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 ⌀ |
|---|---|---|---|---|---|---|
IsClosed.prod {s₁ : Set X} {s₂ : Set Y} (h₁ : IsClosed s₁) (h₂ : IsClosed s₂) :
IsClosed (s₁ ×ˢ s₂) :=
closure_eq_iff_isClosed.mp <| by simp only [h₁.closure_eq, h₂.closure_eq, closure_prod_eq] | theorem | Topology | [
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Maps.Basic",
"Mathlib.Topology.Separation.SeparatedNhds"
] | Mathlib/Topology/Constructions/SumProd.lean | IsClosed.prod | null |
Dense.prod {s : Set X} {t : Set Y} (hs : Dense s) (ht : Dense t) : Dense (s ×ˢ t) :=
fun x => by
rw [closure_prod_eq]
exact ⟨hs x.1, ht x.2⟩ | theorem | Topology | [
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Maps.Basic",
"Mathlib.Topology.Separation.SeparatedNhds"
] | Mathlib/Topology/Constructions/SumProd.lean | Dense.prod | The product of two dense sets is a dense set. |
DenseRange.prodMap {ι : Type*} {κ : Type*} {f : ι → Y} {g : κ → Z} (hf : DenseRange f)
(hg : DenseRange g) : DenseRange (Prod.map f g) := by
simpa only [DenseRange, prod_range_range_eq] using hf.prod hg | theorem | Topology | [
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Maps.Basic",
"Mathlib.Topology.Separation.SeparatedNhds"
] | Mathlib/Topology/Constructions/SumProd.lean | DenseRange.prodMap | If `f` and `g` are maps with dense range, then `Prod.map f g` has dense range. |
Topology.IsInducing.prodMap {f : X → Y} {g : Z → W} (hf : IsInducing f) (hg : IsInducing g) :
IsInducing (Prod.map f g) :=
isInducing_iff_nhds.2 fun (x, z) => by simp_rw [Prod.map_def, nhds_prod_eq, hf.nhds_eq_comap,
hg.nhds_eq_comap, prod_comap_comap_eq]
@[simp] | lemma | Topology | [
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Maps.Basic",
"Mathlib.Topology.Separation.SeparatedNhds"
] | Mathlib/Topology/Constructions/SumProd.lean | Topology.IsInducing.prodMap | null |
Topology.isInducing_const_prod {x : X} {f : Y → Z} :
IsInducing (fun x' => (x, f x')) ↔ IsInducing f := by
simp_rw [isInducing_iff, instTopologicalSpaceProd, induced_inf, induced_compose,
Function.comp_def, induced_const, top_inf_eq]
@[simp] | lemma | Topology | [
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Maps.Basic",
"Mathlib.Topology.Separation.SeparatedNhds"
] | Mathlib/Topology/Constructions/SumProd.lean | Topology.isInducing_const_prod | null |
Topology.isInducing_prod_const {y : Y} {f : X → Z} :
IsInducing (fun x => (f x, y)) ↔ IsInducing f := by
simp_rw [isInducing_iff, instTopologicalSpaceProd, induced_inf, induced_compose,
Function.comp_def, induced_const, inf_top_eq] | lemma | Topology | [
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Maps.Basic",
"Mathlib.Topology.Separation.SeparatedNhds"
] | Mathlib/Topology/Constructions/SumProd.lean | Topology.isInducing_prod_const | null |
isInducing_prodMkLeft (y : Y) : IsInducing (fun x : X ↦ (x, y)) :=
.of_comp (.prodMk_left y) continuous_fst .id | lemma | Topology | [
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Maps.Basic",
"Mathlib.Topology.Separation.SeparatedNhds"
] | Mathlib/Topology/Constructions/SumProd.lean | isInducing_prodMkLeft | null |
isInducing_prodMkRight (x : X) : IsInducing (Prod.mk x : Y → X × Y) :=
.of_comp (.prodMk_right x) continuous_snd .id | lemma | Topology | [
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Maps.Basic",
"Mathlib.Topology.Separation.SeparatedNhds"
] | Mathlib/Topology/Constructions/SumProd.lean | isInducing_prodMkRight | null |
Topology.IsEmbedding.prodMap {f : X → Y} {g : Z → W} (hf : IsEmbedding f)
(hg : IsEmbedding g) : IsEmbedding (Prod.map f g) where
toIsInducing := hf.isInducing.prodMap hg.isInducing
injective := hf.injective.prodMap hg.injective | lemma | Topology | [
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Maps.Basic",
"Mathlib.Topology.Separation.SeparatedNhds"
] | Mathlib/Topology/Constructions/SumProd.lean | Topology.IsEmbedding.prodMap | null |
protected IsOpenMap.prodMap {f : X → Y} {g : Z → W} (hf : IsOpenMap f) (hg : IsOpenMap g) :
IsOpenMap (Prod.map f g) := by
rw [isOpenMap_iff_nhds_le]
rintro ⟨a, b⟩
rw [nhds_prod_eq, nhds_prod_eq, ← Filter.prod_map_map_eq']
exact Filter.prod_mono (hf.nhds_le a) (hg.nhds_le b) | theorem | Topology | [
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Maps.Basic",
"Mathlib.Topology.Separation.SeparatedNhds"
] | Mathlib/Topology/Constructions/SumProd.lean | IsOpenMap.prodMap | null |
protected Topology.IsOpenEmbedding.prodMap {f : X → Y} {g : Z → W} (hf : IsOpenEmbedding f)
(hg : IsOpenEmbedding g) : IsOpenEmbedding (Prod.map f g) :=
.of_isEmbedding_isOpenMap (hf.1.prodMap hg.1) (hf.isOpenMap.prodMap hg.isOpenMap) | lemma | Topology | [
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Maps.Basic",
"Mathlib.Topology.Separation.SeparatedNhds"
] | Mathlib/Topology/Constructions/SumProd.lean | Topology.IsOpenEmbedding.prodMap | null |
isEmbedding_graph {f : X → Y} (hf : Continuous f) : IsEmbedding fun x => (x, f x) :=
.of_comp (continuous_id.prodMk hf) continuous_fst .id | lemma | Topology | [
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Maps.Basic",
"Mathlib.Topology.Separation.SeparatedNhds"
] | Mathlib/Topology/Constructions/SumProd.lean | isEmbedding_graph | null |
isEmbedding_prodMkLeft (y : Y) : IsEmbedding (fun x : X ↦ (x, y)) :=
.of_comp (.prodMk_left y) continuous_fst .id | lemma | Topology | [
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Maps.Basic",
"Mathlib.Topology.Separation.SeparatedNhds"
] | Mathlib/Topology/Constructions/SumProd.lean | isEmbedding_prodMkLeft | null |
isEmbedding_prodMkRight (x : X) : IsEmbedding (Prod.mk x : Y → X × Y) :=
.of_comp (.prodMk_right x) continuous_snd .id
@[deprecated (since := "2025-06-12")] alias isEmbedding_prodMk := isEmbedding_prodMkRight | lemma | Topology | [
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Maps.Basic",
"Mathlib.Topology.Separation.SeparatedNhds"
] | Mathlib/Topology/Constructions/SumProd.lean | isEmbedding_prodMkRight | null |
IsOpenQuotientMap.prodMap {f : X → Y} {g : Z → W} (hf : IsOpenQuotientMap f)
(hg : IsOpenQuotientMap g) : IsOpenQuotientMap (Prod.map f g) :=
⟨.prodMap hf.1 hg.1, .prodMap hf.2 hg.2, .prodMap hf.3 hg.3⟩ | theorem | Topology | [
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Maps.Basic",
"Mathlib.Topology.Separation.SeparatedNhds"
] | Mathlib/Topology/Constructions/SumProd.lean | IsOpenQuotientMap.prodMap | null |
prodCongr (h₁ : X ≃ₜ X') (h₂ : Y ≃ₜ Y') : X × Y ≃ₜ X' × Y' where
toEquiv := h₁.toEquiv.prodCongr h₂.toEquiv
@[simp] | def | Topology | [
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Maps.Basic",
"Mathlib.Topology.Separation.SeparatedNhds"
] | Mathlib/Topology/Constructions/SumProd.lean | prodCongr | Product of two homeomorphisms. |
prodCongr_symm (h₁ : X ≃ₜ X') (h₂ : Y ≃ₜ Y') :
(h₁.prodCongr h₂).symm = h₁.symm.prodCongr h₂.symm :=
rfl
@[simp] | theorem | Topology | [
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Maps.Basic",
"Mathlib.Topology.Separation.SeparatedNhds"
] | Mathlib/Topology/Constructions/SumProd.lean | prodCongr_symm | null |
coe_prodCongr (h₁ : X ≃ₜ X') (h₂ : Y ≃ₜ Y') : ⇑(h₁.prodCongr h₂) = Prod.map h₁ h₂ :=
rfl
variable (W X Y Z) | theorem | Topology | [
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Maps.Basic",
"Mathlib.Topology.Separation.SeparatedNhds"
] | Mathlib/Topology/Constructions/SumProd.lean | coe_prodCongr | null |
prodComm : X × Y ≃ₜ Y × X where
continuous_toFun := continuous_snd.prodMk continuous_fst
continuous_invFun := continuous_snd.prodMk continuous_fst
toEquiv := Equiv.prodComm X Y
@[simp] | def | Topology | [
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Maps.Basic",
"Mathlib.Topology.Separation.SeparatedNhds"
] | Mathlib/Topology/Constructions/SumProd.lean | prodComm | `X × Y` is homeomorphic to `Y × X`. |
prodComm_symm : (prodComm X Y).symm = prodComm Y X :=
rfl
@[simp] | theorem | Topology | [
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Maps.Basic",
"Mathlib.Topology.Separation.SeparatedNhds"
] | Mathlib/Topology/Constructions/SumProd.lean | prodComm_symm | null |
coe_prodComm : ⇑(prodComm X Y) = Prod.swap :=
rfl | theorem | Topology | [
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Maps.Basic",
"Mathlib.Topology.Separation.SeparatedNhds"
] | Mathlib/Topology/Constructions/SumProd.lean | coe_prodComm | null |
prodAssoc : (X × Y) × Z ≃ₜ X × Y × Z where
continuous_toFun := continuous_fst.fst.prodMk (continuous_fst.snd.prodMk continuous_snd)
continuous_invFun := (continuous_fst.prodMk continuous_snd.fst).prodMk continuous_snd.snd
toEquiv := Equiv.prodAssoc X Y Z
@[simp] | def | Topology | [
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Maps.Basic",
"Mathlib.Topology.Separation.SeparatedNhds"
] | Mathlib/Topology/Constructions/SumProd.lean | prodAssoc | `(X × Y) × Z` is homeomorphic to `X × (Y × Z)`. |
prodAssoc_toEquiv : (prodAssoc X Y Z).toEquiv = Equiv.prodAssoc X Y Z := rfl | lemma | Topology | [
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Maps.Basic",
"Mathlib.Topology.Separation.SeparatedNhds"
] | Mathlib/Topology/Constructions/SumProd.lean | prodAssoc_toEquiv | null |
prodProdProdComm : (X × Y) × W × Z ≃ₜ (X × W) × Y × Z where
toEquiv := Equiv.prodProdProdComm X Y W Z
continuous_toFun := by
unfold Equiv.prodProdProdComm
dsimp only
fun_prop
continuous_invFun := by
unfold Equiv.prodProdProdComm
dsimp only
fun_prop
@[simp] | def | Topology | [
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Maps.Basic",
"Mathlib.Topology.Separation.SeparatedNhds"
] | Mathlib/Topology/Constructions/SumProd.lean | prodProdProdComm | Four-way commutativity of `prod`. The name matches `mul_mul_mul_comm`. |
prodProdProdComm_symm : (prodProdProdComm X Y W Z).symm = prodProdProdComm X W Y Z :=
rfl | theorem | Topology | [
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Maps.Basic",
"Mathlib.Topology.Separation.SeparatedNhds"
] | Mathlib/Topology/Constructions/SumProd.lean | prodProdProdComm_symm | null |
@[simps! -fullyApplied apply]
prodPUnit : X × PUnit ≃ₜ X where
toEquiv := Equiv.prodPUnit X
continuous_toFun := continuous_fst
continuous_invFun := .prodMk_left _ | def | Topology | [
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Maps.Basic",
"Mathlib.Topology.Separation.SeparatedNhds"
] | Mathlib/Topology/Constructions/SumProd.lean | prodPUnit | `X × {*}` is homeomorphic to `X`. |
punitProd : PUnit × X ≃ₜ X :=
(prodComm _ _).trans (prodPUnit _)
@[simp] theorem coe_punitProd : ⇑(punitProd X) = Prod.snd := rfl | def | Topology | [
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Maps.Basic",
"Mathlib.Topology.Separation.SeparatedNhds"
] | Mathlib/Topology/Constructions/SumProd.lean | punitProd | `{*} × X` is homeomorphic to `X`. |
continuous_sum_dom {f : X ⊕ Y → Z} :
Continuous f ↔ Continuous (f ∘ Sum.inl) ∧ Continuous (f ∘ Sum.inr) :=
(continuous_sup_dom (t₁ := TopologicalSpace.coinduced Sum.inl _)
(t₂ := TopologicalSpace.coinduced Sum.inr _)).trans <|
continuous_coinduced_dom.and continuous_coinduced_dom | theorem | Topology | [
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Maps.Basic",
"Mathlib.Topology.Separation.SeparatedNhds"
] | Mathlib/Topology/Constructions/SumProd.lean | continuous_sum_dom | null |
continuous_sumElim {f : X → Z} {g : Y → Z} :
Continuous (Sum.elim f g) ↔ Continuous f ∧ Continuous g :=
continuous_sum_dom
@[continuity, fun_prop] | theorem | Topology | [
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Maps.Basic",
"Mathlib.Topology.Separation.SeparatedNhds"
] | Mathlib/Topology/Constructions/SumProd.lean | continuous_sumElim | null |
Continuous.sumElim {f : X → Z} {g : Y → Z} (hf : Continuous f) (hg : Continuous g) :
Continuous (Sum.elim f g) :=
continuous_sumElim.2 ⟨hf, hg⟩
@[continuity, fun_prop] | theorem | Topology | [
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Maps.Basic",
"Mathlib.Topology.Separation.SeparatedNhds"
] | Mathlib/Topology/Constructions/SumProd.lean | Continuous.sumElim | null |
continuous_isLeft : Continuous (isLeft : X ⊕ Y → Bool) :=
continuous_sum_dom.2 ⟨continuous_const, continuous_const⟩
@[continuity, fun_prop] | theorem | Topology | [
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Maps.Basic",
"Mathlib.Topology.Separation.SeparatedNhds"
] | Mathlib/Topology/Constructions/SumProd.lean | continuous_isLeft | null |
continuous_isRight : Continuous (isRight : X ⊕ Y → Bool) :=
continuous_sum_dom.2 ⟨continuous_const, continuous_const⟩
@[continuity, fun_prop] | theorem | Topology | [
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Maps.Basic",
"Mathlib.Topology.Separation.SeparatedNhds"
] | Mathlib/Topology/Constructions/SumProd.lean | continuous_isRight | null |
continuous_inl : Continuous (@inl X Y) := ⟨fun _ => And.left⟩
@[continuity, fun_prop] | theorem | Topology | [
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Maps.Basic",
"Mathlib.Topology.Separation.SeparatedNhds"
] | Mathlib/Topology/Constructions/SumProd.lean | continuous_inl | null |
continuous_inr : Continuous (@inr X Y) := ⟨fun _ => And.right⟩
@[fun_prop, continuity] | theorem | Topology | [
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Maps.Basic",
"Mathlib.Topology.Separation.SeparatedNhds"
] | Mathlib/Topology/Constructions/SumProd.lean | continuous_inr | null |
continuous_sum_swap : Continuous (@Sum.swap X Y) :=
Continuous.sumElim continuous_inr continuous_inl | lemma | Topology | [
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Maps.Basic",
"Mathlib.Topology.Separation.SeparatedNhds"
] | Mathlib/Topology/Constructions/SumProd.lean | continuous_sum_swap | null |
isOpen_sum_iff {s : Set (X ⊕ Y)} : IsOpen s ↔ IsOpen (inl ⁻¹' s) ∧ IsOpen (inr ⁻¹' s) :=
Iff.rfl | theorem | Topology | [
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Maps.Basic",
"Mathlib.Topology.Separation.SeparatedNhds"
] | Mathlib/Topology/Constructions/SumProd.lean | isOpen_sum_iff | null |
isClosed_sum_iff {s : Set (X ⊕ Y)} :
IsClosed s ↔ IsClosed (inl ⁻¹' s) ∧ IsClosed (inr ⁻¹' s) := by
simp only [← isOpen_compl_iff, isOpen_sum_iff, preimage_compl] | theorem | Topology | [
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Maps.Basic",
"Mathlib.Topology.Separation.SeparatedNhds"
] | Mathlib/Topology/Constructions/SumProd.lean | isClosed_sum_iff | null |
isOpenMap_inl : IsOpenMap (@inl X Y) := fun u hu => by
simpa [isOpen_sum_iff, preimage_image_eq u Sum.inl_injective] | theorem | Topology | [
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Maps.Basic",
"Mathlib.Topology.Separation.SeparatedNhds"
] | Mathlib/Topology/Constructions/SumProd.lean | isOpenMap_inl | null |
isOpenMap_inr : IsOpenMap (@inr X Y) := fun u hu => by
simpa [isOpen_sum_iff, preimage_image_eq u Sum.inr_injective] | theorem | Topology | [
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Maps.Basic",
"Mathlib.Topology.Separation.SeparatedNhds"
] | Mathlib/Topology/Constructions/SumProd.lean | isOpenMap_inr | null |
isClosedMap_inl : IsClosedMap (@inl X Y) := fun u hu ↦ by
simpa [isClosed_sum_iff, preimage_image_eq u Sum.inl_injective] | theorem | Topology | [
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Maps.Basic",
"Mathlib.Topology.Separation.SeparatedNhds"
] | Mathlib/Topology/Constructions/SumProd.lean | isClosedMap_inl | null |
isClosedMap_inr : IsClosedMap (@inr X Y) := fun u hu ↦ by
simpa [isClosed_sum_iff, preimage_image_eq u Sum.inr_injective] | theorem | Topology | [
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Maps.Basic",
"Mathlib.Topology.Separation.SeparatedNhds"
] | Mathlib/Topology/Constructions/SumProd.lean | isClosedMap_inr | null |
protected Topology.IsOpenEmbedding.inl : IsOpenEmbedding (@inl X Y) :=
.of_continuous_injective_isOpenMap continuous_inl inl_injective isOpenMap_inl | lemma | Topology | [
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Maps.Basic",
"Mathlib.Topology.Separation.SeparatedNhds"
] | Mathlib/Topology/Constructions/SumProd.lean | Topology.IsOpenEmbedding.inl | null |
protected Topology.IsOpenEmbedding.inr : IsOpenEmbedding (@inr X Y) :=
.of_continuous_injective_isOpenMap continuous_inr inr_injective isOpenMap_inr | lemma | Topology | [
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Maps.Basic",
"Mathlib.Topology.Separation.SeparatedNhds"
] | Mathlib/Topology/Constructions/SumProd.lean | Topology.IsOpenEmbedding.inr | null |
protected Topology.IsEmbedding.inl : IsEmbedding (@inl X Y) := IsOpenEmbedding.inl.1 | lemma | Topology | [
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Maps.Basic",
"Mathlib.Topology.Separation.SeparatedNhds"
] | Mathlib/Topology/Constructions/SumProd.lean | Topology.IsEmbedding.inl | null |
protected Topology.IsEmbedding.inr : IsEmbedding (@inr X Y) := IsOpenEmbedding.inr.1 | lemma | Topology | [
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Maps.Basic",
"Mathlib.Topology.Separation.SeparatedNhds"
] | Mathlib/Topology/Constructions/SumProd.lean | Topology.IsEmbedding.inr | null |
isOpen_range_inl : IsOpen (range (inl : X → X ⊕ Y)) := IsOpenEmbedding.inl.2 | lemma | Topology | [
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Maps.Basic",
"Mathlib.Topology.Separation.SeparatedNhds"
] | Mathlib/Topology/Constructions/SumProd.lean | isOpen_range_inl | null |
isOpen_range_inr : IsOpen (range (inr : Y → X ⊕ Y)) := IsOpenEmbedding.inr.2 | lemma | Topology | [
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Maps.Basic",
"Mathlib.Topology.Separation.SeparatedNhds"
] | Mathlib/Topology/Constructions/SumProd.lean | isOpen_range_inr | null |
isClosed_range_inl : IsClosed (range (inl : X → X ⊕ Y)) := by
rw [← isOpen_compl_iff, compl_range_inl]
exact isOpen_range_inr | theorem | Topology | [
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Maps.Basic",
"Mathlib.Topology.Separation.SeparatedNhds"
] | Mathlib/Topology/Constructions/SumProd.lean | isClosed_range_inl | null |
isClosed_range_inr : IsClosed (range (inr : Y → X ⊕ Y)) := by
rw [← isOpen_compl_iff, compl_range_inr]
exact isOpen_range_inl | theorem | Topology | [
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Maps.Basic",
"Mathlib.Topology.Separation.SeparatedNhds"
] | Mathlib/Topology/Constructions/SumProd.lean | isClosed_range_inr | null |
Topology.IsClosedEmbedding.inl : IsClosedEmbedding (inl : X → X ⊕ Y) :=
⟨.inl, isClosed_range_inl⟩ | theorem | Topology | [
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Maps.Basic",
"Mathlib.Topology.Separation.SeparatedNhds"
] | Mathlib/Topology/Constructions/SumProd.lean | Topology.IsClosedEmbedding.inl | null |
Topology.IsClosedEmbedding.inr : IsClosedEmbedding (inr : Y → X ⊕ Y) :=
⟨.inr, isClosed_range_inr⟩ | theorem | Topology | [
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Maps.Basic",
"Mathlib.Topology.Separation.SeparatedNhds"
] | Mathlib/Topology/Constructions/SumProd.lean | Topology.IsClosedEmbedding.inr | null |
nhds_inl (x : X) : 𝓝 (inl x : X ⊕ Y) = map inl (𝓝 x) :=
(IsOpenEmbedding.inl.map_nhds_eq _).symm | theorem | Topology | [
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Maps.Basic",
"Mathlib.Topology.Separation.SeparatedNhds"
] | Mathlib/Topology/Constructions/SumProd.lean | nhds_inl | null |
nhds_inr (y : Y) : 𝓝 (inr y : X ⊕ Y) = map inr (𝓝 y) :=
(IsOpenEmbedding.inr.map_nhds_eq _).symm
@[simp] | theorem | Topology | [
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Maps.Basic",
"Mathlib.Topology.Separation.SeparatedNhds"
] | Mathlib/Topology/Constructions/SumProd.lean | nhds_inr | null |
continuous_sumMap {f : X → Y} {g : Z → W} :
Continuous (Sum.map f g) ↔ Continuous f ∧ Continuous g :=
continuous_sumElim.trans <|
IsEmbedding.inl.continuous_iff.symm.and IsEmbedding.inr.continuous_iff.symm
@[continuity, fun_prop] | theorem | Topology | [
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Maps.Basic",
"Mathlib.Topology.Separation.SeparatedNhds"
] | Mathlib/Topology/Constructions/SumProd.lean | continuous_sumMap | null |
Continuous.sumMap {f : X → Y} {g : Z → W} (hf : Continuous f) (hg : Continuous g) :
Continuous (Sum.map f g) :=
continuous_sumMap.2 ⟨hf, hg⟩ | theorem | Topology | [
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Maps.Basic",
"Mathlib.Topology.Separation.SeparatedNhds"
] | Mathlib/Topology/Constructions/SumProd.lean | Continuous.sumMap | null |
isOpenMap_sum {f : X ⊕ Y → Z} :
IsOpenMap f ↔ (IsOpenMap fun a => f (inl a)) ∧ IsOpenMap fun b => f (inr b) := by
simp only [isOpenMap_iff_nhds_le, Sum.forall, nhds_inl, nhds_inr, Filter.map_map, comp_def] | theorem | Topology | [
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Maps.Basic",
"Mathlib.Topology.Separation.SeparatedNhds"
] | Mathlib/Topology/Constructions/SumProd.lean | isOpenMap_sum | null |
IsOpenMap.sumMap {f : X → Y} {g : Z → W} (hf : IsOpenMap f) (hg : IsOpenMap g) :
IsOpenMap (Sum.map f g) :=
isOpenMap_sum.2 ⟨isOpenMap_inl.comp hf, isOpenMap_inr.comp hg⟩
@[simp] | theorem | Topology | [
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Maps.Basic",
"Mathlib.Topology.Separation.SeparatedNhds"
] | Mathlib/Topology/Constructions/SumProd.lean | IsOpenMap.sumMap | null |
isOpenMap_sumElim {f : X → Z} {g : Y → Z} :
IsOpenMap (Sum.elim f g) ↔ IsOpenMap f ∧ IsOpenMap g := by
simp only [isOpenMap_sum, elim_inl, elim_inr] | theorem | Topology | [
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Maps.Basic",
"Mathlib.Topology.Separation.SeparatedNhds"
] | Mathlib/Topology/Constructions/SumProd.lean | isOpenMap_sumElim | null |
IsOpenMap.sumElim {f : X → Z} {g : Y → Z} (hf : IsOpenMap f) (hg : IsOpenMap g) :
IsOpenMap (Sum.elim f g) :=
isOpenMap_sumElim.2 ⟨hf, hg⟩ | theorem | Topology | [
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Maps.Basic",
"Mathlib.Topology.Separation.SeparatedNhds"
] | Mathlib/Topology/Constructions/SumProd.lean | IsOpenMap.sumElim | null |
IsOpenEmbedding.sumElim {f : X → Z} {g : Y → Z}
(hf : IsOpenEmbedding f) (hg : IsOpenEmbedding g) (h : Injective (Sum.elim f g)) :
IsOpenEmbedding (Sum.elim f g) := by
rw [isOpenEmbedding_iff_continuous_injective_isOpenMap] at hf hg ⊢
exact ⟨hf.1.sumElim hg.1, h, hf.2.2.sumElim hg.2.2⟩ | lemma | Topology | [
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Maps.Basic",
"Mathlib.Topology.Separation.SeparatedNhds"
] | Mathlib/Topology/Constructions/SumProd.lean | IsOpenEmbedding.sumElim | null |
isClosedMap_sum {f : X ⊕ Y → Z} :
IsClosedMap f ↔ (IsClosedMap fun a => f (.inl a)) ∧ IsClosedMap fun b => f (.inr b) := by
constructor
· intro h
exact ⟨h.comp IsClosedEmbedding.inl.isClosedMap, h.comp IsClosedEmbedding.inr.isClosedMap⟩
· rintro h Z hZ
rw [isClosed_sum_iff] at hZ
convert (h.1 _ hZ.1).union (h.2 _ hZ.2)
ext
simp only [mem_image, Sum.exists, mem_union, mem_preimage] | theorem | Topology | [
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Maps.Basic",
"Mathlib.Topology.Separation.SeparatedNhds"
] | Mathlib/Topology/Constructions/SumProd.lean | isClosedMap_sum | null |
IsClosedMap.sumMap {f : X → Y} {g : Z → W} (hf : IsClosedMap f) (hg : IsClosedMap g) :
IsClosedMap (Sum.map f g) :=
isClosedMap_sum.2 ⟨isClosedMap_inl.comp hf, isClosedMap_inr.comp hg⟩
@[simp] | theorem | Topology | [
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Maps.Basic",
"Mathlib.Topology.Separation.SeparatedNhds"
] | Mathlib/Topology/Constructions/SumProd.lean | IsClosedMap.sumMap | null |
isClosedMap_sumElim {f : X → Z} {g : Y → Z} :
IsClosedMap (Sum.elim f g) ↔ IsClosedMap f ∧ IsClosedMap g := by
simp only [isClosedMap_sum, Sum.elim_inl, Sum.elim_inr] | theorem | Topology | [
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Maps.Basic",
"Mathlib.Topology.Separation.SeparatedNhds"
] | Mathlib/Topology/Constructions/SumProd.lean | isClosedMap_sumElim | null |
IsClosedMap.sumElim {f : X → Z} {g : Y → Z} (hf : IsClosedMap f) (hg : IsClosedMap g) :
IsClosedMap (Sum.elim f g) :=
isClosedMap_sumElim.2 ⟨hf, hg⟩ | theorem | Topology | [
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Maps.Basic",
"Mathlib.Topology.Separation.SeparatedNhds"
] | Mathlib/Topology/Constructions/SumProd.lean | IsClosedMap.sumElim | null |
IsClosedEmbedding.sumElim {f : X → Z} {g : Y → Z}
(hf : IsClosedEmbedding f) (hg : IsClosedEmbedding g) (h : Injective (Sum.elim f g)) :
IsClosedEmbedding (Sum.elim f g) := by
rw [IsClosedEmbedding.isClosedEmbedding_iff_continuous_injective_isClosedMap] at hf hg ⊢
exact ⟨hf.1.sumElim hg.1, h, hf.2.2.sumElim hg.2.2⟩ | lemma | Topology | [
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Maps.Basic",
"Mathlib.Topology.Separation.SeparatedNhds"
] | Mathlib/Topology/Constructions/SumProd.lean | IsClosedEmbedding.sumElim | null |
sumCongr (h₁ : X ≃ₜ X') (h₂ : Y ≃ₜ Y') : X ⊕ Y ≃ₜ X' ⊕ Y' where
continuous_toFun := h₁.continuous.sumMap h₂.continuous
continuous_invFun := h₁.symm.continuous.sumMap h₂.symm.continuous
toEquiv := h₁.toEquiv.sumCongr h₂.toEquiv
@[simp] | def | Topology | [
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Maps.Basic",
"Mathlib.Topology.Separation.SeparatedNhds"
] | Mathlib/Topology/Constructions/SumProd.lean | sumCongr | Sum of two homeomorphisms. |
sumCongr_symm (h₁ : X ≃ₜ X') (h₂ : Y ≃ₜ Y') :
(sumCongr h₁ h₂).symm = sumCongr h₁.symm h₂.symm := rfl
@[simp] | lemma | Topology | [
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Maps.Basic",
"Mathlib.Topology.Separation.SeparatedNhds"
] | Mathlib/Topology/Constructions/SumProd.lean | sumCongr_symm | null |
sumCongr_refl : sumCongr (.refl X) (.refl Y) = .refl (X ⊕ Y) := by
ext i
cases i <;> rfl
@[simp] | theorem | Topology | [
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Maps.Basic",
"Mathlib.Topology.Separation.SeparatedNhds"
] | Mathlib/Topology/Constructions/SumProd.lean | sumCongr_refl | null |
sumCongr_trans {X'' Y'' : Type*} [TopologicalSpace X''] [TopologicalSpace Y'']
(h₁ : X ≃ₜ X') (h₂ : Y ≃ₜ Y') (h₃ : X' ≃ₜ X'') (h₄ : Y' ≃ₜ Y'') :
(sumCongr h₁ h₂).trans (sumCongr h₃ h₄) = sumCongr (h₁.trans h₃) (h₂.trans h₄) := by
ext i
cases i <;> rfl
variable (W X Y Z) | theorem | Topology | [
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Maps.Basic",
"Mathlib.Topology.Separation.SeparatedNhds"
] | Mathlib/Topology/Constructions/SumProd.lean | sumCongr_trans | null |
sumComm : X ⊕ Y ≃ₜ Y ⊕ X where
toEquiv := Equiv.sumComm X Y
continuous_toFun := continuous_sum_swap
continuous_invFun := continuous_sum_swap
@[simp] | def | Topology | [
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Maps.Basic",
"Mathlib.Topology.Separation.SeparatedNhds"
] | Mathlib/Topology/Constructions/SumProd.lean | sumComm | `X ⊕ Y` is homeomorphic to `Y ⊕ X`. |
sumComm_symm : (sumComm X Y).symm = sumComm Y X :=
rfl
@[simp] | theorem | Topology | [
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Maps.Basic",
"Mathlib.Topology.Separation.SeparatedNhds"
] | Mathlib/Topology/Constructions/SumProd.lean | sumComm_symm | null |
coe_sumComm : ⇑(sumComm X Y) = Sum.swap :=
rfl
@[continuity, fun_prop] | theorem | Topology | [
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Maps.Basic",
"Mathlib.Topology.Separation.SeparatedNhds"
] | Mathlib/Topology/Constructions/SumProd.lean | coe_sumComm | null |
continuous_sumAssoc : Continuous (Equiv.sumAssoc X Y Z) :=
Continuous.sumElim (by fun_prop) (by fun_prop)
@[continuity, fun_prop] | lemma | Topology | [
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Maps.Basic",
"Mathlib.Topology.Separation.SeparatedNhds"
] | Mathlib/Topology/Constructions/SumProd.lean | continuous_sumAssoc | null |
continuous_sumAssoc_symm : Continuous (Equiv.sumAssoc X Y Z).symm :=
Continuous.sumElim (by fun_prop) (by fun_prop) | lemma | Topology | [
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Maps.Basic",
"Mathlib.Topology.Separation.SeparatedNhds"
] | Mathlib/Topology/Constructions/SumProd.lean | continuous_sumAssoc_symm | null |
sumAssoc : (X ⊕ Y) ⊕ Z ≃ₜ X ⊕ Y ⊕ Z where
toEquiv := Equiv.sumAssoc X Y Z
continuous_toFun := continuous_sumAssoc X Y Z
continuous_invFun := continuous_sumAssoc_symm X Y Z
@[simp] | def | Topology | [
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Maps.Basic",
"Mathlib.Topology.Separation.SeparatedNhds"
] | Mathlib/Topology/Constructions/SumProd.lean | sumAssoc | `(X ⊕ Y) ⊕ Z` is homeomorphic to `X ⊕ (Y ⊕ Z)`. |
sumAssoc_toEquiv : (sumAssoc X Y Z).toEquiv = Equiv.sumAssoc X Y Z := rfl | lemma | Topology | [
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Maps.Basic",
"Mathlib.Topology.Separation.SeparatedNhds"
] | Mathlib/Topology/Constructions/SumProd.lean | sumAssoc_toEquiv | null |
sumSumSumComm : (X ⊕ Y) ⊕ W ⊕ Z ≃ₜ (X ⊕ W) ⊕ Y ⊕ Z where
toEquiv := Equiv.sumSumSumComm X Y W Z
continuous_toFun := by
have : Continuous (Sum.map (Sum.map (@id X) ⇑(Homeomorph.sumComm Y W)) (@id Z)) := by fun_prop
fun_prop
continuous_invFun := by
have : Continuous (Sum.map (Sum.map (@id X) (Homeomorph.sumComm Y W).symm) (@id Z)) := by
fun_prop
fun_prop
@[simp] | def | Topology | [
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Maps.Basic",
"Mathlib.Topology.Separation.SeparatedNhds"
] | Mathlib/Topology/Constructions/SumProd.lean | sumSumSumComm | Four-way commutativity of the disjoint union. The name matches `add_add_add_comm`. |
sumSumSumComm_toEquiv : (sumSumSumComm W X Y Z).toEquiv = (Equiv.sumSumSumComm W X Y Z) := rfl
@[simp] | lemma | Topology | [
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Maps.Basic",
"Mathlib.Topology.Separation.SeparatedNhds"
] | Mathlib/Topology/Constructions/SumProd.lean | sumSumSumComm_toEquiv | null |
sumSumSumComm_symm : (sumSumSumComm X Y W Z).symm = (sumSumSumComm X W Y Z) := rfl | lemma | Topology | [
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Maps.Basic",
"Mathlib.Topology.Separation.SeparatedNhds"
] | Mathlib/Topology/Constructions/SumProd.lean | sumSumSumComm_symm | null |
@[simps! -fullyApplied apply]
sumEmpty [IsEmpty Y] : X ⊕ Y ≃ₜ X where
toEquiv := Equiv.sumEmpty X Y
continuous_toFun := Continuous.sumElim continuous_id (by fun_prop)
continuous_invFun := continuous_inl | def | Topology | [
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Maps.Basic",
"Mathlib.Topology.Separation.SeparatedNhds"
] | Mathlib/Topology/Constructions/SumProd.lean | sumEmpty | The sum of `X` with any empty topological space is homeomorphic to `X`. |
emptySum [IsEmpty Y] : Y ⊕ X ≃ₜ X := (sumComm Y X).trans (sumEmpty X Y)
@[simp] theorem coe_emptySum [IsEmpty Y] : (emptySum X Y).toEquiv = Equiv.emptySum Y X := rfl
variable {W X Y Z} | def | Topology | [
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Maps.Basic",
"Mathlib.Topology.Separation.SeparatedNhds"
] | Mathlib/Topology/Constructions/SumProd.lean | emptySum | The sum of `X` with any empty topological space is homeomorphic to `X`. |
@[simps!]
sumProdDistrib : (X ⊕ Y) × Z ≃ₜ (X × Z) ⊕ (Y × Z) :=
Homeomorph.symm <|
(Equiv.sumProdDistrib X Y Z).symm.toHomeomorphOfContinuousOpen
((continuous_inl.prodMap continuous_id).sumElim
(continuous_inr.prodMap continuous_id)) <|
(isOpenMap_inl.prodMap IsOpenMap.id).sumElim (isOpenMap_inr.prodMap IsOpenMap.id) | def | Topology | [
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Maps.Basic",
"Mathlib.Topology.Separation.SeparatedNhds"
] | Mathlib/Topology/Constructions/SumProd.lean | sumProdDistrib | `(X ⊕ Y) × Z` is homeomorphic to `X × Z ⊕ Y × Z`. |
prodSumDistrib : X × (Y ⊕ Z) ≃ₜ (X × Y) ⊕ (X × Z) :=
(prodComm _ _).trans <| sumProdDistrib.trans <| sumCongr (prodComm _ _) (prodComm _ _) | def | Topology | [
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Maps.Basic",
"Mathlib.Topology.Separation.SeparatedNhds"
] | Mathlib/Topology/Constructions/SumProd.lean | prodSumDistrib | `X × (Y ⊕ Z)` is homeomorphic to `X × Y ⊕ X × Z`. |
Topology.IsInducing.sumElim_left (h : IsInducing (Sum.elim f g)) : IsInducing f :=
elim_comp_inl f g ▸ h.comp IsEmbedding.inl.isInducing | lemma | Topology | [
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Maps.Basic",
"Mathlib.Topology.Separation.SeparatedNhds"
] | Mathlib/Topology/Constructions/SumProd.lean | Topology.IsInducing.sumElim_left | If `Sum.elim f g` is an inducing map, then so is `f`. |
Topology.IsInducing.sumElim_right (h : IsInducing (Sum.elim f g)) : IsInducing g :=
elim_comp_inr f g ▸ h.comp IsEmbedding.inr.isInducing | lemma | Topology | [
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Maps.Basic",
"Mathlib.Topology.Separation.SeparatedNhds"
] | Mathlib/Topology/Constructions/SumProd.lean | Topology.IsInducing.sumElim_right | If `Sum.elim f g` is an inducing map, then so is `g`. |
Topology.IsInducing.sumElim (hf : IsInducing f) (hg : IsInducing g)
(hFg : Disjoint (closure (range f)) (range g)) (hfG : Disjoint (range f) (closure (range g))) :
IsInducing (Sum.elim f g) := by
rw [← disjoint_principal_nhdsSet] at hFg
rw [← disjoint_nhdsSet_principal] at hfG
rw [isInducing_iff_nhds]
intro x
apply le_antisymm ((hf.continuous.sumElim hg.continuous).tendsto x).le_comap
obtain x | x := x <;>
simp only [comap_sumElim_eq, nhds_inl, nhds_inr, elim_inl, elim_inr, ← hf.nhds_eq_comap,
← hg.nhds_eq_comap, sup_le_iff, le_rfl, true_and, and_true] <;>
convert bot_le (α := Filter (X ⊕ Y)) <;>
rw [map_eq_bot_iff, comap_eq_bot_iff_compl_range]
· rw [← disjoint_principal_right]
exact hfG.mono_left (nhds_le_nhdsSet (mem_range_self x))
· rw [← disjoint_principal_left]
exact hFg.mono_right (nhds_le_nhdsSet (mem_range_self x)) | theorem | Topology | [
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Maps.Basic",
"Mathlib.Topology.Separation.SeparatedNhds"
] | Mathlib/Topology/Constructions/SumProd.lean | Topology.IsInducing.sumElim | If `f` and `g` are inducing maps whose ranges are separated, then `Sum.elim f g` is inducing. |
Topology.IsInducing.disjoint_of_sumElim_aux (h : IsInducing (Sum.elim f g)) :
Disjoint (closure (range f)) (range g) := by
rcases h.isClosed_iff.mp isClosed_range_inl with ⟨C, C_closed, hC⟩
have A : closure (range f) ⊆ C := by
rw [C_closed.closure_subset_iff, ← elim_comp_inl f g, range_comp, image_subset_iff, hC]
have B : Disjoint C (range g) := by
rw [← image_univ, disjoint_image_right, ← elim_comp_inr f g, preimage_comp, hC,
← disjoint_image_right, ← image_univ]
exact disjoint_image_inl_image_inr
exact B.mono_left A | theorem | Topology | [
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Maps.Basic",
"Mathlib.Topology.Separation.SeparatedNhds"
] | Mathlib/Topology/Constructions/SumProd.lean | Topology.IsInducing.disjoint_of_sumElim_aux | If `Sum.elim f g` is inducing, `closure (range f)` and `range g` must be disjoint.
This is an auxiliary result towards proving `isInducing_sumElim`. |
IsOpenEmbedding.sumSwap : IsOpenEmbedding (@Sum.swap X Y) :=
(Homeomorph.sumComm X Y).isOpenEmbedding | theorem | Topology | [
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Maps.Basic",
"Mathlib.Topology.Separation.SeparatedNhds"
] | Mathlib/Topology/Constructions/SumProd.lean | IsOpenEmbedding.sumSwap | null |
IsInducing.sumSwap : IsInducing (@Sum.swap X Y) := IsOpenEmbedding.sumSwap.isInducing | theorem | Topology | [
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Maps.Basic",
"Mathlib.Topology.Separation.SeparatedNhds"
] | Mathlib/Topology/Constructions/SumProd.lean | IsInducing.sumSwap | null |
isInducing_sumElim :
IsInducing (Sum.elim f g) ↔ IsInducing f ∧ IsInducing g ∧
Disjoint (closure (range f)) (range g) ∧ Disjoint (range f) (closure (range g)) :=
⟨fun h ↦ ⟨h.sumElim_left, h.sumElim_right, h.disjoint_of_sumElim_aux,
((Sum.elim_swap ▸ h.comp IsInducing.sumSwap).disjoint_of_sumElim_aux ).symm⟩,
fun ⟨hf, hg, hFg, hfG⟩ ↦ hf.sumElim hg hFg hfG⟩ | theorem | Topology | [
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Maps.Basic",
"Mathlib.Topology.Separation.SeparatedNhds"
] | Mathlib/Topology/Constructions/SumProd.lean | isInducing_sumElim | null |
Topology.IsInducing.sumElim_of_separatedNhds
(hf : IsInducing f) (hg : IsInducing g) (hsep : SeparatedNhds (range f) (range g)) :
IsInducing (Sum.elim f g) :=
hf.sumElim hg hsep.disjoint_closure_left hsep.disjoint_closure_right | lemma | Topology | [
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Maps.Basic",
"Mathlib.Topology.Separation.SeparatedNhds"
] | Mathlib/Topology/Constructions/SumProd.lean | Topology.IsInducing.sumElim_of_separatedNhds | null |
Topology.IsEmbedding.sumElim_left (h : IsEmbedding (Sum.elim f g)) : IsEmbedding f :=
elim_comp_inl f g ▸ h.comp IsEmbedding.inl | lemma | Topology | [
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Maps.Basic",
"Mathlib.Topology.Separation.SeparatedNhds"
] | Mathlib/Topology/Constructions/SumProd.lean | Topology.IsEmbedding.sumElim_left | If `Sum.elim f g` is an embedding, then so is `f`. |
Topology.IsEmbedding.sumElim_right (h : IsEmbedding (Sum.elim f g)) : IsEmbedding g :=
elim_comp_inr f g ▸ h.comp IsEmbedding.inr | lemma | Topology | [
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Maps.Basic",
"Mathlib.Topology.Separation.SeparatedNhds"
] | Mathlib/Topology/Constructions/SumProd.lean | Topology.IsEmbedding.sumElim_right | If `Sum.elim f g` is an embedding, then so is `g`. |
isEmbedding_sumElim :
IsEmbedding (Sum.elim f g) ↔ IsEmbedding f ∧ IsEmbedding g ∧
Disjoint (closure (range f)) (range g) ∧ Disjoint (range f) (closure (range g)) := by
simp_rw [isEmbedding_iff, isInducing_sumElim, Sum.elim_injective]
constructor
· intro ⟨⟨hf₁, hg₁, hFg, hfG⟩, ⟨hf₂, hg₂, f_ne_g⟩⟩
exact ⟨⟨hf₁, hf₂⟩, ⟨hg₁, hg₂⟩, hFg, hfG⟩
· intro ⟨⟨hf₁, hf₂⟩, ⟨hg₁, hg₂⟩, hFg, hfG⟩
refine ⟨⟨hf₁, hg₁, hFg, hfG⟩, ⟨hf₂, hg₂, ?_⟩⟩
exact fun a b ↦ hfG.ne_of_mem (mem_range_self a) (subset_closure (mem_range_self b)) | theorem | Topology | [
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Maps.Basic",
"Mathlib.Topology.Separation.SeparatedNhds"
] | Mathlib/Topology/Constructions/SumProd.lean | isEmbedding_sumElim | null |
Topology.IsEmbedding.sumElim (hf : IsEmbedding f) (hg : IsEmbedding g)
(hFg : Disjoint (closure (range f)) (range g)) (hfG : Disjoint (range f) (closure (range g))) :
IsEmbedding (Sum.elim f g) :=
isEmbedding_sumElim.mpr ⟨hf, hg, hFg, hfG⟩ | theorem | Topology | [
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Maps.Basic",
"Mathlib.Topology.Separation.SeparatedNhds"
] | Mathlib/Topology/Constructions/SumProd.lean | Topology.IsEmbedding.sumElim | If `f` and `g` are embeddings whose ranges are separated, `Sum.elim f g` is an embedding. |
Topology.IsEmbedding.sumElim_of_separatedNhds
(hf : IsEmbedding f) (hg : IsEmbedding g) (hsep : SeparatedNhds (range f) (range g)) :
IsEmbedding (Sum.elim f g) :=
hf.sumElim hg hsep.disjoint_closure_left hsep.disjoint_closure_right | lemma | Topology | [
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Maps.Basic",
"Mathlib.Topology.Separation.SeparatedNhds"
] | Mathlib/Topology/Constructions/SumProd.lean | Topology.IsEmbedding.sumElim_of_separatedNhds | null |
@[to_additive]
instMul [Mul β] [ContinuousMul β] : Mul C(α, β) :=
⟨fun f g => ⟨f * g, continuous_mul.comp (f.continuous.prodMk g.continuous :)⟩⟩
@[to_additive (attr := norm_cast, simp)] | instance | Topology | [
"Mathlib.Algebra.Algebra.Pi",
"Mathlib.Algebra.Algebra.Subalgebra.Basic",
"Mathlib.Tactic.FieldSimp",
"Mathlib.Topology.Algebra.InfiniteSum.Basic",
"Mathlib.Topology.Algebra.Module.LinearMap",
"Mathlib.Topology.Algebra.Ring.Basic",
"Mathlib.Topology.UniformSpace.CompactConvergence"
] | Mathlib/Topology/ContinuousMap/Algebra.lean | instMul | null |
coe_mul [Mul β] [ContinuousMul β] (f g : C(α, β)) : ⇑(f * g) = f * g :=
rfl
@[to_additive (attr := simp)] | theorem | Topology | [
"Mathlib.Algebra.Algebra.Pi",
"Mathlib.Algebra.Algebra.Subalgebra.Basic",
"Mathlib.Tactic.FieldSimp",
"Mathlib.Topology.Algebra.InfiniteSum.Basic",
"Mathlib.Topology.Algebra.Module.LinearMap",
"Mathlib.Topology.Algebra.Ring.Basic",
"Mathlib.Topology.UniformSpace.CompactConvergence"
] | Mathlib/Topology/ContinuousMap/Algebra.lean | coe_mul | null |
mul_apply [Mul β] [ContinuousMul β] (f g : C(α, β)) (x : α) : (f * g) x = f x * g x :=
rfl
@[to_additive (attr := simp)] | theorem | Topology | [
"Mathlib.Algebra.Algebra.Pi",
"Mathlib.Algebra.Algebra.Subalgebra.Basic",
"Mathlib.Tactic.FieldSimp",
"Mathlib.Topology.Algebra.InfiniteSum.Basic",
"Mathlib.Topology.Algebra.Module.LinearMap",
"Mathlib.Topology.Algebra.Ring.Basic",
"Mathlib.Topology.UniformSpace.CompactConvergence"
] | Mathlib/Topology/ContinuousMap/Algebra.lean | mul_apply | null |
mul_comp [Mul γ] [ContinuousMul γ] (f₁ f₂ : C(β, γ)) (g : C(α, β)) :
(f₁ * f₂).comp g = f₁.comp g * f₂.comp g :=
rfl
/-! ### `one` -/
@[to_additive] | theorem | Topology | [
"Mathlib.Algebra.Algebra.Pi",
"Mathlib.Algebra.Algebra.Subalgebra.Basic",
"Mathlib.Tactic.FieldSimp",
"Mathlib.Topology.Algebra.InfiniteSum.Basic",
"Mathlib.Topology.Algebra.Module.LinearMap",
"Mathlib.Topology.Algebra.Ring.Basic",
"Mathlib.Topology.UniformSpace.CompactConvergence"
] | Mathlib/Topology/ContinuousMap/Algebra.lean | mul_comp | null |
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