Sides of affine subspaces

This file defines notions of two points being on the same or opposite sides of an affine subspace.

Main definitions

• s.WSameSide x y: The points x and y are weakly on the same side of the affine subspace s.
• s.SSameSide x y: The points x and y are strictly on the same side of the affine subspace s.
• s.WOppSide x y: The points x and y are weakly on opposite sides of the affine subspace s.
• s.SOppSide x y: The points x and y are strictly on opposite sides of the affine subspace s.

def AffineSubspace.WSameSide {R : Type u_1} {V : Type u_2} {P : Type u_4} [StrictOrderedCommRing R] [AddCommGroup V] [Module R V] [AddTorsor V P] (s : AffineSubspace R P) (x : P) (y : P) : Prop

The points x and y are weakly on the same side of s.

Equations

• AffineSubspace.WSameSide s x y = ∃ p₁ ∈ s, ∃ p₂ ∈ s, SameRay R (x -ᵥ p₁) (y -ᵥ p₂)

def AffineSubspace.SSameSide {R : Type u_1} {V : Type u_2} {P : Type u_4} [StrictOrderedCommRing R] [AddCommGroup V] [Module R V] [AddTorsor V P] (s : AffineSubspace R P) (x : P) (y : P) : Prop

The points x and y are strictly on the same side of s.

Equations

• AffineSubspace.SSameSide s x y = (AffineSubspace.WSameSide s x y ∧ x ∉ s ∧ y ∉ s)

def AffineSubspace.WOppSide {R : Type u_1} {V : Type u_2} {P : Type u_4} [StrictOrderedCommRing R] [AddCommGroup V] [Module R V] [AddTorsor V P] (s : AffineSubspace R P) (x : P) (y : P) : Prop

The points x and y are weakly on opposite sides of s.

Equations

• AffineSubspace.WOppSide s x y = ∃ p₁ ∈ s, ∃ p₂ ∈ s, SameRay R (x -ᵥ p₁) (p₂ -ᵥ y)

def AffineSubspace.SOppSide {R : Type u_1} {V : Type u_2} {P : Type u_4} [StrictOrderedCommRing R] [AddCommGroup V] [Module R V] [AddTorsor V P] (s : AffineSubspace R P) (x : P) (y : P) : Prop

The points x and y are strictly on opposite sides of s.

Equations

• AffineSubspace.SOppSide s x y = (AffineSubspace.WOppSide s x y ∧ x ∉ s ∧ y ∉ s)

theorem AffineSubspace.WSameSide.map {R : Type u_1} {V : Type u_2} {V' : Type u_3} {P : Type u_4} {P' : Type u_5} [StrictOrderedCommRing R] [AddCommGroup V] [Module R V] [AddTorsor V P] [AddCommGroup V'] [Module R V'] [AddTorsor V' P'] {s : AffineSubspace R P} {x : P} {y : P} (h : AffineSubspace.WSameSide s x y) (f : P →ᵃ[R] P') : AffineSubspace.WSameSide (AffineSubspace.map f s) (f x) (f y)

theorem Function.Injective.wSameSide_map_iff {R : Type u_1} {V : Type u_2} {V' : Type u_3} {P : Type u_4} {P' : Type u_5} [StrictOrderedCommRing R] [AddCommGroup V] [Module R V] [AddTorsor V P] [AddCommGroup V'] [Module R V'] [AddTorsor V' P'] {s : AffineSubspace R P} {x : P} {y : P} {f : P →ᵃ[R] P'} (hf : Function.Injective ⇑f) : AffineSubspace.WSameSide (AffineSubspace.map f s) (f x) (f y) ↔ AffineSubspace.WSameSide s x y

theorem Function.Injective.sSameSide_map_iff {R : Type u_1} {V : Type u_2} {V' : Type u_3} {P : Type u_4} {P' : Type u_5} [StrictOrderedCommRing R] [AddCommGroup V] [Module R V] [AddTorsor V P] [AddCommGroup V'] [Module R V'] [AddTorsor V' P'] {s : AffineSubspace R P} {x : P} {y : P} {f : P →ᵃ[R] P'} (hf : Function.Injective ⇑f) : AffineSubspace.SSameSide (AffineSubspace.map f s) (f x) (f y) ↔ AffineSubspace.SSameSide s x y

@[simp]

theorem AffineEquiv.wSameSide_map_iff {R : Type u_1} {V : Type u_2} {V' : Type u_3} {P : Type u_4} {P' : Type u_5} [StrictOrderedCommRing R] [AddCommGroup V] [Module R V] [AddTorsor V P] [AddCommGroup V'] [Module R V'] [AddTorsor V' P'] {s : AffineSubspace R P} {x : P} {y : P} (f : P ≃ᵃ[R] P') : AffineSubspace.WSameSide (AffineSubspace.map (↑f) s) (f x) (f y) ↔ AffineSubspace.WSameSide s x y

@[simp]

theorem AffineEquiv.sSameSide_map_iff {R : Type u_1} {V : Type u_2} {V' : Type u_3} {P : Type u_4} {P' : Type u_5} [StrictOrderedCommRing R] [AddCommGroup V] [Module R V] [AddTorsor V P] [AddCommGroup V'] [Module R V'] [AddTorsor V' P'] {s : AffineSubspace R P} {x : P} {y : P} (f : P ≃ᵃ[R] P') : AffineSubspace.SSameSide (AffineSubspace.map (↑f) s) (f x) (f y) ↔ AffineSubspace.SSameSide s x y

theorem AffineSubspace.WOppSide.map {R : Type u_1} {V : Type u_2} {V' : Type u_3} {P : Type u_4} {P' : Type u_5} [StrictOrderedCommRing R] [AddCommGroup V] [Module R V] [AddTorsor V P] [AddCommGroup V'] [Module R V'] [AddTorsor V' P'] {s : AffineSubspace R P} {x : P} {y : P} (h : AffineSubspace.WOppSide s x y) (f : P →ᵃ[R] P') : AffineSubspace.WOppSide (AffineSubspace.map f s) (f x) (f y)

theorem Function.Injective.wOppSide_map_iff {R : Type u_1} {V : Type u_2} {V' : Type u_3} {P : Type u_4} {P' : Type u_5} [StrictOrderedCommRing R] [AddCommGroup V] [Module R V] [AddTorsor V P] [AddCommGroup V'] [Module R V'] [AddTorsor V' P'] {s : AffineSubspace R P} {x : P} {y : P} {f : P →ᵃ[R] P'} (hf : Function.Injective ⇑f) : AffineSubspace.WOppSide (AffineSubspace.map f s) (f x) (f y) ↔ AffineSubspace.WOppSide s x y

theorem Function.Injective.sOppSide_map_iff {R : Type u_1} {V : Type u_2} {V' : Type u_3} {P : Type u_4} {P' : Type u_5} [StrictOrderedCommRing R] [AddCommGroup V] [Module R V] [AddTorsor V P] [AddCommGroup V'] [Module R V'] [AddTorsor V' P'] {s : AffineSubspace R P} {x : P} {y : P} {f : P →ᵃ[R] P'} (hf : Function.Injective ⇑f) : AffineSubspace.SOppSide (AffineSubspace.map f s) (f x) (f y) ↔ AffineSubspace.SOppSide s x y

@[simp]

theorem AffineEquiv.wOppSide_map_iff {R : Type u_1} {V : Type u_2} {V' : Type u_3} {P : Type u_4} {P' : Type u_5} [StrictOrderedCommRing R] [AddCommGroup V] [Module R V] [AddTorsor V P] [AddCommGroup V'] [Module R V'] [AddTorsor V' P'] {s : AffineSubspace R P} {x : P} {y : P} (f : P ≃ᵃ[R] P') : AffineSubspace.WOppSide (AffineSubspace.map (↑f) s) (f x) (f y) ↔ AffineSubspace.WOppSide s x y

@[simp]

theorem AffineEquiv.sOppSide_map_iff {R : Type u_1} {V : Type u_2} {V' : Type u_3} {P : Type u_4} {P' : Type u_5} [StrictOrderedCommRing R] [AddCommGroup V] [Module R V] [AddTorsor V P] [AddCommGroup V'] [Module R V'] [AddTorsor V' P'] {s : AffineSubspace R P} {x : P} {y : P} (f : P ≃ᵃ[R] P') : AffineSubspace.SOppSide (AffineSubspace.map (↑f) s) (f x) (f y) ↔ AffineSubspace.SOppSide s x y

theorem AffineSubspace.WSameSide.nonempty {R : Type u_1} {V : Type u_2} {P : Type u_4} [StrictOrderedCommRing R] [AddCommGroup V] [Module R V] [AddTorsor V P] {s : AffineSubspace R P} {x : P} {y : P} (h : AffineSubspace.WSameSide s x y) : Set.Nonempty ↑s

theorem AffineSubspace.SSameSide.nonempty {R : Type u_1} {V : Type u_2} {P : Type u_4} [StrictOrderedCommRing R] [AddCommGroup V] [Module R V] [AddTorsor V P] {s : AffineSubspace R P} {x : P} {y : P} (h : AffineSubspace.SSameSide s x y) : Set.Nonempty ↑s

theorem AffineSubspace.WOppSide.nonempty {R : Type u_1} {V : Type u_2} {P : Type u_4} [StrictOrderedCommRing R] [AddCommGroup V] [Module R V] [AddTorsor V P] {s : AffineSubspace R P} {x : P} {y : P} (h : AffineSubspace.WOppSide s x y) : Set.Nonempty ↑s

theorem AffineSubspace.SOppSide.nonempty {R : Type u_1} {V : Type u_2} {P : Type u_4} [StrictOrderedCommRing R] [AddCommGroup V] [Module R V] [AddTorsor V P] {s : AffineSubspace R P} {x : P} {y : P} (h : AffineSubspace.SOppSide s x y) : Set.Nonempty ↑s

theorem AffineSubspace.SSameSide.wSameSide {R : Type u_1} {V : Type u_2} {P : Type u_4} [StrictOrderedCommRing R] [AddCommGroup V] [Module R V] [AddTorsor V P] {s : AffineSubspace R P} {x : P} {y : P} (h : AffineSubspace.SSameSide s x y) : AffineSubspace.WSameSide s x y

theorem AffineSubspace.SSameSide.left_not_mem {R : Type u_1} {V : Type u_2} {P : Type u_4} [StrictOrderedCommRing R] [AddCommGroup V] [Module R V] [AddTorsor V P] {s : AffineSubspace R P} {x : P} {y : P} (h : AffineSubspace.SSameSide s x y) : x ∉ s

theorem AffineSubspace.SSameSide.right_not_mem {R : Type u_1} {V : Type u_2} {P : Type u_4} [StrictOrderedCommRing R] [AddCommGroup V] [Module R V] [AddTorsor V P] {s : AffineSubspace R P} {x : P} {y : P} (h : AffineSubspace.SSameSide s x y) : y ∉ s

theorem AffineSubspace.SOppSide.wOppSide {R : Type u_1} {V : Type u_2} {P : Type u_4} [StrictOrderedCommRing R] [AddCommGroup V] [Module R V] [AddTorsor V P] {s : AffineSubspace R P} {x : P} {y : P} (h : AffineSubspace.SOppSide s x y) : AffineSubspace.WOppSide s x y

theorem AffineSubspace.SOppSide.left_not_mem {R : Type u_1} {V : Type u_2} {P : Type u_4} [StrictOrderedCommRing R] [AddCommGroup V] [Module R V] [AddTorsor V P] {s : AffineSubspace R P} {x : P} {y : P} (h : AffineSubspace.SOppSide s x y) : x ∉ s

theorem AffineSubspace.SOppSide.right_not_mem {R : Type u_1} {V : Type u_2} {P : Type u_4} [StrictOrderedCommRing R] [AddCommGroup V] [Module R V] [AddTorsor V P] {s : AffineSubspace R P} {x : P} {y : P} (h : AffineSubspace.SOppSide s x y) : y ∉ s

theorem AffineSubspace.wSameSide_comm {R : Type u_1} {V : Type u_2} {P : Type u_4} [StrictOrderedCommRing R] [AddCommGroup V] [Module R V] [AddTorsor V P] {s : AffineSubspace R P} {x : P} {y : P} : AffineSubspace.WSameSide s x y ↔ AffineSubspace.WSameSide s y x

theorem AffineSubspace.WSameSide.symm {R : Type u_1} {V : Type u_2} {P : Type u_4} [StrictOrderedCommRing R] [AddCommGroup V] [Module R V] [AddTorsor V P] {s : AffineSubspace R P} {x : P} {y : P} : AffineSubspace.WSameSide s x y → AffineSubspace.WSameSide s y x

Alias of the forward direction of AffineSubspace.wSameSide_comm.

theorem AffineSubspace.sSameSide_comm {R : Type u_1} {V : Type u_2} {P : Type u_4} [StrictOrderedCommRing R] [AddCommGroup V] [Module R V] [AddTorsor V P] {s : AffineSubspace R P} {x : P} {y : P} : AffineSubspace.SSameSide s x y ↔ AffineSubspace.SSameSide s y x

theorem AffineSubspace.SSameSide.symm {R : Type u_1} {V : Type u_2} {P : Type u_4} [StrictOrderedCommRing R] [AddCommGroup V] [Module R V] [AddTorsor V P] {s : AffineSubspace R P} {x : P} {y : P} : AffineSubspace.SSameSide s x y → AffineSubspace.SSameSide s y x

Alias of the forward direction of AffineSubspace.sSameSide_comm.

theorem AffineSubspace.wOppSide_comm {R : Type u_1} {V : Type u_2} {P : Type u_4} [StrictOrderedCommRing R] [AddCommGroup V] [Module R V] [AddTorsor V P] {s : AffineSubspace R P} {x : P} {y : P} : AffineSubspace.WOppSide s x y ↔ AffineSubspace.WOppSide s y x

theorem AffineSubspace.WOppSide.symm {R : Type u_1} {V : Type u_2} {P : Type u_4} [StrictOrderedCommRing R] [AddCommGroup V] [Module R V] [AddTorsor V P] {s : AffineSubspace R P} {x : P} {y : P} : AffineSubspace.WOppSide s x y → AffineSubspace.WOppSide s y x

Alias of the forward direction of AffineSubspace.wOppSide_comm.

theorem AffineSubspace.sOppSide_comm {R : Type u_1} {V : Type u_2} {P : Type u_4} [StrictOrderedCommRing R] [AddCommGroup V] [Module R V] [AddTorsor V P] {s : AffineSubspace R P} {x : P} {y : P} : AffineSubspace.SOppSide s x y ↔ AffineSubspace.SOppSide s y x

theorem AffineSubspace.SOppSide.symm {R : Type u_1} {V : Type u_2} {P : Type u_4} [StrictOrderedCommRing R] [AddCommGroup V] [Module R V] [AddTorsor V P] {s : AffineSubspace R P} {x : P} {y : P} : AffineSubspace.SOppSide s x y → AffineSubspace.SOppSide s y x

Alias of the forward direction of AffineSubspace.sOppSide_comm.

theorem AffineSubspace.not_wSameSide_bot {R : Type u_1} {V : Type u_2} {P : Type u_4} [StrictOrderedCommRing R] [AddCommGroup V] [Module R V] [AddTorsor V P] (x : P) (y : P) : ¬AffineSubspace.WSameSide ⊥ x y

theorem AffineSubspace.not_sSameSide_bot {R : Type u_1} {V : Type u_2} {P : Type u_4} [StrictOrderedCommRing R] [AddCommGroup V] [Module R V] [AddTorsor V P] (x : P) (y : P) : ¬AffineSubspace.SSameSide ⊥ x y

theorem AffineSubspace.not_wOppSide_bot {R : Type u_1} {V : Type u_2} {P : Type u_4} [StrictOrderedCommRing R] [AddCommGroup V] [Module R V] [AddTorsor V P] (x : P) (y : P) : ¬AffineSubspace.WOppSide ⊥ x y

theorem AffineSubspace.not_sOppSide_bot {R : Type u_1} {V : Type u_2} {P : Type u_4} [StrictOrderedCommRing R] [AddCommGroup V] [Module R V] [AddTorsor V P] (x : P) (y : P) : ¬AffineSubspace.SOppSide ⊥ x y

@[simp]

theorem AffineSubspace.wSameSide_self_iff {R : Type u_1} {V : Type u_2} {P : Type u_4} [StrictOrderedCommRing R] [AddCommGroup V] [Module R V] [AddTorsor V P] {s : AffineSubspace R P} {x : P} : AffineSubspace.WSameSide s x x ↔ Set.Nonempty ↑s

theorem AffineSubspace.sSameSide_self_iff {R : Type u_1} {V : Type u_2} {P : Type u_4} [StrictOrderedCommRing R] [AddCommGroup V] [Module R V] [AddTorsor V P] {s : AffineSubspace R P} {x : P} : AffineSubspace.SSameSide s x x ↔ Set.Nonempty ↑s ∧ x ∉ s

theorem AffineSubspace.wSameSide_of_left_mem {R : Type u_1} {V : Type u_2} {P : Type u_4} [StrictOrderedCommRing R] [AddCommGroup V] [Module R V] [AddTorsor V P] {s : AffineSubspace R P} {x : P} (y : P) (hx : x ∈ s) : AffineSubspace.WSameSide s x y

theorem AffineSubspace.wSameSide_of_right_mem {R : Type u_1} {V : Type u_2} {P : Type u_4} [StrictOrderedCommRing R] [AddCommGroup V] [Module R V] [AddTorsor V P] {s : AffineSubspace R P} (x : P) {y : P} (hy : y ∈ s) : AffineSubspace.WSameSide s x y

theorem AffineSubspace.wOppSide_of_left_mem {R : Type u_1} {V : Type u_2} {P : Type u_4} [StrictOrderedCommRing R] [AddCommGroup V] [Module R V] [AddTorsor V P] {s : AffineSubspace R P} {x : P} (y : P) (hx : x ∈ s) : AffineSubspace.WOppSide s x y

theorem AffineSubspace.wOppSide_of_right_mem {R : Type u_1} {V : Type u_2} {P : Type u_4} [StrictOrderedCommRing R] [AddCommGroup V] [Module R V] [AddTorsor V P] {s : AffineSubspace R P} (x : P) {y : P} (hy : y ∈ s) : AffineSubspace.WOppSide s x y

theorem AffineSubspace.wSameSide_vadd_left_iff {R : Type u_1} {V : Type u_2} {P : Type u_4} [StrictOrderedCommRing R] [AddCommGroup V] [Module R V] [AddTorsor V P] {s : AffineSubspace R P} {x : P} {y : P} {v : V} (hv : v ∈ AffineSubspace.direction s) : AffineSubspace.WSameSide s (v +ᵥ x) y ↔ AffineSubspace.WSameSide s x y

theorem AffineSubspace.wSameSide_vadd_right_iff {R : Type u_1} {V : Type u_2} {P : Type u_4} [StrictOrderedCommRing R] [AddCommGroup V] [Module R V] [AddTorsor V P] {s : AffineSubspace R P} {x : P} {y : P} {v : V} (hv : v ∈ AffineSubspace.direction s) : AffineSubspace.WSameSide s x (v +ᵥ y) ↔ AffineSubspace.WSameSide s x y

theorem AffineSubspace.sSameSide_vadd_left_iff {R : Type u_1} {V : Type u_2} {P : Type u_4} [StrictOrderedCommRing R] [AddCommGroup V] [Module R V] [AddTorsor V P] {s : AffineSubspace R P} {x : P} {y : P} {v : V} (hv : v ∈ AffineSubspace.direction s) : AffineSubspace.SSameSide s (v +ᵥ x) y ↔ AffineSubspace.SSameSide s x y

theorem AffineSubspace.sSameSide_vadd_right_iff {R : Type u_1} {V : Type u_2} {P : Type u_4} [StrictOrderedCommRing R] [AddCommGroup V] [Module R V] [AddTorsor V P] {s : AffineSubspace R P} {x : P} {y : P} {v : V} (hv : v ∈ AffineSubspace.direction s) : AffineSubspace.SSameSide s x (v +ᵥ y) ↔ AffineSubspace.SSameSide s x y

theorem AffineSubspace.wOppSide_vadd_left_iff {R : Type u_1} {V : Type u_2} {P : Type u_4} [StrictOrderedCommRing R] [AddCommGroup V] [Module R V] [AddTorsor V P] {s : AffineSubspace R P} {x : P} {y : P} {v : V} (hv : v ∈ AffineSubspace.direction s) : AffineSubspace.WOppSide s (v +ᵥ x) y ↔ AffineSubspace.WOppSide s x y

theorem AffineSubspace.wOppSide_vadd_right_iff {R : Type u_1} {V : Type u_2} {P : Type u_4} [StrictOrderedCommRing R] [AddCommGroup V] [Module R V] [AddTorsor V P] {s : AffineSubspace R P} {x : P} {y : P} {v : V} (hv : v ∈ AffineSubspace.direction s) : AffineSubspace.WOppSide s x (v +ᵥ y) ↔ AffineSubspace.WOppSide s x y

theorem AffineSubspace.sOppSide_vadd_left_iff {R : Type u_1} {V : Type u_2} {P : Type u_4} [StrictOrderedCommRing R] [AddCommGroup V] [Module R V] [AddTorsor V P] {s : AffineSubspace R P} {x : P} {y : P} {v : V} (hv : v ∈ AffineSubspace.direction s) : AffineSubspace.SOppSide s (v +ᵥ x) y ↔ AffineSubspace.SOppSide s x y

theorem AffineSubspace.sOppSide_vadd_right_iff {R : Type u_1} {V : Type u_2} {P : Type u_4} [StrictOrderedCommRing R] [AddCommGroup V] [Module R V] [AddTorsor V P] {s : AffineSubspace R P} {x : P} {y : P} {v : V} (hv : v ∈ AffineSubspace.direction s) : AffineSubspace.SOppSide s x (v +ᵥ y) ↔ AffineSubspace.SOppSide s x y

theorem AffineSubspace.wSameSide_smul_vsub_vadd_left {R : Type u_1} {V : Type u_2} {P : Type u_4} [StrictOrderedCommRing R] [AddCommGroup V] [Module R V] [AddTorsor V P] {s : AffineSubspace R P} {p₁ : P} {p₂ : P} (x : P) (hp₁ : p₁ ∈ s) (hp₂ : p₂ ∈ s) {t : R} (ht : 0 ≤ t) : AffineSubspace.WSameSide s (t • (x -ᵥ p₁) +ᵥ p₂) x

theorem AffineSubspace.wSameSide_smul_vsub_vadd_right {R : Type u_1} {V : Type u_2} {P : Type u_4} [StrictOrderedCommRing R] [AddCommGroup V] [Module R V] [AddTorsor V P] {s : AffineSubspace R P} {p₁ : P} {p₂ : P} (x : P) (hp₁ : p₁ ∈ s) (hp₂ : p₂ ∈ s) {t : R} (ht : 0 ≤ t) : AffineSubspace.WSameSide s x (t • (x -ᵥ p₁) +ᵥ p₂)

theorem AffineSubspace.wSameSide_lineMap_left {R : Type u_1} {V : Type u_2} {P : Type u_4} [StrictOrderedCommRing R] [AddCommGroup V] [Module R V] [AddTorsor V P] {s : AffineSubspace R P} {x : P} (y : P) (h : x ∈ s) {t : R} (ht : 0 ≤ t) : AffineSubspace.WSameSide s ((AffineMap.lineMap x y) t) y

theorem AffineSubspace.wSameSide_lineMap_right {R : Type u_1} {V : Type u_2} {P : Type u_4} [StrictOrderedCommRing R] [AddCommGroup V] [Module R V] [AddTorsor V P] {s : AffineSubspace R P} {x : P} (y : P) (h : x ∈ s) {t : R} (ht : 0 ≤ t) : AffineSubspace.WSameSide s y ((AffineMap.lineMap x y) t)

theorem AffineSubspace.wOppSide_smul_vsub_vadd_left {R : Type u_1} {V : Type u_2} {P : Type u_4} [StrictOrderedCommRing R] [AddCommGroup V] [Module R V] [AddTorsor V P] {s : AffineSubspace R P} {p₁ : P} {p₂ : P} (x : P) (hp₁ : p₁ ∈ s) (hp₂ : p₂ ∈ s) {t : R} (ht : t ≤ 0) : AffineSubspace.WOppSide s (t • (x -ᵥ p₁) +ᵥ p₂) x

theorem AffineSubspace.wOppSide_smul_vsub_vadd_right {R : Type u_1} {V : Type u_2} {P : Type u_4} [StrictOrderedCommRing R] [AddCommGroup V] [Module R V] [AddTorsor V P] {s : AffineSubspace R P} {p₁ : P} {p₂ : P} (x : P) (hp₁ : p₁ ∈ s) (hp₂ : p₂ ∈ s) {t : R} (ht : t ≤ 0) : AffineSubspace.WOppSide s x (t • (x -ᵥ p₁) +ᵥ p₂)

theorem AffineSubspace.wOppSide_lineMap_left {R : Type u_1} {V : Type u_2} {P : Type u_4} [StrictOrderedCommRing R] [AddCommGroup V] [Module R V] [AddTorsor V P] {s : AffineSubspace R P} {x : P} (y : P) (h : x ∈ s) {t : R} (ht : t ≤ 0) : AffineSubspace.WOppSide s ((AffineMap.lineMap x y) t) y

theorem AffineSubspace.wOppSide_lineMap_right {R : Type u_1} {V : Type u_2} {P : Type u_4} [StrictOrderedCommRing R] [AddCommGroup V] [Module R V] [AddTorsor V P] {s : AffineSubspace R P} {x : P} (y : P) (h : x ∈ s) {t : R} (ht : t ≤ 0) : AffineSubspace.WOppSide s y ((AffineMap.lineMap x y) t)

theorem Wbtw.wSameSide₂₃ {R : Type u_1} {V : Type u_2} {P : Type u_4} [StrictOrderedCommRing R] [AddCommGroup V] [Module R V] [AddTorsor V P] {s : AffineSubspace R P} {x : P} {y : P} {z : P} (h : Wbtw R x y z) (hx : x ∈ s) : AffineSubspace.WSameSide s y z

theorem Wbtw.wSameSide₃₂ {R : Type u_1} {V : Type u_2} {P : Type u_4} [StrictOrderedCommRing R] [AddCommGroup V] [Module R V] [AddTorsor V P] {s : AffineSubspace R P} {x : P} {y : P} {z : P} (h : Wbtw R x y z) (hx : x ∈ s) : AffineSubspace.WSameSide s z y

theorem Wbtw.wSameSide₁₂ {R : Type u_1} {V : Type u_2} {P : Type u_4} [StrictOrderedCommRing R] [AddCommGroup V] [Module R V] [AddTorsor V P] {s : AffineSubspace R P} {x : P} {y : P} {z : P} (h : Wbtw R x y z) (hz : z ∈ s) : AffineSubspace.WSameSide s x y

theorem Wbtw.wSameSide₂₁ {R : Type u_1} {V : Type u_2} {P : Type u_4} [StrictOrderedCommRing R] [AddCommGroup V] [Module R V] [AddTorsor V P] {s : AffineSubspace R P} {x : P} {y : P} {z : P} (h : Wbtw R x y z) (hz : z ∈ s) : AffineSubspace.WSameSide s y x

theorem Wbtw.wOppSide₁₃ {R : Type u_1} {V : Type u_2} {P : Type u_4} [StrictOrderedCommRing R] [AddCommGroup V] [Module R V] [AddTorsor V P] {s : AffineSubspace R P} {x : P} {y : P} {z : P} (h : Wbtw R x y z) (hy : y ∈ s) : AffineSubspace.WOppSide s x z

theorem Wbtw.wOppSide₃₁ {R : Type u_1} {V : Type u_2} {P : Type u_4} [StrictOrderedCommRing R] [AddCommGroup V] [Module R V] [AddTorsor V P] {s : AffineSubspace R P} {x : P} {y : P} {z : P} (h : Wbtw R x y z) (hy : y ∈ s) : AffineSubspace.WOppSide s z x

@[simp]

theorem AffineSubspace.wOppSide_self_iff {R : Type u_1} {V : Type u_2} {P : Type u_4} [LinearOrderedField R] [AddCommGroup V] [Module R V] [AddTorsor V P] {s : AffineSubspace R P} {x : P} : AffineSubspace.WOppSide s x x ↔ x ∈ s

theorem AffineSubspace.not_sOppSide_self {R : Type u_1} {V : Type u_2} {P : Type u_4} [LinearOrderedField R] [AddCommGroup V] [Module R V] [AddTorsor V P] (s : AffineSubspace R P) (x : P) : ¬AffineSubspace.SOppSide s x x

theorem AffineSubspace.wSameSide_iff_exists_left {R : Type u_1} {V : Type u_2} {P : Type u_4} [LinearOrderedField R] [AddCommGroup V] [Module R V] [AddTorsor V P] {s : AffineSubspace R P} {x : P} {y : P} {p₁ : P} (h : p₁ ∈ s) : AffineSubspace.WSameSide s x y ↔ x ∈ s ∨ ∃ p₂ ∈ s, SameRay R (x -ᵥ p₁) (y -ᵥ p₂)

theorem AffineSubspace.wSameSide_iff_exists_right {R : Type u_1} {V : Type u_2} {P : Type u_4} [LinearOrderedField R] [AddCommGroup V] [Module R V] [AddTorsor V P] {s : AffineSubspace R P} {x : P} {y : P} {p₂ : P} (h : p₂ ∈ s) : AffineSubspace.WSameSide s x y ↔ y ∈ s ∨ ∃ p₁ ∈ s, SameRay R (x -ᵥ p₁) (y -ᵥ p₂)

theorem AffineSubspace.sSameSide_iff_exists_left {R : Type u_1} {V : Type u_2} {P : Type u_4} [LinearOrderedField R] [AddCommGroup V] [Module R V] [AddTorsor V P] {s : AffineSubspace R P} {x : P} {y : P} {p₁ : P} (h : p₁ ∈ s) : AffineSubspace.SSameSide s x y ↔ x ∉ s ∧ y ∉ s ∧ ∃ p₂ ∈ s, SameRay R (x -ᵥ p₁) (y -ᵥ p₂)

theorem AffineSubspace.sSameSide_iff_exists_right {R : Type u_1} {V : Type u_2} {P : Type u_4} [LinearOrderedField R] [AddCommGroup V] [Module R V] [AddTorsor V P] {s : AffineSubspace R P} {x : P} {y : P} {p₂ : P} (h : p₂ ∈ s) : AffineSubspace.SSameSide s x y ↔ x ∉ s ∧ y ∉ s ∧ ∃ p₁ ∈ s, SameRay R (x -ᵥ p₁) (y -ᵥ p₂)

theorem AffineSubspace.wOppSide_iff_exists_left {R : Type u_1} {V : Type u_2} {P : Type u_4} [LinearOrderedField R] [AddCommGroup V] [Module R V] [AddTorsor V P] {s : AffineSubspace R P} {x : P} {y : P} {p₁ : P} (h : p₁ ∈ s) : AffineSubspace.WOppSide s x y ↔ x ∈ s ∨ ∃ p₂ ∈ s, SameRay R (x -ᵥ p₁) (p₂ -ᵥ y)

theorem AffineSubspace.wOppSide_iff_exists_right {R : Type u_1} {V : Type u_2} {P : Type u_4} [LinearOrderedField R] [AddCommGroup V] [Module R V] [AddTorsor V P] {s : AffineSubspace R P} {x : P} {y : P} {p₂ : P} (h : p₂ ∈ s) : AffineSubspace.WOppSide s x y ↔ y ∈ s ∨ ∃ p₁ ∈ s, SameRay R (x -ᵥ p₁) (p₂ -ᵥ y)

theorem AffineSubspace.sOppSide_iff_exists_left {R : Type u_1} {V : Type u_2} {P : Type u_4} [LinearOrderedField R] [AddCommGroup V] [Module R V] [AddTorsor V P] {s : AffineSubspace R P} {x : P} {y : P} {p₁ : P} (h : p₁ ∈ s) : AffineSubspace.SOppSide s x y ↔ x ∉ s ∧ y ∉ s ∧ ∃ p₂ ∈ s, SameRay R (x -ᵥ p₁) (p₂ -ᵥ y)

theorem AffineSubspace.sOppSide_iff_exists_right {R : Type u_1} {V : Type u_2} {P : Type u_4} [LinearOrderedField R] [AddCommGroup V] [Module R V] [AddTorsor V P] {s : AffineSubspace R P} {x : P} {y : P} {p₂ : P} (h : p₂ ∈ s) : AffineSubspace.SOppSide s x y ↔ x ∉ s ∧ y ∉ s ∧ ∃ p₁ ∈ s, SameRay R (x -ᵥ p₁) (p₂ -ᵥ y)

theorem AffineSubspace.WSameSide.trans {R : Type u_1} {V : Type u_2} {P : Type u_4} [LinearOrderedField R] [AddCommGroup V] [Module R V] [AddTorsor V P] {s : AffineSubspace R P} {x : P} {y : P} {z : P} (hxy : AffineSubspace.WSameSide s x y) (hyz : AffineSubspace.WSameSide s y z) (hy : y ∉ s) : AffineSubspace.WSameSide s x z

theorem AffineSubspace.WSameSide.trans_sSameSide {R : Type u_1} {V : Type u_2} {P : Type u_4} [LinearOrderedField R] [AddCommGroup V] [Module R V] [AddTorsor V P] {s : AffineSubspace R P} {x : P} {y : P} {z : P} (hxy : AffineSubspace.WSameSide s x y) (hyz : AffineSubspace.SSameSide s y z) : AffineSubspace.WSameSide s x z

theorem AffineSubspace.WSameSide.trans_wOppSide {R : Type u_1} {V : Type u_2} {P : Type u_4} [LinearOrderedField R] [AddCommGroup V] [Module R V] [AddTorsor V P] {s : AffineSubspace R P} {x : P} {y : P} {z : P} (hxy : AffineSubspace.WSameSide s x y) (hyz : AffineSubspace.WOppSide s y z) (hy : y ∉ s) : AffineSubspace.WOppSide s x z

theorem AffineSubspace.WSameSide.trans_sOppSide {R : Type u_1} {V : Type u_2} {P : Type u_4} [LinearOrderedField R] [AddCommGroup V] [Module R V] [AddTorsor V P] {s : AffineSubspace R P} {x : P} {y : P} {z : P} (hxy : AffineSubspace.WSameSide s x y) (hyz : AffineSubspace.SOppSide s y z) : AffineSubspace.WOppSide s x z

theorem AffineSubspace.SSameSide.trans_wSameSide {R : Type u_1} {V : Type u_2} {P : Type u_4} [LinearOrderedField R] [AddCommGroup V] [Module R V] [AddTorsor V P] {s : AffineSubspace R P} {x : P} {y : P} {z : P} (hxy : AffineSubspace.SSameSide s x y) (hyz : AffineSubspace.WSameSide s y z) : AffineSubspace.WSameSide s x z

theorem AffineSubspace.SSameSide.trans {R : Type u_1} {V : Type u_2} {P : Type u_4} [LinearOrderedField R] [AddCommGroup V] [Module R V] [AddTorsor V P] {s : AffineSubspace R P} {x : P} {y : P} {z : P} (hxy : AffineSubspace.SSameSide s x y) (hyz : AffineSubspace.SSameSide s y z) : AffineSubspace.SSameSide s x z

theorem AffineSubspace.SSameSide.trans_wOppSide {R : Type u_1} {V : Type u_2} {P : Type u_4} [LinearOrderedField R] [AddCommGroup V] [Module R V] [AddTorsor V P] {s : AffineSubspace R P} {x : P} {y : P} {z : P} (hxy : AffineSubspace.SSameSide s x y) (hyz : AffineSubspace.WOppSide s y z) : AffineSubspace.WOppSide s x z

theorem AffineSubspace.SSameSide.trans_sOppSide {R : Type u_1} {V : Type u_2} {P : Type u_4} [LinearOrderedField R] [AddCommGroup V] [Module R V] [AddTorsor V P] {s : AffineSubspace R P} {x : P} {y : P} {z : P} (hxy : AffineSubspace.SSameSide s x y) (hyz : AffineSubspace.SOppSide s y z) : AffineSubspace.SOppSide s x z

theorem AffineSubspace.WOppSide.trans_wSameSide {R : Type u_1} {V : Type u_2} {P : Type u_4} [LinearOrderedField R] [AddCommGroup V] [Module R V] [AddTorsor V P] {s : AffineSubspace R P} {x : P} {y : P} {z : P} (hxy : AffineSubspace.WOppSide s x y) (hyz : AffineSubspace.WSameSide s y z) (hy : y ∉ s) : AffineSubspace.WOppSide s x z

theorem AffineSubspace.WOppSide.trans_sSameSide {R : Type u_1} {V : Type u_2} {P : Type u_4} [LinearOrderedField R] [AddCommGroup V] [Module R V] [AddTorsor V P] {s : AffineSubspace R P} {x : P} {y : P} {z : P} (hxy : AffineSubspace.WOppSide s x y) (hyz : AffineSubspace.SSameSide s y z) : AffineSubspace.WOppSide s x z

theorem AffineSubspace.WOppSide.trans {R : Type u_1} {V : Type u_2} {P : Type u_4} [LinearOrderedField R] [AddCommGroup V] [Module R V] [AddTorsor V P] {s : AffineSubspace R P} {x : P} {y : P} {z : P} (hxy : AffineSubspace.WOppSide s x y) (hyz : AffineSubspace.WOppSide s y z) (hy : y ∉ s) : AffineSubspace.WSameSide s x z

theorem AffineSubspace.WOppSide.trans_sOppSide {R : Type u_1} {V : Type u_2} {P : Type u_4} [LinearOrderedField R] [AddCommGroup V] [Module R V] [AddTorsor V P] {s : AffineSubspace R P} {x : P} {y : P} {z : P} (hxy : AffineSubspace.WOppSide s x y) (hyz : AffineSubspace.SOppSide s y z) : AffineSubspace.WSameSide s x z

theorem AffineSubspace.SOppSide.trans_wSameSide {R : Type u_1} {V : Type u_2} {P : Type u_4} [LinearOrderedField R] [AddCommGroup V] [Module R V] [AddTorsor V P] {s : AffineSubspace R P} {x : P} {y : P} {z : P} (hxy : AffineSubspace.SOppSide s x y) (hyz : AffineSubspace.WSameSide s y z) : AffineSubspace.WOppSide s x z

theorem AffineSubspace.SOppSide.trans_sSameSide {R : Type u_1} {V : Type u_2} {P : Type u_4} [LinearOrderedField R] [AddCommGroup V] [Module R V] [AddTorsor V P] {s : AffineSubspace R P} {x : P} {y : P} {z : P} (hxy : AffineSubspace.SOppSide s x y) (hyz : AffineSubspace.SSameSide s y z) : AffineSubspace.SOppSide s x z

theorem AffineSubspace.SOppSide.trans_wOppSide {R : Type u_1} {V : Type u_2} {P : Type u_4} [LinearOrderedField R] [AddCommGroup V] [Module R V] [AddTorsor V P] {s : AffineSubspace R P} {x : P} {y : P} {z : P} (hxy : AffineSubspace.SOppSide s x y) (hyz : AffineSubspace.WOppSide s y z) : AffineSubspace.WSameSide s x z

theorem AffineSubspace.SOppSide.trans {R : Type u_1} {V : Type u_2} {P : Type u_4} [LinearOrderedField R] [AddCommGroup V] [Module R V] [AddTorsor V P] {s : AffineSubspace R P} {x : P} {y : P} {z : P} (hxy : AffineSubspace.SOppSide s x y) (hyz : AffineSubspace.SOppSide s y z) : AffineSubspace.SSameSide s x z

theorem AffineSubspace.wSameSide_and_wOppSide_iff {R : Type u_1} {V : Type u_2} {P : Type u_4} [LinearOrderedField R] [AddCommGroup V] [Module R V] [AddTorsor V P] {s : AffineSubspace R P} {x : P} {y : P} : AffineSubspace.WSameSide s x y ∧ AffineSubspace.WOppSide s x y ↔ x ∈ s ∨ y ∈ s

theorem AffineSubspace.WSameSide.not_sOppSide {R : Type u_1} {V : Type u_2} {P : Type u_4} [LinearOrderedField R] [AddCommGroup V] [Module R V] [AddTorsor V P] {s : AffineSubspace R P} {x : P} {y : P} (h : AffineSubspace.WSameSide s x y) : ¬AffineSubspace.SOppSide s x y

theorem AffineSubspace.SSameSide.not_wOppSide {R : Type u_1} {V : Type u_2} {P : Type u_4} [LinearOrderedField R] [AddCommGroup V] [Module R V] [AddTorsor V P] {s : AffineSubspace R P} {x : P} {y : P} (h : AffineSubspace.SSameSide s x y) : ¬AffineSubspace.WOppSide s x y

theorem AffineSubspace.SSameSide.not_sOppSide {R : Type u_1} {V : Type u_2} {P : Type u_4} [LinearOrderedField R] [AddCommGroup V] [Module R V] [AddTorsor V P] {s : AffineSubspace R P} {x : P} {y : P} (h : AffineSubspace.SSameSide s x y) : ¬AffineSubspace.SOppSide s x y

theorem AffineSubspace.WOppSide.not_sSameSide {R : Type u_1} {V : Type u_2} {P : Type u_4} [LinearOrderedField R] [AddCommGroup V] [Module R V] [AddTorsor V P] {s : AffineSubspace R P} {x : P} {y : P} (h : AffineSubspace.WOppSide s x y) : ¬AffineSubspace.SSameSide s x y

theorem AffineSubspace.SOppSide.not_wSameSide {R : Type u_1} {V : Type u_2} {P : Type u_4} [LinearOrderedField R] [AddCommGroup V] [Module R V] [AddTorsor V P] {s : AffineSubspace R P} {x : P} {y : P} (h : AffineSubspace.SOppSide s x y) : ¬AffineSubspace.WSameSide s x y

theorem AffineSubspace.SOppSide.not_sSameSide {R : Type u_1} {V : Type u_2} {P : Type u_4} [LinearOrderedField R] [AddCommGroup V] [Module R V] [AddTorsor V P] {s : AffineSubspace R P} {x : P} {y : P} (h : AffineSubspace.SOppSide s x y) : ¬AffineSubspace.SSameSide s x y

theorem AffineSubspace.wOppSide_iff_exists_wbtw {R : Type u_1} {V : Type u_2} {P : Type u_4} [LinearOrderedField R] [AddCommGroup V] [Module R V] [AddTorsor V P] {s : AffineSubspace R P} {x : P} {y : P} : AffineSubspace.WOppSide s x y ↔ ∃ p ∈ s, Wbtw R x p y

theorem AffineSubspace.SOppSide.exists_sbtw {R : Type u_1} {V : Type u_2} {P : Type u_4} [LinearOrderedField R] [AddCommGroup V] [Module R V] [AddTorsor V P] {s : AffineSubspace R P} {x : P} {y : P} (h : AffineSubspace.SOppSide s x y) : ∃ p ∈ s, Sbtw R x p y

theorem Sbtw.sOppSide_of_not_mem_of_mem {R : Type u_1} {V : Type u_2} {P : Type u_4} [LinearOrderedField R] [AddCommGroup V] [Module R V] [AddTorsor V P] {s : AffineSubspace R P} {x : P} {y : P} {z : P} (h : Sbtw R x y z) (hx : x ∉ s) (hy : y ∈ s) : AffineSubspace.SOppSide s x z

theorem AffineSubspace.sSameSide_smul_vsub_vadd_left {R : Type u_1} {V : Type u_2} {P : Type u_4} [LinearOrderedField R] [AddCommGroup V] [Module R V] [AddTorsor V P] {s : AffineSubspace R P} {x : P} {p₁ : P} {p₂ : P} (hx : x ∉ s) (hp₁ : p₁ ∈ s) (hp₂ : p₂ ∈ s) {t : R} (ht : 0 < t) : AffineSubspace.SSameSide s (t • (x -ᵥ p₁) +ᵥ p₂) x

theorem AffineSubspace.sSameSide_smul_vsub_vadd_right {R : Type u_1} {V : Type u_2} {P : Type u_4} [LinearOrderedField R] [AddCommGroup V] [Module R V] [AddTorsor V P] {s : AffineSubspace R P} {x : P} {p₁ : P} {p₂ : P} (hx : x ∉ s) (hp₁ : p₁ ∈ s) (hp₂ : p₂ ∈ s) {t : R} (ht : 0 < t) : AffineSubspace.SSameSide s x (t • (x -ᵥ p₁) +ᵥ p₂)

theorem AffineSubspace.sSameSide_lineMap_left {R : Type u_1} {V : Type u_2} {P : Type u_4} [LinearOrderedField R] [AddCommGroup V] [Module R V] [AddTorsor V P] {s : AffineSubspace R P} {x : P} {y : P} (hx : x ∈ s) (hy : y ∉ s) {t : R} (ht : 0 < t) : AffineSubspace.SSameSide s ((AffineMap.lineMap x y) t) y

theorem AffineSubspace.sSameSide_lineMap_right {R : Type u_1} {V : Type u_2} {P : Type u_4} [LinearOrderedField R] [AddCommGroup V] [Module R V] [AddTorsor V P] {s : AffineSubspace R P} {x : P} {y : P} (hx : x ∈ s) (hy : y ∉ s) {t : R} (ht : 0 < t) : AffineSubspace.SSameSide s y ((AffineMap.lineMap x y) t)

theorem AffineSubspace.sOppSide_smul_vsub_vadd_left {R : Type u_1} {V : Type u_2} {P : Type u_4} [LinearOrderedField R] [AddCommGroup V] [Module R V] [AddTorsor V P] {s : AffineSubspace R P} {x : P} {p₁ : P} {p₂ : P} (hx : x ∉ s) (hp₁ : p₁ ∈ s) (hp₂ : p₂ ∈ s) {t : R} (ht : t < 0) : AffineSubspace.SOppSide s (t • (x -ᵥ p₁) +ᵥ p₂) x

theorem AffineSubspace.sOppSide_smul_vsub_vadd_right {R : Type u_1} {V : Type u_2} {P : Type u_4} [LinearOrderedField R] [AddCommGroup V] [Module R V] [AddTorsor V P] {s : AffineSubspace R P} {x : P} {p₁ : P} {p₂ : P} (hx : x ∉ s) (hp₁ : p₁ ∈ s) (hp₂ : p₂ ∈ s) {t : R} (ht : t < 0) : AffineSubspace.SOppSide s x (t • (x -ᵥ p₁) +ᵥ p₂)

theorem AffineSubspace.sOppSide_lineMap_left {R : Type u_1} {V : Type u_2} {P : Type u_4} [LinearOrderedField R] [AddCommGroup V] [Module R V] [AddTorsor V P] {s : AffineSubspace R P} {x : P} {y : P} (hx : x ∈ s) (hy : y ∉ s) {t : R} (ht : t < 0) : AffineSubspace.SOppSide s ((AffineMap.lineMap x y) t) y

theorem AffineSubspace.sOppSide_lineMap_right {R : Type u_1} {V : Type u_2} {P : Type u_4} [LinearOrderedField R] [AddCommGroup V] [Module R V] [AddTorsor V P] {s : AffineSubspace R P} {x : P} {y : P} (hx : x ∈ s) (hy : y ∉ s) {t : R} (ht : t < 0) : AffineSubspace.SOppSide s y ((AffineMap.lineMap x y) t)

theorem AffineSubspace.setOf_wSameSide_eq_image2 {R : Type u_1} {V : Type u_2} {P : Type u_4} [LinearOrderedField R] [AddCommGroup V] [Module R V] [AddTorsor V P] {s : AffineSubspace R P} {x : P} {p : P} (hx : x ∉ s) (hp : p ∈ s) : {y : P | AffineSubspace.WSameSide s x y} = Set.image2 (fun (t : R) (q : P) => t • (x -ᵥ p) +ᵥ q) (Set.Ici 0) ↑s

theorem AffineSubspace.setOf_sSameSide_eq_image2 {R : Type u_1} {V : Type u_2} {P : Type u_4} [LinearOrderedField R] [AddCommGroup V] [Module R V] [AddTorsor V P] {s : AffineSubspace R P} {x : P} {p : P} (hx : x ∉ s) (hp : p ∈ s) : {y : P | AffineSubspace.SSameSide s x y} = Set.image2 (fun (t : R) (q : P) => t • (x -ᵥ p) +ᵥ q) (Set.Ioi 0) ↑s

theorem AffineSubspace.setOf_wOppSide_eq_image2 {R : Type u_1} {V : Type u_2} {P : Type u_4} [LinearOrderedField R] [AddCommGroup V] [Module R V] [AddTorsor V P] {s : AffineSubspace R P} {x : P} {p : P} (hx : x ∉ s) (hp : p ∈ s) : {y : P | AffineSubspace.WOppSide s x y} = Set.image2 (fun (t : R) (q : P) => t • (x -ᵥ p) +ᵥ q) (Set.Iic 0) ↑s

theorem AffineSubspace.setOf_sOppSide_eq_image2 {R : Type u_1} {V : Type u_2} {P : Type u_4} [LinearOrderedField R] [AddCommGroup V] [Module R V] [AddTorsor V P] {s : AffineSubspace R P} {x : P} {p : P} (hx : x ∉ s) (hp : p ∈ s) : {y : P | AffineSubspace.SOppSide s x y} = Set.image2 (fun (t : R) (q : P) => t • (x -ᵥ p) +ᵥ q) (Set.Iio 0) ↑s

theorem AffineSubspace.wOppSide_pointReflection {R : Type u_1} {V : Type u_2} {P : Type u_4} [LinearOrderedField R] [AddCommGroup V] [Module R V] [AddTorsor V P] {s : AffineSubspace R P} {x : P} (y : P) (hx : x ∈ s) : AffineSubspace.WOppSide s y ((AffineEquiv.pointReflection R x) y)

theorem AffineSubspace.sOppSide_pointReflection {R : Type u_1} {V : Type u_2} {P : Type u_4} [LinearOrderedField R] [AddCommGroup V] [Module R V] [AddTorsor V P] {s : AffineSubspace R P} {x : P} {y : P} (hx : x ∈ s) (hy : y ∉ s) : AffineSubspace.SOppSide s y ((AffineEquiv.pointReflection R x) y)

theorem AffineSubspace.isConnected_setOf_wSameSide {V : Type u_2} {P : Type u_4} [SeminormedAddCommGroup V] [NormedSpace ℝ V] [PseudoMetricSpace P] [NormedAddTorsor V P] {s : AffineSubspace ℝ P} (x : P) (h : Set.Nonempty ↑s) : IsConnected {y : P | AffineSubspace.WSameSide s x y}

theorem AffineSubspace.isPreconnected_setOf_wSameSide {V : Type u_2} {P : Type u_4} [SeminormedAddCommGroup V] [NormedSpace ℝ V] [PseudoMetricSpace P] [NormedAddTorsor V P] (s : AffineSubspace ℝ P) (x : P) : IsPreconnected {y : P | AffineSubspace.WSameSide s x y}

theorem AffineSubspace.isConnected_setOf_sSameSide {V : Type u_2} {P : Type u_4} [SeminormedAddCommGroup V] [NormedSpace ℝ V] [PseudoMetricSpace P] [NormedAddTorsor V P] {s : AffineSubspace ℝ P} {x : P} (hx : x ∉ s) (h : Set.Nonempty ↑s) : IsConnected {y : P | AffineSubspace.SSameSide s x y}

theorem AffineSubspace.isPreconnected_setOf_sSameSide {V : Type u_2} {P : Type u_4} [SeminormedAddCommGroup V] [NormedSpace ℝ V] [PseudoMetricSpace P] [NormedAddTorsor V P] (s : AffineSubspace ℝ P) (x : P) : IsPreconnected {y : P | AffineSubspace.SSameSide s x y}

theorem AffineSubspace.isConnected_setOf_wOppSide {V : Type u_2} {P : Type u_4} [SeminormedAddCommGroup V] [NormedSpace ℝ V] [PseudoMetricSpace P] [NormedAddTorsor V P] {s : AffineSubspace ℝ P} (x : P) (h : Set.Nonempty ↑s) : IsConnected {y : P | AffineSubspace.WOppSide s x y}

theorem AffineSubspace.isPreconnected_setOf_wOppSide {V : Type u_2} {P : Type u_4} [SeminormedAddCommGroup V] [NormedSpace ℝ V] [PseudoMetricSpace P] [NormedAddTorsor V P] (s : AffineSubspace ℝ P) (x : P) : IsPreconnected {y : P | AffineSubspace.WOppSide s x y}

theorem AffineSubspace.isConnected_setOf_sOppSide {V : Type u_2} {P : Type u_4} [SeminormedAddCommGroup V] [NormedSpace ℝ V] [PseudoMetricSpace P] [NormedAddTorsor V P] {s : AffineSubspace ℝ P} {x : P} (hx : x ∉ s) (h : Set.Nonempty ↑s) : IsConnected {y : P | AffineSubspace.SOppSide s x y}

theorem AffineSubspace.isPreconnected_setOf_sOppSide {V : Type u_2} {P : Type u_4} [SeminormedAddCommGroup V] [NormedSpace ℝ V] [PseudoMetricSpace P] [NormedAddTorsor V P] (s : AffineSubspace ℝ P) (x : P) : IsPreconnected {y : P | AffineSubspace.SOppSide s x y}