Theorem: Matroid.Indep.cl_inter_eq_inter_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
h : M.Indep (I ∪ J)
⊢ M.cl (I ∩ J) = M.cl I ∩ M.cl J
Tactic:
rw [inter_eq_iInter, cl_iInter_eq_iInter_cl_of_iUnion_indep, inter_eq_iInter]
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
h : M.Indep (I ∪ J)
⊢ ⋂ i, M.cl (bif i then I else J) = ⋂ b, bif b then M.cl I else M.cl J

case h
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
h : M.Indep (I ∪ J)
⊢ M.Indep (⋃ i, bif i then I else J)

State before:
case h
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
h : M.Indep (I ∪ J)
⊢ M.Indep (⋃ i, bif i then I else J)
Tactic:
rwa [← union_eq_iUnion]
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
h : M.Indep (I ∪ J)
⊢ ⋂ i, M.cl (bif i then I else J) = ⋂ b, bif b then M.cl I else M.cl J
Tactic:
exact iInter_congr (by 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
h : M.Indep (I ∪ J)
⊢ ∀ (i : Bool), M.cl (bif i then I else J) = bif i then M.cl I else M.cl J
Tactic:
simp
State after:
no goals