File: Mathlib/Algebra/Homology/ShortComplex/Linear.lean

full_name: CategoryTheory.ShortComplex.instSMulHom
code: instance : SMul R (S₁ ⟶ S₂) 
goal: SMul R (S₁ ⟶ S₂)

full_name: CategoryTheory.ShortComplex.smul_τ₁
code: @[simp] lemma smul_τ₁ (a : R) (φ : S₁ ⟶ S₂) : (a • φ).τ₁ = a • φ.τ₁ 
goal: (a • φ).τ₁ = a • φ.τ₁

full_name: CategoryTheory.ShortComplex.smul_τ₂
code: @[simp] lemma smul_τ₂ (a : R) (φ : S₁ ⟶ S₂) : (a • φ).τ₂ = a • φ.τ₂ 
goal: (a • φ).τ₂ = a • φ.τ₂

full_name: CategoryTheory.ShortComplex.smul_τ₃
code: @[simp] lemma smul_τ₃ (a : R) (φ : S₁ ⟶ S₂) : (a • φ).τ₃ = a • φ.τ₃ 
goal: (a • φ).τ₃ = a • φ.τ₃

full_name: CategoryTheory.ShortComplex.instModuleHom
code: instance : Module R (S₁ ⟶ S₂) 
goal: Module R (S₁ ⟶ S₂)

full_name: CategoryTheory.ShortComplex.instLinear
code: instance : Linear R (ShortComplex C) 
goal: CategoryTheory.Linear R (CategoryTheory.ShortComplex C)

full_name: CategoryTheory.ShortComplex.LeftHomologyMapData.smul_φH
code: /-- Given a left homology map data for morphism `φ`, this is the induced left homology
map data for `a • φ`. -/
@[simps]
def smul (a : R) : LeftHomologyMapData (a • φ) h₁ h₂ 
goal: (γ.smul a).φH = a • γ.φH

full_name: CategoryTheory.ShortComplex.LeftHomologyMapData.smul_φK
code: /-- Given a left homology map data for morphism `φ`, this is the induced left homology
map data for `a • φ`. -/
@[simps]
def smul (a : R) : LeftHomologyMapData (a • φ) h₁ h₂ 
goal: (γ.smul a).φK = a • γ.φK

full_name: CategoryTheory.ShortComplex.LeftHomologyMapData.smul
code: /-- Given a left homology map data for morphism `φ`, this is the induced left homology
map data for `a • φ`. -/
@[simps]
def smul (a : R) : LeftHomologyMapData (a • φ) h₁ h₂ 
goal: CategoryTheory.ShortComplex.LeftHomologyMapData (a • φ) h₁ h₂

full_name: CategoryTheory.ShortComplex.leftHomologyMap'_smul
code: @[simp]
lemma leftHomologyMap'_smul :
    leftHomologyMap' (a • φ) h₁ h₂ = a • leftHomologyMap' φ h₁ h₂ 
goal: CategoryTheory.ShortComplex.leftHomologyMap' (a • φ) h₁ h₂ = a • CategoryTheory.ShortComplex.leftHomologyMap' φ h₁ h₂

full_name: CategoryTheory.ShortComplex.cyclesMap'_smul
code: @[simp]
lemma cyclesMap'_smul :
    cyclesMap' (a • φ) h₁ h₂ = a • cyclesMap' φ h₁ h₂ 
goal: CategoryTheory.ShortComplex.cyclesMap' (a • φ) h₁ h₂ = a • CategoryTheory.ShortComplex.cyclesMap' φ h₁ h₂

full_name: CategoryTheory.ShortComplex.leftHomologyMap_smul
code: @[simp]
lemma leftHomologyMap_smul : leftHomologyMap (a • φ) = a • leftHomologyMap φ 
goal: CategoryTheory.ShortComplex.leftHomologyMap (a • φ) = a • CategoryTheory.ShortComplex.leftHomologyMap φ

full_name: CategoryTheory.ShortComplex.cyclesMap_smul
code: @[simp]
lemma cyclesMap_smul : cyclesMap (a • φ) = a • cyclesMap φ 
goal: CategoryTheory.ShortComplex.cyclesMap (a • φ) = a • CategoryTheory.ShortComplex.cyclesMap φ

full_name: CategoryTheory.ShortComplex.leftHomologyFunctor_linear
code: instance leftHomologyFunctor_linear [HasKernels C] [HasCokernels C] :
    Functor.Linear R (leftHomologyFunctor C) 
goal: CategoryTheory.Functor.Linear R (CategoryTheory.ShortComplex.leftHomologyFunctor C)

full_name: CategoryTheory.ShortComplex.cyclesFunctor_linear
code: instance cyclesFunctor_linear [HasKernels C] [HasCokernels C] :
    Functor.Linear R (cyclesFunctor C) 
goal: CategoryTheory.Functor.Linear R (CategoryTheory.ShortComplex.cyclesFunctor C)

full_name: CategoryTheory.ShortComplex.RightHomologyMapData.smul_φH
code: /-- Given a right homology map data for morphism `φ`, this is the induced right homology
map data for `a • φ`. -/
@[simps]
def smul (a : R) : RightHomologyMapData (a • φ) h₁ h₂ 
goal: (γ.smul a).φH = a • γ.φH

full_name: CategoryTheory.ShortComplex.RightHomologyMapData.smul
code: /-- Given a right homology map data for morphism `φ`, this is the induced right homology
map data for `a • φ`. -/
@[simps]
def smul (a : R) : RightHomologyMapData (a • φ) h₁ h₂ 
goal: CategoryTheory.ShortComplex.RightHomologyMapData (a • φ) h₁ h₂

full_name: CategoryTheory.ShortComplex.RightHomologyMapData.smul_φQ
code: /-- Given a right homology map data for morphism `φ`, this is the induced right homology
map data for `a • φ`. -/
@[simps]
def smul (a : R) : RightHomologyMapData (a • φ) h₁ h₂ 
goal: (γ.smul a).φQ = a • γ.φQ

full_name: CategoryTheory.ShortComplex.rightHomologyMap'_smul
code: @[simp]
lemma rightHomologyMap'_smul :
    rightHomologyMap' (a • φ) h₁ h₂ = a • rightHomologyMap' φ h₁ h₂ 
goal: CategoryTheory.ShortComplex.rightHomologyMap' (a • φ) h₁ h₂ = a • CategoryTheory.ShortComplex.rightHomologyMap' φ h₁ h₂

full_name: CategoryTheory.ShortComplex.opcyclesMap'_smul
code: @[simp]
lemma opcyclesMap'_smul :
    opcyclesMap' (a • φ) h₁ h₂ = a • opcyclesMap' φ h₁ h₂ 
goal: CategoryTheory.ShortComplex.opcyclesMap' (a • φ) h₁ h₂ = a • CategoryTheory.ShortComplex.opcyclesMap' φ h₁ h₂

full_name: CategoryTheory.ShortComplex.rightHomologyMap_smul
code: @[simp]
lemma rightHomologyMap_smul : rightHomologyMap (a • φ) = a • rightHomologyMap φ 
goal: CategoryTheory.ShortComplex.rightHomologyMap (a • φ) = a • CategoryTheory.ShortComplex.rightHomologyMap φ

full_name: CategoryTheory.ShortComplex.opcyclesMap_smul
code: @[simp]
lemma opcyclesMap_smul : opcyclesMap (a • φ) = a • opcyclesMap φ 
goal: CategoryTheory.ShortComplex.opcyclesMap (a • φ) = a • CategoryTheory.ShortComplex.opcyclesMap φ

full_name: CategoryTheory.ShortComplex.rightHomologyFunctor_linear
code: instance rightHomologyFunctor_linear [HasKernels C] [HasCokernels C] :
    Functor.Linear R (rightHomologyFunctor C) 
goal: CategoryTheory.Functor.Linear R (CategoryTheory.ShortComplex.rightHomologyFunctor C)

full_name: CategoryTheory.ShortComplex.opcyclesFunctor_linear
code: instance opcyclesFunctor_linear [HasKernels C] [HasCokernels C] :
    Functor.Linear R (opcyclesFunctor C) 
goal: CategoryTheory.Functor.Linear R (CategoryTheory.ShortComplex.opcyclesFunctor C)

full_name: CategoryTheory.ShortComplex.HomologyMapData.smul_left
code: /-- Given a homology map data for a morphism `φ`, this is the induced homology
map data for `a • φ`. -/
@[simps]
def smul (a : R) : HomologyMapData (a • φ) h₁ h₂ 
goal: (γ.smul a).left = γ.left.smul a

full_name: CategoryTheory.ShortComplex.HomologyMapData.smul_right
code: /-- Given a homology map data for a morphism `φ`, this is the induced homology
map data for `a • φ`. -/
@[simps]
def smul (a : R) : HomologyMapData (a • φ) h₁ h₂ 
goal: (γ.smul a).right = γ.right.smul a

full_name: CategoryTheory.ShortComplex.HomologyMapData.smul
code: /-- Given a homology map data for a morphism `φ`, this is the induced homology
map data for `a • φ`. -/
@[simps]
def smul (a : R) : HomologyMapData (a • φ) h₁ h₂ 
goal: CategoryTheory.ShortComplex.HomologyMapData (a • φ) h₁ h₂

full_name: CategoryTheory.ShortComplex.homologyMap'_smul
code: @[simp]
lemma homologyMap'_smul :
    homologyMap' (a • φ) h₁ h₂ = a • homologyMap' φ h₁ h₂ 
goal: CategoryTheory.ShortComplex.homologyMap' (a • φ) h₁ h₂ = a • CategoryTheory.ShortComplex.homologyMap' φ h₁ h₂

full_name: CategoryTheory.ShortComplex.homologyMap_smul
code: @[simp]
lemma homologyMap_smul [S₁.HasHomology] [S₂.HasHomology] :
    homologyMap (a • φ) = a • homologyMap φ 
goal: CategoryTheory.ShortComplex.homologyMap (a • φ) = a • CategoryTheory.ShortComplex.homologyMap φ

full_name: CategoryTheory.ShortComplex.homologyFunctor_linear
code: instance homologyFunctor_linear [CategoryWithHomology C] :
    Functor.Linear R (homologyFunctor C) 
goal: CategoryTheory.Functor.Linear R (CategoryTheory.ShortComplex.homologyFunctor C)

full_name: CategoryTheory.ShortComplex.Homotopy.smul_h₃
code: /-- Homotopy between morphisms of short complexes is compatible with the scalar multiplication. -/
@[simps]
def Homotopy.smul {φ₁ φ₂ : S₁ ⟶ S₂} (h : Homotopy φ₁ φ₂) (a : R) :
    Homotopy (a • φ₁) (a • φ₂) 
goal: (h.smul a).h₃ = a • h.h₃

full_name: CategoryTheory.ShortComplex.Homotopy.smul_h₀
code: /-- Homotopy between morphisms of short complexes is compatible with the scalar multiplication. -/
@[simps]
def Homotopy.smul {φ₁ φ₂ : S₁ ⟶ S₂} (h : Homotopy φ₁ φ₂) (a : R) :
    Homotopy (a • φ₁) (a • φ₂) 
goal: (h.smul a).h₀ = a • h.h₀

full_name: CategoryTheory.ShortComplex.Homotopy.smul
code: /-- Homotopy between morphisms of short complexes is compatible with the scalar multiplication. -/
@[simps]
def Homotopy.smul {φ₁ φ₂ : S₁ ⟶ S₂} (h : Homotopy φ₁ φ₂) (a : R) :
    Homotopy (a • φ₁) (a • φ₂) 
goal: CategoryTheory.ShortComplex.Homotopy (a • φ₁) (a • φ₂)

full_name: CategoryTheory.ShortComplex.Homotopy.smul_h₁
code: /-- Homotopy between morphisms of short complexes is compatible with the scalar multiplication. -/
@[simps]
def Homotopy.smul {φ₁ φ₂ : S₁ ⟶ S₂} (h : Homotopy φ₁ φ₂) (a : R) :
    Homotopy (a • φ₁) (a • φ₂) 
goal: (h.smul a).h₁ = a • h.h₁

full_name: CategoryTheory.ShortComplex.Homotopy.smul_h₂
code: /-- Homotopy between morphisms of short complexes is compatible with the scalar multiplication. -/
@[simps]
def Homotopy.smul {φ₁ φ₂ : S₁ ⟶ S₂} (h : Homotopy φ₁ φ₂) (a : R) :
    Homotopy (a • φ₁) (a • φ₂) 
goal: (h.smul a).h₂ = a • h.h₂