# 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} [] [Module R V] [] (s : ) (x : P) (y : P) :

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

Equations
• s.WSameSide x y = p₁s, p₂s, SameRay R (x -ᵥ p₁) (y -ᵥ p₂)
Instances For
def AffineSubspace.SSameSide {R : Type u_1} {V : Type u_2} {P : Type u_4} [] [Module R V] [] (s : ) (x : P) (y : P) :

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

Equations
• s.SSameSide x y = (s.WSameSide x y xs ys)
Instances For
def AffineSubspace.WOppSide {R : Type u_1} {V : Type u_2} {P : Type u_4} [] [Module R V] [] (s : ) (x : P) (y : P) :

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

Equations
• s.WOppSide x y = p₁s, p₂s, SameRay R (x -ᵥ p₁) (p₂ -ᵥ y)
Instances For
def AffineSubspace.SOppSide {R : Type u_1} {V : Type u_2} {P : Type u_4} [] [Module R V] [] (s : ) (x : P) (y : P) :

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

Equations
• s.SOppSide x y = (s.WOppSide x y xs ys)
Instances For
theorem AffineSubspace.WSameSide.map {R : Type u_1} {V : Type u_2} {V' : Type u_3} {P : Type u_4} {P' : Type u_5} [] [Module R V] [] [] [Module R V'] [AddTorsor V' P'] {s : } {x : P} {y : P} (h : s.WSameSide x y) (f : P →ᵃ[R] P') :
().WSameSide (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} [] [Module R V] [] [] [Module R V'] [AddTorsor V' P'] {s : } {x : P} {y : P} {f : P →ᵃ[R] P'} (hf : ) :
().WSameSide (f x) (f y) s.WSameSide 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} [] [Module R V] [] [] [Module R V'] [AddTorsor V' P'] {s : } {x : P} {y : P} {f : P →ᵃ[R] P'} (hf : ) :
().SSameSide (f x) (f y) s.SSameSide 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} [] [Module R V] [] [] [Module R V'] [AddTorsor V' P'] {s : } {x : P} {y : P} (f : P ≃ᵃ[R] P') :
(AffineSubspace.map (f) s).WSameSide (f x) (f y) s.WSameSide 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} [] [Module R V] [] [] [Module R V'] [AddTorsor V' P'] {s : } {x : P} {y : P} (f : P ≃ᵃ[R] P') :
(AffineSubspace.map (f) s).SSameSide (f x) (f y) s.SSameSide 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} [] [Module R V] [] [] [Module R V'] [AddTorsor V' P'] {s : } {x : P} {y : P} (h : s.WOppSide x y) (f : P →ᵃ[R] P') :
().WOppSide (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} [] [Module R V] [] [] [Module R V'] [AddTorsor V' P'] {s : } {x : P} {y : P} {f : P →ᵃ[R] P'} (hf : ) :
().WOppSide (f x) (f y) s.WOppSide 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} [] [Module R V] [] [] [Module R V'] [AddTorsor V' P'] {s : } {x : P} {y : P} {f : P →ᵃ[R] P'} (hf : ) :
().SOppSide (f x) (f y) s.SOppSide 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} [] [Module R V] [] [] [Module R V'] [AddTorsor V' P'] {s : } {x : P} {y : P} (f : P ≃ᵃ[R] P') :
(AffineSubspace.map (f) s).WOppSide (f x) (f y) s.WOppSide 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} [] [Module R V] [] [] [Module R V'] [AddTorsor V' P'] {s : } {x : P} {y : P} (f : P ≃ᵃ[R] P') :
(AffineSubspace.map (f) s).SOppSide (f x) (f y) s.SOppSide x y
theorem AffineSubspace.WSameSide.nonempty {R : Type u_1} {V : Type u_2} {P : Type u_4} [] [Module R V] [] {s : } {x : P} {y : P} (h : s.WSameSide x y) :
(s).Nonempty
theorem AffineSubspace.SSameSide.nonempty {R : Type u_1} {V : Type u_2} {P : Type u_4} [] [Module R V] [] {s : } {x : P} {y : P} (h : s.SSameSide x y) :
(s).Nonempty
theorem AffineSubspace.WOppSide.nonempty {R : Type u_1} {V : Type u_2} {P : Type u_4} [] [Module R V] [] {s : } {x : P} {y : P} (h : s.WOppSide x y) :
(s).Nonempty
theorem AffineSubspace.SOppSide.nonempty {R : Type u_1} {V : Type u_2} {P : Type u_4} [] [Module R V] [] {s : } {x : P} {y : P} (h : s.SOppSide x y) :
(s).Nonempty
theorem AffineSubspace.SSameSide.wSameSide {R : Type u_1} {V : Type u_2} {P : Type u_4} [] [Module R V] [] {s : } {x : P} {y : P} (h : s.SSameSide x y) :
s.WSameSide x y
theorem AffineSubspace.SSameSide.left_not_mem {R : Type u_1} {V : Type u_2} {P : Type u_4} [] [Module R V] [] {s : } {x : P} {y : P} (h : s.SSameSide x y) :
xs
theorem AffineSubspace.SSameSide.right_not_mem {R : Type u_1} {V : Type u_2} {P : Type u_4} [] [Module R V] [] {s : } {x : P} {y : P} (h : s.SSameSide x y) :
ys
theorem AffineSubspace.SOppSide.wOppSide {R : Type u_1} {V : Type u_2} {P : Type u_4} [] [Module R V] [] {s : } {x : P} {y : P} (h : s.SOppSide x y) :
s.WOppSide x y
theorem AffineSubspace.SOppSide.left_not_mem {R : Type u_1} {V : Type u_2} {P : Type u_4} [] [Module R V] [] {s : } {x : P} {y : P} (h : s.SOppSide x y) :
xs
theorem AffineSubspace.SOppSide.right_not_mem {R : Type u_1} {V : Type u_2} {P : Type u_4} [] [Module R V] [] {s : } {x : P} {y : P} (h : s.SOppSide x y) :
ys
theorem AffineSubspace.wSameSide_comm {R : Type u_1} {V : Type u_2} {P : Type u_4} [] [Module R V] [] {s : } {x : P} {y : P} :
s.WSameSide x y s.WSameSide y x
theorem AffineSubspace.WSameSide.symm {R : Type u_1} {V : Type u_2} {P : Type u_4} [] [Module R V] [] {s : } {x : P} {y : P} :
s.WSameSide x ys.WSameSide 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} [] [Module R V] [] {s : } {x : P} {y : P} :
s.SSameSide x y s.SSameSide y x
theorem AffineSubspace.SSameSide.symm {R : Type u_1} {V : Type u_2} {P : Type u_4} [] [Module R V] [] {s : } {x : P} {y : P} :
s.SSameSide x ys.SSameSide 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} [] [Module R V] [] {s : } {x : P} {y : P} :
s.WOppSide x y s.WOppSide y x
theorem AffineSubspace.WOppSide.symm {R : Type u_1} {V : Type u_2} {P : Type u_4} [] [Module R V] [] {s : } {x : P} {y : P} :
s.WOppSide x ys.WOppSide 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} [] [Module R V] [] {s : } {x : P} {y : P} :
s.SOppSide x y s.SOppSide y x
theorem AffineSubspace.SOppSide.symm {R : Type u_1} {V : Type u_2} {P : Type u_4} [] [Module R V] [] {s : } {x : P} {y : P} :
s.SOppSide x ys.SOppSide 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} [] [Module R V] [] (x : P) (y : P) :
¬.WSameSide x y
theorem AffineSubspace.not_sSameSide_bot {R : Type u_1} {V : Type u_2} {P : Type u_4} [] [Module R V] [] (x : P) (y : P) :
¬.SSameSide x y
theorem AffineSubspace.not_wOppSide_bot {R : Type u_1} {V : Type u_2} {P : Type u_4} [] [Module R V] [] (x : P) (y : P) :
¬.WOppSide x y
theorem AffineSubspace.not_sOppSide_bot {R : Type u_1} {V : Type u_2} {P : Type u_4} [] [Module R V] [] (x : P) (y : P) :
¬.SOppSide x y
@[simp]
theorem AffineSubspace.wSameSide_self_iff {R : Type u_1} {V : Type u_2} {P : Type u_4} [] [Module R V] [] {s : } {x : P} :
s.WSameSide x x (s).Nonempty
theorem AffineSubspace.sSameSide_self_iff {R : Type u_1} {V : Type u_2} {P : Type u_4} [] [Module R V] [] {s : } {x : P} :
s.SSameSide x x (s).Nonempty xs
theorem AffineSubspace.wSameSide_of_left_mem {R : Type u_1} {V : Type u_2} {P : Type u_4} [] [Module R V] [] {s : } {x : P} (y : P) (hx : x s) :
s.WSameSide x y
theorem AffineSubspace.wSameSide_of_right_mem {R : Type u_1} {V : Type u_2} {P : Type u_4} [] [Module R V] [] {s : } (x : P) {y : P} (hy : y s) :
s.WSameSide x y
theorem AffineSubspace.wOppSide_of_left_mem {R : Type u_1} {V : Type u_2} {P : Type u_4} [] [Module R V] [] {s : } {x : P} (y : P) (hx : x s) :
s.WOppSide x y
theorem AffineSubspace.wOppSide_of_right_mem {R : Type u_1} {V : Type u_2} {P : Type u_4} [] [Module R V] [] {s : } (x : P) {y : P} (hy : y s) :
s.WOppSide x y
theorem AffineSubspace.wSameSide_vadd_left_iff {R : Type u_1} {V : Type u_2} {P : Type u_4} [] [Module R V] [] {s : } {x : P} {y : P} {v : V} (hv : v s.direction) :
s.WSameSide (v +ᵥ x) y s.WSameSide x y
theorem AffineSubspace.wSameSide_vadd_right_iff {R : Type u_1} {V : Type u_2} {P : Type u_4} [] [Module R V] [] {s : } {x : P} {y : P} {v : V} (hv : v s.direction) :
s.WSameSide x (v +ᵥ y) s.WSameSide x y
theorem AffineSubspace.sSameSide_vadd_left_iff {R : Type u_1} {V : Type u_2} {P : Type u_4} [] [Module R V] [] {s : } {x : P} {y : P} {v : V} (hv : v s.direction) :
s.SSameSide (v +ᵥ x) y s.SSameSide x y
theorem AffineSubspace.sSameSide_vadd_right_iff {R : Type u_1} {V : Type u_2} {P : Type u_4} [] [Module R V] [] {s : } {x : P} {y : P} {v : V} (hv : v s.direction) :
s.SSameSide x (v +ᵥ y) s.SSameSide x y
theorem AffineSubspace.wOppSide_vadd_left_iff {R : Type u_1} {V : Type u_2} {P : Type u_4} [] [Module R V] [] {s : } {x : P} {y : P} {v : V} (hv : v s.direction) :
s.WOppSide (v +ᵥ x) y s.WOppSide x y
theorem AffineSubspace.wOppSide_vadd_right_iff {R : Type u_1} {V : Type u_2} {P : Type u_4} [] [Module R V] [] {s : } {x : P} {y : P} {v : V} (hv : v s.direction) :
s.WOppSide x (v +ᵥ y) s.WOppSide x y
theorem AffineSubspace.sOppSide_vadd_left_iff {R : Type u_1} {V : Type u_2} {P : Type u_4} [] [Module R V] [] {s : } {x : P} {y : P} {v : V} (hv : v s.direction) :
s.SOppSide (v +ᵥ x) y s.SOppSide x y
theorem AffineSubspace.sOppSide_vadd_right_iff {R : Type u_1} {V : Type u_2} {P : Type u_4} [] [Module R V] [] {s : } {x : P} {y : P} {v : V} (hv : v s.direction) :
s.SOppSide x (v +ᵥ y) s.SOppSide x y
theorem AffineSubspace.wSameSide_smul_vsub_vadd_left {R : Type u_1} {V : Type u_2} {P : Type u_4} [] [Module R V] [] {s : } {p₁ : P} {p₂ : P} (x : P) (hp₁ : p₁ s) (hp₂ : p₂ s) {t : R} (ht : 0 t) :
s.WSameSide (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} [] [Module R V] [] {s : } {p₁ : P} {p₂ : P} (x : P) (hp₁ : p₁ s) (hp₂ : p₂ s) {t : R} (ht : 0 t) :
s.WSameSide x (t (x -ᵥ p₁) +ᵥ p₂)
theorem AffineSubspace.wSameSide_lineMap_left {R : Type u_1} {V : Type u_2} {P : Type u_4} [] [Module R V] [] {s : } {x : P} (y : P) (h : x s) {t : R} (ht : 0 t) :
s.WSameSide (() t) y
theorem AffineSubspace.wSameSide_lineMap_right {R : Type u_1} {V : Type u_2} {P : Type u_4} [] [Module R V] [] {s : } {x : P} (y : P) (h : x s) {t : R} (ht : 0 t) :
s.WSameSide y (() t)
theorem AffineSubspace.wOppSide_smul_vsub_vadd_left {R : Type u_1} {V : Type u_2} {P : Type u_4} [] [Module R V] [] {s : } {p₁ : P} {p₂ : P} (x : P) (hp₁ : p₁ s) (hp₂ : p₂ s) {t : R} (ht : t 0) :
s.WOppSide (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} [] [Module R V] [] {s : } {p₁ : P} {p₂ : P} (x : P) (hp₁ : p₁ s) (hp₂ : p₂ s) {t : R} (ht : t 0) :
s.WOppSide x (t (x -ᵥ p₁) +ᵥ p₂)
theorem AffineSubspace.wOppSide_lineMap_left {R : Type u_1} {V : Type u_2} {P : Type u_4} [] [Module R V] [] {s : } {x : P} (y : P) (h : x s) {t : R} (ht : t 0) :
s.WOppSide (() t) y
theorem AffineSubspace.wOppSide_lineMap_right {R : Type u_1} {V : Type u_2} {P : Type u_4} [] [Module R V] [] {s : } {x : P} (y : P) (h : x s) {t : R} (ht : t 0) :
s.WOppSide y (() t)
theorem Wbtw.wSameSide₂₃ {R : Type u_1} {V : Type u_2} {P : Type u_4} [] [Module R V] [] {s : } {x : P} {y : P} {z : P} (h : Wbtw R x y z) (hx : x s) :
s.WSameSide y z
theorem Wbtw.wSameSide₃₂ {R : Type u_1} {V : Type u_2} {P : Type u_4} [] [Module R V] [] {s : } {x : P} {y : P} {z : P} (h : Wbtw R x y z) (hx : x s) :
s.WSameSide z y
theorem Wbtw.wSameSide₁₂ {R : Type u_1} {V : Type u_2} {P : Type u_4} [] [Module R V] [] {s : } {x : P} {y : P} {z : P} (h : Wbtw R x y z) (hz : z s) :
s.WSameSide x y
theorem Wbtw.wSameSide₂₁ {R : Type u_1} {V : Type u_2} {P : Type u_4} [] [Module R V] [] {s : } {x : P} {y : P} {z : P} (h : Wbtw R x y z) (hz : z s) :
s.WSameSide y x
theorem Wbtw.wOppSide₁₃ {R : Type u_1} {V : Type u_2} {P : Type u_4} [] [Module R V] [] {s : } {x : P} {y : P} {z : P} (h : Wbtw R x y z) (hy : y s) :
s.WOppSide x z
theorem Wbtw.wOppSide₃₁ {R : Type u_1} {V : Type u_2} {P : Type u_4} [] [Module R V] [] {s : } {x : P} {y : P} {z : P} (h : Wbtw R x y z) (hy : y s) :
s.WOppSide z x
@[simp]
theorem AffineSubspace.wOppSide_self_iff {R : Type u_1} {V : Type u_2} {P : Type u_4} [] [Module R V] [] {s : } {x : P} :
s.WOppSide x x x s
theorem AffineSubspace.not_sOppSide_self {R : Type u_1} {V : Type u_2} {P : Type u_4} [] [Module R V] [] (s : ) (x : P) :
¬s.SOppSide x x
theorem AffineSubspace.wSameSide_iff_exists_left {R : Type u_1} {V : Type u_2} {P : Type u_4} [] [Module R V] [] {s : } {x : P} {y : P} {p₁ : P} (h : p₁ s) :
s.WSameSide x y x s p₂s, 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} [] [Module R V] [] {s : } {x : P} {y : P} {p₂ : P} (h : p₂ s) :
s.WSameSide x y y s p₁s, 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} [] [Module R V] [] {s : } {x : P} {y : P} {p₁ : P} (h : p₁ s) :
s.SSameSide x y xs ys p₂s, 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} [] [Module R V] [] {s : } {x : P} {y : P} {p₂ : P} (h : p₂ s) :
s.SSameSide x y xs ys p₁s, 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} [] [Module R V] [] {s : } {x : P} {y : P} {p₁ : P} (h : p₁ s) :
s.WOppSide x y x s p₂s, 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} [] [Module R V] [] {s : } {x : P} {y : P} {p₂ : P} (h : p₂ s) :
s.WOppSide x y y s p₁s, 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} [] [Module R V] [] {s : } {x : P} {y : P} {p₁ : P} (h : p₁ s) :
s.SOppSide x y xs ys p₂s, 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} [] [Module R V] [] {s : } {x : P} {y : P} {p₂ : P} (h : p₂ s) :
s.SOppSide x y xs ys p₁s, SameRay R (x -ᵥ p₁) (p₂ -ᵥ y)
theorem AffineSubspace.WSameSide.trans {R : Type u_1} {V : Type u_2} {P : Type u_4} [] [Module R V] [] {s : } {x : P} {y : P} {z : P} (hxy : s.WSameSide x y) (hyz : s.WSameSide y z) (hy : ys) :
s.WSameSide x z
theorem AffineSubspace.WSameSide.trans_sSameSide {R : Type u_1} {V : Type u_2} {P : Type u_4} [] [Module R V] [] {s : } {x : P} {y : P} {z : P} (hxy : s.WSameSide x y) (hyz : s.SSameSide y z) :
s.WSameSide x z
theorem AffineSubspace.WSameSide.trans_wOppSide {R : Type u_1} {V : Type u_2} {P : Type u_4} [] [Module R V] [] {s : } {x : P} {y : P} {z : P} (hxy : s.WSameSide x y) (hyz : s.WOppSide y z) (hy : ys) :
s.WOppSide x z
theorem AffineSubspace.WSameSide.trans_sOppSide {R : Type u_1} {V : Type u_2} {P : Type u_4} [] [Module R V] [] {s : } {x : P} {y : P} {z : P} (hxy : s.WSameSide x y) (hyz : s.SOppSide y z) :
s.WOppSide x z
theorem AffineSubspace.SSameSide.trans_wSameSide {R : Type u_1} {V : Type u_2} {P : Type u_4} [] [Module R V] [] {s : } {x : P} {y : P} {z : P} (hxy : s.SSameSide x y) (hyz : s.WSameSide y z) :
s.WSameSide x z
theorem AffineSubspace.SSameSide.trans {R : Type u_1} {V : Type u_2} {P : Type u_4} [] [Module R V] [] {s : } {x : P} {y : P} {z : P} (hxy : s.SSameSide x y) (hyz : s.SSameSide y z) :
s.SSameSide x z
theorem AffineSubspace.SSameSide.trans_wOppSide {R : Type u_1} {V : Type u_2} {P : Type u_4} [] [Module R V] [] {s : } {x : P} {y : P} {z : P} (hxy : s.SSameSide x y) (hyz : s.WOppSide y z) :
s.WOppSide x z
theorem AffineSubspace.SSameSide.trans_sOppSide {R : Type u_1} {V : Type u_2} {P : Type u_4} [] [Module R V] [] {s : } {x : P} {y : P} {z : P} (hxy : s.SSameSide x y) (hyz : s.SOppSide y z) :
s.SOppSide x z
theorem AffineSubspace.WOppSide.trans_wSameSide {R : Type u_1} {V : Type u_2} {P : Type u_4} [] [Module R V] [] {s : } {x : P} {y : P} {z : P} (hxy : s.WOppSide x y) (hyz : s.WSameSide y z) (hy : ys) :
s.WOppSide x z
theorem AffineSubspace.WOppSide.trans_sSameSide {R : Type u_1} {V : Type u_2} {P : Type u_4} [] [Module R V] [] {s : } {x : P} {y : P} {z : P} (hxy : s.WOppSide x y) (hyz : s.SSameSide y z) :
s.WOppSide x z
theorem AffineSubspace.WOppSide.trans {R : Type u_1} {V : Type u_2} {P : Type u_4} [] [Module R V] [] {s : } {x : P} {y : P} {z : P} (hxy : s.WOppSide x y) (hyz : s.WOppSide y z) (hy : ys) :
s.WSameSide x z
theorem AffineSubspace.WOppSide.trans_sOppSide {R : Type u_1} {V : Type u_2} {P : Type u_4} [] [Module R V] [] {s : } {x : P} {y : P} {z : P} (hxy : s.WOppSide x y) (hyz : s.SOppSide y z) :
s.WSameSide x z
theorem AffineSubspace.SOppSide.trans_wSameSide {R : Type u_1} {V : Type u_2} {P : Type u_4} [] [Module R V] [] {s : } {x : P} {y : P} {z : P} (hxy : s.SOppSide x y) (hyz : s.WSameSide y z) :
s.WOppSide x z
theorem AffineSubspace.SOppSide.trans_sSameSide {R : Type u_1} {V : Type u_2} {P : Type u_4} [] [Module R V] [] {s : } {x : P} {y : P} {z : P} (hxy : s.SOppSide x y) (hyz : s.SSameSide y z) :
s.SOppSide x z
theorem AffineSubspace.SOppSide.trans_wOppSide {R : Type u_1} {V : Type u_2} {P : Type u_4} [] [Module R V] [] {s : } {x : P} {y : P} {z : P} (hxy : s.SOppSide x y) (hyz : s.WOppSide y z) :
s.WSameSide x z
theorem AffineSubspace.SOppSide.trans {R : Type u_1} {V : Type u_2} {P : Type u_4} [] [Module R V] [] {s : } {x : P} {y : P} {z : P} (hxy : s.SOppSide x y) (hyz : s.SOppSide y z) :
s.SSameSide x z
theorem AffineSubspace.wSameSide_and_wOppSide_iff {R : Type u_1} {V : Type u_2} {P : Type u_4} [] [Module R V] [] {s : } {x : P} {y : P} :
s.WSameSide x y s.WOppSide x y x s y s
theorem AffineSubspace.WSameSide.not_sOppSide {R : Type u_1} {V : Type u_2} {P : Type u_4} [] [Module R V] [] {s : } {x : P} {y : P} (h : s.WSameSide x y) :
¬s.SOppSide x y
theorem AffineSubspace.SSameSide.not_wOppSide {R : Type u_1} {V : Type u_2} {P : Type u_4} [] [Module R V] [] {s : } {x : P} {y : P} (h : s.SSameSide x y) :
¬s.WOppSide x y
theorem AffineSubspace.SSameSide.not_sOppSide {R : Type u_1} {V : Type u_2} {P : Type u_4} [] [Module R V] [] {s : } {x : P} {y : P} (h : s.SSameSide x y) :
¬s.SOppSide x y
theorem AffineSubspace.WOppSide.not_sSameSide {R : Type u_1} {V : Type u_2} {P : Type u_4} [] [Module R V] [] {s : } {x : P} {y : P} (h : s.WOppSide x y) :
¬s.SSameSide x y
theorem AffineSubspace.SOppSide.not_wSameSide {R : Type u_1} {V : Type u_2} {P : Type u_4} [] [Module R V] [] {s : } {x : P} {y : P} (h : s.SOppSide x y) :
¬s.WSameSide x y
theorem AffineSubspace.SOppSide.not_sSameSide {R : Type u_1} {V : Type u_2} {P : Type u_4} [] [Module R V] [] {s : } {x : P} {y : P} (h : s.SOppSide x y) :
¬s.SSameSide x y
theorem AffineSubspace.wOppSide_iff_exists_wbtw {R : Type u_1} {V : Type u_2} {P : Type u_4} [] [Module R V] [] {s : } {x : P} {y : P} :
s.WOppSide x y ps, Wbtw R x p y
theorem AffineSubspace.SOppSide.exists_sbtw {R : Type u_1} {V : Type u_2} {P : Type u_4} [] [Module R V] [] {s : } {x : P} {y : P} (h : s.SOppSide x y) :
ps, 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} [] [Module R V] [] {s : } {x : P} {y : P} {z : P} (h : Sbtw R x y z) (hx : xs) (hy : y s) :
s.SOppSide x z
theorem AffineSubspace.sSameSide_smul_vsub_vadd_left {R : Type u_1} {V : Type u_2} {P : Type u_4} [] [Module R V] [] {s : } {x : P} {p₁ : P} {p₂ : P} (hx : xs) (hp₁ : p₁ s) (hp₂ : p₂ s) {t : R} (ht : 0 < t) :
s.SSameSide (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} [] [Module R V] [] {s : } {x : P} {p₁ : P} {p₂ : P} (hx : xs) (hp₁ : p₁ s) (hp₂ : p₂ s) {t : R} (ht : 0 < t) :
s.SSameSide x (t (x -ᵥ p₁) +ᵥ p₂)
theorem AffineSubspace.sSameSide_lineMap_left {R : Type u_1} {V : Type u_2} {P : Type u_4} [] [Module R V] [] {s : } {x : P} {y : P} (hx : x s) (hy : ys) {t : R} (ht : 0 < t) :
s.SSameSide (() t) y
theorem AffineSubspace.sSameSide_lineMap_right {R : Type u_1} {V : Type u_2} {P : Type u_4} [] [Module R V] [] {s : } {x : P} {y : P} (hx : x s) (hy : ys) {t : R} (ht : 0 < t) :
s.SSameSide y (() t)
theorem AffineSubspace.sOppSide_smul_vsub_vadd_left {R : Type u_1} {V : Type u_2} {P : Type u_4} [] [Module R V] [] {s : } {x : P} {p₁ : P} {p₂ : P} (hx : xs) (hp₁ : p₁ s) (hp₂ : p₂ s) {t : R} (ht : t < 0) :
s.SOppSide (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} [] [Module R V] [] {s : } {x : P} {p₁ : P} {p₂ : P} (hx : xs) (hp₁ : p₁ s) (hp₂ : p₂ s) {t : R} (ht : t < 0) :
s.SOppSide x (t (x -ᵥ p₁) +ᵥ p₂)
theorem AffineSubspace.sOppSide_lineMap_left {R : Type u_1} {V : Type u_2} {P : Type u_4} [] [Module R V] [] {s : } {x : P} {y : P} (hx : x s) (hy : ys) {t : R} (ht : t < 0) :
s.SOppSide (() t) y
theorem AffineSubspace.sOppSide_lineMap_right {R : Type u_1} {V : Type u_2} {P : Type u_4} [] [Module R V] [] {s : } {x : P} {y : P} (hx : x s) (hy : ys) {t : R} (ht : t < 0) :
s.SOppSide y (() t)
theorem AffineSubspace.setOf_wSameSide_eq_image2 {R : Type u_1} {V : Type u_2} {P : Type u_4} [] [Module R V] [] {s : } {x : P} {p : P} (hx : xs) (hp : p s) :
{y : P | s.WSameSide x y} = Set.image2 (fun (t : R) (q : P) => t (x -ᵥ p) +ᵥ q) () s
theorem AffineSubspace.setOf_sSameSide_eq_image2 {R : Type u_1} {V : Type u_2} {P : Type u_4} [] [Module R V] [] {s : } {x : P} {p : P} (hx : xs) (hp : p s) :
{y : P | s.SSameSide x y} = Set.image2 (fun (t : R) (q : P) => t (x -ᵥ p) +ᵥ q) () s
theorem AffineSubspace.setOf_wOppSide_eq_image2 {R : Type u_1} {V : Type u_2} {P : Type u_4} [] [Module R V] [] {s : } {x : P} {p : P} (hx : xs) (hp : p s) :
{y : P | s.WOppSide x y} = Set.image2 (fun (t : R) (q : P) => t (x -ᵥ p) +ᵥ q) () s
theorem AffineSubspace.setOf_sOppSide_eq_image2 {R : Type u_1} {V : Type u_2} {P : Type u_4} [] [Module R V] [] {s : } {x : P} {p : P} (hx : xs) (hp : p s) :
{y : P | s.SOppSide x y} = Set.image2 (fun (t : R) (q : P) => t (x -ᵥ p) +ᵥ q) () s
theorem AffineSubspace.wOppSide_pointReflection {R : Type u_1} {V : Type u_2} {P : Type u_4} [] [Module R V] [] {s : } {x : P} (y : P) (hx : x s) :
s.WOppSide y ( y)
theorem AffineSubspace.sOppSide_pointReflection {R : Type u_1} {V : Type u_2} {P : Type u_4} [] [Module R V] [] {s : } {x : P} {y : P} (hx : x s) (hy : ys) :
s.SOppSide y ( y)
theorem AffineSubspace.isConnected_setOf_wSameSide {V : Type u_2} {P : Type u_4} [] [] {s : } (x : P) (h : (s).Nonempty) :
IsConnected {y : P | s.WSameSide x y}
theorem AffineSubspace.isPreconnected_setOf_wSameSide {V : Type u_2} {P : Type u_4} [] [] (s : ) (x : P) :
IsPreconnected {y : P | s.WSameSide x y}
theorem AffineSubspace.isConnected_setOf_sSameSide {V : Type u_2} {P : Type u_4} [] [] {s : } {x : P} (hx : xs) (h : (s).Nonempty) :
IsConnected {y : P | s.SSameSide x y}
theorem AffineSubspace.isPreconnected_setOf_sSameSide {V : Type u_2} {P : Type u_4} [] [] (s : ) (x : P) :
IsPreconnected {y : P | s.SSameSide x y}
theorem AffineSubspace.isConnected_setOf_wOppSide {V : Type u_2} {P : Type u_4} [] [] {s : } (x : P) (h : (s).Nonempty) :
IsConnected {y : P | s.WOppSide x y}
theorem AffineSubspace.isPreconnected_setOf_wOppSide {V : Type u_2} {P : Type u_4} [] [] (s : ) (x : P) :
IsPreconnected {y : P | s.WOppSide x y}
theorem AffineSubspace.isConnected_setOf_sOppSide {V : Type u_2} {P : Type u_4} [] [] {s : } {x : P} (hx : xs) (h : (s).Nonempty) :
IsConnected {y : P | s.SOppSide x y}
theorem AffineSubspace.isPreconnected_setOf_sOppSide {V : Type u_2} {P : Type u_4} [] [] (s : ) (x : P) :
IsPreconnected {y : P | s.SOppSide x y}