Affine spaces

This file defines affine subspaces (over modules) and the affine span of a set of points.

Main definitions

• AffineSubspace k P is the type of affine subspaces. Unlike affine spaces, affine subspaces are allowed to be empty, and lemmas that do not apply to empty affine subspaces have Nonempty hypotheses. There is a CompleteLattice structure on affine subspaces.
• AffineSubspace.direction gives the Submodule spanned by the pairwise differences of points in an AffineSubspace. There are various lemmas relating to the set of vectors in the direction, and relating the lattice structure on affine subspaces to that on their directions.
• affineSpan gives the affine subspace spanned by a set of points, with vectorSpan giving its direction. The affineSpan is defined in terms of spanPoints, which gives an explicit description of the points contained in the affine span; spanPoints itself should generally only be used when that description is required, with affineSpan being the main definition for other purposes. Two other descriptions of the affine span are proved equivalent: it is the sInf of affine subspaces containing the points, and (if [Nontrivial k]) it contains exactly those points that are affine combinations of points in the given set.

Implementation notes

outParam is used in the definition of AddTorsor V P to make V an implicit argument (deduced from P) in most cases. As for modules, k is an explicit argument rather than implied by P or V.

This file only provides purely algebraic definitions and results. Those depending on analysis or topology are defined elsewhere; see Analysis.Normed.Affine.AddTorsor and Topology.Algebra.Affine.

References

• https://en.wikipedia.org/wiki/Affine_space
• https://en.wikipedia.org/wiki/Principal_homogeneous_space

def vectorSpan (k : Type u_1) {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [AddTorsor V P] (s : Set P) : Submodule k V

The submodule spanning the differences of a (possibly empty) set of points.

Equations

• vectorSpan k s = Submodule.span k (s -ᵥ s)

theorem vectorSpan_def (k : Type u_1) {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [AddTorsor V P] (s : Set P) : vectorSpan k s = Submodule.span k (s -ᵥ s)

The definition of vectorSpan, for rewriting.

theorem vectorSpan_mono (k : Type u_1) {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [AddTorsor V P] {s₁ s₂ : Set P} (h : s₁ ⊆ s₂) : vectorSpan k s₁ ≤ vectorSpan k s₂

vectorSpan is monotone.

@[simp]

theorem vectorSpan_empty (k : Type u_1) {V : Type u_2} (P : Type u_3) [Ring k] [AddCommGroup V] [Module k V] [AddTorsor V P] : vectorSpan k ∅ = ⊥

The vectorSpan of the empty set is ⊥.

@[simp]

theorem vectorSpan_singleton (k : Type u_1) {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [AddTorsor V P] (p : P) : vectorSpan k {p} = ⊥

The vectorSpan of a single point is ⊥.

theorem vsub_set_subset_vectorSpan (k : Type u_1) {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [AddTorsor V P] (s : Set P) : s -ᵥ s ⊆ ↑(vectorSpan k s)

The s -ᵥ s lies within the vectorSpan k s.

theorem vsub_mem_vectorSpan (k : Type u_1) {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [AddTorsor V P] {s : Set P} {p₁ p₂ : P} (hp₁ : p₁ ∈ s) (hp₂ : p₂ ∈ s) : p₁ -ᵥ p₂ ∈ vectorSpan k s

Each pairwise difference is in the vectorSpan.

def spanPoints (k : Type u_1) {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [AddTorsor V P] (s : Set P) : Set P

The points in the affine span of a (possibly empty) set of points. Use affineSpan instead to get an AffineSubspace k P.

Equations

• spanPoints k s = {p : P | ∃ p₁ ∈ s, ∃ v ∈ vectorSpan k s, p = v +ᵥ p₁}

theorem mem_spanPoints (k : Type u_1) {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [AddTorsor V P] (p : P) (s : Set P) : p ∈ s → p ∈ spanPoints k s

A point in a set is in its affine span.

theorem subset_spanPoints (k : Type u_1) {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [AddTorsor V P] (s : Set P) : s ⊆ spanPoints k s

A set is contained in its spanPoints.

@[simp]

theorem spanPoints_nonempty (k : Type u_1) {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [AddTorsor V P] (s : Set P) : (spanPoints k s).Nonempty ↔ s.Nonempty

The spanPoints of a set is nonempty if and only if that set is.

theorem vadd_mem_spanPoints_of_mem_spanPoints_of_mem_vectorSpan (k : Type u_1) {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [AddTorsor V P] {s : Set P} {p : P} {v : V} (hp : p ∈ spanPoints k s) (hv : v ∈ vectorSpan k s) : v +ᵥ p ∈ spanPoints k s

Adding a point in the affine span and a vector in the spanning submodule produces a point in the affine span.

theorem vsub_mem_vectorSpan_of_mem_spanPoints_of_mem_spanPoints (k : Type u_1) {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [AddTorsor V P] {s : Set P} {p₁ p₂ : P} (hp₁ : p₁ ∈ spanPoints k s) (hp₂ : p₂ ∈ spanPoints k s) : p₁ -ᵥ p₂ ∈ vectorSpan k s

Subtracting two points in the affine span produces a vector in the spanning submodule.

structure AffineSubspace (k : Type u_1) {V : Type u_2} (P : Type u_3) [Ring k] [AddCommGroup V] [Module k V] [AddTorsor V P] : Type u_3

An AffineSubspace k P is a subset of an AffineSpace V P that, if not empty, has an affine space structure induced by a corresponding subspace of the Module k V.

• carrier : Set P

  The affine subspace seen as a subset.

• smul_vsub_vadd_mem (c : k) {p₁ p₂ p₃ : P} : p₁ ∈ self.carrier → p₂ ∈ self.carrier → p₃ ∈ self.carrier → c • (p₁ -ᵥ p₂) +ᵥ p₃ ∈ self.carrier

def Submodule.toAffineSubspace {k : Type u_1} {V : Type u_2} [Ring k] [AddCommGroup V] [Module k V] (p : Submodule k V) : AffineSubspace k V

Reinterpret p : Submodule k V as an AffineSubspace k V.

Equations

• p.toAffineSubspace = { carrier := ↑p, smul_vsub_vadd_mem := ⋯ }

instance AffineSubspace.instSetLike (k : Type u_1) {V : Type u_2} (P : Type u_3) [Ring k] [AddCommGroup V] [Module k V] [AddTorsor V P] : SetLike (AffineSubspace k P) P

Equations

• AffineSubspace.instSetLike k P = { coe := AffineSubspace.carrier, coe_injective' := ⋯ }

theorem AffineSubspace.mem_coe (k : Type u_1) {V : Type u_2} (P : Type u_3) [Ring k] [AddCommGroup V] [Module k V] [AddTorsor V P] (p : P) (s : AffineSubspace k P) : p ∈ ↑s ↔ p ∈ s

A point is in an affine subspace coerced to a set if and only if it is in that affine subspace.

def AffineSubspace.direction {k : Type u_1} {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [AddTorsor V P] (s : AffineSubspace k P) : Submodule k V

The direction of an affine subspace is the submodule spanned by the pairwise differences of points. (Except in the case of an empty affine subspace, where the direction is the zero submodule, every vector in the direction is the difference of two points in the affine subspace.)

Equations

• s.direction = vectorSpan k ↑s

theorem AffineSubspace.direction_eq_vectorSpan {k : Type u_1} {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [AddTorsor V P] (s : AffineSubspace k P) : s.direction = vectorSpan k ↑s

The direction equals the vectorSpan.

def AffineSubspace.directionOfNonempty {k : Type u_1} {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [AddTorsor V P] {s : AffineSubspace k P} (h : (↑s).Nonempty) : Submodule k V

Alternative definition of the direction when the affine subspace is nonempty. This is defined so that the order on submodules (as used in the definition of Submodule.span) can be used in the proof of coe_direction_eq_vsub_set, and is not intended to be used beyond that proof.

Equations

• AffineSubspace.directionOfNonempty h = { carrier := ↑s -ᵥ ↑s, add_mem' := ⋯, zero_mem' := ⋯, smul_mem' := ⋯ }

theorem AffineSubspace.directionOfNonempty_eq_direction {k : Type u_1} {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [AddTorsor V P] {s : AffineSubspace k P} (h : (↑s).Nonempty) : directionOfNonempty h = s.direction

direction_of_nonempty gives the same submodule as direction.

theorem AffineSubspace.coe_direction_eq_vsub_set {k : Type u_1} {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [AddTorsor V P] {s : AffineSubspace k P} (h : (↑s).Nonempty) : ↑s.direction = ↑s -ᵥ ↑s

The set of vectors in the direction of a nonempty affine subspace is given by vsub_set.

theorem AffineSubspace.mem_direction_iff_eq_vsub {k : Type u_1} {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [AddTorsor V P] {s : AffineSubspace k P} (h : (↑s).Nonempty) (v : V) : v ∈ s.direction ↔ ∃ p₁ ∈ s, ∃ p₂ ∈ s, v = p₁ -ᵥ p₂

A vector is in the direction of a nonempty affine subspace if and only if it is the subtraction of two vectors in the subspace.

theorem AffineSubspace.vadd_mem_of_mem_direction {k : Type u_1} {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [AddTorsor V P] {s : AffineSubspace k P} {v : V} (hv : v ∈ s.direction) {p : P} (hp : p ∈ s) : v +ᵥ p ∈ s

Adding a vector in the direction to a point in the subspace produces a point in the subspace.

theorem AffineSubspace.vsub_mem_direction {k : Type u_1} {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [AddTorsor V P] {s : AffineSubspace k P} {p₁ p₂ : P} (hp₁ : p₁ ∈ s) (hp₂ : p₂ ∈ s) : p₁ -ᵥ p₂ ∈ s.direction

Subtracting two points in the subspace produces a vector in the direction.

theorem AffineSubspace.vadd_mem_iff_mem_direction {k : Type u_1} {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [AddTorsor V P] {s : AffineSubspace k P} (v : V) {p : P} (hp : p ∈ s) : v +ᵥ p ∈ s ↔ v ∈ s.direction

Adding a vector to a point in a subspace produces a point in the subspace if and only if the vector is in the direction.

theorem AffineSubspace.vadd_mem_iff_mem_of_mem_direction {k : Type u_1} {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [AddTorsor V P] {s : AffineSubspace k P} {v : V} (hv : v ∈ s.direction) {p : P} : v +ᵥ p ∈ s ↔ p ∈ s

Adding a vector in the direction to a point produces a point in the subspace if and only if the original point is in the subspace.

theorem AffineSubspace.coe_direction_eq_vsub_set_right {k : Type u_1} {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [AddTorsor V P] {s : AffineSubspace k P} {p : P} (hp : p ∈ s) : ↑s.direction = (fun (x : P) => x -ᵥ p) '' ↑s

Given a point in an affine subspace, the set of vectors in its direction equals the set of vectors subtracting that point on the right.

theorem AffineSubspace.coe_direction_eq_vsub_set_left {k : Type u_1} {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [AddTorsor V P] {s : AffineSubspace k P} {p : P} (hp : p ∈ s) : ↑s.direction = (fun (x : P) => p -ᵥ x) '' ↑s

Given a point in an affine subspace, the set of vectors in its direction equals the set of vectors subtracting that point on the left.

theorem AffineSubspace.mem_direction_iff_eq_vsub_right {k : Type u_1} {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [AddTorsor V P] {s : AffineSubspace k P} {p : P} (hp : p ∈ s) (v : V) : v ∈ s.direction ↔ ∃ p₂ ∈ s, v = p₂ -ᵥ p

Given a point in an affine subspace, a vector is in its direction if and only if it results from subtracting that point on the right.

theorem AffineSubspace.mem_direction_iff_eq_vsub_left {k : Type u_1} {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [AddTorsor V P] {s : AffineSubspace k P} {p : P} (hp : p ∈ s) (v : V) : v ∈ s.direction ↔ ∃ p₂ ∈ s, v = p -ᵥ p₂

Given a point in an affine subspace, a vector is in its direction if and only if it results from subtracting that point on the left.

theorem AffineSubspace.coe_injective {k : Type u_1} {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [AddTorsor V P] : Function.Injective SetLike.coe

Two affine subspaces are equal if they have the same points.

theorem AffineSubspace.ext {k : Type u_1} {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [AddTorsor V P] {p q : AffineSubspace k P} (h : ∀ (x : P), x ∈ p ↔ x ∈ q) : p = q

theorem AffineSubspace.ext_iff {k : Type u_1} {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [AddTorsor V P] (s₁ s₂ : AffineSubspace k P) : s₁ = s₂ ↔ ↑s₁ = ↑s₂

theorem AffineSubspace.ext_of_direction_eq {k : Type u_1} {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [AddTorsor V P] {s₁ s₂ : AffineSubspace k P} (hd : s₁.direction = s₂.direction) (hn : (↑s₁ ∩ ↑s₂).Nonempty) : s₁ = s₂

Two affine subspaces with the same direction and nonempty intersection are equal.

theorem AffineSubspace.eq_iff_direction_eq_of_mem {k : Type u_1} {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [AddTorsor V P] {s₁ s₂ : AffineSubspace k P} {p : P} (h₁ : p ∈ s₁) (h₂ : p ∈ s₂) : s₁ = s₂ ↔ s₁.direction = s₂.direction

Two affine subspaces with nonempty intersection are equal if and only if their directions are equal.

def AffineSubspace.mk' {k : Type u_1} {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [AddTorsor V P] (p : P) (direction : Submodule k V) : AffineSubspace k P

Construct an affine subspace from a point and a direction.

Equations

• AffineSubspace.mk' p direction = { carrier := {q : P | ∃ v ∈ direction, q = v +ᵥ p}, smul_vsub_vadd_mem := ⋯ }

theorem AffineSubspace.self_mem_mk' {k : Type u_1} {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [AddTorsor V P] (p : P) (direction : Submodule k V) : p ∈ mk' p direction

An affine subspace constructed from a point and a direction contains that point.

theorem AffineSubspace.vadd_mem_mk' {k : Type u_1} {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [AddTorsor V P] {v : V} (p : P) {direction : Submodule k V} (hv : v ∈ direction) : v +ᵥ p ∈ mk' p direction

An affine subspace constructed from a point and a direction contains the result of adding a vector in that direction to that point.

theorem AffineSubspace.mk'_nonempty {k : Type u_1} {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [AddTorsor V P] (p : P) (direction : Submodule k V) : (↑(mk' p direction)).Nonempty

An affine subspace constructed from a point and a direction is nonempty.

@[simp]

theorem AffineSubspace.direction_mk' {k : Type u_1} {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [AddTorsor V P] (p : P) (direction : Submodule k V) : (mk' p direction).direction = direction

The direction of an affine subspace constructed from a point and a direction.

theorem AffineSubspace.mem_mk'_iff_vsub_mem {k : Type u_1} {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [AddTorsor V P] {p₁ p₂ : P} {direction : Submodule k V} : p₂ ∈ mk' p₁ direction ↔ p₂ -ᵥ p₁ ∈ direction

A point lies in an affine subspace constructed from another point and a direction if and only if their difference is in that direction.

@[simp]

theorem AffineSubspace.mk'_eq {k : Type u_1} {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [AddTorsor V P] {s : AffineSubspace k P} {p : P} (hp : p ∈ s) : mk' p s.direction = s

Constructing an affine subspace from a point in a subspace and that subspace's direction yields the original subspace.

theorem AffineSubspace.spanPoints_subset_coe_of_subset_coe {k : Type u_1} {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [AddTorsor V P] {s : Set P} {s₁ : AffineSubspace k P} (h : s ⊆ ↑s₁) : spanPoints k s ⊆ ↑s₁

If an affine subspace contains a set of points, it contains the spanPoints of that set.

@[simp]

theorem Submodule.mem_toAffineSubspace {k : Type u_1} {V : Type u_2} [Ring k] [AddCommGroup V] [Module k V] {p : Submodule k V} {x : V} : x ∈ p.toAffineSubspace ↔ x ∈ p

@[simp]

theorem Submodule.toAffineSubspace_direction {k : Type u_1} {V : Type u_2} [Ring k] [AddCommGroup V] [Module k V] (s : Submodule k V) : s.toAffineSubspace.direction = s

def affineSpan (k : Type u_1) {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [AddTorsor V P] (s : Set P) : AffineSubspace k P

The affine span of a set of points is the smallest affine subspace containing those points. (Actually defined here in terms of spans in modules.)

Equations

• affineSpan k s = { carrier := spanPoints k s, smul_vsub_vadd_mem := ⋯ }

@[simp]

theorem coe_affineSpan (k : Type u_1) {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [AddTorsor V P] (s : Set P) : ↑(affineSpan k s) = spanPoints k s

The affine span, converted to a set, is spanPoints.

theorem subset_affineSpan (k : Type u_1) {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [AddTorsor V P] (s : Set P) : s ⊆ ↑(affineSpan k s)

A set is contained in its affine span.

theorem direction_affineSpan (k : Type u_1) {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [AddTorsor V P] (s : Set P) : (affineSpan k s).direction = vectorSpan k s

The direction of the affine span is the vectorSpan.

theorem mem_affineSpan (k : Type u_1) {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [AddTorsor V P] {p : P} {s : Set P} (hp : p ∈ s) : p ∈ affineSpan k s

A point in a set is in its affine span.

@[simp]

theorem vectorSpan_add_self (k : Type u_1) {V : Type u_2} [Ring k] [AddCommGroup V] [Module k V] (s : Set V) : ↑(vectorSpan k s) + s = ↑(affineSpan k s)

instance AffineSubspace.instCompleteLattice {k : Type u_1} {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [S : AddTorsor V P] : CompleteLattice (AffineSubspace k P)

Equations

• AffineSubspace.instCompleteLattice = CompleteLattice.mk ⋯ ⋯ ⋯ ⋯ ⋯ ⋯

instance AffineSubspace.instInhabited {k : Type u_1} {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [S : AddTorsor V P] : Inhabited (AffineSubspace k P)

Equations

• AffineSubspace.instInhabited = { default := ⊤ }

theorem AffineSubspace.le_def {k : Type u_1} {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [S : AddTorsor V P] (s₁ s₂ : AffineSubspace k P) : s₁ ≤ s₂ ↔ ↑s₁ ⊆ ↑s₂

The ≤ order on subspaces is the same as that on the corresponding sets.

theorem AffineSubspace.le_def' {k : Type u_1} {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [S : AddTorsor V P] (s₁ s₂ : AffineSubspace k P) : s₁ ≤ s₂ ↔ ∀ p ∈ s₁, p ∈ s₂

One subspace is less than or equal to another if and only if all its points are in the second subspace.

theorem AffineSubspace.lt_def {k : Type u_1} {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [S : AddTorsor V P] (s₁ s₂ : AffineSubspace k P) : s₁ < s₂ ↔ ↑s₁ ⊂ ↑s₂

The < order on subspaces is the same as that on the corresponding sets.

theorem AffineSubspace.not_le_iff_exists {k : Type u_1} {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [S : AddTorsor V P] (s₁ s₂ : AffineSubspace k P) : ¬s₁ ≤ s₂ ↔ ∃ p ∈ s₁, p ∉ s₂

One subspace is not less than or equal to another if and only if it has a point not in the second subspace.

theorem AffineSubspace.exists_of_lt {k : Type u_1} {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [S : AddTorsor V P] {s₁ s₂ : AffineSubspace k P} (h : s₁ < s₂) : ∃ p ∈ s₂, p ∉ s₁

If a subspace is less than another, there is a point only in the second.

theorem AffineSubspace.lt_iff_le_and_exists {k : Type u_1} {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [S : AddTorsor V P] (s₁ s₂ : AffineSubspace k P) : s₁ < s₂ ↔ s₁ ≤ s₂ ∧ ∃ p ∈ s₂, p ∉ s₁

A subspace is less than another if and only if it is less than or equal to the second subspace and there is a point only in the second.

theorem AffineSubspace.eq_of_direction_eq_of_nonempty_of_le {k : Type u_1} {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [S : AddTorsor V P] {s₁ s₂ : AffineSubspace k P} (hd : s₁.direction = s₂.direction) (hn : (↑s₁).Nonempty) (hle : s₁ ≤ s₂) : s₁ = s₂

If an affine subspace is nonempty and contained in another with the same direction, they are equal.

theorem AffineSubspace.affineSpan_eq_sInf (k : Type u_1) (V : Type u_2) {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [S : AddTorsor V P] (s : Set P) : affineSpan k s = sInf {s' : AffineSubspace k P | s ⊆ ↑s'}

The affine span is the sInf of subspaces containing the given points.

def AffineSubspace.gi (k : Type u_1) (V : Type u_2) (P : Type u_3) [Ring k] [AddCommGroup V] [Module k V] [S : AddTorsor V P] : GaloisInsertion (affineSpan k) SetLike.coe

The Galois insertion formed by affineSpan and coercion back to a set.

Equations

• AffineSubspace.gi k V P = { choice := fun (s : Set P) (x : ↑(affineSpan k s) ≤ s) => affineSpan k s, gc := ⋯, le_l_u := ⋯, choice_eq := ⋯ }

@[simp]

theorem AffineSubspace.span_empty (k : Type u_1) (V : Type u_2) (P : Type u_3) [Ring k] [AddCommGroup V] [Module k V] [S : AddTorsor V P] : affineSpan k ∅ = ⊥

The span of the empty set is ⊥.

@[simp]

theorem AffineSubspace.span_univ (k : Type u_1) (V : Type u_2) (P : Type u_3) [Ring k] [AddCommGroup V] [Module k V] [S : AddTorsor V P] : affineSpan k Set.univ = ⊤

The span of univ is ⊤.

theorem affineSpan_le {k : Type u_1} {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [S : AddTorsor V P] {s : Set P} {Q : AffineSubspace k P} : affineSpan k s ≤ Q ↔ s ⊆ ↑Q

theorem AffineSubspace.span_union (k : Type u_1) (V : Type u_2) {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [S : AddTorsor V P] (s t : Set P) : affineSpan k (s ∪ t) = affineSpan k s ⊔ affineSpan k t

The span of a union of sets is the sup of their spans.

theorem AffineSubspace.span_iUnion (k : Type u_1) (V : Type u_2) {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [S : AddTorsor V P] {ι : Type u_4} (s : ι → Set P) : affineSpan k (⋃ (i : ι), s i) = ⨆ (i : ι), affineSpan k (s i)

The span of a union of an indexed family of sets is the sup of their spans.

@[simp]

theorem AffineSubspace.top_coe (k : Type u_1) (V : Type u_2) (P : Type u_3) [Ring k] [AddCommGroup V] [Module k V] [S : AddTorsor V P] : ↑⊤ = Set.univ

⊤, coerced to a set, is the whole set of points.

@[simp]

theorem AffineSubspace.mem_top (k : Type u_1) (V : Type u_2) {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [S : AddTorsor V P] (p : P) : p ∈ ⊤

All points are in ⊤.

@[simp]

theorem AffineSubspace.mk'_top (k : Type u_1) (V : Type u_2) {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [S : AddTorsor V P] (p : P) : mk' p ⊤ = ⊤

@[simp]

theorem AffineSubspace.direction_top (k : Type u_1) (V : Type u_2) (P : Type u_3) [Ring k] [AddCommGroup V] [Module k V] [S : AddTorsor V P] : ⊤.direction = ⊤

The direction of ⊤ is the whole module as a submodule.

@[simp]

theorem AffineSubspace.bot_coe (k : Type u_1) (V : Type u_2) (P : Type u_3) [Ring k] [AddCommGroup V] [Module k V] [S : AddTorsor V P] : ↑⊥ = ∅

⊥, coerced to a set, is the empty set.

theorem AffineSubspace.bot_ne_top (k : Type u_1) (V : Type u_2) (P : Type u_3) [Ring k] [AddCommGroup V] [Module k V] [S : AddTorsor V P] : ⊥ ≠ ⊤

instance AffineSubspace.instNontrivial (k : Type u_1) (V : Type u_2) (P : Type u_3) [Ring k] [AddCommGroup V] [Module k V] [S : AddTorsor V P] : Nontrivial (AffineSubspace k P)

theorem AffineSubspace.nonempty_of_affineSpan_eq_top (k : Type u_1) (V : Type u_2) (P : Type u_3) [Ring k] [AddCommGroup V] [Module k V] [S : AddTorsor V P] {s : Set P} (h : affineSpan k s = ⊤) : s.Nonempty

theorem AffineSubspace.vectorSpan_eq_top_of_affineSpan_eq_top (k : Type u_1) (V : Type u_2) (P : Type u_3) [Ring k] [AddCommGroup V] [Module k V] [S : AddTorsor V P] {s : Set P} (h : affineSpan k s = ⊤) : vectorSpan k s = ⊤

If the affine span of a set is ⊤, then the vector span of the same set is the ⊤.

theorem AffineSubspace.affineSpan_eq_top_iff_vectorSpan_eq_top_of_nonempty (k : Type u_1) (V : Type u_2) (P : Type u_3) [Ring k] [AddCommGroup V] [Module k V] [S : AddTorsor V P] {s : Set P} (hs : s.Nonempty) : affineSpan k s = ⊤ ↔ vectorSpan k s = ⊤

For a nonempty set, the affine span is ⊤ iff its vector span is ⊤.

theorem AffineSubspace.affineSpan_eq_top_iff_vectorSpan_eq_top_of_nontrivial (k : Type u_1) (V : Type u_2) (P : Type u_3) [Ring k] [AddCommGroup V] [Module k V] [S : AddTorsor V P] {s : Set P} [Nontrivial P] : affineSpan k s = ⊤ ↔ vectorSpan k s = ⊤

For a non-trivial space, the affine span of a set is ⊤ iff its vector span is ⊤.

theorem AffineSubspace.card_pos_of_affineSpan_eq_top (k : Type u_1) (V : Type u_2) (P : Type u_3) [Ring k] [AddCommGroup V] [Module k V] [S : AddTorsor V P] {ι : Type u_4} [Fintype ι] {p : ι → P} (h : affineSpan k (Set.range p) = ⊤) : 0 < Fintype.card ι

instance AffineSubspace.instNonemptySubtypeMemTop (k : Type u_1) (V : Type u_2) (P : Type u_3) [Ring k] [AddCommGroup V] [Module k V] [S : AddTorsor V P] : Nonempty ↥⊤

theorem AffineSubspace.not_mem_bot (k : Type u_1) (V : Type u_2) {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [S : AddTorsor V P] (p : P) : p ∉ ⊥

No points are in ⊥.

instance AffineSubspace.isEmpty_bot (k : Type u_1) (V : Type u_2) {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [S : AddTorsor V P] : IsEmpty ↥⊥

@[simp]

theorem AffineSubspace.direction_bot (k : Type u_1) (V : Type u_2) (P : Type u_3) [Ring k] [AddCommGroup V] [Module k V] [S : AddTorsor V P] : ⊥.direction = ⊥

The direction of ⊥ is the submodule ⊥.

@[simp]

theorem AffineSubspace.coe_eq_bot_iff {k : Type u_1} {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [S : AddTorsor V P] (Q : AffineSubspace k P) : ↑Q = ∅ ↔ Q = ⊥

@[simp]

theorem AffineSubspace.coe_eq_univ_iff {k : Type u_1} {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [S : AddTorsor V P] (Q : AffineSubspace k P) : ↑Q = Set.univ ↔ Q = ⊤

theorem AffineSubspace.nonempty_iff_ne_bot {k : Type u_1} {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [S : AddTorsor V P] (Q : AffineSubspace k P) : (↑Q).Nonempty ↔ Q ≠ ⊥

theorem AffineSubspace.eq_bot_or_nonempty {k : Type u_1} {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [S : AddTorsor V P] (Q : AffineSubspace k P) : Q = ⊥ ∨ (↑Q).Nonempty

@[simp]

theorem AffineSubspace.direction_eq_top_iff_of_nonempty {k : Type u_1} {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [S : AddTorsor V P] {s : AffineSubspace k P} (h : (↑s).Nonempty) : s.direction = ⊤ ↔ s = ⊤

A nonempty affine subspace is ⊤ if and only if its direction is ⊤.

@[simp]

theorem AffineSubspace.inf_coe {k : Type u_1} {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [S : AddTorsor V P] (s₁ s₂ : AffineSubspace k P) : ↑s₁ ⊓ ↑s₂ = ↑s₁ ∩ ↑s₂

The inf of two affine subspaces, coerced to a set, is the intersection of the two sets of points.

theorem AffineSubspace.mem_inf_iff {k : Type u_1} {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [S : AddTorsor V P] (p : P) (s₁ s₂ : AffineSubspace k P) : p ∈ s₁ ⊓ s₂ ↔ p ∈ s₁ ∧ p ∈ s₂

A point is in the inf of two affine subspaces if and only if it is in both of them.

theorem AffineSubspace.direction_inf {k : Type u_1} {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [S : AddTorsor V P] (s₁ s₂ : AffineSubspace k P) : (s₁ ⊓ s₂).direction ≤ s₁.direction ⊓ s₂.direction

The direction of the inf of two affine subspaces is less than or equal to the inf of their directions.

theorem AffineSubspace.direction_inf_of_mem {k : Type u_1} {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [S : AddTorsor V P] {s₁ s₂ : AffineSubspace k P} {p : P} (h₁ : p ∈ s₁) (h₂ : p ∈ s₂) : (s₁ ⊓ s₂).direction = s₁.direction ⊓ s₂.direction

If two affine subspaces have a point in common, the direction of their inf equals the inf of their directions.

theorem AffineSubspace.direction_inf_of_mem_inf {k : Type u_1} {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [S : AddTorsor V P] {s₁ s₂ : AffineSubspace k P} {p : P} (h : p ∈ s₁ ⊓ s₂) : (s₁ ⊓ s₂).direction = s₁.direction ⊓ s₂.direction

If two affine subspaces have a point in their inf, the direction of their inf equals the inf of their directions.

theorem AffineSubspace.direction_le {k : Type u_1} {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [S : AddTorsor V P] {s₁ s₂ : AffineSubspace k P} (h : s₁ ≤ s₂) : s₁.direction ≤ s₂.direction

If one affine subspace is less than or equal to another, the same applies to their directions.

theorem AffineSubspace.sup_direction_le {k : Type u_1} {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [S : AddTorsor V P] (s₁ s₂ : AffineSubspace k P) : s₁.direction ⊔ s₂.direction ≤ (s₁ ⊔ s₂).direction

The sup of the directions of two affine subspaces is less than or equal to the direction of their sup.

theorem AffineSubspace.sup_direction_lt_of_nonempty_of_inter_empty {k : Type u_1} {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [S : AddTorsor V P] {s₁ s₂ : AffineSubspace k P} (h1 : (↑s₁).Nonempty) (h2 : (↑s₂).Nonempty) (he : ↑s₁ ∩ ↑s₂ = ∅) : s₁.direction ⊔ s₂.direction < (s₁ ⊔ s₂).direction

The sup of the directions of two nonempty affine subspaces with empty intersection is less than the direction of their sup.

theorem AffineSubspace.inter_nonempty_of_nonempty_of_sup_direction_eq_top {k : Type u_1} {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [S : AddTorsor V P] {s₁ s₂ : AffineSubspace k P} (h1 : (↑s₁).Nonempty) (h2 : (↑s₂).Nonempty) (hd : s₁.direction ⊔ s₂.direction = ⊤) : (↑s₁ ∩ ↑s₂).Nonempty

If the directions of two nonempty affine subspaces span the whole module, they have nonempty intersection.

theorem AffineSubspace.inter_eq_singleton_of_nonempty_of_isCompl {k : Type u_1} {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [S : AddTorsor V P] {s₁ s₂ : AffineSubspace k P} (h1 : (↑s₁).Nonempty) (h2 : (↑s₂).Nonempty) (hd : IsCompl s₁.direction s₂.direction) : ∃ (p : P), ↑s₁ ∩ ↑s₂ = {p}

If the directions of two nonempty affine subspaces are complements of each other, they intersect in exactly one point.

@[simp]

theorem AffineSubspace.affineSpan_coe {k : Type u_1} {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [S : AddTorsor V P] (s : AffineSubspace k P) : affineSpan k ↑s = s

Coercing a subspace to a set then taking the affine span produces the original subspace.

theorem affineSpan_nonempty (k : Type u_1) {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [AddTorsor V P] {s : Set P} : (↑(affineSpan k s)).Nonempty ↔ s.Nonempty

The affine span of a set is nonempty if and only if that set is.

theorem Set.Nonempty.affineSpan (k : Type u_1) {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [AddTorsor V P] {s : Set P} : s.Nonempty → (↑(_root_.affineSpan k s)).Nonempty

Alias of the reverse direction of affineSpan_nonempty.

---

The affine span of a set is nonempty if and only if that set is.

instance instNonemptySubtypeMemAffineSubspaceAffineSpanOfElem (k : Type u_1) {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [AddTorsor V P] {s : Set P} [Nonempty ↑s] : Nonempty ↥(affineSpan k s)

The affine span of a nonempty set is nonempty.

@[simp]

theorem affineSpan_eq_bot (k : Type u_1) {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [AddTorsor V P] {s : Set P} : affineSpan k s = ⊥ ↔ s = ∅

The affine span of a set is ⊥ if and only if that set is empty.

@[simp]

theorem bot_lt_affineSpan (k : Type u_1) {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [AddTorsor V P] {s : Set P} : ⊥ < affineSpan k s ↔ s.Nonempty

theorem affineSpan_induction {k : Type u_1} {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [AddTorsor V P] {x : P} {s : Set P} {p : P → Prop} (h : x ∈ affineSpan k s) (mem : ∀ x ∈ s, p x) (smul_vsub_vadd : ∀ (c : k) (u v w : P), p u → p v → p w → p (c • (u -ᵥ v) +ᵥ w)) : p x

An induction principle for span membership. If p holds for all elements of s and is preserved under certain affine combinations, then p holds for all elements of the span of s.

theorem affineSpan_induction' {k : Type u_1} {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [AddTorsor V P] {s : Set P} {p : (x : P) → x ∈ affineSpan k s → Prop} (mem : ∀ (y : P) (hys : y ∈ s), p y ⋯) (smul_vsub_vadd : ∀ (c : k) (u : P) (hu : u ∈ affineSpan k s) (v : P) (hv : v ∈ affineSpan k s) (w : P) (hw : w ∈ affineSpan k s), p u hu → p v hv → p w hw → p (c • (u -ᵥ v) +ᵥ w) ⋯) {x : P} (h : x ∈ affineSpan k s) : p x h

A dependent version of affineSpan_induction.

theorem vsub_mem_vectorSpan_pair (k : Type u_1) {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [AddTorsor V P] (p₁ p₂ : P) : p₁ -ᵥ p₂ ∈ vectorSpan k {p₁, p₂}

The difference between two points lies in their vectorSpan.

theorem vsub_rev_mem_vectorSpan_pair (k : Type u_1) {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [AddTorsor V P] (p₁ p₂ : P) : p₂ -ᵥ p₁ ∈ vectorSpan k {p₁, p₂}

The difference between two points (reversed) lies in their vectorSpan.

theorem smul_vsub_mem_vectorSpan_pair {k : Type u_1} {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [AddTorsor V P] (r : k) (p₁ p₂ : P) : r • (p₁ -ᵥ p₂) ∈ vectorSpan k {p₁, p₂}

A multiple of the difference between two points lies in their vectorSpan.

theorem smul_vsub_rev_mem_vectorSpan_pair {k : Type u_1} {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [AddTorsor V P] (r : k) (p₁ p₂ : P) : r • (p₂ -ᵥ p₁) ∈ vectorSpan k {p₁, p₂}

A multiple of the difference between two points (reversed) lies in their vectorSpan.

def «termLine[_,_,_]» : Lean.ParserDescr

The line between two points, as an affine subspace.

Equations

• One or more equations did not get rendered due to their size.

theorem left_mem_affineSpan_pair (k : Type u_1) {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [AddTorsor V P] (p₁ p₂ : P) : p₁ ∈ affineSpan k {p₁, p₂}

The first of two points lies in their affine span.

theorem right_mem_affineSpan_pair (k : Type u_1) {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [AddTorsor V P] (p₁ p₂ : P) : p₂ ∈ affineSpan k {p₁, p₂}

The second of two points lies in their affine span.

theorem affineSpan_pair_le_of_mem_of_mem {k : Type u_1} {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [AddTorsor V P] {p₁ p₂ : P} {s : AffineSubspace k P} (hp₁ : p₁ ∈ s) (hp₂ : p₂ ∈ s) : affineSpan k {p₁, p₂} ≤ s

The span of two points that lie in an affine subspace is contained in that subspace.

theorem affineSpan_pair_le_of_left_mem {k : Type u_1} {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [AddTorsor V P] {p₁ p₂ p₃ : P} (h : p₁ ∈ affineSpan k {p₂, p₃}) : affineSpan k {p₁, p₃} ≤ affineSpan k {p₂, p₃}

One line is contained in another differing in the first point if the first point of the first line is contained in the second line.

theorem affineSpan_pair_le_of_right_mem {k : Type u_1} {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [AddTorsor V P] {p₁ p₂ p₃ : P} (h : p₁ ∈ affineSpan k {p₂, p₃}) : affineSpan k {p₂, p₁} ≤ affineSpan k {p₂, p₃}

One line is contained in another differing in the second point if the second point of the first line is contained in the second line.

theorem affineSpan_mono (k : Type u_1) {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [AddTorsor V P] {s₁ s₂ : Set P} (h : s₁ ⊆ s₂) : affineSpan k s₁ ≤ affineSpan k s₂

affineSpan is monotone.

theorem affineSpan_insert_affineSpan (k : Type u_1) {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [AddTorsor V P] (p : P) (ps : Set P) : affineSpan k (insert p ↑(affineSpan k ps)) = affineSpan k (insert p ps)

Taking the affine span of a set, adding a point and taking the span again produces the same results as adding the point to the set and taking the span.

theorem affineSpan_insert_eq_affineSpan (k : Type u_1) {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [AddTorsor V P] {p : P} {ps : Set P} (h : p ∈ affineSpan k ps) : affineSpan k (insert p ps) = affineSpan k ps

If a point is in the affine span of a set, adding it to that set does not change the affine span.

theorem vectorSpan_insert_eq_vectorSpan {k : Type u_1} {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [AddTorsor V P] {p : P} {ps : Set P} (h : p ∈ affineSpan k ps) : vectorSpan k (insert p ps) = vectorSpan k ps

If a point is in the affine span of a set, adding it to that set does not change the vector span.

theorem affineSpan_le_toAffineSubspace_span {k : Type u_1} {V : Type u_2} [Ring k] [AddCommGroup V] [Module k V] {s : Set V} : affineSpan k s ≤ (Submodule.span k s).toAffineSubspace

When the affine space is also a vector space, the affine span is contained within the linear span.

theorem affineSpan_subset_span {k : Type u_1} {V : Type u_2} [Ring k] [AddCommGroup V] [Module k V] {s : Set V} : ↑(affineSpan k s) ⊆ ↑(Submodule.span k s)

theorem affineSpan_insert_zero {k : Type u_1} {V : Type u_2} [Ring k] [AddCommGroup V] [Module k V] (s : Set V) : ↑(affineSpan k (insert 0 s)) = ↑(Submodule.span k s)