Theorem: Matroid.ModularPair.of_subset_cl_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
h : M.ModularPair X Y
hXX' : X ⊆ X'
hX' : X' ⊆ M.cl X
hYY' : Y ⊆ Y'
hY' : Y' ⊆ M.cl Y
⊢ M.ModularPair X' Y'
Tactic:
apply ModularFamily.modularFamily_of_forall_subset_cl h
State after:
case hXY
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
h : M.ModularPair X Y
hXX' : X ⊆ X'
hX' : X' ⊆ M.cl X
hYY' : Y ⊆ Y'
hY' : Y' ⊆ M.cl Y
⊢ ∀ (i : Bool), (bif i then X else Y) ⊆ bif i then X' else Y'

case hYX
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
h : M.ModularPair X Y
hXX' : X ⊆ X'
hX' : X' ⊆ M.cl X
hYY' : Y ⊆ Y'
hY' : Y' ⊆ M.cl Y
⊢ ∀ (i : Bool), (bif i then X' else Y') ⊆ M.cl (bif i then X else Y)

State before:
case hYX
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
h : M.ModularPair X Y
hXX' : X ⊆ X'
hX' : X' ⊆ M.cl X
hYY' : Y ⊆ Y'
hY' : Y' ⊆ M.cl Y
⊢ ∀ (i : Bool), (bif i then X' else Y') ⊆ M.cl (bif i then X else Y)
Tactic:
simp [hY', hX']
State after:
no goals

State before:
case hXY
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
h : M.ModularPair X Y
hXX' : X ⊆ X'
hX' : X' ⊆ M.cl X
hYY' : Y ⊆ Y'
hY' : Y' ⊆ M.cl Y
⊢ ∀ (i : Bool), (bif i then X else Y) ⊆ bif i then X' else Y'
Tactic:
simp [hYY', hXX']
State after:
no goals