Theorem: Matroid.Indep.interBasis_iInter (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
inst✝ : Nonempty ι
X : ι → Set α
hI : M.Indep I
h : ∀ (i : ι), M.Basis (X i ∩ I) (X i)
⊢ M.Basis ((⋂ i, X i) ∩ I) (⋂ i, X i)
Tactic:
rw [← biInter_univ]
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
inst✝ : Nonempty ι
X : ι → Set α
hI : M.Indep I
h : ∀ (i : ι), M.Basis (X i ∩ I) (X i)
⊢ M.Basis ((⋂ x ∈ univ, X x) ∩ I) (⋂ x ∈ univ, X x)

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
inst✝ : Nonempty ι
X : ι → Set α
hI : M.Indep I
h : ∀ (i : ι), M.Basis (X i ∩ I) (X i)
⊢ M.Basis ((⋂ x ∈ univ, X x) ∩ I) (⋂ x ∈ univ, X x)
Tactic:
exact hI.interBasis_biInter (by simp) (by simpa)
State after:
no goals

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
inst✝ : Nonempty ι
X : ι → Set α
hI : M.Indep I
h : ∀ (i : ι), M.Basis (X i ∩ I) (X i)
⊢ univ.Nonempty
Tactic:
simp
State after:
no goals

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
inst✝ : Nonempty ι
X : ι → Set α
hI : M.Indep I
h : ∀ (i : ι), M.Basis (X i ∩ I) (X i)
⊢ ∀ i ∈ univ, M.Basis (X i ∩ I) (X i)
Tactic:
simpa
State after:
no goals