-- Mathlib/Data/Matroid/Sum.lean
@[simp] lemma sigma_isBasis_iff {I X} :
    (Matroid.sigma M).IsBasis I X ↔ ∀ i, (M i).IsBasis (Sigma.mk i ⁻¹' I) (Sigma.mk i ⁻¹' X)

Proof state:
case refine_3
ι : Type u_1
α : ι → Type u_2
M : (i : ι) → Matroid (α i)
I X : Set ((i : ι) × α i)
hI : ∀ (i : ι), (M i).Indep (Sigma.mk i ⁻¹' I)
x✝ :
  (∀ (x : ι), Sigma.mk x ⁻¹' I ⊆ Sigma.mk x ⁻¹' X) ∧
    (∀ (x : ι) ⦃t : Set (α x)⦄, (M x).Indep t → t ⊆ Sigma.mk x ⁻¹' X → Sigma.mk x ⁻¹' I ⊆ t → Sigma.mk x ⁻¹' I = t) ∧
      ∀ (x : ι), Sigma.mk x ⁻¹' X ⊆ (M x).E
hIX : ∀ (x : ι), Sigma.mk x ⁻¹' I ⊆ Sigma.mk x ⁻¹' X
h' : ∀ (x : ι) ⦃t : Set (α x)⦄, (M x).Indep t → t ⊆ Sigma.mk x ⁻¹' X → Sigma.mk x ⁻¹' I ⊆ t → Sigma.mk x ⁻¹' I = t
h'' : ∀ (x : ι), Sigma.mk x ⁻¹' X ⊆ (M x).E
⊢ I ⊆ X

Tactic:
exact fun ⟨i, x⟩ hx ↦ by simpa using hIX i hx

Result:
no goals

Proof state:
case refine_1
ι : Type u_1
α : ι → Type u_2
M : (i : ι) → Matroid (α i)
I X : Set ((i : ι) × α i)
hI : ∀ (i : ι), (M i).Indep (Sigma.mk i ⁻¹' I)
x✝ :
  I ⊆ X ∧
    (∀ ⦃t : Set ((i : ι) × α i)⦄, (∀ (i : ι), (M i).Indep (Sigma.mk i ⁻¹' t)) → t ⊆ X → I ⊆ t → I = t) ∧
      X ⊆ univ.sigma fun i => (M i).E
hIX : I ⊆ X
h : ∀ ⦃t : Set ((i : ι) × α i)⦄, (∀ (i : ι), (M i).Indep (Sigma.mk i ⁻¹' t)) → t ⊆ X → I ⊆ t → I = t
h' : X ⊆ univ.sigma fun i => (M i).E
i : ι
I₀ : Set (α i)
hI₀ : (M i).Indep I₀
hI₀X : I₀ ⊆ Sigma.mk i ⁻¹' X
hII₀ : Sigma.mk i ⁻¹' I ⊆ I₀
⊢ I₀ ⊆ Sigma.mk i ⁻¹' I

Tactic:
specialize h (t := I ∪ Sigma.mk i '' I₀)

Result:
case refine_1
ι : Type u_1
α : ι → Type u_2
M : (i : ι) → Matroid (α i)
I X : Set ((i : ι) × α i)
hI : ∀ (i : ι), (M i).Indep (Sigma.mk i ⁻¹' I)
x✝ :
  I ⊆ X ∧
    (∀ ⦃t : Set ((i : ι) × α i)⦄, (∀ (i : ι), (M i).Indep (Sigma.mk i ⁻¹' t)) → t ⊆ X → I ⊆ t → I = t) ∧
      X ⊆ univ.sigma fun i => (M i).E
hIX : I ⊆ X
h' : X ⊆ univ.sigma fun i => (M i).E
i : ι
I₀ : Set (α i)
hI₀ : (M i).Indep I₀
hI₀X : I₀ ⊆ Sigma.mk i ⁻¹' X
hII₀ : Sigma.mk i ⁻¹' I ⊆ I₀
h :
  (∀ (i_1 : ι), (M i_1).Indep (Sigma.mk i_1 ⁻¹' (I ∪ Sigma.mk i '' I₀))) →
    I ∪ Sigma.mk i '' I₀ ⊆ X → I ⊆ I ∪ Sigma.mk i '' I₀ → I = I ∪ Sigma.mk i '' I₀
⊢ I₀ ⊆ Sigma.mk i ⁻¹' I

Proof state:
case refine_1
ι : Type u_1
α : ι → Type u_2
M : (i : ι) → Matroid (α i)
I X : Set ((i : ι) × α i)
hI : ∀ (i : ι), (M i).Indep (Sigma.mk i ⁻¹' I)
x✝ :
  I ⊆ X ∧
    (∀ ⦃t : Set ((i : ι) × α i)⦄, (∀ (i : ι), (M i).Indep (Sigma.mk i ⁻¹' t)) → t ⊆ X → I ⊆ t → I = t) ∧
      X ⊆ univ.sigma fun i => (M i).E
hIX : I ⊆ X
h' : X ⊆ univ.sigma fun i => (M i).E
i : ι
I₀ : Set (α i)
hI₀ : (M i).Indep I₀
hI₀X : I₀ ⊆ Sigma.mk i ⁻¹' X
hII₀ : Sigma.mk i ⁻¹' I ⊆ I₀
h :
  (∀ (i_1 : ι), (M i_1).Indep (Sigma.mk i_1 ⁻¹' (I ∪ Sigma.mk i '' I₀))) →
    I ∪ Sigma.mk i '' I₀ ⊆ X → I ⊆ I ∪ Sigma.mk i '' I₀ → I = I ∪ Sigma.mk i '' I₀
⊢ I₀ ⊆ Sigma.mk i ⁻¹' I

Tactic:
simp only [preimage_union, union_subset_iff, hIX, image_subset_iff, hI₀X, and_self,
  subset_union_left, true_implies] at h

Result:
case refine_1
ι : Type u_1
α : ι → Type u_2
M : (i : ι) → Matroid (α i)
I X : Set ((i : ι) × α i)
hI : ∀ (i : ι), (M i).Indep (Sigma.mk i ⁻¹' I)
x✝ :
  I ⊆ X ∧
    (∀ ⦃t : Set ((i : ι) × α i)⦄, (∀ (i : ι), (M i).Indep (Sigma.mk i ⁻¹' t)) → t ⊆ X → I ⊆ t → I = t) ∧
      X ⊆ univ.sigma fun i => (M i).E
hIX : I ⊆ X
h' : X ⊆ univ.sigma fun i => (M i).E
i : ι
I₀ : Set (α i)
hI₀ : (M i).Indep I₀
hI₀X : I₀ ⊆ Sigma.mk i ⁻¹' X
hII₀ : Sigma.mk i ⁻¹' I ⊆ I₀
h : (∀ (i_1 : ι), (M i_1).Indep (Sigma.mk i_1 ⁻¹' I ∪ Sigma.mk i_1 ⁻¹' (Sigma.mk i '' I₀))) → I = I ∪ Sigma.mk i '' I₀
⊢ I₀ ⊆ Sigma.mk i ⁻¹' I

Proof state:
ι : Type u_1
α : ι → Type u_2
M : (i : ι) → Matroid (α i)
I X : Set ((i : ι) × α i)
hI : ∀ (i : ι), (M i).Indep (Sigma.mk i ⁻¹' I)
x✝ :
  I ⊆ X ∧
    (∀ ⦃t : Set ((i : ι) × α i)⦄, (∀ (i : ι), (M i).Indep (Sigma.mk i ⁻¹' t)) → t ⊆ X → I ⊆ t → I = t) ∧
      X ⊆ univ.sigma fun i => (M i).E
hIX : I ⊆ X
h : ∀ ⦃t : Set ((i : ι) × α i)⦄, (∀ (i : ι), (M i).Indep (Sigma.mk i ⁻¹' t)) → t ⊆ X → I ⊆ t → I = t
h' : X ⊆ univ.sigma fun i => (M i).E
i : ι
⊢ Sigma.mk i ⁻¹' X ⊆ (M i).E

Tactic:
simpa using preimage_mono (f := Sigma.mk i) h'

Result:
no goals

Proof state:
ι : Type u_1
α : ι → Type u_2
M : (i : ι) → Matroid (α i)
I X : Set ((i : ι) × α i)
hI : ∀ (i : ι), (M i).Indep (Sigma.mk i ⁻¹' I)
x✝¹ :
  (∀ (x : ι), Sigma.mk x ⁻¹' I ⊆ Sigma.mk x ⁻¹' X) ∧
    (∀ (x : ι) ⦃t : Set (α x)⦄, (M x).Indep t → t ⊆ Sigma.mk x ⁻¹' X → Sigma.mk x ⁻¹' I ⊆ t → Sigma.mk x ⁻¹' I = t) ∧
      ∀ (x : ι), Sigma.mk x ⁻¹' X ⊆ (M x).E
hIX : ∀ (x : ι), Sigma.mk x ⁻¹' I ⊆ Sigma.mk x ⁻¹' X
h' : ∀ (x : ι) ⦃t : Set (α x)⦄, (M x).Indep t → t ⊆ Sigma.mk x ⁻¹' X → Sigma.mk x ⁻¹' I ⊆ t → Sigma.mk x ⁻¹' I = t
h'' : ∀ (x : ι), Sigma.mk x ⁻¹' X ⊆ (M x).E
x✝ : (i : ι) × α i
i : ι
x : α i
hx : ⟨i, x⟩ ∈ I
⊢ ⟨i, x⟩ ∈ X

Tactic:
simpa using hIX i hx

Result:
no goals

Proof state:
case refine_1
ι : Type u_1
α : ι → Type u_2
M : (i : ι) → Matroid (α i)
I X : Set ((i : ι) × α i)
hI : ∀ (i : ι), (M i).Indep (Sigma.mk i ⁻¹' I)
x✝ :
  I ⊆ X ∧
    (∀ ⦃t : Set ((i : ι) × α i)⦄, (∀ (i : ι), (M i).Indep (Sigma.mk i ⁻¹' t)) → t ⊆ X → I ⊆ t → I = t) ∧
      X ⊆ univ.sigma fun i => (M i).E
hIX : I ⊆ X
h' : X ⊆ univ.sigma fun i => (M i).E
i : ι
I₀ : Set (α i)
hI₀ : (M i).Indep I₀
hI₀X : I₀ ⊆ Sigma.mk i ⁻¹' X
hII₀ : Sigma.mk i ⁻¹' I ⊆ I₀
h : (∀ (i_1 : ι), (M i_1).Indep (Sigma.mk i_1 ⁻¹' I ∪ Sigma.mk i_1 ⁻¹' (Sigma.mk i '' I₀))) → I = I ∪ Sigma.mk i '' I₀
⊢ I₀ ⊆ Sigma.mk i ⁻¹' I

Tactic:
rw [h, preimage_union, sigma_mk_preimage_image_eq_self]

Result:
case refine_1
ι : Type u_1
α : ι → Type u_2
M : (i : ι) → Matroid (α i)
I X : Set ((i : ι) × α i)
hI : ∀ (i : ι), (M i).Indep (Sigma.mk i ⁻¹' I)
x✝ :
  I ⊆ X ∧
    (∀ ⦃t : Set ((i : ι) × α i)⦄, (∀ (i : ι), (M i).Indep (Sigma.mk i ⁻¹' t)) → t ⊆ X → I ⊆ t → I = t) ∧
      X ⊆ univ.sigma fun i => (M i).E
hIX : I ⊆ X
h' : X ⊆ univ.sigma fun i => (M i).E
i : ι
I₀ : Set (α i)
hI₀ : (M i).Indep I₀
hI₀X : I₀ ⊆ Sigma.mk i ⁻¹' X
hII₀ : Sigma.mk i ⁻¹' I ⊆ I₀
h : (∀ (i_1 : ι), (M i_1).Indep (Sigma.mk i_1 ⁻¹' I ∪ Sigma.mk i_1 ⁻¹' (Sigma.mk i '' I₀))) → I = I ∪ Sigma.mk i '' I₀
⊢ I₀ ⊆ Sigma.mk i ⁻¹' I ∪ I₀

case refine_1
ι : Type u_1
α : ι → Type u_2
M : (i : ι) → Matroid (α i)
I X : Set ((i : ι) × α i)
hI : ∀ (i : ι), (M i).Indep (Sigma.mk i ⁻¹' I)
x✝ :
  I ⊆ X ∧
    (∀ ⦃t : Set ((i : ι) × α i)⦄, (∀ (i : ι), (M i).Indep (Sigma.mk i ⁻¹' t)) → t ⊆ X → I ⊆ t → I = t) ∧
      X ⊆ univ.sigma fun i => (M i).E
hIX : I ⊆ X
h' : X ⊆ univ.sigma fun i => (M i).E
i : ι
I₀ : Set (α i)
hI₀ : (M i).Indep I₀
hI₀X : I₀ ⊆ Sigma.mk i ⁻¹' X
hII₀ : Sigma.mk i ⁻¹' I ⊆ I₀
h : (∀ (i_1 : ι), (M i_1).Indep (Sigma.mk i_1 ⁻¹' I ∪ Sigma.mk i_1 ⁻¹' (Sigma.mk i '' I₀))) → I = I ∪ Sigma.mk i '' I₀
⊢ ∀ (i_1 : ι), (M i_1).Indep (Sigma.mk i_1 ⁻¹' I ∪ Sigma.mk i_1 ⁻¹' (Sigma.mk i '' I₀))