Theorem: Matroid.modularPair_iff_exists_subsets_cl_inter (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
⊢ M.ModularPair X Y ↔ ∃ I, M.Indep I ∧ X ⊆ M.cl (X ∩ I) ∧ Y ⊆ M.cl (Y ∩ I)
Tactic:
rw [modularPair_iff]
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
⊢ (∃ I, M.Indep I ∧ M.Basis (X ∩ I) X ∧ M.Basis (Y ∩ I) Y) ↔ ∃ I, M.Indep I ∧ X ⊆ M.cl (X ∩ I) ∧ Y ⊆ M.cl (Y ∩ I)

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
⊢ (∃ I, M.Indep I ∧ M.Basis (X ∩ I) X ∧ M.Basis (Y ∩ I) Y) ↔ ∃ I, M.Indep I ∧ X ⊆ M.cl (X ∩ I) ∧ Y ⊆ M.cl (Y ∩ I)
Tactic:
refine ⟨fun ⟨I,hI,h⟩ ↦ ⟨I,hI,?_⟩, fun ⟨I,hI,h⟩ ↦ ⟨I,hI,?_⟩⟩
State after:
case refine_1
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
x✝ : ∃ I, M.Indep I ∧ M.Basis (X ∩ I) X ∧ M.Basis (Y ∩ I) Y
I : Set α
hI : M.Indep I
h : M.Basis (X ∩ I) X ∧ M.Basis (Y ∩ I) Y
⊢ X ⊆ M.cl (X ∩ I) ∧ Y ⊆ M.cl (Y ∩ I)

case refine_2
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
x✝ : ∃ I, M.Indep I ∧ X ⊆ M.cl (X ∩ I) ∧ Y ⊆ M.cl (Y ∩ I)
I : Set α
hI : M.Indep I
h : X ⊆ M.cl (X ∩ I) ∧ Y ⊆ M.cl (Y ∩ I)
⊢ M.Basis (X ∩ I) X ∧ M.Basis (Y ∩ I) Y

State before:
case refine_2
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
x✝ : ∃ I, M.Indep I ∧ X ⊆ M.cl (X ∩ I) ∧ Y ⊆ M.cl (Y ∩ I)
I : Set α
hI : M.Indep I
h : X ⊆ M.cl (X ∩ I) ∧ Y ⊆ M.cl (Y ∩ I)
⊢ M.Basis (X ∩ I) X ∧ M.Basis (Y ∩ I) Y
Tactic:
rwa [hI.inter_basis_cl_iff_subset_cl_inter, hI.inter_basis_cl_iff_subset_cl_inter]
State after:
no goals

State before:
case refine_1
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
x✝ : ∃ I, M.Indep I ∧ M.Basis (X ∩ I) X ∧ M.Basis (Y ∩ I) Y
I : Set α
hI : M.Indep I
h : M.Basis (X ∩ I) X ∧ M.Basis (Y ∩ I) Y
⊢ X ⊆ M.cl (X ∩ I) ∧ Y ⊆ M.cl (Y ∩ I)
Tactic:
rwa [← hI.inter_basis_cl_iff_subset_cl_inter, ← hI.inter_basis_cl_iff_subset_cl_inter]
State after:
no goals