Theorem: Matroid.conn_subset (Matroid/Connectivity.lean)

State before:
α : Type u_1
M✝ : Matroid α
B B' I I' J K X Y : Set α
M : Matroid α
hXY : X ⊆ Y
⊢ M.conn X Y = M.er X
Tactic:
obtain ⟨I, hI⟩ := M.exists_basis' X
State after:
case intro
α : Type u_1
M✝ : Matroid α
B B' I✝ I' J K X Y : Set α
M : Matroid α
hXY : X ⊆ Y
I : Set α
hI : M.Basis' I X
⊢ M.conn X Y = M.er X

State before:
case intro
α : Type u_1
M✝ : Matroid α
B B' I✝ I' J K X Y : Set α
M : Matroid α
hXY : X ⊆ Y
I : Set α
hI : M.Basis' I X
⊢ M.conn X Y = M.er X
Tactic:
obtain ⟨J, hJ, hIJ⟩ := hI.indep.subset_basis'_of_subset (hI.subset.trans hXY)
State after:
case intro.intro.intro
α : Type u_1
M✝ : Matroid α
B B' I✝ I' J✝ K X Y : Set α
M : Matroid α
hXY : X ⊆ Y
I : Set α
hI : M.Basis' I X
J : Set α
hJ : M.Basis' J Y
hIJ : I ⊆ J
⊢ M.conn X Y = M.er X

State before:
case intro.intro.intro
α : Type u_1
M✝ : Matroid α
B B' I✝ I' J✝ K X Y : Set α
M : Matroid α
hXY : X ⊆ Y
I : Set α
hI : M.Basis' I X
J : Set α
hJ : M.Basis' J Y
hIJ : I ⊆ J
⊢ M.conn X Y = M.er X
Tactic:
rw [hI.conn_eq hJ, inter_eq_self_of_subset_left hIJ, union_eq_self_of_subset_left hIJ,
  hJ.indep.erk_dual_restrict_eq, ← hI.encard, add_zero]
State after:
no goals