Theorem: Matroid.Indep.modularFamily (Matroid/Modular.lean)

State before:
ι : Type u_1
α : Type u_2
M : Matroid α
B I J X X' Y Y' F : Set α
Xs Ys Js Is : ι → Set α
i j : ι
e f : α
hI : M.Indep I
hXs : ∀ (i : ι), M.Basis (Xs i ∩ I) (Xs i)
⊢ M.ModularFamily Xs
Tactic:
simp_rw [hI.inter_basis_cl_iff_subset_cl_inter] at hXs
State after:
ι : Type u_1
α : Type u_2
M : Matroid α
B I J X X' Y Y' F : Set α
Xs Ys Js Is : ι → Set α
i j : ι
e f : α
hI : M.Indep I
hXs : ∀ (i : ι), Xs i ⊆ M.cl (Xs i ∩ I)
⊢ M.ModularFamily Xs

State before:
ι : Type u_1
α : Type u_2
M : Matroid α
B I J X X' Y Y' F : Set α
Xs Ys Js Is : ι → Set α
i j : ι
e f : α
hI : M.Indep I
hXs : ∀ (i : ι), Xs i ⊆ M.cl (Xs i ∩ I)
⊢ M.ModularFamily Xs
Tactic:
obtain ⟨B, hB, hIB⟩ := hI.exists_base_superset
State after:
case intro.intro
ι : Type u_1
α : Type u_2
M : Matroid α
B✝ I J X X' Y Y' F : Set α
Xs Ys Js Is : ι → Set α
i j : ι
e f : α
hI : M.Indep I
hXs : ∀ (i : ι), Xs i ⊆ M.cl (Xs i ∩ I)
B : Set α
hB : M.Base B
hIB : I ⊆ B
⊢ M.ModularFamily Xs

State before:
case intro.intro
ι : Type u_1
α : Type u_2
M : Matroid α
B✝ I J X X' Y Y' F : Set α
Xs Ys Js Is : ι → Set α
i j : ι
e f : α
hI : M.Indep I
hXs : ∀ (i : ι), Xs i ⊆ M.cl (Xs i ∩ I)
B : Set α
hB : M.Base B
hIB : I ⊆ B
⊢ M.ModularFamily Xs
Tactic:
refine ⟨B, hB, ?_⟩
State after:
case intro.intro
ι : Type u_1
α : Type u_2
M : Matroid α
B✝ I J X X' Y Y' F : Set α
Xs Ys Js Is : ι → Set α
i j : ι
e f : α
hI : M.Indep I
hXs : ∀ (i : ι), Xs i ⊆ M.cl (Xs i ∩ I)
B : Set α
hB : M.Base B
hIB : I ⊆ B
⊢ ∀ (i : ι), M.Basis (Xs i ∩ B) (Xs i)

State before:
case intro.intro
ι : Type u_1
α : Type u_2
M : Matroid α
B✝ I J X X' Y Y' F : Set α
Xs Ys Js Is : ι → Set α
i j : ι
e f : α
hI : M.Indep I
hXs : ∀ (i : ι), Xs i ⊆ M.cl (Xs i ∩ I)
B : Set α
hB : M.Base B
hIB : I ⊆ B
⊢ ∀ (i : ι), M.Basis (Xs i ∩ B) (Xs i)
Tactic:
simp_rw [hB.indep.inter_basis_cl_iff_subset_cl_inter]
State after:
case intro.intro
ι : Type u_1
α : Type u_2
M : Matroid α
B✝ I J X X' Y Y' F : Set α
Xs Ys Js Is : ι → Set α
i j : ι
e f : α
hI : M.Indep I
hXs : ∀ (i : ι), Xs i ⊆ M.cl (Xs i ∩ I)
B : Set α
hB : M.Base B
hIB : I ⊆ B
⊢ ∀ (i : ι), Xs i ⊆ M.cl (Xs i ∩ B)

State before:
case intro.intro
ι : Type u_1
α : Type u_2
M : Matroid α
B✝ I J X X' Y Y' F : Set α
Xs Ys Js Is : ι → Set α
i j : ι
e f : α
hI : M.Indep I
hXs : ∀ (i : ι), Xs i ⊆ M.cl (Xs i ∩ I)
B : Set α
hB : M.Base B
hIB : I ⊆ B
⊢ ∀ (i : ι), Xs i ⊆ M.cl (Xs i ∩ B)
Tactic:
exact fun i ↦ (hXs i).trans (M.cl_subset_cl (inter_subset_inter_right _ hIB))
State after:
no goals