Theorem: Matroid.Indep.mem_cl_iff' (Matroid/Closure.lean)

State before:
α : Type u_1
ι : Type u_2
M : Matroid α
F I J X Y B C R : Set α
e f x y : α
hI : M.Indep I
⊢ x ∈ M.cl I ↔ x ∈ M.E ∧ (M.Indep (insert x I) → x ∈ I)
Tactic:
rw [hI.mem_cl_iff, dep_iff, insert_subset_iff, and_iff_left hI.subset_ground,
  imp_iff_not_or]
State after:
α : Type u_1
ι : Type u_2
M : Matroid α
F I J X Y B C R : Set α
e f x y : α
hI : M.Indep I
⊢ ¬M.Indep (insert x I) ∧ x ∈ M.E ∨ x ∈ I ↔ x ∈ M.E ∧ (¬M.Indep (insert x I) ∨ x ∈ I)

State before:
α : Type u_1
ι : Type u_2
M : Matroid α
F I J X Y B C R : Set α
e f x y : α
hI : M.Indep I
⊢ ¬M.Indep (insert x I) ∧ x ∈ M.E ∨ x ∈ I ↔ x ∈ M.E ∧ (¬M.Indep (insert x I) ∨ x ∈ I)
Tactic:
have := hI.subset_ground
State after:
α : Type u_1
ι : Type u_2
M : Matroid α
F I J X Y B C R : Set α
e f x y : α
hI : M.Indep I
this : I ⊆ M.E
⊢ ¬M.Indep (insert x I) ∧ x ∈ M.E ∨ x ∈ I ↔ x ∈ M.E ∧ (¬M.Indep (insert x I) ∨ x ∈ I)

State before:
α : Type u_1
ι : Type u_2
M : Matroid α
F I J X Y B C R : Set α
e f x y : α
hI : M.Indep I
this : I ⊆ M.E
⊢ ¬M.Indep (insert x I) ∧ x ∈ M.E ∨ x ∈ I ↔ x ∈ M.E ∧ (¬M.Indep (insert x I) ∨ x ∈ I)
Tactic:
aesop
State after:
no goals