Theorem: Profinite.NobelingProof.GoodProducts.span_iff_products
File: Mathlib/Topology/Category/Profinite/Nobeling.lean

Proof steps:
state_before: I : Type u
inst✝¹ : LinearOrder I
inst✝ : IsWellOrder I fun x x_1 => x < x_1
C : Set (I → Bool)
⊢ ⊤ ≤ Submodule.span ℤ (Set.range (eval C)) ↔ ⊤ ≤ Submodule.span ℤ (Set.range (Products.eval C))
state_after: I : Type u
inst✝¹ : LinearOrder I
inst✝ : IsWellOrder I fun x x_1 => x < x_1
C : Set (I → Bool)
h : ⊤ ≤ Submodule.span ℤ (Set.range (Products.eval C))
⊢ Submodule.span ℤ (Set.range (Products.eval C)) ≤ Submodule.span ℤ (Set.range (eval C))
tactic: refine ⟨fun h ↦ le_trans h (span_mono (fun a ⟨b, hb⟩ ↦ ⟨b.val, hb⟩)), fun h ↦ le_trans h ?_⟩
state_before: I : Type u
inst✝¹ : LinearOrder I
inst✝ : IsWellOrder I fun x x_1 => x < x_1
C : Set (I → Bool)
h : ⊤ ≤ Submodule.span ℤ (Set.range (Products.eval C))
⊢ Submodule.span ℤ (Set.range (Products.eval C)) ≤ Submodule.span ℤ (Set.range (eval C))
state_after: I : Type u
inst✝¹ : LinearOrder I
inst✝ : IsWellOrder I fun x x_1 => x < x_1
C : Set (I → Bool)
h : ⊤ ≤ Submodule.span ℤ (Set.range (Products.eval C))
⊢ Set.range (Products.eval C) ⊆ ↑(Submodule.span ℤ (Set.range (eval C)))
tactic: rw [span_le]
state_before: I : Type u
inst✝¹ : LinearOrder I
inst✝ : IsWellOrder I fun x x_1 => x < x_1
C : Set (I → Bool)
h : ⊤ ≤ Submodule.span ℤ (Set.range (Products.eval C))
⊢ Set.range (Products.eval C) ⊆ ↑(Submodule.span ℤ (Set.range (eval C)))
state_after: case intro
I : Type u
inst✝¹ : LinearOrder I
inst✝ : IsWellOrder I fun x x_1 => x < x_1
C : Set (I → Bool)
h : ⊤ ≤ Submodule.span ℤ (Set.range (Products.eval C))
l : Products I
⊢ Products.eval C l ∈ ↑(Submodule.span ℤ (Set.range (eval C)))
tactic: rintro f ⟨l, rfl⟩
state_before: case intro
I : Type u
inst✝¹ : LinearOrder I
inst✝ : IsWellOrder I fun x x_1 => x < x_1
C : Set (I → Bool)
h : ⊤ ≤ Submodule.span ℤ (Set.range (Products.eval C))
l : Products I
⊢ Products.eval C l ∈ ↑(Submodule.span ℤ (Set.range (eval C)))
state_after: case intro
I : Type u
inst✝¹ : LinearOrder I
inst✝ : IsWellOrder I fun x x_1 => x < x_1
C : Set (I → Bool)
h : ⊤ ≤ Submodule.span ℤ (Set.range (Products.eval C))
l : Products I
L : Products I → Prop := fun m => Products.eval C m ∈ Submodule.span ℤ (Set.range (eval C))
⊢ Products.eval C l ∈ ↑(Submodule.span ℤ (Set.range (eval C)))
tactic: let L : Products I → Prop := fun m ↦ m.eval C ∈ span ℤ (Set.range (GoodProducts.eval C))
state_before: case intro
I : Type u
inst✝¹ : LinearOrder I
inst✝ : IsWellOrder I fun x x_1 => x < x_1
C : Set (I → Bool)
h : ⊤ ≤ Submodule.span ℤ (Set.range (Products.eval C))
l : Products I
L : Products I → Prop := fun m => Products.eval C m ∈ Submodule.span ℤ (Set.range (eval C))
⊢ Products.eval C l ∈ ↑(Submodule.span ℤ (Set.range (eval C)))
state_after: case intro
I : Type u
inst✝¹ : LinearOrder I
inst✝ : IsWellOrder I fun x x_1 => x < x_1
C : Set (I → Bool)
h : ⊤ ≤ Submodule.span ℤ (Set.range (Products.eval C))
l : Products I
L : Products I → Prop := fun m => Products.eval C m ∈ Submodule.span ℤ (Set.range (eval C))
⊢ L l
tactic: suffices L l by assumption
state_before: case intro.a
I : Type u
inst✝¹ : LinearOrder I
inst✝ : IsWellOrder I fun x x_1 => x < x_1
C : Set (I → Bool)
h : ⊤ ≤ Submodule.span ℤ (Set.range (Products.eval C))
l : Products I
L : Products I → Prop := fun m => Products.eval C m ∈ Submodule.span ℤ (Set.range (eval C))
⊢ ∀ (x : Products I), (∀ y < x, L y) → L x
state_after: case intro.a
I : Type u
inst✝¹ : LinearOrder I
inst✝ : IsWellOrder I fun x x_1 => x < x_1
C : Set (I → Bool)
h✝ : ⊤ ≤ Submodule.span ℤ (Set.range (Products.eval C))
l✝ : Products I
L : Products I → Prop := fun m => Products.eval C m ∈ Submodule.span ℤ (Set.range (eval C))
l : Products I
h : ∀ y < l, L y
⊢ L l
tactic: intro l h
state_before: case intro.a
I : Type u
inst✝¹ : LinearOrder I
inst✝ : IsWellOrder I fun x x_1 => x < x_1
C : Set (I → Bool)
h✝ : ⊤ ≤ Submodule.span ℤ (Set.range (Products.eval C))
l✝ : Products I
L : Products I → Prop := fun m => Products.eval C m ∈ Submodule.span ℤ (Set.range (eval C))
l : Products I
h : ∀ y < l, L y
⊢ L l
state_after: case pos
I : Type u
inst✝¹ : LinearOrder I
inst✝ : IsWellOrder I fun x x_1 => x < x_1
C : Set (I → Bool)
h✝ : ⊤ ≤ Submodule.span ℤ (Set.range (Products.eval C))
l✝ : Products I
L : Products I → Prop := fun m => Products.eval C m ∈ Submodule.span ℤ (Set.range (eval C))
l : Products I
h : ∀ y < l, L y
hl : Products.isGood C l
⊢ L l

case neg
I : Type u
inst✝¹ : LinearOrder I
inst✝ : IsWellOrder I fun x x_1 => x < x_1
C : Set (I → Bool)
h✝ : ⊤ ≤ Submodule.span ℤ (Set.range (Products.eval C))
l✝ : Products I
L : Products I → Prop := fun m => Products.eval C m ∈ Submodule.span ℤ (Set.range (eval C))
l : Products I
h : ∀ y < l, L y
hl : ¬Products.isGood C l
⊢ L l
tactic: by_cases hl : l.isGood C