Theorem: Matroid.eq_restrict_removeLoops (Matroid/Loop.lean)

State before:
α : Type u_1
β : Type u_2
M✝ N : Matroid α
e f : α
B L L' I X Y Z F C K : Set α
M : Matroid α
⊢ M = M.removeLoops ↾ M.E
Tactic:
rw [removeLoops, eq_iff_indep_iff_indep_forall]
State after:
α : Type u_1
β : Type u_2
M✝ N : Matroid α
e f : α
B L L' I X Y Z F C K : Set α
M : Matroid α
⊢ M.E = (M ↾ {e | M.Nonloop e} ↾ M.E).E ∧ ∀ I ⊆ M.E, M.Indep I ↔ (M ↾ {e | M.Nonloop e} ↾ M.E).Indep I

State before:
α : Type u_1
β : Type u_2
M✝ N : Matroid α
e f : α
B L L' I X Y Z F C K : Set α
M : Matroid α
⊢ M.E = (M ↾ {e | M.Nonloop e} ↾ M.E).E ∧ ∀ I ⊆ M.E, M.Indep I ↔ (M ↾ {e | M.Nonloop e} ↾ M.E).Indep I
Tactic:
simp only [restrict_ground_eq, restrict_indep_iff, true_and]
State after:
α : Type u_1
β : Type u_2
M✝ N : Matroid α
e f : α
B L L' I X Y Z F C K : Set α
M : Matroid α
⊢ ∀ I ⊆ M.E, M.Indep I ↔ (M.Indep I ∧ I ⊆ {e | M.Nonloop e}) ∧ I ⊆ M.E

State before:
α : Type u_1
β : Type u_2
M✝ N : Matroid α
e f : α
B L L' I X Y Z F C K : Set α
M : Matroid α
⊢ ∀ I ⊆ M.E, M.Indep I ↔ (M.Indep I ∧ I ⊆ {e | M.Nonloop e}) ∧ I ⊆ M.E
Tactic:
exact fun I hIE ↦ ⟨fun hI ↦ ⟨⟨hI,fun e heI ↦ hI.nonloop_of_mem heI⟩, hIE⟩, fun hI ↦ hI.1.1⟩
State after:
no goals