hom

Tactic:
;

Goal state after:
C✝ : Type u
inst✝⁴ : Category.{v, u} C✝
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : CartesianMonoidalCategory C
P : ObjectProperty C
inst✝¹ : P.IsClosedUnderLimitsOfShape (Discrete PEmpty.{1})
inst✝ : P.IsClosedUnderLimitsOfShape (Discrete WalkingPair)
X Y : P.FullSubcategory
⊢ (ObjectProperty.homMk (snd X.obj Y.obj)).hom =
    ((IsTerminal.ofUniqueHom (fun X ↦ ObjectProperty.homMk (toUnit X.obj)) ⋯).from X ▷ Y ≫ (λ_ Y).hom).hom
Module: Mathlib.RingTheory.Coalgebra.Convolution

Goal state before:
pp: R : Type u_1
S : Type u_2
A : Type u_3
B : Type u_4
C : Type u_5
ι : Type u_6
inst✝⁷ : CommSemiring R
inst✝⁶ : Semiring A
inst✝⁵ : Algebra R A
inst✝⁴ : Semiring B
inst✝³ : Algebra R B
inst✝² : AddCommMonoid C
inst✝¹ : Module R C
inst✝ : Coalgebra R C
f : WithConv (C →ₗ[R] A)
x✝ : C
⊢ (1 * f).ofConv x✝ = f.ofConv x✝

Tactic:
;

Goal state after:
R : Type u_1
S : Type u_2
A : Type u_3
B : Type u_4
C : Type u_5
ι : Type u_6
inst✝⁷ : CommSemiring R
inst✝⁶ : Semiring A
inst✝⁵ : Algebra R A
inst✝⁴ : Semiring B
inst✝³ : Algebra R B
inst✝² : AddCommMonoid C
inst✝¹ : Module R C
inst✝ : Coalgebra R C
f : WithConv (C →ₗ[R] A)
x✝ : C
⊢ (1 * f).ofConv x✝ = f.ofConv x✝
Module: Mathlib.RingTheory.Coalgebra.Convolution

Goal state before:
pp: R : Type u_1
S : Type u_2
A : Type u_3
B : Type u_4
C : Type u_5
ι : Type u_6
inst✝⁷ : CommSemiring R
inst✝⁶ : Semiring A
inst✝⁵ : Algebra R A
inst✝⁴ : Semiring B
inst✝³ : Algebra R B
inst✝² : AddCommMonoid C
inst✝¹ : Module R C
inst✝ : Coalgebra R C
f : WithConv (C →ₗ[R] A)
x✝ : C
⊢ (f * 1).ofConv x✝ = f.ofConv x✝

Tactic:
;

Goal state after:
R : Type u_1
S : Type u_2
A : Type u_3
B : Type u_4
C : Type u_5
ι : Type u_6
inst✝⁷ : CommSemiring R
inst✝⁶ : Semiring A
inst✝⁵ : Algebra R A
inst✝⁴ : Semiring B
inst✝³ : Algebra R B
inst✝² : AddCommMonoid C
inst✝¹ : Module R C
inst✝ : Coalgebra R C
f : WithConv (C →ₗ[R] A)
x✝ : C
⊢ (f * 1).ofConv x✝ = f.ofConv x✝
Module: Mathlib.RingTheory.Coalgebra.Convolution

Goal state before:
pp: R : Type u_1
S : Type u_2
A : Type u_3
B : Type u_4
C : Type u_5
ι : Type u_6
inst✝⁶ : CommSemiring R
inst✝⁵ : CommSemiring A
inst✝⁴ : AddCommMonoid C
inst✝³ : Algebra R A
inst✝² : Module R C
inst✝¹ : Coalgebra R C
inst✝ : IsCocomm R C
f g : WithConv (C →ₗ[R] A)
x : C
⊢ (f * g).ofConv x = (g * f).ofConv x

Tactic:
;

Goal state after:
R : Type u_1
S : Type u_2
A : Type u_3
B : Type u_4
C : Type u_5
ι : Type u_6
inst✝⁶ : CommSemiring R
inst✝⁵ : CommSemiring A
inst✝⁴ : AddCommMonoid C
inst✝³ : Algebra R A
inst✝² : Module R C
inst✝¹ : Coalgebra R C
inst✝ : IsCocomm R C
f g : WithConv (C →ₗ[R] A)
x : C
⊢ (f * g).ofConv x = (g * f).ofConv x
Module: Mathlib.Algebra.Category.CommAlgCat.Monoidal

Goal state before:
pp: R : Type u
inst✝ : CommRing R
A B C D : CommAlgCat R
S T : (CommAlgCat R)ᵒᵖ
x : ↑(unop (op ((pair (unop S) (unop T)).obj { as := WalkingPair.left })))
⊢ (Hom.hom ((unop S).binaryCofan (unop T)).inl.op.unop) x =
    (Hom.hom (S ◁ (terminalOpOfInitial isInitialSelf).from T ≫ (ρ_ S).hom).unop) x

Tactic:
;

Goal state after:
R : Type u
inst✝ : CommRing R
A B C D : CommAlgCat R
S T : (CommAlgCat R)ᵒᵖ
x : ↑(unop (op ((pair (unop S) (unop T)).obj { as := WalkingPair.left })))
⊢ (Hom.hom ((unop S).binaryCofan (unop T)).inl.op.unop) x =
    (Hom.hom (S ◁ (terminalOpOfInitial isInitialSelf).from T ≫ (ρ_ S).hom).unop) x
Module: Mathlib.Algebra.Category.CommAlgCat.Monoidal

Goal state before:
pp: R : Type u
inst✝ : CommRing R
A B C D : CommAlgCat R
S T : (CommAlgCat R)ᵒᵖ
x : ↑(unop (op ((pair (unop S) (unop T)).obj { as := WalkingPair.left })))
⊢ x ⊗ₜ[R] 1 = x ⊗ₜ[R] (algebraMap R ↑(unop T)) 1

Tactic:
;

Goal state after:
R : Type u
inst✝ : CommRing R
A B C D : CommAlgCat R
S T : (CommAlgCat R)ᵒᵖ
x : ↑(unop (op ((pair (unop S) (unop T)).obj { as := WalkingPair.left })))
⊢ x ⊗ₜ[R] 1 = x ⊗ₜ[R] (algebraMap R ↑(unop T)) 1
Module: Mathlib.Algebra.Category.CommAlgCat.Monoidal

Goal state before:
pp: R : Type u
inst✝ : CommRing R
A B C D : CommAlgCat R
S T : (CommAlgCat R)ᵒᵖ
x : ↑(unop (op ((pair (unop S) (unop T)).obj { as := WalkingPair.right })))
⊢ (Hom.hom ((unop S).binaryCofan (unop T)).inr.op.unop) x =
    (Hom.hom ((terminalOpOfInitial isInitialSelf).from S ▷ T ≫ (λ_ T).hom).unop) x

Tactic:
;

Goal state after:
R : Type u
inst✝ : CommRing R
A B C D : CommAlgCat R
S T : (CommAlgCat R)ᵒᵖ
x : ↑(unop (op ((pair (unop S) (unop T)).obj { as := WalkingPair.right })))
⊢ (Hom.hom ((unop S).binaryCofan (unop T)).inr.op.unop) x =
    (Hom.hom ((terminalOpOfInitial isInitialSelf).from S ▷ T ≫ (λ_ T).hom).unop) x
Module: Mathlib.Algebra.Category.CommAlgCat.Monoidal

Goal state before:
pp: R : Type u
inst✝ : CommRing R
A B C D : CommAlgCat R
S T : (CommAlgCat R)ᵒᵖ
x : ↑(unop (op ((pair (unop S) (unop T)).obj { as := WalkingPair.right })))
⊢ 1 ⊗ₜ[R] x = (algebraMap R ↑(unop S)) 1 ⊗ₜ[R] x

Tactic:
;

Goal state after:
R : Type u
inst✝ : CommRing R
A B C D : CommAlgCat R
S T : (CommAlgCat R)ᵒᵖ
x : ↑(unop (op ((pair (unop S) (unop T)).obj { as := WalkingPair.right })))
⊢ 1 ⊗ₜ[R] x = (algebraMap R ↑(unop S)) 1 ⊗ₜ[R] x
Module: Mathlib.CategoryTheory.Monoidal.Cartesian.Basic

Goal state before:
pp: case h_snd
C : Type u₁
inst✝⁷ : Category.{v₁, u₁} C
inst✝⁶ : CartesianMonoidalCategory C
D : Type u₂
inst✝⁵ : Category.{v₂, u₂} D
inst✝⁴ : CartesianMonoidalCategory D
E : Type u₃
inst✝³ : Category.{v₃, u₃} E
inst✝² : CartesianMonoidalCategory E
F✝ : C ⥤ D
G : D ⥤ E
X Y Z : C
inst✝¹ : F✝.OplaxMonoidal
inst✝ : PreservesFiniteProducts F✝
F : C ⥤ D
x✝ : C
⊢ (λ_ (F.obj x✝)).inv ≫ snd (𝟙_ D) (F.obj x✝) =
    (F.map (λ_ x✝).inv ≫
        CartesianMonoidalCategory.prodComparison F (𝟙_ C) x✝ ≫
          CartesianMonoidalCategory.terminalComparison F ▷ F.obj x✝) ≫
      snd (𝟙_ D) (F.obj x✝)

Tactic:
;

Goal state after:
case h_snd
C : Type u₁
inst✝⁷ : Category.{v₁, u₁} C
inst✝⁶ : CartesianMonoidalCategory C
D : Type u₂
inst✝⁵ : Category.{v₂, u₂} D
inst✝⁴ : CartesianMonoidalCategory D
E : Type u₃
inst✝³ : Category.{v₃, u₃} E
inst✝² : CartesianMonoidalCategory E
F✝ : C ⥤ D
G : D ⥤ E
X Y Z : C
inst✝¹ : F✝.OplaxMonoidal
inst✝ : PreservesFiniteProducts F✝
F : C ⥤ D
x✝ : C
⊢ (λ_ (F.obj x✝)).inv ≫ snd (𝟙_ D) (F.obj x✝) =
    (F.map (λ_ x✝).inv ≫
        CartesianMonoidalCategory.prodComparison F (𝟙_ C) x✝ ≫
          CartesianMonoidalCategory.terminalComparison F ▷ F.obj x✝) ≫
      snd (𝟙_ D) (F.obj x✝)
Module: Mathlib.CategoryTheory.Monoidal.Cartesian.Basic

Goal state before:
pp: case h_fst
C : Type u₁
inst✝⁷ : Category.{v₁, u₁} C
inst✝⁶ : CartesianMonoidalCategory C
D : Type u₂
inst✝⁵ : Category.{v₂, u₂} D
inst✝⁴ : CartesianMonoidalCategory D
E : Type u₃
inst✝³ : Category.{v₃, u₃} E
inst✝² : CartesianMonoidalCategory E
F✝ : C ⥤ D
G : D ⥤ E
X Y Z : C
inst✝¹ : F✝.OplaxMonoidal
inst✝ : PreservesFiniteProducts F✝
F : C ⥤ D
x✝ : C
⊢ (ρ_ (F.obj x✝)).inv ≫ fst (F.obj x✝) (𝟙_ D) =
    (F.map (ρ_ x✝).inv ≫
        CartesianMonoidalCategory.prodComparison F x✝ (𝟙_ C) ≫
          F.obj x✝ ◁ CartesianMonoidalCategory.terminalComparison F) ≫
      fst (F.obj x✝) (𝟙_ D)

Tactic:
;

Goal state after:
case h_fst
C : Type u₁
inst✝⁷ : Category.{v₁, u₁} C
inst✝⁶ : CartesianMonoidalCategory C
D : Type u₂
inst✝⁵ : Category.{v₂, u₂} D
inst✝⁴ : CartesianMonoidalCategory D
E : Type u₃
inst✝³ : Category.{v₃, u₃} E
inst✝² : CartesianMonoidalCategory E
F✝ : C ⥤ D
G : D ⥤ E
X Y Z : C
inst✝¹ : F✝.OplaxMonoidal
inst✝ : PreservesFiniteProducts F✝
F : C ⥤ D
x✝ : C
⊢ (ρ_ (F.obj x✝)).inv ≫ fst (F.obj x✝) (𝟙_ D) =
    (F.map (ρ_ x✝).inv ≫
        CartesianMonoidalCategory.prodComparison F x✝ (𝟙_ C) ≫
          F.obj x✝ ◁ CartesianMonoidalCategory.terminalComparison F) ≫
      fst (F.obj x✝) (𝟙_ D)
Module: Mathlib.CategoryTheory.Monoidal.Cartesian.Basic

Goal state before:
pp: case δ
C : Type u₁
inst✝⁷ : Category.{v₁, u₁} C
inst✝⁶ : CartesianMonoidalCategory C
D : Type u₂
inst✝⁵ : Category.{v₂, u₂} D
inst✝⁴ : CartesianMonoidalCategory D
E : Type u₃
inst✝³ : Category.{v₃, u₃} E
inst✝² : CartesianMonoidalCategory E
F : C ⥤ D
G : D ⥤ E
X Y Z : C
inst✝¹ : F.OplaxMonoidal
inst✝ : PreservesFiniteProducts F
a b : F.OplaxMonoidal
x✝ : C
⊢ δ F x✝ = δ F x✝

Tactic:
;

Goal state after:
case δ
C : Type u₁
inst✝⁷ : Category.{v₁, u₁} C
inst✝⁶ : CartesianMonoidalCategory C
D : Type u₂
inst✝⁵ : Category.{v₂, u₂} D
inst✝⁴ : CartesianMonoidalCategory D
E : Type u₃
inst✝³ : Category.{v₃, u₃} E
inst✝² : CartesianMonoidalCategory E
F : C ⥤ D
G : D ⥤ E
X Y Z : C
inst✝¹ : F.OplaxMonoidal
inst✝ : PreservesFiniteProducts F
a b : F.OplaxMonoidal
x✝ : C
⊢ δ F x✝ = δ F x✝
Module: Mathlib.CategoryTheory.Monoidal.Cartesian.Basic

Goal state before:
pp: case δ
C : Type u₁
inst✝⁷ : Category.{v₁, u₁} C
inst✝⁶ : CartesianMonoidalCategory C
D : Type u₂
inst✝⁵ : Category.{v₂, u₂} D
inst✝⁴ : CartesianMonoidalCategory D
E : Type u₃
inst✝³ : Category.{v₃, u₃} E
inst✝² : CartesianMonoidalCategory E
F : C ⥤ D
G : D ⥤ E
X Y Z : C
inst✝¹ : F.OplaxMonoidal
inst✝ : PreservesFiniteProducts F
a b : F.OplaxMonoidal
x✝¹ x✝ : C
⊢ δ F x✝¹ x✝ = δ F x✝¹ x✝

Tactic:
;

Goal state after:
case δ
C : Type u₁
inst✝⁷ : Category.{v₁, u₁} C
inst✝⁶ : CartesianMonoidalCategory C
D : Type u₂
inst✝⁵ : Category.{v₂, u₂} D
inst✝⁴ : CartesianMonoidalCategory D
E : Type u₃
inst✝³ : Category.{v₃, u₃} E
inst✝² : CartesianMonoidalCategory E
F : C ⥤ D
G : D ⥤ E
X Y Z : C
inst✝¹ : F.OplaxMonoidal
inst✝ : PreservesFiniteProducts F
a b : F.OplaxMonoidal
x✝¹ x✝ : C
⊢ δ F x✝¹ x✝ = δ F x✝¹ x✝
Module: Mathlib.CategoryTheory.Monoidal.Cartesian.Basic

Goal state before:
pp: C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
inst✝³ : CartesianMonoidalCategory C
D : Type u₂
inst✝² : Category.{v₂, u₂} D
inst✝¹ : CartesianMonoidalCategory D
F : C ⥤ D
inst✝ : F.Monoidal
X : C
⊢ (toUnit (F.obj X) ≫ ε F) ≫ (εIso F).inv = F.map (toUnit X) ≫ (εIso F).inv

Tactic:
;

Goal state after:
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
inst✝³ : CartesianMonoidalCategory C
D : Type u₂
inst✝² : Category.{v₂, u₂} D
inst✝¹ : CartesianMonoidalCategory D
F : C ⥤ D
inst✝ : F.Monoidal
X : C
⊢ (toUnit (F.obj X) ≫ ε F) ≫ (εIso F).inv = F.map (toUnit X) ≫ (εIso F).inv
Module: Mathlib.CategoryTheory.Monoidal.Cartesian.Basic

Goal state before:
pp: C : Type u₁
inst✝⁸ : Category.{v₁, u₁} C
inst✝⁷ : CartesianMonoidalCategory C
D : Type u₂
inst✝⁶ : Category.{v₂, u₂} D
inst✝⁵ : CartesianMonoidalCategory D
E : Type u₃
inst✝⁴ : Category.{v₃, u₃} E
inst✝³ : CartesianMonoidalCategory E
F : C ⥤ D
G : D ⥤ E
inst✝² : F.Monoidal
inst✝¹ : G.Monoidal
inst✝ : (F ⋙ G).Monoidal
X Y : C
⊢ μ (F ⋙ G) X Y ≫ (μIso (F ⋙ G) X Y).inv = (μ G (F.obj X) (F.obj Y) ≫ G.map (μ F X Y)) ≫ (μIso (F ⋙ G) X Y).inv

Tactic:
;

Goal state after:
C : Type u₁
inst✝⁸ : Category.{v₁, u₁} C
inst✝⁷ : CartesianMonoidalCategory C
D : Type u₂
inst✝⁶ : Category.{v₂, u₂} D
inst✝⁵ : CartesianMonoidalCategory D
E : Type u₃
inst✝⁴ : Category.{v₃, u₃} E
inst✝³ : CartesianMonoidalCategory E
F : C ⥤ D
G : D ⥤ E
inst✝² : F.Monoidal
inst✝¹ : G.Monoidal
inst✝ : (F ⋙ G).Monoidal
X Y : C
⊢ μ (F ⋙ G) X Y ≫ (μIso (F ⋙ G) X Y).inv = (μ G (F.obj X) (F.obj Y) ≫ G.map (μ F X Y)) ≫ (μIso (F ⋙ G) X Y).inv
Module: Mathlib.CategoryTheory.Monoidal.Cartesian.Basic

Goal state before:
pp: C : Type u₁
inst✝⁵ : Category.{v₁, u₁} C
inst✝⁴ : CartesianMonoidalCategory C
D : Type u₂
inst✝³ : Category.{v₂, u₂} D
inst✝² : CartesianMonoidalCategory D
F : C ⥤ D
inst✝¹ : F.Monoidal
inst✝ : PreservesFiniteProducts F
⊢ (εIso F).symm.inv = (preservesTerminalIso F).inv

Tactic:
;

Goal state after:
C : Type u₁
inst✝⁵ : Category.{v₁, u₁} C
inst✝⁴ : CartesianMonoidalCategory C
D : Type u₂
inst✝³ : Category.{v₂, u₂} D
inst✝² : CartesianMonoidalCategory D
F : C ⥤ D
inst✝¹ : F.Monoidal
inst✝ : PreservesFiniteProducts F
⊢ (εIso F).symm.inv = (preservesTerminalIso F).inv
Module: Mathlib.CategoryTheory.Monoidal.Cartesian.Basic

Goal state before:
pp: C : Type u₁
inst✝⁵ : Category.{v₁, u₁} C
inst✝⁴ : CartesianMonoidalCategory C
D : Type u₂
inst✝³ : Category.{v₂, u₂} D
inst✝² : CartesianMonoidalCategory D
F : C ⥤ D
inst✝¹ : F.Monoidal
inst✝ : PreservesFiniteProducts F
⊢ (εIso F).symm = preservesTerminalIso F

Tactic:
;

Goal state after:
C : Type u₁
inst✝⁵ : Category.{v₁, u₁} C
inst✝⁴ : CartesianMonoidalCategory C
D : Type u₂
inst✝³ : Category.{v₂, u₂} D
inst✝² : CartesianMonoidalCategory D
F : C ⥤ D
inst✝¹ : F.Monoidal
inst✝ : PreservesFiniteProducts F
⊢ (εIso F).symm = preservesTerminalIso F