Documentation

Mathlib.LinearAlgebra.AffineSpace.AffineEquiv

Affine equivalences #

In this file we define AffineEquiv k P₁ P₂ (notation: P₁ ≃ᵃ[k] P₂) to be the type of affine equivalences between P₁ and P₂, i.e., equivalences such that both forward and inverse maps are affine maps.

We define the following equivalences:

We equip AffineEquiv k P P with a Group structure with multiplication corresponding to composition in AffineEquiv.group.

Tags #

affine space, affine equivalence

structure AffineEquiv (k : Type u_1) (P₁ : Type u_2) (P₂ : Type u_3) {V₁ : Type u_4} {V₂ : Type u_5} [Ring k] [AddCommGroup V₁] [Module k V₁] [AddTorsor V₁ P₁] [AddCommGroup V₂] [Module k V₂] [AddTorsor V₂ P₂] extends Equiv :
Type (max (max (max u_2 u_3) u_4) u_5)

An affine equivalence is an equivalence between affine spaces such that both forward and inverse maps are affine.

We define it using an Equiv for the map and a LinearEquiv for the linear part in order to allow affine equivalences with good definitional equalities.

Instances For
    def AffineEquiv.toAffineMap {k : Type u_1} {P₁ : Type u_2} {P₂ : Type u_3} {V₁ : Type u_6} {V₂ : Type u_7} [Ring k] [AddCommGroup V₁] [Module k V₁] [AddTorsor V₁ P₁] [AddCommGroup V₂] [Module k V₂] [AddTorsor V₂ P₂] (e : P₁ ≃ᵃ[k] P₂) :
    P₁ →ᵃ[k] P₂

    Reinterpret an AffineEquiv as an AffineMap.

    Instances For
      @[simp]
      theorem AffineEquiv.toAffineMap_mk {k : Type u_1} {P₁ : Type u_2} {P₂ : Type u_3} {V₁ : Type u_6} {V₂ : Type u_7} [Ring k] [AddCommGroup V₁] [Module k V₁] [AddTorsor V₁ P₁] [AddCommGroup V₂] [Module k V₂] [AddTorsor V₂ P₂] (f : P₁ P₂) (f' : V₁ ≃ₗ[k] V₂) (h : ∀ (p : P₁) (v : V₁), f (v +ᵥ p) = f' v +ᵥ f p) :
      { toEquiv := f, linear := f', map_vadd' := h } = { toFun := f, linear := f', map_vadd' := h }
      @[simp]
      theorem AffineEquiv.linear_toAffineMap {k : Type u_1} {P₁ : Type u_2} {P₂ : Type u_3} {V₁ : Type u_6} {V₂ : Type u_7} [Ring k] [AddCommGroup V₁] [Module k V₁] [AddTorsor V₁ P₁] [AddCommGroup V₂] [Module k V₂] [AddTorsor V₂ P₂] (e : P₁ ≃ᵃ[k] P₂) :
      (e).linear = e.linear
      theorem AffineEquiv.toAffineMap_injective {k : Type u_1} {P₁ : Type u_2} {P₂ : Type u_3} {V₁ : Type u_6} {V₂ : Type u_7} [Ring k] [AddCommGroup V₁] [Module k V₁] [AddTorsor V₁ P₁] [AddCommGroup V₂] [Module k V₂] [AddTorsor V₂ P₂] :
      Function.Injective AffineEquiv.toAffineMap
      @[simp]
      theorem AffineEquiv.toAffineMap_inj {k : Type u_1} {P₁ : Type u_2} {P₂ : Type u_3} {V₁ : Type u_6} {V₂ : Type u_7} [Ring k] [AddCommGroup V₁] [Module k V₁] [AddTorsor V₁ P₁] [AddCommGroup V₂] [Module k V₂] [AddTorsor V₂ P₂] {e : P₁ ≃ᵃ[k] P₂} {e' : P₁ ≃ᵃ[k] P₂} :
      e = e' e = e'
      instance AffineEquiv.equivLike {k : Type u_1} {P₁ : Type u_2} {P₂ : Type u_3} {V₁ : Type u_6} {V₂ : Type u_7} [Ring k] [AddCommGroup V₁] [Module k V₁] [AddTorsor V₁ P₁] [AddCommGroup V₂] [Module k V₂] [AddTorsor V₂ P₂] :
      EquivLike (P₁ ≃ᵃ[k] P₂) P₁ P₂
      instance AffineEquiv.instCoeFunAffineEquivForAll {k : Type u_1} {P₁ : Type u_2} {P₂ : Type u_3} {V₁ : Type u_6} {V₂ : Type u_7} [Ring k] [AddCommGroup V₁] [Module k V₁] [AddTorsor V₁ P₁] [AddCommGroup V₂] [Module k V₂] [AddTorsor V₂ P₂] :
      CoeFun (P₁ ≃ᵃ[k] P₂) fun x => P₁P₂
      instance AffineEquiv.instCoeOutAffineEquivEquiv {k : Type u_1} {P₁ : Type u_2} {P₂ : Type u_3} {V₁ : Type u_6} {V₂ : Type u_7} [Ring k] [AddCommGroup V₁] [Module k V₁] [AddTorsor V₁ P₁] [AddCommGroup V₂] [Module k V₂] [AddTorsor V₂ P₂] :
      CoeOut (P₁ ≃ᵃ[k] P₂) (P₁ P₂)
      @[simp]
      theorem AffineEquiv.map_vadd {k : Type u_1} {P₁ : Type u_2} {P₂ : Type u_3} {V₁ : Type u_6} {V₂ : Type u_7} [Ring k] [AddCommGroup V₁] [Module k V₁] [AddTorsor V₁ P₁] [AddCommGroup V₂] [Module k V₂] [AddTorsor V₂ P₂] (e : P₁ ≃ᵃ[k] P₂) (p : P₁) (v : V₁) :
      e (v +ᵥ p) = e.linear v +ᵥ e p
      @[simp]
      theorem AffineEquiv.coe_toEquiv {k : Type u_1} {P₁ : Type u_2} {P₂ : Type u_3} {V₁ : Type u_6} {V₂ : Type u_7} [Ring k] [AddCommGroup V₁] [Module k V₁] [AddTorsor V₁ P₁] [AddCommGroup V₂] [Module k V₂] [AddTorsor V₂ P₂] (e : P₁ ≃ᵃ[k] P₂) :
      e.toEquiv = e
      instance AffineEquiv.instCoeAffineEquivAffineMap {k : Type u_1} {P₁ : Type u_2} {P₂ : Type u_3} {V₁ : Type u_6} {V₂ : Type u_7} [Ring k] [AddCommGroup V₁] [Module k V₁] [AddTorsor V₁ P₁] [AddCommGroup V₂] [Module k V₂] [AddTorsor V₂ P₂] :
      Coe (P₁ ≃ᵃ[k] P₂) (P₁ →ᵃ[k] P₂)
      @[simp]
      theorem AffineEquiv.coe_toAffineMap {k : Type u_1} {P₁ : Type u_2} {P₂ : Type u_3} {V₁ : Type u_6} {V₂ : Type u_7} [Ring k] [AddCommGroup V₁] [Module k V₁] [AddTorsor V₁ P₁] [AddCommGroup V₂] [Module k V₂] [AddTorsor V₂ P₂] (e : P₁ ≃ᵃ[k] P₂) :
      e = e
      @[simp]
      theorem AffineEquiv.coe_coe {k : Type u_1} {P₁ : Type u_2} {P₂ : Type u_3} {V₁ : Type u_6} {V₂ : Type u_7} [Ring k] [AddCommGroup V₁] [Module k V₁] [AddTorsor V₁ P₁] [AddCommGroup V₂] [Module k V₂] [AddTorsor V₂ P₂] (e : P₁ ≃ᵃ[k] P₂) :
      e = e
      @[simp]
      theorem AffineEquiv.coe_linear {k : Type u_1} {P₁ : Type u_2} {P₂ : Type u_3} {V₁ : Type u_6} {V₂ : Type u_7} [Ring k] [AddCommGroup V₁] [Module k V₁] [AddTorsor V₁ P₁] [AddCommGroup V₂] [Module k V₂] [AddTorsor V₂ P₂] (e : P₁ ≃ᵃ[k] P₂) :
      (e).linear = e.linear
      theorem AffineEquiv.ext {k : Type u_1} {P₁ : Type u_2} {P₂ : Type u_3} {V₁ : Type u_6} {V₂ : Type u_7} [Ring k] [AddCommGroup V₁] [Module k V₁] [AddTorsor V₁ P₁] [AddCommGroup V₂] [Module k V₂] [AddTorsor V₂ P₂] {e : P₁ ≃ᵃ[k] P₂} {e' : P₁ ≃ᵃ[k] P₂} (h : ∀ (x : P₁), e x = e' x) :
      e = e'
      theorem AffineEquiv.coeFn_injective {k : Type u_1} {P₁ : Type u_2} {P₂ : Type u_3} {V₁ : Type u_6} {V₂ : Type u_7} [Ring k] [AddCommGroup V₁] [Module k V₁] [AddTorsor V₁ P₁] [AddCommGroup V₂] [Module k V₂] [AddTorsor V₂ P₂] :
      Function.Injective FunLike.coe
      theorem AffineEquiv.coeFn_inj {k : Type u_1} {P₁ : Type u_2} {P₂ : Type u_3} {V₁ : Type u_6} {V₂ : Type u_7} [Ring k] [AddCommGroup V₁] [Module k V₁] [AddTorsor V₁ P₁] [AddCommGroup V₂] [Module k V₂] [AddTorsor V₂ P₂] {e : P₁ ≃ᵃ[k] P₂} {e' : P₁ ≃ᵃ[k] P₂} :
      e = e' e = e'
      theorem AffineEquiv.toEquiv_injective {k : Type u_1} {P₁ : Type u_2} {P₂ : Type u_3} {V₁ : Type u_6} {V₂ : Type u_7} [Ring k] [AddCommGroup V₁] [Module k V₁] [AddTorsor V₁ P₁] [AddCommGroup V₂] [Module k V₂] [AddTorsor V₂ P₂] :
      Function.Injective AffineEquiv.toEquiv
      @[simp]
      theorem AffineEquiv.toEquiv_inj {k : Type u_1} {P₁ : Type u_2} {P₂ : Type u_3} {V₁ : Type u_6} {V₂ : Type u_7} [Ring k] [AddCommGroup V₁] [Module k V₁] [AddTorsor V₁ P₁] [AddCommGroup V₂] [Module k V₂] [AddTorsor V₂ P₂] {e : P₁ ≃ᵃ[k] P₂} {e' : P₁ ≃ᵃ[k] P₂} :
      e.toEquiv = e'.toEquiv e = e'
      @[simp]
      theorem AffineEquiv.coe_mk {k : Type u_1} {P₁ : Type u_2} {P₂ : Type u_3} {V₁ : Type u_6} {V₂ : Type u_7} [Ring k] [AddCommGroup V₁] [Module k V₁] [AddTorsor V₁ P₁] [AddCommGroup V₂] [Module k V₂] [AddTorsor V₂ P₂] (e : P₁ P₂) (e' : V₁ ≃ₗ[k] V₂) (h : ∀ (p : P₁) (v : V₁), e (v +ᵥ p) = e' v +ᵥ e p) :
      { toEquiv := e, linear := e', map_vadd' := h } = e
      def AffineEquiv.mk' {k : Type u_1} {P₁ : Type u_2} {P₂ : Type u_3} {V₁ : Type u_6} {V₂ : Type u_7} [Ring k] [AddCommGroup V₁] [Module k V₁] [AddTorsor V₁ P₁] [AddCommGroup V₂] [Module k V₂] [AddTorsor V₂ P₂] (e : P₁P₂) (e' : V₁ ≃ₗ[k] V₂) (p : P₁) (h : ∀ (p' : P₁), e p' = e' (p' -ᵥ p) +ᵥ e p) :
      P₁ ≃ᵃ[k] P₂

      Construct an affine equivalence by verifying the relation between the map and its linear part at one base point. Namely, this function takes a map e : P₁ → P₂, a linear equivalence e' : V₁ ≃ₗ[k] V₂, and a point p such that for any other point p' we have e p' = e' (p' -ᵥ p) +ᵥ e p.

      Instances For
        @[simp]
        theorem AffineEquiv.coe_mk' {k : Type u_1} {P₁ : Type u_2} {P₂ : Type u_3} {V₁ : Type u_6} {V₂ : Type u_7} [Ring k] [AddCommGroup V₁] [Module k V₁] [AddTorsor V₁ P₁] [AddCommGroup V₂] [Module k V₂] [AddTorsor V₂ P₂] (e : P₁ P₂) (e' : V₁ ≃ₗ[k] V₂) (p : P₁) (h : ∀ (p' : P₁), e p' = e' (p' -ᵥ p) +ᵥ e p) :
        ↑(AffineEquiv.mk' (e) e' p h) = e
        @[simp]
        theorem AffineEquiv.linear_mk' {k : Type u_1} {P₁ : Type u_2} {P₂ : Type u_3} {V₁ : Type u_6} {V₂ : Type u_7} [Ring k] [AddCommGroup V₁] [Module k V₁] [AddTorsor V₁ P₁] [AddCommGroup V₂] [Module k V₂] [AddTorsor V₂ P₂] (e : P₁ P₂) (e' : V₁ ≃ₗ[k] V₂) (p : P₁) (h : ∀ (p' : P₁), e p' = e' (p' -ᵥ p) +ᵥ e p) :
        (AffineEquiv.mk' (e) e' p h).linear = e'
        def AffineEquiv.symm {k : Type u_1} {P₁ : Type u_2} {P₂ : Type u_3} {V₁ : Type u_6} {V₂ : Type u_7} [Ring k] [AddCommGroup V₁] [Module k V₁] [AddTorsor V₁ P₁] [AddCommGroup V₂] [Module k V₂] [AddTorsor V₂ P₂] (e : P₁ ≃ᵃ[k] P₂) :
        P₂ ≃ᵃ[k] P₁

        Inverse of an affine equivalence as an affine equivalence.

        Instances For
          @[simp]
          theorem AffineEquiv.symm_toEquiv {k : Type u_1} {P₁ : Type u_2} {P₂ : Type u_3} {V₁ : Type u_6} {V₂ : Type u_7} [Ring k] [AddCommGroup V₁] [Module k V₁] [AddTorsor V₁ P₁] [AddCommGroup V₂] [Module k V₂] [AddTorsor V₂ P₂] (e : P₁ ≃ᵃ[k] P₂) :
          e.symm = (AffineEquiv.symm e).toEquiv
          @[simp]
          theorem AffineEquiv.symm_linear {k : Type u_1} {P₁ : Type u_2} {P₂ : Type u_3} {V₁ : Type u_6} {V₂ : Type u_7} [Ring k] [AddCommGroup V₁] [Module k V₁] [AddTorsor V₁ P₁] [AddCommGroup V₂] [Module k V₂] [AddTorsor V₂ P₂] (e : P₁ ≃ᵃ[k] P₂) :
          LinearEquiv.symm e.linear = (AffineEquiv.symm e).linear
          def AffineEquiv.Simps.apply {k : Type u_1} {P₁ : Type u_2} {P₂ : Type u_3} {V₁ : Type u_6} {V₂ : Type u_7} [Ring k] [AddCommGroup V₁] [Module k V₁] [AddTorsor V₁ P₁] [AddCommGroup V₂] [Module k V₂] [AddTorsor V₂ P₂] (e : P₁ ≃ᵃ[k] P₂) :
          P₁P₂

          See Note [custom simps projection]

          Instances For
            def AffineEquiv.Simps.symm_apply {k : Type u_1} {P₁ : Type u_2} {P₂ : Type u_3} {V₁ : Type u_6} {V₂ : Type u_7} [Ring k] [AddCommGroup V₁] [Module k V₁] [AddTorsor V₁ P₁] [AddCommGroup V₂] [Module k V₂] [AddTorsor V₂ P₂] (e : P₁ ≃ᵃ[k] P₂) :
            P₂P₁

            See Note [custom simps projection]

            Instances For
              theorem AffineEquiv.bijective {k : Type u_1} {P₁ : Type u_2} {P₂ : Type u_3} {V₁ : Type u_6} {V₂ : Type u_7} [Ring k] [AddCommGroup V₁] [Module k V₁] [AddTorsor V₁ P₁] [AddCommGroup V₂] [Module k V₂] [AddTorsor V₂ P₂] (e : P₁ ≃ᵃ[k] P₂) :
              theorem AffineEquiv.surjective {k : Type u_1} {P₁ : Type u_2} {P₂ : Type u_3} {V₁ : Type u_6} {V₂ : Type u_7} [Ring k] [AddCommGroup V₁] [Module k V₁] [AddTorsor V₁ P₁] [AddCommGroup V₂] [Module k V₂] [AddTorsor V₂ P₂] (e : P₁ ≃ᵃ[k] P₂) :
              theorem AffineEquiv.injective {k : Type u_1} {P₁ : Type u_2} {P₂ : Type u_3} {V₁ : Type u_6} {V₂ : Type u_7} [Ring k] [AddCommGroup V₁] [Module k V₁] [AddTorsor V₁ P₁] [AddCommGroup V₂] [Module k V₂] [AddTorsor V₂ P₂] (e : P₁ ≃ᵃ[k] P₂) :
              @[simp]
              theorem AffineEquiv.ofBijective_apply {k : Type u_1} {P₁ : Type u_2} {P₂ : Type u_3} {V₁ : Type u_6} {V₂ : Type u_7} [Ring k] [AddCommGroup V₁] [Module k V₁] [AddTorsor V₁ P₁] [AddCommGroup V₂] [Module k V₂] [AddTorsor V₂ P₂] {φ : P₁ →ᵃ[k] P₂} (hφ : Function.Bijective φ) (a : P₁) :
              ↑(AffineEquiv.ofBijective ) a = φ a
              @[simp]
              theorem AffineEquiv.linear_ofBijective {k : Type u_1} {P₁ : Type u_2} {P₂ : Type u_3} {V₁ : Type u_6} {V₂ : Type u_7} [Ring k] [AddCommGroup V₁] [Module k V₁] [AddTorsor V₁ P₁] [AddCommGroup V₂] [Module k V₂] [AddTorsor V₂ P₂] {φ : P₁ →ᵃ[k] P₂} (hφ : Function.Bijective φ) :
              (AffineEquiv.ofBijective ).linear = LinearEquiv.ofBijective φ.linear (_ : Function.Bijective φ.linear)
              noncomputable def AffineEquiv.ofBijective {k : Type u_1} {P₁ : Type u_2} {P₂ : Type u_3} {V₁ : Type u_6} {V₂ : Type u_7} [Ring k] [AddCommGroup V₁] [Module k V₁] [AddTorsor V₁ P₁] [AddCommGroup V₂] [Module k V₂] [AddTorsor V₂ P₂] {φ : P₁ →ᵃ[k] P₂} (hφ : Function.Bijective φ) :
              P₁ ≃ᵃ[k] P₂

              Bijective affine maps are affine isomorphisms.

              Instances For
                theorem AffineEquiv.ofBijective.symm_eq {k : Type u_1} {P₁ : Type u_2} {P₂ : Type u_3} {V₁ : Type u_6} {V₂ : Type u_7} [Ring k] [AddCommGroup V₁] [Module k V₁] [AddTorsor V₁ P₁] [AddCommGroup V₂] [Module k V₂] [AddTorsor V₂ P₂] {φ : P₁ →ᵃ[k] P₂} (hφ : Function.Bijective φ) :
                (AffineEquiv.symm (AffineEquiv.ofBijective )).toEquiv = (Equiv.ofBijective (φ) ).symm
                @[simp]
                theorem AffineEquiv.range_eq {k : Type u_1} {P₁ : Type u_2} {P₂ : Type u_3} {V₁ : Type u_6} {V₂ : Type u_7} [Ring k] [AddCommGroup V₁] [Module k V₁] [AddTorsor V₁ P₁] [AddCommGroup V₂] [Module k V₂] [AddTorsor V₂ P₂] (e : P₁ ≃ᵃ[k] P₂) :
                Set.range e = Set.univ
                @[simp]
                theorem AffineEquiv.apply_symm_apply {k : Type u_1} {P₁ : Type u_2} {P₂ : Type u_3} {V₁ : Type u_6} {V₂ : Type u_7} [Ring k] [AddCommGroup V₁] [Module k V₁] [AddTorsor V₁ P₁] [AddCommGroup V₂] [Module k V₂] [AddTorsor V₂ P₂] (e : P₁ ≃ᵃ[k] P₂) (p : P₂) :
                e (↑(AffineEquiv.symm e) p) = p
                @[simp]
                theorem AffineEquiv.symm_apply_apply {k : Type u_1} {P₁ : Type u_2} {P₂ : Type u_3} {V₁ : Type u_6} {V₂ : Type u_7} [Ring k] [AddCommGroup V₁] [Module k V₁] [AddTorsor V₁ P₁] [AddCommGroup V₂] [Module k V₂] [AddTorsor V₂ P₂] (e : P₁ ≃ᵃ[k] P₂) (p : P₁) :
                ↑(AffineEquiv.symm e) (e p) = p
                theorem AffineEquiv.apply_eq_iff_eq_symm_apply {k : Type u_1} {P₁ : Type u_2} {P₂ : Type u_3} {V₁ : Type u_6} {V₂ : Type u_7} [Ring k] [AddCommGroup V₁] [Module k V₁] [AddTorsor V₁ P₁] [AddCommGroup V₂] [Module k V₂] [AddTorsor V₂ P₂] (e : P₁ ≃ᵃ[k] P₂) {p₁ : P₁} {p₂ : (fun a => P₂) p₁} :
                e p₁ = p₂ p₁ = ↑(AffineEquiv.symm e) p₂
                theorem AffineEquiv.apply_eq_iff_eq {k : Type u_1} {P₁ : Type u_2} {P₂ : Type u_3} {V₁ : Type u_6} {V₂ : Type u_7} [Ring k] [AddCommGroup V₁] [Module k V₁] [AddTorsor V₁ P₁] [AddCommGroup V₂] [Module k V₂] [AddTorsor V₂ P₂] (e : P₁ ≃ᵃ[k] P₂) {p₁ : P₁} {p₂ : P₁} :
                e p₁ = e p₂ p₁ = p₂
                @[simp]
                theorem AffineEquiv.image_symm {k : Type u_1} {P₁ : Type u_2} {P₂ : Type u_3} {V₁ : Type u_6} {V₂ : Type u_7} [Ring k] [AddCommGroup V₁] [Module k V₁] [AddTorsor V₁ P₁] [AddCommGroup V₂] [Module k V₂] [AddTorsor V₂ P₂] (f : P₁ ≃ᵃ[k] P₂) (s : Set P₂) :
                ↑(AffineEquiv.symm f) '' s = f ⁻¹' s
                @[simp]
                theorem AffineEquiv.preimage_symm {k : Type u_1} {P₁ : Type u_2} {P₂ : Type u_3} {V₁ : Type u_6} {V₂ : Type u_7} [Ring k] [AddCommGroup V₁] [Module k V₁] [AddTorsor V₁ P₁] [AddCommGroup V₂] [Module k V₂] [AddTorsor V₂ P₂] (f : P₁ ≃ᵃ[k] P₂) (s : Set P₁) :
                ↑(AffineEquiv.symm f) ⁻¹' s = f '' s
                def AffineEquiv.refl (k : Type u_1) (P₁ : Type u_2) {V₁ : Type u_6} [Ring k] [AddCommGroup V₁] [Module k V₁] [AddTorsor V₁ P₁] :
                P₁ ≃ᵃ[k] P₁

                Identity map as an AffineEquiv.

                Instances For
                  @[simp]
                  theorem AffineEquiv.coe_refl (k : Type u_1) (P₁ : Type u_2) {V₁ : Type u_6} [Ring k] [AddCommGroup V₁] [Module k V₁] [AddTorsor V₁ P₁] :
                  ↑(AffineEquiv.refl k P₁) = id
                  @[simp]
                  theorem AffineEquiv.coe_refl_to_affineMap (k : Type u_1) (P₁ : Type u_2) {V₁ : Type u_6} [Ring k] [AddCommGroup V₁] [Module k V₁] [AddTorsor V₁ P₁] :
                  ↑(AffineEquiv.refl k P₁) = AffineMap.id k P₁
                  @[simp]
                  theorem AffineEquiv.refl_apply (k : Type u_1) (P₁ : Type u_2) {V₁ : Type u_6} [Ring k] [AddCommGroup V₁] [Module k V₁] [AddTorsor V₁ P₁] (x : P₁) :
                  ↑(AffineEquiv.refl k P₁) x = x
                  @[simp]
                  theorem AffineEquiv.toEquiv_refl (k : Type u_1) (P₁ : Type u_2) {V₁ : Type u_6} [Ring k] [AddCommGroup V₁] [Module k V₁] [AddTorsor V₁ P₁] :
                  (AffineEquiv.refl k P₁).toEquiv = Equiv.refl P₁
                  @[simp]
                  theorem AffineEquiv.linear_refl (k : Type u_1) (P₁ : Type u_2) {V₁ : Type u_6} [Ring k] [AddCommGroup V₁] [Module k V₁] [AddTorsor V₁ P₁] :
                  (AffineEquiv.refl k P₁).linear = LinearEquiv.refl k V₁
                  @[simp]
                  theorem AffineEquiv.symm_refl (k : Type u_1) (P₁ : Type u_2) {V₁ : Type u_6} [Ring k] [AddCommGroup V₁] [Module k V₁] [AddTorsor V₁ P₁] :
                  def AffineEquiv.trans {k : Type u_1} {P₁ : Type u_2} {P₂ : Type u_3} {P₃ : Type u_4} {V₁ : Type u_6} {V₂ : Type u_7} {V₃ : Type u_8} [Ring k] [AddCommGroup V₁] [Module k V₁] [AddTorsor V₁ P₁] [AddCommGroup V₂] [Module k V₂] [AddTorsor V₂ P₂] [AddCommGroup V₃] [Module k V₃] [AddTorsor V₃ P₃] (e : P₁ ≃ᵃ[k] P₂) (e' : P₂ ≃ᵃ[k] P₃) :
                  P₁ ≃ᵃ[k] P₃

                  Composition of two AffineEquivalences, applied left to right.

                  Instances For
                    @[simp]
                    theorem AffineEquiv.coe_trans {k : Type u_1} {P₁ : Type u_2} {P₂ : Type u_3} {P₃ : Type u_4} {V₁ : Type u_6} {V₂ : Type u_7} {V₃ : Type u_8} [Ring k] [AddCommGroup V₁] [Module k V₁] [AddTorsor V₁ P₁] [AddCommGroup V₂] [Module k V₂] [AddTorsor V₂ P₂] [AddCommGroup V₃] [Module k V₃] [AddTorsor V₃ P₃] (e : P₁ ≃ᵃ[k] P₂) (e' : P₂ ≃ᵃ[k] P₃) :
                    ↑(AffineEquiv.trans e e') = e' e
                    @[simp]
                    theorem AffineEquiv.coe_trans_to_affineMap {k : Type u_1} {P₁ : Type u_2} {P₂ : Type u_3} {P₃ : Type u_4} {V₁ : Type u_6} {V₂ : Type u_7} {V₃ : Type u_8} [Ring k] [AddCommGroup V₁] [Module k V₁] [AddTorsor V₁ P₁] [AddCommGroup V₂] [Module k V₂] [AddTorsor V₂ P₂] [AddCommGroup V₃] [Module k V₃] [AddTorsor V₃ P₃] (e : P₁ ≃ᵃ[k] P₂) (e' : P₂ ≃ᵃ[k] P₃) :
                    ↑(AffineEquiv.trans e e') = AffineMap.comp e' e
                    @[simp]
                    theorem AffineEquiv.trans_apply {k : Type u_1} {P₁ : Type u_2} {P₂ : Type u_3} {P₃ : Type u_4} {V₁ : Type u_6} {V₂ : Type u_7} {V₃ : Type u_8} [Ring k] [AddCommGroup V₁] [Module k V₁] [AddTorsor V₁ P₁] [AddCommGroup V₂] [Module k V₂] [AddTorsor V₂ P₂] [AddCommGroup V₃] [Module k V₃] [AddTorsor V₃ P₃] (e : P₁ ≃ᵃ[k] P₂) (e' : P₂ ≃ᵃ[k] P₃) (p : P₁) :
                    ↑(AffineEquiv.trans e e') p = e' (e p)
                    theorem AffineEquiv.trans_assoc {k : Type u_1} {P₁ : Type u_2} {P₂ : Type u_3} {P₃ : Type u_4} {P₄ : Type u_5} {V₁ : Type u_6} {V₂ : Type u_7} {V₃ : Type u_8} {V₄ : Type u_9} [Ring k] [AddCommGroup V₁] [Module k V₁] [AddTorsor V₁ P₁] [AddCommGroup V₂] [Module k V₂] [AddTorsor V₂ P₂] [AddCommGroup V₃] [Module k V₃] [AddTorsor V₃ P₃] [AddCommGroup V₄] [Module k V₄] [AddTorsor V₄ P₄] (e₁ : P₁ ≃ᵃ[k] P₂) (e₂ : P₂ ≃ᵃ[k] P₃) (e₃ : P₃ ≃ᵃ[k] P₄) :
                    @[simp]
                    theorem AffineEquiv.trans_refl {k : Type u_1} {P₁ : Type u_2} {P₂ : Type u_3} {V₁ : Type u_6} {V₂ : Type u_7} [Ring k] [AddCommGroup V₁] [Module k V₁] [AddTorsor V₁ P₁] [AddCommGroup V₂] [Module k V₂] [AddTorsor V₂ P₂] (e : P₁ ≃ᵃ[k] P₂) :
                    @[simp]
                    theorem AffineEquiv.refl_trans {k : Type u_1} {P₁ : Type u_2} {P₂ : Type u_3} {V₁ : Type u_6} {V₂ : Type u_7} [Ring k] [AddCommGroup V₁] [Module k V₁] [AddTorsor V₁ P₁] [AddCommGroup V₂] [Module k V₂] [AddTorsor V₂ P₂] (e : P₁ ≃ᵃ[k] P₂) :
                    @[simp]
                    theorem AffineEquiv.self_trans_symm {k : Type u_1} {P₁ : Type u_2} {P₂ : Type u_3} {V₁ : Type u_6} {V₂ : Type u_7} [Ring k] [AddCommGroup V₁] [Module k V₁] [AddTorsor V₁ P₁] [AddCommGroup V₂] [Module k V₂] [AddTorsor V₂ P₂] (e : P₁ ≃ᵃ[k] P₂) :
                    @[simp]
                    theorem AffineEquiv.symm_trans_self {k : Type u_1} {P₁ : Type u_2} {P₂ : Type u_3} {V₁ : Type u_6} {V₂ : Type u_7} [Ring k] [AddCommGroup V₁] [Module k V₁] [AddTorsor V₁ P₁] [AddCommGroup V₂] [Module k V₂] [AddTorsor V₂ P₂] (e : P₁ ≃ᵃ[k] P₂) :
                    @[simp]
                    theorem AffineEquiv.apply_lineMap {k : Type u_1} {P₁ : Type u_2} {P₂ : Type u_3} {V₁ : Type u_6} {V₂ : Type u_7} [Ring k] [AddCommGroup V₁] [Module k V₁] [AddTorsor V₁ P₁] [AddCommGroup V₂] [Module k V₂] [AddTorsor V₂ P₂] (e : P₁ ≃ᵃ[k] P₂) (a : P₁) (b : P₁) (c : k) :
                    e (↑(AffineMap.lineMap a b) c) = ↑(AffineMap.lineMap (e a) (e b)) c
                    instance AffineEquiv.group {k : Type u_1} {P₁ : Type u_2} {V₁ : Type u_6} [Ring k] [AddCommGroup V₁] [Module k V₁] [AddTorsor V₁ P₁] :
                    Group (P₁ ≃ᵃ[k] P₁)
                    theorem AffineEquiv.one_def {k : Type u_1} {P₁ : Type u_2} {V₁ : Type u_6} [Ring k] [AddCommGroup V₁] [Module k V₁] [AddTorsor V₁ P₁] :
                    @[simp]
                    theorem AffineEquiv.coe_one {k : Type u_1} {P₁ : Type u_2} {V₁ : Type u_6} [Ring k] [AddCommGroup V₁] [Module k V₁] [AddTorsor V₁ P₁] :
                    1 = id
                    theorem AffineEquiv.mul_def {k : Type u_1} {P₁ : Type u_2} {V₁ : Type u_6} [Ring k] [AddCommGroup V₁] [Module k V₁] [AddTorsor V₁ P₁] (e : P₁ ≃ᵃ[k] P₁) (e' : P₁ ≃ᵃ[k] P₁) :
                    @[simp]
                    theorem AffineEquiv.coe_mul {k : Type u_1} {P₁ : Type u_2} {V₁ : Type u_6} [Ring k] [AddCommGroup V₁] [Module k V₁] [AddTorsor V₁ P₁] (e : P₁ ≃ᵃ[k] P₁) (e' : P₁ ≃ᵃ[k] P₁) :
                    ↑(e * e') = e e'
                    theorem AffineEquiv.inv_def {k : Type u_1} {P₁ : Type u_2} {V₁ : Type u_6} [Ring k] [AddCommGroup V₁] [Module k V₁] [AddTorsor V₁ P₁] (e : P₁ ≃ᵃ[k] P₁) :
                    @[simp]
                    theorem AffineEquiv.linearHom_apply {k : Type u_1} {P₁ : Type u_2} {V₁ : Type u_6} [Ring k] [AddCommGroup V₁] [Module k V₁] [AddTorsor V₁ P₁] (self : P₁ ≃ᵃ[k] P₁) :
                    AffineEquiv.linearHom self = self.linear
                    def AffineEquiv.linearHom {k : Type u_1} {P₁ : Type u_2} {V₁ : Type u_6} [Ring k] [AddCommGroup V₁] [Module k V₁] [AddTorsor V₁ P₁] :
                    (P₁ ≃ᵃ[k] P₁) →* V₁ ≃ₗ[k] V₁

                    AffineEquiv.linear on automorphisms is a MonoidHom.

                    Instances For
                      @[simp]
                      theorem AffineEquiv.val_equivUnitsAffineMap_apply {k : Type u_1} {P₁ : Type u_2} {V₁ : Type u_6} [Ring k] [AddCommGroup V₁] [Module k V₁] [AddTorsor V₁ P₁] (e : P₁ ≃ᵃ[k] P₁) :
                      ↑(AffineEquiv.equivUnitsAffineMap e) = e
                      @[simp]
                      theorem AffineEquiv.linear_equivUnitsAffineMap_symm_apply {k : Type u_1} {P₁ : Type u_2} {V₁ : Type u_6} [Ring k] [AddCommGroup V₁] [Module k V₁] [AddTorsor V₁ P₁] (u : (P₁ →ᵃ[k] P₁)ˣ) :
                      (↑(MulEquiv.symm AffineEquiv.equivUnitsAffineMap) u).linear = ↑(LinearMap.GeneralLinearGroup.generalLinearEquiv k V₁) (↑(Units.map AffineMap.linearHom) u)
                      @[simp]
                      theorem AffineEquiv.equivUnitsAffineMap_symm_apply_apply {k : Type u_1} {P₁ : Type u_2} {V₁ : Type u_6} [Ring k] [AddCommGroup V₁] [Module k V₁] [AddTorsor V₁ P₁] (u : (P₁ →ᵃ[k] P₁)ˣ) (a : P₁) :
                      ↑(↑(MulEquiv.symm AffineEquiv.equivUnitsAffineMap) u) a = u a
                      @[simp]
                      theorem AffineEquiv.equivUnitsAffineMap_symm_apply_toFun {k : Type u_1} {P₁ : Type u_2} {V₁ : Type u_6} [Ring k] [AddCommGroup V₁] [Module k V₁] [AddTorsor V₁ P₁] (u : (P₁ →ᵃ[k] P₁)ˣ) (a : P₁) :
                      ↑(↑(MulEquiv.symm AffineEquiv.equivUnitsAffineMap) u) a = u a
                      @[simp]
                      theorem AffineEquiv.equivUnitsAffineMap_symm_apply_invFun {k : Type u_1} {P₁ : Type u_2} {V₁ : Type u_6} [Ring k] [AddCommGroup V₁] [Module k V₁] [AddTorsor V₁ P₁] (u : (P₁ →ᵃ[k] P₁)ˣ) (a : P₁) :
                      Equiv.invFun (↑(MulEquiv.symm AffineEquiv.equivUnitsAffineMap) u).toEquiv a = u⁻¹ a
                      @[simp]
                      theorem AffineEquiv.equivUnitsAffineMap_symm_apply_symm_apply {k : Type u_1} {P₁ : Type u_2} {V₁ : Type u_6} [Ring k] [AddCommGroup V₁] [Module k V₁] [AddTorsor V₁ P₁] (u : (P₁ →ᵃ[k] P₁)ˣ) (a : P₁) :
                      ↑(AffineEquiv.symm (↑(MulEquiv.symm AffineEquiv.equivUnitsAffineMap) u)) a = u⁻¹ a
                      @[simp]
                      theorem AffineEquiv.val_inv_equivUnitsAffineMap_apply {k : Type u_1} {P₁ : Type u_2} {V₁ : Type u_6} [Ring k] [AddCommGroup V₁] [Module k V₁] [AddTorsor V₁ P₁] (e : P₁ ≃ᵃ[k] P₁) :
                      (AffineEquiv.equivUnitsAffineMap e)⁻¹ = ↑(AffineEquiv.symm e)
                      def AffineEquiv.equivUnitsAffineMap {k : Type u_1} {P₁ : Type u_2} {V₁ : Type u_6} [Ring k] [AddCommGroup V₁] [Module k V₁] [AddTorsor V₁ P₁] :
                      (P₁ ≃ᵃ[k] P₁) ≃* (P₁ →ᵃ[k] P₁)ˣ

                      The group of AffineEquivs are equivalent to the group of units of AffineMap.

                      This is the affine version of LinearMap.GeneralLinearGroup.generalLinearEquiv.

                      Instances For
                        @[simp]
                        theorem AffineEquiv.linear_vaddConst (k : Type u_1) {P₁ : Type u_2} {V₁ : Type u_6} [Ring k] [AddCommGroup V₁] [Module k V₁] [AddTorsor V₁ P₁] (b : P₁) :
                        @[simp]
                        theorem AffineEquiv.vaddConst_symm_apply (k : Type u_1) {P₁ : Type u_2} {V₁ : Type u_6} [Ring k] [AddCommGroup V₁] [Module k V₁] [AddTorsor V₁ P₁] (b : P₁) (p' : P₁) :
                        @[simp]
                        theorem AffineEquiv.vaddConst_apply (k : Type u_1) {P₁ : Type u_2} {V₁ : Type u_6} [Ring k] [AddCommGroup V₁] [Module k V₁] [AddTorsor V₁ P₁] (b : P₁) (v : V₁) :
                        def AffineEquiv.vaddConst (k : Type u_1) {P₁ : Type u_2} {V₁ : Type u_6} [Ring k] [AddCommGroup V₁] [Module k V₁] [AddTorsor V₁ P₁] (b : P₁) :
                        V₁ ≃ᵃ[k] P₁

                        The map v ↦ v +ᵥ b as an affine equivalence between a module V and an affine space P with tangent space V.

                        Instances For
                          def AffineEquiv.constVSub (k : Type u_1) {P₁ : Type u_2} {V₁ : Type u_6} [Ring k] [AddCommGroup V₁] [Module k V₁] [AddTorsor V₁ P₁] (p : P₁) :
                          P₁ ≃ᵃ[k] V₁

                          p' ↦ p -ᵥ p' as an equivalence.

                          Instances For
                            @[simp]
                            theorem AffineEquiv.coe_constVSub (k : Type u_1) {P₁ : Type u_2} {V₁ : Type u_6} [Ring k] [AddCommGroup V₁] [Module k V₁] [AddTorsor V₁ P₁] (p : P₁) :
                            ↑(AffineEquiv.constVSub k p) = (fun x x_1 => x -ᵥ x_1) p
                            @[simp]
                            theorem AffineEquiv.coe_constVSub_symm (k : Type u_1) {P₁ : Type u_2} {V₁ : Type u_6} [Ring k] [AddCommGroup V₁] [Module k V₁] [AddTorsor V₁ P₁] (p : P₁) :
                            @[simp]
                            theorem AffineEquiv.constVAdd_apply (k : Type u_1) (P₁ : Type u_2) {V₁ : Type u_6} [Ring k] [AddCommGroup V₁] [Module k V₁] [AddTorsor V₁ P₁] (v : V₁) :
                            ∀ (x : P₁), ↑(AffineEquiv.constVAdd k P₁ v) x = v +ᵥ x
                            @[simp]
                            theorem AffineEquiv.linear_constVAdd (k : Type u_1) (P₁ : Type u_2) {V₁ : Type u_6} [Ring k] [AddCommGroup V₁] [Module k V₁] [AddTorsor V₁ P₁] (v : V₁) :
                            (AffineEquiv.constVAdd k P₁ v).linear = LinearEquiv.refl k V₁
                            def AffineEquiv.constVAdd (k : Type u_1) (P₁ : Type u_2) {V₁ : Type u_6} [Ring k] [AddCommGroup V₁] [Module k V₁] [AddTorsor V₁ P₁] (v : V₁) :
                            P₁ ≃ᵃ[k] P₁

                            The map p ↦ v +ᵥ p as an affine automorphism of an affine space.

                            Note that there is no need for an AffineMap.constVAdd as it is always an equivalence. This is roughly to DistribMulAction.toLinearEquiv as +ᵥ is to .

                            Instances For
                              @[simp]
                              theorem AffineEquiv.constVAdd_zero (k : Type u_1) (P₁ : Type u_2) {V₁ : Type u_6} [Ring k] [AddCommGroup V₁] [Module k V₁] [AddTorsor V₁ P₁] :
                              @[simp]
                              theorem AffineEquiv.constVAdd_add (k : Type u_1) (P₁ : Type u_2) {V₁ : Type u_6} [Ring k] [AddCommGroup V₁] [Module k V₁] [AddTorsor V₁ P₁] (v : V₁) (w : V₁) :
                              @[simp]
                              theorem AffineEquiv.constVAdd_symm (k : Type u_1) (P₁ : Type u_2) {V₁ : Type u_6} [Ring k] [AddCommGroup V₁] [Module k V₁] [AddTorsor V₁ P₁] (v : V₁) :
                              @[simp]
                              theorem AffineEquiv.constVAddHom_apply (k : Type u_1) (P₁ : Type u_2) {V₁ : Type u_6} [Ring k] [AddCommGroup V₁] [Module k V₁] [AddTorsor V₁ P₁] (v : Multiplicative V₁) :
                              ↑(AffineEquiv.constVAddHom k P₁) v = AffineEquiv.constVAdd k P₁ (Multiplicative.toAdd v)
                              def AffineEquiv.constVAddHom (k : Type u_1) (P₁ : Type u_2) {V₁ : Type u_6} [Ring k] [AddCommGroup V₁] [Module k V₁] [AddTorsor V₁ P₁] :
                              Multiplicative V₁ →* P₁ ≃ᵃ[k] P₁

                              A more bundled version of AffineEquiv.constVAdd.

                              Instances For
                                theorem AffineEquiv.constVAdd_nsmul (k : Type u_1) (P₁ : Type u_2) {V₁ : Type u_6} [Ring k] [AddCommGroup V₁] [Module k V₁] [AddTorsor V₁ P₁] (n : ) (v : V₁) :
                                theorem AffineEquiv.constVAdd_zsmul (k : Type u_1) (P₁ : Type u_2) {V₁ : Type u_6} [Ring k] [AddCommGroup V₁] [Module k V₁] [AddTorsor V₁ P₁] (z : ) (v : V₁) :
                                def AffineEquiv.homothetyUnitsMulHom {R : Type u_10} {V : Type u_11} {P : Type u_12} [CommRing R] [AddCommGroup V] [Module R V] [AddTorsor V P] (p : P) :

                                Fixing a point in affine space, homothety about this point gives a group homomorphism from (the centre of) the units of the scalars into the group of affine equivalences.

                                Instances For
                                  @[simp]
                                  theorem AffineEquiv.coe_homothetyUnitsMulHom_apply {R : Type u_10} {V : Type u_11} {P : Type u_12} [CommRing R] [AddCommGroup V] [Module R V] [AddTorsor V P] (p : P) (t : Rˣ) :
                                  @[simp]
                                  theorem AffineEquiv.coe_homothetyUnitsMulHom_apply_symm {R : Type u_10} {V : Type u_11} {P : Type u_12} [CommRing R] [AddCommGroup V] [Module R V] [AddTorsor V P] (p : P) (t : Rˣ) :
                                  @[simp]
                                  theorem AffineEquiv.coe_homothetyUnitsMulHom_eq_homothetyHom_coe {R : Type u_10} {V : Type u_11} {P : Type u_12} [CommRing R] [AddCommGroup V] [Module R V] [AddTorsor V P] (p : P) :
                                  AffineEquiv.toAffineMap ↑(AffineEquiv.homothetyUnitsMulHom p) = ↑(AffineMap.homothetyHom p) Units.val
                                  def AffineEquiv.pointReflection (k : Type u_1) {P₁ : Type u_2} {V₁ : Type u_6} [Ring k] [AddCommGroup V₁] [Module k V₁] [AddTorsor V₁ P₁] (x : P₁) :
                                  P₁ ≃ᵃ[k] P₁

                                  Point reflection in x as a permutation.

                                  Instances For
                                    theorem AffineEquiv.pointReflection_apply (k : Type u_1) {P₁ : Type u_2} {V₁ : Type u_6} [Ring k] [AddCommGroup V₁] [Module k V₁] [AddTorsor V₁ P₁] (x : P₁) (y : P₁) :
                                    @[simp]
                                    theorem AffineEquiv.pointReflection_symm (k : Type u_1) {P₁ : Type u_2} {V₁ : Type u_6} [Ring k] [AddCommGroup V₁] [Module k V₁] [AddTorsor V₁ P₁] (x : P₁) :
                                    @[simp]
                                    theorem AffineEquiv.toEquiv_pointReflection (k : Type u_1) {P₁ : Type u_2} {V₁ : Type u_6} [Ring k] [AddCommGroup V₁] [Module k V₁] [AddTorsor V₁ P₁] (x : P₁) :
                                    @[simp]
                                    theorem AffineEquiv.pointReflection_self (k : Type u_1) {P₁ : Type u_2} {V₁ : Type u_6} [Ring k] [AddCommGroup V₁] [Module k V₁] [AddTorsor V₁ P₁] (x : P₁) :
                                    theorem AffineEquiv.pointReflection_involutive (k : Type u_1) {P₁ : Type u_2} {V₁ : Type u_6} [Ring k] [AddCommGroup V₁] [Module k V₁] [AddTorsor V₁ P₁] (x : P₁) :
                                    theorem AffineEquiv.pointReflection_fixed_iff_of_injective_bit0 (k : Type u_1) {P₁ : Type u_2} {V₁ : Type u_6} [Ring k] [AddCommGroup V₁] [Module k V₁] [AddTorsor V₁ P₁] {x : P₁} {y : P₁} (h : Function.Injective bit0) :

                                    x is the only fixed point of pointReflection x. This lemma requires x + x = y + y ↔ x = y. There is no typeclass to use here, so we add it as an explicit argument.

                                    theorem AffineEquiv.injective_pointReflection_left_of_injective_bit0 (k : Type u_1) {P₁ : Type u_2} {V₁ : Type u_6} [Ring k] [AddCommGroup V₁] [Module k V₁] [AddTorsor V₁ P₁] (h : Function.Injective bit0) (y : P₁) :
                                    theorem AffineEquiv.injective_pointReflection_left_of_module (k : Type u_1) {P₁ : Type u_2} {V₁ : Type u_6} [Ring k] [AddCommGroup V₁] [Module k V₁] [AddTorsor V₁ P₁] [Invertible 2] (y : P₁) :
                                    theorem AffineEquiv.pointReflection_fixed_iff_of_module (k : Type u_1) {P₁ : Type u_2} {V₁ : Type u_6} [Ring k] [AddCommGroup V₁] [Module k V₁] [AddTorsor V₁ P₁] [Invertible 2] {x : P₁} {y : P₁} :
                                    def LinearEquiv.toAffineEquiv {k : Type u_1} {V₁ : Type u_6} {V₂ : Type u_7} [Ring k] [AddCommGroup V₁] [Module k V₁] [AddCommGroup V₂] [Module k V₂] (e : V₁ ≃ₗ[k] V₂) :
                                    V₁ ≃ᵃ[k] V₂

                                    Interpret a linear equivalence between modules as an affine equivalence.

                                    Instances For
                                      @[simp]
                                      theorem LinearEquiv.coe_toAffineEquiv {k : Type u_1} {V₁ : Type u_6} {V₂ : Type u_7} [Ring k] [AddCommGroup V₁] [Module k V₁] [AddCommGroup V₂] [Module k V₂] (e : V₁ ≃ₗ[k] V₂) :
                                      theorem AffineMap.lineMap_vadd {k : Type u_1} {P₁ : Type u_2} {V₁ : Type u_6} [Ring k] [AddCommGroup V₁] [Module k V₁] [AddTorsor V₁ P₁] (v : V₁) (v' : V₁) (p : P₁) (c : k) :
                                      ↑(AffineMap.lineMap v v') c +ᵥ p = ↑(AffineMap.lineMap (v +ᵥ p) (v' +ᵥ p)) c
                                      theorem AffineMap.lineMap_vsub {k : Type u_1} {P₁ : Type u_2} {V₁ : Type u_6} [Ring k] [AddCommGroup V₁] [Module k V₁] [AddTorsor V₁ P₁] (p₁ : P₁) (p₂ : P₁) (p₃ : P₁) (c : k) :
                                      ↑(AffineMap.lineMap p₁ p₂) c -ᵥ p₃ = ↑(AffineMap.lineMap (p₁ -ᵥ p₃) (p₂ -ᵥ p₃)) c
                                      theorem AffineMap.vsub_lineMap {k : Type u_1} {P₁ : Type u_2} {V₁ : Type u_6} [Ring k] [AddCommGroup V₁] [Module k V₁] [AddTorsor V₁ P₁] (p₁ : P₁) (p₂ : P₁) (p₃ : P₁) (c : k) :
                                      p₁ -ᵥ ↑(AffineMap.lineMap p₂ p₃) c = ↑(AffineMap.lineMap (p₁ -ᵥ p₂) (p₁ -ᵥ p₃)) c
                                      theorem AffineMap.vadd_lineMap {k : Type u_1} {P₁ : Type u_2} {V₁ : Type u_6} [Ring k] [AddCommGroup V₁] [Module k V₁] [AddTorsor V₁ P₁] (v : V₁) (p₁ : P₁) (p₂ : P₁) (c : k) :
                                      v +ᵥ ↑(AffineMap.lineMap p₁ p₂) c = ↑(AffineMap.lineMap (v +ᵥ p₁) (v +ᵥ p₂)) c
                                      theorem AffineMap.homothety_neg_one_apply {P₁ : Type u_2} {V₁ : Type u_6} [AddCommGroup V₁] [AddTorsor V₁ P₁] {R' : Type u_10} [CommRing R'] [Module R' V₁] (c : P₁) (p : P₁) :