Theorem: Matroid.ModularBase.modularBase_of_forall_subset_subset_cl (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
hB : M.ModularBase B Xs
hXY : ∀ (i : ι), Xs i ⊆ Ys i
hYX : ∀ (i : ι), Ys i ⊆ M.cl (Xs i)
⊢ M.ModularBase B Ys
Tactic:
refine ⟨hB.base, fun i ↦ hB.indep.inter_basis_cl_iff_subset_cl_inter.2 ?_⟩
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
hB : M.ModularBase B Xs
hXY : ∀ (i : ι), Xs i ⊆ Ys i
hYX : ∀ (i : ι), Ys i ⊆ M.cl (Xs i)
i : ι
⊢ Ys i ⊆ M.cl (Ys i ∩ B)

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
hB : M.ModularBase B Xs
hXY : ∀ (i : ι), Xs i ⊆ Ys i
hYX : ∀ (i : ι), Ys i ⊆ M.cl (Xs i)
i : ι
⊢ Ys i ⊆ M.cl (Ys i ∩ B)
Tactic:
refine (hYX i).trans (M.cl_subset_cl_of_subset_cl ?_)
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
hB : M.ModularBase B Xs
hXY : ∀ (i : ι), Xs i ⊆ Ys i
hYX : ∀ (i : ι), Ys i ⊆ M.cl (Xs i)
i : ι
⊢ Xs i ⊆ M.cl (Ys i ∩ B)

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
hB : M.ModularBase B Xs
hXY : ∀ (i : ι), Xs i ⊆ Ys i
hYX : ∀ (i : ι), Ys i ⊆ M.cl (Xs i)
i : ι
⊢ Xs i ⊆ M.cl (Ys i ∩ B)
Tactic:
exact (hB.2 i).subset_cl.trans (M.cl_subset_cl (inter_subset_inter_left B (hXY i)))
State after:
no goals