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 ⌀ |
|---|---|---|---|---|---|---|
continuousWithinAt_hom_bundle {M : Type*} [TopologicalSpace M]
(f : M → TotalSpace (F₁ →SL[σ] F₂) (fun x ↦ E₁ x →SL[σ] E₂ x)) {s : Set M} {x₀ : M} :
ContinuousWithinAt f s x₀ ↔
ContinuousWithinAt (fun x ↦ (f x).1) s x₀ ∧
ContinuousWithinAt
(fun x ↦ inCoordinates F₁ E₁ F₂ E₂ (f x₀).1 (f x... | theorem | Topology | [
"Mathlib.Topology.VectorBundle.Basic"
] | Mathlib/Topology/VectorBundle/Hom.lean | continuousWithinAt_hom_bundle | null |
continuousAt_hom_bundle {M : Type*} [TopologicalSpace M]
(f : M → TotalSpace (F₁ →SL[σ] F₂) (fun x ↦ E₁ x →SL[σ] E₂ x)) {x₀ : M} :
ContinuousAt f x₀ ↔
ContinuousAt (fun x ↦ (f x).1) x₀ ∧
ContinuousAt
(fun x ↦ inCoordinates F₁ E₁ F₂ E₂ (f x₀).1 (f x).1 (f x₀).1 (f x).1 (f x).2) x₀ :=
Fi... | theorem | Topology | [
"Mathlib.Topology.VectorBundle.Basic"
] | Mathlib/Topology/VectorBundle/Hom.lean | continuousAt_hom_bundle | null |
ContinuousWithinAt.clm_apply_of_inCoordinates
(hϕ : ContinuousWithinAt
(fun m ↦ inCoordinates F₁ E₁ F₂ E₂ (b₁ m₀) (b₁ m) (b₂ m₀) (b₂ m) (ϕ m)) s m₀)
(hv : ContinuousWithinAt (fun m ↦ (v m : TotalSpace F₁ E₁)) s m₀)
(hb₂ : ContinuousWithinAt b₂ s m₀) :
ContinuousWithinAt (fun m ↦ (ϕ m (v m) : Total... | lemma | Topology | [
"Mathlib.Topology.VectorBundle.Basic"
] | Mathlib/Topology/VectorBundle/Hom.lean | ContinuousWithinAt.clm_apply_of_inCoordinates | Consider a continuous map `v : M → E₁` to a vector bundle, over a base map `b₁ : M → B₁`, and
another basemap `b₂ : M → B₂`. Given linear maps `ϕ m : E₁ (b₁ m) → E₂ (b₂ m)` depending
continuously on `m`, one can apply `ϕ m` to `g m`, and the resulting map is continuous.
Note that the continuity of `ϕ` cannot be always... |
ContinuousAt.clm_apply_of_inCoordinates
(hϕ : ContinuousAt
(fun m ↦ inCoordinates F₁ E₁ F₂ E₂ (b₁ m₀) (b₁ m) (b₂ m₀) (b₂ m) (ϕ m)) m₀)
(hv : ContinuousAt (fun m ↦ (v m : TotalSpace F₁ E₁)) m₀)
(hb₂ : ContinuousAt b₂ m₀) :
ContinuousAt (fun m ↦ (ϕ m (v m) : TotalSpace F₂ E₂)) m₀ := by
rw [← conti... | lemma | Topology | [
"Mathlib.Topology.VectorBundle.Basic"
] | Mathlib/Topology/VectorBundle/Hom.lean | ContinuousAt.clm_apply_of_inCoordinates | Consider a continuous map `v : M → E₁` to a vector bundle, over a base map `b₁ : M → B₁`, and
another basemap `b₂ : M → B₂`. Given linear maps `ϕ m : E₁ (b₁ m) → E₂ (b₂ m)` depending
continuously on `m`, one can apply `ϕ m` to `g m`, and the resulting map is continuous.
Note that the continuity of `ϕ` cannot be always... |
ContinuousWithinAt.clm_bundle_apply
(hϕ : ContinuousWithinAt
(fun m ↦ TotalSpace.mk' (F₁ →L[𝕜] F₂) (E := fun (x : B) ↦ (E₁ x →L[𝕜] E₂ x)) (b m) (ϕ m))
s x)
(hv : ContinuousWithinAt (fun m ↦ TotalSpace.mk' F₁ (b m) (v m)) s x) :
ContinuousWithinAt
(fun m ↦ TotalSpace.mk' F₂ (b m) (ϕ m (v ... | lemma | Topology | [
"Mathlib.Topology.VectorBundle.Basic"
] | Mathlib/Topology/VectorBundle/Hom.lean | ContinuousWithinAt.clm_bundle_apply | Consider a `C^n` map `v : M → E₁` to a vector bundle, over a basemap `b : M → B`, and
linear maps `ϕ m : E₁ (b m) → E₂ (b m)` depending smoothly on `m`.
One can apply `ϕ m` to `v m`, and the resulting map is `C^n`. |
ContinuousAt.clm_bundle_apply
(hϕ : ContinuousAt
(fun m ↦ TotalSpace.mk' (F₁ →L[𝕜] F₂) (E := fun (x : B) ↦ (E₁ x →L[𝕜] E₂ x)) (b m) (ϕ m)) x)
(hv : ContinuousAt (fun m ↦ TotalSpace.mk' F₁ (b m) (v m)) x) :
ContinuousAt (fun m ↦ TotalSpace.mk' F₂ (b m) (ϕ m (v m))) x := by
simp only [← continuousWi... | lemma | Topology | [
"Mathlib.Topology.VectorBundle.Basic"
] | Mathlib/Topology/VectorBundle/Hom.lean | ContinuousAt.clm_bundle_apply | Consider a `C^n` map `v : M → E₁` to a vector bundle, over a basemap `b : M → B`, and
linear maps `ϕ m : E₁ (b m) → E₂ (b m)` depending smoothly on `m`.
One can apply `ϕ m` to `v m`, and the resulting map is `C^n`. |
ContinuousOn.clm_bundle_apply
(hϕ : ContinuousOn
(fun m ↦ TotalSpace.mk' (F₁ →L[𝕜] F₂) (E := fun (x : B) ↦ (E₁ x →L[𝕜] E₂ x)) (b m) (ϕ m)) s)
(hv : ContinuousOn (fun m ↦ TotalSpace.mk' F₁ (b m) (v m)) s) :
ContinuousOn (fun m ↦ TotalSpace.mk' F₂ (b m) (ϕ m (v m))) s :=
fun x hx ↦ (hϕ x hx).clm_bun... | lemma | Topology | [
"Mathlib.Topology.VectorBundle.Basic"
] | Mathlib/Topology/VectorBundle/Hom.lean | ContinuousOn.clm_bundle_apply | Consider a `C^n` map `v : M → E₁` to a vector bundle, over a basemap `b : M → B`, and
linear maps `ϕ m : E₁ (b m) → E₂ (b m)` depending smoothly on `m`.
One can apply `ϕ m` to `v m`, and the resulting map is `C^n`. |
Continuous.clm_bundle_apply
(hϕ : Continuous
(fun m ↦ TotalSpace.mk' (F₁ →L[𝕜] F₂) (E := fun (x : B) ↦ (E₁ x →L[𝕜] E₂ x)) (b m) (ϕ m)))
(hv : Continuous (fun m ↦ TotalSpace.mk' F₁ (b m) (v m))) :
Continuous (fun m ↦ TotalSpace.mk' F₂ (b m) (ϕ m (v m))) := by
simp only [← continuousOn_univ] at hϕ h... | lemma | Topology | [
"Mathlib.Topology.VectorBundle.Basic"
] | Mathlib/Topology/VectorBundle/Hom.lean | Continuous.clm_bundle_apply | Consider a `C^n` map `v : M → E₁` to a vector bundle, over a basemap `b : M → B`, and
linear maps `ϕ m : E₁ (b m) → E₂ (b m)` depending smoothly on `m`.
One can apply `ϕ m` to `v m`, and the resulting map is `C^n`. |
ContinuousWithinAt.clm_bundle_apply₂
(hψ : ContinuousWithinAt (fun m ↦ TotalSpace.mk' (F₁ →L[𝕜] F₂ →L[𝕜] F₃)
(E := fun (x : B) ↦ (E₁ x →L[𝕜] E₂ x →L[𝕜] E₃ x)) (b m) (ψ m)) s x)
(hv : ContinuousWithinAt (fun m ↦ TotalSpace.mk' F₁ (b m) (v m)) s x)
(hw : ContinuousWithinAt (fun m ↦ TotalSpace.mk' F₂... | lemma | Topology | [
"Mathlib.Topology.VectorBundle.Basic"
] | Mathlib/Topology/VectorBundle/Hom.lean | ContinuousWithinAt.clm_bundle_apply₂ | Consider `C^n` maps `v : M → E₁` and `v : M → E₂` to vector bundles, over a basemap
`b : M → B`, and bilinear maps `ψ m : E₁ (b m) → E₂ (b m) → E₃ (b m)` depending smoothly on `m`.
One can apply `ψ m` to `v m` and `w m`, and the resulting map is `C^n`. |
ContinuousAt.clm_bundle_apply₂
(hψ : ContinuousAt (fun m ↦ TotalSpace.mk' (F₁ →L[𝕜] F₂ →L[𝕜] F₃)
(E := fun (x : B) ↦ (E₁ x →L[𝕜] E₂ x →L[𝕜] E₃ x)) (b m) (ψ m)) x)
(hv : ContinuousAt (fun m ↦ TotalSpace.mk' F₁ (b m) (v m)) x)
(hw : ContinuousAt (fun m ↦ TotalSpace.mk' F₂ (b m) (w m)) x) :
Conti... | lemma | Topology | [
"Mathlib.Topology.VectorBundle.Basic"
] | Mathlib/Topology/VectorBundle/Hom.lean | ContinuousAt.clm_bundle_apply₂ | Consider `C^n` maps `v : M → E₁` and `v : M → E₂` to vector bundles, over a basemap
`b : M → B`, and bilinear maps `ψ m : E₁ (b m) → E₂ (b m) → E₃ (b m)` depending smoothly on `m`.
One can apply `ψ m` to `v m` and `w m`, and the resulting map is `C^n`. |
ContinuousOn.clm_bundle_apply₂
(hψ : ContinuousOn
(fun m ↦ TotalSpace.mk' (F₁ →L[𝕜] F₂ →L[𝕜] F₃)
(E := fun (x : B) ↦ (E₁ x →L[𝕜] E₂ x →L[𝕜] E₃ x)) (b m) (ψ m)) s)
(hv : ContinuousOn (fun m ↦ TotalSpace.mk' F₁ (b m) (v m)) s)
(hw : ContinuousOn (fun m ↦ TotalSpace.mk' F₂ (b m) (w m)) s) :
... | lemma | Topology | [
"Mathlib.Topology.VectorBundle.Basic"
] | Mathlib/Topology/VectorBundle/Hom.lean | ContinuousOn.clm_bundle_apply₂ | Consider `C^n` maps `v : M → E₁` and `v : M → E₂` to vector bundles, over a basemap
`b : M → B`, and bilinear maps `ψ m : E₁ (b m) → E₂ (b m) → E₃ (b m)` depending smoothly on `m`.
One can apply `ψ m` to `v m` and `w m`, and the resulting map is `C^n`. |
Continuous.clm_bundle_apply₂
(hψ : Continuous (fun m ↦ TotalSpace.mk' (F₁ →L[𝕜] F₂ →L[𝕜] F₃)
(E := fun (x : B) ↦ (E₁ x →L[𝕜] E₂ x →L[𝕜] E₃ x)) (b m) (ψ m)))
(hv : Continuous (fun m ↦ TotalSpace.mk' F₁ (b m) (v m)))
(hw : Continuous (fun m ↦ TotalSpace.mk' F₂ (b m) (w m))) :
Continuous (fun m ↦... | lemma | Topology | [
"Mathlib.Topology.VectorBundle.Basic"
] | Mathlib/Topology/VectorBundle/Hom.lean | Continuous.clm_bundle_apply₂ | Consider `C^n` maps `v : M → E₁` and `v : M → E₂` to vector bundles, over a basemap
`b : M → B`, and bilinear maps `ψ m : E₁ (b m) → E₂ (b m) → E₃ (b m)` depending smoothly on `m`.
One can apply `ψ m` to `v m` and `w m`, and the resulting map is `C^n`. |
inCoordinates_apply_eq₂
{x₀ x : B} {ϕ : E₁ x →L[𝕜] E₂ x →L[𝕜] E₃ x} {v : F₁} {w : F₂}
(h₁x : x ∈ (trivializationAt F₁ E₁ x₀).baseSet)
(h₂x : x ∈ (trivializationAt F₂ E₂ x₀).baseSet)
(h₃x : x ∈ (trivializationAt F₃ E₃ x₀).baseSet) :
inCoordinates F₁ E₁ (F₂ →L[𝕜] F₃) (fun x ↦ E₂ x →L[𝕜] E₃ x) x₀ x... | theorem | Topology | [
"Mathlib.Topology.VectorBundle.Basic"
] | Mathlib/Topology/VectorBundle/Hom.lean | inCoordinates_apply_eq₂ | Rewrite `ContinuousLinearMap.inCoordinates` using continuous linear equivalences, in the
bundle of bilinear maps. |
IsContinuousRiemannianBundle : Prop where
/-- There exists a bilinear form, depending continuously on the basepoint and defining the
inner product in the fibers. This is expressed as an existence statement so that it is Prop-valued
in terms of existing data, the inner product on the fibers and the fiber bundle st... | class | Topology | [
"Mathlib.Analysis.InnerProductSpace.LinearMap",
"Mathlib.Topology.VectorBundle.Constructions",
"Mathlib.Topology.VectorBundle.Hom"
] | Mathlib/Topology/VectorBundle/Riemannian.lean | IsContinuousRiemannianBundle | Consider a real vector bundle in which each fiber is endowed with an inner product.
We say that the bundle is *Riemannian* if the inner product depends continuously on the base point.
This assumption is spelled `IsContinuousRiemannianBundle F E` where `F` is the model fiber,
and `E : B → Type*` is the bundle. |
ContinuousWithinAt.inner_bundle
(hv : ContinuousWithinAt (fun m ↦ (v m : TotalSpace F E)) s x)
(hw : ContinuousWithinAt (fun m ↦ (w m : TotalSpace F E)) s x) :
ContinuousWithinAt (fun m ↦ ⟪v m, w m⟫) s x := by
rcases h.exists_continuous with ⟨g, g_cont, hg⟩
have hf : ContinuousWithinAt b s x := by
s... | lemma | Topology | [
"Mathlib.Analysis.InnerProductSpace.LinearMap",
"Mathlib.Topology.VectorBundle.Constructions",
"Mathlib.Topology.VectorBundle.Hom"
] | Mathlib/Topology/VectorBundle/Riemannian.lean | ContinuousWithinAt.inner_bundle | A trivial vector bundle, in which the model fiber has a inner product,
is a Riemannian bundle. -/
instance : IsContinuousRiemannianBundle F₁ (Bundle.Trivial B F₁) := by
refine ⟨fun x ↦ innerSL ℝ, ?_, fun x v w ↦ rfl⟩
rw [continuous_iff_continuousAt]
intro x
rw [FiberBundle.continuousAt_totalSpace]
refine ⟨con... |
ContinuousAt.inner_bundle
(hv : ContinuousAt (fun m ↦ (v m : TotalSpace F E)) x)
(hw : ContinuousAt (fun m ↦ (w m : TotalSpace F E)) x) :
ContinuousAt (fun b ↦ ⟪v b, w b⟫) x := by
simp only [← continuousWithinAt_univ] at hv hw ⊢
exact ContinuousWithinAt.inner_bundle hv hw | lemma | Topology | [
"Mathlib.Analysis.InnerProductSpace.LinearMap",
"Mathlib.Topology.VectorBundle.Constructions",
"Mathlib.Topology.VectorBundle.Hom"
] | Mathlib/Topology/VectorBundle/Riemannian.lean | ContinuousAt.inner_bundle | Given two continuous maps into the same fibers of a continuous Riemannian bundle,
their inner product is continuous. Version with `ContinuousAt`. |
ContinuousOn.inner_bundle
(hv : ContinuousOn (fun m ↦ (v m : TotalSpace F E)) s)
(hw : ContinuousOn (fun m ↦ (w m : TotalSpace F E)) s) :
ContinuousOn (fun b ↦ ⟪v b, w b⟫) s :=
fun x hx ↦ (hv x hx).inner_bundle (hw x hx) | lemma | Topology | [
"Mathlib.Analysis.InnerProductSpace.LinearMap",
"Mathlib.Topology.VectorBundle.Constructions",
"Mathlib.Topology.VectorBundle.Hom"
] | Mathlib/Topology/VectorBundle/Riemannian.lean | ContinuousOn.inner_bundle | Given two continuous maps into the same fibers of a continuous Riemannian bundle,
their inner product is continuous. Version with `ContinuousOn`. |
Continuous.inner_bundle
(hv : Continuous (fun m ↦ (v m : TotalSpace F E)))
(hw : Continuous (fun m ↦ (w m : TotalSpace F E))) :
Continuous (fun b ↦ ⟪v b, w b⟫) := by
simp only [continuous_iff_continuousAt] at hv hw ⊢
exact fun x ↦ (hv x).inner_bundle (hw x)
variable (F E) | lemma | Topology | [
"Mathlib.Analysis.InnerProductSpace.LinearMap",
"Mathlib.Topology.VectorBundle.Constructions",
"Mathlib.Topology.VectorBundle.Hom"
] | Mathlib/Topology/VectorBundle/Riemannian.lean | Continuous.inner_bundle | Given two continuous maps into the same fibers of a continuous Riemannian bundle,
their inner product is continuous. |
eventually_norm_symmL_trivializationAt_self_comp_lt (x : B) {r : ℝ} (hr : 1 < r) :
∀ᶠ y in 𝓝 x, ‖((trivializationAt F E x).symmL ℝ x)
∘L ((trivializationAt F E x).continuousLinearMapAt ℝ y)‖ < r := by
/- We will expand the definition of continuity of the inner product structure, in the chart.
Denote `g' ... | lemma | Topology | [
"Mathlib.Analysis.InnerProductSpace.LinearMap",
"Mathlib.Topology.VectorBundle.Constructions",
"Mathlib.Topology.VectorBundle.Hom"
] | Mathlib/Topology/VectorBundle/Riemannian.lean | eventually_norm_symmL_trivializationAt_self_comp_lt | In a continuous Riemannian bundle, local changes of coordinates given by the trivialization at
a point distort the norm by a factor arbitrarily close to 1. |
eventually_norm_trivializationAt_lt (x : B) :
∃ C > 0, ∀ᶠ y in 𝓝 x, ‖(trivializationAt F E x).continuousLinearMapAt ℝ y‖ < C := by
refine ⟨(1 + ‖(trivializationAt F E x).continuousLinearMapAt ℝ x‖) * 2, by positivity, ?_⟩
filter_upwards [eventually_norm_symmL_trivializationAt_self_comp_lt F E x one_lt_two] wi... | lemma | Topology | [
"Mathlib.Analysis.InnerProductSpace.LinearMap",
"Mathlib.Topology.VectorBundle.Constructions",
"Mathlib.Topology.VectorBundle.Hom"
] | Mathlib/Topology/VectorBundle/Riemannian.lean | eventually_norm_trivializationAt_lt | In a continuous Riemannian bundle, the trivialization at a point is locally bounded in norm. |
eventually_norm_symmL_trivializationAt_comp_self_lt (x : B) {r : ℝ} (hr : 1 < r) :
∀ᶠ y in 𝓝 x, ‖((trivializationAt F E x).symmL ℝ y)
∘L ((trivializationAt F E x).continuousLinearMapAt ℝ x)‖ < r := by
/- We will expand the definition of continuity of the inner product structure, in the chart.
Denote `g' ... | lemma | Topology | [
"Mathlib.Analysis.InnerProductSpace.LinearMap",
"Mathlib.Topology.VectorBundle.Constructions",
"Mathlib.Topology.VectorBundle.Hom"
] | Mathlib/Topology/VectorBundle/Riemannian.lean | eventually_norm_symmL_trivializationAt_comp_self_lt | In a continuous Riemannian bundle, local changes of coordinates given by the trivialization at
a point distort the norm by a factor arbitrarily close to 1. |
eventually_norm_symmL_trivializationAt_lt (x : B) :
∃ C > 0, ∀ᶠ y in 𝓝 x, ‖(trivializationAt F E x).symmL ℝ y‖ < C := by
refine ⟨2 * (1 + ‖(trivializationAt F E x).symmL ℝ x‖), by positivity, ?_⟩
filter_upwards [eventually_norm_symmL_trivializationAt_comp_self_lt F E x one_lt_two] with y hy
have A : ((trivia... | lemma | Topology | [
"Mathlib.Analysis.InnerProductSpace.LinearMap",
"Mathlib.Topology.VectorBundle.Constructions",
"Mathlib.Topology.VectorBundle.Hom"
] | Mathlib/Topology/VectorBundle/Riemannian.lean | eventually_norm_symmL_trivializationAt_lt | In a continuous Riemannian bundle, the inverse of the trivialization at a point is locally
bounded in norm. |
RiemannianMetric where
/-- The inner product along the fibers of the bundle. -/
inner (b : B) : E b →L[ℝ] E b →L[ℝ] ℝ
symm (b : B) (v w : E b) : inner b v w = inner b w v
pos (b : B) (v : E b) (hv : v ≠ 0) : 0 < inner b v v
/-- The continuity at `0` is automatic when `E b` is isomorphic to a normed space, but... | structure | Topology | [
"Mathlib.Analysis.InnerProductSpace.LinearMap",
"Mathlib.Topology.VectorBundle.Constructions",
"Mathlib.Topology.VectorBundle.Hom"
] | Mathlib/Topology/VectorBundle/Riemannian.lean | RiemannianMetric | A family of inner product space structures on the fibers of a fiber bundle, defining the same
topology as the already existing one. This family is not assumed to be continuous or smooth: to
guarantee continuity, resp. smoothness, of the inner product as a function of the base point,
use `ContinuousRiemannianMetric` or ... |
@[reducible] noncomputable RiemannianMetric.toCore (g : RiemannianMetric E) (b : B) :
InnerProductSpace.Core ℝ (E b) where
inner v w := g.inner b v w
conj_inner_symm v w := g.symm b w v
re_inner_nonneg v := by
rcases eq_or_ne v 0 with rfl | hv
· simp
· simpa using (g.pos b v hv).le
add_left v w ... | def | Topology | [
"Mathlib.Analysis.InnerProductSpace.LinearMap",
"Mathlib.Topology.VectorBundle.Constructions",
"Mathlib.Topology.VectorBundle.Hom"
] | Mathlib/Topology/VectorBundle/Riemannian.lean | RiemannianMetric.toCore | `Core structure associated to a family of inner products on the fibers of a fiber bundle. This
is an auxiliary construction to endow the fibers with an inner product space structure without
creating diamonds.
Warning: Do not use this `Core` structure if the space you are interested in already has a norm
instance defin... |
RiemannianBundle where
/-- The family of inner products on the fibers -/
g : RiemannianMetric E | class | Topology | [
"Mathlib.Analysis.InnerProductSpace.LinearMap",
"Mathlib.Topology.VectorBundle.Constructions",
"Mathlib.Topology.VectorBundle.Hom"
] | Mathlib/Topology/VectorBundle/Riemannian.lean | RiemannianBundle | Class used to create an inner product structure space on the fibers of a fiber bundle, without
creating diamonds. Use as follows:
* `instance : RiemannianBundle E := ⟨g⟩` where `g : RiemannianMetric E` registers the inner product
space on the fibers;
* `instance : RiemannianBundle E := ⟨g.toRiemannianMetric⟩` where
... |
ContinuousRiemannianMetric where
/-- The inner product along the fibers of the bundle. -/
inner (b : B) : E b →L[ℝ] E b →L[ℝ] ℝ
symm (b : B) (v w : E b) : inner b v w = inner b w v
pos (b : B) (v : E b) (hv : v ≠ 0) : 0 < inner b v v
isVonNBounded (b : B) : IsVonNBounded ℝ {v : E b | inner b v v < 1}
contin... | structure | Topology | [
"Mathlib.Analysis.InnerProductSpace.LinearMap",
"Mathlib.Topology.VectorBundle.Constructions",
"Mathlib.Topology.VectorBundle.Hom"
] | Mathlib/Topology/VectorBundle/Riemannian.lean | ContinuousRiemannianMetric | A fiber in a bundle satisfying the `[RiemannianBundle E]` typeclass inherits
a `NormedAddCommGroup` structure.
The normal priority for an instance which always applies like this one should be 100.
We use 80 as this is rather specialized, so we want other paths to be tried first typically.
As this instance is quite spe... |
ContinuousRiemannianMetric.toRiemannianMetric (g : ContinuousRiemannianMetric F E) :
RiemannianMetric E where
inner := g.inner
symm := g.symm
pos := g.pos
isVonNBounded := g.isVonNBounded
continuousAt b := by
let e : E b ≃L[ℝ] F := Trivialization.continuousLinearEquivAt ℝ (trivializationAt F E b) _
... | def | Topology | [
"Mathlib.Analysis.InnerProductSpace.LinearMap",
"Mathlib.Topology.VectorBundle.Constructions",
"Mathlib.Topology.VectorBundle.Hom"
] | Mathlib/Topology/VectorBundle/Riemannian.lean | ContinuousRiemannianMetric.toRiemannianMetric | A continuous Riemannian metric is in particular a Riemannian metric. |
ContinuousAlgEquiv (R A B : Type*) [CommSemiring R]
[Semiring A] [TopologicalSpace A] [Semiring B] [TopologicalSpace B] [Algebra R A]
[Algebra R B] extends A ≃ₐ[R] B, A ≃ₜ B
@[inherit_doc]
notation:50 A " ≃A[" R "] " B => ContinuousAlgEquiv R A B
attribute [nolint docBlame] ContinuousAlgEquiv.toHomeomorph | structure | Topology | [
"Mathlib.Topology.Algebra.Algebra"
] | Mathlib/Topology/Algebra/Algebra/Equiv.lean | ContinuousAlgEquiv | `ContinuousAlgEquiv R A B`, with notation `A ≃A[R] B`, is the type of bijections
between the topological `R`-algebras `A` and `B` which are both homeomorphisms
and `R`-algebra isomorphisms. |
ContinuousAlgEquivClass (F : Type*) (R A B : outParam Type*) [CommSemiring R]
[Semiring A] [TopologicalSpace A] [Semiring B] [TopologicalSpace B]
[Algebra R A] [Algebra R B] [EquivLike F A B] : Prop
extends AlgEquivClass F R A B, HomeomorphClass F A B | class | Topology | [
"Mathlib.Topology.Algebra.Algebra"
] | Mathlib/Topology/Algebra/Algebra/Equiv.lean | ContinuousAlgEquivClass | `ContinuousAlgEquivClass F R A B` states that `F` is a type of topological algebra
structure-preserving equivalences. You should extend this class when you
extend `ContinuousAlgEquiv`. |
@[coe]
toContinuousAlgHom (e : A ≃A[R] B) : A →A[R] B where
__ := e.toAlgHom
cont := e.continuous_toFun | def | Topology | [
"Mathlib.Topology.Algebra.Algebra"
] | Mathlib/Topology/Algebra/Algebra/Equiv.lean | toContinuousAlgHom | The natural coercion from a continuous algebra isomorphism to a continuous
algebra morphism. |
coe : Coe (A ≃A[R] B) (A →A[R] B) := ⟨toContinuousAlgHom⟩ | instance | Topology | [
"Mathlib.Topology.Algebra.Algebra"
] | Mathlib/Topology/Algebra/Algebra/Equiv.lean | coe | null |
equivLike : EquivLike (A ≃A[R] B) A B where
coe f := f.toFun
inv f := f.invFun
coe_injective' f g h₁ h₂ := by
obtain ⟨f', _⟩ := f
obtain ⟨g', _⟩ := g
rcases f' with ⟨⟨_, _⟩, _⟩
rcases g' with ⟨⟨_, _⟩, _⟩
congr
left_inv f := f.left_inv
right_inv f := f.right_inv | instance | Topology | [
"Mathlib.Topology.Algebra.Algebra"
] | Mathlib/Topology/Algebra/Algebra/Equiv.lean | equivLike | null |
continuousAlgEquivClass : ContinuousAlgEquivClass (A ≃A[R] B) R A B where
map_add f := f.map_add'
map_mul f := f.map_mul'
commutes f := f.commutes'
map_continuous := continuous_toFun
inv_continuous := continuous_invFun | instance | Topology | [
"Mathlib.Topology.Algebra.Algebra"
] | Mathlib/Topology/Algebra/Algebra/Equiv.lean | continuousAlgEquivClass | null |
coe_apply (e : A ≃A[R] B) (a : A) : (e : A →A[R] B) a = e a := rfl
@[simp] | theorem | Topology | [
"Mathlib.Topology.Algebra.Algebra"
] | Mathlib/Topology/Algebra/Algebra/Equiv.lean | coe_apply | null |
coe_coe (e : A ≃A[R] B) : ⇑(e : A →A[R] B) = e := rfl | theorem | Topology | [
"Mathlib.Topology.Algebra.Algebra"
] | Mathlib/Topology/Algebra/Algebra/Equiv.lean | coe_coe | null |
toAlgEquiv_injective : Function.Injective (toAlgEquiv : (A ≃A[R] B) → A ≃ₐ[R] B) := by
rintro ⟨e, _, _⟩ ⟨e', _, _⟩ rfl
rfl
@[ext] | theorem | Topology | [
"Mathlib.Topology.Algebra.Algebra"
] | Mathlib/Topology/Algebra/Algebra/Equiv.lean | toAlgEquiv_injective | null |
ext {f g : A ≃A[R] B} (h : ⇑f = ⇑g) : f = g :=
toAlgEquiv_injective <| AlgEquiv.ext <| congr_fun h | theorem | Topology | [
"Mathlib.Topology.Algebra.Algebra"
] | Mathlib/Topology/Algebra/Algebra/Equiv.lean | ext | null |
coe_injective : Function.Injective ((↑) : (A ≃A[R] B) → A →A[R] B) :=
fun _ _ h => ext <| funext <| ContinuousAlgHom.ext_iff.1 h
@[simp] | theorem | Topology | [
"Mathlib.Topology.Algebra.Algebra"
] | Mathlib/Topology/Algebra/Algebra/Equiv.lean | coe_injective | null |
coe_inj {f g : A ≃A[R] B} : (f : A →A[R] B) = g ↔ f = g :=
coe_injective.eq_iff
@[simp] | theorem | Topology | [
"Mathlib.Topology.Algebra.Algebra"
] | Mathlib/Topology/Algebra/Algebra/Equiv.lean | coe_inj | null |
coe_toAlgEquiv (e : A ≃A[R] B) : ⇑e.toAlgEquiv = e := rfl | theorem | Topology | [
"Mathlib.Topology.Algebra.Algebra"
] | Mathlib/Topology/Algebra/Algebra/Equiv.lean | coe_toAlgEquiv | null |
isOpenMap (e : A ≃A[R] B) : IsOpenMap e :=
e.toHomeomorph.isOpenMap | theorem | Topology | [
"Mathlib.Topology.Algebra.Algebra"
] | Mathlib/Topology/Algebra/Algebra/Equiv.lean | isOpenMap | null |
image_closure (e : A ≃A[R] B) (S : Set A) : e '' closure S = closure (e '' S) :=
e.toHomeomorph.image_closure S | theorem | Topology | [
"Mathlib.Topology.Algebra.Algebra"
] | Mathlib/Topology/Algebra/Algebra/Equiv.lean | image_closure | null |
preimage_closure (e : A ≃A[R] B) (S : Set B) : e ⁻¹' closure S = closure (e ⁻¹' S) :=
e.toHomeomorph.preimage_closure S
@[simp] | theorem | Topology | [
"Mathlib.Topology.Algebra.Algebra"
] | Mathlib/Topology/Algebra/Algebra/Equiv.lean | preimage_closure | null |
isClosed_image (e : A ≃A[R] B) {S : Set A} : IsClosed (e '' S) ↔ IsClosed S :=
e.toHomeomorph.isClosed_image | theorem | Topology | [
"Mathlib.Topology.Algebra.Algebra"
] | Mathlib/Topology/Algebra/Algebra/Equiv.lean | isClosed_image | null |
map_nhds_eq (e : A ≃A[R] B) (a : A) : Filter.map e (𝓝 a) = 𝓝 (e a) :=
e.toHomeomorph.map_nhds_eq a | theorem | Topology | [
"Mathlib.Topology.Algebra.Algebra"
] | Mathlib/Topology/Algebra/Algebra/Equiv.lean | map_nhds_eq | null |
map_eq_zero_iff (e : A ≃A[R] B) {a : A} : e a = 0 ↔ a = 0 :=
e.toAlgEquiv.toLinearEquiv.map_eq_zero_iff
attribute [continuity]
ContinuousAlgEquiv.continuous_invFun ContinuousAlgEquiv.continuous_toFun
@[fun_prop] | theorem | Topology | [
"Mathlib.Topology.Algebra.Algebra"
] | Mathlib/Topology/Algebra/Algebra/Equiv.lean | map_eq_zero_iff | null |
continuous (e : A ≃A[R] B) : Continuous e := e.continuous_toFun | theorem | Topology | [
"Mathlib.Topology.Algebra.Algebra"
] | Mathlib/Topology/Algebra/Algebra/Equiv.lean | continuous | null |
continuousOn (e : A ≃A[R] B) {S : Set A} : ContinuousOn e S :=
e.continuous.continuousOn | theorem | Topology | [
"Mathlib.Topology.Algebra.Algebra"
] | Mathlib/Topology/Algebra/Algebra/Equiv.lean | continuousOn | null |
continuousAt (e : A ≃A[R] B) {a : A} : ContinuousAt e a :=
e.continuous.continuousAt | theorem | Topology | [
"Mathlib.Topology.Algebra.Algebra"
] | Mathlib/Topology/Algebra/Algebra/Equiv.lean | continuousAt | null |
continuousWithinAt (e : A ≃A[R] B) {S : Set A} {a : A} :
ContinuousWithinAt e S a :=
e.continuous.continuousWithinAt | theorem | Topology | [
"Mathlib.Topology.Algebra.Algebra"
] | Mathlib/Topology/Algebra/Algebra/Equiv.lean | continuousWithinAt | null |
comp_continuous_iff {α : Type*} [TopologicalSpace α] (e : A ≃A[R] B) {f : α → A} :
Continuous (e ∘ f) ↔ Continuous f :=
e.toHomeomorph.comp_continuous_iff | theorem | Topology | [
"Mathlib.Topology.Algebra.Algebra"
] | Mathlib/Topology/Algebra/Algebra/Equiv.lean | comp_continuous_iff | null |
comp_continuous_iff' {β : Type*} [TopologicalSpace β] (e : A ≃A[R] B) {g : B → β} :
Continuous (g ∘ e) ↔ Continuous g :=
e.toHomeomorph.comp_continuous_iff'
variable (R A) | theorem | Topology | [
"Mathlib.Topology.Algebra.Algebra"
] | Mathlib/Topology/Algebra/Algebra/Equiv.lean | comp_continuous_iff' | null |
@[refl]
refl : A ≃A[R] A where
__ := AlgEquiv.refl
continuous_toFun := continuous_id
continuous_invFun := continuous_id
@[simp] | def | Topology | [
"Mathlib.Topology.Algebra.Algebra"
] | Mathlib/Topology/Algebra/Algebra/Equiv.lean | refl | The identity isomorphism as a continuous `R`-algebra equivalence. |
refl_apply (a : A) : refl R A a = a := rfl
@[simp] | theorem | Topology | [
"Mathlib.Topology.Algebra.Algebra"
] | Mathlib/Topology/Algebra/Algebra/Equiv.lean | refl_apply | null |
coe_refl : refl R A = ContinuousAlgHom.id R A := rfl
@[simp] | theorem | Topology | [
"Mathlib.Topology.Algebra.Algebra"
] | Mathlib/Topology/Algebra/Algebra/Equiv.lean | coe_refl | null |
coe_refl' : ⇑(refl R A) = id := rfl
variable {R A} | theorem | Topology | [
"Mathlib.Topology.Algebra.Algebra"
] | Mathlib/Topology/Algebra/Algebra/Equiv.lean | coe_refl' | null |
@[symm]
symm (e : A ≃A[R] B) : B ≃A[R] A where
__ := e.toAlgEquiv.symm
continuous_toFun := e.continuous_invFun
continuous_invFun := e.continuous_toFun
@[simp] | def | Topology | [
"Mathlib.Topology.Algebra.Algebra"
] | Mathlib/Topology/Algebra/Algebra/Equiv.lean | symm | The inverse of a continuous algebra equivalence. |
apply_symm_apply (e : A ≃A[R] B) (b : B) : e (e.symm b) = b :=
e.1.right_inv b
@[simp] | theorem | Topology | [
"Mathlib.Topology.Algebra.Algebra"
] | Mathlib/Topology/Algebra/Algebra/Equiv.lean | apply_symm_apply | null |
symm_apply_apply (e : A ≃A[R] B) (a : A) : e.symm (e a) = a :=
e.1.left_inv a
@[simp] | theorem | Topology | [
"Mathlib.Topology.Algebra.Algebra"
] | Mathlib/Topology/Algebra/Algebra/Equiv.lean | symm_apply_apply | null |
symm_image_image (e : A ≃A[R] B) (S : Set A) : e.symm '' (e '' S) = S :=
e.toEquiv.symm_image_image S
@[simp] | theorem | Topology | [
"Mathlib.Topology.Algebra.Algebra"
] | Mathlib/Topology/Algebra/Algebra/Equiv.lean | symm_image_image | null |
image_symm_image (e : A ≃A[R] B) (S : Set B) : e '' (e.symm '' S) = S :=
e.symm.symm_image_image S
@[simp] | theorem | Topology | [
"Mathlib.Topology.Algebra.Algebra"
] | Mathlib/Topology/Algebra/Algebra/Equiv.lean | image_symm_image | null |
symm_toAlgEquiv (e : A ≃A[R] B) : e.symm.toAlgEquiv = e.toAlgEquiv.symm := rfl
@[simp] | theorem | Topology | [
"Mathlib.Topology.Algebra.Algebra"
] | Mathlib/Topology/Algebra/Algebra/Equiv.lean | symm_toAlgEquiv | null |
symm_toHomeomorph (e : A ≃A[R] B) : e.symm.toHomeomorph = e.toHomeomorph.symm := rfl | theorem | Topology | [
"Mathlib.Topology.Algebra.Algebra"
] | Mathlib/Topology/Algebra/Algebra/Equiv.lean | symm_toHomeomorph | null |
symm_map_nhds_eq (e : A ≃A[R] B) (a : A) : Filter.map e.symm (𝓝 (e a)) = 𝓝 a :=
e.toHomeomorph.symm_map_nhds_eq a | theorem | Topology | [
"Mathlib.Topology.Algebra.Algebra"
] | Mathlib/Topology/Algebra/Algebra/Equiv.lean | symm_map_nhds_eq | null |
@[trans]
trans (e₁ : A ≃A[R] B) (e₂ : B ≃A[R] C) : A ≃A[R] C where
__ := e₁.toAlgEquiv.trans e₂.toAlgEquiv
continuous_toFun := e₂.continuous_toFun.comp e₁.continuous_toFun
continuous_invFun := e₁.continuous_invFun.comp e₂.continuous_invFun
@[simp] | def | Topology | [
"Mathlib.Topology.Algebra.Algebra"
] | Mathlib/Topology/Algebra/Algebra/Equiv.lean | trans | The composition of two continuous algebra equivalences. |
trans_toAlgEquiv (e₁ : A ≃A[R] B) (e₂ : B ≃A[R] C) :
(e₁.trans e₂).toAlgEquiv = e₁.toAlgEquiv.trans e₂.toAlgEquiv :=
rfl
@[simp] | theorem | Topology | [
"Mathlib.Topology.Algebra.Algebra"
] | Mathlib/Topology/Algebra/Algebra/Equiv.lean | trans_toAlgEquiv | null |
trans_apply (e₁ : A ≃A[R] B) (e₂ : B ≃A[R] C) (a : A) :
(e₁.trans e₂) a = e₂ (e₁ a) :=
rfl
@[simp] | theorem | Topology | [
"Mathlib.Topology.Algebra.Algebra"
] | Mathlib/Topology/Algebra/Algebra/Equiv.lean | trans_apply | null |
symm_trans_apply (e₁ : B ≃A[R] A) (e₂ : C ≃A[R] B) (a : A) :
(e₂.trans e₁).symm a = e₂.symm (e₁.symm a) :=
rfl | theorem | Topology | [
"Mathlib.Topology.Algebra.Algebra"
] | Mathlib/Topology/Algebra/Algebra/Equiv.lean | symm_trans_apply | null |
comp_coe (e₁ : A ≃A[R] B) (e₂ : B ≃A[R] C) :
e₂.toAlgHom.comp e₁.toAlgHom = e₁.trans e₂ := by
rfl
@[simp high] | theorem | Topology | [
"Mathlib.Topology.Algebra.Algebra"
] | Mathlib/Topology/Algebra/Algebra/Equiv.lean | comp_coe | null |
coe_comp_coe_symm (e : A ≃A[R] B) :
e.toContinuousAlgHom.comp e.symm = ContinuousAlgHom.id R B :=
ContinuousAlgHom.ext e.apply_symm_apply
@[simp high] | theorem | Topology | [
"Mathlib.Topology.Algebra.Algebra"
] | Mathlib/Topology/Algebra/Algebra/Equiv.lean | coe_comp_coe_symm | null |
coe_symm_comp_coe (e : A ≃A[R] B) :
e.symm.toContinuousAlgHom.comp e = ContinuousAlgHom.id R A :=
ContinuousAlgHom.ext e.symm_apply_apply
@[simp] | theorem | Topology | [
"Mathlib.Topology.Algebra.Algebra"
] | Mathlib/Topology/Algebra/Algebra/Equiv.lean | coe_symm_comp_coe | null |
symm_comp_self (e : A ≃A[R] B) : (e.symm : B → A) ∘ e = id := by
exact funext <| e.symm_apply_apply
@[simp] | theorem | Topology | [
"Mathlib.Topology.Algebra.Algebra"
] | Mathlib/Topology/Algebra/Algebra/Equiv.lean | symm_comp_self | null |
self_comp_symm (e : A ≃A[R] B) : (e : A → B) ∘ e.symm = id :=
funext <| e.apply_symm_apply
@[simp] | theorem | Topology | [
"Mathlib.Topology.Algebra.Algebra"
] | Mathlib/Topology/Algebra/Algebra/Equiv.lean | self_comp_symm | null |
symm_symm (e : A ≃A[R] B) : e.symm.symm = e := rfl | theorem | Topology | [
"Mathlib.Topology.Algebra.Algebra"
] | Mathlib/Topology/Algebra/Algebra/Equiv.lean | symm_symm | null |
symm_bijective : Function.Bijective (symm : (A ≃A[R] B) → _) :=
Function.bijective_iff_has_inverse.mpr ⟨_, symm_symm, symm_symm⟩
@[simp] | theorem | Topology | [
"Mathlib.Topology.Algebra.Algebra"
] | Mathlib/Topology/Algebra/Algebra/Equiv.lean | symm_bijective | null |
refl_symm : (refl R A).symm = refl R A := rfl | theorem | Topology | [
"Mathlib.Topology.Algebra.Algebra"
] | Mathlib/Topology/Algebra/Algebra/Equiv.lean | refl_symm | null |
symm_symm_apply (e : A ≃A[R] B) (a : A) : e.symm.symm a = e a := rfl | theorem | Topology | [
"Mathlib.Topology.Algebra.Algebra"
] | Mathlib/Topology/Algebra/Algebra/Equiv.lean | symm_symm_apply | null |
symm_apply_eq (e : A ≃A[R] B) {a : A} {b : B} : e.symm b = a ↔ b = e a :=
e.toEquiv.symm_apply_eq | theorem | Topology | [
"Mathlib.Topology.Algebra.Algebra"
] | Mathlib/Topology/Algebra/Algebra/Equiv.lean | symm_apply_eq | null |
eq_symm_apply (e : A ≃A[R] B) {a : A} {b : B} : a = e.symm b ↔ e a = b :=
e.toEquiv.eq_symm_apply | theorem | Topology | [
"Mathlib.Topology.Algebra.Algebra"
] | Mathlib/Topology/Algebra/Algebra/Equiv.lean | eq_symm_apply | null |
image_eq_preimage (e : A ≃A[R] B) (S : Set A) : e '' S = e.symm ⁻¹' S :=
e.toEquiv.image_eq_preimage S | theorem | Topology | [
"Mathlib.Topology.Algebra.Algebra"
] | Mathlib/Topology/Algebra/Algebra/Equiv.lean | image_eq_preimage | null |
image_symm_eq_preimage (e : A ≃A[R] B) (S : Set B) : e.symm '' S = e ⁻¹' S := by
rw [e.symm.image_eq_preimage, e.symm_symm]
@[simp] | theorem | Topology | [
"Mathlib.Topology.Algebra.Algebra"
] | Mathlib/Topology/Algebra/Algebra/Equiv.lean | image_symm_eq_preimage | null |
symm_preimage_preimage (e : A ≃A[R] B) (S : Set B) : e.symm ⁻¹' (e ⁻¹' S) = S :=
e.toEquiv.symm_preimage_preimage S
@[simp] | theorem | Topology | [
"Mathlib.Topology.Algebra.Algebra"
] | Mathlib/Topology/Algebra/Algebra/Equiv.lean | symm_preimage_preimage | null |
preimage_symm_preimage (e : A ≃A[R] B) (S : Set A) : e ⁻¹' (e.symm ⁻¹' S) = S :=
e.symm.symm_preimage_preimage S | theorem | Topology | [
"Mathlib.Topology.Algebra.Algebra"
] | Mathlib/Topology/Algebra/Algebra/Equiv.lean | preimage_symm_preimage | null |
isUniformEmbedding {E₁ E₂ : Type*} [UniformSpace E₁] [UniformSpace E₂] [Ring E₁]
[IsUniformAddGroup E₁] [Algebra R E₁] [Ring E₂] [IsUniformAddGroup E₂] [Algebra R E₂]
(e : E₁ ≃A[R] E₂) : IsUniformEmbedding e :=
e.toAlgEquiv.isUniformEmbedding e.toContinuousAlgHom.uniformContinuous
e.symm.toContinuousAlgHo... | theorem | Topology | [
"Mathlib.Topology.Algebra.Algebra"
] | Mathlib/Topology/Algebra/Algebra/Equiv.lean | isUniformEmbedding | null |
_root_.AlgEquiv.isUniformEmbedding {E₁ E₂ : Type*} [UniformSpace E₁] [UniformSpace E₂]
[Ring E₁] [IsUniformAddGroup E₁] [Algebra R E₁] [Ring E₂] [IsUniformAddGroup E₂] [Algebra R E₂]
(e : E₁ ≃ₐ[R] E₂) (h₁ : Continuous e) (h₂ : Continuous e.symm) :
IsUniformEmbedding e :=
ContinuousAlgEquiv.isUniformEmbedd... | theorem | Topology | [
"Mathlib.Topology.Algebra.Algebra"
] | Mathlib/Topology/Algebra/Algebra/Equiv.lean | _root_.AlgEquiv.isUniformEmbedding | null |
DivisionRing.continuousConstSMul_rat {A} [DivisionRing A] [TopologicalSpace A]
[ContinuousMul A] [CharZero A] : ContinuousConstSMul ℚ A :=
⟨fun r => by simpa only [Algebra.smul_def] using continuous_const.mul continuous_id⟩ | instance | Topology | [
"Mathlib.Algebra.Algebra.Rat",
"Mathlib.Topology.Algebra.ConstMulAction",
"Mathlib.Topology.Algebra.Monoid.Defs"
] | Mathlib/Topology/Algebra/Algebra/Rat.lean | DivisionRing.continuousConstSMul_rat | The action induced by `DivisionRing.toRatAlgebra` is continuous. |
@[to_additive /-- Put the same topological space structure on `Mᵈᵃᵃ` as on the original space. -/]
instTopologicalSpace : TopologicalSpace Mᵈᵐᵃ := .induced mk.symm ‹_›
@[to_additive (attr := continuity, fun_prop)] | instance | Topology | [
"Mathlib.Topology.Homeomorph.Lemmas",
"Mathlib.GroupTheory.GroupAction.DomAct.Basic"
] | Mathlib/Topology/Algebra/Constructions/DomMulAct.lean | instTopologicalSpace | Put the same topological space structure on `Mᵈᵐᵃ` as on the original space. |
continuous_mk : Continuous (@mk M) := continuous_induced_rng.2 continuous_id
@[to_additive (attr := continuity, fun_prop)] | theorem | Topology | [
"Mathlib.Topology.Homeomorph.Lemmas",
"Mathlib.GroupTheory.GroupAction.DomAct.Basic"
] | Mathlib/Topology/Algebra/Constructions/DomMulAct.lean | continuous_mk | null |
continuous_mk_symm : Continuous (@mk M).symm := continuous_induced_dom | theorem | Topology | [
"Mathlib.Topology.Homeomorph.Lemmas",
"Mathlib.GroupTheory.GroupAction.DomAct.Basic"
] | Mathlib/Topology/Algebra/Constructions/DomMulAct.lean | continuous_mk_symm | null |
@[to_additive (attr := simps toEquiv) /-- `DomAddAct.mk` as a homeomorphism. -/]
mkHomeomorph : M ≃ₜ Mᵈᵐᵃ where
toEquiv := mk
@[to_additive (attr := simp)] theorem coe_mkHomeomorph : ⇑(mkHomeomorph : M ≃ₜ Mᵈᵐᵃ) = mk := rfl
@[to_additive (attr := simp)] | def | Topology | [
"Mathlib.Topology.Homeomorph.Lemmas",
"Mathlib.GroupTheory.GroupAction.DomAct.Basic"
] | Mathlib/Topology/Algebra/Constructions/DomMulAct.lean | mkHomeomorph | `DomMulAct.mk` as a homeomorphism. |
coe_mkHomeomorph_symm : ⇑(mkHomeomorph : M ≃ₜ Mᵈᵐᵃ).symm = mk.symm := rfl
@[to_additive] theorem isInducing_mk : IsInducing (@mk M) := mkHomeomorph.isInducing
@[to_additive] theorem isEmbedding_mk : IsEmbedding (@mk M) := mkHomeomorph.isEmbedding
@[to_additive] theorem isOpenEmbedding_mk : IsOpenEmbedding (@mk M) := mk... | theorem | Topology | [
"Mathlib.Topology.Homeomorph.Lemmas",
"Mathlib.GroupTheory.GroupAction.DomAct.Basic"
] | Mathlib/Topology/Algebra/Constructions/DomMulAct.lean | coe_mkHomeomorph_symm | null |
isOpenEmbedding_mk_symm : IsOpenEmbedding (@mk M).symm := mkHomeomorph.symm.isOpenEmbedding
@[to_additive] | theorem | Topology | [
"Mathlib.Topology.Homeomorph.Lemmas",
"Mathlib.GroupTheory.GroupAction.DomAct.Basic"
] | Mathlib/Topology/Algebra/Constructions/DomMulAct.lean | isOpenEmbedding_mk_symm | null |
isClosedEmbedding_mk_symm : IsClosedEmbedding (@mk M).symm :=
mkHomeomorph.symm.isClosedEmbedding
@[to_additive] | theorem | Topology | [
"Mathlib.Topology.Homeomorph.Lemmas",
"Mathlib.GroupTheory.GroupAction.DomAct.Basic"
] | Mathlib/Topology/Algebra/Constructions/DomMulAct.lean | isClosedEmbedding_mk_symm | null |
isQuotientMap_mk_symm : IsQuotientMap (@mk M).symm := mkHomeomorph.symm.isQuotientMap
@[to_additive] instance instT0Space [T0Space M] : T0Space Mᵈᵐᵃ := mkHomeomorph.t0Space
@[to_additive] instance instT1Space [T1Space M] : T1Space Mᵈᵐᵃ := mkHomeomorph.t1Space
@[to_additive] instance instT2Space [T2Space M] : T2Space Mᵈ... | theorem | Topology | [
"Mathlib.Topology.Homeomorph.Lemmas",
"Mathlib.GroupTheory.GroupAction.DomAct.Basic"
] | Mathlib/Topology/Algebra/Constructions/DomMulAct.lean | isQuotientMap_mk_symm | null |
instRegularSpace [RegularSpace M] : RegularSpace Mᵈᵐᵃ := isEmbedding_mk_symm.regularSpace
@[to_additive] | instance | Topology | [
"Mathlib.Topology.Homeomorph.Lemmas",
"Mathlib.GroupTheory.GroupAction.DomAct.Basic"
] | Mathlib/Topology/Algebra/Constructions/DomMulAct.lean | instRegularSpace | null |
instNormalSpace [NormalSpace M] : NormalSpace Mᵈᵐᵃ := mkHomeomorph.normalSpace
@[to_additive] | instance | Topology | [
"Mathlib.Topology.Homeomorph.Lemmas",
"Mathlib.GroupTheory.GroupAction.DomAct.Basic"
] | Mathlib/Topology/Algebra/Constructions/DomMulAct.lean | instNormalSpace | null |
instCompletelyNormalSpace [CompletelyNormalSpace M] : CompletelyNormalSpace Mᵈᵐᵃ :=
isEmbedding_mk_symm.completelyNormalSpace
@[to_additive] | instance | Topology | [
"Mathlib.Topology.Homeomorph.Lemmas",
"Mathlib.GroupTheory.GroupAction.DomAct.Basic"
] | Mathlib/Topology/Algebra/Constructions/DomMulAct.lean | instCompletelyNormalSpace | null |
instDiscreteTopology [DiscreteTopology M] : DiscreteTopology Mᵈᵐᵃ :=
isEmbedding_mk_symm.discreteTopology
@[to_additive] | instance | Topology | [
"Mathlib.Topology.Homeomorph.Lemmas",
"Mathlib.GroupTheory.GroupAction.DomAct.Basic"
] | Mathlib/Topology/Algebra/Constructions/DomMulAct.lean | instDiscreteTopology | null |
instSeparableSpace [SeparableSpace M] : SeparableSpace Mᵈᵐᵃ :=
isQuotientMap_mk.separableSpace
@[to_additive] | instance | Topology | [
"Mathlib.Topology.Homeomorph.Lemmas",
"Mathlib.GroupTheory.GroupAction.DomAct.Basic"
] | Mathlib/Topology/Algebra/Constructions/DomMulAct.lean | instSeparableSpace | null |
instFirstCountableTopology [FirstCountableTopology M] : FirstCountableTopology Mᵈᵐᵃ :=
isInducing_mk_symm.firstCountableTopology
@[to_additive] | instance | Topology | [
"Mathlib.Topology.Homeomorph.Lemmas",
"Mathlib.GroupTheory.GroupAction.DomAct.Basic"
] | Mathlib/Topology/Algebra/Constructions/DomMulAct.lean | instFirstCountableTopology | null |
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