Theorem: PresheafOfModules.Sheafify.mul_smul
File: Mathlib/Algebra/Category/ModuleCat/Presheaf/Sheafify.lean

Proof steps:
state_before: C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
J : GrothendieckTopology C
R₀ : Cᵒᵖ ⥤ RingCat
R : Sheaf J RingCat
α : R₀ ⟶ R.val
inst✝³ : Presheaf.IsLocallyInjective J α
inst✝² : Presheaf.IsLocallySurjective J α
M₀ : PresheafOfModules R₀
A : Sheaf J AddCommGrp
φ : M₀.presheaf ⟶ A.val
inst✝¹ : Presheaf.IsLocallyInjective J φ
inst✝ : Presheaf.IsLocallySurjective J φ
X Y : Cᵒᵖ
π : X ⟶ Y
r r' : ↑(R.val.obj X)
m m' : ↑(A.val.obj X)
⊢ smul α φ (r * r') m = smul α φ r (smul α φ r' m)
state_after: C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
J : GrothendieckTopology C
R₀ : Cᵒᵖ ⥤ RingCat
R : Sheaf J RingCat
α : R₀ ⟶ R.val
inst✝³ : Presheaf.IsLocallyInjective J α
inst✝² : Presheaf.IsLocallySurjective J α
M₀ : PresheafOfModules R₀
A : Sheaf J AddCommGrp
φ : M₀.presheaf ⟶ A.val
inst✝¹ : Presheaf.IsLocallyInjective J φ
inst✝ : Presheaf.IsLocallySurjective J φ
X Y : Cᵒᵖ
π : X ⟶ Y
r r' : ↑(R.val.obj X)
m m' : ↑(A.val.obj X)
S : Sieve (Opposite.unop X) := Presheaf.imageSieve α r ⊓ Presheaf.imageSieve α r' ⊓ Presheaf.imageSieve φ m
⊢ smul α φ (r * r') m = smul α φ r (smul α φ r' m)
tactic: let S := Presheaf.imageSieve α r ⊓ Presheaf.imageSieve α r' ⊓ Presheaf.imageSieve φ m
state_before: C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
J : GrothendieckTopology C
R₀ : Cᵒᵖ ⥤ RingCat
R : Sheaf J RingCat
α : R₀ ⟶ R.val
inst✝³ : Presheaf.IsLocallyInjective J α
inst✝² : Presheaf.IsLocallySurjective J α
M₀ : PresheafOfModules R₀
A : Sheaf J AddCommGrp
φ : M₀.presheaf ⟶ A.val
inst✝¹ : Presheaf.IsLocallyInjective J φ
inst✝ : Presheaf.IsLocallySurjective J φ
X Y : Cᵒᵖ
π : X ⟶ Y
r r' : ↑(R.val.obj X)
m m' : ↑(A.val.obj X)
S : Sieve (Opposite.unop X) := Presheaf.imageSieve α r ⊓ Presheaf.imageSieve α r' ⊓ Presheaf.imageSieve φ m
⊢ smul α φ (r * r') m = smul α φ r (smul α φ r' m)
state_after: C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
J : GrothendieckTopology C
R₀ : Cᵒᵖ ⥤ RingCat
R : Sheaf J RingCat
α : R₀ ⟶ R.val
inst✝³ : Presheaf.IsLocallyInjective J α
inst✝² : Presheaf.IsLocallySurjective J α
M₀ : PresheafOfModules R₀
A : Sheaf J AddCommGrp
φ : M₀.presheaf ⟶ A.val
inst✝¹ : Presheaf.IsLocallyInjective J φ
inst✝ : Presheaf.IsLocallySurjective J φ
X Y : Cᵒᵖ
π : X ⟶ Y
r r' : ↑(R.val.obj X)
m m' : ↑(A.val.obj X)
S : Sieve (Opposite.unop X) := Presheaf.imageSieve α r ⊓ Presheaf.imageSieve α r' ⊓ Presheaf.imageSieve φ m
hS : S ∈ J.sieves (Opposite.unop X)
⊢ smul α φ (r * r') m = smul α φ r (smul α φ r' m)
tactic: have hS : S ∈ J X.unop := by
    refine J.intersection_covering (J.intersection_covering ?_ ?_) ?_
    all_goals apply Presheaf.imageSieve_mem
state_before: C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
J : GrothendieckTopology C
R₀ : Cᵒᵖ ⥤ RingCat
R : Sheaf J RingCat
α : R₀ ⟶ R.val
inst✝³ : Presheaf.IsLocallyInjective J α
inst✝² : Presheaf.IsLocallySurjective J α
M₀ : PresheafOfModules R₀
A : Sheaf J AddCommGrp
φ : M₀.presheaf ⟶ A.val
inst✝¹ : Presheaf.IsLocallyInjective J φ
inst✝ : Presheaf.IsLocallySurjective J φ
X Y : Cᵒᵖ
π : X ⟶ Y
r r' : ↑(R.val.obj X)
m m' : ↑(A.val.obj X)
S : Sieve (Opposite.unop X) := Presheaf.imageSieve α r ⊓ Presheaf.imageSieve α r' ⊓ Presheaf.imageSieve φ m
hS : S ∈ J.sieves (Opposite.unop X)
⊢ smul α φ (r * r') m = smul α φ r (smul α φ r' m)
state_after: case a
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
J : GrothendieckTopology C
R₀ : Cᵒᵖ ⥤ RingCat
R : Sheaf J RingCat
α : R₀ ⟶ R.val
inst✝³ : Presheaf.IsLocallyInjective J α
inst✝² : Presheaf.IsLocallySurjective J α
M₀ : PresheafOfModules R₀
A : Sheaf J AddCommGrp
φ : M₀.presheaf ⟶ A.val
inst✝¹ : Presheaf.IsLocallyInjective J φ
inst✝ : Presheaf.IsLocallySurjective J φ
X Y : Cᵒᵖ
π : X ⟶ Y
r r' : ↑(R.val.obj X)
m m' : ↑(A.val.obj X)
S : Sieve (Opposite.unop X) := Presheaf.imageSieve α r ⊓ Presheaf.imageSieve α r' ⊓ Presheaf.imageSieve φ m
hS : S ∈ J.sieves (Opposite.unop X)
⊢ ∀ (Y : C) (f : Y ⟶ Opposite.unop X),
    S.arrows f → (A.val.map f.op) (smul α φ (r * r') m) = (A.val.map f.op) (smul α φ r (smul α φ r' m))
tactic: apply A.isSeparated _ _ hS
state_before: case a
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
J : GrothendieckTopology C
R₀ : Cᵒᵖ ⥤ RingCat
R : Sheaf J RingCat
α : R₀ ⟶ R.val
inst✝³ : Presheaf.IsLocallyInjective J α
inst✝² : Presheaf.IsLocallySurjective J α
M₀ : PresheafOfModules R₀
A : Sheaf J AddCommGrp
φ : M₀.presheaf ⟶ A.val
inst✝¹ : Presheaf.IsLocallyInjective J φ
inst✝ : Presheaf.IsLocallySurjective J φ
X Y : Cᵒᵖ
π : X ⟶ Y
r r' : ↑(R.val.obj X)
m m' : ↑(A.val.obj X)
S : Sieve (Opposite.unop X) := Presheaf.imageSieve α r ⊓ Presheaf.imageSieve α r' ⊓ Presheaf.imageSieve φ m
hS : S ∈ J.sieves (Opposite.unop X)
⊢ ∀ (Y : C) (f : Y ⟶ Opposite.unop X),
    S.arrows f → (A.val.map f.op) (smul α φ (r * r') m) = (A.val.map f.op) (smul α φ r (smul α φ r' m))
state_after: case a.intro.intro.intro.intro.intro
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
J : GrothendieckTopology C
R₀ : Cᵒᵖ ⥤ RingCat
R : Sheaf J RingCat
α : R₀ ⟶ R.val
inst✝³ : Presheaf.IsLocallyInjective J α
inst✝² : Presheaf.IsLocallySurjective J α
M₀ : PresheafOfModules R₀
A : Sheaf J AddCommGrp
φ : M₀.presheaf ⟶ A.val
inst✝¹ : Presheaf.IsLocallyInjective J φ
inst✝ : Presheaf.IsLocallySurjective J φ
X Y✝ : Cᵒᵖ
π : X ⟶ Y✝
r r' : ↑(R.val.obj X)
m m' : ↑(A.val.obj X)
S : Sieve (Opposite.unop X) := Presheaf.imageSieve α r ⊓ Presheaf.imageSieve α r' ⊓ Presheaf.imageSieve φ m
hS : S ∈ J.sieves (Opposite.unop X)
Y : C
f : Y ⟶ Opposite.unop X
r₀ : ↑(R₀.obj (Opposite.op Y))
hr₀ : (α.app (Opposite.op Y)) r₀ = (R.val.map f.op) r
r₀' : ↑(R₀.obj (Opposite.op Y))
hr₀' : (α.app (Opposite.op Y)) r₀' = (R.val.map f.op) r'
m₀ : ↑(M₀.presheaf.obj (Opposite.op Y))
hm₀ : (φ.app (Opposite.op Y)) m₀ = (A.val.map f.op) m
⊢ (A.val.map f.op) (smul α φ (r * r') m) = (A.val.map f.op) (smul α φ r (smul α φ r' m))
tactic: rintro Y f ⟨⟨⟨r₀ : R₀.obj _, hr₀⟩, ⟨r₀' : R₀.obj _, hr₀'⟩⟩, ⟨m₀ : M₀.presheaf.obj _, hm₀⟩⟩
state_before: case a.intro.intro.intro.intro.intro
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
J : GrothendieckTopology C
R₀ : Cᵒᵖ ⥤ RingCat
R : Sheaf J RingCat
α : R₀ ⟶ R.val
inst✝³ : Presheaf.IsLocallyInjective J α
inst✝² : Presheaf.IsLocallySurjective J α
M₀ : PresheafOfModules R₀
A : Sheaf J AddCommGrp
φ : M₀.presheaf ⟶ A.val
inst✝¹ : Presheaf.IsLocallyInjective J φ
inst✝ : Presheaf.IsLocallySurjective J φ
X Y✝ : Cᵒᵖ
π : X ⟶ Y✝
r r' : ↑(R.val.obj X)
m m' : ↑(A.val.obj X)
S : Sieve (Opposite.unop X) := Presheaf.imageSieve α r ⊓ Presheaf.imageSieve α r' ⊓ Presheaf.imageSieve φ m
hS : S ∈ J.sieves (Opposite.unop X)
Y : C
f : Y ⟶ Opposite.unop X
r₀ : ↑(R₀.obj (Opposite.op Y))
hr₀ : (α.app (Opposite.op Y)) r₀ = (R.val.map f.op) r
r₀' : ↑(R₀.obj (Opposite.op Y))
hr₀' : (α.app (Opposite.op Y)) r₀' = (R.val.map f.op) r'
m₀ : ↑(M₀.presheaf.obj (Opposite.op Y))
hm₀ : (φ.app (Opposite.op Y)) m₀ = (A.val.map f.op) m
⊢ (A.val.map f.op) (smul α φ (r * r') m) = (A.val.map f.op) (smul α φ r (smul α φ r' m))
state_after: no goals
tactic: erw [map_smul_eq α φ (r * r') m f.op (r₀ * r₀')
    (by rw [map_mul, map_mul, hr₀, hr₀']) m₀ hm₀, mul_smul,
    map_smul_eq α φ r (smul α φ r' m) f.op r₀ hr₀ (r₀' • m₀)
      (map_smul_eq α φ r' m f.op r₀' hr₀' m₀ hm₀).symm]