Theorem: Matroid.Basis.cl_inter_basis_cl (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
Xs Ys : Set (Set α)
ι : Type u_3
h : M.Basis (X ∩ I) X
hI : M.Indep I
⊢ M.Basis (M.cl X ∩ I) (M.cl X)
Tactic:
rw [hI.inter_basis_cl_iff_subset_cl_inter] at h ⊢
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
Xs Ys : Set (Set α)
ι : Type u_3
h : X ⊆ M.cl (X ∩ I)
hI : M.Indep I
⊢ M.cl X ⊆ M.cl (M.cl 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
Xs Ys : Set (Set α)
ι : Type u_3
h : X ⊆ M.cl (X ∩ I)
hI : M.Indep I
⊢ M.cl X ⊆ M.cl (M.cl X ∩ I)
Tactic:
exact (M.cl_subset_cl_of_subset_cl h).trans (M.cl_subset_cl
  (inter_subset_inter_left _ (h.trans (M.cl_subset_cl inter_subset_left))))
State after:
no goals