Theorem: groupCohomology.resolution.forget₂ToModuleCatHomotopyEquiv_f_0_eq
File: Mathlib/RepresentationTheory/GroupCohomology/Resolution.lean

Proof steps:
state_before: k G : Type u
inst✝¹ : CommRing k
n : ℕ
inst✝ : Monoid G
⊢ (forget₂ToModuleCatHomotopyEquiv k G).hom.f 0 = (forget₂ (Rep k G) (ModuleCat k)).map (ε k G)
state_after: k G : Type u
inst✝¹ : CommRing k
n : ℕ
inst✝ : Monoid G
⊢ ((HomotopyEquiv.ofIso (compForgetAugmentedIso k G).symm).hom ≫
          (extraDegeneracyCompForgetAugmentedToModule k G).homotopyEquiv.hom ≫
            (HomotopyEquiv.ofIso
                ((ChainComplex.single₀ (ModuleCat k)).mapIso
                  (Finsupp.LinearEquiv.finsuppUnique k k (⊤_ Type u)).toModuleIso)).hom).f
      0 =
    (forget₂ (Rep k G) (ModuleCat k)).map (ε k G)
tactic: show (HomotopyEquiv.hom _ ≫ HomotopyEquiv.hom _ ≫ HomotopyEquiv.hom _).f 0 = _
state_before: k G : Type u
inst✝¹ : CommRing k
n : ℕ
inst✝ : Monoid G
⊢ ((HomotopyEquiv.ofIso (compForgetAugmentedIso k G).symm).hom ≫
          (extraDegeneracyCompForgetAugmentedToModule k G).homotopyEquiv.hom ≫
            (HomotopyEquiv.ofIso
                ((ChainComplex.single₀ (ModuleCat k)).mapIso
                  (Finsupp.LinearEquiv.finsuppUnique k k (⊤_ Type u)).toModuleIso)).hom).f
      0 =
    (forget₂ (Rep k G) (ModuleCat k)).map (ε k G)
state_after: k G : Type u
inst✝¹ : CommRing k
n : ℕ
inst✝ : Monoid G
⊢ (HomotopyEquiv.ofIso (compForgetAugmentedIso k G).symm).hom.f 0 ≫
      (extraDegeneracyCompForgetAugmentedToModule k G).homotopyEquiv.hom.f 0 ≫
        (HomotopyEquiv.ofIso
                ((ChainComplex.single₀ (ModuleCat k)).mapIso
                  (Finsupp.LinearEquiv.finsuppUnique k k (⊤_ Type u)).toModuleIso)).hom.f
          0 =
    (forget₂ (Rep k G) (ModuleCat k)).map (ε k G)
tactic: simp only [HomologicalComplex.comp_f]
state_before: k G : Type u
inst✝¹ : CommRing k
n : ℕ
inst✝ : Monoid G
⊢ (HomotopyEquiv.ofIso (compForgetAugmentedIso k G).symm).hom.f 0 ≫
      (extraDegeneracyCompForgetAugmentedToModule k G).homotopyEquiv.hom.f 0 ≫
        (HomotopyEquiv.ofIso
                ((ChainComplex.single₀ (ModuleCat k)).mapIso
                  (Finsupp.LinearEquiv.finsuppUnique k k (⊤_ Type u)).toModuleIso)).hom.f
          0 =
    (forget₂ (Rep k G) (ModuleCat k)).map (ε k G)
state_after: case h.e'_2.h.e'_6.h
k G : Type u
inst✝¹ : CommRing k
n : ℕ
inst✝ : Monoid G
e_4✝ : (compForgetAugmented.toModule k G).left.obj (Opposite.op (SimplexCategory.mk 0)) = (forget₂ToModuleCat k G).X 0
⊢ (HomotopyEquiv.ofIso (compForgetAugmentedIso k G).symm).hom.f 0 = 𝟙 ((forget₂ToModuleCat k G).X 0)

case h.e'_3
k G : Type u
inst✝¹ : CommRing k
n : ℕ
inst✝ : Monoid G
⊢ (forget₂ (Rep k G) (ModuleCat k)).map (ε k G) =
    (extraDegeneracyCompForgetAugmentedToModule k G).homotopyEquiv.hom.f 0 ≫
      (HomotopyEquiv.ofIso
              ((ChainComplex.single₀ (ModuleCat k)).mapIso
                (Finsupp.LinearEquiv.finsuppUnique k k (⊤_ Type u)).toModuleIso)).hom.f
        0
tactic: convert Category.id_comp (X := (forget₂ToModuleCat k G).X 0) _
state_before: case h.e'_3
k G : Type u
inst✝¹ : CommRing k
n : ℕ
inst✝ : Monoid G
⊢ (forget₂ (Rep k G) (ModuleCat k)).map (ε k G) =
    (extraDegeneracyCompForgetAugmentedToModule k G).homotopyEquiv.hom.f 0 ≫
      (HomotopyEquiv.ofIso
              ((ChainComplex.single₀ (ModuleCat k)).mapIso
                (Finsupp.LinearEquiv.finsuppUnique k k (⊤_ Type u)).toModuleIso)).hom.f
        0
state_after: k G : Type u
inst✝¹ : CommRing k
n : ℕ
inst✝ : Monoid G
⊢ (forget₂ (Rep k G) (ModuleCat k)).map (ε k G) =
    (Finsupp.total (⊤_ Type u) k k fun x => 1) ∘ₗ (ModuleCat.free k).map (terminal.from (Fin 1 → G))

k G : Type u
inst✝¹ : CommRing k
n : ℕ
inst✝ : Monoid G
⊢ (Finsupp.total (⊤_ Type u) k k fun x => 1) ∘ₗ (ModuleCat.free k).map (terminal.from (Fin 1 → G)) =
    (extraDegeneracyCompForgetAugmentedToModule k G).homotopyEquiv.hom.f 0 ≫
      (HomotopyEquiv.ofIso
              ((ChainComplex.single₀ (ModuleCat k)).mapIso
                (Finsupp.LinearEquiv.finsuppUnique k k (⊤_ Type u)).toModuleIso)).hom.f
        0
tactic: trans (Finsupp.total _ _ _ fun _ => (1 : k)).comp ((ModuleCat.free k).map (terminal.from _))