Theorem: Matroid.restrict_er_eq' (Matroid/Rank.lean)

State before:
α : Type u_1
ι : Type u_2
M✝ N : Matroid α
I B X✝ X' Y Y' Z R✝ : Set α
n : ℕ∞
e f : α
M : Matroid α
R X : Set α
⊢ (M ↾ R).er X = M.er (X ∩ R)
Tactic:
obtain ⟨I, hI⟩ := (M ↾ R).exists_basis' X
State after:
case intro
α : Type u_1
ι : Type u_2
M✝ N : Matroid α
I✝ B X✝ X' Y Y' Z R✝ : Set α
n : ℕ∞
e f : α
M : Matroid α
R X I : Set α
hI : (M ↾ R).Basis' I X
⊢ (M ↾ R).er X = M.er (X ∩ R)

State before:
case intro
α : Type u_1
ι : Type u_2
M✝ N : Matroid α
I✝ B X✝ X' Y Y' Z R✝ : Set α
n : ℕ∞
e f : α
M : Matroid α
R X I : Set α
hI : (M ↾ R).Basis' I X
⊢ (M ↾ R).er X = M.er (X ∩ R)
Tactic:
rw [hI.er_eq_encard]
State after:
case intro
α : Type u_1
ι : Type u_2
M✝ N : Matroid α
I✝ B X✝ X' Y Y' Z R✝ : Set α
n : ℕ∞
e f : α
M : Matroid α
R X I : Set α
hI : (M ↾ R).Basis' I X
⊢ I.encard = M.er (X ∩ R)

State before:
case intro
α : Type u_1
ι : Type u_2
M✝ N : Matroid α
I✝ B X✝ X' Y Y' Z R✝ : Set α
n : ℕ∞
e f : α
M : Matroid α
R X I : Set α
hI : (M ↾ R).Basis' I X
⊢ I.encard = M.er (X ∩ R)
Tactic:
rw [basis'_iff_basis_inter_ground, basis_restrict_iff', restrict_ground_eq] at hI
State after:
case intro
α : Type u_1
ι : Type u_2
M✝ N : Matroid α
I✝ B X✝ X' Y Y' Z R✝ : Set α
n : ℕ∞
e f : α
M : Matroid α
R X I : Set α
hI : M.Basis I (X ∩ R ∩ M.E) ∧ X ∩ R ⊆ R
⊢ I.encard = M.er (X ∩ R)

State before:
case intro
α : Type u_1
ι : Type u_2
M✝ N : Matroid α
I✝ B X✝ X' Y Y' Z R✝ : Set α
n : ℕ∞
e f : α
M : Matroid α
R X I : Set α
hI : M.Basis I (X ∩ R ∩ M.E) ∧ X ∩ R ⊆ R
⊢ I.encard = M.er (X ∩ R)
Tactic:
rw [← er_inter_ground_eq, ← hI.1.er_eq_encard]
State after:
no goals