File: Mathlib/Analysis/NormedSpace/AddTorsor.lean

full_name: AffineSubspace.isClosed_direction_iff
code: theorem AffineSubspace.isClosed_direction_iff (s : AffineSubspace 𝕜 Q) :
    IsClosed (s.direction : Set W) ↔ IsClosed (s : Set Q) 
goal: IsClosed ↑s.direction ↔ IsClosed ↑s

full_name: dist_center_homothety
code: @[simp]
theorem dist_center_homothety (p₁ p₂ : P) (c : 𝕜) :
    dist p₁ (homothety p₁ c p₂) = ‖c‖ * dist p₁ p₂ 
goal: dist p₁ ((AffineMap.homothety p₁ c) p₂) = ‖c‖ * dist p₁ p₂

full_name: nndist_center_homothety
code: @[simp]
theorem nndist_center_homothety (p₁ p₂ : P) (c : 𝕜) :
    nndist p₁ (homothety p₁ c p₂) = ‖c‖₊ * nndist p₁ p₂ 
goal: nndist p₁ ((AffineMap.homothety p₁ c) p₂) = ‖c‖₊ * nndist p₁ p₂

full_name: dist_homothety_center
code: @[simp]
theorem dist_homothety_center (p₁ p₂ : P) (c : 𝕜) :
    dist (homothety p₁ c p₂) p₁ = ‖c‖ * dist p₁ p₂ 
goal: dist ((AffineMap.homothety p₁ c) p₂) p₁ = ‖c‖ * dist p₁ p₂

full_name: nndist_homothety_center
code: @[simp]
theorem nndist_homothety_center (p₁ p₂ : P) (c : 𝕜) :
    nndist (homothety p₁ c p₂) p₁ = ‖c‖₊ * nndist p₁ p₂ 
goal: nndist ((AffineMap.homothety p₁ c) p₂) p₁ = ‖c‖₊ * nndist p₁ p₂

full_name: dist_lineMap_lineMap
code: @[simp]
theorem dist_lineMap_lineMap (p₁ p₂ : P) (c₁ c₂ : 𝕜) :
    dist (lineMap p₁ p₂ c₁) (lineMap p₁ p₂ c₂) = dist c₁ c₂ * dist p₁ p₂ 
goal: dist ((AffineMap.lineMap p₁ p₂) c₁) ((AffineMap.lineMap p₁ p₂) c₂) = dist c₁ c₂ * dist p₁ p₂

full_name: nndist_lineMap_lineMap
code: @[simp]
theorem nndist_lineMap_lineMap (p₁ p₂ : P) (c₁ c₂ : 𝕜) :
    nndist (lineMap p₁ p₂ c₁) (lineMap p₁ p₂ c₂) = nndist c₁ c₂ * nndist p₁ p₂ 
goal: nndist ((AffineMap.lineMap p₁ p₂) c₁) ((AffineMap.lineMap p₁ p₂) c₂) = nndist c₁ c₂ * nndist p₁ p₂

full_name: lipschitzWith_lineMap
code: theorem lipschitzWith_lineMap (p₁ p₂ : P) : LipschitzWith (nndist p₁ p₂) (lineMap p₁ p₂ : 𝕜 → P) 
goal: LipschitzWith (nndist p₁ p₂) ⇑(AffineMap.lineMap p₁ p₂)

full_name: dist_lineMap_left
code: @[simp]
theorem dist_lineMap_left (p₁ p₂ : P) (c : 𝕜) : dist (lineMap p₁ p₂ c) p₁ = ‖c‖ * dist p₁ p₂ 
goal: dist ((AffineMap.lineMap p₁ p₂) c) p₁ = ‖c‖ * dist p₁ p₂

full_name: nndist_lineMap_left
code: @[simp]
theorem nndist_lineMap_left (p₁ p₂ : P) (c : 𝕜) :
    nndist (lineMap p₁ p₂ c) p₁ = ‖c‖₊ * nndist p₁ p₂ 
goal: nndist ((AffineMap.lineMap p₁ p₂) c) p₁ = ‖c‖₊ * nndist p₁ p₂

full_name: dist_left_lineMap
code: @[simp]
theorem dist_left_lineMap (p₁ p₂ : P) (c : 𝕜) : dist p₁ (lineMap p₁ p₂ c) = ‖c‖ * dist p₁ p₂ 
goal: dist p₁ ((AffineMap.lineMap p₁ p₂) c) = ‖c‖ * dist p₁ p₂

full_name: nndist_left_lineMap
code: @[simp]
theorem nndist_left_lineMap (p₁ p₂ : P) (c : 𝕜) :
    nndist p₁ (lineMap p₁ p₂ c) = ‖c‖₊ * nndist p₁ p₂ 
goal: nndist p₁ ((AffineMap.lineMap p₁ p₂) c) = ‖c‖₊ * nndist p₁ p₂

full_name: dist_lineMap_right
code: @[simp]
theorem dist_lineMap_right (p₁ p₂ : P) (c : 𝕜) :
    dist (lineMap p₁ p₂ c) p₂ = ‖1 - c‖ * dist p₁ p₂ 
goal: dist ((AffineMap.lineMap p₁ p₂) c) p₂ = ‖1 - c‖ * dist p₁ p₂

full_name: nndist_lineMap_right
code: @[simp]
theorem nndist_lineMap_right (p₁ p₂ : P) (c : 𝕜) :
    nndist (lineMap p₁ p₂ c) p₂ = ‖1 - c‖₊ * nndist p₁ p₂ 
goal: nndist ((AffineMap.lineMap p₁ p₂) c) p₂ = ‖1 - c‖₊ * nndist p₁ p₂

full_name: dist_right_lineMap
code: @[simp]
theorem dist_right_lineMap (p₁ p₂ : P) (c : 𝕜) : dist p₂ (lineMap p₁ p₂ c) = ‖1 - c‖ * dist p₁ p₂ 
goal: dist p₂ ((AffineMap.lineMap p₁ p₂) c) = ‖1 - c‖ * dist p₁ p₂

full_name: nndist_right_lineMap
code: @[simp]
theorem nndist_right_lineMap (p₁ p₂ : P) (c : 𝕜) :
    nndist p₂ (lineMap p₁ p₂ c) = ‖1 - c‖₊ * nndist p₁ p₂ 
goal: nndist p₂ ((AffineMap.lineMap p₁ p₂) c) = ‖1 - c‖₊ * nndist p₁ p₂

full_name: dist_homothety_self
code: @[simp]
theorem dist_homothety_self (p₁ p₂ : P) (c : 𝕜) :
    dist (homothety p₁ c p₂) p₂ = ‖1 - c‖ * dist p₁ p₂ 
goal: dist ((AffineMap.homothety p₁ c) p₂) p₂ = ‖1 - c‖ * dist p₁ p₂

full_name: nndist_homothety_self
code: @[simp]
theorem nndist_homothety_self (p₁ p₂ : P) (c : 𝕜) :
    nndist (homothety p₁ c p₂) p₂ = ‖1 - c‖₊ * nndist p₁ p₂ 
goal: nndist ((AffineMap.homothety p₁ c) p₂) p₂ = ‖1 - c‖₊ * nndist p₁ p₂

full_name: dist_self_homothety
code: @[simp]
theorem dist_self_homothety (p₁ p₂ : P) (c : 𝕜) :
    dist p₂ (homothety p₁ c p₂) = ‖1 - c‖ * dist p₁ p₂ 
goal: dist p₂ ((AffineMap.homothety p₁ c) p₂) = ‖1 - c‖ * dist p₁ p₂

full_name: nndist_self_homothety
code: @[simp]
theorem nndist_self_homothety (p₁ p₂ : P) (c : 𝕜) :
    nndist p₂ (homothety p₁ c p₂) = ‖1 - c‖₊ * nndist p₁ p₂ 
goal: nndist p₂ ((AffineMap.homothety p₁ c) p₂) = ‖1 - c‖₊ * nndist p₁ p₂

full_name: dist_left_midpoint
code: @[simp]
theorem dist_left_midpoint (p₁ p₂ : P) : dist p₁ (midpoint 𝕜 p₁ p₂) = ‖(2 : 𝕜)‖⁻¹ * dist p₁ p₂ 
goal: dist p₁ (midpoint 𝕜 p₁ p₂) = ‖2‖⁻¹ * dist p₁ p₂

full_name: nndist_left_midpoint
code: @[simp]
theorem nndist_left_midpoint (p₁ p₂ : P) :
    nndist p₁ (midpoint 𝕜 p₁ p₂) = ‖(2 : 𝕜)‖₊⁻¹ * nndist p₁ p₂ 
goal: nndist p₁ (midpoint 𝕜 p₁ p₂) = ‖2‖₊⁻¹ * nndist p₁ p₂

full_name: dist_midpoint_left
code: @[simp]
theorem dist_midpoint_left (p₁ p₂ : P) : dist (midpoint 𝕜 p₁ p₂) p₁ = ‖(2 : 𝕜)‖⁻¹ * dist p₁ p₂ 
goal: dist (midpoint 𝕜 p₁ p₂) p₁ = ‖2‖⁻¹ * dist p₁ p₂

full_name: nndist_midpoint_left
code: @[simp]
theorem nndist_midpoint_left (p₁ p₂ : P) :
    nndist (midpoint 𝕜 p₁ p₂) p₁ = ‖(2 : 𝕜)‖₊⁻¹ * nndist p₁ p₂ 
goal: nndist (midpoint 𝕜 p₁ p₂) p₁ = ‖2‖₊⁻¹ * nndist p₁ p₂

full_name: dist_midpoint_right
code: @[simp]
theorem dist_midpoint_right (p₁ p₂ : P) :
    dist (midpoint 𝕜 p₁ p₂) p₂ = ‖(2 : 𝕜)‖⁻¹ * dist p₁ p₂ 
goal: dist (midpoint 𝕜 p₁ p₂) p₂ = ‖2‖⁻¹ * dist p₁ p₂

full_name: nndist_midpoint_right
code: @[simp]
theorem nndist_midpoint_right (p₁ p₂ : P) :
    nndist (midpoint 𝕜 p₁ p₂) p₂ = ‖(2 : 𝕜)‖₊⁻¹ * nndist p₁ p₂ 
goal: nndist (midpoint 𝕜 p₁ p₂) p₂ = ‖2‖₊⁻¹ * nndist p₁ p₂

full_name: dist_right_midpoint
code: @[simp]
theorem dist_right_midpoint (p₁ p₂ : P) :
    dist p₂ (midpoint 𝕜 p₁ p₂) = ‖(2 : 𝕜)‖⁻¹ * dist p₁ p₂ 
goal: dist p₂ (midpoint 𝕜 p₁ p₂) = ‖2‖⁻¹ * dist p₁ p₂

full_name: nndist_right_midpoint
code: @[simp]
theorem nndist_right_midpoint (p₁ p₂ : P) :
    nndist p₂ (midpoint 𝕜 p₁ p₂) = ‖(2 : 𝕜)‖₊⁻¹ * nndist p₁ p₂ 
goal: nndist p₂ (midpoint 𝕜 p₁ p₂) = ‖2‖₊⁻¹ * nndist p₁ p₂

full_name: dist_midpoint_midpoint_le'
code: theorem dist_midpoint_midpoint_le' (p₁ p₂ p₃ p₄ : P) :
    dist (midpoint 𝕜 p₁ p₂) (midpoint 𝕜 p₃ p₄) ≤ (dist p₁ p₃ + dist p₂ p₄) / ‖(2 : 𝕜)‖ 
goal: dist (midpoint 𝕜 p₁ p₂) (midpoint 𝕜 p₃ p₄) ≤ (dist p₁ p₃ + dist p₂ p₄) / ‖2‖

full_name: nndist_midpoint_midpoint_le'
code: theorem nndist_midpoint_midpoint_le' (p₁ p₂ p₃ p₄ : P) :
    nndist (midpoint 𝕜 p₁ p₂) (midpoint 𝕜 p₃ p₄) ≤ (nndist p₁ p₃ + nndist p₂ p₄) / ‖(2 : 𝕜)‖₊ 
goal: nndist (midpoint 𝕜 p₁ p₂) (midpoint 𝕜 p₃ p₄) ≤ (nndist p₁ p₃ + nndist p₂ p₄) / ‖2‖₊

full_name: dist_pointReflection_left
code: @[simp] theorem dist_pointReflection_left (p q : P) :
    dist (Equiv.pointReflection p q) p = dist p q 
goal: dist ((Equiv.pointReflection p) q) p = dist p q

full_name: dist_left_pointReflection
code: @[simp] theorem dist_left_pointReflection (p q : P) :
    dist p (Equiv.pointReflection p q) = dist p q 
goal: dist p ((Equiv.pointReflection p) q) = dist p q

full_name: dist_pointReflection_right
code: theorem dist_pointReflection_right (p q : P) :
    dist (Equiv.pointReflection p q) q = ‖(2 : 𝕜)‖ * dist p q 
goal: dist ((Equiv.pointReflection p) q) q = ‖2‖ * dist p q

full_name: dist_right_pointReflection
code: theorem dist_right_pointReflection (p q : P) :
    dist q (Equiv.pointReflection p q) = ‖(2 : 𝕜)‖ * dist p q 
goal: dist q ((Equiv.pointReflection p) q) = ‖2‖ * dist p q

full_name: antilipschitzWith_lineMap
code: theorem antilipschitzWith_lineMap {p₁ p₂ : Q} (h : p₁ ≠ p₂) :
    AntilipschitzWith (nndist p₁ p₂)⁻¹ (lineMap p₁ p₂ : 𝕜 → Q) 
goal: AntilipschitzWith (nndist p₁ p₂)⁻¹ ⇑(AffineMap.lineMap p₁ p₂)

full_name: eventually_homothety_mem_of_mem_interior
code: theorem eventually_homothety_mem_of_mem_interior (x : Q) {s : Set Q} {y : Q} (hy : y ∈ interior s) :
    ∀ᶠ δ in 𝓝 (1 : 𝕜), homothety x δ y ∈ s 
goal: ∀ᶠ (δ : 𝕜) in nhds 1, (AffineMap.homothety x δ) y ∈ s

full_name: eventually_homothety_image_subset_of_finite_subset_interior
code: theorem eventually_homothety_image_subset_of_finite_subset_interior (x : Q) {s : Set Q} {t : Set Q}
    (ht : t.Finite) (h : t ⊆ interior s) : ∀ᶠ δ in 𝓝 (1 : 𝕜), homothety x δ '' t ⊆ s 
goal: ∀ᶠ (δ : 𝕜) in nhds 1, ⇑(AffineMap.homothety x δ) '' t ⊆ s

full_name: dist_midpoint_midpoint_le
code: theorem dist_midpoint_midpoint_le (p₁ p₂ p₃ p₄ : V) :
    dist (midpoint ℝ p₁ p₂) (midpoint ℝ p₃ p₄) ≤ (dist p₁ p₃ + dist p₂ p₄) / 2 
goal: dist (midpoint ℝ p₁ p₂) (midpoint ℝ p₃ p₄) ≤ (dist p₁ p₃ + dist p₂ p₄) / 2

full_name: nndist_midpoint_midpoint_le
code: theorem nndist_midpoint_midpoint_le (p₁ p₂ p₃ p₄ : V) :
    nndist (midpoint ℝ p₁ p₂) (midpoint ℝ p₃ p₄) ≤ (nndist p₁ p₃ + nndist p₂ p₄) / 2 
goal: nndist (midpoint ℝ p₁ p₂) (midpoint ℝ p₃ p₄) ≤ (nndist p₁ p₃ + nndist p₂ p₄) / 2

full_name: AffineMap.ofMapMidpoint
code: /-- A continuous map between two normed affine spaces is an affine map provided that
it sends midpoints to midpoints. -/
def AffineMap.ofMapMidpoint (f : P → Q) (h : ∀ x y, f (midpoint ℝ x y) = midpoint ℝ (f x) (f y))
    (hfc : Continuous f) : P →ᵃ[ℝ] Q 
goal: P →ᵃ[ℝ] Q

full_name: DilationEquiv.smulTorsor
code: /-- Scaling by an element `k` of the scalar ring as a `DilationEquiv` with ratio `‖k‖₊`, mapping
from a normed space to a normed torsor over that space sending `0` to `c`. -/
@[simps]
def DilationEquiv.smulTorsor (c : P) {k : 𝕜} (hk : k ≠ 0) : E ≃ᵈ P 
goal: E ≃ᵈ P

full_name: DilationEquiv.smulTorsor_apply
code: /-- Scaling by an element `k` of the scalar ring as a `DilationEquiv` with ratio `‖k‖₊`, mapping
from a normed space to a normed torsor over that space sending `0` to `c`. -/
@[simps]
def DilationEquiv.smulTorsor (c : P) {k : 𝕜} (hk : k ≠ 0) : E ≃ᵈ P 
goal: ∀ (x : E), (DilationEquiv.smulTorsor c hk) x = k • x +ᵥ c

full_name: DilationEquiv.smulTorsor_symm_apply
code: /-- Scaling by an element `k` of the scalar ring as a `DilationEquiv` with ratio `‖k‖₊`, mapping
from a normed space to a normed torsor over that space sending `0` to `c`. -/
@[simps]
def DilationEquiv.smulTorsor (c : P) {k : 𝕜} (hk : k ≠ 0) : E ≃ᵈ P 
goal: ∀ (x : P), (DilationEquiv.smulTorsor c hk).symm x = (k⁻¹ • fun x => x -ᵥ c) x

full_name: DilationEquiv.smulTorsor_ratio
code: @[simp]
lemma DilationEquiv.smulTorsor_ratio {c : P} {k : 𝕜} (hk : k ≠ 0) {x y : E}
    (h : dist x y ≠ 0) : ratio (smulTorsor c hk) = ‖k‖₊ 
goal: Dilation.ratio (DilationEquiv.smulTorsor c hk) = ‖k‖₊

full_name: DilationEquiv.smulTorsor_preimage_ball
code: @[simp]
lemma DilationEquiv.smulTorsor_preimage_ball {c : P} {k : 𝕜} (hk : k ≠ 0) :
    smulTorsor c hk ⁻¹' (Metric.ball c ‖k‖) = Metric.ball (0 : E) 1 
goal: ⇑(DilationEquiv.smulTorsor c hk) ⁻¹' Metric.ball c ‖k‖ = Metric.ball 0 1