File: Mathlib/LinearAlgebra/Projectivization/Independence.lean

full_name: Projectivization.Independent.mk
code: /-- A linearly independent family of nonzero vectors gives an independent family of points
in projective space. -/
inductive Independent : (ι → ℙ K V) → Prop
  | mk (f : ι → V) (hf : ∀ i : ι, f i ≠ 0) (hl : LinearIndependent K f) :
    Independent fun i => mk K (f i) (hf i)
goal: Projectivization.Independent fun i => Projectivization.mk K (f i) ⋯

full_name: Projectivization.Independent
code: /-- A linearly independent family of nonzero vectors gives an independent family of points
in projective space. -/
inductive Independent : (ι → ℙ K V) → Prop
  | mk (f : ι → V) (hf : ∀ i : ι, f i ≠ 0) (hl : LinearIndependent K f) :
    Independent fun i => mk K (f i) (hf i)
goal: (ι → ℙ K V) → Prop

full_name: Projectivization.independent_iff
code: /-- A family of points in a projective space is independent if and only if the representative
vectors determined by the family are linearly independent. -/
theorem independent_iff : Independent f ↔ LinearIndependent K (Projectivization.rep ∘ f) 
goal: Projectivization.Independent f ↔ LinearIndependent K (Projectivization.rep ∘ f)

full_name: Projectivization.independent_iff_completeLattice_independent
code: /-- A family of points in projective space is independent if and only if the family of
submodules which the points determine is independent in the lattice-theoretic sense. -/
theorem independent_iff_completeLattice_independent :
    Independent f ↔ CompleteLattice.Independent fun i => (f i).submodule 
goal: Projectivization.Independent f ↔ CompleteLattice.Independent fun i => (f i).submodule

full_name: Projectivization.Dependent.mk
code: /-- A linearly dependent family of nonzero vectors gives a dependent family of points
in projective space. -/
inductive Dependent : (ι → ℙ K V) → Prop
  | mk (f : ι → V) (hf : ∀ i : ι, f i ≠ 0) (h : ¬LinearIndependent K f) :
    Dependent fun i => mk K (f i) (hf i)
goal: Projectivization.Dependent fun i => Projectivization.mk K (f i) ⋯

full_name: Projectivization.Dependent
code: /-- A linearly dependent family of nonzero vectors gives a dependent family of points
in projective space. -/
inductive Dependent : (ι → ℙ K V) → Prop
  | mk (f : ι → V) (hf : ∀ i : ι, f i ≠ 0) (h : ¬LinearIndependent K f) :
    Dependent fun i => mk K (f i) (hf i)
goal: (ι → ℙ K V) → Prop

full_name: Projectivization.dependent_iff
code: /-- A family of points in a projective space is dependent if and only if their
representatives are linearly dependent. -/
theorem dependent_iff : Dependent f ↔ ¬LinearIndependent K (Projectivization.rep ∘ f) 
goal: Projectivization.Dependent f ↔ ¬LinearIndependent K (Projectivization.rep ∘ f)

full_name: Projectivization.dependent_iff_not_independent
code: /-- Dependence is the negation of independence. -/
theorem dependent_iff_not_independent : Dependent f ↔ ¬Independent f 
goal: Projectivization.Dependent f ↔ ¬Projectivization.Independent f

full_name: Projectivization.independent_iff_not_dependent
code: /-- Independence is the negation of dependence. -/
theorem independent_iff_not_dependent : Independent f ↔ ¬Dependent f 
goal: Projectivization.Independent f ↔ ¬Projectivization.Dependent f

full_name: Projectivization.dependent_pair_iff_eq
code: /-- Two points in a projective space are dependent if and only if they are equal. -/
@[simp]
theorem dependent_pair_iff_eq (u v : ℙ K V) : Dependent ![u, v] ↔ u = v 
goal: Projectivization.Dependent ![u, v] ↔ u = v

full_name: Projectivization.independent_pair_iff_neq
code: /-- Two points in a projective space are independent if and only if the points are not equal. -/
@[simp]
theorem independent_pair_iff_neq (u v : ℙ K V) : Independent ![u, v] ↔ u ≠ v 
goal: Projectivization.Independent ![u, v] ↔ u ≠ v