fact stringlengths 6 3.84k | type stringclasses 11 values | library stringclasses 32 values | imports listlengths 1 14 | filename stringlengths 20 95 | symbolic_name stringlengths 1 90 | docstring stringlengths 7 20k ⌀ |
|---|---|---|---|---|---|---|
@[to_additive (attr := simps! apply toEquiv) /-- Subtraction `y ↦ y - x` as an `IsometryEquiv`. -/]
divRight [IsIsometricSMul Gᵐᵒᵖ G] (c : G) : G ≃ᵢ G where
toEquiv := Equiv.divRight c
isometry_toFun a b := edist_div_right a b c
@[to_additive (attr := simp)] | def | Topology | [
"Mathlib.Algebra.GroupWithZero.Pointwise.Set.Basic",
"Mathlib.Algebra.Ring.Pointwise.Set",
"Mathlib.Topology.Algebra.ConstMulAction",
"Mathlib.Topology.MetricSpace.Isometry",
"Mathlib.Topology.MetricSpace.Lipschitz"
] | Mathlib/Topology/MetricSpace/IsometricSMul.lean | divRight | Division `y ↦ y / x` as an `IsometryEquiv`. |
divRight_symm [IsIsometricSMul Gᵐᵒᵖ G] (c : G) : (divRight c).symm = mulRight c :=
ext fun _ => rfl
variable [IsIsometricSMul G G] [IsIsometricSMul Gᵐᵒᵖ G] | theorem | Topology | [
"Mathlib.Algebra.GroupWithZero.Pointwise.Set.Basic",
"Mathlib.Algebra.Ring.Pointwise.Set",
"Mathlib.Topology.Algebra.ConstMulAction",
"Mathlib.Topology.MetricSpace.Isometry",
"Mathlib.Topology.MetricSpace.Lipschitz"
] | Mathlib/Topology/MetricSpace/IsometricSMul.lean | divRight_symm | null |
@[to_additive (attr := simps! apply symm_apply toEquiv)
/-- Subtraction `y ↦ x - y` as an `IsometryEquiv`. -/]
divLeft (c : G) : G ≃ᵢ G where
toEquiv := Equiv.divLeft c
isometry_toFun := edist_div_left c
variable (G) | def | Topology | [
"Mathlib.Algebra.GroupWithZero.Pointwise.Set.Basic",
"Mathlib.Algebra.Ring.Pointwise.Set",
"Mathlib.Topology.Algebra.ConstMulAction",
"Mathlib.Topology.MetricSpace.Isometry",
"Mathlib.Topology.MetricSpace.Lipschitz"
] | Mathlib/Topology/MetricSpace/IsometricSMul.lean | divLeft | Division `y ↦ x / y` as an `IsometryEquiv`. |
@[to_additive (attr := simps! apply toEquiv) /-- Negation `x ↦ -x` as an `IsometryEquiv`. -/]
inv : G ≃ᵢ G where
toEquiv := Equiv.inv G
isometry_toFun := edist_inv_inv
@[to_additive (attr := simp)] theorem inv_symm : (inv G).symm = inv G := rfl | def | Topology | [
"Mathlib.Algebra.GroupWithZero.Pointwise.Set.Basic",
"Mathlib.Algebra.Ring.Pointwise.Set",
"Mathlib.Topology.Algebra.ConstMulAction",
"Mathlib.Topology.MetricSpace.Isometry",
"Mathlib.Topology.MetricSpace.Lipschitz"
] | Mathlib/Topology/MetricSpace/IsometricSMul.lean | inv | Inversion `x ↦ x⁻¹` as an `IsometryEquiv`. |
@[to_additive (attr := simp)]
smul_ball (c : G) (x : X) (r : ℝ≥0∞) : c • ball x r = ball (c • x) r :=
(IsometryEquiv.constSMul c).image_emetric_ball _ _
@[to_additive (attr := simp)] | theorem | Topology | [
"Mathlib.Algebra.GroupWithZero.Pointwise.Set.Basic",
"Mathlib.Algebra.Ring.Pointwise.Set",
"Mathlib.Topology.Algebra.ConstMulAction",
"Mathlib.Topology.MetricSpace.Isometry",
"Mathlib.Topology.MetricSpace.Lipschitz"
] | Mathlib/Topology/MetricSpace/IsometricSMul.lean | smul_ball | null |
preimage_smul_ball (c : G) (x : X) (r : ℝ≥0∞) :
(c • ·) ⁻¹' ball x r = ball (c⁻¹ • x) r := by
rw [preimage_smul, smul_ball]
@[to_additive (attr := simp)] | theorem | Topology | [
"Mathlib.Algebra.GroupWithZero.Pointwise.Set.Basic",
"Mathlib.Algebra.Ring.Pointwise.Set",
"Mathlib.Topology.Algebra.ConstMulAction",
"Mathlib.Topology.MetricSpace.Isometry",
"Mathlib.Topology.MetricSpace.Lipschitz"
] | Mathlib/Topology/MetricSpace/IsometricSMul.lean | preimage_smul_ball | null |
smul_closedBall (c : G) (x : X) (r : ℝ≥0∞) : c • closedBall x r = closedBall (c • x) r :=
(IsometryEquiv.constSMul c).image_emetric_closedBall _ _
@[to_additive (attr := simp)] | theorem | Topology | [
"Mathlib.Algebra.GroupWithZero.Pointwise.Set.Basic",
"Mathlib.Algebra.Ring.Pointwise.Set",
"Mathlib.Topology.Algebra.ConstMulAction",
"Mathlib.Topology.MetricSpace.Isometry",
"Mathlib.Topology.MetricSpace.Lipschitz"
] | Mathlib/Topology/MetricSpace/IsometricSMul.lean | smul_closedBall | null |
preimage_smul_closedBall (c : G) (x : X) (r : ℝ≥0∞) :
(c • ·) ⁻¹' closedBall x r = closedBall (c⁻¹ • x) r := by
rw [preimage_smul, smul_closedBall]
variable [PseudoEMetricSpace G]
@[to_additive (attr := simp)] | theorem | Topology | [
"Mathlib.Algebra.GroupWithZero.Pointwise.Set.Basic",
"Mathlib.Algebra.Ring.Pointwise.Set",
"Mathlib.Topology.Algebra.ConstMulAction",
"Mathlib.Topology.MetricSpace.Isometry",
"Mathlib.Topology.MetricSpace.Lipschitz"
] | Mathlib/Topology/MetricSpace/IsometricSMul.lean | preimage_smul_closedBall | null |
preimage_mul_left_ball [IsIsometricSMul G G] (a b : G) (r : ℝ≥0∞) :
(a * ·) ⁻¹' ball b r = ball (a⁻¹ * b) r :=
preimage_smul_ball a b r
@[to_additive (attr := simp)] | theorem | Topology | [
"Mathlib.Algebra.GroupWithZero.Pointwise.Set.Basic",
"Mathlib.Algebra.Ring.Pointwise.Set",
"Mathlib.Topology.Algebra.ConstMulAction",
"Mathlib.Topology.MetricSpace.Isometry",
"Mathlib.Topology.MetricSpace.Lipschitz"
] | Mathlib/Topology/MetricSpace/IsometricSMul.lean | preimage_mul_left_ball | null |
preimage_mul_right_ball [IsIsometricSMul Gᵐᵒᵖ G] (a b : G) (r : ℝ≥0∞) :
(fun x => x * a) ⁻¹' ball b r = ball (b / a) r := by
rw [div_eq_mul_inv]
exact preimage_smul_ball (MulOpposite.op a) b r
@[to_additive (attr := simp)] | theorem | Topology | [
"Mathlib.Algebra.GroupWithZero.Pointwise.Set.Basic",
"Mathlib.Algebra.Ring.Pointwise.Set",
"Mathlib.Topology.Algebra.ConstMulAction",
"Mathlib.Topology.MetricSpace.Isometry",
"Mathlib.Topology.MetricSpace.Lipschitz"
] | Mathlib/Topology/MetricSpace/IsometricSMul.lean | preimage_mul_right_ball | null |
preimage_mul_left_closedBall [IsIsometricSMul G G] (a b : G) (r : ℝ≥0∞) :
(a * ·) ⁻¹' closedBall b r = closedBall (a⁻¹ * b) r :=
preimage_smul_closedBall a b r
@[to_additive (attr := simp)] | theorem | Topology | [
"Mathlib.Algebra.GroupWithZero.Pointwise.Set.Basic",
"Mathlib.Algebra.Ring.Pointwise.Set",
"Mathlib.Topology.Algebra.ConstMulAction",
"Mathlib.Topology.MetricSpace.Isometry",
"Mathlib.Topology.MetricSpace.Lipschitz"
] | Mathlib/Topology/MetricSpace/IsometricSMul.lean | preimage_mul_left_closedBall | null |
preimage_mul_right_closedBall [IsIsometricSMul Gᵐᵒᵖ G] (a b : G) (r : ℝ≥0∞) :
(fun x => x * a) ⁻¹' closedBall b r = closedBall (b / a) r := by
rw [div_eq_mul_inv]
exact preimage_smul_closedBall (MulOpposite.op a) b r | theorem | Topology | [
"Mathlib.Algebra.GroupWithZero.Pointwise.Set.Basic",
"Mathlib.Algebra.Ring.Pointwise.Set",
"Mathlib.Topology.Algebra.ConstMulAction",
"Mathlib.Topology.MetricSpace.Isometry",
"Mathlib.Topology.MetricSpace.Lipschitz"
] | Mathlib/Topology/MetricSpace/IsometricSMul.lean | preimage_mul_right_closedBall | null |
@[to_additive (attr := simp)]
dist_smul [PseudoMetricSpace X] [SMul M X] [IsIsometricSMul M X] (c : M) (x y : X) :
dist (c • x) (c • y) = dist x y :=
(isometry_smul X c).dist_eq x y
@[to_additive (attr := simp)] | theorem | Topology | [
"Mathlib.Algebra.GroupWithZero.Pointwise.Set.Basic",
"Mathlib.Algebra.Ring.Pointwise.Set",
"Mathlib.Topology.Algebra.ConstMulAction",
"Mathlib.Topology.MetricSpace.Isometry",
"Mathlib.Topology.MetricSpace.Lipschitz"
] | Mathlib/Topology/MetricSpace/IsometricSMul.lean | dist_smul | null |
nndist_smul [PseudoMetricSpace X] [SMul M X] [IsIsometricSMul M X] (c : M) (x y : X) :
nndist (c • x) (c • y) = nndist x y :=
(isometry_smul X c).nndist_eq x y
@[to_additive (attr := simp)] | theorem | Topology | [
"Mathlib.Algebra.GroupWithZero.Pointwise.Set.Basic",
"Mathlib.Algebra.Ring.Pointwise.Set",
"Mathlib.Topology.Algebra.ConstMulAction",
"Mathlib.Topology.MetricSpace.Isometry",
"Mathlib.Topology.MetricSpace.Lipschitz"
] | Mathlib/Topology/MetricSpace/IsometricSMul.lean | nndist_smul | null |
diam_smul [PseudoMetricSpace X] [SMul M X] [IsIsometricSMul M X] (c : M) (s : Set X) :
Metric.diam (c • s) = Metric.diam s :=
(isometry_smul _ _).diam_image s
@[to_additive (attr := simp)] | theorem | Topology | [
"Mathlib.Algebra.GroupWithZero.Pointwise.Set.Basic",
"Mathlib.Algebra.Ring.Pointwise.Set",
"Mathlib.Topology.Algebra.ConstMulAction",
"Mathlib.Topology.MetricSpace.Isometry",
"Mathlib.Topology.MetricSpace.Lipschitz"
] | Mathlib/Topology/MetricSpace/IsometricSMul.lean | diam_smul | null |
dist_mul_left [PseudoMetricSpace M] [Mul M] [IsIsometricSMul M M] (a b c : M) :
dist (a * b) (a * c) = dist b c :=
dist_smul a b c
@[to_additive (attr := simp)] | theorem | Topology | [
"Mathlib.Algebra.GroupWithZero.Pointwise.Set.Basic",
"Mathlib.Algebra.Ring.Pointwise.Set",
"Mathlib.Topology.Algebra.ConstMulAction",
"Mathlib.Topology.MetricSpace.Isometry",
"Mathlib.Topology.MetricSpace.Lipschitz"
] | Mathlib/Topology/MetricSpace/IsometricSMul.lean | dist_mul_left | null |
nndist_mul_left [PseudoMetricSpace M] [Mul M] [IsIsometricSMul M M] (a b c : M) :
nndist (a * b) (a * c) = nndist b c :=
nndist_smul a b c
@[to_additive (attr := simp)] | theorem | Topology | [
"Mathlib.Algebra.GroupWithZero.Pointwise.Set.Basic",
"Mathlib.Algebra.Ring.Pointwise.Set",
"Mathlib.Topology.Algebra.ConstMulAction",
"Mathlib.Topology.MetricSpace.Isometry",
"Mathlib.Topology.MetricSpace.Lipschitz"
] | Mathlib/Topology/MetricSpace/IsometricSMul.lean | nndist_mul_left | null |
dist_mul_right [Mul M] [PseudoMetricSpace M] [IsIsometricSMul Mᵐᵒᵖ M] (a b c : M) :
dist (a * c) (b * c) = dist a b :=
dist_smul (MulOpposite.op c) a b
@[to_additive (attr := simp)] | theorem | Topology | [
"Mathlib.Algebra.GroupWithZero.Pointwise.Set.Basic",
"Mathlib.Algebra.Ring.Pointwise.Set",
"Mathlib.Topology.Algebra.ConstMulAction",
"Mathlib.Topology.MetricSpace.Isometry",
"Mathlib.Topology.MetricSpace.Lipschitz"
] | Mathlib/Topology/MetricSpace/IsometricSMul.lean | dist_mul_right | null |
nndist_mul_right [PseudoMetricSpace M] [Mul M] [IsIsometricSMul Mᵐᵒᵖ M] (a b c : M) :
nndist (a * c) (b * c) = nndist a b :=
nndist_smul (MulOpposite.op c) a b
@[to_additive (attr := simp)] | theorem | Topology | [
"Mathlib.Algebra.GroupWithZero.Pointwise.Set.Basic",
"Mathlib.Algebra.Ring.Pointwise.Set",
"Mathlib.Topology.Algebra.ConstMulAction",
"Mathlib.Topology.MetricSpace.Isometry",
"Mathlib.Topology.MetricSpace.Lipschitz"
] | Mathlib/Topology/MetricSpace/IsometricSMul.lean | nndist_mul_right | null |
dist_div_right [DivInvMonoid M] [PseudoMetricSpace M] [IsIsometricSMul Mᵐᵒᵖ M]
(a b c : M) : dist (a / c) (b / c) = dist a b := by simp only [div_eq_mul_inv, dist_mul_right]
@[to_additive (attr := simp)] | theorem | Topology | [
"Mathlib.Algebra.GroupWithZero.Pointwise.Set.Basic",
"Mathlib.Algebra.Ring.Pointwise.Set",
"Mathlib.Topology.Algebra.ConstMulAction",
"Mathlib.Topology.MetricSpace.Isometry",
"Mathlib.Topology.MetricSpace.Lipschitz"
] | Mathlib/Topology/MetricSpace/IsometricSMul.lean | dist_div_right | null |
nndist_div_right [DivInvMonoid M] [PseudoMetricSpace M] [IsIsometricSMul Mᵐᵒᵖ M]
(a b c : M) : nndist (a / c) (b / c) = nndist a b := by
simp only [div_eq_mul_inv, nndist_mul_right]
@[to_additive (attr := simp)] | theorem | Topology | [
"Mathlib.Algebra.GroupWithZero.Pointwise.Set.Basic",
"Mathlib.Algebra.Ring.Pointwise.Set",
"Mathlib.Topology.Algebra.ConstMulAction",
"Mathlib.Topology.MetricSpace.Isometry",
"Mathlib.Topology.MetricSpace.Lipschitz"
] | Mathlib/Topology/MetricSpace/IsometricSMul.lean | nndist_div_right | null |
dist_inv_inv [Group G] [PseudoMetricSpace G] [IsIsometricSMul G G]
[IsIsometricSMul Gᵐᵒᵖ G] (a b : G) : dist a⁻¹ b⁻¹ = dist a b :=
(IsometryEquiv.inv G).dist_eq a b
@[to_additive (attr := simp)] | theorem | Topology | [
"Mathlib.Algebra.GroupWithZero.Pointwise.Set.Basic",
"Mathlib.Algebra.Ring.Pointwise.Set",
"Mathlib.Topology.Algebra.ConstMulAction",
"Mathlib.Topology.MetricSpace.Isometry",
"Mathlib.Topology.MetricSpace.Lipschitz"
] | Mathlib/Topology/MetricSpace/IsometricSMul.lean | dist_inv_inv | null |
nndist_inv_inv [Group G] [PseudoMetricSpace G] [IsIsometricSMul G G]
[IsIsometricSMul Gᵐᵒᵖ G] (a b : G) : nndist a⁻¹ b⁻¹ = nndist a b :=
(IsometryEquiv.inv G).nndist_eq a b
@[to_additive (attr := simp)] | theorem | Topology | [
"Mathlib.Algebra.GroupWithZero.Pointwise.Set.Basic",
"Mathlib.Algebra.Ring.Pointwise.Set",
"Mathlib.Topology.Algebra.ConstMulAction",
"Mathlib.Topology.MetricSpace.Isometry",
"Mathlib.Topology.MetricSpace.Lipschitz"
] | Mathlib/Topology/MetricSpace/IsometricSMul.lean | nndist_inv_inv | null |
dist_div_left [Group G] [PseudoMetricSpace G] [IsIsometricSMul G G]
[IsIsometricSMul Gᵐᵒᵖ G] (a b c : G) : dist (a / b) (a / c) = dist b c := by
simp [div_eq_mul_inv]
@[to_additive (attr := simp)] | theorem | Topology | [
"Mathlib.Algebra.GroupWithZero.Pointwise.Set.Basic",
"Mathlib.Algebra.Ring.Pointwise.Set",
"Mathlib.Topology.Algebra.ConstMulAction",
"Mathlib.Topology.MetricSpace.Isometry",
"Mathlib.Topology.MetricSpace.Lipschitz"
] | Mathlib/Topology/MetricSpace/IsometricSMul.lean | dist_div_left | null |
nndist_div_left [Group G] [PseudoMetricSpace G] [IsIsometricSMul G G]
[IsIsometricSMul Gᵐᵒᵖ G] (a b c : G) : nndist (a / b) (a / c) = nndist b c := by
simp [div_eq_mul_inv] | theorem | Topology | [
"Mathlib.Algebra.GroupWithZero.Pointwise.Set.Basic",
"Mathlib.Algebra.Ring.Pointwise.Set",
"Mathlib.Topology.Algebra.ConstMulAction",
"Mathlib.Topology.MetricSpace.Isometry",
"Mathlib.Topology.MetricSpace.Lipschitz"
] | Mathlib/Topology/MetricSpace/IsometricSMul.lean | nndist_div_left | null |
@[to_additive /-- Given an additive isometric action of `G` on `X`, the image of a bounded set in
`X` under translation by `c : G` is bounded. -/]
Bornology.IsBounded.smul [PseudoMetricSpace X] [SMul G X] [IsIsometricSMul G X] {s : Set X}
(hs : IsBounded s) (c : G) : IsBounded (c • s) :=
(isometry_smul X c).lipschitz.isBounded_image hs | theorem | Topology | [
"Mathlib.Algebra.GroupWithZero.Pointwise.Set.Basic",
"Mathlib.Algebra.Ring.Pointwise.Set",
"Mathlib.Topology.Algebra.ConstMulAction",
"Mathlib.Topology.MetricSpace.Isometry",
"Mathlib.Topology.MetricSpace.Lipschitz"
] | Mathlib/Topology/MetricSpace/IsometricSMul.lean | Bornology.IsBounded.smul | If `G` acts isometrically on `X`, then the image of a bounded set in `X` under scalar
multiplication by `c : G` is bounded. See also `Bornology.IsBounded.smul₀` for a similar lemma about
normed spaces. |
@[to_additive (attr := simp)]
smul_ball (c : G) (x : X) (r : ℝ) : c • ball x r = ball (c • x) r :=
(IsometryEquiv.constSMul c).image_ball _ _
@[to_additive (attr := simp)] | theorem | Topology | [
"Mathlib.Algebra.GroupWithZero.Pointwise.Set.Basic",
"Mathlib.Algebra.Ring.Pointwise.Set",
"Mathlib.Topology.Algebra.ConstMulAction",
"Mathlib.Topology.MetricSpace.Isometry",
"Mathlib.Topology.MetricSpace.Lipschitz"
] | Mathlib/Topology/MetricSpace/IsometricSMul.lean | smul_ball | null |
preimage_smul_ball (c : G) (x : X) (r : ℝ) : (c • ·) ⁻¹' ball x r = ball (c⁻¹ • x) r := by
rw [preimage_smul, smul_ball]
@[to_additive (attr := simp)] | theorem | Topology | [
"Mathlib.Algebra.GroupWithZero.Pointwise.Set.Basic",
"Mathlib.Algebra.Ring.Pointwise.Set",
"Mathlib.Topology.Algebra.ConstMulAction",
"Mathlib.Topology.MetricSpace.Isometry",
"Mathlib.Topology.MetricSpace.Lipschitz"
] | Mathlib/Topology/MetricSpace/IsometricSMul.lean | preimage_smul_ball | null |
smul_closedBall (c : G) (x : X) (r : ℝ) : c • closedBall x r = closedBall (c • x) r :=
(IsometryEquiv.constSMul c).image_closedBall _ _
@[to_additive (attr := simp)] | theorem | Topology | [
"Mathlib.Algebra.GroupWithZero.Pointwise.Set.Basic",
"Mathlib.Algebra.Ring.Pointwise.Set",
"Mathlib.Topology.Algebra.ConstMulAction",
"Mathlib.Topology.MetricSpace.Isometry",
"Mathlib.Topology.MetricSpace.Lipschitz"
] | Mathlib/Topology/MetricSpace/IsometricSMul.lean | smul_closedBall | null |
preimage_smul_closedBall (c : G) (x : X) (r : ℝ) :
(c • ·) ⁻¹' closedBall x r = closedBall (c⁻¹ • x) r := by rw [preimage_smul, smul_closedBall]
@[to_additive (attr := simp)] | theorem | Topology | [
"Mathlib.Algebra.GroupWithZero.Pointwise.Set.Basic",
"Mathlib.Algebra.Ring.Pointwise.Set",
"Mathlib.Topology.Algebra.ConstMulAction",
"Mathlib.Topology.MetricSpace.Isometry",
"Mathlib.Topology.MetricSpace.Lipschitz"
] | Mathlib/Topology/MetricSpace/IsometricSMul.lean | preimage_smul_closedBall | null |
smul_sphere (c : G) (x : X) (r : ℝ) : c • sphere x r = sphere (c • x) r :=
(IsometryEquiv.constSMul c).image_sphere _ _
@[to_additive (attr := simp)] | theorem | Topology | [
"Mathlib.Algebra.GroupWithZero.Pointwise.Set.Basic",
"Mathlib.Algebra.Ring.Pointwise.Set",
"Mathlib.Topology.Algebra.ConstMulAction",
"Mathlib.Topology.MetricSpace.Isometry",
"Mathlib.Topology.MetricSpace.Lipschitz"
] | Mathlib/Topology/MetricSpace/IsometricSMul.lean | smul_sphere | null |
preimage_smul_sphere (c : G) (x : X) (r : ℝ) :
(c • ·) ⁻¹' sphere x r = sphere (c⁻¹ • x) r := by rw [preimage_smul, smul_sphere]
variable [PseudoMetricSpace G]
@[to_additive (attr := simp)] | theorem | Topology | [
"Mathlib.Algebra.GroupWithZero.Pointwise.Set.Basic",
"Mathlib.Algebra.Ring.Pointwise.Set",
"Mathlib.Topology.Algebra.ConstMulAction",
"Mathlib.Topology.MetricSpace.Isometry",
"Mathlib.Topology.MetricSpace.Lipschitz"
] | Mathlib/Topology/MetricSpace/IsometricSMul.lean | preimage_smul_sphere | null |
preimage_mul_left_ball [IsIsometricSMul G G] (a b : G) (r : ℝ) :
(a * ·) ⁻¹' ball b r = ball (a⁻¹ * b) r :=
preimage_smul_ball a b r
@[to_additive (attr := simp)] | theorem | Topology | [
"Mathlib.Algebra.GroupWithZero.Pointwise.Set.Basic",
"Mathlib.Algebra.Ring.Pointwise.Set",
"Mathlib.Topology.Algebra.ConstMulAction",
"Mathlib.Topology.MetricSpace.Isometry",
"Mathlib.Topology.MetricSpace.Lipschitz"
] | Mathlib/Topology/MetricSpace/IsometricSMul.lean | preimage_mul_left_ball | null |
preimage_mul_right_ball [IsIsometricSMul Gᵐᵒᵖ G] (a b : G) (r : ℝ) :
(fun x => x * a) ⁻¹' ball b r = ball (b / a) r := by
rw [div_eq_mul_inv]
exact preimage_smul_ball (MulOpposite.op a) b r
@[to_additive (attr := simp)] | theorem | Topology | [
"Mathlib.Algebra.GroupWithZero.Pointwise.Set.Basic",
"Mathlib.Algebra.Ring.Pointwise.Set",
"Mathlib.Topology.Algebra.ConstMulAction",
"Mathlib.Topology.MetricSpace.Isometry",
"Mathlib.Topology.MetricSpace.Lipschitz"
] | Mathlib/Topology/MetricSpace/IsometricSMul.lean | preimage_mul_right_ball | null |
preimage_mul_left_closedBall [IsIsometricSMul G G] (a b : G) (r : ℝ) :
(a * ·) ⁻¹' closedBall b r = closedBall (a⁻¹ * b) r :=
preimage_smul_closedBall a b r
@[to_additive (attr := simp)] | theorem | Topology | [
"Mathlib.Algebra.GroupWithZero.Pointwise.Set.Basic",
"Mathlib.Algebra.Ring.Pointwise.Set",
"Mathlib.Topology.Algebra.ConstMulAction",
"Mathlib.Topology.MetricSpace.Isometry",
"Mathlib.Topology.MetricSpace.Lipschitz"
] | Mathlib/Topology/MetricSpace/IsometricSMul.lean | preimage_mul_left_closedBall | null |
preimage_mul_right_closedBall [IsIsometricSMul Gᵐᵒᵖ G] (a b : G) (r : ℝ) :
(fun x => x * a) ⁻¹' closedBall b r = closedBall (b / a) r := by
rw [div_eq_mul_inv]
exact preimage_smul_closedBall (MulOpposite.op a) b r | theorem | Topology | [
"Mathlib.Algebra.GroupWithZero.Pointwise.Set.Basic",
"Mathlib.Algebra.Ring.Pointwise.Set",
"Mathlib.Topology.Algebra.ConstMulAction",
"Mathlib.Topology.MetricSpace.Isometry",
"Mathlib.Topology.MetricSpace.Lipschitz"
] | Mathlib/Topology/MetricSpace/IsometricSMul.lean | preimage_mul_right_closedBall | null |
@[to_additive]
Prod.instIsIsometricSMul [SMul M Y] [IsIsometricSMul M Y] : IsIsometricSMul M (X × Y) :=
⟨fun c => (isometry_smul X c).prodMap (isometry_smul Y c)⟩
@[to_additive] | instance | Topology | [
"Mathlib.Algebra.GroupWithZero.Pointwise.Set.Basic",
"Mathlib.Algebra.Ring.Pointwise.Set",
"Mathlib.Topology.Algebra.ConstMulAction",
"Mathlib.Topology.MetricSpace.Isometry",
"Mathlib.Topology.MetricSpace.Lipschitz"
] | Mathlib/Topology/MetricSpace/IsometricSMul.lean | Prod.instIsIsometricSMul | null |
Prod.isIsometricSMul' {N} [Mul M] [PseudoEMetricSpace M] [IsIsometricSMul M M] [Mul N]
[PseudoEMetricSpace N] [IsIsometricSMul N N] : IsIsometricSMul (M × N) (M × N) :=
⟨fun c => (isometry_smul M c.1).prodMap (isometry_smul N c.2)⟩
@[to_additive] | instance | Topology | [
"Mathlib.Algebra.GroupWithZero.Pointwise.Set.Basic",
"Mathlib.Algebra.Ring.Pointwise.Set",
"Mathlib.Topology.Algebra.ConstMulAction",
"Mathlib.Topology.MetricSpace.Isometry",
"Mathlib.Topology.MetricSpace.Lipschitz"
] | Mathlib/Topology/MetricSpace/IsometricSMul.lean | Prod.isIsometricSMul' | null |
Prod.isIsometricSMul'' {N} [Mul M] [PseudoEMetricSpace M] [IsIsometricSMul Mᵐᵒᵖ M]
[Mul N] [PseudoEMetricSpace N] [IsIsometricSMul Nᵐᵒᵖ N] :
IsIsometricSMul (M × N)ᵐᵒᵖ (M × N) :=
⟨fun c => (isometry_mul_right c.unop.1).prodMap (isometry_mul_right c.unop.2)⟩
@[to_additive] | instance | Topology | [
"Mathlib.Algebra.GroupWithZero.Pointwise.Set.Basic",
"Mathlib.Algebra.Ring.Pointwise.Set",
"Mathlib.Topology.Algebra.ConstMulAction",
"Mathlib.Topology.MetricSpace.Isometry",
"Mathlib.Topology.MetricSpace.Lipschitz"
] | Mathlib/Topology/MetricSpace/IsometricSMul.lean | Prod.isIsometricSMul'' | null |
Units.isIsometricSMul [Monoid M] : IsIsometricSMul Mˣ X :=
⟨fun c => isometry_smul X (c : M)⟩
@[to_additive] | instance | Topology | [
"Mathlib.Algebra.GroupWithZero.Pointwise.Set.Basic",
"Mathlib.Algebra.Ring.Pointwise.Set",
"Mathlib.Topology.Algebra.ConstMulAction",
"Mathlib.Topology.MetricSpace.Isometry",
"Mathlib.Topology.MetricSpace.Lipschitz"
] | Mathlib/Topology/MetricSpace/IsometricSMul.lean | Units.isIsometricSMul | null |
@[to_additive]
ULift.isIsometricSMul : IsIsometricSMul (ULift M) X :=
⟨fun c => by simpa only using isometry_smul X c.down⟩
@[to_additive] | instance | Topology | [
"Mathlib.Algebra.GroupWithZero.Pointwise.Set.Basic",
"Mathlib.Algebra.Ring.Pointwise.Set",
"Mathlib.Topology.Algebra.ConstMulAction",
"Mathlib.Topology.MetricSpace.Isometry",
"Mathlib.Topology.MetricSpace.Lipschitz"
] | Mathlib/Topology/MetricSpace/IsometricSMul.lean | ULift.isIsometricSMul | null |
ULift.isIsometricSMul' : IsIsometricSMul M (ULift X) :=
⟨fun c x y => by simpa only using edist_smul_left c x.1 y.1⟩
@[to_additive] | instance | Topology | [
"Mathlib.Algebra.GroupWithZero.Pointwise.Set.Basic",
"Mathlib.Algebra.Ring.Pointwise.Set",
"Mathlib.Topology.Algebra.ConstMulAction",
"Mathlib.Topology.MetricSpace.Isometry",
"Mathlib.Topology.MetricSpace.Lipschitz"
] | Mathlib/Topology/MetricSpace/IsometricSMul.lean | ULift.isIsometricSMul' | null |
@[to_additive]
Pi.isIsometricSMul' {ι} {M X : ι → Type*} [Fintype ι] [∀ i, SMul (M i) (X i)]
[∀ i, PseudoEMetricSpace (X i)] [∀ i, IsIsometricSMul (M i) (X i)] :
IsIsometricSMul (∀ i, M i) (∀ i, X i) :=
⟨fun c => .piMap (fun i => (c i • ·)) fun _ => isometry_smul _ _⟩
@[to_additive] | instance | Topology | [
"Mathlib.Algebra.GroupWithZero.Pointwise.Set.Basic",
"Mathlib.Algebra.Ring.Pointwise.Set",
"Mathlib.Topology.Algebra.ConstMulAction",
"Mathlib.Topology.MetricSpace.Isometry",
"Mathlib.Topology.MetricSpace.Lipschitz"
] | Mathlib/Topology/MetricSpace/IsometricSMul.lean | Pi.isIsometricSMul' | null |
Pi.isIsometricSMul'' {ι} {M : ι → Type*} [Fintype ι] [∀ i, Mul (M i)]
[∀ i, PseudoEMetricSpace (M i)] [∀ i, IsIsometricSMul (M i)ᵐᵒᵖ (M i)] :
IsIsometricSMul (∀ i, M i)ᵐᵒᵖ (∀ i, M i) :=
⟨fun c => .piMap (fun i (x : M i) => x * c.unop i) fun _ => isometry_mul_right _⟩ | instance | Topology | [
"Mathlib.Algebra.GroupWithZero.Pointwise.Set.Basic",
"Mathlib.Algebra.Ring.Pointwise.Set",
"Mathlib.Topology.Algebra.ConstMulAction",
"Mathlib.Topology.MetricSpace.Isometry",
"Mathlib.Topology.MetricSpace.Lipschitz"
] | Mathlib/Topology/MetricSpace/IsometricSMul.lean | Pi.isIsometricSMul'' | null |
Additive.isIsIsometricVAdd : IsIsometricVAdd (Additive M) X :=
⟨fun c => isometry_smul X c.toMul⟩ | instance | Topology | [
"Mathlib.Algebra.GroupWithZero.Pointwise.Set.Basic",
"Mathlib.Algebra.Ring.Pointwise.Set",
"Mathlib.Topology.Algebra.ConstMulAction",
"Mathlib.Topology.MetricSpace.Isometry",
"Mathlib.Topology.MetricSpace.Lipschitz"
] | Mathlib/Topology/MetricSpace/IsometricSMul.lean | Additive.isIsIsometricVAdd | null |
Additive.isIsIsometricVAdd' [Mul M] [PseudoEMetricSpace M] [IsIsometricSMul M M] :
IsIsometricVAdd (Additive M) (Additive M) :=
⟨fun c x y => edist_smul_left c.toMul x.toMul y.toMul⟩ | instance | Topology | [
"Mathlib.Algebra.GroupWithZero.Pointwise.Set.Basic",
"Mathlib.Algebra.Ring.Pointwise.Set",
"Mathlib.Topology.Algebra.ConstMulAction",
"Mathlib.Topology.MetricSpace.Isometry",
"Mathlib.Topology.MetricSpace.Lipschitz"
] | Mathlib/Topology/MetricSpace/IsometricSMul.lean | Additive.isIsIsometricVAdd' | null |
Additive.isIsIsometricVAdd'' [Mul M] [PseudoEMetricSpace M] [IsIsometricSMul Mᵐᵒᵖ M] :
IsIsometricVAdd (Additive M)ᵃᵒᵖ (Additive M) :=
⟨fun c x y => edist_smul_left (MulOpposite.op c.unop.toMul) x.toMul y.toMul⟩ | instance | Topology | [
"Mathlib.Algebra.GroupWithZero.Pointwise.Set.Basic",
"Mathlib.Algebra.Ring.Pointwise.Set",
"Mathlib.Topology.Algebra.ConstMulAction",
"Mathlib.Topology.MetricSpace.Isometry",
"Mathlib.Topology.MetricSpace.Lipschitz"
] | Mathlib/Topology/MetricSpace/IsometricSMul.lean | Additive.isIsIsometricVAdd'' | null |
Multiplicative.isIsometricSMul {M X} [VAdd M X] [PseudoEMetricSpace X]
[IsIsometricVAdd M X] : IsIsometricSMul (Multiplicative M) X :=
⟨fun c => isometry_vadd X c.toAdd⟩ | instance | Topology | [
"Mathlib.Algebra.GroupWithZero.Pointwise.Set.Basic",
"Mathlib.Algebra.Ring.Pointwise.Set",
"Mathlib.Topology.Algebra.ConstMulAction",
"Mathlib.Topology.MetricSpace.Isometry",
"Mathlib.Topology.MetricSpace.Lipschitz"
] | Mathlib/Topology/MetricSpace/IsometricSMul.lean | Multiplicative.isIsometricSMul | null |
Multiplicative.isIsometricSMul' [Add M] [PseudoEMetricSpace M] [IsIsometricVAdd M M] :
IsIsometricSMul (Multiplicative M) (Multiplicative M) :=
⟨fun c x y => edist_vadd_left c.toAdd x.toAdd y.toAdd⟩ | instance | Topology | [
"Mathlib.Algebra.GroupWithZero.Pointwise.Set.Basic",
"Mathlib.Algebra.Ring.Pointwise.Set",
"Mathlib.Topology.Algebra.ConstMulAction",
"Mathlib.Topology.MetricSpace.Isometry",
"Mathlib.Topology.MetricSpace.Lipschitz"
] | Mathlib/Topology/MetricSpace/IsometricSMul.lean | Multiplicative.isIsometricSMul' | null |
Multiplicative.isIsIsometricVAdd'' [Add M] [PseudoEMetricSpace M]
[IsIsometricVAdd Mᵃᵒᵖ M] : IsIsometricSMul (Multiplicative M)ᵐᵒᵖ (Multiplicative M) :=
⟨fun c x y => edist_vadd_left (AddOpposite.op c.unop.toAdd) x.toAdd y.toAdd⟩ | instance | Topology | [
"Mathlib.Algebra.GroupWithZero.Pointwise.Set.Basic",
"Mathlib.Algebra.Ring.Pointwise.Set",
"Mathlib.Topology.Algebra.ConstMulAction",
"Mathlib.Topology.MetricSpace.Isometry",
"Mathlib.Topology.MetricSpace.Lipschitz"
] | Mathlib/Topology/MetricSpace/IsometricSMul.lean | Multiplicative.isIsIsometricVAdd'' | null |
Isometry [PseudoEMetricSpace α] [PseudoEMetricSpace β] (f : α → β) : Prop :=
∀ x1 x2 : α, edist (f x1) (f x2) = edist x1 x2 | def | Topology | [
"Mathlib.Data.Fintype.Lattice",
"Mathlib.Data.Fintype.Sum",
"Mathlib.Topology.Homeomorph.Lemmas",
"Mathlib.Topology.MetricSpace.Antilipschitz"
] | Mathlib/Topology/MetricSpace/Isometry.lean | Isometry | An isometry (also known as isometric embedding) is a map preserving the edistance
between pseudoemetric spaces, or equivalently the distance between pseudometric space. |
isometry_iff_nndist_eq [PseudoMetricSpace α] [PseudoMetricSpace β] {f : α → β} :
Isometry f ↔ ∀ x y, nndist (f x) (f y) = nndist x y := by
simp only [Isometry, edist_nndist, ENNReal.coe_inj] | theorem | Topology | [
"Mathlib.Data.Fintype.Lattice",
"Mathlib.Data.Fintype.Sum",
"Mathlib.Topology.Homeomorph.Lemmas",
"Mathlib.Topology.MetricSpace.Antilipschitz"
] | Mathlib/Topology/MetricSpace/Isometry.lean | isometry_iff_nndist_eq | On pseudometric spaces, a map is an isometry if and only if it preserves nonnegative
distances. |
isometry_iff_dist_eq [PseudoMetricSpace α] [PseudoMetricSpace β] {f : α → β} :
Isometry f ↔ ∀ x y, dist (f x) (f y) = dist x y := by
simp only [isometry_iff_nndist_eq, ← coe_nndist, NNReal.coe_inj] | theorem | Topology | [
"Mathlib.Data.Fintype.Lattice",
"Mathlib.Data.Fintype.Sum",
"Mathlib.Topology.Homeomorph.Lemmas",
"Mathlib.Topology.MetricSpace.Antilipschitz"
] | Mathlib/Topology/MetricSpace/Isometry.lean | isometry_iff_dist_eq | On pseudometric spaces, a map is an isometry if and only if it preserves distances. |
edist_eq (hf : Isometry f) (x y : α) : edist (f x) (f y) = edist x y :=
hf x y | theorem | Topology | [
"Mathlib.Data.Fintype.Lattice",
"Mathlib.Data.Fintype.Sum",
"Mathlib.Topology.Homeomorph.Lemmas",
"Mathlib.Topology.MetricSpace.Antilipschitz"
] | Mathlib/Topology/MetricSpace/Isometry.lean | edist_eq | An isometry preserves distances. -/
alias ⟨Isometry.dist_eq, _⟩ := isometry_iff_dist_eq
/-- A map that preserves distances is an isometry -/
alias ⟨_, Isometry.of_dist_eq⟩ := isometry_iff_dist_eq
/-- An isometry preserves non-negative distances. -/
alias ⟨Isometry.nndist_eq, _⟩ := isometry_iff_nndist_eq
/-- A map that preserves non-negative distances is an isometry. -/
alias ⟨_, Isometry.of_nndist_eq⟩ := isometry_iff_nndist_eq
namespace Isometry
section PseudoEmetricIsometry
variable [PseudoEMetricSpace α] [PseudoEMetricSpace β] [PseudoEMetricSpace γ]
variable {f : α → β} {x : α}
/-- An isometry preserves edistances. |
lipschitz (h : Isometry f) : LipschitzWith 1 f :=
LipschitzWith.of_edist_le fun x y => (h x y).le | theorem | Topology | [
"Mathlib.Data.Fintype.Lattice",
"Mathlib.Data.Fintype.Sum",
"Mathlib.Topology.Homeomorph.Lemmas",
"Mathlib.Topology.MetricSpace.Antilipschitz"
] | Mathlib/Topology/MetricSpace/Isometry.lean | lipschitz | null |
antilipschitz (h : Isometry f) : AntilipschitzWith 1 f := fun x y => by
simp only [h x y, ENNReal.coe_one, one_mul, le_refl] | theorem | Topology | [
"Mathlib.Data.Fintype.Lattice",
"Mathlib.Data.Fintype.Sum",
"Mathlib.Topology.Homeomorph.Lemmas",
"Mathlib.Topology.MetricSpace.Antilipschitz"
] | Mathlib/Topology/MetricSpace/Isometry.lean | antilipschitz | null |
@[nontriviality]
_root_.isometry_subsingleton [Subsingleton α] : Isometry f := fun x y => by
rw [Subsingleton.elim x y]; simp | theorem | Topology | [
"Mathlib.Data.Fintype.Lattice",
"Mathlib.Data.Fintype.Sum",
"Mathlib.Topology.Homeomorph.Lemmas",
"Mathlib.Topology.MetricSpace.Antilipschitz"
] | Mathlib/Topology/MetricSpace/Isometry.lean | _root_.isometry_subsingleton | Any map on a subsingleton is an isometry |
_root_.isometry_id : Isometry (id : α → α) := fun _ _ => rfl | theorem | Topology | [
"Mathlib.Data.Fintype.Lattice",
"Mathlib.Data.Fintype.Sum",
"Mathlib.Topology.Homeomorph.Lemmas",
"Mathlib.Topology.MetricSpace.Antilipschitz"
] | Mathlib/Topology/MetricSpace/Isometry.lean | _root_.isometry_id | The identity is an isometry |
prodMap {δ} [PseudoEMetricSpace δ] {f : α → β} {g : γ → δ} (hf : Isometry f)
(hg : Isometry g) : Isometry (Prod.map f g) := fun x y => by
simp only [Prod.edist_eq, Prod.map_fst, hf.edist_eq, Prod.map_snd, hg.edist_eq]
@[deprecated (since := "2025-04-18")]
alias prod_map := prodMap | theorem | Topology | [
"Mathlib.Data.Fintype.Lattice",
"Mathlib.Data.Fintype.Sum",
"Mathlib.Topology.Homeomorph.Lemmas",
"Mathlib.Topology.MetricSpace.Antilipschitz"
] | Mathlib/Topology/MetricSpace/Isometry.lean | prodMap | null |
protected piMap {ι} [Fintype ι] {α β : ι → Type*} [∀ i, PseudoEMetricSpace (α i)]
[∀ i, PseudoEMetricSpace (β i)] (f : ∀ i, α i → β i) (hf : ∀ i, Isometry (f i)) :
Isometry (Pi.map f) := fun x y => by
simp only [edist_pi_def, (hf _).edist_eq, Pi.map_apply] | theorem | Topology | [
"Mathlib.Data.Fintype.Lattice",
"Mathlib.Data.Fintype.Sum",
"Mathlib.Topology.Homeomorph.Lemmas",
"Mathlib.Topology.MetricSpace.Antilipschitz"
] | Mathlib/Topology/MetricSpace/Isometry.lean | piMap | null |
protected single [Fintype ι] [DecidableEq ι] {E : ι → Type*} [∀ i, PseudoEMetricSpace (E i)]
[∀ i, Zero (E i)] (i : ι) :
Isometry (Pi.single (M := E) i) := by
intro x y
rw [edist_pi_def]
refine le_antisymm (Finset.sup_le fun j ↦ ?_) (Finset.le_sup_of_le (Finset.mem_univ i) (by simp))
obtain rfl | h := eq_or_ne i j
· simp
· simp [h] | lemma | Topology | [
"Mathlib.Data.Fintype.Lattice",
"Mathlib.Data.Fintype.Sum",
"Mathlib.Topology.Homeomorph.Lemmas",
"Mathlib.Topology.MetricSpace.Antilipschitz"
] | Mathlib/Topology/MetricSpace/Isometry.lean | single | null |
protected inl [AddZeroClass α] [AddZeroClass β] : Isometry (AddMonoidHom.inl α β) := by
intro x y
rw [Prod.edist_eq]
simp | lemma | Topology | [
"Mathlib.Data.Fintype.Lattice",
"Mathlib.Data.Fintype.Sum",
"Mathlib.Topology.Homeomorph.Lemmas",
"Mathlib.Topology.MetricSpace.Antilipschitz"
] | Mathlib/Topology/MetricSpace/Isometry.lean | inl | null |
protected inr [AddZeroClass α] [AddZeroClass β] : Isometry (AddMonoidHom.inr α β) := by
intro x y
rw [Prod.edist_eq]
simp | lemma | Topology | [
"Mathlib.Data.Fintype.Lattice",
"Mathlib.Data.Fintype.Sum",
"Mathlib.Topology.Homeomorph.Lemmas",
"Mathlib.Topology.MetricSpace.Antilipschitz"
] | Mathlib/Topology/MetricSpace/Isometry.lean | inr | null |
comp {g : β → γ} {f : α → β} (hg : Isometry g) (hf : Isometry f) : Isometry (g ∘ f) :=
fun _ _ => (hg _ _).trans (hf _ _) | theorem | Topology | [
"Mathlib.Data.Fintype.Lattice",
"Mathlib.Data.Fintype.Sum",
"Mathlib.Topology.Homeomorph.Lemmas",
"Mathlib.Topology.MetricSpace.Antilipschitz"
] | Mathlib/Topology/MetricSpace/Isometry.lean | comp | The composition of isometries is an isometry. |
protected uniformContinuous (hf : Isometry f) : UniformContinuous f :=
hf.lipschitz.uniformContinuous | theorem | Topology | [
"Mathlib.Data.Fintype.Lattice",
"Mathlib.Data.Fintype.Sum",
"Mathlib.Topology.Homeomorph.Lemmas",
"Mathlib.Topology.MetricSpace.Antilipschitz"
] | Mathlib/Topology/MetricSpace/Isometry.lean | uniformContinuous | An isometry from a metric space is a uniform continuous map |
isUniformInducing (hf : Isometry f) : IsUniformInducing f :=
hf.antilipschitz.isUniformInducing hf.uniformContinuous | theorem | Topology | [
"Mathlib.Data.Fintype.Lattice",
"Mathlib.Data.Fintype.Sum",
"Mathlib.Topology.Homeomorph.Lemmas",
"Mathlib.Topology.MetricSpace.Antilipschitz"
] | Mathlib/Topology/MetricSpace/Isometry.lean | isUniformInducing | An isometry from a metric space is a uniform inducing map |
tendsto_nhds_iff {ι : Type*} {f : α → β} {g : ι → α} {a : Filter ι} {b : α}
(hf : Isometry f) : Filter.Tendsto g a (𝓝 b) ↔ Filter.Tendsto (f ∘ g) a (𝓝 (f b)) :=
hf.isUniformInducing.isInducing.tendsto_nhds_iff | theorem | Topology | [
"Mathlib.Data.Fintype.Lattice",
"Mathlib.Data.Fintype.Sum",
"Mathlib.Topology.Homeomorph.Lemmas",
"Mathlib.Topology.MetricSpace.Antilipschitz"
] | Mathlib/Topology/MetricSpace/Isometry.lean | tendsto_nhds_iff | null |
protected continuous (hf : Isometry f) : Continuous f :=
hf.lipschitz.continuous | theorem | Topology | [
"Mathlib.Data.Fintype.Lattice",
"Mathlib.Data.Fintype.Sum",
"Mathlib.Topology.Homeomorph.Lemmas",
"Mathlib.Topology.MetricSpace.Antilipschitz"
] | Mathlib/Topology/MetricSpace/Isometry.lean | continuous | An isometry is continuous. |
right_inv {f : α → β} {g : β → α} (h : Isometry f) (hg : RightInverse g f) : Isometry g :=
fun x y => by rw [← h, hg _, hg _] | theorem | Topology | [
"Mathlib.Data.Fintype.Lattice",
"Mathlib.Data.Fintype.Sum",
"Mathlib.Topology.Homeomorph.Lemmas",
"Mathlib.Topology.MetricSpace.Antilipschitz"
] | Mathlib/Topology/MetricSpace/Isometry.lean | right_inv | The right inverse of an isometry is an isometry. |
preimage_emetric_closedBall (h : Isometry f) (x : α) (r : ℝ≥0∞) :
f ⁻¹' EMetric.closedBall (f x) r = EMetric.closedBall x r := by
ext y
simp [h.edist_eq] | theorem | Topology | [
"Mathlib.Data.Fintype.Lattice",
"Mathlib.Data.Fintype.Sum",
"Mathlib.Topology.Homeomorph.Lemmas",
"Mathlib.Topology.MetricSpace.Antilipschitz"
] | Mathlib/Topology/MetricSpace/Isometry.lean | preimage_emetric_closedBall | null |
preimage_emetric_ball (h : Isometry f) (x : α) (r : ℝ≥0∞) :
f ⁻¹' EMetric.ball (f x) r = EMetric.ball x r := by
ext y
simp [h.edist_eq] | theorem | Topology | [
"Mathlib.Data.Fintype.Lattice",
"Mathlib.Data.Fintype.Sum",
"Mathlib.Topology.Homeomorph.Lemmas",
"Mathlib.Topology.MetricSpace.Antilipschitz"
] | Mathlib/Topology/MetricSpace/Isometry.lean | preimage_emetric_ball | null |
ediam_image (hf : Isometry f) (s : Set α) : EMetric.diam (f '' s) = EMetric.diam s :=
eq_of_forall_ge_iff fun d => by simp only [EMetric.diam_le_iff, forall_mem_image, hf.edist_eq] | theorem | Topology | [
"Mathlib.Data.Fintype.Lattice",
"Mathlib.Data.Fintype.Sum",
"Mathlib.Topology.Homeomorph.Lemmas",
"Mathlib.Topology.MetricSpace.Antilipschitz"
] | Mathlib/Topology/MetricSpace/Isometry.lean | ediam_image | Isometries preserve the diameter in pseudoemetric spaces. |
ediam_range (hf : Isometry f) : EMetric.diam (range f) = EMetric.diam (univ : Set α) := by
rw [← image_univ]
exact hf.ediam_image univ | theorem | Topology | [
"Mathlib.Data.Fintype.Lattice",
"Mathlib.Data.Fintype.Sum",
"Mathlib.Topology.Homeomorph.Lemmas",
"Mathlib.Topology.MetricSpace.Antilipschitz"
] | Mathlib/Topology/MetricSpace/Isometry.lean | ediam_range | null |
mapsTo_emetric_ball (hf : Isometry f) (x : α) (r : ℝ≥0∞) :
MapsTo f (EMetric.ball x r) (EMetric.ball (f x) r) :=
(hf.preimage_emetric_ball x r).ge | theorem | Topology | [
"Mathlib.Data.Fintype.Lattice",
"Mathlib.Data.Fintype.Sum",
"Mathlib.Topology.Homeomorph.Lemmas",
"Mathlib.Topology.MetricSpace.Antilipschitz"
] | Mathlib/Topology/MetricSpace/Isometry.lean | mapsTo_emetric_ball | null |
mapsTo_emetric_closedBall (hf : Isometry f) (x : α) (r : ℝ≥0∞) :
MapsTo f (EMetric.closedBall x r) (EMetric.closedBall (f x) r) :=
(hf.preimage_emetric_closedBall x r).ge | theorem | Topology | [
"Mathlib.Data.Fintype.Lattice",
"Mathlib.Data.Fintype.Sum",
"Mathlib.Topology.Homeomorph.Lemmas",
"Mathlib.Topology.MetricSpace.Antilipschitz"
] | Mathlib/Topology/MetricSpace/Isometry.lean | mapsTo_emetric_closedBall | null |
_root_.isometry_subtype_coe {s : Set α} : Isometry ((↑) : s → α) := fun _ _ => rfl | theorem | Topology | [
"Mathlib.Data.Fintype.Lattice",
"Mathlib.Data.Fintype.Sum",
"Mathlib.Topology.Homeomorph.Lemmas",
"Mathlib.Topology.MetricSpace.Antilipschitz"
] | Mathlib/Topology/MetricSpace/Isometry.lean | _root_.isometry_subtype_coe | The injection from a subtype is an isometry |
comp_continuousOn_iff {γ} [TopologicalSpace γ] (hf : Isometry f) {g : γ → α} {s : Set γ} :
ContinuousOn (f ∘ g) s ↔ ContinuousOn g s :=
hf.isUniformInducing.isInducing.continuousOn_iff.symm | theorem | Topology | [
"Mathlib.Data.Fintype.Lattice",
"Mathlib.Data.Fintype.Sum",
"Mathlib.Topology.Homeomorph.Lemmas",
"Mathlib.Topology.MetricSpace.Antilipschitz"
] | Mathlib/Topology/MetricSpace/Isometry.lean | comp_continuousOn_iff | null |
comp_continuous_iff {γ} [TopologicalSpace γ] (hf : Isometry f) {g : γ → α} :
Continuous (f ∘ g) ↔ Continuous g :=
hf.isUniformInducing.isInducing.continuous_iff.symm | theorem | Topology | [
"Mathlib.Data.Fintype.Lattice",
"Mathlib.Data.Fintype.Sum",
"Mathlib.Topology.Homeomorph.Lemmas",
"Mathlib.Topology.MetricSpace.Antilipschitz"
] | Mathlib/Topology/MetricSpace/Isometry.lean | comp_continuous_iff | null |
protected injective (h : Isometry f) : Injective f :=
h.antilipschitz.injective | theorem | Topology | [
"Mathlib.Data.Fintype.Lattice",
"Mathlib.Data.Fintype.Sum",
"Mathlib.Topology.Homeomorph.Lemmas",
"Mathlib.Topology.MetricSpace.Antilipschitz"
] | Mathlib/Topology/MetricSpace/Isometry.lean | injective | An isometry from an emetric space is injective |
isUniformEmbedding (hf : Isometry f) : IsUniformEmbedding f :=
hf.antilipschitz.isUniformEmbedding hf.lipschitz.uniformContinuous | lemma | Topology | [
"Mathlib.Data.Fintype.Lattice",
"Mathlib.Data.Fintype.Sum",
"Mathlib.Topology.Homeomorph.Lemmas",
"Mathlib.Topology.MetricSpace.Antilipschitz"
] | Mathlib/Topology/MetricSpace/Isometry.lean | isUniformEmbedding | An isometry from an emetric space is a uniform embedding |
isEmbedding (hf : Isometry f) : IsEmbedding f := hf.isUniformEmbedding.isEmbedding | theorem | Topology | [
"Mathlib.Data.Fintype.Lattice",
"Mathlib.Data.Fintype.Sum",
"Mathlib.Topology.Homeomorph.Lemmas",
"Mathlib.Topology.MetricSpace.Antilipschitz"
] | Mathlib/Topology/MetricSpace/Isometry.lean | isEmbedding | An isometry from an emetric space is an embedding |
isClosedEmbedding [CompleteSpace α] [EMetricSpace γ] {f : α → γ} (hf : Isometry f) :
IsClosedEmbedding f :=
hf.antilipschitz.isClosedEmbedding hf.lipschitz.uniformContinuous | theorem | Topology | [
"Mathlib.Data.Fintype.Lattice",
"Mathlib.Data.Fintype.Sum",
"Mathlib.Topology.Homeomorph.Lemmas",
"Mathlib.Topology.MetricSpace.Antilipschitz"
] | Mathlib/Topology/MetricSpace/Isometry.lean | isClosedEmbedding | An isometry from a complete emetric space is a closed embedding |
diam_image (hf : Isometry f) (s : Set α) : Metric.diam (f '' s) = Metric.diam s := by
rw [Metric.diam, Metric.diam, hf.ediam_image] | theorem | Topology | [
"Mathlib.Data.Fintype.Lattice",
"Mathlib.Data.Fintype.Sum",
"Mathlib.Topology.Homeomorph.Lemmas",
"Mathlib.Topology.MetricSpace.Antilipschitz"
] | Mathlib/Topology/MetricSpace/Isometry.lean | diam_image | An isometry preserves the diameter in pseudometric spaces. |
diam_range (hf : Isometry f) : Metric.diam (range f) = Metric.diam (univ : Set α) := by
rw [← image_univ]
exact hf.diam_image univ | theorem | Topology | [
"Mathlib.Data.Fintype.Lattice",
"Mathlib.Data.Fintype.Sum",
"Mathlib.Topology.Homeomorph.Lemmas",
"Mathlib.Topology.MetricSpace.Antilipschitz"
] | Mathlib/Topology/MetricSpace/Isometry.lean | diam_range | null |
preimage_setOf_dist (hf : Isometry f) (x : α) (p : ℝ → Prop) :
f ⁻¹' { y | p (dist y (f x)) } = { y | p (dist y x) } := by
simp [hf.dist_eq] | theorem | Topology | [
"Mathlib.Data.Fintype.Lattice",
"Mathlib.Data.Fintype.Sum",
"Mathlib.Topology.Homeomorph.Lemmas",
"Mathlib.Topology.MetricSpace.Antilipschitz"
] | Mathlib/Topology/MetricSpace/Isometry.lean | preimage_setOf_dist | null |
preimage_closedBall (hf : Isometry f) (x : α) (r : ℝ) :
f ⁻¹' Metric.closedBall (f x) r = Metric.closedBall x r :=
hf.preimage_setOf_dist x (· ≤ r) | theorem | Topology | [
"Mathlib.Data.Fintype.Lattice",
"Mathlib.Data.Fintype.Sum",
"Mathlib.Topology.Homeomorph.Lemmas",
"Mathlib.Topology.MetricSpace.Antilipschitz"
] | Mathlib/Topology/MetricSpace/Isometry.lean | preimage_closedBall | null |
preimage_ball (hf : Isometry f) (x : α) (r : ℝ) :
f ⁻¹' Metric.ball (f x) r = Metric.ball x r :=
hf.preimage_setOf_dist x (· < r) | theorem | Topology | [
"Mathlib.Data.Fintype.Lattice",
"Mathlib.Data.Fintype.Sum",
"Mathlib.Topology.Homeomorph.Lemmas",
"Mathlib.Topology.MetricSpace.Antilipschitz"
] | Mathlib/Topology/MetricSpace/Isometry.lean | preimage_ball | null |
preimage_sphere (hf : Isometry f) (x : α) (r : ℝ) :
f ⁻¹' Metric.sphere (f x) r = Metric.sphere x r :=
hf.preimage_setOf_dist x (· = r) | theorem | Topology | [
"Mathlib.Data.Fintype.Lattice",
"Mathlib.Data.Fintype.Sum",
"Mathlib.Topology.Homeomorph.Lemmas",
"Mathlib.Topology.MetricSpace.Antilipschitz"
] | Mathlib/Topology/MetricSpace/Isometry.lean | preimage_sphere | null |
mapsTo_ball (hf : Isometry f) (x : α) (r : ℝ) :
MapsTo f (Metric.ball x r) (Metric.ball (f x) r) :=
(hf.preimage_ball x r).ge | theorem | Topology | [
"Mathlib.Data.Fintype.Lattice",
"Mathlib.Data.Fintype.Sum",
"Mathlib.Topology.Homeomorph.Lemmas",
"Mathlib.Topology.MetricSpace.Antilipschitz"
] | Mathlib/Topology/MetricSpace/Isometry.lean | mapsTo_ball | null |
mapsTo_sphere (hf : Isometry f) (x : α) (r : ℝ) :
MapsTo f (Metric.sphere x r) (Metric.sphere (f x) r) :=
(hf.preimage_sphere x r).ge | theorem | Topology | [
"Mathlib.Data.Fintype.Lattice",
"Mathlib.Data.Fintype.Sum",
"Mathlib.Topology.Homeomorph.Lemmas",
"Mathlib.Topology.MetricSpace.Antilipschitz"
] | Mathlib/Topology/MetricSpace/Isometry.lean | mapsTo_sphere | null |
mapsTo_closedBall (hf : Isometry f) (x : α) (r : ℝ) :
MapsTo f (Metric.closedBall x r) (Metric.closedBall (f x) r) :=
(hf.preimage_closedBall x r).ge | theorem | Topology | [
"Mathlib.Data.Fintype.Lattice",
"Mathlib.Data.Fintype.Sum",
"Mathlib.Topology.Homeomorph.Lemmas",
"Mathlib.Topology.MetricSpace.Antilipschitz"
] | Mathlib/Topology/MetricSpace/Isometry.lean | mapsTo_closedBall | null |
IsUniformEmbedding.to_isometry {α β} [UniformSpace α] [MetricSpace β] {f : α → β}
(h : IsUniformEmbedding f) : (letI := h.comapMetricSpace f; Isometry f) :=
let _ := h.comapMetricSpace f
Isometry.of_dist_eq fun _ _ => rfl | theorem | Topology | [
"Mathlib.Data.Fintype.Lattice",
"Mathlib.Data.Fintype.Sum",
"Mathlib.Topology.Homeomorph.Lemmas",
"Mathlib.Topology.MetricSpace.Antilipschitz"
] | Mathlib/Topology/MetricSpace/Isometry.lean | IsUniformEmbedding.to_isometry | A uniform embedding from a uniform space to a metric space is an isometry with respect to the
induced metric space structure on the source space. |
Topology.IsEmbedding.to_isometry {α β} [TopologicalSpace α] [MetricSpace β] {f : α → β}
(h : IsEmbedding f) : (letI := h.comapMetricSpace f; Isometry f) :=
let _ := h.comapMetricSpace f
Isometry.of_dist_eq fun _ _ => rfl | theorem | Topology | [
"Mathlib.Data.Fintype.Lattice",
"Mathlib.Data.Fintype.Sum",
"Mathlib.Topology.Homeomorph.Lemmas",
"Mathlib.Topology.MetricSpace.Antilipschitz"
] | Mathlib/Topology/MetricSpace/Isometry.lean | Topology.IsEmbedding.to_isometry | An embedding from a topological space to a metric space is an isometry with respect to the
induced metric space structure on the source space. |
PseudoEMetricSpace.isometry_induced (f : α → β) [m : PseudoEMetricSpace β] :
letI := m.induced f; Isometry f := fun _ _ ↦ rfl | theorem | Topology | [
"Mathlib.Data.Fintype.Lattice",
"Mathlib.Data.Fintype.Sum",
"Mathlib.Topology.Homeomorph.Lemmas",
"Mathlib.Topology.MetricSpace.Antilipschitz"
] | Mathlib/Topology/MetricSpace/Isometry.lean | PseudoEMetricSpace.isometry_induced | null |
PseudoMetricSpace.isometry_induced (f : α → β) [m : PseudoMetricSpace β] :
letI := m.induced f; Isometry f := fun _ _ ↦ rfl
@[deprecated (since := "2025-07-27")]
alias PsuedoMetricSpace.isometry_induced := PseudoMetricSpace.isometry_induced | theorem | Topology | [
"Mathlib.Data.Fintype.Lattice",
"Mathlib.Data.Fintype.Sum",
"Mathlib.Topology.Homeomorph.Lemmas",
"Mathlib.Topology.MetricSpace.Antilipschitz"
] | Mathlib/Topology/MetricSpace/Isometry.lean | PseudoMetricSpace.isometry_induced | null |
EMetricSpace.isometry_induced (f : α → β) (hf : f.Injective) [m : EMetricSpace β] :
letI := m.induced f hf; Isometry f := fun _ _ ↦ rfl | theorem | Topology | [
"Mathlib.Data.Fintype.Lattice",
"Mathlib.Data.Fintype.Sum",
"Mathlib.Topology.Homeomorph.Lemmas",
"Mathlib.Topology.MetricSpace.Antilipschitz"
] | Mathlib/Topology/MetricSpace/Isometry.lean | EMetricSpace.isometry_induced | null |
MetricSpace.isometry_induced (f : α → β) (hf : f.Injective) [m : MetricSpace β] :
letI := m.induced f hf; Isometry f := fun _ _ ↦ rfl | theorem | Topology | [
"Mathlib.Data.Fintype.Lattice",
"Mathlib.Data.Fintype.Sum",
"Mathlib.Topology.Homeomorph.Lemmas",
"Mathlib.Topology.MetricSpace.Antilipschitz"
] | Mathlib/Topology/MetricSpace/Isometry.lean | MetricSpace.isometry_induced | null |
IsometryClass (F : Type*) (α β : outParam Type*)
[PseudoEMetricSpace α] [PseudoEMetricSpace β] [FunLike F α β] : Prop where
protected isometry (f : F) : Isometry f | class | Topology | [
"Mathlib.Data.Fintype.Lattice",
"Mathlib.Data.Fintype.Sum",
"Mathlib.Topology.Homeomorph.Lemmas",
"Mathlib.Topology.MetricSpace.Antilipschitz"
] | Mathlib/Topology/MetricSpace/Isometry.lean | IsometryClass | `IsometryClass F α β` states that `F` is a type of isometries. |
protected edist_eq (x y : α) : edist (f x) (f y) = edist x y :=
(IsometryClass.isometry f).edist_eq x y | theorem | Topology | [
"Mathlib.Data.Fintype.Lattice",
"Mathlib.Data.Fintype.Sum",
"Mathlib.Topology.Homeomorph.Lemmas",
"Mathlib.Topology.MetricSpace.Antilipschitz"
] | Mathlib/Topology/MetricSpace/Isometry.lean | edist_eq | null |
protected continuous : Continuous f :=
(IsometryClass.isometry f).continuous | theorem | Topology | [
"Mathlib.Data.Fintype.Lattice",
"Mathlib.Data.Fintype.Sum",
"Mathlib.Topology.Homeomorph.Lemmas",
"Mathlib.Topology.MetricSpace.Antilipschitz"
] | Mathlib/Topology/MetricSpace/Isometry.lean | continuous | null |
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