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Mathlib.LinearAlgebra.AffineSpace.Pointwise

Pointwise instances on AffineSubspaces #

This file provides the additive action AffineSubspace.pointwiseAddAction in the Pointwise locale.

def AffineSubspace.pointwiseVAdd {k : Type u_2} {V : Type u_3} {P : Type u_4} [Ring k] [AddCommGroup V] [Module k V] [AddTorsor V P] :

The additive action on an affine subspace corresponding to applying the action to every element.

This is available as an instance in the Pointwise locale.

Equations
Instances For
    @[simp]
    theorem AffineSubspace.coe_pointwise_vadd {k : Type u_2} {V : Type u_3} {P : Type u_4} [Ring k] [AddCommGroup V] [Module k V] [AddTorsor V P] (v : V) (s : AffineSubspace k P) :
    (v +ᵥ s) = v +ᵥ s
    def AffineSubspace.pointwiseAddAction {k : Type u_2} {V : Type u_3} {P : Type u_4} [Ring k] [AddCommGroup V] [Module k V] [AddTorsor V P] :

    The additive action on an affine subspace corresponding to applying the action to every element.

    This is available as an instance in the Pointwise locale.

    Equations
    Instances For
      theorem AffineSubspace.pointwise_vadd_eq_map {k : Type u_2} {V : Type u_3} {P : Type u_4} [Ring k] [AddCommGroup V] [Module k V] [AddTorsor V P] (v : V) (s : AffineSubspace k P) :
      theorem AffineSubspace.vadd_mem_pointwise_vadd_iff {k : Type u_2} {V : Type u_3} {P : Type u_4} [Ring k] [AddCommGroup V] [Module k V] [AddTorsor V P] {v : V} {s : AffineSubspace k P} {p : P} :
      v +ᵥ p v +ᵥ s p s
      @[simp]
      theorem AffineSubspace.pointwise_vadd_bot {k : Type u_2} {V : Type u_3} {P : Type u_4} [Ring k] [AddCommGroup V] [Module k V] [AddTorsor V P] (v : V) :
      @[simp]
      theorem AffineSubspace.pointwise_vadd_top {k : Type u_2} {V : Type u_3} {P : Type u_4} [Ring k] [AddCommGroup V] [Module k V] [AddTorsor V P] (v : V) :
      theorem AffineSubspace.pointwise_vadd_direction {k : Type u_2} {V : Type u_3} {P : Type u_4} [Ring k] [AddCommGroup V] [Module k V] [AddTorsor V P] (v : V) (s : AffineSubspace k P) :
      (v +ᵥ s).direction = s.direction
      theorem AffineSubspace.pointwise_vadd_span {k : Type u_2} {V : Type u_3} {P : Type u_4} [Ring k] [AddCommGroup V] [Module k V] [AddTorsor V P] (v : V) (s : Set P) :
      theorem AffineSubspace.map_pointwise_vadd {k : Type u_2} {V₁ : Type u_5} {P₁ : Type u_6} {V₂ : Type u_7} {P₂ : Type u_8} [Ring k] [AddCommGroup V₁] [Module k V₁] [AddTorsor V₁ P₁] [AddCommGroup V₂] [Module k V₂] [AddTorsor V₂ P₂] (f : P₁ →ᵃ[k] P₂) (v : V₁) (s : AffineSubspace k P₁) :
      def AffineSubspace.pointwiseSMul {M : Type u_1} {k : Type u_2} {V : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [Monoid M] [DistribMulAction M V] [SMulCommClass M k V] :

      The multiplicative action on an affine subspace corresponding to applying the action to every element.

      This is available as an instance in the Pointwise locale.

      TODO: generalize to include SMul (P ≃ᵃ[k] P) (AffineSubspace k P), which acts on P with a VAdd version of a DistribMulAction.

      Equations
      Instances For
        @[simp]
        theorem AffineSubspace.coe_smul {M : Type u_1} {k : Type u_2} {V : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [Monoid M] [DistribMulAction M V] [SMulCommClass M k V] (a : M) (s : AffineSubspace k V) :
        (a s) = a s
        def AffineSubspace.mulAction {M : Type u_1} {k : Type u_2} {V : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [Monoid M] [DistribMulAction M V] [SMulCommClass M k V] :

        The multiplicative action on an affine subspace corresponding to applying the action to every element.

        This is available as an instance in the Pointwise locale.

        TODO: generalize to include SMul (P ≃ᵃ[k] P) (AffineSubspace k P), which acts on P with a VAdd version of a DistribMulAction.

        Equations
        Instances For
          theorem AffineSubspace.smul_eq_map {M : Type u_1} {k : Type u_2} {V : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [Monoid M] [DistribMulAction M V] [SMulCommClass M k V] (a : M) (s : AffineSubspace k V) :
          theorem AffineSubspace.smul_mem_smul_iff {k : Type u_2} {V : Type u_3} [Ring k] [AddCommGroup V] [Module k V] {s : AffineSubspace k V} {p : V} {G : Type u_9} [Group G] [DistribMulAction G V] [SMulCommClass G k V] {a : G} :
          a p a s p s
          theorem AffineSubspace.smul_mem_smul_iff_of_isUnit {M : Type u_1} {k : Type u_2} {V : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [Monoid M] [DistribMulAction M V] [SMulCommClass M k V] {a : M} {s : AffineSubspace k V} {p : V} (ha : IsUnit a) :
          a p a s p s
          theorem AffineSubspace.smul_mem_smul_iff₀ {k : Type u_2} {V : Type u_3} [Ring k] [AddCommGroup V] [Module k V] {s : AffineSubspace k V} {p : V} {G₀ : Type u_9} [GroupWithZero G₀] [DistribMulAction G₀ V] [SMulCommClass G₀ k V] {a : G₀} (ha : a 0) :
          a p a s p s
          @[simp]
          theorem AffineSubspace.smul_bot {M : Type u_1} {k : Type u_2} {V : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [Monoid M] [DistribMulAction M V] [SMulCommClass M k V] (a : M) :
          @[simp]
          theorem AffineSubspace.smul_top {M : Type u_1} {k : Type u_2} {V : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [Monoid M] [DistribMulAction M V] [SMulCommClass M k V] {a : M} (ha : IsUnit a) :
          theorem AffineSubspace.smul_span {M : Type u_1} {k : Type u_2} {V : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [Monoid M] [DistribMulAction M V] [SMulCommClass M k V] (a : M) (s : Set V) :
          a affineSpan k s = affineSpan k (a s)
          @[simp]
          theorem AffineSubspace.direction_smul {k : Type u_2} {V : Type u_3} [Field k] [AddCommGroup V] [Module k V] {a : k} (ha : a 0) (s : AffineSubspace k V) :
          (a s).direction = s.direction