# mathlib3documentation

analysis.convex.between

# Betweenness in affine spaces #

THIS FILE IS SYNCHRONIZED WITH MATHLIB4. Any changes to this file require a corresponding PR to mathlib4.

This file defines notions of a point in an affine space being between two given points.

## Main definitions #

• affine_segment R x y: The segment of points weakly between x and y.
• wbtw R x y z: The point y is weakly between x and z.
• sbtw R x y z: The point y is strictly between x and z.
def affine_segment (R : Type u_1) {V : Type u_2} {P : Type u_4} [ordered_ring R] [ V] [ P] (x y : P) :
set P

The segment of points weakly between x and y. When convexity is refactored to support abstract affine combination spaces, this will no longer need to be a separate definition from segment. However, lemmas involving +ᵥ or -ᵥ will still be relevant after such a refactoring, as distinct from versions involving + or - in a module.

Equations
theorem affine_segment_eq_segment (R : Type u_1) {V : Type u_2} [ordered_ring R] [ V] (x y : V) :
y = x y
theorem affine_segment_comm (R : Type u_1) {V : Type u_2} {P : Type u_4} [ordered_ring R] [ V] [ P] (x y : P) :
y = x
theorem left_mem_affine_segment (R : Type u_1) {V : Type u_2} {P : Type u_4} [ordered_ring R] [ V] [ P] (x y : P) :
x y
theorem right_mem_affine_segment (R : Type u_1) {V : Type u_2} {P : Type u_4} [ordered_ring R] [ V] [ P] (x y : P) :
y y
@[simp]
theorem affine_segment_same (R : Type u_1) {V : Type u_2} {P : Type u_4} [ordered_ring R] [ V] [ P] (x : P) :
x = {x}
@[simp]
theorem affine_segment_image {R : Type u_1} {V : Type u_2} {V' : Type u_3} {P : Type u_4} {P' : Type u_5} [ordered_ring R] [ V] [ P] [add_comm_group V'] [ V'] [ P'] (f : P →ᵃ[R] P') (x y : P) :
f '' y = (f x) (f y)
@[simp]
theorem affine_segment_const_vadd_image (R : Type u_1) {V : Type u_2} {P : Type u_4} [ordered_ring R] [ V] [ P] (x y : P) (v : V) :
'' y = (v +ᵥ x) (v +ᵥ y)
@[simp]
theorem affine_segment_vadd_const_image (R : Type u_1) {V : Type u_2} {P : Type u_4} [ordered_ring R] [ V] [ P] (x y : V) (p : P) :
(λ (_x : V), _x +ᵥ p) '' y = (x +ᵥ p) (y +ᵥ p)
@[simp]
theorem affine_segment_const_vsub_image (R : Type u_1) {V : Type u_2} {P : Type u_4} [ordered_ring R] [ V] [ P] (x y p : P) :
'' y = (p -ᵥ x) (p -ᵥ y)
@[simp]
theorem affine_segment_vsub_const_image (R : Type u_1) {V : Type u_2} {P : Type u_4} [ordered_ring R] [ V] [ P] (x y p : P) :
(λ (_x : P), _x -ᵥ p) '' y = (x -ᵥ p) (y -ᵥ p)
@[simp]
theorem mem_const_vadd_affine_segment {R : Type u_1} {V : Type u_2} {P : Type u_4} [ordered_ring R] [ V] [ P] {x y z : P} (v : V) :
v +ᵥ z (v +ᵥ x) (v +ᵥ y) z y
@[simp]
theorem mem_vadd_const_affine_segment {R : Type u_1} {V : Type u_2} {P : Type u_4} [ordered_ring R] [ V] [ P] {x y z : V} (p : P) :
z +ᵥ p (x +ᵥ p) (y +ᵥ p) z y
@[simp]
theorem mem_const_vsub_affine_segment {R : Type u_1} {V : Type u_2} {P : Type u_4} [ordered_ring R] [ V] [ P] {x y z : P} (p : P) :
p -ᵥ z (p -ᵥ x) (p -ᵥ y) z y
@[simp]
theorem mem_vsub_const_affine_segment {R : Type u_1} {V : Type u_2} {P : Type u_4} [ordered_ring R] [ V] [ P] {x y z : P} (p : P) :
z -ᵥ p (x -ᵥ p) (y -ᵥ p) z y
def wbtw (R : Type u_1) {V : Type u_2} {P : Type u_4} [ordered_ring R] [ V] [ P] (x y z : P) :
Prop

The point y is weakly between x and z.

Equations
def sbtw (R : Type u_1) {V : Type u_2} {P : Type u_4} [ordered_ring R] [ V] [ P] (x y z : P) :
Prop

The point y is strictly between x and z.

Equations
theorem wbtw.map {R : Type u_1} {V : Type u_2} {V' : Type u_3} {P : Type u_4} {P' : Type u_5} [ordered_ring R] [ V] [ P] [add_comm_group V'] [ V'] [ P'] {x y z : P} (h : wbtw R x y z) (f : P →ᵃ[R] P') :
wbtw R (f x) (f y) (f z)
theorem function.injective.wbtw_map_iff {R : Type u_1} {V : Type u_2} {V' : Type u_3} {P : Type u_4} {P' : Type u_5} [ordered_ring R] [ V] [ P] [add_comm_group V'] [ V'] [ P'] {x y z : P} {f : P →ᵃ[R] P'} (hf : function.injective f) :
wbtw R (f x) (f y) (f z) wbtw R x y z
theorem function.injective.sbtw_map_iff {R : Type u_1} {V : Type u_2} {V' : Type u_3} {P : Type u_4} {P' : Type u_5} [ordered_ring R] [ V] [ P] [add_comm_group V'] [ V'] [ P'] {x y z : P} {f : P →ᵃ[R] P'} (hf : function.injective f) :
sbtw R (f x) (f y) (f z) sbtw R x y z
@[simp]
theorem affine_equiv.wbtw_map_iff {R : Type u_1} {V : Type u_2} {V' : Type u_3} {P : Type u_4} {P' : Type u_5} [ordered_ring R] [ V] [ P] [add_comm_group V'] [ V'] [ P'] {x y z : P} (f : P ≃ᵃ[R] P') :
wbtw R (f x) (f y) (f z) wbtw R x y z
@[simp]
theorem affine_equiv.sbtw_map_iff {R : Type u_1} {V : Type u_2} {V' : Type u_3} {P : Type u_4} {P' : Type u_5} [ordered_ring R] [ V] [ P] [add_comm_group V'] [ V'] [ P'] {x y z : P} (f : P ≃ᵃ[R] P') :
sbtw R (f x) (f y) (f z) sbtw R x y z
@[simp]
theorem wbtw_const_vadd_iff {R : Type u_1} {V : Type u_2} {P : Type u_4} [ordered_ring R] [ V] [ P] {x y z : P} (v : V) :
wbtw R (v +ᵥ x) (v +ᵥ y) (v +ᵥ z) wbtw R x y z
@[simp]
theorem wbtw_vadd_const_iff {R : Type u_1} {V : Type u_2} {P : Type u_4} [ordered_ring R] [ V] [ P] {x y z : V} (p : P) :
wbtw R (x +ᵥ p) (y +ᵥ p) (z +ᵥ p) wbtw R x y z
@[simp]
theorem wbtw_const_vsub_iff {R : Type u_1} {V : Type u_2} {P : Type u_4} [ordered_ring R] [ V] [ P] {x y z : P} (p : P) :
wbtw R (p -ᵥ x) (p -ᵥ y) (p -ᵥ z) wbtw R x y z
@[simp]
theorem wbtw_vsub_const_iff {R : Type u_1} {V : Type u_2} {P : Type u_4} [ordered_ring R] [ V] [ P] {x y z : P} (p : P) :
wbtw R (x -ᵥ p) (y -ᵥ p) (z -ᵥ p) wbtw R x y z
@[simp]
theorem sbtw_const_vadd_iff {R : Type u_1} {V : Type u_2} {P : Type u_4} [ordered_ring R] [ V] [ P] {x y z : P} (v : V) :
sbtw R (v +ᵥ x) (v +ᵥ y) (v +ᵥ z) sbtw R x y z
@[simp]
theorem sbtw_vadd_const_iff {R : Type u_1} {V : Type u_2} {P : Type u_4} [ordered_ring R] [ V] [ P] {x y z : V} (p : P) :
sbtw R (x +ᵥ p) (y +ᵥ p) (z +ᵥ p) sbtw R x y z
@[simp]
theorem sbtw_const_vsub_iff {R : Type u_1} {V : Type u_2} {P : Type u_4} [ordered_ring R] [ V] [ P] {x y z : P} (p : P) :
sbtw R (p -ᵥ x) (p -ᵥ y) (p -ᵥ z) sbtw R x y z
@[simp]
theorem sbtw_vsub_const_iff {R : Type u_1} {V : Type u_2} {P : Type u_4} [ordered_ring R] [ V] [ P] {x y z : P} (p : P) :
sbtw R (x -ᵥ p) (y -ᵥ p) (z -ᵥ p) sbtw R x y z
theorem sbtw.wbtw {R : Type u_1} {V : Type u_2} {P : Type u_4} [ordered_ring R] [ V] [ P] {x y z : P} (h : sbtw R x y z) :
wbtw R x y z
theorem sbtw.ne_left {R : Type u_1} {V : Type u_2} {P : Type u_4} [ordered_ring R] [ V] [ P] {x y z : P} (h : sbtw R x y z) :
y x
theorem sbtw.left_ne {R : Type u_1} {V : Type u_2} {P : Type u_4} [ordered_ring R] [ V] [ P] {x y z : P} (h : sbtw R x y z) :
x y
theorem sbtw.ne_right {R : Type u_1} {V : Type u_2} {P : Type u_4} [ordered_ring R] [ V] [ P] {x y z : P} (h : sbtw R x y z) :
y z
theorem sbtw.right_ne {R : Type u_1} {V : Type u_2} {P : Type u_4} [ordered_ring R] [ V] [ P] {x y z : P} (h : sbtw R x y z) :
z y
theorem sbtw.mem_image_Ioo {R : Type u_1} {V : Type u_2} {P : Type u_4} [ordered_ring R] [ V] [ P] {x y z : P} (h : sbtw R x y z) :
y z) '' 1
theorem wbtw.mem_affine_span {R : Type u_1} {V : Type u_2} {P : Type u_4} [ordered_ring R] [ V] [ P] {x y z : P} (h : wbtw R x y z) :
y {x, z}
theorem wbtw_comm {R : Type u_1} {V : Type u_2} {P : Type u_4} [ordered_ring R] [ V] [ P] {x y z : P} :
wbtw R x y z wbtw R z y x
theorem wbtw.symm {R : Type u_1} {V : Type u_2} {P : Type u_4} [ordered_ring R] [ V] [ P] {x y z : P} :
wbtw R x y z wbtw R z y x

Alias of the forward direction of wbtw_comm.

theorem sbtw_comm {R : Type u_1} {V : Type u_2} {P : Type u_4} [ordered_ring R] [ V] [ P] {x y z : P} :
sbtw R x y z sbtw R z y x
theorem sbtw.symm {R : Type u_1} {V : Type u_2} {P : Type u_4} [ordered_ring R] [ V] [ P] {x y z : P} :
sbtw R x y z sbtw R z y x

Alias of the forward direction of sbtw_comm.

@[simp]
theorem wbtw_self_left (R : Type u_1) {V : Type u_2} {P : Type u_4} [ordered_ring R] [ V] [ P] (x y : P) :
wbtw R x x y
@[simp]
theorem wbtw_self_right (R : Type u_1) {V : Type u_2} {P : Type u_4} [ordered_ring R] [ V] [ P] (x y : P) :
wbtw R x y y
@[simp]
theorem wbtw_self_iff (R : Type u_1) {V : Type u_2} {P : Type u_4} [ordered_ring R] [ V] [ P] {x y : P} :
wbtw R x y x y = x
@[simp]
theorem not_sbtw_self_left (R : Type u_1) {V : Type u_2} {P : Type u_4} [ordered_ring R] [ V] [ P] (x y : P) :
¬sbtw R x x y
@[simp]
theorem not_sbtw_self_right (R : Type u_1) {V : Type u_2} {P : Type u_4} [ordered_ring R] [ V] [ P] (x y : P) :
¬sbtw R x y y
theorem wbtw.left_ne_right_of_ne_left {R : Type u_1} {V : Type u_2} {P : Type u_4} [ordered_ring R] [ V] [ P] {x y z : P} (h : wbtw R x y z) (hne : y x) :
x z
theorem wbtw.left_ne_right_of_ne_right {R : Type u_1} {V : Type u_2} {P : Type u_4} [ordered_ring R] [ V] [ P] {x y z : P} (h : wbtw R x y z) (hne : y z) :
x z
theorem sbtw.left_ne_right {R : Type u_1} {V : Type u_2} {P : Type u_4} [ordered_ring R] [ V] [ P] {x y z : P} (h : sbtw R x y z) :
x z
theorem sbtw_iff_mem_image_Ioo_and_ne {R : Type u_1} {V : Type u_2} {P : Type u_4} [ordered_ring R] [ V] [ P] {x y z : P} :
sbtw R x y z y z) '' 1 x z
@[simp]
theorem not_sbtw_self (R : Type u_1) {V : Type u_2} {P : Type u_4} [ordered_ring R] [ V] [ P] (x y : P) :
¬sbtw R x y x
theorem wbtw_swap_left_iff (R : Type u_1) {V : Type u_2} {P : Type u_4} [ordered_ring R] [ V] [ P] {x y : P} (z : P) :
wbtw R x y z wbtw R y x z x = y
theorem wbtw_swap_right_iff (R : Type u_1) {V : Type u_2} {P : Type u_4} [ordered_ring R] [ V] [ P] (x : P) {y z : P} :
wbtw R x y z wbtw R x z y y = z
theorem wbtw_rotate_iff (R : Type u_1) {V : Type u_2} {P : Type u_4} [ordered_ring R] [ V] [ P] (x : P) {y z : P} :
wbtw R x y z wbtw R z x y x = y
theorem wbtw.swap_left_iff {R : Type u_1} {V : Type u_2} {P : Type u_4} [ordered_ring R] [ V] [ P] {x y z : P} (h : wbtw R x y z) :
wbtw R y x z x = y
theorem wbtw.swap_right_iff {R : Type u_1} {V : Type u_2} {P : Type u_4} [ordered_ring R] [ V] [ P] {x y z : P} (h : wbtw R x y z) :
wbtw R x z y y = z
theorem wbtw.rotate_iff {R : Type u_1} {V : Type u_2} {P : Type u_4} [ordered_ring R] [ V] [ P] {x y z : P} (h : wbtw R x y z) :
wbtw R z x y x = y
theorem sbtw.not_swap_left {R : Type u_1} {V : Type u_2} {P : Type u_4} [ordered_ring R] [ V] [ P] {x y z : P} (h : sbtw R x y z) :
¬wbtw R y x z
theorem sbtw.not_swap_right {R : Type u_1} {V : Type u_2} {P : Type u_4} [ordered_ring R] [ V] [ P] {x y z : P} (h : sbtw R x y z) :
¬wbtw R x z y
theorem sbtw.not_rotate {R : Type u_1} {V : Type u_2} {P : Type u_4} [ordered_ring R] [ V] [ P] {x y z : P} (h : sbtw R x y z) :
¬wbtw R z x y
@[simp]
theorem wbtw_line_map_iff {R : Type u_1} {V : Type u_2} {P : Type u_4} [ordered_ring R] [ V] [ P] {x y : P} {r : R} :
wbtw R x ( y) r) y x = y r 1
@[simp]
theorem sbtw_line_map_iff {R : Type u_1} {V : Type u_2} {P : Type u_4} [ordered_ring R] [ V] [ P] {x y : P} {r : R} :
sbtw R x ( y) r) y x y r 1
@[simp]
theorem wbtw_mul_sub_add_iff {R : Type u_1} [ordered_ring R] {x y r : R} :
wbtw R x (r * (y - x) + x) y x = y r 1
@[simp]
theorem sbtw_mul_sub_add_iff {R : Type u_1} [ordered_ring R] {x y r : R} :
sbtw R x (r * (y - x) + x) y x y r 1
@[simp]
theorem wbtw_zero_one_iff {R : Type u_1} [ordered_ring R] {x : R} :
wbtw R 0 x 1 x 1
@[simp]
theorem wbtw_one_zero_iff {R : Type u_1} [ordered_ring R] {x : R} :
wbtw R 1 x 0 x 1
@[simp]
theorem sbtw_zero_one_iff {R : Type u_1} [ordered_ring R] {x : R} :
sbtw R 0 x 1 x 1
@[simp]
theorem sbtw_one_zero_iff {R : Type u_1} [ordered_ring R] {x : R} :
sbtw R 1 x 0 x 1
theorem wbtw.trans_left {R : Type u_1} {V : Type u_2} {P : Type u_4} [ordered_ring R] [ V] [ P] {w x y z : P} (h₁ : wbtw R w y z) (h₂ : wbtw R w x y) :
wbtw R w x z
theorem wbtw.trans_right {R : Type u_1} {V : Type u_2} {P : Type u_4} [ordered_ring R] [ V] [ P] {w x y z : P} (h₁ : wbtw R w x z) (h₂ : wbtw R x y z) :
wbtw R w y z
theorem wbtw.trans_sbtw_left {R : Type u_1} {V : Type u_2} {P : Type u_4} [ordered_ring R] [ V] [ P] {w x y z : P} (h₁ : wbtw R w y z) (h₂ : sbtw R w x y) :
sbtw R w x z
theorem wbtw.trans_sbtw_right {R : Type u_1} {V : Type u_2} {P : Type u_4} [ordered_ring R] [ V] [ P] {w x y z : P} (h₁ : wbtw R w x z) (h₂ : sbtw R x y z) :
sbtw R w y z
theorem sbtw.trans_left {R : Type u_1} {V : Type u_2} {P : Type u_4} [ordered_ring R] [ V] [ P] {w x y z : P} (h₁ : sbtw R w y z) (h₂ : sbtw R w x y) :
sbtw R w x z
theorem sbtw.trans_right {R : Type u_1} {V : Type u_2} {P : Type u_4} [ordered_ring R] [ V] [ P] {w x y z : P} (h₁ : sbtw R w x z) (h₂ : sbtw R x y z) :
sbtw R w y z
theorem wbtw.trans_left_ne {R : Type u_1} {V : Type u_2} {P : Type u_4} [ordered_ring R] [ V] [ P] {w x y z : P} (h₁ : wbtw R w y z) (h₂ : wbtw R w x y) (h : y z) :
x z
theorem wbtw.trans_right_ne {R : Type u_1} {V : Type u_2} {P : Type u_4} [ordered_ring R] [ V] [ P] {w x y z : P} (h₁ : wbtw R w x z) (h₂ : wbtw R x y z) (h : w x) :
w y
theorem sbtw.trans_wbtw_left_ne {R : Type u_1} {V : Type u_2} {P : Type u_4} [ordered_ring R] [ V] [ P] {w x y z : P} (h₁ : sbtw R w y z) (h₂ : wbtw R w x y) :
x z
theorem sbtw.trans_wbtw_right_ne {R : Type u_1} {V : Type u_2} {P : Type u_4} [ordered_ring R] [ V] [ P] {w x y z : P} (h₁ : sbtw R w x z) (h₂ : wbtw R x y z) :
w y
theorem sbtw.affine_combination_of_mem_affine_span_pair {R : Type u_1} {V : Type u_2} {P : Type u_4} [ordered_ring R] [ V] [ P] {ι : Type u_3} {p : ι P} (ha : p) {w w₁ w₂ : ι R} {s : finset ι} (hw : s.sum (λ (i : ι), w i) = 1) (hw₁ : s.sum (λ (i : ι), w₁ i) = 1) (hw₂ : s.sum (λ (i : ι), w₂ i) = 1) (h : p) w { p) w₁, p) w₂}) {i : ι} (his : i s) (hs : sbtw R (w₁ i) (w i) (w₂ i)) :
sbtw R ( p) w₁) ( p) w) ( p) w₂)
theorem wbtw.same_ray_vsub {R : Type u_1} {V : Type u_2} {P : Type u_4} [ V] [ P] {x y z : P} (h : wbtw R x y z) :
(y -ᵥ x) (z -ᵥ y)
theorem wbtw.same_ray_vsub_left {R : Type u_1} {V : Type u_2} {P : Type u_4} [ V] [ P] {x y z : P} (h : wbtw R x y z) :
(y -ᵥ x) (z -ᵥ x)
theorem wbtw.same_ray_vsub_right {R : Type u_1} {V : Type u_2} {P : Type u_4} [ V] [ P] {x y z : P} (h : wbtw R x y z) :
(z -ᵥ x) (z -ᵥ y)
theorem sbtw_of_sbtw_of_sbtw_of_mem_affine_span_pair {R : Type u_1} {V : Type u_2} {P : Type u_4} [ V] [ P] {t : P} {i₁ i₂ i₃ : fin 3} (h₁₂ : i₁ i₂) {p₁ p₂ p : P} (h₁ : sbtw R (t.points i₂) p₁ (t.points i₃)) (h₂ : sbtw R (t.points i₁) p₂ (t.points i₃)) (h₁' : p {t.points i₁, p₁}) (h₂' : p {t.points i₂, p₂}) :
sbtw R (t.points i₁) p p₁

Suppose lines from two vertices of a triangle to interior points of the opposite side meet at p. Then p lies in the interior of the first (and by symmetry the other) segment from a vertex to the point on the opposite side.

theorem wbtw_iff_left_eq_or_right_mem_image_Ici {R : Type u_1} {V : Type u_2} {P : Type u_4} [ V] [ P] {x y z : P} :
wbtw R x y z x = y z y) ''
theorem wbtw.right_mem_image_Ici_of_left_ne {R : Type u_1} {V : Type u_2} {P : Type u_4} [ V] [ P] {x y z : P} (h : wbtw R x y z) (hne : x y) :
z y) ''
theorem wbtw.right_mem_affine_span_of_left_ne {R : Type u_1} {V : Type u_2} {P : Type u_4} [ V] [ P] {x y z : P} (h : wbtw R x y z) (hne : x y) :
z {x, y}
theorem sbtw_iff_left_ne_and_right_mem_image_IoI {R : Type u_1} {V : Type u_2} {P : Type u_4} [ V] [ P] {x y z : P} :
sbtw R x y z x y z y) ''
theorem sbtw.right_mem_image_Ioi {R : Type u_1} {V : Type u_2} {P : Type u_4} [ V] [ P] {x y z : P} (h : sbtw R x y z) :
z y) ''
theorem sbtw.right_mem_affine_span {R : Type u_1} {V : Type u_2} {P : Type u_4} [ V] [ P] {x y z : P} (h : sbtw R x y z) :
z {x, y}
theorem wbtw_iff_right_eq_or_left_mem_image_Ici {R : Type u_1} {V : Type u_2} {P : Type u_4} [ V] [ P] {x y z : P} :
wbtw R x y z z = y x y) ''
theorem wbtw.left_mem_image_Ici_of_right_ne {R : Type u_1} {V : Type u_2} {P : Type u_4} [ V] [ P] {x y z : P} (h : wbtw R x y z) (hne : z y) :
x y) ''
theorem wbtw.left_mem_affine_span_of_right_ne {R : Type u_1} {V : Type u_2} {P : Type u_4} [ V] [ P] {x y z : P} (h : wbtw R x y z) (hne : z y) :
x {z, y}
theorem sbtw_iff_right_ne_and_left_mem_image_IoI {R : Type u_1} {V : Type u_2} {P : Type u_4} [ V] [ P] {x y z : P} :
sbtw R x y z z y x y) ''
theorem sbtw.left_mem_image_Ioi {R : Type u_1} {V : Type u_2} {P : Type u_4} [ V] [ P] {x y z : P} (h : sbtw R x y z) :
x y) ''
theorem sbtw.left_mem_affine_span {R : Type u_1} {V : Type u_2} {P : Type u_4} [ V] [ P] {x y z : P} (h : sbtw R x y z) :
x {z, y}
theorem wbtw_smul_vadd_smul_vadd_of_nonneg_of_le {R : Type u_1} {V : Type u_2} {P : Type u_4} [ V] [ P] (x : P) (v : V) {r₁ r₂ : R} (hr₁ : 0 r₁) (hr₂ : r₁ r₂) :
wbtw R x (r₁ v +ᵥ x) (r₂ v +ᵥ x)
theorem wbtw_or_wbtw_smul_vadd_of_nonneg {R : Type u_1} {V : Type u_2} {P : Type u_4} [ V] [ P] (x : P) (v : V) {r₁ r₂ : R} (hr₁ : 0 r₁) (hr₂ : 0 r₂) :
wbtw R x (r₁ v +ᵥ x) (r₂ v +ᵥ x) wbtw R x (r₂ v +ᵥ x) (r₁ v +ᵥ x)
theorem wbtw_smul_vadd_smul_vadd_of_nonpos_of_le {R : Type u_1} {V : Type u_2} {P : Type u_4} [ V] [ P] (x : P) (v : V) {r₁ r₂ : R} (hr₁ : r₁ 0) (hr₂ : r₂ r₁) :
wbtw R x (r₁ v +ᵥ x) (r₂ v +ᵥ x)
theorem wbtw_or_wbtw_smul_vadd_of_nonpos {R : Type u_1} {V : Type u_2} {P : Type u_4} [ V] [ P] (x : P) (v : V) {r₁ r₂ : R} (hr₁ : r₁ 0) (hr₂ : r₂ 0) :
wbtw R x (r₁ v +ᵥ x) (r₂ v +ᵥ x) wbtw R x (r₂ v +ᵥ x) (r₁ v +ᵥ x)
theorem wbtw_smul_vadd_smul_vadd_of_nonpos_of_nonneg {R : Type u_1} {V : Type u_2} {P : Type u_4} [ V] [ P] (x : P) (v : V) {r₁ r₂ : R} (hr₁ : r₁ 0) (hr₂ : 0 r₂) :
wbtw R (r₁ v +ᵥ x) x (r₂ v +ᵥ x)
theorem wbtw_smul_vadd_smul_vadd_of_nonneg_of_nonpos {R : Type u_1} {V : Type u_2} {P : Type u_4} [ V] [ P] (x : P) (v : V) {r₁ r₂ : R} (hr₁ : 0 r₁) (hr₂ : r₂ 0) :
wbtw R (r₁ v +ᵥ x) x (r₂ v +ᵥ x)
theorem wbtw.trans_left_right {R : Type u_1} {V : Type u_2} {P : Type u_4} [ V] [ P] {w x y z : P} (h₁ : wbtw R w y z) (h₂ : wbtw R w x y) :
wbtw R x y z
theorem wbtw.trans_right_left {R : Type u_1} {V : Type u_2} {P : Type u_4} [ V] [ P] {w x y z : P} (h₁ : wbtw R w x z) (h₂ : wbtw R x y z) :
wbtw R w x y
theorem sbtw.trans_left_right {R : Type u_1} {V : Type u_2} {P : Type u_4} [ V] [ P] {w x y z : P} (h₁ : sbtw R w y z) (h₂ : sbtw R w x y) :
sbtw R x y z
theorem sbtw.trans_right_left {R : Type u_1} {V : Type u_2} {P : Type u_4} [ V] [ P] {w x y z : P} (h₁ : sbtw R w x z) (h₂ : sbtw R x y z) :
sbtw R w x y
theorem wbtw.collinear {R : Type u_1} {V : Type u_2} {P : Type u_4} [ V] [ P] {x y z : P} (h : wbtw R x y z) :
{x, y, z}
theorem collinear.wbtw_or_wbtw_or_wbtw {R : Type u_1} {V : Type u_2} {P : Type u_4} [ V] [ P] {x y z : P} (h : {x, y, z}) :
wbtw R x y z wbtw R y z x wbtw R z x y
theorem wbtw_iff_same_ray_vsub {R : Type u_1} {V : Type u_2} {P : Type u_4} [ V] [ P] {x y z : P} :
wbtw R x y z (y -ᵥ x) (z -ᵥ y)
theorem wbtw_point_reflection (R : Type u_1) {V : Type u_2} {P : Type u_4} [ V] [ P] (x y : P) :
wbtw R y x y)
theorem sbtw_point_reflection_of_ne (R : Type u_1) {V : Type u_2} {P : Type u_4} [ V] [ P] {x y : P} (h : x y) :
sbtw R y x y)
theorem wbtw_midpoint (R : Type u_1) {V : Type u_2} {P : Type u_4} [ V] [ P] (x y : P) :
wbtw R x (midpoint R x y) y
theorem sbtw_midpoint_of_ne (R : Type u_1) {V : Type u_2} {P : Type u_4} [ V] [ P] {x y : P} (h : x y) :
sbtw R x (midpoint R x y) y