Pointwise instances on AffineSubspaces

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

def AffineSubspace.pointwiseAddAction {k : Type u_1} [Ring k] {V : Type u_2} {P : Type u_3} [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 = AddAction.mk ⋯ ⋯

theorem AffineSubspace.pointwise_vadd_eq_map {k : Type u_1} [Ring k] {V : Type u_2} {P : Type u_3} [AddCommGroup V] [Module k V] [AddTorsor V P] (v : V) (s : AffineSubspace k P) : v +ᵥ s = AffineSubspace.map (↑(AffineEquiv.constVAdd k P v)) s

@[simp]

theorem AffineSubspace.coe_pointwise_vadd {k : Type u_1} [Ring k] {V : Type u_2} {P : Type u_3} [AddCommGroup V] [Module k V] [AddTorsor V P] (v : V) (s : AffineSubspace k P) : ↑(v +ᵥ s) = v +ᵥ ↑s

theorem AffineSubspace.vadd_mem_pointwise_vadd_iff {k : Type u_1} [Ring k] {V : Type u_2} {P : Type u_3} [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_1} [Ring k] {V : Type u_2} {P : Type u_3} [AddCommGroup V] [Module k V] [AddTorsor V P] (v : V) : v +ᵥ ⊥ = ⊥

@[simp]

theorem AffineSubspace.pointwise_vadd_top {k : Type u_1} [Ring k] {V : Type u_2} {P : Type u_3} [AddCommGroup V] [Module k V] [AddTorsor V P] (v : V) : v +ᵥ ⊤ = ⊤

theorem AffineSubspace.pointwise_vadd_direction {k : Type u_1} [Ring k] {V : Type u_2} {P : Type u_3} [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_1} [Ring k] {V : Type u_2} {P : Type u_3} [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_1} [Ring k] {V₁ : Type u_4} {P₁ : Type u_5} {V₂ : Type u_6} {P₂ : Type u_7} [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) : AffineSubspace.map f (v +ᵥ s) = f.linear v +ᵥ AffineSubspace.map f s