Midpoint of a segment #

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Main definitions #

midpoint R x y: midpoint of the segment [x, y]. We define it for x and y in a module over a ring R with invertible 2.
add_monoid_hom.of_map_midpoint: construct an add_monoid_hom given a map f such that f sends zero to zero and midpoints to midpoints.

Main theorems #

midpoint_eq_iff: z is the midpoint of [x, y] if and only if x + y = z + z,
midpoint_unique: midpoint R x y does not depend on R;
midpoint x y is linear both in x and y;
point_reflection_midpoint_left, point_reflection_midpoint_right: equiv.point_reflection (midpoint R x y) swaps x and y.

We do not mark most lemmas as @[simp] because it is hard to tell which side is simpler.

Tags #

midpoint, add_monoid_hom

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def midpoint (R : Type u_1) {V : Type u_2} {P : Type u_4} [ring R] [invertible 2] [add_comm_group V] [module R V] [add_torsor V P] (x y : P) :

midpoint x y is the midpoint of the segment [x, y].

Equations

midpoint R x y = ⇑(affine_map.line_map x y) (⅟ 2)

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@[simp]

theorem affine_map.map_midpoint {R : Type u_1} {V : Type u_2} {V' : Type u_3} {P : Type u_4} {P' : Type u_5} [ring R] [invertible 2] [add_comm_group V] [module R V] [add_torsor V P] [add_comm_group V'] [module R V'] [add_torsor V' P'] (f : P →ᵃ[R] P') (a b : P) :

⇑f (midpoint R a b) = midpoint R (⇑f a) (⇑f b)

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@[simp]

theorem affine_equiv.map_midpoint {R : Type u_1} {V : Type u_2} {V' : Type u_3} {P : Type u_4} {P' : Type u_5} [ring R] [invertible 2] [add_comm_group V] [module R V] [add_torsor V P] [add_comm_group V'] [module R V'] [add_torsor V' P'] (f : P ≃ᵃ[R] P') (a b : P) :

⇑f (midpoint R a b) = midpoint R (⇑f a) (⇑f b)

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@[simp]

theorem affine_equiv.point_reflection_midpoint_left {R : Type u_1} {V : Type u_2} {P : Type u_4} [ring R] [invertible 2] [add_comm_group V] [module R V] [add_torsor V P] (x y : P) :

⇑(affine_equiv.point_reflection R (midpoint R x y)) x = y

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theorem midpoint_comm {R : Type u_1} {V : Type u_2} {P : Type u_4} [ring R] [invertible 2] [add_comm_group V] [module R V] [add_torsor V P] (x y : P) :

midpoint R x y = midpoint R y x

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@[simp]

theorem affine_equiv.point_reflection_midpoint_right {R : Type u_1} {V : Type u_2} {P : Type u_4} [ring R] [invertible 2] [add_comm_group V] [module R V] [add_torsor V P] (x y : P) :

⇑(affine_equiv.point_reflection R (midpoint R x y)) y = x

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theorem midpoint_vsub_midpoint {R : Type u_1} {V : Type u_2} {P : Type u_4} [ring R] [invertible 2] [add_comm_group V] [module R V] [add_torsor V P] (p₁ p₂ p₃ p₄ : P) :

midpoint R p₁ p₂ -ᵥ midpoint R p₃ p₄ = midpoint R (p₁ -ᵥ p₃) (p₂ -ᵥ p₄)

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theorem midpoint_vadd_midpoint {R : Type u_1} {V : Type u_2} {P : Type u_4} [ring R] [invertible 2] [add_comm_group V] [module R V] [add_torsor V P] (v v' : V) (p p' : P) :

midpoint R v v' +ᵥ midpoint R p p' = midpoint R (v +ᵥ p) (v' +ᵥ p')

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theorem midpoint_eq_iff {R : Type u_1} {V : Type u_2} {P : Type u_4} [ring R] [invertible 2] [add_comm_group V] [module R V] [add_torsor V P] {x y z : P} :

midpoint R x y = z ↔ ⇑(affine_equiv.point_reflection R z) x = y

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@[simp]

theorem midpoint_vsub_left {R : Type u_1} {V : Type u_2} {P : Type u_4} [ring R] [invertible 2] [add_comm_group V] [module R V] [add_torsor V P] (p₁ p₂ : P) :

midpoint R p₁ p₂ -ᵥ p₁ = ⅟ 2 • (p₂ -ᵥ p₁)

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@[simp]

theorem midpoint_vsub_right {R : Type u_1} {V : Type u_2} {P : Type u_4} [ring R] [invertible 2] [add_comm_group V] [module R V] [add_torsor V P] (p₁ p₂ : P) :

midpoint R p₁ p₂ -ᵥ p₂ = ⅟ 2 • (p₁ -ᵥ p₂)

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@[simp]

theorem left_vsub_midpoint {R : Type u_1} {V : Type u_2} {P : Type u_4} [ring R] [invertible 2] [add_comm_group V] [module R V] [add_torsor V P] (p₁ p₂ : P) :

p₁ -ᵥ midpoint R p₁ p₂ = ⅟ 2 • (p₁ -ᵥ p₂)

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@[simp]

theorem right_vsub_midpoint {R : Type u_1} {V : Type u_2} {P : Type u_4} [ring R] [invertible 2] [add_comm_group V] [module R V] [add_torsor V P] (p₁ p₂ : P) :

p₂ -ᵥ midpoint R p₁ p₂ = ⅟ 2 • (p₂ -ᵥ p₁)

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theorem midpoint_vsub {R : Type u_1} {V : Type u_2} {P : Type u_4} [ring R] [invertible 2] [add_comm_group V] [module R V] [add_torsor V P] (p₁ p₂ p : P) :

midpoint R p₁ p₂ -ᵥ p = ⅟ 2 • (p₁ -ᵥ p) + ⅟ 2 • (p₂ -ᵥ p)

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theorem vsub_midpoint {R : Type u_1} {V : Type u_2} {P : Type u_4} [ring R] [invertible 2] [add_comm_group V] [module R V] [add_torsor V P] (p₁ p₂ p : P) :

p -ᵥ midpoint R p₁ p₂ = ⅟ 2 • (p -ᵥ p₁) + ⅟ 2 • (p -ᵥ p₂)

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@[simp]

theorem midpoint_sub_left {R : Type u_1} {V : Type u_2} [ring R] [invertible 2] [add_comm_group V] [module R V] (v₁ v₂ : V) :

midpoint R v₁ v₂ - v₁ = ⅟ 2 • (v₂ - v₁)

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@[simp]

theorem midpoint_sub_right {R : Type u_1} {V : Type u_2} [ring R] [invertible 2] [add_comm_group V] [module R V] (v₁ v₂ : V) :

midpoint R v₁ v₂ - v₂ = ⅟ 2 • (v₁ - v₂)

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@[simp]

theorem left_sub_midpoint {R : Type u_1} {V : Type u_2} [ring R] [invertible 2] [add_comm_group V] [module R V] (v₁ v₂ : V) :

v₁ - midpoint R v₁ v₂ = ⅟ 2 • (v₁ - v₂)

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@[simp]

theorem right_sub_midpoint {R : Type u_1} {V : Type u_2} [ring R] [invertible 2] [add_comm_group V] [module R V] (v₁ v₂ : V) :

v₂ - midpoint R v₁ v₂ = ⅟ 2 • (v₂ - v₁)

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@[simp]

theorem midpoint_eq_left_iff (R : Type u_1) {V : Type u_2} {P : Type u_4} [ring R] [invertible 2] [add_comm_group V] [module R V] [add_torsor V P] {x y : P} :

midpoint R x y = x ↔ x = y

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@[simp]

theorem left_eq_midpoint_iff (R : Type u_1) {V : Type u_2} {P : Type u_4} [ring R] [invertible 2] [add_comm_group V] [module R V] [add_torsor V P] {x y : P} :

x = midpoint R x y ↔ x = y

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@[simp]

theorem midpoint_eq_right_iff (R : Type u_1) {V : Type u_2} {P : Type u_4} [ring R] [invertible 2] [add_comm_group V] [module R V] [add_torsor V P] {x y : P} :

midpoint R x y = y ↔ x = y

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@[simp]

theorem right_eq_midpoint_iff (R : Type u_1) {V : Type u_2} {P : Type u_4} [ring R] [invertible 2] [add_comm_group V] [module R V] [add_torsor V P] {x y : P} :

y = midpoint R x y ↔ x = y

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theorem midpoint_eq_midpoint_iff_vsub_eq_vsub (R : Type u_1) {V : Type u_2} {P : Type u_4} [ring R] [invertible 2] [add_comm_group V] [module R V] [add_torsor V P] {x x' y y' : P} :

midpoint R x y = midpoint R x' y' ↔ x -ᵥ x' = y' -ᵥ y

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theorem midpoint_eq_iff' (R : Type u_1) {V : Type u_2} {P : Type u_4} [ring R] [invertible 2] [add_comm_group V] [module R V] [add_torsor V P] {x y z : P} :

midpoint R x y = z ↔ ⇑(equiv.point_reflection z) x = y

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theorem midpoint_unique (R : Type u_1) {V : Type u_2} {P : Type u_4} [ring R] [invertible 2] [add_comm_group V] [module R V] [add_torsor V P] (R' : Type u_3) [ring R'] [invertible 2] [module R' V] (x y : P) :

midpoint R x y = midpoint R' x y

midpoint does not depend on the ring R.

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@[simp]

theorem midpoint_self (R : Type u_1) {V : Type u_2} {P : Type u_4} [ring R] [invertible 2] [add_comm_group V] [module R V] [add_torsor V P] (x : P) :

midpoint R x x = x

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@[simp]

theorem midpoint_add_self (R : Type u_1) {V : Type u_2} [ring R] [invertible 2] [add_comm_group V] [module R V] (x y : V) :

midpoint R x y + midpoint R x y = x + y

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theorem midpoint_zero_add (R : Type u_1) {V : Type u_2} [ring R] [invertible 2] [add_comm_group V] [module R V] (x y : V) :

midpoint R 0 (x + y) = midpoint R x y

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theorem midpoint_eq_smul_add (R : Type u_1) {V : Type u_2} [ring R] [invertible 2] [add_comm_group V] [module R V] (x y : V) :

midpoint R x y = ⅟ 2 • (x + y)

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@[simp]

theorem midpoint_self_neg (R : Type u_1) {V : Type u_2} [ring R] [invertible 2] [add_comm_group V] [module R V] (x : V) :

midpoint R x (-x) = 0

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@[simp]

theorem midpoint_neg_self (R : Type u_1) {V : Type u_2} [ring R] [invertible 2] [add_comm_group V] [module R V] (x : V) :

midpoint R (-x) x = 0

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@[simp]

theorem midpoint_sub_add (R : Type u_1) {V : Type u_2} [ring R] [invertible 2] [add_comm_group V] [module R V] (x y : V) :

midpoint R (x - y) (x + y) = x

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@[simp]

theorem midpoint_add_sub (R : Type u_1) {V : Type u_2} [ring R] [invertible 2] [add_comm_group V] [module R V] (x y : V) :

midpoint R (x + y) (x - y) = x

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def add_monoid_hom.of_map_midpoint (R : Type u_1) (R' : Type u_2) {E : Type u_3} {F : Type u_4} [ring R] [invertible 2] [add_comm_group E] [module R E] [ring R'] [invertible 2] [add_comm_group F] [module R' F] (f : E → F) (h0 : f 0 = 0) (hm : ∀ (x y : E), f (midpoint R x y) = midpoint R' (f x) (f y)) :

E →+ F

A map f : E → F sending zero to zero and midpoints to midpoints is an add_monoid_hom.

Equations

add_monoid_hom.of_map_midpoint R R' f h0 hm = {to_fun := f, map_zero' := h0, map_add' := _}

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@[simp]

theorem add_monoid_hom.coe_of_map_midpoint (R : Type u_1) (R' : Type u_2) {E : Type u_3} {F : Type u_4} [ring R] [invertible 2] [add_comm_group E] [module R E] [ring R'] [invertible 2] [add_comm_group F] [module R' F] (f : E → F) (h0 : f 0 = 0) (hm : ∀ (x y : E), f (midpoint R x y) = midpoint R' (f x) (f y)) :

⇑(add_monoid_hom.of_map_midpoint R R' f h0 hm) = f