Theorem: AddSubgroup.toSubgroup_symm_apply_coe
File: Mathlib/Algebra/Group/Subgroup/Basic.lean

Proof steps:
state_before: G : Type u_1
G' : Type u_2
G'' : Type u_3
inst✝³ : Group G
inst✝² : Group G'
inst✝¹ : Group G''
A : Type u_4
inst✝ : AddGroup A
x : AddSubgroup A
⊢ (fun S =>
        let __src := Submonoid.toAddSubmonoid S.toSubmonoid;
        { toAddSubmonoid := __src, neg_mem' := ⋯ })
      ((fun S =>
          let __src := AddSubmonoid.toSubmonoid S.toAddSubmonoid;
          { toSubmonoid := __src, inv_mem' := ⋯ })
        x) =
    x
state_after: case mk
G : Type u_1
G' : Type u_2
G'' : Type u_3
inst✝³ : Group G
inst✝² : Group G'
inst✝¹ : Group G''
A : Type u_4
inst✝ : AddGroup A
toAddSubmonoid✝ : AddSubmonoid A
neg_mem'✝ : ∀ {x : A}, x ∈ toAddSubmonoid✝.carrier → -x ∈ toAddSubmonoid✝.carrier
⊢ (fun S =>
        let __src := Submonoid.toAddSubmonoid S.toSubmonoid;
        { toAddSubmonoid := __src, neg_mem' := ⋯ })
      ((fun S =>
          let __src := AddSubmonoid.toSubmonoid S.toAddSubmonoid;
          { toSubmonoid := __src, inv_mem' := ⋯ })
        { toAddSubmonoid := toAddSubmonoid✝, neg_mem' := neg_mem'✝ }) =
    { toAddSubmonoid := toAddSubmonoid✝, neg_mem' := neg_mem'✝ }
tactic: cases x
state_before: case mk
G : Type u_1
G' : Type u_2
G'' : Type u_3
inst✝³ : Group G
inst✝² : Group G'
inst✝¹ : Group G''
A : Type u_4
inst✝ : AddGroup A
toAddSubmonoid✝ : AddSubmonoid A
neg_mem'✝ : ∀ {x : A}, x ∈ toAddSubmonoid✝.carrier → -x ∈ toAddSubmonoid✝.carrier
⊢ (fun S =>
        let __src := Submonoid.toAddSubmonoid S.toSubmonoid;
        { toAddSubmonoid := __src, neg_mem' := ⋯ })
      ((fun S =>
          let __src := AddSubmonoid.toSubmonoid S.toAddSubmonoid;
          { toSubmonoid := __src, inv_mem' := ⋯ })
        { toAddSubmonoid := toAddSubmonoid✝, neg_mem' := neg_mem'✝ }) =
    { toAddSubmonoid := toAddSubmonoid✝, neg_mem' := neg_mem'✝ }
state_after: no goals
tactic: rfl
state_before: G : Type u_1
G' : Type u_2
G'' : Type u_3
inst✝³ : Group G
inst✝² : Group G'
inst✝¹ : Group G''
A : Type u_4
inst✝ : AddGroup A
x : Subgroup (Multiplicative A)
⊢ (fun S =>
        let __src := AddSubmonoid.toSubmonoid S.toAddSubmonoid;
        { toSubmonoid := __src, inv_mem' := ⋯ })
      ((fun S =>
          let __src := Submonoid.toAddSubmonoid S.toSubmonoid;
          { toAddSubmonoid := __src, neg_mem' := ⋯ })
        x) =
    x
state_after: case mk
G : Type u_1
G' : Type u_2
G'' : Type u_3
inst✝³ : Group G
inst✝² : Group G'
inst✝¹ : Group G''
A : Type u_4
inst✝ : AddGroup A
toSubmonoid✝ : Submonoid (Multiplicative A)
inv_mem'✝ : ∀ {x : Multiplicative A}, x ∈ toSubmonoid✝.carrier → x⁻¹ ∈ toSubmonoid✝.carrier
⊢ (fun S =>
        let __src := AddSubmonoid.toSubmonoid S.toAddSubmonoid;
        { toSubmonoid := __src, inv_mem' := ⋯ })
      ((fun S =>
          let __src := Submonoid.toAddSubmonoid S.toSubmonoid;
          { toAddSubmonoid := __src, neg_mem' := ⋯ })
        { toSubmonoid := toSubmonoid✝, inv_mem' := inv_mem'✝ }) =
    { toSubmonoid := toSubmonoid✝, inv_mem' := inv_mem'✝ }
tactic: cases x
state_before: case mk
G : Type u_1
G' : Type u_2
G'' : Type u_3
inst✝³ : Group G
inst✝² : Group G'
inst✝¹ : Group G''
A : Type u_4
inst✝ : AddGroup A
toSubmonoid✝ : Submonoid (Multiplicative A)
inv_mem'✝ : ∀ {x : Multiplicative A}, x ∈ toSubmonoid✝.carrier → x⁻¹ ∈ toSubmonoid✝.carrier
⊢ (fun S =>
        let __src := AddSubmonoid.toSubmonoid S.toAddSubmonoid;
        { toSubmonoid := __src, inv_mem' := ⋯ })
      ((fun S =>
          let __src := Submonoid.toAddSubmonoid S.toSubmonoid;
          { toAddSubmonoid := __src, neg_mem' := ⋯ })
        { toSubmonoid := toSubmonoid✝, inv_mem' := inv_mem'✝ }) =
    { toSubmonoid := toSubmonoid✝, inv_mem' := inv_mem'✝ }
state_after: no goals
tactic: rfl