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] : VAdd V (AffineSubspace k 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

• AffineSubspace.pointwiseVAdd = { vadd := fun (x : V) (s : AffineSubspace k P) => AffineSubspace.map (↑(AffineEquiv.constVAdd k P x)) s }

@[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] : AddAction V (AffineSubspace k 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

• AffineSubspace.pointwiseAddAction = Function.Injective.addAction SetLike.coe ⋯ ⋯

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) : v +ᵥ s = map (↑(AffineEquiv.constVAdd k P v)) s

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) : 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) : 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) : v +ᵥ affineSpan k s = affineSpan k (v +ᵥ s)

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) : map f (v +ᵥ s) = f.linear v +ᵥ map f s

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] : SMul M (AffineSubspace 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

• AffineSubspace.pointwiseSMul = { smul := fun (a : M) (s : AffineSubspace k V) => AffineSubspace.map (DistribMulAction.toLinearMap k V a).toAffineMap s }

@[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] : MulAction M (AffineSubspace 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

• AffineSubspace.mulAction = Function.Injective.mulAction SetLike.coe ⋯ ⋯

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) : a • s = map (DistribMulAction.toLinearMap k V a).toAffineMap s

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) : a • ⊥ = ⊥

@[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) : 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