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Found 328 declarations mentioning AffineSubspace.direction. Of these, only the first 200 are shown.

• AffineSubspace.direction 📋 Mathlib.LinearAlgebra.AffineSpace.AffineSubspace.Defs
  {k : Type u_1} {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [AddTorsor V P] (s : AffineSubspace k P) : Submodule k V
• Submodule.toAffineSubspace_direction 📋 Mathlib.LinearAlgebra.AffineSpace.AffineSubspace.Defs
  {k : Type u_1} {V : Type u_2} [Ring k] [AddCommGroup V] [Module k V] (s : Submodule k V) : (↑s).direction = s
• direction_affineSpan 📋 Mathlib.LinearAlgebra.AffineSpace.AffineSubspace.Defs
  (k : Type u_1) {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [AddTorsor V P] (s : Set P) : (affineSpan k s).direction = vectorSpan k s
• AffineSubspace.direction_mk' 📋 Mathlib.LinearAlgebra.AffineSpace.AffineSubspace.Defs
  {k : Type u_1} {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [AddTorsor V P] (p : P) (direction : Submodule k V) : (AffineSubspace.mk' p direction).direction = direction
• AffineSubspace.direction_eq_vectorSpan 📋 Mathlib.LinearAlgebra.AffineSpace.AffineSubspace.Defs
  {k : Type u_1} {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [AddTorsor V P] (s : AffineSubspace k P) : s.direction = vectorSpan k ↑s
• AffineSubspace.directionOfNonempty_eq_direction 📋 Mathlib.LinearAlgebra.AffineSpace.AffineSubspace.Defs
  {k : Type u_1} {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [AddTorsor V P] {s : AffineSubspace k P} (h : (↑s).Nonempty) : AffineSubspace.directionOfNonempty h = s.direction
• AffineSubspace.direction_singleton 📋 Mathlib.LinearAlgebra.AffineSpace.AffineSubspace.Defs
  {k : Type u_1} {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [AddTorsor V P] (x : P) : {x}.direction = ⊥
• AffineSubspace.mk'_eq 📋 Mathlib.LinearAlgebra.AffineSpace.AffineSubspace.Defs
  {k : Type u_1} {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [AddTorsor V P] {s : AffineSubspace k P} {p : P} (hp : p ∈ s) : AffineSubspace.mk' p s.direction = s
• AffineSubspace.direction_eq_self_iff_zero_mem 📋 Mathlib.LinearAlgebra.AffineSpace.AffineSubspace.Defs
  {k : Type u_1} {V : Type u_2} [Ring k] [AddCommGroup V] [Module k V] {s : AffineSubspace k V} : ↑s.direction = s ↔ 0 ∈ s
• AffineSubspace.ext_of_direction_eq 📋 Mathlib.LinearAlgebra.AffineSpace.AffineSubspace.Defs
  {k : Type u_1} {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [AddTorsor V P] {s₁ s₂ : AffineSubspace k P} (hd : s₁.direction = s₂.direction) (hn : (↑s₁ ∩ ↑s₂).Nonempty) : s₁ = s₂
• AffineSubspace.coe_direction_eq_vsub_set_left 📋 Mathlib.LinearAlgebra.AffineSpace.AffineSubspace.Defs
  {k : Type u_1} {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [AddTorsor V P] {s : AffineSubspace k P} {p : P} (hp : p ∈ s) : ↑s.direction = (fun x => p -ᵥ x) '' ↑s
• AffineSubspace.coe_direction_eq_vsub_set_right 📋 Mathlib.LinearAlgebra.AffineSpace.AffineSubspace.Defs
  {k : Type u_1} {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [AddTorsor V P] {s : AffineSubspace k P} {p : P} (hp : p ∈ s) : ↑s.direction = (fun x => x -ᵥ p) '' ↑s
• AffineSubspace.direction_le 📋 Mathlib.LinearAlgebra.AffineSpace.AffineSubspace.Defs
  {k : Type u_1} {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [S : AddTorsor V P] {s₁ s₂ : AffineSubspace k P} (h : s₁ ≤ s₂) : s₁.direction ≤ s₂.direction
• AffineSubspace.eq_of_direction_eq_of_nonempty_of_le 📋 Mathlib.LinearAlgebra.AffineSpace.AffineSubspace.Defs
  {k : Type u_1} {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [S : AddTorsor V P] {s₁ s₂ : AffineSubspace k P} (hd : s₁.direction = s₂.direction) (hn : (↑s₁).Nonempty) (hle : s₁ ≤ s₂) : s₁ = s₂
• AffineSubspace.coe_direction_eq_vsub_set 📋 Mathlib.LinearAlgebra.AffineSpace.AffineSubspace.Defs
  {k : Type u_1} {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [AddTorsor V P] {s : AffineSubspace k P} (h : (↑s).Nonempty) : ↑s.direction = ↑s -ᵥ ↑s
• AffineSubspace.eq_iff_direction_eq_of_mem 📋 Mathlib.LinearAlgebra.AffineSpace.AffineSubspace.Defs
  {k : Type u_1} {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [AddTorsor V P] {s₁ s₂ : AffineSubspace k P} {p : P} (h₁ : p ∈ s₁) (h₂ : p ∈ s₂) : s₁ = s₂ ↔ s₁.direction = s₂.direction
• AffineSubspace.vsub_mem_direction 📋 Mathlib.LinearAlgebra.AffineSpace.AffineSubspace.Defs
  {k : Type u_1} {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [AddTorsor V P] {s : AffineSubspace k P} {p₁ p₂ : P} (hp₁ : p₁ ∈ s) (hp₂ : p₂ ∈ s) : p₁ -ᵥ p₂ ∈ s.direction
• AffineSubspace.direction_iInf 📋 Mathlib.LinearAlgebra.AffineSpace.AffineSubspace.Defs
  {k : Type u_1} {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [S : AddTorsor V P] {ι : Sort u_4} (s : ι → AffineSubspace k P) : (iInf s).direction ≤ ⨅ i, (s i).direction
• AffineSubspace.direction_iInf_of_mem 📋 Mathlib.LinearAlgebra.AffineSpace.AffineSubspace.Defs
  {k : Type u_1} {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [S : AddTorsor V P] {ι : Sort u_4} (s : ι → AffineSubspace k P) (p : P) (h : ∀ (i : ι), p ∈ s i) : (iInf s).direction = ⨅ i, (s i).direction
• AffineSubspace.mem_direction_iff_eq_vsub_left 📋 Mathlib.LinearAlgebra.AffineSpace.AffineSubspace.Defs
  {k : Type u_1} {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [AddTorsor V P] {s : AffineSubspace k P} {p : P} (hp : p ∈ s) (v : V) : v ∈ s.direction ↔ ∃ p₂ ∈ s, v = p -ᵥ p₂
• AffineSubspace.mem_direction_iff_eq_vsub_right 📋 Mathlib.LinearAlgebra.AffineSpace.AffineSubspace.Defs
  {k : Type u_1} {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [AddTorsor V P] {s : AffineSubspace k P} {p : P} (hp : p ∈ s) (v : V) : v ∈ s.direction ↔ ∃ p₂ ∈ s, v = p₂ -ᵥ p
• AffineSubspace.vadd_mem_of_mem_direction 📋 Mathlib.LinearAlgebra.AffineSpace.AffineSubspace.Defs
  {k : Type u_1} {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [AddTorsor V P] {s : AffineSubspace k P} {v : V} (hv : v ∈ s.direction) {p : P} (hp : p ∈ s) : v +ᵥ p ∈ s
• AffineSubspace.vadd_mem_iff_mem_direction 📋 Mathlib.LinearAlgebra.AffineSpace.AffineSubspace.Defs
  {k : Type u_1} {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [AddTorsor V P] {s : AffineSubspace k P} (v : V) {p : P} (hp : p ∈ s) : v +ᵥ p ∈ s ↔ v ∈ s.direction
• AffineSubspace.vadd_mem_iff_mem_of_mem_direction 📋 Mathlib.LinearAlgebra.AffineSpace.AffineSubspace.Defs
  {k : Type u_1} {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [AddTorsor V P] {s : AffineSubspace k P} {v : V} (hv : v ∈ s.direction) {p : P} : v +ᵥ p ∈ s ↔ p ∈ s
• AffineSubspace.mem_direction_iff_eq_vsub 📋 Mathlib.LinearAlgebra.AffineSpace.AffineSubspace.Defs
  {k : Type u_1} {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [AddTorsor V P] {s : AffineSubspace k P} (h : (↑s).Nonempty) (v : V) : v ∈ s.direction ↔ ∃ p₁ ∈ s, ∃ p₂ ∈ s, v = p₁ -ᵥ p₂
• AffineSubspace.direction_inf 📋 Mathlib.LinearAlgebra.AffineSpace.AffineSubspace.Defs
  {k : Type u_1} {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [S : AddTorsor V P] (s₁ s₂ : AffineSubspace k P) : (s₁ ⊓ s₂).direction ≤ s₁.direction ⊓ s₂.direction
• AffineSubspace.direction_iInf_of_mem_iInf 📋 Mathlib.LinearAlgebra.AffineSpace.AffineSubspace.Defs
  {k : Type u_1} {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [S : AddTorsor V P] {ι : Sort u_4} (s : ι → AffineSubspace k P) (p : P) (h : p ∈ iInf s) : (iInf s).direction = ⨅ i, (s i).direction
• AffineSubspace.inter_eq_singleton_of_nonempty_of_isCompl 📋 Mathlib.LinearAlgebra.AffineSpace.AffineSubspace.Defs
  {k : Type u_1} {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [S : AddTorsor V P] {s₁ s₂ : AffineSubspace k P} (h1 : (↑s₁).Nonempty) (h2 : (↑s₂).Nonempty) (hd : IsCompl s₁.direction s₂.direction) : ∃ p, ↑s₁ ∩ ↑s₂ = {p}
• AffineSubspace.direction_bot 📋 Mathlib.LinearAlgebra.AffineSpace.AffineSubspace.Defs
  (k : Type u_1) (V : Type u_2) (P : Type u_3) [Ring k] [AddCommGroup V] [Module k V] [S : AddTorsor V P] : ⊥.direction = ⊥
• AffineSubspace.direction_inf_of_mem 📋 Mathlib.LinearAlgebra.AffineSpace.AffineSubspace.Defs
  {k : Type u_1} {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [S : AddTorsor V P] {s₁ s₂ : AffineSubspace k P} {p : P} (h₁ : p ∈ s₁) (h₂ : p ∈ s₂) : (s₁ ⊓ s₂).direction = s₁.direction ⊓ s₂.direction
• AffineSubspace.direction_top 📋 Mathlib.LinearAlgebra.AffineSpace.AffineSubspace.Defs
  (k : Type u_1) (V : Type u_2) (P : Type u_3) [Ring k] [AddCommGroup V] [Module k V] [S : AddTorsor V P] : ⊤.direction = ⊤
• AffineSubspace.sup_direction_le 📋 Mathlib.LinearAlgebra.AffineSpace.AffineSubspace.Defs
  {k : Type u_1} {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [S : AddTorsor V P] (s₁ s₂ : AffineSubspace k P) : s₁.direction ⊔ s₂.direction ≤ (s₁ ⊔ s₂).direction
• AffineSubspace.direction_inf_of_mem_inf 📋 Mathlib.LinearAlgebra.AffineSpace.AffineSubspace.Defs
  {k : Type u_1} {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [S : AddTorsor V P] {s₁ s₂ : AffineSubspace k P} {p : P} (h : p ∈ s₁ ⊓ s₂) : (s₁ ⊓ s₂).direction = s₁.direction ⊓ s₂.direction
• AffineSubspace.inter_nonempty_of_nonempty_of_sup_direction_eq_top 📋 Mathlib.LinearAlgebra.AffineSpace.AffineSubspace.Defs
  {k : Type u_1} {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [S : AddTorsor V P] {s₁ s₂ : AffineSubspace k P} (h1 : (↑s₁).Nonempty) (h2 : (↑s₂).Nonempty) (hd : s₁.direction ⊔ s₂.direction = ⊤) : (↑s₁ ∩ ↑s₂).Nonempty
• AffineSubspace.direction_sInf 📋 Mathlib.LinearAlgebra.AffineSpace.AffineSubspace.Defs
  {k : Type u_1} {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [S : AddTorsor V P] (t : Set (AffineSubspace k P)) : (sInf t).direction ≤ ⨅ s ∈ t, s.direction
• AffineSubspace.direction_eq_top_iff_of_nonempty 📋 Mathlib.LinearAlgebra.AffineSpace.AffineSubspace.Defs
  {k : Type u_1} {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [S : AddTorsor V P] {s : AffineSubspace k P} (h : (↑s).Nonempty) : s.direction = ⊤ ↔ s = ⊤
• AffineSubspace.direction_sInf_of_mem_sInf 📋 Mathlib.LinearAlgebra.AffineSpace.AffineSubspace.Defs
  {k : Type u_1} {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [S : AddTorsor V P] (t : Set (AffineSubspace k P)) (p : P) (h : p ∈ sInf t) : (sInf t).direction = ⨅ s ∈ t, s.direction
• AffineSubspace.direction_sInf_of_mem 📋 Mathlib.LinearAlgebra.AffineSpace.AffineSubspace.Defs
  {k : Type u_1} {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [S : AddTorsor V P] (t : Set (AffineSubspace k P)) (p : P) (h : ∀ s ∈ t, p ∈ s) : (sInf t).direction = ⨅ s ∈ t, s.direction
• AffineSubspace.sup_direction_lt_of_nonempty_of_inter_empty 📋 Mathlib.LinearAlgebra.AffineSpace.AffineSubspace.Defs
  {k : Type u_1} {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [S : AddTorsor V P] {s₁ s₂ : AffineSubspace k P} (h1 : (↑s₁).Nonempty) (h2 : (↑s₂).Nonempty) (he : ↑s₁ ∩ ↑s₂ = ∅) : s₁.direction ⊔ s₂.direction < (s₁ ⊔ s₂).direction
• AffineSubspace.Parallel.direction_eq 📋 Mathlib.LinearAlgebra.AffineSpace.AffineSubspace.Basic
  {k : Type u_1} {V : Type u_2} {P : Type u_4} [Ring k] [AddCommGroup V] [Module k V] [AddTorsor V P] {s₁ s₂ : AffineSubspace k P} (h : s₁.Parallel s₂) : s₁.direction = s₂.direction
• AffineSubspace.vsub_left_mem_direction_iff_mem 📋 Mathlib.LinearAlgebra.AffineSpace.AffineSubspace.Basic
  {k : Type u_1} {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [AddTorsor V P] {s : AffineSubspace k P} {p : P} (hp : p ∈ s) (p₂ : P) : p -ᵥ p₂ ∈ s.direction ↔ p₂ ∈ s
• AffineSubspace.vsub_right_mem_direction_iff_mem 📋 Mathlib.LinearAlgebra.AffineSpace.AffineSubspace.Basic
  {k : Type u_1} {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [AddTorsor V P] {s : AffineSubspace k P} {p : P} (hp : p ∈ s) (p₂ : P) : p₂ -ᵥ p ∈ s.direction ↔ p₂ ∈ s
• AffineSubspace.map_direction 📋 Mathlib.LinearAlgebra.AffineSpace.AffineSubspace.Basic
  {k : Type u_1} {V₁ : Type u_2} {P₁ : Type u_3} {V₂ : Type u_4} {P₂ : Type u_5} [Ring k] [AddCommGroup V₁] [Module k V₁] [AddTorsor V₁ P₁] [AddCommGroup V₂] [Module k V₂] [AddTorsor V₂ P₂] (f : P₁ →ᵃ[k] P₂) (s : AffineSubspace k P₁) : (AffineSubspace.map f s).direction = Submodule.map f.linear s.direction
• AffineSubspace.direction_lt_of_nonempty 📋 Mathlib.LinearAlgebra.AffineSpace.AffineSubspace.Basic
  {k : Type u_1} {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [S : AddTorsor V P] {s₁ s₂ : AffineSubspace k P} (h : s₁ < s₂) (hn : (↑s₁).Nonempty) : s₁.direction < s₂.direction
• AffineSubspace.subtype 📋 Mathlib.LinearAlgebra.AffineSpace.AffineSubspace.Basic
  {k : Type u_1} {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [AddTorsor V P] (s : AffineSubspace k P) [Nonempty ↥s] : ↥s →ᵃ[k] P
• AffineSubspace.toAddTorsor 📋 Mathlib.LinearAlgebra.AffineSpace.AffineSubspace.Basic
  {k : Type u_1} {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [AddTorsor V P] (s : AffineSubspace k P) [Nonempty ↥s] : AddTorsor ↥s.direction ↥s
• AffineSubspace.direction_affineSpan_insert 📋 Mathlib.LinearAlgebra.AffineSpace.AffineSubspace.Basic
  {k : Type u_1} {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [AddTorsor V P] {s : AffineSubspace k P} {p₁ p₂ : P} (hp₁ : p₁ ∈ s) : (affineSpan k (insert p₂ ↑s)).direction = k ∙ (p₂ -ᵥ p₁) ⊔ s.direction
• AffineSubspace.direction_affineSpan_pair_le_iff_exists_smul 📋 Mathlib.LinearAlgebra.AffineSpace.AffineSubspace.Basic
  {k : Type u_1} {V : Type u_2} {P : Type u_4} [Ring k] [AddCommGroup V] [Module k V] [AddTorsor V P] {p₁ q₁ p₂ q₂ : P} : line[k, p₁, q₁].direction ≤ line[k, p₂, q₂].direction ↔ ∃ z, z • (q₂ -ᵥ p₂) = q₁ -ᵥ p₁
• AffineSubspace.direction_sup_eq_sup_direction 📋 Mathlib.LinearAlgebra.AffineSpace.AffineSubspace.Basic
  {k : Type u_1} {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [AddTorsor V P] {s₁ s₂ : AffineSubspace k P} {p : P} (hp₁ : p ∈ s₁) (hp₂ : p ∈ s₂) : (s₁ ⊔ s₂).direction = s₁.direction ⊔ s₂.direction
• AffineSubspace.direction_prod_le 📋 Mathlib.LinearAlgebra.AffineSpace.AffineSubspace.Basic
  {k : Type u_1} {V : Type u_2} {W : Type u_3} {P : Type u_4} {Q : Type u_5} [Ring k] [AddCommGroup V] [Module k V] [AddTorsor V P] [AddCommGroup W] [Module k W] [AddTorsor W Q] (s : AffineSubspace k P) (t : AffineSubspace k Q) : (s.prod t).direction ≤ s.direction.prod t.direction
• AffineSubspace.direction_sup 📋 Mathlib.LinearAlgebra.AffineSpace.AffineSubspace.Basic
  {k : Type u_1} {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [AddTorsor V P] {s₁ s₂ : AffineSubspace k P} {p₁ p₂ : P} (hp₁ : p₁ ∈ s₁) (hp₂ : p₂ ∈ s₂) : (s₁ ⊔ s₂).direction = s₁.direction ⊔ s₂.direction ⊔ k ∙ (p₂ -ᵥ p₁)
• AffineEquiv.ofEq 📋 Mathlib.LinearAlgebra.AffineSpace.AffineSubspace.Basic
  {k : Type u_1} {V₁ : Type u_2} {P₁ : Type u_3} [Ring k] [AddCommGroup V₁] [Module k V₁] [AddTorsor V₁ P₁] (S₁ S₂ : AffineSubspace k P₁) [Nonempty ↥S₁] [Nonempty ↥S₂] (h : S₁ = S₂) : ↥S₁ ≃ᵃ[k] ↥S₂
• AffineSubspace.subtype_linear 📋 Mathlib.LinearAlgebra.AffineSpace.AffineSubspace.Basic
  {k : Type u_1} {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [AddTorsor V P] (s : AffineSubspace k P) [Nonempty ↥s] : s.subtype.linear = s.direction.subtype
• AffineSubspace.parallel_iff_direction_eq_and_eq_bot_iff_eq_bot 📋 Mathlib.LinearAlgebra.AffineSpace.AffineSubspace.Basic
  {k : Type u_1} {V : Type u_2} {P : Type u_4} [Ring k] [AddCommGroup V] [Module k V] [AddTorsor V P] {s₁ s₂ : AffineSubspace k P} : s₁.Parallel s₂ ↔ s₁.direction = s₂.direction ∧ (s₁ = ⊥ ↔ s₂ = ⊥)
• AffineSubspace.subtype_injective 📋 Mathlib.LinearAlgebra.AffineSpace.AffineSubspace.Basic
  {k : Type u_1} {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [AddTorsor V P] (s : AffineSubspace k P) [Nonempty ↥s] : Function.Injective ⇑s.subtype
• AffineSubspace.inclusion 📋 Mathlib.LinearAlgebra.AffineSpace.AffineSubspace.Basic
  {k : Type u_1} {V₁ : Type u_2} {P₁ : Type u_3} [Ring k] [AddCommGroup V₁] [Module k V₁] [AddTorsor V₁ P₁] {S₁ S₂ : AffineSubspace k P₁} [Nonempty ↥S₁] (h : S₁ ≤ S₂) : ↥S₁ →ᵃ[k] ↥S₂
• AffineEquiv.ofEq_rfl 📋 Mathlib.LinearAlgebra.AffineSpace.AffineSubspace.Basic
  {k : Type u_1} {V₁ : Type u_2} {P₁ : Type u_3} [Ring k] [AddCommGroup V₁] [Module k V₁] [AddTorsor V₁ P₁] (S₁ : AffineSubspace k P₁) [Nonempty ↥S₁] : AffineEquiv.ofEq S₁ S₁ ⋯ = AffineEquiv.refl k ↥S₁
• AffineSubspace.coe_subtype 📋 Mathlib.LinearAlgebra.AffineSpace.AffineSubspace.Basic
  {k : Type u_1} {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [AddTorsor V P] (s : AffineSubspace k P) [Nonempty ↥s] : ⇑s.subtype = Subtype.val
• AffineSubspace.subtype_apply 📋 Mathlib.LinearAlgebra.AffineSpace.AffineSubspace.Basic
  {k : Type u_1} {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [AddTorsor V P] {s : AffineSubspace k P} [Nonempty ↥s] (p : ↥s) : s.subtype p = ↑p
• AffineSubspace.coe_vsub 📋 Mathlib.LinearAlgebra.AffineSpace.AffineSubspace.Basic
  {k : Type u_1} {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [AddTorsor V P] (s : AffineSubspace k P) [Nonempty ↥s] (a b : ↥s) : ↑(a -ᵥ b) = ↑a -ᵥ ↑b
• AffineSubspace.direction_prod_eq 📋 Mathlib.LinearAlgebra.AffineSpace.AffineSubspace.Basic
  {k : Type u_1} {V : Type u_2} {W : Type u_3} {P : Type u_4} {Q : Type u_5} [Ring k] [AddCommGroup V] [Module k V] [AddTorsor V P] [AddCommGroup W] [Module k W] [AddTorsor W Q] {s : AffineSubspace k P} {t : AffineSubspace k Q} (hs : s ≠ ⊥) (ht : t ≠ ⊥) : (s.prod t).direction = s.direction.prod t.direction
• AffineSubspace.inclusion_rfl 📋 Mathlib.LinearAlgebra.AffineSpace.AffineSubspace.Basic
  {k : Type u_1} {V₁ : Type u_2} {P₁ : Type u_3} [Ring k] [AddCommGroup V₁] [Module k V₁] [AddTorsor V₁ P₁] {S₁ : AffineSubspace k P₁} [Nonempty ↥S₁] : AffineSubspace.inclusion ⋯ = AffineMap.id k ↥S₁
• AffineEquiv.linear_ofEq 📋 Mathlib.LinearAlgebra.AffineSpace.AffineSubspace.Basic
  {k : Type u_1} {V₁ : Type u_2} {P₁ : Type u_3} [Ring k] [AddCommGroup V₁] [Module k V₁] [AddTorsor V₁ P₁] (S₁ S₂ : AffineSubspace k P₁) [Nonempty ↥S₁] [Nonempty ↥S₂] (h : S₁ = S₂) : (AffineEquiv.ofEq S₁ S₂ h).linear = LinearEquiv.ofEq S₁.direction S₂.direction ⋯
• AffineEquiv.ofEq_symm 📋 Mathlib.LinearAlgebra.AffineSpace.AffineSubspace.Basic
  {k : Type u_1} {V₁ : Type u_2} {P₁ : Type u_3} [Ring k] [AddCommGroup V₁] [Module k V₁] [AddTorsor V₁ P₁] (S₁ S₂ : AffineSubspace k P₁) [Nonempty ↥S₁] [Nonempty ↥S₂] (h : S₁ = S₂) : (AffineEquiv.ofEq S₁ S₂ h).symm = AffineEquiv.ofEq S₂ S₁ ⋯
• exists_eq_smul_of_parallel 📋 Mathlib.LinearAlgebra.AffineSpace.AffineSubspace.Basic
  {k : Type u_1} {V : Type u_2} {P : Type u_3} [DivisionRing k] [AddCommGroup V] [Module k V] [AddTorsor V P] {p₁ p₂ p₃ p₄ p₅ p₆ : P} (h₂ : p₂ ∉ line[k, p₁, p₃]) (h₁₂₄₅ : line[k, p₁, p₂].Parallel line[k, p₄, p₅]) (h₂₃₅₆ : line[k, p₅, p₆].direction ≤ line[k, p₂, p₃].direction) (h₃₁₆₄ : line[k, p₆, p₄].direction ≤ line[k, p₃, p₁].direction) : ∃ r, r ≠ 0 ∧ p₅ -ᵥ p₄ = r • (p₂ -ᵥ p₁) ∧ p₆ -ᵥ p₅ = r • (p₃ -ᵥ p₂) ∧ p₄ -ᵥ p₆ = r • (p₁ -ᵥ p₃)
• AffineSubspace.inclusion_linear 📋 Mathlib.LinearAlgebra.AffineSpace.AffineSubspace.Basic
  {k : Type u_1} {V₁ : Type u_2} {P₁ : Type u_3} [Ring k] [AddCommGroup V₁] [Module k V₁] [AddTorsor V₁ P₁] {S₁ S₂ : AffineSubspace k P₁} [Nonempty ↥S₁] (h : S₁ ≤ S₂) : (AffineSubspace.inclusion h).linear = Submodule.inclusion ⋯
• AffineSubspace.topEquiv 📋 Mathlib.LinearAlgebra.AffineSpace.AffineSubspace.Basic
  (k : Type u_1) (V : Type u_2) (P : Type u_3) [Ring k] [AddCommGroup V] [Module k V] [S : AddTorsor V P] : ↥⊤ ≃ᵃ[k] P
• AffineSubspace.coe_vadd 📋 Mathlib.LinearAlgebra.AffineSpace.AffineSubspace.Basic
  {k : Type u_1} {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [AddTorsor V P] (s : AffineSubspace k P) [Nonempty ↥s] (a : ↥s.direction) (b : ↥s) : ↑(a +ᵥ b) = ↑a +ᵥ ↑b
• AffineSubspace.coe_inclusion_apply 📋 Mathlib.LinearAlgebra.AffineSpace.AffineSubspace.Basic
  {k : Type u_1} {V₁ : Type u_2} {P₁ : Type u_3} [Ring k] [AddCommGroup V₁] [Module k V₁] [AddTorsor V₁ P₁] {S₁ S₂ : AffineSubspace k P₁} [Nonempty ↥S₁] (h : S₁ ≤ S₂) (x : ↥S₁) : ↑((AffineSubspace.inclusion h) x) = ↑x
• AffineEquiv.coe_ofEq_apply 📋 Mathlib.LinearAlgebra.AffineSpace.AffineSubspace.Basic
  {k : Type u_1} {V₁ : Type u_2} {P₁ : Type u_3} [Ring k] [AddCommGroup V₁] [Module k V₁] [AddTorsor V₁ P₁] (S₁ S₂ : AffineSubspace k P₁) [Nonempty ↥S₁] [Nonempty ↥S₂] (h : S₁ = S₂) (x : ↥S₁) : ↑((AffineEquiv.ofEq S₁ S₂ h) x) = ↑x
• AffineSubspace.linear_topEquiv 📋 Mathlib.LinearAlgebra.AffineSpace.AffineSubspace.Basic
  (k : Type u_1) (V : Type u_2) (P : Type u_3) [Ring k] [AddCommGroup V] [Module k V] [S : AddTorsor V P] : (AffineSubspace.topEquiv k V P).linear = (LinearEquiv.ofEq ⊤.direction ⊤ ⋯).trans Submodule.topEquiv
• AffineSubspace.topEquiv_apply 📋 Mathlib.LinearAlgebra.AffineSpace.AffineSubspace.Basic
  (k : Type u_1) (V : Type u_2) (P : Type u_3) [Ring k] [AddCommGroup V] [Module k V] [S : AddTorsor V P] (self : { x // x ∈ Set.univ }) : (AffineSubspace.topEquiv k V P) self = ↑self
• affineSpan_coe_preimage_eq_top 📋 Mathlib.LinearAlgebra.AffineSpace.AffineSubspace.Basic
  {k : Type u_1} {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [AddTorsor V P] (A : Set P) [Nonempty ↑A] : affineSpan k (Subtype.val ⁻¹' A) = ⊤
• AffineSubspace.topEquiv_symm_apply_coe 📋 Mathlib.LinearAlgebra.AffineSpace.AffineSubspace.Basic
  (k : Type u_1) (V : Type u_2) (P : Type u_3) [Ring k] [AddCommGroup V] [Module k V] [S : AddTorsor V P] (a : P) : ↑((AffineSubspace.topEquiv k V P).symm a) = a
• AffineSubspace.pointwise_vadd_direction 📋 Mathlib.LinearAlgebra.AffineSpace.Pointwise
  {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
• AffineSubspace.direction_smul 📋 Mathlib.LinearAlgebra.AffineSpace.Pointwise
  {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
• AffineMap.restrict.linear_aux 📋 Mathlib.LinearAlgebra.AffineSpace.Restrict
  {k : Type u_1} {V₁ : Type u_2} {P₁ : Type u_3} {V₂ : Type u_4} {P₂ : Type u_5} [Ring k] [AddCommGroup V₁] [AddCommGroup V₂] [Module k V₁] [Module k V₂] [AddTorsor V₁ P₁] [AddTorsor V₂ P₂] {φ : P₁ →ᵃ[k] P₂} {E : AffineSubspace k P₁} {F : AffineSubspace k P₂} (hEF : AffineSubspace.map φ E ≤ F) : E.direction ≤ Submodule.comap φ.linear F.direction
• AffineMap.restrict 📋 Mathlib.LinearAlgebra.AffineSpace.Restrict
  {k : Type u_1} {V₁ : Type u_2} {P₁ : Type u_3} {V₂ : Type u_4} {P₂ : Type u_5} [Ring k] [AddCommGroup V₁] [AddCommGroup V₂] [Module k V₁] [Module k V₂] [AddTorsor V₁ P₁] [AddTorsor V₂ P₂] (φ : P₁ →ᵃ[k] P₂) {E : AffineSubspace k P₁} {F : AffineSubspace k P₂} [Nonempty ↥E] [Nonempty ↥F] (hEF : AffineSubspace.map φ E ≤ F) : ↥E →ᵃ[k] ↥F
• AffineEquiv.affineSubspaceMap 📋 Mathlib.LinearAlgebra.AffineSpace.Restrict
  {k : Type u_1} {V₁ : Type u_2} {P₁ : Type u_3} {V₂ : Type u_4} {P₂ : Type u_5} [Ring k] [AddCommGroup V₁] [AddCommGroup V₂] [Module k V₁] [Module k V₂] [AddTorsor V₁ P₁] [AddTorsor V₂ P₂] (e : P₁ ≃ᵃ[k] P₂) (s : AffineSubspace k P₁) [Nonempty ↥s] : ↥s ≃ᵃ[k] ↥(AffineSubspace.map (↑e) s)
• AffineMap.restrict.linear 📋 Mathlib.LinearAlgebra.AffineSpace.Restrict
  {k : Type u_1} {V₁ : Type u_2} {P₁ : Type u_3} {V₂ : Type u_4} {P₂ : Type u_5} [Ring k] [AddCommGroup V₁] [AddCommGroup V₂] [Module k V₁] [Module k V₂] [AddTorsor V₁ P₁] [AddTorsor V₂ P₂] (φ : P₁ →ᵃ[k] P₂) {E : AffineSubspace k P₁} {F : AffineSubspace k P₂} [Nonempty ↥E] [Nonempty ↥F] (hEF : AffineSubspace.map φ E ≤ F) : (φ.restrict hEF).linear = φ.linear.restrict ⋯
• AffineMap.restrict.surjective 📋 Mathlib.LinearAlgebra.AffineSpace.Restrict
  {k : Type u_1} {V₁ : Type u_2} {P₁ : Type u_3} {V₂ : Type u_4} {P₂ : Type u_5} [Ring k] [AddCommGroup V₁] [AddCommGroup V₂] [Module k V₁] [Module k V₂] [AddTorsor V₁ P₁] [AddTorsor V₂ P₂] (φ : P₁ →ᵃ[k] P₂) {E : AffineSubspace k P₁} {F : AffineSubspace k P₂} [Nonempty ↥E] [Nonempty ↥F] (h : AffineSubspace.map φ E = F) : Function.Surjective ⇑(φ.restrict ⋯)
• AffineMap.restrict.injective 📋 Mathlib.LinearAlgebra.AffineSpace.Restrict
  {k : Type u_1} {V₁ : Type u_2} {P₁ : Type u_3} {V₂ : Type u_4} {P₂ : Type u_5} [Ring k] [AddCommGroup V₁] [AddCommGroup V₂] [Module k V₁] [Module k V₂] [AddTorsor V₁ P₁] [AddTorsor V₂ P₂] {φ : P₁ →ᵃ[k] P₂} (hφ : Function.Injective ⇑φ) {E : AffineSubspace k P₁} {F : AffineSubspace k P₂} [Nonempty ↥E] [Nonempty ↥F] (hEF : AffineSubspace.map φ E ≤ F) : Function.Injective ⇑(φ.restrict hEF)
• AffineMap.restrict.coe_apply 📋 Mathlib.LinearAlgebra.AffineSpace.Restrict
  {k : Type u_1} {V₁ : Type u_2} {P₁ : Type u_3} {V₂ : Type u_4} {P₂ : Type u_5} [Ring k] [AddCommGroup V₁] [AddCommGroup V₂] [Module k V₁] [Module k V₂] [AddTorsor V₁ P₁] [AddTorsor V₂ P₂] (φ : P₁ →ᵃ[k] P₂) {E : AffineSubspace k P₁} {F : AffineSubspace k P₂} [Nonempty ↥E] [Nonempty ↥F] (hEF : AffineSubspace.map φ E ≤ F) (x : ↥E) : ↑((φ.restrict hEF) x) = φ ↑x
• AffineMap.restrict.bijective 📋 Mathlib.LinearAlgebra.AffineSpace.Restrict
  {k : Type u_1} {V₁ : Type u_2} {P₁ : Type u_3} {V₂ : Type u_4} {P₂ : Type u_5} [Ring k] [AddCommGroup V₁] [AddCommGroup V₂] [Module k V₁] [Module k V₂] [AddTorsor V₁ P₁] [AddTorsor V₂ P₂] {E : AffineSubspace k P₁} [Nonempty ↥E] {φ : P₁ →ᵃ[k] P₂} (hφ : Function.Injective ⇑φ) : Function.Bijective ⇑(φ.restrict ⋯)
• AffineEquiv.affineSubspaceMap_apply 📋 Mathlib.LinearAlgebra.AffineSpace.Restrict
  {k : Type u_1} {V₁ : Type u_2} {P₁ : Type u_3} {V₂ : Type u_4} {P₂ : Type u_5} [Ring k] [AddCommGroup V₁] [AddCommGroup V₂] [Module k V₁] [Module k V₂] [AddTorsor V₁ P₁] [AddTorsor V₂ P₂] (e : P₁ ≃ᵃ[k] P₂) (s : AffineSubspace k P₁) [Nonempty ↥s] (x : ↥s) : ↑((e.affineSubspaceMap s) x) = e ↑x
• AffineEquiv.affineSubspaceMap_apply_symm_apply 📋 Mathlib.LinearAlgebra.AffineSpace.Restrict
  {k : Type u_1} {V₁ : Type u_2} {P₁ : Type u_3} {V₂ : Type u_4} {P₂ : Type u_5} [Ring k] [AddCommGroup V₁] [AddCommGroup V₂] [Module k V₁] [Module k V₂] [AddTorsor V₁ P₁] [AddTorsor V₂ P₂] (e : P₁ ≃ᵃ[k] P₂) (s : AffineSubspace k P₁) [Nonempty ↥s] (x : ↥(AffineSubspace.map (↑e) s)) : e ↑((e.affineSubspaceMap s).symm x) = ↑x
• Affine.Simplex.restrict 📋 Mathlib.LinearAlgebra.AffineSpace.Simplex.Basic
  {k : Type u_1} {V : Type u_2} {P : Type u_5} [Ring k] [AddCommGroup V] [Module k V] [AddTorsor V P] {n : ℕ} (s : Affine.Simplex k P n) (S : AffineSubspace k P) (hS : affineSpan k (Set.range s.points) ≤ S) : Affine.Simplex k (↥S) n
• Affine.Simplex.restrict_points_coe 📋 Mathlib.LinearAlgebra.AffineSpace.Simplex.Basic
  {k : Type u_1} {V : Type u_2} {P : Type u_5} [Ring k] [AddCommGroup V] [Module k V] [AddTorsor V P] {n : ℕ} (s : Affine.Simplex k P n) (S : AffineSubspace k P) (hS : affineSpan k (Set.range s.points) ≤ S) (i : Fin (n + 1)) : ↑((s.restrict S hS).points i) = s.points i
• Affine.Simplex.face_restrict 📋 Mathlib.LinearAlgebra.AffineSpace.Simplex.Basic
  {k : Type u_1} {V : Type u_2} {P : Type u_5} [Ring k] [AddCommGroup V] [Module k V] [AddTorsor V P] {n : ℕ} (s : Affine.Simplex k P n) {S : AffineSubspace k P} (hS : affineSpan k (Set.range s.points) ≤ S) {fs : Finset (Fin (n + 1))} {m : ℕ} (h : fs.card = m + 1) : (s.restrict S hS).face h = (s.face h).restrict S ⋯
• Affine.Simplex.faceOpposite_restrict 📋 Mathlib.LinearAlgebra.AffineSpace.Simplex.Basic
  {k : Type u_1} {V : Type u_2} {P : Type u_5} [Ring k] [AddCommGroup V] [Module k V] [AddTorsor V P] {n : ℕ} [NeZero n] (s : Affine.Simplex k P n) {S : AffineSubspace k P} (hS : affineSpan k (Set.range s.points) ≤ S) (i : Fin (n + 1)) : (s.restrict S hS).faceOpposite i = (s.faceOpposite i).restrict S ⋯
• Affine.Simplex.restrict_reindex 📋 Mathlib.LinearAlgebra.AffineSpace.Simplex.Basic
  {k : Type u_1} {V : Type u_2} {P : Type u_5} [Ring k] [AddCommGroup V] [Module k V] [AddTorsor V P] {m n : ℕ} (s : Affine.Simplex k P n) (e : Fin (n + 1) ≃ Fin (m + 1)) {S : AffineSubspace k P} (hS : affineSpan k (Set.range s.points) ≤ S) : (s.reindex e).restrict S ⋯ = (s.restrict S hS).reindex e
• Affine.Simplex.restrict_map_subtype 📋 Mathlib.LinearAlgebra.AffineSpace.Simplex.Basic
  {k : Type u_1} {V : Type u_2} {P : Type u_5} [Ring k] [AddCommGroup V] [Module k V] [AddTorsor V P] {n : ℕ} (s : Affine.Simplex k P n) : (s.restrict (affineSpan k (Set.range s.points)) ⋯).map (affineSpan k (Set.range s.points)).subtype ⋯ = s
• Affine.Simplex.closedInterior_restrict 📋 Mathlib.LinearAlgebra.AffineSpace.Simplex.Basic
  {k : Type u_1} {V : Type u_2} {P : Type u_4} [Ring k] [AddCommGroup V] [Module k V] [AddTorsor V P] [PartialOrder k] {n : ℕ} (s : Affine.Simplex k P n) {S : AffineSubspace k P} (hS : affineSpan k (Set.range s.points) ≤ S) : (s.restrict S hS).closedInterior = ⇑S.subtype ⁻¹' s.closedInterior
• Affine.Simplex.interior_restrict 📋 Mathlib.LinearAlgebra.AffineSpace.Simplex.Basic
  {k : Type u_1} {V : Type u_2} {P : Type u_4} [Ring k] [AddCommGroup V] [Module k V] [AddTorsor V P] [PartialOrder k] {n : ℕ} (s : Affine.Simplex k P n) {S : AffineSubspace k P} (hS : affineSpan k (Set.range s.points) ≤ S) : (s.restrict S hS).interior = ⇑S.subtype ⁻¹' s.interior
• Affine.Simplex.setInterior_restrict 📋 Mathlib.LinearAlgebra.AffineSpace.Simplex.Basic
  {k : Type u_1} {V : Type u_2} {P : Type u_4} [Ring k] [AddCommGroup V] [Module k V] [AddTorsor V P] (I : Set k) {n : ℕ} (s : Affine.Simplex k P n) {S : AffineSubspace k P} (hS : affineSpan k (Set.range s.points) ≤ S) : Affine.Simplex.setInterior I (s.restrict S hS) = ⇑S.subtype ⁻¹' Affine.Simplex.setInterior I s
• Affine.Simplex.restrict_map_inclusion 📋 Mathlib.LinearAlgebra.AffineSpace.Simplex.Basic
  {k : Type u_1} {V : Type u_2} {P : Type u_5} [Ring k] [AddCommGroup V] [Module k V] [AddTorsor V P] {n : ℕ} (s : Affine.Simplex k P n) (S₁ S₂ : AffineSubspace k P) (hS₁ : affineSpan k (Set.range s.points) ≤ S₁) (hS₂ : S₁ ≤ S₂) : (s.restrict S₁ hS₁).map (AffineSubspace.inclusion hS₂) ⋯ = s.restrict S₂ ⋯
• Affine.Simplex.map_subtype_restrict 📋 Mathlib.LinearAlgebra.AffineSpace.Simplex.Basic
  {k : Type u_1} {V : Type u_2} {P : Type u_5} [Ring k] [AddCommGroup V] [Module k V] [AddTorsor V P] {n : ℕ} (S : AffineSubspace k P) [Nonempty ↥S] (s : Affine.Simplex k (↥S) n) : (s.map S.subtype ⋯).restrict S ⋯ = s
• Affine.Simplex.restrict_map_restrict 📋 Mathlib.LinearAlgebra.AffineSpace.Simplex.Basic
  {k : Type u_1} {V : Type u_2} {V₂ : Type u_3} {P : Type u_5} {P₂ : Type u_6} [Ring k] [AddCommGroup V] [AddCommGroup V₂] [Module k V] [Module k V₂] [AddTorsor V P] [AddTorsor V₂ P₂] {n : ℕ} (s : Affine.Simplex k P n) (f : P →ᵃ[k] P₂) (hf : Function.Injective ⇑f) (S₁ : AffineSubspace k P) (S₂ : AffineSubspace k P₂) (hS₁ : affineSpan k (Set.range s.points) ≤ S₁) (hfS : AffineSubspace.map f S₁ ≤ S₂) : (s.restrict S₁ hS₁).map (f.restrict hfS) ⋯ = (s.map f hf).restrict S₂ ⋯
• AffineSubspace.toNormedAddTorsor 📋 Mathlib.Analysis.Normed.Group.AddTorsor
  {V : Type u_2} {P : Type u_3} [SeminormedAddCommGroup V] [PseudoMetricSpace P] [NormedAddTorsor V P] {R : Type u_6} [Ring R] [Module R V] (s : AffineSubspace R P) [Nonempty ↥s] : NormedAddTorsor ↥s.direction ↥s
• AffineSubspace.isClosed_direction_iff 📋 Mathlib.Topology.Algebra.AffineSubspace
  {R : Type u_1} {V : Type u_2} {P : Type u_3} [Ring R] [AddCommGroup V] [Module R V] [TopologicalSpace P] [AddTorsor V P] [TopologicalSpace V] [IsTopologicalAddTorsor P] [T1Space V] (s : AffineSubspace R P) : IsClosed ↑s.direction ↔ IsClosed ↑s
• AffineSubspace.subtypeA 📋 Mathlib.Topology.Algebra.AffineSubspace
  {R : Type u_1} {V : Type u_2} {P : Type u_3} [Ring R] [AddCommGroup V] [Module R V] [TopologicalSpace P] [AddTorsor V P] (s : AffineSubspace R P) [Nonempty ↥s] : ↥s →ᴬ[R] P
• AffineSubspace.instIsTopologicalAddTorsorSubtypeMemSubmoduleDirection 📋 Mathlib.Topology.Algebra.AffineSubspace
  {R : Type u_1} {V : Type u_2} {P : Type u_3} [Ring R] [AddCommGroup V] [Module R V] [TopologicalSpace P] [AddTorsor V P] [TopologicalSpace V] [IsTopologicalAddTorsor P] {s : AffineSubspace R P} [Nonempty ↥s] : IsTopologicalAddTorsor ↥s
• AffineSubspace.subtypeA_toAffineMap 📋 Mathlib.Topology.Algebra.AffineSubspace
  {R : Type u_1} {V : Type u_2} {P : Type u_3} [Ring R] [AddCommGroup V] [Module R V] [TopologicalSpace P] [AddTorsor V P] (s : AffineSubspace R P) [Nonempty ↥s] : ↑s.subtypeA = s.subtype
• AffineSubspace.ofEq 📋 Mathlib.Topology.Algebra.AffineSubspace
  {R : Type u_1} {V : Type u_2} {P : Type u_3} [Ring R] [AddCommGroup V] [Module R V] [TopologicalSpace P] [AddTorsor V P] {s t : AffineSubspace R P} [Nonempty ↥s] [Nonempty ↥t] (h : s = t) : ↥s ≃ᴬ[R] ↥t
• AffineSubspace.coe_subtypeA 📋 Mathlib.Topology.Algebra.AffineSubspace
  {R : Type u_1} {V : Type u_2} {P : Type u_3} [Ring R] [AddCommGroup V] [Module R V] [TopologicalSpace P] [AddTorsor V P] (s : AffineSubspace R P) [Nonempty ↥s] : ⇑s.subtypeA = Subtype.val
• ContinuousAffineEquiv.affineSubspaceMap 📋 Mathlib.Topology.Algebra.AffineSubspace
  {R : Type u_1} {V : Type u_2} {P : Type u_3} {W : Type u_4} {Q : Type u_5} [Ring R] [AddCommGroup V] [Module R V] [TopologicalSpace P] [AddTorsor V P] [AddCommGroup W] [Module R W] [TopologicalSpace Q] [AddTorsor W Q] (e : P ≃ᴬ[R] Q) (s : AffineSubspace R P) [Nonempty ↥s] : ↥s ≃ᴬ[R] ↥(AffineSubspace.map (↑↑e) s)
• AffineSubspace.coe_ofEq_apply 📋 Mathlib.Topology.Algebra.AffineSubspace
  {R : Type u_1} {V : Type u_2} {P : Type u_3} [Ring R] [AddCommGroup V] [Module R V] [TopologicalSpace P] [AddTorsor V P] {s t : AffineSubspace R P} [Nonempty ↥s] [Nonempty ↥t] (h : s = t) (x : ↥s) : ↑((AffineSubspace.ofEq h) x) = ↑x
• ContinuousAffineEquiv.affineSubspaceMap_apply 📋 Mathlib.Topology.Algebra.AffineSubspace
  {R : Type u_1} {V : Type u_2} {P : Type u_3} {W : Type u_4} {Q : Type u_5} [Ring R] [AddCommGroup V] [Module R V] [TopologicalSpace P] [AddTorsor V P] [AddCommGroup W] [Module R W] [TopologicalSpace Q] [AddTorsor W Q] (e : P ≃ᴬ[R] Q) (s : AffineSubspace R P) [Nonempty ↥s] (x : ↥s) : ↑((e.affineSubspaceMap s) x) = e ↑x
• ContinuousAffineEquiv.affineSubspaceMap_apply_symm_apply 📋 Mathlib.Topology.Algebra.AffineSubspace
  {R : Type u_1} {V : Type u_2} {P : Type u_3} {W : Type u_4} {Q : Type u_5} [Ring R] [AddCommGroup V] [Module R V] [TopologicalSpace P] [AddTorsor V P] [AddCommGroup W] [Module R W] [TopologicalSpace Q] [AddTorsor W Q] (e : P ≃ᴬ[R] Q) (s : AffineSubspace R P) [Nonempty ↥s] (x : ↥(AffineSubspace.map (↑↑e) s)) : e ↑((e.affineSubspaceMap s).symm x) = ↑x
• AffineSubspace.subtypeₐᵢ 📋 Mathlib.Analysis.Normed.Affine.Isometry
  {𝕜 : Type u_1} {V : Type u_2} {P : Type u_10} [NormedField 𝕜] [SeminormedAddCommGroup V] [NormedSpace 𝕜 V] [PseudoMetricSpace P] [NormedAddTorsor V P] (s : AffineSubspace 𝕜 P) [Nonempty ↥s] : ↥s →ᵃⁱ[𝕜] P
• AffineIsometryEquiv.ofTop 📋 Mathlib.Analysis.Normed.Affine.Isometry
  {𝕜 : Type u_1} {V : Type u_2} {P : Type u_10} [NormedField 𝕜] [SeminormedAddCommGroup V] [NormedSpace 𝕜 V] [PseudoMetricSpace P] [NormedAddTorsor V P] (s₁ : AffineSubspace 𝕜 P) [Nonempty ↥s₁] (h : s₁ = ⊤) : ↥s₁ ≃ᵃⁱ[𝕜] P
• AffineSubspace.subtypeₐᵢ_linearIsometry 📋 Mathlib.Analysis.Normed.Affine.Isometry
  {𝕜 : Type u_1} {V : Type u_2} {P : Type u_10} [NormedField 𝕜] [SeminormedAddCommGroup V] [NormedSpace 𝕜 V] [PseudoMetricSpace P] [NormedAddTorsor V P] (s : AffineSubspace 𝕜 P) [Nonempty ↥s] : s.subtypeₐᵢ.linearIsometry = s.direction.subtypeₗᵢ
• AffineSubspace.equivMapOfInjective 📋 Mathlib.Analysis.Normed.Affine.Isometry
  {𝕜 : Type u_1} {V₁ : Type u_3} {V₂ : Type u_5} {P₁ : Type u_8} {P₂ : Type u_11} [NormedField 𝕜] [SeminormedAddCommGroup V₁] [NormedSpace 𝕜 V₁] [PseudoMetricSpace P₁] [NormedAddTorsor V₁ P₁] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V₂] [PseudoMetricSpace P₂] [NormedAddTorsor V₂ P₂] (E : AffineSubspace 𝕜 P₁) [Nonempty ↥E] (φ : P₁ →ᵃ[𝕜] P₂) (hφ : Function.Injective ⇑φ) : ↥E ≃ᵃ[𝕜] ↥(AffineSubspace.map φ E)
• AffineIsometryEquiv.ofEq 📋 Mathlib.Analysis.Normed.Affine.Isometry
  {𝕜 : Type u_1} {V : Type u_2} {P : Type u_10} [NormedField 𝕜] [SeminormedAddCommGroup V] [NormedSpace 𝕜 V] [PseudoMetricSpace P] [NormedAddTorsor V P] (s₁ s₂ : AffineSubspace 𝕜 P) [Nonempty ↥s₁] [Nonempty ↥s₂] (h : s₁ = s₂) : ↥s₁ ≃ᵃⁱ[𝕜] ↥s₂
• AffineSubspace.isometryEquivMap 📋 Mathlib.Analysis.Normed.Affine.Isometry
  {𝕜 : Type u_1} {V₁' : Type u_4} {V₂ : Type u_5} {P₁' : Type u_9} {P₂ : Type u_11} [NormedField 𝕜] [SeminormedAddCommGroup V₁'] [NormedSpace 𝕜 V₁'] [MetricSpace P₁'] [NormedAddTorsor V₁' P₁'] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V₂] [PseudoMetricSpace P₂] [NormedAddTorsor V₂ P₂] (φ : P₁' →ᵃⁱ[𝕜] P₂) (E : AffineSubspace 𝕜 P₁') [Nonempty ↥E] : ↥E ≃ᵃⁱ[𝕜] ↥(AffineSubspace.map φ.toAffineMap E)
• AffineIsometryEquiv.ofEq_rfl 📋 Mathlib.Analysis.Normed.Affine.Isometry
  {𝕜 : Type u_1} {V : Type u_2} {P : Type u_10} [NormedField 𝕜] [SeminormedAddCommGroup V] [NormedSpace 𝕜 V] [PseudoMetricSpace P] [NormedAddTorsor V P] {s₁ : AffineSubspace 𝕜 P} [Nonempty ↥s₁] : AffineIsometryEquiv.ofEq s₁ s₁ ⋯ = AffineIsometryEquiv.refl 𝕜 ↥s₁
• AffineSubspace.subtypeₐᵢ_toAffineMap 📋 Mathlib.Analysis.Normed.Affine.Isometry
  {𝕜 : Type u_1} {V : Type u_2} {P : Type u_10} [NormedField 𝕜] [SeminormedAddCommGroup V] [NormedSpace 𝕜 V] [PseudoMetricSpace P] [NormedAddTorsor V P] (s : AffineSubspace 𝕜 P) [Nonempty ↥s] : s.subtypeₐᵢ.toAffineMap = s.subtype
• AffineSubspace.toContinuousAffineMap_subtypeₐᵢ 📋 Mathlib.Analysis.Normed.Affine.Isometry
  {𝕜 : Type u_1} {V : Type u_2} {P : Type u_10} [NormedField 𝕜] [SeminormedAddCommGroup V] [NormedSpace 𝕜 V] [PseudoMetricSpace P] [NormedAddTorsor V P] (s : AffineSubspace 𝕜 P) [Nonempty ↥s] : s.subtypeₐᵢ.toContinuousAffineMap = s.subtypeA
• AffineIsometryEquiv.ofEq_symm 📋 Mathlib.Analysis.Normed.Affine.Isometry
  {𝕜 : Type u_1} {V : Type u_2} {P : Type u_10} [NormedField 𝕜] [SeminormedAddCommGroup V] [NormedSpace 𝕜 V] [PseudoMetricSpace P] [NormedAddTorsor V P] {s₁ s₂ : AffineSubspace 𝕜 P} [Nonempty ↥s₁] [Nonempty ↥s₂] (h : s₁ = s₂) : (AffineIsometryEquiv.ofEq s₁ s₂ h).symm = AffineIsometryEquiv.ofEq s₂ s₁ ⋯
• AffineSubspace.coe_subtypeₐᵢ 📋 Mathlib.Analysis.Normed.Affine.Isometry
  {𝕜 : Type u_1} {V : Type u_2} {P : Type u_10} [NormedField 𝕜] [SeminormedAddCommGroup V] [NormedSpace 𝕜 V] [PseudoMetricSpace P] [NormedAddTorsor V P] (s : AffineSubspace 𝕜 P) [Nonempty ↥s] : ⇑s.subtypeₐᵢ = ⇑s.subtype
• AffineSubspace.subtypeₐᵢ_linear 📋 Mathlib.Analysis.Normed.Affine.Isometry
  {𝕜 : Type u_1} {V : Type u_2} {P : Type u_10} [NormedField 𝕜] [SeminormedAddCommGroup V] [NormedSpace 𝕜 V] [PseudoMetricSpace P] [NormedAddTorsor V P] (s : AffineSubspace 𝕜 P) [Nonempty ↥s] : s.subtypeₐᵢ.linear = s.direction.subtype
• AffineIsometryEquiv.ofTop_apply 📋 Mathlib.Analysis.Normed.Affine.Isometry
  {𝕜 : Type u_1} {V : Type u_2} {P : Type u_10} [NormedField 𝕜] [SeminormedAddCommGroup V] [NormedSpace 𝕜 V] [PseudoMetricSpace P] [NormedAddTorsor V P] {s₁ : AffineSubspace 𝕜 P} [Nonempty ↥s₁] (h : s₁ = ⊤) (x : ↥s₁) : (AffineIsometryEquiv.ofTop s₁ h) x = ↑x
• AffineIsometryEquiv.ofTop_symm_apply_coe 📋 Mathlib.Analysis.Normed.Affine.Isometry
  {𝕜 : Type u_1} {V : Type u_2} {P : Type u_10} [NormedField 𝕜] [SeminormedAddCommGroup V] [NormedSpace 𝕜 V] [PseudoMetricSpace P] [NormedAddTorsor V P] {s₁ : AffineSubspace 𝕜 P} [Nonempty ↥s₁] (h : s₁ = ⊤) (x : P) : ↑((AffineIsometryEquiv.ofTop s₁ h).symm x) = x
• AffineSubspace.linear_equivMapOfInjective 📋 Mathlib.Analysis.Normed.Affine.Isometry
  {𝕜 : Type u_1} {V₁ : Type u_3} {V₂ : Type u_5} {P₁ : Type u_8} {P₂ : Type u_11} [NormedField 𝕜] [SeminormedAddCommGroup V₁] [NormedSpace 𝕜 V₁] [PseudoMetricSpace P₁] [NormedAddTorsor V₁ P₁] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V₂] [PseudoMetricSpace P₂] [NormedAddTorsor V₂ P₂] (E : AffineSubspace 𝕜 P₁) [Nonempty ↥E] (φ : P₁ →ᵃ[𝕜] P₂) (hφ : Function.Injective ⇑φ) : (E.equivMapOfInjective φ hφ).linear = (Submodule.equivMapOfInjective φ.linear ⋯ E.direction).trans (LinearEquiv.ofEq (Submodule.map φ.linear E.direction) (AffineSubspace.map φ E).direction ⋯)
• AffineIsometryEquiv.coe_ofEq_apply 📋 Mathlib.Analysis.Normed.Affine.Isometry
  {𝕜 : Type u_1} {V : Type u_2} {P : Type u_10} [NormedField 𝕜] [SeminormedAddCommGroup V] [NormedSpace 𝕜 V] [PseudoMetricSpace P] [NormedAddTorsor V P] {s₁ s₂ : AffineSubspace 𝕜 P} [Nonempty ↥s₁] [Nonempty ↥s₂] (h : s₁ = s₂) (x : ↥s₁) : ↑((AffineIsometryEquiv.ofEq s₁ s₂ h) x) = ↑x
• AffineSubspace.equivMapOfInjective_toFun 📋 Mathlib.Analysis.Normed.Affine.Isometry
  {𝕜 : Type u_1} {V₁ : Type u_3} {V₂ : Type u_5} {P₁ : Type u_8} {P₂ : Type u_11} [NormedField 𝕜] [SeminormedAddCommGroup V₁] [NormedSpace 𝕜 V₁] [PseudoMetricSpace P₁] [NormedAddTorsor V₁ P₁] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V₂] [PseudoMetricSpace P₂] [NormedAddTorsor V₂ P₂] (E : AffineSubspace 𝕜 P₁) [Nonempty ↥E] (φ : P₁ →ᵃ[𝕜] P₂) (hφ : Function.Injective ⇑φ) (p : ↑↑E) : (E.equivMapOfInjective φ hφ) p = ⟨φ ↑p, ⋯⟩
• AffineSubspace.isometryEquivMap.coe_apply 📋 Mathlib.Analysis.Normed.Affine.Isometry
  {𝕜 : Type u_1} {V₁' : Type u_4} {V₂ : Type u_5} {P₁' : Type u_9} {P₂ : Type u_11} [NormedField 𝕜] [SeminormedAddCommGroup V₁'] [NormedSpace 𝕜 V₁'] [MetricSpace P₁'] [NormedAddTorsor V₁' P₁'] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V₂] [PseudoMetricSpace P₂] [NormedAddTorsor V₂ P₂] (φ : P₁' →ᵃⁱ[𝕜] P₂) (E : AffineSubspace 𝕜 P₁') [Nonempty ↥E] (g : ↥E) : ↑((AffineSubspace.isometryEquivMap φ E) g) = φ ↑g
• AffineSubspace.isometryEquivMap.apply_symm_apply 📋 Mathlib.Analysis.Normed.Affine.Isometry
  {𝕜 : Type u_1} {V₁' : Type u_4} {V₂ : Type u_5} {P₁' : Type u_9} {P₂ : Type u_11} [NormedField 𝕜] [SeminormedAddCommGroup V₁'] [NormedSpace 𝕜 V₁'] [MetricSpace P₁'] [NormedAddTorsor V₁' P₁'] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V₂] [PseudoMetricSpace P₂] [NormedAddTorsor V₂ P₂] {E : AffineSubspace 𝕜 P₁'} [Nonempty ↥E] {φ : P₁' →ᵃⁱ[𝕜] P₂} (x : ↥(AffineSubspace.map φ.toAffineMap E)) : φ ↑((AffineSubspace.isometryEquivMap φ E).symm x) = ↑x
• AffineSubspace.isometryEquivMap.toAffineMap_eq 📋 Mathlib.Analysis.Normed.Affine.Isometry
  {𝕜 : Type u_1} {V₁' : Type u_4} {V₂ : Type u_5} {P₁' : Type u_9} {P₂ : Type u_11} [NormedField 𝕜] [SeminormedAddCommGroup V₁'] [NormedSpace 𝕜 V₁'] [MetricSpace P₁'] [NormedAddTorsor V₁' P₁'] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V₂] [PseudoMetricSpace P₂] [NormedAddTorsor V₂ P₂] (φ : P₁' →ᵃⁱ[𝕜] P₂) (E : AffineSubspace 𝕜 P₁') [Nonempty ↥E] : ↑(AffineSubspace.isometryEquivMap φ E).toAffineEquiv = ↑(E.equivMapOfInjective φ.toAffineMap ⋯)
• AffineSubspace.closed_of_finiteDimensional 📋 Mathlib.Analysis.Normed.Module.FiniteDimension
  {𝕜 : Type u} [NontriviallyNormedField 𝕜] {E : Type v} [NormedAddCommGroup E] [NormedSpace 𝕜 E] [CompleteSpace 𝕜] {P : Type u_1} [MetricSpace P] [NormedAddTorsor E P] (s : AffineSubspace 𝕜 P) [FiniteDimensional 𝕜 ↥s.direction] : IsClosed ↑s
• Affine.Simplex.centroid_restrict 📋 Mathlib.LinearAlgebra.AffineSpace.Simplex.Centroid
  {k : Type u_1} {V : Type u_2} {P : Type u_3} [DivisionRing k] [AddCommGroup V] [Module k V] [AddTorsor V P] [CharZero k] {n : ℕ} (s : Affine.Simplex k P n) (S : AffineSubspace k P) (hS : affineSpan k (Set.range s.points) ≤ S) : ↑(s.restrict S hS).centroid = s.centroid
• Affine.Simplex.faceOppositeCentroid_restrict 📋 Mathlib.LinearAlgebra.AffineSpace.Simplex.Centroid
  {k : Type u_1} {V : Type u_2} {P : Type u_3} [DivisionRing k] [AddCommGroup V] [Module k V] [AddTorsor V P] {n : ℕ} [NeZero n] [CharZero k] (s : Affine.Simplex k P n) (S : AffineSubspace k P) (hS : affineSpan k (Set.range s.points) ≤ S) {i : Fin (n + 1)} : ↑((s.restrict S hS).faceOppositeCentroid i) = s.faceOppositeCentroid i
• Affine.Simplex.medial_restrict 📋 Mathlib.LinearAlgebra.AffineSpace.Simplex.Centroid
  {k : Type u_1} {V : Type u_2} {P : Type u_3} [DivisionRing k] [AddCommGroup V] [Module k V] [AddTorsor V P] {n : ℕ} [NeZero n] [CharZero k] (s : Affine.Simplex k P n) (S : AffineSubspace k P) (hS : affineSpan k (Set.range s.points) ≤ S) : (s.restrict S hS).medial = s.medial.restrict S ⋯
• Affine.Simplex.median_restrict 📋 Mathlib.LinearAlgebra.AffineSpace.Simplex.Centroid
  {k : Type u_1} {V : Type u_2} {P : Type u_3} [DivisionRing k] [AddCommGroup V] [Module k V] [AddTorsor V P] {n : ℕ} [NeZero n] [CharZero k] (s : Affine.Simplex k P n) (i : Fin (n + 1)) (S : AffineSubspace k P) (hS : affineSpan k (Set.range s.points) ≤ S) : AffineSubspace.map S.subtype ((s.restrict S hS).median i) = s.median i
• finiteDimensional_direction_affineSpan_of_finite 📋 Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional
  (k : Type u_1) {V : Type u_2} {P : Type u_3} [DivisionRing k] [AddCommGroup V] [Module k V] [AddTorsor V P] {s : Set P} (h : s.Finite) : FiniteDimensional k ↥(affineSpan k s).direction
• Collinear.finiteDimensional_direction_affineSpan 📋 Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional
  {k : Type u_1} {V : Type u_2} {P : Type u_3} [DivisionRing k] [AddCommGroup V] [Module k V] [AddTorsor V P] {s : Set P} (h : Collinear k s) : FiniteDimensional k ↥(affineSpan k s).direction
• Coplanar.finiteDimensional_direction_affineSpan 📋 Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional
  {k : Type u_1} {V : Type u_2} {P : Type u_3} [DivisionRing k] [AddCommGroup V] [Module k V] [AddTorsor V P] {s : Set P} (h : Coplanar k s) : FiniteDimensional k ↥(affineSpan k s).direction
• finiteDimensional_direction_affineSpan_range 📋 Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional
  (k : Type u_1) {V : Type u_2} {P : Type u_3} {ι : Type u_4} [DivisionRing k] [AddCommGroup V] [Module k V] [AddTorsor V P] [Finite ι] (p : ι → P) : FiniteDimensional k ↥(affineSpan k (Set.range p)).direction
• finiteDimensional_direction_affineSpan_image_of_finite 📋 Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional
  (k : Type u_1) {V : Type u_2} {P : Type u_3} {ι : Type u_4} [DivisionRing k] [AddCommGroup V] [Module k V] [AddTorsor V P] [Finite ι] (p : ι → P) (s : Set ι) : FiniteDimensional k ↥(affineSpan k (p '' s)).direction
• finiteDimensional_direction_affineSpan_singleton 📋 Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional
  (k : Type u_1) {V : Type u_2} {P : Type u_3} [DivisionRing k] [AddCommGroup V] [Module k V] [AddTorsor V P] (p : P) : FiniteDimensional k ↥(affineSpan k {p}).direction
• finiteDimensional_direction_singleton 📋 Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional
  (k : Type u_1) {V : Type u_2} {P : Type u_3} [DivisionRing k] [AddCommGroup V] [Module k V] [AddTorsor V P] (p : P) : FiniteDimensional k ↥{p}.direction
• Affine.Simplex.fact_finrank_direction_affineSpan_eq 📋 Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional
  {k : Type u_1} {V : Type u_2} {P : Type u_3} [DivisionRing k] [AddCommGroup V] [Module k V] [AddTorsor V P] {n : ℕ} {s : Affine.Simplex k P n} : Fact (Module.finrank k ↥(affineSpan k (Set.range s.points)).direction = n)
• finiteDimensional_direction_affineSpan_insert_set 📋 Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional
  (k : Type u_1) {V : Type u_2} {P : Type u_3} [DivisionRing k] [AddCommGroup V] [Module k V] [AddTorsor V P] (s : Set P) [FiniteDimensional k ↥(affineSpan k s).direction] (p : P) : FiniteDimensional k ↥(affineSpan k (insert p s)).direction
• finiteDimensional_direction_map 📋 Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional
  {k : Type u_1} {V : Type u_2} {P : Type u_3} [DivisionRing k] [AddCommGroup V] [Module k V] [AddTorsor V P] {V₂ : Type u_5} {P₂ : Type u_6} [AddCommGroup V₂] [Module k V₂] [AddTorsor V₂ P₂] (s : AffineSubspace k P) [FiniteDimensional k ↥s.direction] (f : P →ᵃ[k] P₂) : FiniteDimensional k ↥(AffineSubspace.map f s).direction
• finiteDimensional_vectorSpan_insert 📋 Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional
  {k : Type u_1} {V : Type u_2} {P : Type u_3} [DivisionRing k] [AddCommGroup V] [Module k V] [AddTorsor V P] (s : AffineSubspace k P) [FiniteDimensional k ↥s.direction] (p : P) : FiniteDimensional k ↥(vectorSpan k (insert p ↑s))
• finiteDimensional_direction_affineSpan_insert 📋 Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional
  {k : Type u_1} {V : Type u_2} {P : Type u_3} [DivisionRing k] [AddCommGroup V] [Module k V] [AddTorsor V P] (s : AffineSubspace k P) [FiniteDimensional k ↥s.direction] (p : P) : FiniteDimensional k ↥(affineSpan k (insert p ↑s)).direction
• finrank_vectorSpan_insert_le 📋 Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional
  {k : Type u_1} {V : Type u_2} {P : Type u_3} [DivisionRing k] [AddCommGroup V] [Module k V] [AddTorsor V P] (s : AffineSubspace k P) (p : P) : Module.finrank k ↥(vectorSpan k (insert p ↑s)) ≤ Module.finrank k ↥s.direction + 1
• AffineIndependent.affineSpan_eq_of_le_of_card_eq_finrank_add_one 📋 Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional
  {k : Type u_1} {V : Type u_2} {P : Type u_3} {ι : Type u_4} [DivisionRing k] [AddCommGroup V] [Module k V] [AddTorsor V P] [Fintype ι] {p : ι → P} (hi : AffineIndependent k p) {sp : AffineSubspace k P} [FiniteDimensional k ↥sp.direction] (hle : affineSpan k (Set.range p) ≤ sp) (hc : Fintype.card ι = Module.finrank k ↥sp.direction + 1) : affineSpan k (Set.range p) = sp
• AffineIndependent.affineSpan_image_finset_eq_of_le_of_card_eq_finrank_add_one 📋 Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional
  {k : Type u_1} {V : Type u_2} {P : Type u_3} {ι : Type u_4} [DivisionRing k] [AddCommGroup V] [Module k V] [AddTorsor V P] [DecidableEq P] {p : ι → P} (hi : AffineIndependent k p) {s : Finset ι} {sp : AffineSubspace k P} [FiniteDimensional k ↥sp.direction] (hle : affineSpan k ↑(Finset.image p s) ≤ sp) (hc : s.card = Module.finrank k ↥sp.direction + 1) : affineSpan k ↑(Finset.image p s) = sp
• AffineSubspace.finiteDimensional_sup 📋 Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional
  {k : Type u_1} {V : Type u_2} {P : Type u_3} [DivisionRing k] [AddCommGroup V] [Module k V] [AddTorsor V P] (s₁ s₂ : AffineSubspace k P) [FiniteDimensional k ↥s₁.direction] [FiniteDimensional k ↥s₂.direction] : FiniteDimensional k ↥(s₁ ⊔ s₂).direction
• AffineSubspace.sOppSide_vadd_left_iff 📋 Mathlib.Analysis.Convex.Side
  {R : Type u_1} {V : Type u_2} {P : Type u_4} [CommRing R] [PartialOrder R] [IsStrictOrderedRing R] [AddCommGroup V] [Module R V] [AddTorsor V P] {s : AffineSubspace R P} {x y : P} {v : V} (hv : v ∈ s.direction) : s.SOppSide (v +ᵥ x) y ↔ s.SOppSide x y
• AffineSubspace.sOppSide_vadd_right_iff 📋 Mathlib.Analysis.Convex.Side
  {R : Type u_1} {V : Type u_2} {P : Type u_4} [CommRing R] [PartialOrder R] [IsStrictOrderedRing R] [AddCommGroup V] [Module R V] [AddTorsor V P] {s : AffineSubspace R P} {x y : P} {v : V} (hv : v ∈ s.direction) : s.SOppSide x (v +ᵥ y) ↔ s.SOppSide x y
• AffineSubspace.sSameSide_vadd_left_iff 📋 Mathlib.Analysis.Convex.Side
  {R : Type u_1} {V : Type u_2} {P : Type u_4} [CommRing R] [PartialOrder R] [IsStrictOrderedRing R] [AddCommGroup V] [Module R V] [AddTorsor V P] {s : AffineSubspace R P} {x y : P} {v : V} (hv : v ∈ s.direction) : s.SSameSide (v +ᵥ x) y ↔ s.SSameSide x y
• AffineSubspace.sSameSide_vadd_right_iff 📋 Mathlib.Analysis.Convex.Side
  {R : Type u_1} {V : Type u_2} {P : Type u_4} [CommRing R] [PartialOrder R] [IsStrictOrderedRing R] [AddCommGroup V] [Module R V] [AddTorsor V P] {s : AffineSubspace R P} {x y : P} {v : V} (hv : v ∈ s.direction) : s.SSameSide x (v +ᵥ y) ↔ s.SSameSide x y
• AffineSubspace.wOppSide_vadd_left_iff 📋 Mathlib.Analysis.Convex.Side
  {R : Type u_1} {V : Type u_2} {P : Type u_4} [CommRing R] [PartialOrder R] [IsStrictOrderedRing R] [AddCommGroup V] [Module R V] [AddTorsor V P] {s : AffineSubspace R P} {x y : P} {v : V} (hv : v ∈ s.direction) : s.WOppSide (v +ᵥ x) y ↔ s.WOppSide x y
• AffineSubspace.wOppSide_vadd_right_iff 📋 Mathlib.Analysis.Convex.Side
  {R : Type u_1} {V : Type u_2} {P : Type u_4} [CommRing R] [PartialOrder R] [IsStrictOrderedRing R] [AddCommGroup V] [Module R V] [AddTorsor V P] {s : AffineSubspace R P} {x y : P} {v : V} (hv : v ∈ s.direction) : s.WOppSide x (v +ᵥ y) ↔ s.WOppSide x y
• AffineSubspace.wSameSide_vadd_left_iff 📋 Mathlib.Analysis.Convex.Side
  {R : Type u_1} {V : Type u_2} {P : Type u_4} [CommRing R] [PartialOrder R] [IsStrictOrderedRing R] [AddCommGroup V] [Module R V] [AddTorsor V P] {s : AffineSubspace R P} {x y : P} {v : V} (hv : v ∈ s.direction) : s.WSameSide (v +ᵥ x) y ↔ s.WSameSide x y
• AffineSubspace.wSameSide_vadd_right_iff 📋 Mathlib.Analysis.Convex.Side
  {R : Type u_1} {V : Type u_2} {P : Type u_4} [CommRing R] [PartialOrder R] [IsStrictOrderedRing R] [AddCommGroup V] [Module R V] [AddTorsor V P] {s : AffineSubspace R P} {x y : P} {v : V} (hv : v ∈ s.direction) : s.WSameSide x (v +ᵥ y) ↔ s.WSameSide x y
• AffineSubspace.euclideanHausdorffMeasure_eq_lintegral 📋 Mathlib.Geometry.Euclidean.Volume.Measure
  {V : Type u_3} {P : Type u_4} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [MeasurableSpace V] [BorelSpace V] [FiniteDimensional ℝ V] [MetricSpace P] [MeasurableSpace P] [BorelSpace P] [NormedAddTorsor V P] (s : AffineSubspace ℝ P) [hs : Nonempty ↥s] {t : Set P} (ht : MeasurableSet t) : (MeasureTheory.Measure.euclideanHausdorffMeasure (Module.finrank ℝ V)) t = ∫⁻ (x : ↥s), (MeasureTheory.Measure.euclideanHausdorffMeasure (Module.finrank ℝ ↥s.directionᗮ)) (t ∩ ↑(AffineSubspace.mk' (↑x) s.directionᗮ)) ∂MeasureTheory.Measure.euclideanHausdorffMeasure (Module.finrank ℝ ↥s.direction)
• asymptoticCone_affineSubspace 📋 Mathlib.Topology.Algebra.AsymptoticCone
  {k : Type u_1} {V : Type u_2} {P : Type u_3} [Field k] [LinearOrder k] [AddCommGroup V] [Module k V] [AddTorsor V P] [TopologicalSpace V] [TopologicalSpace k] [OrderTopology k] [IsStrictOrderedRing k] [IsTopologicalAddGroup V] [ContinuousSMul k V] {s : AffineSubspace k P} (hs : (↑s).Nonempty) : asymptoticCone k ↑s = closure ↑s.direction
• EuclideanGeometry.reflection 📋 Mathlib.Geometry.Euclidean.Projection
  {𝕜 : Type u_1} {V : Type u_2} {P : Type u_3} [RCLike 𝕜] [NormedAddCommGroup V] [InnerProductSpace 𝕜 V] [MetricSpace P] [NormedAddTorsor V P] (s : AffineSubspace 𝕜 P) [Nonempty ↥s] [s.direction.HasOrthogonalProjection] : P ≃ᵃⁱ[𝕜] P
• EuclideanGeometry.reflection_symm 📋 Mathlib.Geometry.Euclidean.Projection
  {𝕜 : Type u_1} {V : Type u_2} {P : Type u_3} [RCLike 𝕜] [NormedAddCommGroup V] [InnerProductSpace 𝕜 V] [MetricSpace P] [NormedAddTorsor V P] (s : AffineSubspace 𝕜 P) [Nonempty ↥s] [s.direction.HasOrthogonalProjection] : (EuclideanGeometry.reflection s).symm = EuclideanGeometry.reflection s
• EuclideanGeometry.reflection_involutive 📋 Mathlib.Geometry.Euclidean.Projection
  {𝕜 : Type u_1} {V : Type u_2} {P : Type u_3} [RCLike 𝕜] [NormedAddCommGroup V] [InnerProductSpace 𝕜 V] [MetricSpace P] [NormedAddTorsor V P] (s : AffineSubspace 𝕜 P) [Nonempty ↥s] [s.direction.HasOrthogonalProjection] : Function.Involutive ⇑(EuclideanGeometry.reflection s)
• EuclideanGeometry.reflection_reflection 📋 Mathlib.Geometry.Euclidean.Projection
  {𝕜 : Type u_1} {V : Type u_2} {P : Type u_3} [RCLike 𝕜] [NormedAddCommGroup V] [InnerProductSpace 𝕜 V] [MetricSpace P] [NormedAddTorsor V P] (s : AffineSubspace 𝕜 P) [Nonempty ↥s] [s.direction.HasOrthogonalProjection] (p : P) : (EuclideanGeometry.reflection s) ((EuclideanGeometry.reflection s) p) = p
• EuclideanGeometry.reflection_eq_self_iff 📋 Mathlib.Geometry.Euclidean.Projection
  {𝕜 : Type u_1} {V : Type u_2} {P : Type u_3} [RCLike 𝕜] [NormedAddCommGroup V] [InnerProductSpace 𝕜 V] [MetricSpace P] [NormedAddTorsor V P] {s : AffineSubspace 𝕜 P} [Nonempty ↥s] [s.direction.HasOrthogonalProjection] (p : P) : (EuclideanGeometry.reflection s) p = p ↔ p ∈ s
• EuclideanGeometry.dist_reflection 📋 Mathlib.Geometry.Euclidean.Projection
  {𝕜 : Type u_1} {V : Type u_2} {P : Type u_3} [RCLike 𝕜] [NormedAddCommGroup V] [InnerProductSpace 𝕜 V] [MetricSpace P] [NormedAddTorsor V P] (s : AffineSubspace 𝕜 P) [Nonempty ↥s] [s.direction.HasOrthogonalProjection] (p₁ p₂ : P) : dist p₁ ((EuclideanGeometry.reflection s) p₂) = dist ((EuclideanGeometry.reflection s) p₁) p₂
• EuclideanGeometry.dist_reflection_eq_of_mem 📋 Mathlib.Geometry.Euclidean.Projection
  {𝕜 : Type u_1} {V : Type u_2} {P : Type u_3} [RCLike 𝕜] [NormedAddCommGroup V] [InnerProductSpace 𝕜 V] [MetricSpace P] [NormedAddTorsor V P] (s : AffineSubspace 𝕜 P) [Nonempty ↥s] [s.direction.HasOrthogonalProjection] {p₁ : P} (hp₁ : p₁ ∈ s) (p₂ : P) : dist p₁ ((EuclideanGeometry.reflection s) p₂) = dist p₁ p₂
• EuclideanGeometry.eq_reflection_of_eq_subspace 📋 Mathlib.Geometry.Euclidean.Projection
  {𝕜 : Type u_1} {V : Type u_2} {P : Type u_3} [RCLike 𝕜] [NormedAddCommGroup V] [InnerProductSpace 𝕜 V] [MetricSpace P] [NormedAddTorsor V P] {s s' : AffineSubspace 𝕜 P} [Nonempty ↥s] [Nonempty ↥s'] [s.direction.HasOrthogonalProjection] [s'.direction.HasOrthogonalProjection] (h : s = s') (p : P) : (EuclideanGeometry.reflection s) p = (EuclideanGeometry.reflection s') p
• EuclideanGeometry.reflection_orthogonal_vadd 📋 Mathlib.Geometry.Euclidean.Projection
  {𝕜 : Type u_1} {V : Type u_2} {P : Type u_3} [RCLike 𝕜] [NormedAddCommGroup V] [InnerProductSpace 𝕜 V] [MetricSpace P] [NormedAddTorsor V P] {s : AffineSubspace 𝕜 P} [Nonempty ↥s] [s.direction.HasOrthogonalProjection] {p : P} (hp : p ∈ s) {v : V} (hv : v ∈ s.directionᗮ) : (EuclideanGeometry.reflection s) (v +ᵥ p) = -v +ᵥ p
• EuclideanGeometry.reflection_mem_of_le_of_mem 📋 Mathlib.Geometry.Euclidean.Projection
  {𝕜 : Type u_1} {V : Type u_2} {P : Type u_3} [RCLike 𝕜] [NormedAddCommGroup V] [InnerProductSpace 𝕜 V] [MetricSpace P] [NormedAddTorsor V P] {s₁ s₂ : AffineSubspace 𝕜 P} [Nonempty ↥s₁] [s₁.direction.HasOrthogonalProjection] (hle : s₁ ≤ s₂) {p : P} (hp : p ∈ s₂) : (EuclideanGeometry.reflection s₁) p ∈ s₂
• EuclideanGeometry.orthogonalProjection 📋 Mathlib.Geometry.Euclidean.Projection
  {𝕜 : Type u_1} {V : Type u_2} {P : Type u_3} [RCLike 𝕜] [NormedAddCommGroup V] [InnerProductSpace 𝕜 V] [MetricSpace P] [NormedAddTorsor V P] (s : AffineSubspace 𝕜 P) [Nonempty ↥s] [s.direction.HasOrthogonalProjection] : P →ᴬ[𝕜] ↥s
• EuclideanGeometry.reflection_apply_of_mem 📋 Mathlib.Geometry.Euclidean.Projection
  {𝕜 : Type u_1} {V : Type u_2} {P : Type u_3} [RCLike 𝕜] [NormedAddCommGroup V] [InnerProductSpace 𝕜 V] [MetricSpace P] [NormedAddTorsor V P] (s : AffineSubspace 𝕜 P) [Nonempty ↥s] [s.direction.HasOrthogonalProjection] (p : P) {x : P} (hx : x ∈ s) : (EuclideanGeometry.reflection s) p = s.direction.reflection (p -ᵥ x) +ᵥ x
• EuclideanGeometry.reflection_map 📋 Mathlib.Geometry.Euclidean.Projection
  {𝕜 : Type u_1} {V : Type u_2} {P : Type u_3} [RCLike 𝕜] [NormedAddCommGroup V] [InnerProductSpace 𝕜 V] {V₂ : Type u_4} {P₂ : Type u_5} [NormedAddCommGroup V₂] [InnerProductSpace 𝕜 V₂] [MetricSpace P] [NormedAddTorsor V P] [MetricSpace P₂] [NormedAddTorsor V₂ P₂] (s : AffineSubspace 𝕜 P) [Nonempty ↥s] [s.direction.HasOrthogonalProjection] (f : P →ᵃⁱ[𝕜] P₂) [(AffineSubspace.map f.toAffineMap s).direction.HasOrthogonalProjection] (p : P) : (EuclideanGeometry.reflection (AffineSubspace.map f.toAffineMap s)) (f p) = f ((EuclideanGeometry.reflection s) p)
• EuclideanGeometry.dist_sq_smul_orthogonal_vadd_smul_orthogonal_vadd 📋 Mathlib.Geometry.Euclidean.Projection
  {𝕜 : Type u_1} {V : Type u_2} {P : Type u_3} [RCLike 𝕜] [NormedAddCommGroup V] [InnerProductSpace 𝕜 V] [MetricSpace P] [NormedAddTorsor V P] {s : AffineSubspace 𝕜 P} {p₁ p₂ : P} (hp₁ : p₁ ∈ s) (hp₂ : p₂ ∈ s) (r₁ r₂ : 𝕜) {v : V} (hv : v ∈ s.directionᗮ) : dist (r₁ • v +ᵥ p₁) (r₂ • v +ᵥ p₂) * dist (r₁ • v +ᵥ p₁) (r₂ • v +ᵥ p₂) = dist p₁ p₂ * dist p₁ p₂ + ‖r₁ - r₂‖ * ‖r₁ - r₂‖ * (‖v‖ * ‖v‖)
• EuclideanGeometry.reflection_apply 📋 Mathlib.Geometry.Euclidean.Projection
  {𝕜 : Type u_1} {V : Type u_2} {P : Type u_3} [RCLike 𝕜] [NormedAddCommGroup V] [InnerProductSpace 𝕜 V] [MetricSpace P] [NormedAddTorsor V P] (s : AffineSubspace 𝕜 P) [Nonempty ↥s] [s.direction.HasOrthogonalProjection] (p : P) : (EuclideanGeometry.reflection s) p = s.direction.reflection (p -ᵥ ↑(Classical.arbitrary ↥s)) +ᵥ ↑(Classical.arbitrary ↥s)
• Affine.Simplex.orthogonalProjectionSpan 📋 Mathlib.Geometry.Euclidean.Projection
  {𝕜 : Type u_1} {V : Type u_2} {P : Type u_3} [RCLike 𝕜] [NormedAddCommGroup V] [InnerProductSpace 𝕜 V] [MetricSpace P] [NormedAddTorsor V P] {n : ℕ} (s : Affine.Simplex 𝕜 P n) : P →ᴬ[𝕜] ↥(affineSpan 𝕜 (Set.range s.points))
• EuclideanGeometry.orthogonalProjection_contLinear 📋 Mathlib.Geometry.Euclidean.Projection
  {𝕜 : Type u_1} {V : Type u_2} {P : Type u_3} [RCLike 𝕜] [NormedAddCommGroup V] [InnerProductSpace 𝕜 V] [MetricSpace P] [NormedAddTorsor V P] {s : AffineSubspace 𝕜 P} [Nonempty ↥s] [s.direction.HasOrthogonalProjection] : (EuclideanGeometry.orthogonalProjection s).contLinear = s.direction.orthogonalProjectionOnto
• EuclideanGeometry.dist_orthogonalProjection_eq_infDist 📋 Mathlib.Geometry.Euclidean.Projection
  {𝕜 : Type u_1} {V : Type u_2} {P : Type u_3} [RCLike 𝕜] [NormedAddCommGroup V] [InnerProductSpace 𝕜 V] [MetricSpace P] [NormedAddTorsor V P] (s : AffineSubspace 𝕜 P) [Nonempty ↥s] [s.direction.HasOrthogonalProjection] (p : P) : dist p ↑((EuclideanGeometry.orthogonalProjection s) p) = Metric.infDist p ↑s
• EuclideanGeometry.dist_orthogonalProjection_eq_infNndist 📋 Mathlib.Geometry.Euclidean.Projection
  {𝕜 : Type u_1} {V : Type u_2} {P : Type u_3} [RCLike 𝕜] [NormedAddCommGroup V] [InnerProductSpace 𝕜 V] [MetricSpace P] [NormedAddTorsor V P] (s : AffineSubspace 𝕜 P) [Nonempty ↥s] [s.direction.HasOrthogonalProjection] (p : P) : nndist p ↑((EuclideanGeometry.orthogonalProjection s) p) = Metric.infNndist p ↑s
• EuclideanGeometry.orthogonalProjection_mem 📋 Mathlib.Geometry.Euclidean.Projection
  {𝕜 : Type u_1} {V : Type u_2} {P : Type u_3} [RCLike 𝕜] [NormedAddCommGroup V] [InnerProductSpace 𝕜 V] [MetricSpace P] [NormedAddTorsor V P] {s : AffineSubspace 𝕜 P} [Nonempty ↥s] [s.direction.HasOrthogonalProjection] (p : P) : ↑((EuclideanGeometry.orthogonalProjection s) p) ∈ s
• EuclideanGeometry.orthogonalProjection_eq_self_iff 📋 Mathlib.Geometry.Euclidean.Projection
  {𝕜 : Type u_1} {V : Type u_2} {P : Type u_3} [RCLike 𝕜] [NormedAddCommGroup V] [InnerProductSpace 𝕜 V] [MetricSpace P] [NormedAddTorsor V P] {s : AffineSubspace 𝕜 P} [Nonempty ↥s] [s.direction.HasOrthogonalProjection] {p : P} : ↑((EuclideanGeometry.orthogonalProjection s) p) = p ↔ p ∈ s
• EuclideanGeometry.exists_dist_eq_iff_exists_dist_orthogonalProjection_eq 📋 Mathlib.Geometry.Euclidean.Projection
  {𝕜 : Type u_1} {V : Type u_2} {P : Type u_3} [RCLike 𝕜] [NormedAddCommGroup V] [InnerProductSpace 𝕜 V] [MetricSpace P] [NormedAddTorsor V P] {s : AffineSubspace 𝕜 P} [Nonempty ↥s] [s.direction.HasOrthogonalProjection] {ps : Set P} (hps : ps ⊆ ↑s) (p : P) : (∃ r, ∀ p₁ ∈ ps, dist p₁ p = r) ↔ ∃ r, ∀ p₁ ∈ ps, dist p₁ ↑((EuclideanGeometry.orthogonalProjection s) p) = r
• EuclideanGeometry.dist_orthogonalProjection_eq_zero_iff 📋 Mathlib.Geometry.Euclidean.Projection
  {𝕜 : Type u_1} {V : Type u_2} {P : Type u_3} [RCLike 𝕜] [NormedAddCommGroup V] [InnerProductSpace 𝕜 V] [MetricSpace P] [NormedAddTorsor V P] {s : AffineSubspace 𝕜 P} [Nonempty ↥s] [s.direction.HasOrthogonalProjection] {p : P} : dist p ↑((EuclideanGeometry.orthogonalProjection s) p) = 0 ↔ p ∈ s
• EuclideanGeometry.dist_orthogonalProjection_ne_zero_of_notMem 📋 Mathlib.Geometry.Euclidean.Projection
  {𝕜 : Type u_1} {V : Type u_2} {P : Type u_3} [RCLike 𝕜] [NormedAddCommGroup V] [InnerProductSpace 𝕜 V] [MetricSpace P] [NormedAddTorsor V P] {s : AffineSubspace 𝕜 P} [Nonempty ↥s] [s.direction.HasOrthogonalProjection] {p : P} (hp : p ∉ s) : dist p ↑((EuclideanGeometry.orthogonalProjection s) p) ≠ 0
• EuclideanGeometry.orthogonalProjection_vsub_mem_direction_orthogonal 📋 Mathlib.Geometry.Euclidean.Projection
  {𝕜 : Type u_1} {V : Type u_2} {P : Type u_3} [RCLike 𝕜] [NormedAddCommGroup V] [InnerProductSpace 𝕜 V] [MetricSpace P] [NormedAddTorsor V P] (s : AffineSubspace 𝕜 P) [Nonempty ↥s] [s.direction.HasOrthogonalProjection] (p : P) : ↑((EuclideanGeometry.orthogonalProjection s) p) -ᵥ p ∈ s.directionᗮ
• EuclideanGeometry.vsub_orthogonalProjection_mem_direction_orthogonal 📋 Mathlib.Geometry.Euclidean.Projection
  {𝕜 : Type u_1} {V : Type u_2} {P : Type u_3} [RCLike 𝕜] [NormedAddCommGroup V] [InnerProductSpace 𝕜 V] [MetricSpace P] [NormedAddTorsor V P] (s : AffineSubspace 𝕜 P) [Nonempty ↥s] [s.direction.HasOrthogonalProjection] (p : P) : p -ᵥ ↑((EuclideanGeometry.orthogonalProjection s) p) ∈ s.directionᗮ
• EuclideanGeometry.orthogonalProjection_mem_orthogonal 📋 Mathlib.Geometry.Euclidean.Projection
  {𝕜 : Type u_1} {V : Type u_2} {P : Type u_3} [RCLike 𝕜] [NormedAddCommGroup V] [InnerProductSpace 𝕜 V] [MetricSpace P] [NormedAddTorsor V P] (s : AffineSubspace 𝕜 P) [Nonempty ↥s] [s.direction.HasOrthogonalProjection] (p : P) : ↑((EuclideanGeometry.orthogonalProjection s) p) ∈ AffineSubspace.mk' p s.directionᗮ
• EuclideanGeometry.orthogonalProjection_mem_subspace_eq_self 📋 Mathlib.Geometry.Euclidean.Projection
  {𝕜 : Type u_1} {V : Type u_2} {P : Type u_3} [RCLike 𝕜] [NormedAddCommGroup V] [InnerProductSpace 𝕜 V] [MetricSpace P] [NormedAddTorsor V P] {s : AffineSubspace 𝕜 P} [Nonempty ↥s] [s.direction.HasOrthogonalProjection] (p : ↥s) : (EuclideanGeometry.orthogonalProjection s) ↑p = p
• EuclideanGeometry.inter_eq_singleton_orthogonalProjection 📋 Mathlib.Geometry.Euclidean.Projection
  {𝕜 : Type u_1} {V : Type u_2} {P : Type u_3} [RCLike 𝕜] [NormedAddCommGroup V] [InnerProductSpace 𝕜 V] [MetricSpace P] [NormedAddTorsor V P] {s : AffineSubspace 𝕜 P} [Nonempty ↥s] [s.direction.HasOrthogonalProjection] (p : P) : ↑s ∩ ↑(AffineSubspace.mk' p s.directionᗮ) = {↑((EuclideanGeometry.orthogonalProjection s) p)}
• EuclideanGeometry.coe_orthogonalProjection_eq_iff_mem 📋 Mathlib.Geometry.Euclidean.Projection
  {𝕜 : Type u_1} {V : Type u_2} {P : Type u_3} [RCLike 𝕜] [NormedAddCommGroup V] [InnerProductSpace 𝕜 V] [MetricSpace P] [NormedAddTorsor V P] {s : AffineSubspace 𝕜 P} [Nonempty ↥s] [s.direction.HasOrthogonalProjection] {p q : P} : ↑((EuclideanGeometry.orthogonalProjection s) p) = q ↔ q ∈ s ∧ p -ᵥ q ∈ s.directionᗮ
• EuclideanGeometry.orthogonalProjection_eq_iff_mem 📋 Mathlib.Geometry.Euclidean.Projection
  {𝕜 : Type u_1} {V : Type u_2} {P : Type u_3} [RCLike 𝕜] [NormedAddCommGroup V] [InnerProductSpace 𝕜 V] [MetricSpace P] [NormedAddTorsor V P] {s : AffineSubspace 𝕜 P} [Nonempty ↥s] [s.direction.HasOrthogonalProjection] {p : P} {q : ↥s} : (EuclideanGeometry.orthogonalProjection s) p = q ↔ p -ᵥ ↑q ∈ s.directionᗮ
• EuclideanGeometry.orthogonalProjection_vadd_eq_self 📋 Mathlib.Geometry.Euclidean.Projection
  {𝕜 : Type u_1} {V : Type u_2} {P : Type u_3} [RCLike 𝕜] [NormedAddCommGroup V] [InnerProductSpace 𝕜 V] [MetricSpace P] [NormedAddTorsor V P] {s : AffineSubspace 𝕜 P} [Nonempty ↥s] [s.direction.HasOrthogonalProjection] {p : P} (hp : p ∈ s) {v : V} (hv : v ∈ s.directionᗮ) : (EuclideanGeometry.orthogonalProjection s) (v +ᵥ p) = ⟨p, hp⟩
• EuclideanGeometry.orthogonalProjection_linear 📋 Mathlib.Geometry.Euclidean.Projection
  {𝕜 : Type u_1} {V : Type u_2} {P : Type u_3} [RCLike 𝕜] [NormedAddCommGroup V] [InnerProductSpace 𝕜 V] [MetricSpace P] [NormedAddTorsor V P] {s : AffineSubspace 𝕜 P} [Nonempty ↥s] [s.direction.HasOrthogonalProjection] : (↑(EuclideanGeometry.orthogonalProjection s)).linear = ↑s.direction.orthogonalProjectionOnto
• EuclideanGeometry.orthogonalProjection_vsub_mem_direction 📋 Mathlib.Geometry.Euclidean.Projection
  {𝕜 : Type u_1} {V : Type u_2} {P : Type u_3} [RCLike 𝕜] [NormedAddCommGroup V] [InnerProductSpace 𝕜 V] [MetricSpace P] [NormedAddTorsor V P] {s : AffineSubspace 𝕜 P} [Nonempty ↥s] [s.direction.HasOrthogonalProjection] {p₁ : P} (p₂ : P) (hp₁ : p₁ ∈ s) : ↑((EuclideanGeometry.orthogonalProjection s) p₂ -ᵥ ⟨p₁, hp₁⟩) ∈ s.direction
• EuclideanGeometry.vsub_orthogonalProjection_mem_direction 📋 Mathlib.Geometry.Euclidean.Projection
  {𝕜 : Type u_1} {V : Type u_2} {P : Type u_3} [RCLike 𝕜] [NormedAddCommGroup V] [InnerProductSpace 𝕜 V] [MetricSpace P] [NormedAddTorsor V P] {s : AffineSubspace 𝕜 P} [Nonempty ↥s] [s.direction.HasOrthogonalProjection] {p₁ : P} (p₂ : P) (hp₁ : p₁ ∈ s) : ↑(⟨p₁, hp₁⟩ -ᵥ (EuclideanGeometry.orthogonalProjection s) p₂) ∈ s.direction
• EuclideanGeometry.orthogonalProjection_affineSpan_singleton 📋 Mathlib.Geometry.Euclidean.Projection
  {𝕜 : Type u_1} {V : Type u_2} {P : Type u_3} [RCLike 𝕜] [NormedAddCommGroup V] [InnerProductSpace 𝕜 V] [MetricSpace P] [NormedAddTorsor V P] (p₁ p₂ : P) : ↑((EuclideanGeometry.orthogonalProjection (affineSpan 𝕜 {p₁})) p₂) = p₁
• EuclideanGeometry.orthogonalProjection_apply_mem 📋 Mathlib.Geometry.Euclidean.Projection
  {𝕜 : Type u_1} {V : Type u_2} {P : Type u_3} [RCLike 𝕜] [NormedAddCommGroup V] [InnerProductSpace 𝕜 V] [MetricSpace P] [NormedAddTorsor V P] (s : AffineSubspace 𝕜 P) [Nonempty ↥s] [s.direction.HasOrthogonalProjection] {p x : P} (hx : x ∈ s) : ↑((EuclideanGeometry.orthogonalProjection s) p) = ↑(s.direction.orthogonalProjectionOnto (p -ᵥ x)) +ᵥ x
• EuclideanGeometry.orthogonalProjection_eq_orthogonalProjection_iff_vsub_mem 📋 Mathlib.Geometry.Euclidean.Projection
  {𝕜 : Type u_1} {V : Type u_2} {P : Type u_3} [RCLike 𝕜] [NormedAddCommGroup V] [InnerProductSpace 𝕜 V] [MetricSpace P] [NormedAddTorsor V P] {s : AffineSubspace 𝕜 P} [Nonempty ↥s] [s.direction.HasOrthogonalProjection] {p q : P} : (EuclideanGeometry.orthogonalProjection s) p = (EuclideanGeometry.orthogonalProjection s) q ↔ p -ᵥ q ∈ s.directionᗮ
• EuclideanGeometry.dist_set_eq_iff_dist_orthogonalProjection_eq 📋 Mathlib.Geometry.Euclidean.Projection
  {𝕜 : Type u_1} {V : Type u_2} {P : Type u_3} [RCLike 𝕜] [NormedAddCommGroup V] [InnerProductSpace 𝕜 V] [MetricSpace P] [NormedAddTorsor V P] {s : AffineSubspace 𝕜 P} [Nonempty ↥s] [s.direction.HasOrthogonalProjection] {ps : Set P} (hps : ps ⊆ ↑s) (p : P) : (ps.Pairwise fun p₁ p₂ => dist p₁ p = dist p₂ p) ↔ ps.Pairwise fun p₁ p₂ => dist p₁ ↑((EuclideanGeometry.orthogonalProjection s) p) = dist p₂ ↑((EuclideanGeometry.orthogonalProjection s) p)
• EuclideanGeometry.reflection_apply' 📋 Mathlib.Geometry.Euclidean.Projection
  {𝕜 : Type u_1} {V : Type u_2} {P : Type u_3} [RCLike 𝕜] [NormedAddCommGroup V] [InnerProductSpace 𝕜 V] [MetricSpace P] [NormedAddTorsor V P] (s : AffineSubspace 𝕜 P) [Nonempty ↥s] [s.direction.HasOrthogonalProjection] (p : P) : (EuclideanGeometry.reflection s) p = (↑((EuclideanGeometry.orthogonalProjection s) p) -ᵥ p) +ᵥ ↑((EuclideanGeometry.orthogonalProjection s) p)

About

Loogle searches Lean and Mathlib definitions and theorems.

You can use Loogle from within the Lean4 VSCode language extension using the Loogle command from the command palette. You can also try the #loogle command from LeanSearchClient, the CLI version, the Loogle VS Code extension, the lean.nvim integration or the Zulip bot.

Usage

Loogle finds definitions and lemmas in various ways:

1. By constant:
   🔍 Real.sin
   finds all lemmas whose statement somehow mentions the sine function.

2. By lemma name substring:
   🔍 "differ"
   finds all lemmas that have "differ" somewhere in their lemma name.

3. By subexpression:
   🔍 _ * (_ ^ _)
   finds all lemmas whose statements somewhere include a product where the second argument is raised to some power.

   The pattern can also be non-linear, as in
   🔍 Real.sqrt ?a * Real.sqrt ?a

   If the pattern has parameters, they are matched in any order. Both of these will find List.map:
   🔍 (?a -> ?b) -> List ?a -> List ?b
   🔍 List ?a -> (?a -> ?b) -> List ?b

4. By main conclusion:
   🔍 |- tsum _ = _ * tsum _
   finds all lemmas where the conclusion (the subexpression to the right of all → and ∀) has the given shape.

   As before, if the pattern has parameters, they are matched against the hypotheses of the lemma in any order; for example,
   🔍 |- _ < _ → tsum _ < tsum _
   will find tsum_lt_tsum even though the hypothesis f i < g i is not the last.

5. You can filter for definitions vs theorems: Using ⊢ (_ : Type _) finds all definitions which provide data while ⊢ (_ : Prop) finds all theorems (and definitions of proofs).

If you pass more than one such search filter, separated by commas Loogle will return lemmas which match all of them. The search
🔍 Real.sin, "two", tsum, _ * _, _ ^ _, |- _ < _ → _
would find all lemmas which mention the constants Real.sin and tsum, have "two" as a substring of the lemma name, include a product and a power somewhere in the type, and have a hypothesis of the form _ < _ (if there were any such lemmas). Metavariables (?a) are assigned independently in each filter.

The #lucky button will directly send you to the documentation of the first hit.

Source code

You can find the source code for this service at https://github.com/nomeata/loogle. The https://loogle.lean-lang.org/ service is provided by the Lean FRO. Please review the Lean FRO Terms of Use and Privacy Policy.

This is Loogle revision 9f11169 serving mathlib revision ce5dd8c