Theorem: Matroid.Parallel.mem_cl_iff_mem_cl (Matroid/Parallel.lean)

State before:
α : Type u_1
M N : Matroid α
e f g : α
I X✝ P D : Set α
h : M.Parallel e f
X : Set α
⊢ e ∈ M.cl X ↔ f ∈ M.cl X
Tactic:
refine ⟨fun h' ↦ ?_, fun h' ↦ ?_⟩
State after:
case refine_1
α : Type u_1
M N : Matroid α
e f g : α
I X✝ P D : Set α
h : M.Parallel e f
X : Set α
h' : e ∈ M.cl X
⊢ f ∈ M.cl X

case refine_2
α : Type u_1
M N : Matroid α
e f g : α
I X✝ P D : Set α
h : M.Parallel e f
X : Set α
h' : f ∈ M.cl X
⊢ e ∈ M.cl X

State before:
case refine_2
α : Type u_1
M N : Matroid α
e f g : α
I X✝ P D : Set α
h : M.Parallel e f
X : Set α
h' : f ∈ M.cl X
⊢ e ∈ M.cl X
Tactic:
exact mem_of_mem_of_subset h.mem_cl (M.cl_subset_cl_of_subset_cl (by simpa))
State after:
no goals

State before:
case refine_1
α : Type u_1
M N : Matroid α
e f g : α
I X✝ P D : Set α
h : M.Parallel e f
X : Set α
h' : e ∈ M.cl X
⊢ f ∈ M.cl X
Tactic:
exact mem_of_mem_of_subset h.symm.mem_cl (M.cl_subset_cl_of_subset_cl (by simpa))
State after:
no goals

State before:
α : Type u_1
M N : Matroid α
e f g : α
I X✝ P D : Set α
h : M.Parallel e f
X : Set α
h' : e ∈ M.cl X
⊢ {e} ⊆ M.cl X
Tactic:
simpa
State after:
no goals

State before:
α : Type u_1
M N : Matroid α
e f g : α
I X✝ P D : Set α
h : M.Parallel e f
X : Set α
h' : f ∈ M.cl X
⊢ {f} ⊆ M.cl X
Tactic:
simpa
State after:
no goals