statement stringlengths 1 2.93k | proof stringlengths 0 19.2k | type stringclasses 13
values | symbolic_name stringlengths 1 131 | library stringlengths 4 62 | filename stringlengths 20 95 | imports listlengths 0 10 | deps listlengths 0 64 | docstring stringlengths 0 4.95k | source_url stringclasses 1
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image_eq_target_inter_inv_preimage {s : Set X} (h : s ⊆ e.source) :
e '' s = e.target ∩ e.symm ⁻¹' s | e.toPartialEquiv.image_eq_target_inter_inv_preimage h | theorem | OpenPartialHomeomorph.image_eq_target_inter_inv_preimage | Topology.OpenPartialHomeomorph | Mathlib/Topology/OpenPartialHomeomorph/Basic.lean | [] | [
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
image_source_inter_eq' (s : Set X) : e '' (e.source ∩ s) = e.target ∩ e.symm ⁻¹' s | e.toPartialEquiv.image_source_inter_eq' s | theorem | OpenPartialHomeomorph.image_source_inter_eq' | Topology.OpenPartialHomeomorph | Mathlib/Topology/OpenPartialHomeomorph/Basic.lean | [] | [
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
image_source_inter_eq (s : Set X) :
e '' (e.source ∩ s) = e.target ∩ e.symm ⁻¹' (e.source ∩ s) | e.toPartialEquiv.image_source_inter_eq s | theorem | OpenPartialHomeomorph.image_source_inter_eq | Topology.OpenPartialHomeomorph | Mathlib/Topology/OpenPartialHomeomorph/Basic.lean | [] | [
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
symm_image_eq_source_inter_preimage {s : Set Y} (h : s ⊆ e.target) :
e.symm '' s = e.source ∩ e ⁻¹' s | e.symm.image_eq_target_inter_inv_preimage h | theorem | OpenPartialHomeomorph.symm_image_eq_source_inter_preimage | Topology.OpenPartialHomeomorph | Mathlib/Topology/OpenPartialHomeomorph/Basic.lean | [] | [
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
symm_image_target_inter_eq (s : Set Y) :
e.symm '' (e.target ∩ s) = e.source ∩ e ⁻¹' (e.target ∩ s) | e.symm.image_source_inter_eq _ | theorem | OpenPartialHomeomorph.symm_image_target_inter_eq | Topology.OpenPartialHomeomorph | Mathlib/Topology/OpenPartialHomeomorph/Basic.lean | [] | [
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
source_inter_preimage_inv_preimage (s : Set X) :
e.source ∩ e ⁻¹' e.symm ⁻¹' s = e.source ∩ s | e.toPartialEquiv.source_inter_preimage_inv_preimage s | theorem | OpenPartialHomeomorph.source_inter_preimage_inv_preimage | Topology.OpenPartialHomeomorph | Mathlib/Topology/OpenPartialHomeomorph/Basic.lean | [] | [
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
target_inter_inv_preimage_preimage (s : Set Y) :
e.target ∩ e.symm ⁻¹' e ⁻¹' s = e.target ∩ s | e.symm.source_inter_preimage_inv_preimage _ | theorem | OpenPartialHomeomorph.target_inter_inv_preimage_preimage | Topology.OpenPartialHomeomorph | Mathlib/Topology/OpenPartialHomeomorph/Basic.lean | [] | [
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
source_inter_preimage_target_inter (s : Set Y) :
e.source ∩ e ⁻¹' (e.target ∩ s) = e.source ∩ e ⁻¹' s | e.toPartialEquiv.source_inter_preimage_target_inter s | theorem | OpenPartialHomeomorph.source_inter_preimage_target_inter | Topology.OpenPartialHomeomorph | Mathlib/Topology/OpenPartialHomeomorph/Basic.lean | [] | [
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
image_source_eq_target : e '' e.source = e.target | e.toPartialEquiv.image_source_eq_target | theorem | OpenPartialHomeomorph.image_source_eq_target | Topology.OpenPartialHomeomorph | Mathlib/Topology/OpenPartialHomeomorph/Basic.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isOpen_inter_preimage {s : Set Y} (hs : IsOpen s) : IsOpen (e.source ∩ e ⁻¹' s) | e.continuousOn.isOpen_inter_preimage e.open_source hs | theorem | OpenPartialHomeomorph.isOpen_inter_preimage | Topology.OpenPartialHomeomorph | Mathlib/Topology/OpenPartialHomeomorph/Basic.lean | [] | [
"IsOpen",
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isOpen_inter_preimage_symm {s : Set X} (hs : IsOpen s) : IsOpen (e.target ∩ e.symm ⁻¹' s) | e.symm.continuousOn.isOpen_inter_preimage e.open_target hs | theorem | OpenPartialHomeomorph.isOpen_inter_preimage_symm | Topology.OpenPartialHomeomorph | Mathlib/Topology/OpenPartialHomeomorph/Basic.lean | [] | [
"IsOpen",
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isOpen_image_of_subset_source {s : Set X} (hs : IsOpen s) (hse : s ⊆ e.source) :
IsOpen (e '' s) | by
rw [(image_eq_target_inter_inv_preimage (e := e) hse)]
exact e.continuousOn_invFun.isOpen_inter_preimage e.open_target hs | lemma | OpenPartialHomeomorph.isOpen_image_of_subset_source | Topology.OpenPartialHomeomorph | Mathlib/Topology/OpenPartialHomeomorph/Basic.lean | [] | [
"IsOpen",
"Set"
] | An open partial homeomorphism is an open map on its source:
the image of an open subset of the source is open. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
isOpen_image_source_inter {s : Set X} (hs : IsOpen s) :
IsOpen (e '' (e.source ∩ s)) | e.isOpen_image_of_subset_source (e.open_source.inter hs) inter_subset_left | theorem | OpenPartialHomeomorph.isOpen_image_source_inter | Topology.OpenPartialHomeomorph | Mathlib/Topology/OpenPartialHomeomorph/Basic.lean | [] | [
"IsOpen",
"Set"
] | The image of the restriction of an open set to the source is open. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
isOpen_image_symm_of_subset_target {t : Set Y} (ht : IsOpen t) (hte : t ⊆ e.target) :
IsOpen (e.symm '' t) | isOpen_image_of_subset_source e.symm ht (e.symm_source ▸ hte) | lemma | OpenPartialHomeomorph.isOpen_image_symm_of_subset_target | Topology.OpenPartialHomeomorph | Mathlib/Topology/OpenPartialHomeomorph/Basic.lean | [] | [
"IsOpen",
"Set"
] | The inverse of an open partial homeomorphism `e` is an open map on `e.target`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
isOpen_symm_image_iff_of_subset_target {t : Set Y} (hs : t ⊆ e.target) :
IsOpen (e.symm '' t) ↔ IsOpen t | by
refine ⟨fun h ↦ ?_, fun h ↦ e.symm.isOpen_image_of_subset_source h hs⟩
have hs' : e.symm '' t ⊆ e.source := by
rw [e.symm_image_eq_source_inter_preimage hs]
apply Set.inter_subset_left
rw [← e.image_symm_image_of_subset_target hs]
exact e.isOpen_image_of_subset_source h hs' | lemma | OpenPartialHomeomorph.isOpen_symm_image_iff_of_subset_target | Topology.OpenPartialHomeomorph | Mathlib/Topology/OpenPartialHomeomorph/Basic.lean | [] | [
"IsOpen",
"Set",
"Set.inter_subset_left"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isOpen_image_iff_of_subset_source {s : Set X} (hs : s ⊆ e.source) :
IsOpen (e '' s) ↔ IsOpen s | by
rw [← e.symm.isOpen_symm_image_iff_of_subset_target hs, e.symm_symm] | theorem | OpenPartialHomeomorph.isOpen_image_iff_of_subset_source | Topology.OpenPartialHomeomorph | Mathlib/Topology/OpenPartialHomeomorph/Basic.lean | [] | [
"IsOpen",
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
ofContinuousOpenRestrict (e : PartialEquiv X Y) (hc : ContinuousOn e e.source)
(ho : IsOpenMap (e.source.restrict e)) (hs : IsOpen e.source) : OpenPartialHomeomorph X Y | where
toPartialEquiv := e
open_source := hs
open_target := by simpa only [range_restrict, e.image_source_eq_target] using ho.isOpen_range
continuousOn_toFun := hc
continuousOn_invFun := e.image_source_eq_target ▸ ho.continuousOn_image_of_leftInvOn e.leftInvOn | def | OpenPartialHomeomorph.ofContinuousOpenRestrict | Topology.OpenPartialHomeomorph | Mathlib/Topology/OpenPartialHomeomorph/Basic.lean | [] | [
"ContinuousOn",
"IsOpen",
"IsOpenMap",
"OpenPartialHomeomorph",
"PartialEquiv"
] | A `PartialEquiv` with continuous open forward map and open source is a
`OpenPartialHomeomorph`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
coe_ofContinuousOpenRestrict (e : PartialEquiv X Y) (hc : ContinuousOn e e.source)
(ho : IsOpenMap (e.source.restrict e)) (hs : IsOpen e.source) :
⇑(ofContinuousOpenRestrict e hc ho hs) = e | rfl | theorem | OpenPartialHomeomorph.coe_ofContinuousOpenRestrict | Topology.OpenPartialHomeomorph | Mathlib/Topology/OpenPartialHomeomorph/Basic.lean | [] | [
"ContinuousOn",
"IsOpen",
"IsOpenMap",
"PartialEquiv"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
coe_ofContinuousOpenRestrict_symm (e : PartialEquiv X Y) (hc : ContinuousOn e e.source)
(ho : IsOpenMap (e.source.restrict e)) (hs : IsOpen e.source) :
⇑(ofContinuousOpenRestrict e hc ho hs).symm = e.symm | rfl | theorem | OpenPartialHomeomorph.coe_ofContinuousOpenRestrict_symm | Topology.OpenPartialHomeomorph | Mathlib/Topology/OpenPartialHomeomorph/Basic.lean | [] | [
"ContinuousOn",
"IsOpen",
"IsOpenMap",
"PartialEquiv",
"symm"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
ofContinuousOpen (e : PartialEquiv X Y) (hc : ContinuousOn e e.source) (ho : IsOpenMap e)
(hs : IsOpen e.source) : OpenPartialHomeomorph X Y | ofContinuousOpenRestrict e hc (ho.restrict hs) hs | def | OpenPartialHomeomorph.ofContinuousOpen | Topology.OpenPartialHomeomorph | Mathlib/Topology/OpenPartialHomeomorph/Basic.lean | [] | [
"ContinuousOn",
"IsOpen",
"IsOpenMap",
"OpenPartialHomeomorph",
"PartialEquiv"
] | A `PartialEquiv` with continuous open forward map and open source is a
`OpenPartialHomeomorph`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
coe_ofContinuousOpen (e : PartialEquiv X Y) (hc : ContinuousOn e e.source)
(ho : IsOpenMap e) (hs : IsOpen e.source) :
⇑(ofContinuousOpen e hc ho hs) = e | rfl | theorem | OpenPartialHomeomorph.coe_ofContinuousOpen | Topology.OpenPartialHomeomorph | Mathlib/Topology/OpenPartialHomeomorph/Basic.lean | [] | [
"ContinuousOn",
"IsOpen",
"IsOpenMap",
"PartialEquiv"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
coe_ofContinuousOpen_symm (e : PartialEquiv X Y) (hc : ContinuousOn e e.source)
(ho : IsOpenMap e) (hs : IsOpen e.source) :
⇑(ofContinuousOpen e hc ho hs).symm = e.symm | rfl | theorem | OpenPartialHomeomorph.coe_ofContinuousOpen_symm | Topology.OpenPartialHomeomorph | Mathlib/Topology/OpenPartialHomeomorph/Basic.lean | [] | [
"ContinuousOn",
"IsOpen",
"IsOpenMap",
"PartialEquiv",
"symm"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
homeomorphOfImageSubsetSource {s : Set X} {t : Set Y} (hs : s ⊆ e.source) (ht : e '' s = t) :
s ≃ₜ t | have h₁ : MapsTo e s t := mapsTo_iff_image_subset.2 ht.subset
have h₂ : t ⊆ e.target := ht ▸ e.image_source_eq_target ▸ image_mono hs
have h₃ : MapsTo e.symm t s := ht ▸ forall_mem_image.2 fun _x hx =>
(e.left_inv (hs hx)).symm ▸ hx
{ toFun := MapsTo.restrict e s t h₁
invFun := MapsTo.restrict e.symm t ... | def | OpenPartialHomeomorph.homeomorphOfImageSubsetSource | Topology.OpenPartialHomeomorph | Mathlib/Topology/OpenPartialHomeomorph/Basic.lean | [] | [
"Set",
"symm"
] | The homeomorphism obtained by restricting an `OpenPartialHomeomorph` to a subset of the source. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
toHomeomorphSourceTarget : e.source ≃ₜ e.target | e.homeomorphOfImageSubsetSource subset_rfl e.image_source_eq_target | def | OpenPartialHomeomorph.toHomeomorphSourceTarget | Topology.OpenPartialHomeomorph | Mathlib/Topology/OpenPartialHomeomorph/Basic.lean | [] | [
"subset_rfl"
] | An open partial homeomorphism defines a homeomorphism between its source and target. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
secondCountableTopology_source [SecondCountableTopology Y] :
SecondCountableTopology e.source | e.toHomeomorphSourceTarget.secondCountableTopology | theorem | OpenPartialHomeomorph.secondCountableTopology_source | Topology.OpenPartialHomeomorph | Mathlib/Topology/OpenPartialHomeomorph/Basic.lean | [] | [
"SecondCountableTopology"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
nhds_eq_comap_inf_principal {x} (hx : x ∈ e.source) :
𝓝 x = comap e (𝓝 (e x)) ⊓ 𝓟 e.source | by
lift x to e.source using hx
rw [← e.open_source.nhdsWithin_eq x.2, ← map_nhds_subtype_val, ← map_comap_setCoe_val,
e.toHomeomorphSourceTarget.nhds_eq_comap, nhds_subtype_eq_comap]
simp only [Function.comp_def, toHomeomorphSourceTarget_apply_coe, comap_comap] | theorem | OpenPartialHomeomorph.nhds_eq_comap_inf_principal | Topology.OpenPartialHomeomorph | Mathlib/Topology/OpenPartialHomeomorph/Basic.lean | [] | [
"comap",
"map_nhds_subtype_val",
"nhds_subtype_eq_comap"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
toHomeomorphOfSourceEqUnivTargetEqUniv (h : e.source = (univ : Set X)) (h' : e.target = univ) :
X ≃ₜ Y | where
toFun := e
invFun := e.symm
left_inv x :=
e.left_inv <| by
rw [h]
exact mem_univ _
right_inv x :=
e.right_inv <| by
rw [h']
exact mem_univ _
continuous_toFun := by
simpa only [continuousOn_univ, h] using e.continuousOn
continuous_invFun := by
simpa only [continu... | def | OpenPartialHomeomorph.toHomeomorphOfSourceEqUnivTargetEqUniv | Topology.OpenPartialHomeomorph | Mathlib/Topology/OpenPartialHomeomorph/Basic.lean | [] | [
"Set",
"continuousOn_univ"
] | If an open partial homeomorphism has source and target equal to univ, then it induces a
homeomorphism between the whole spaces, expressed in this definition. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
isOpenEmbedding_restrict : IsOpenEmbedding (e.source.restrict e) | by
refine .of_continuous_injective_isOpenMap (e.continuousOn.comp_continuous
continuous_subtype_val Subtype.prop) e.injOn.injective fun V hV ↦ ?_
rw [Set.restrict_eq, Set.image_comp]
exact e.isOpen_image_of_subset_source (e.open_source.isOpenMap_subtype_val V hV)
fun _ ⟨x, _, h⟩ ↦ h ▸ x.2 | theorem | OpenPartialHomeomorph.isOpenEmbedding_restrict | Topology.OpenPartialHomeomorph | Mathlib/Topology/OpenPartialHomeomorph/Basic.lean | [] | [
"Set.image_comp",
"Set.restrict_eq",
"Subtype.prop",
"continuous_subtype_val"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
to_isOpenEmbedding (h : e.source = Set.univ) : IsOpenEmbedding e | e.isOpenEmbedding_restrict.comp
((Homeomorph.setCongr h).trans <| Homeomorph.Set.univ X).symm.isOpenEmbedding | theorem | OpenPartialHomeomorph.to_isOpenEmbedding | Topology.OpenPartialHomeomorph | Mathlib/Topology/OpenPartialHomeomorph/Basic.lean | [] | [
"Homeomorph.Set.univ",
"Homeomorph.setCongr",
"Set.univ",
"trans"
] | An open partial homeomorphism whose source is all of `X` defines an open embedding of `X` into
`Y`. The converse is also true; see `IsOpenEmbedding.toOpenPartialHomeomorph`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
toOpenPartialHomeomorph [Nonempty X] : OpenPartialHomeomorph X Y | OpenPartialHomeomorph.ofContinuousOpen (h.isEmbedding.injective.injOn.toPartialEquiv f univ)
h.continuous.continuousOn h.isOpenMap isOpen_univ | def | Topology.IsOpenEmbedding.toOpenPartialHomeomorph | Topology.OpenPartialHomeomorph | Mathlib/Topology/OpenPartialHomeomorph/Basic.lean | [] | [
"OpenPartialHomeomorph",
"OpenPartialHomeomorph.ofContinuousOpen",
"isOpen_univ"
] | An open embedding of `X` into `Y`, with `X` nonempty, defines an open partial homeomorphism
whose source is all of `X`. The converse is also true; see
`OpenPartialHomeomorph.to_isOpenEmbedding`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
toOpenPartialHomeomorph_left_inv {x : X} : (h.toOpenPartialHomeomorph f).symm (f x) = x | by
rw [← congr_fun (h.toOpenPartialHomeomorph_apply f), OpenPartialHomeomorph.left_inv]
exact Set.mem_univ _ | lemma | Topology.IsOpenEmbedding.toOpenPartialHomeomorph_left_inv | Topology.OpenPartialHomeomorph | Mathlib/Topology/OpenPartialHomeomorph/Basic.lean | [] | [
"OpenPartialHomeomorph.left_inv",
"Set.mem_univ",
"symm"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
toOpenPartialHomeomorph_right_inv {x : Y} (hx : x ∈ Set.range f) :
f ((h.toOpenPartialHomeomorph f).symm x) = x | by
rw [← congr_fun (h.toOpenPartialHomeomorph_apply f), OpenPartialHomeomorph.right_inv]
rwa [toOpenPartialHomeomorph_target] | lemma | Topology.IsOpenEmbedding.toOpenPartialHomeomorph_right_inv | Topology.OpenPartialHomeomorph | Mathlib/Topology/OpenPartialHomeomorph/Basic.lean | [] | [
"OpenPartialHomeomorph.right_inv",
"Set.range",
"symm"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
openPartialHomeomorphSubtypeCoe : OpenPartialHomeomorph s X | IsOpenEmbedding.toOpenPartialHomeomorph _ s.2.isOpenEmbedding_subtypeVal | def | TopologicalSpace.Opens.openPartialHomeomorphSubtypeCoe | Topology.OpenPartialHomeomorph | Mathlib/Topology/OpenPartialHomeomorph/Basic.lean | [] | [
"OpenPartialHomeomorph"
] | The inclusion of an open subset `s` of a space `X` into `X` is an open partial homeomorphism
from the subtype `s` to `X`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
openPartialHomeomorphSubtypeCoe_coe :
(s.openPartialHomeomorphSubtypeCoe hs : s → X) = (↑) | rfl | theorem | TopologicalSpace.Opens.openPartialHomeomorphSubtypeCoe_coe | Topology.OpenPartialHomeomorph | Mathlib/Topology/OpenPartialHomeomorph/Basic.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
openPartialHomeomorphSubtypeCoe_source :
(s.openPartialHomeomorphSubtypeCoe hs).source = Set.univ | rfl | theorem | TopologicalSpace.Opens.openPartialHomeomorphSubtypeCoe_source | Topology.OpenPartialHomeomorph | Mathlib/Topology/OpenPartialHomeomorph/Basic.lean | [] | [
"Set.univ"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
openPartialHomeomorphSubtypeCoe_target :
(s.openPartialHomeomorphSubtypeCoe hs).target = s | by
simp only [openPartialHomeomorphSubtypeCoe, Subtype.range_coe_subtype, mfld_simps]
rfl | theorem | TopologicalSpace.Opens.openPartialHomeomorphSubtypeCoe_target | Topology.OpenPartialHomeomorph | Mathlib/Topology/OpenPartialHomeomorph/Basic.lean | [] | [
"Subtype.range_coe_subtype"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
trans' (h : e.target = e'.source) : OpenPartialHomeomorph X Z | where
toPartialEquiv := PartialEquiv.trans' e.toPartialEquiv e'.toPartialEquiv h
open_source := e.open_source
open_target := e'.open_target
continuousOn_toFun := e'.continuousOn.comp e.continuousOn <| h ▸ e.mapsTo
continuousOn_invFun := e.continuousOn_symm.comp e'.continuousOn_symm <| h.symm ▸ e'.mapsTo_symm | def | OpenPartialHomeomorph.trans' | Topology.OpenPartialHomeomorph | Mathlib/Topology/OpenPartialHomeomorph/Composition.lean | [] | [
"OpenPartialHomeomorph",
"PartialEquiv.trans'"
] | Composition of two open partial homeomorphisms when the target of the first and the source of
the second coincide. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
trans : OpenPartialHomeomorph X Z | OpenPartialHomeomorph.trans' (e.symm.restrOpen e'.source e'.open_source).symm
(e'.restrOpen e.target e.open_target) (by simp [inter_comm]) | def | OpenPartialHomeomorph.trans | Topology.OpenPartialHomeomorph | Mathlib/Topology/OpenPartialHomeomorph/Composition.lean | [] | [
"OpenPartialHomeomorph",
"OpenPartialHomeomorph.trans'",
"symm",
"trans"
] | Composing two open partial homeomorphisms, by restricting to the maximal domain where their
composition is well defined.
Within the `Manifold` namespace, there is the notation `e ≫ₕ f` for this. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
trans_toPartialEquiv :
(e.trans e').toPartialEquiv = e.toPartialEquiv.trans e'.toPartialEquiv | rfl | theorem | OpenPartialHomeomorph.trans_toPartialEquiv | Topology.OpenPartialHomeomorph | Mathlib/Topology/OpenPartialHomeomorph/Composition.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
coe_trans : (e.trans e' : X → Z) = e' ∘ e | rfl | theorem | OpenPartialHomeomorph.coe_trans | Topology.OpenPartialHomeomorph | Mathlib/Topology/OpenPartialHomeomorph/Composition.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
coe_trans_symm : ((e.trans e').symm : Z → X) = e.symm ∘ e'.symm | rfl | theorem | OpenPartialHomeomorph.coe_trans_symm | Topology.OpenPartialHomeomorph | Mathlib/Topology/OpenPartialHomeomorph/Composition.lean | [] | [
"symm"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
trans_apply {x : X} : (e.trans e') x = e' (e x) | rfl | theorem | OpenPartialHomeomorph.trans_apply | Topology.OpenPartialHomeomorph | Mathlib/Topology/OpenPartialHomeomorph/Composition.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
trans_source : (e.trans e').source = e.source ∩ e ⁻¹' e'.source | PartialEquiv.trans_source e.toPartialEquiv e'.toPartialEquiv | theorem | OpenPartialHomeomorph.trans_source | Topology.OpenPartialHomeomorph | Mathlib/Topology/OpenPartialHomeomorph/Composition.lean | [] | [
"PartialEquiv.trans_source"
] | This could be considered as a simp lemma, but there are many situations where it makes something
simple into something more complicated. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
trans_source' : (e.trans e').source = e.source ∩ e ⁻¹' (e.target ∩ e'.source) | PartialEquiv.trans_source' e.toPartialEquiv e'.toPartialEquiv | theorem | OpenPartialHomeomorph.trans_source' | Topology.OpenPartialHomeomorph | Mathlib/Topology/OpenPartialHomeomorph/Composition.lean | [] | [
"PartialEquiv.trans_source'"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
trans_source'' : (e.trans e').source = e.symm '' (e.target ∩ e'.source) | PartialEquiv.trans_source'' e.toPartialEquiv e'.toPartialEquiv | theorem | OpenPartialHomeomorph.trans_source'' | Topology.OpenPartialHomeomorph | Mathlib/Topology/OpenPartialHomeomorph/Composition.lean | [] | [
"PartialEquiv.trans_source''"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
image_trans_source : e '' (e.trans e').source = e.target ∩ e'.source | PartialEquiv.image_trans_source e.toPartialEquiv e'.toPartialEquiv | theorem | OpenPartialHomeomorph.image_trans_source | Topology.OpenPartialHomeomorph | Mathlib/Topology/OpenPartialHomeomorph/Composition.lean | [] | [
"PartialEquiv.image_trans_source"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
trans_assoc (e'' : OpenPartialHomeomorph Z Z') :
(e.trans e').trans e'' = e.trans (e'.trans e'') | toPartialEquiv_injective <| e.1.trans_assoc _ _ | theorem | OpenPartialHomeomorph.trans_assoc | Topology.OpenPartialHomeomorph | Mathlib/Topology/OpenPartialHomeomorph/Composition.lean | [] | [
"OpenPartialHomeomorph",
"trans"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
trans_refl : e.trans (OpenPartialHomeomorph.refl Y) = e | toPartialEquiv_injective e.1.trans_refl | theorem | OpenPartialHomeomorph.trans_refl | Topology.OpenPartialHomeomorph | Mathlib/Topology/OpenPartialHomeomorph/Composition.lean | [] | [
"OpenPartialHomeomorph.refl"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
refl_trans : (OpenPartialHomeomorph.refl X).trans e = e | toPartialEquiv_injective e.1.refl_trans | theorem | OpenPartialHomeomorph.refl_trans | Topology.OpenPartialHomeomorph | Mathlib/Topology/OpenPartialHomeomorph/Composition.lean | [] | [
"OpenPartialHomeomorph.refl",
"trans"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
trans_ofSet {s : Set Y} (hs : IsOpen s) : e.trans (ofSet s hs) = e.restr (e ⁻¹' s) | OpenPartialHomeomorph.ext _ _ (fun _ => rfl) (fun _ => rfl) <| by
rw [trans_source, restr_source, ofSet_source, ← preimage_interior, hs.interior_eq] | theorem | OpenPartialHomeomorph.trans_ofSet | Topology.OpenPartialHomeomorph | Mathlib/Topology/OpenPartialHomeomorph/Composition.lean | [] | [
"IsOpen",
"OpenPartialHomeomorph.ext",
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
trans_of_set' {s : Set Y} (hs : IsOpen s) :
e.trans (ofSet s hs) = e.restr (e.source ∩ e ⁻¹' s) | by rw [trans_ofSet, restr_source_inter] | theorem | OpenPartialHomeomorph.trans_of_set' | Topology.OpenPartialHomeomorph | Mathlib/Topology/OpenPartialHomeomorph/Composition.lean | [] | [
"IsOpen",
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
ofSet_trans {s : Set X} (hs : IsOpen s) : (ofSet s hs).trans e = e.restr s | OpenPartialHomeomorph.ext _ _ (fun _ => rfl) (fun _ => rfl) <|
by simp [hs.interior_eq, inter_comm] | theorem | OpenPartialHomeomorph.ofSet_trans | Topology.OpenPartialHomeomorph | Mathlib/Topology/OpenPartialHomeomorph/Composition.lean | [] | [
"IsOpen",
"OpenPartialHomeomorph.ext",
"Set",
"trans"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
ofSet_trans' {s : Set X} (hs : IsOpen s) :
(ofSet s hs).trans e = e.restr (e.source ∩ s) | by
rw [ofSet_trans, restr_source_inter] | theorem | OpenPartialHomeomorph.ofSet_trans' | Topology.OpenPartialHomeomorph | Mathlib/Topology/OpenPartialHomeomorph/Composition.lean | [] | [
"IsOpen",
"Set",
"trans"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
ofSet_trans_ofSet {s : Set X} (hs : IsOpen s) {s' : Set X} (hs' : IsOpen s') :
(ofSet s hs).trans (ofSet s' hs') = ofSet (s ∩ s') (IsOpen.inter hs hs') | by
rw [(ofSet s hs).trans_ofSet hs']
ext <;> simp [hs'.interior_eq] | theorem | OpenPartialHomeomorph.ofSet_trans_ofSet | Topology.OpenPartialHomeomorph | Mathlib/Topology/OpenPartialHomeomorph/Composition.lean | [] | [
"IsOpen",
"IsOpen.inter",
"Set",
"trans"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
restr_trans (s : Set X) : (e.restr s).trans e' = (e.trans e').restr s | toPartialEquiv_injective <|
PartialEquiv.restr_trans e.toPartialEquiv e'.toPartialEquiv (interior s) | theorem | OpenPartialHomeomorph.restr_trans | Topology.OpenPartialHomeomorph | Mathlib/Topology/OpenPartialHomeomorph/Composition.lean | [] | [
"PartialEquiv.restr_trans",
"Set",
"interior",
"trans"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
EqOnSource.trans' {e e' : OpenPartialHomeomorph X Y} {f f' : OpenPartialHomeomorph Y Z}
(he : e ≈ e') (hf : f ≈ f') : e.trans f ≈ e'.trans f' | PartialEquiv.EqOnSource.trans' he hf | theorem | OpenPartialHomeomorph.EqOnSource.trans' | Topology.OpenPartialHomeomorph | Mathlib/Topology/OpenPartialHomeomorph/Composition.lean | [] | [
"OpenPartialHomeomorph",
"PartialEquiv.EqOnSource.trans'"
] | Composition of open partial homeomorphisms respects equivalence. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
self_trans_symm : e.trans e.symm ≈ OpenPartialHomeomorph.ofSet e.source e.open_source | PartialEquiv.self_trans_symm _ | theorem | OpenPartialHomeomorph.self_trans_symm | Topology.OpenPartialHomeomorph | Mathlib/Topology/OpenPartialHomeomorph/Composition.lean | [] | [
"OpenPartialHomeomorph.ofSet",
"PartialEquiv.self_trans_symm"
] | Composition of an open partial homeomorphism and its inverse is equivalent to the restriction of
the identity to the source | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
symm_trans_self : e.symm.trans e ≈ OpenPartialHomeomorph.ofSet e.target e.open_target | e.symm.self_trans_symm | theorem | OpenPartialHomeomorph.symm_trans_self | Topology.OpenPartialHomeomorph | Mathlib/Topology/OpenPartialHomeomorph/Composition.lean | [] | [
"OpenPartialHomeomorph.ofSet"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
restr_symm_trans {e' : OpenPartialHomeomorph X Y}
(hs : IsOpen s) (hs' : IsOpen (e '' s)) (hs'' : s ⊆ e.source) :
(e.restr s).symm.trans e' ≈ (e.symm.trans e').restr (e '' s) | by
refine ⟨?_, ?_⟩
· simp only [trans_toPartialEquiv, symm_toPartialEquiv, restr_toPartialEquiv,
PartialEquiv.trans_source, PartialEquiv.symm_source, PartialEquiv.restr_target,
coe_toPartialEquiv_symm, PartialEquiv.restr_coe_symm, PartialEquiv.restr_source]
rw [interior_eq_iff_isOpen.mpr hs', interi... | theorem | OpenPartialHomeomorph.restr_symm_trans | Topology.OpenPartialHomeomorph | Mathlib/Topology/OpenPartialHomeomorph/Composition.lean | [] | [
"IsOpen",
"OpenPartialHomeomorph",
"PartialEquiv.restr_coe_symm",
"PartialEquiv.restr_source",
"PartialEquiv.restr_target",
"PartialEquiv.symm_source",
"PartialEquiv.trans_source"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
symm_trans_restr (e' : OpenPartialHomeomorph X Y) (hs : IsOpen s) :
e'.symm.trans (e.restr s) ≈ (e'.symm.trans e).restr (e'.target ∩ e'.symm ⁻¹' s) | by
have ht : IsOpen (e'.target ∩ e'.symm ⁻¹' s) := by
rw [← image_source_inter_eq']
exact isOpen_image_source_inter e' hs
refine ⟨?_, ?_⟩
· simp only [trans_toPartialEquiv, symm_toPartialEquiv, restr_toPartialEquiv,
PartialEquiv.trans_source, PartialEquiv.symm_source, coe_toPartialEquiv_symm,
... | theorem | OpenPartialHomeomorph.symm_trans_restr | Topology.OpenPartialHomeomorph | Mathlib/Topology/OpenPartialHomeomorph/Composition.lean | [] | [
"IsOpen",
"OpenPartialHomeomorph",
"PartialEquiv.restr_source",
"PartialEquiv.symm_source",
"PartialEquiv.trans_source",
"Set.eqOn_refl",
"interior_inter"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
trans_toOpenPartialHomeomorph : (e.trans e').toOpenPartialHomeomorph =
e.toOpenPartialHomeomorph.trans e'.toOpenPartialHomeomorph | OpenPartialHomeomorph.toPartialEquiv_injective <| Equiv.trans_toPartialEquiv _ _ | theorem | Homeomorph.trans_toOpenPartialHomeomorph | Topology.OpenPartialHomeomorph | Mathlib/Topology/OpenPartialHomeomorph/Composition.lean | [] | [
"Equiv.trans_toPartialEquiv",
"OpenPartialHomeomorph.toPartialEquiv_injective"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
transOpenPartialHomeomorph (e : X ≃ₜ Y) (f' : OpenPartialHomeomorph Y Z) :
OpenPartialHomeomorph X Z | where
toPartialEquiv := e.toEquiv.transPartialEquiv f'.toPartialEquiv
open_source := f'.open_source.preimage e.continuous
open_target := f'.open_target
continuousOn_toFun := f'.continuousOn.comp e.continuous.continuousOn fun _ => id
continuousOn_invFun := e.symm.continuous.comp_continuousOn f'.symm.continuous... | def | Homeomorph.transOpenPartialHomeomorph | Topology.OpenPartialHomeomorph | Mathlib/Topology/OpenPartialHomeomorph/Composition.lean | [] | [
"OpenPartialHomeomorph"
] | Precompose an open partial homeomorphism with a homeomorphism.
We modify the source and target to have better definitional behavior. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
transOpenPartialHomeomorph_eq_trans (e : X ≃ₜ Y) (f' : OpenPartialHomeomorph Y Z) :
e.transOpenPartialHomeomorph f' = e.toOpenPartialHomeomorph.trans f' | OpenPartialHomeomorph.toPartialEquiv_injective <| Equiv.transPartialEquiv_eq_trans _ _ | theorem | Homeomorph.transOpenPartialHomeomorph_eq_trans | Topology.OpenPartialHomeomorph | Mathlib/Topology/OpenPartialHomeomorph/Composition.lean | [] | [
"Equiv.transPartialEquiv_eq_trans",
"OpenPartialHomeomorph",
"OpenPartialHomeomorph.toPartialEquiv_injective"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
transOpenPartialHomeomorph_trans (e : X ≃ₜ Y) (f : OpenPartialHomeomorph Y Z)
(f' : OpenPartialHomeomorph Z Z') :
(e.transOpenPartialHomeomorph f).trans f' = e.transOpenPartialHomeomorph (f.trans f') | by
simp only [transOpenPartialHomeomorph_eq_trans, OpenPartialHomeomorph.trans_assoc] | theorem | Homeomorph.transOpenPartialHomeomorph_trans | Topology.OpenPartialHomeomorph | Mathlib/Topology/OpenPartialHomeomorph/Composition.lean | [] | [
"OpenPartialHomeomorph",
"OpenPartialHomeomorph.trans_assoc",
"trans"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
trans_transOpenPartialHomeomorph (e : X ≃ₜ Y) (e' : Y ≃ₜ Z)
(f'' : OpenPartialHomeomorph Z Z') : (e.trans e').transOpenPartialHomeomorph f'' =
e.transOpenPartialHomeomorph (e'.transOpenPartialHomeomorph f'') | by
simp only [transOpenPartialHomeomorph_eq_trans, OpenPartialHomeomorph.trans_assoc,
trans_toOpenPartialHomeomorph] | theorem | Homeomorph.trans_transOpenPartialHomeomorph | Topology.OpenPartialHomeomorph | Mathlib/Topology/OpenPartialHomeomorph/Composition.lean | [] | [
"OpenPartialHomeomorph",
"OpenPartialHomeomorph.trans_assoc"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
const (ha : IsOpen {a}) (hb : IsOpen {b}) : OpenPartialHomeomorph X Y | where
toPartialEquiv := PartialEquiv.single a b
open_source := ha
open_target := hb
continuousOn_toFun := by simp
continuousOn_invFun := by simp | def | OpenPartialHomeomorph.const | Topology.OpenPartialHomeomorph | Mathlib/Topology/OpenPartialHomeomorph/Constructions.lean | [] | [
"IsOpen",
"OpenPartialHomeomorph",
"PartialEquiv.single"
] | This is `PartialEquiv.single` as an open partial homeomorphism: a constant map,
whose source and target are necessarily singleton sets. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
const_apply (ha : IsOpen {a}) (hb : IsOpen {b}) (x : X) : (const ha hb) x = b | rfl | lemma | OpenPartialHomeomorph.const_apply | Topology.OpenPartialHomeomorph | Mathlib/Topology/OpenPartialHomeomorph/Constructions.lean | [] | [
"IsOpen"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
const_source (ha : IsOpen {a}) (hb : IsOpen {b}) : (const ha hb).source = {a} | rfl | lemma | OpenPartialHomeomorph.const_source | Topology.OpenPartialHomeomorph | Mathlib/Topology/OpenPartialHomeomorph/Constructions.lean | [] | [
"IsOpen"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
const_target (ha : IsOpen {a}) (hb : IsOpen {b}) : (const ha hb).target = {b} | rfl | lemma | OpenPartialHomeomorph.const_target | Topology.OpenPartialHomeomorph | Mathlib/Topology/OpenPartialHomeomorph/Constructions.lean | [] | [
"IsOpen"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
prod (eX : OpenPartialHomeomorph X X') (eY : OpenPartialHomeomorph Y Y') :
OpenPartialHomeomorph (X × Y) (X' × Y') | where
open_source := eX.open_source.prod eY.open_source
open_target := eX.open_target.prod eY.open_target
continuousOn_toFun := eX.continuousOn.prodMap eY.continuousOn
continuousOn_invFun := eX.continuousOn_symm.prodMap eY.continuousOn_symm
toPartialEquiv := eX.toPartialEquiv.prod eY.toPartialEquiv | def | OpenPartialHomeomorph.prod | Topology.OpenPartialHomeomorph | Mathlib/Topology/OpenPartialHomeomorph/Constructions.lean | [] | [
"OpenPartialHomeomorph"
] | The product of two open partial homeomorphisms, as an open partial homeomorphism on the product
space. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
prod_symm (eX : OpenPartialHomeomorph X X') (eY : OpenPartialHomeomorph Y Y') :
(eX.prod eY).symm = eX.symm.prod eY.symm | rfl | theorem | OpenPartialHomeomorph.prod_symm | Topology.OpenPartialHomeomorph | Mathlib/Topology/OpenPartialHomeomorph/Constructions.lean | [] | [
"OpenPartialHomeomorph",
"symm"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
refl_prod_refl : (OpenPartialHomeomorph.refl X).prod (OpenPartialHomeomorph.refl Y) =
OpenPartialHomeomorph.refl (X × Y) | OpenPartialHomeomorph.ext _ _ (fun _ => rfl) (fun _ => rfl) univ_prod_univ | theorem | OpenPartialHomeomorph.refl_prod_refl | Topology.OpenPartialHomeomorph | Mathlib/Topology/OpenPartialHomeomorph/Constructions.lean | [] | [
"OpenPartialHomeomorph.ext",
"OpenPartialHomeomorph.refl"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
prod_trans (e : OpenPartialHomeomorph X Y) (f : OpenPartialHomeomorph Y Z)
(e' : OpenPartialHomeomorph X' Y') (f' : OpenPartialHomeomorph Y' Z') :
(e.prod e').trans (f.prod f') = (e.trans f).prod (e'.trans f') | toPartialEquiv_injective <| e.1.prod_trans .. | theorem | OpenPartialHomeomorph.prod_trans | Topology.OpenPartialHomeomorph | Mathlib/Topology/OpenPartialHomeomorph/Constructions.lean | [] | [
"OpenPartialHomeomorph",
"trans"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
prod_eq_prod_of_nonempty {eX eX' : OpenPartialHomeomorph X X'}
{eY eY' : OpenPartialHomeomorph Y Y'} (h : (eX.prod eY).source.Nonempty) :
eX.prod eY = eX'.prod eY' ↔ eX = eX' ∧ eY = eY' | by
obtain ⟨⟨x, y⟩, -⟩ := id h
haveI : Nonempty X := ⟨x⟩
haveI : Nonempty X' := ⟨eX x⟩
haveI : Nonempty Y := ⟨y⟩
haveI : Nonempty Y' := ⟨eY y⟩
simp_rw [OpenPartialHomeomorph.ext_iff, prod_apply, prod_symm_apply, prod_source, Prod.ext_iff,
Set.prod_eq_prod_iff_of_nonempty h, forall_and, Prod.forall, foral... | theorem | OpenPartialHomeomorph.prod_eq_prod_of_nonempty | Topology.OpenPartialHomeomorph | Mathlib/Topology/OpenPartialHomeomorph/Constructions.lean | [] | [
"OpenPartialHomeomorph",
"Set.prod_eq_prod_iff_of_nonempty",
"prod_apply"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
prod_eq_prod_of_nonempty'
{eX eX' : OpenPartialHomeomorph X X'} {eY eY' : OpenPartialHomeomorph Y Y'}
(h : (eX'.prod eY').source.Nonempty) : eX.prod eY = eX'.prod eY' ↔ eX = eX' ∧ eY = eY' | by
rw [eq_comm, prod_eq_prod_of_nonempty h, eq_comm, @eq_comm _ eY'] | theorem | OpenPartialHomeomorph.prod_eq_prod_of_nonempty' | Topology.OpenPartialHomeomorph | Mathlib/Topology/OpenPartialHomeomorph/Constructions.lean | [] | [
"OpenPartialHomeomorph"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
prod_symm_trans_prod
(e f : OpenPartialHomeomorph X Y) (e' f' : OpenPartialHomeomorph X' Y') :
(e.prod e').symm.trans (f.prod f') = (e.symm.trans f).prod (e'.symm.trans f') | by
simp | theorem | OpenPartialHomeomorph.prod_symm_trans_prod | Topology.OpenPartialHomeomorph | Mathlib/Topology/OpenPartialHomeomorph/Constructions.lean | [] | [
"OpenPartialHomeomorph"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
pi : OpenPartialHomeomorph (∀ i, X i) (∀ i, Y i) | where
toPartialEquiv := PartialEquiv.pi fun i => (ei i).toPartialEquiv
open_source := isOpen_set_pi finite_univ fun i _ => (ei i).open_source
open_target := isOpen_set_pi finite_univ fun i _ => (ei i).open_target
continuousOn_toFun := continuousOn_pi.2 fun i =>
(ei i).continuousOn.comp (continuous_apply _).... | def | OpenPartialHomeomorph.pi | Topology.OpenPartialHomeomorph | Mathlib/Topology/OpenPartialHomeomorph/Constructions.lean | [] | [
"OpenPartialHomeomorph",
"PartialEquiv.pi",
"continuous_apply",
"isOpen_set_pi"
] | The product of a finite family of `OpenPartialHomeomorph`s. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
piecewise (e e' : OpenPartialHomeomorph X Y) (s : Set X) (t : Set Y) [∀ x, Decidable (x ∈ s)]
[∀ y, Decidable (y ∈ t)] (H : e.IsImage s t) (H' : e'.IsImage s t)
(Hs : e.source ∩ frontier s = e'.source ∩ frontier s)
(Heq : EqOn e e' (e.source ∩ frontier s)) : OpenPartialHomeomorph X Y | where
toPartialEquiv := e.toPartialEquiv.piecewise e'.toPartialEquiv s t H H'
open_source := e.open_source.ite e'.open_source Hs
open_target :=
e.open_target.ite e'.open_target <| H.frontier.inter_eq_of_inter_eq_of_eqOn H'.frontier Hs Heq
continuousOn_toFun := continuousOn_piecewise_ite e.continuousOn e'.co... | def | OpenPartialHomeomorph.piecewise | Topology.OpenPartialHomeomorph | Mathlib/Topology/OpenPartialHomeomorph/Constructions.lean | [] | [
"OpenPartialHomeomorph",
"Set",
"continuousOn_piecewise_ite",
"frontier"
] | Combine two `OpenPartialHomeomorph`s using `Set.piecewise`. The source of the new
`OpenPartialHomeomorph` is `s.ite e.source e'.source = e.source ∩ s ∪ e'.source \ s`, and similarly
for target. The function sends `e.source ∩ s` to `e.target ∩ t` using `e` and
`e'.source \ s` to `e'.target \ t` using `e'`, and similarl... | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
symm_piecewise (e e' : OpenPartialHomeomorph X Y) {s : Set X} {t : Set Y}
[∀ x, Decidable (x ∈ s)] [∀ y, Decidable (y ∈ t)] (H : e.IsImage s t) (H' : e'.IsImage s t)
(Hs : e.source ∩ frontier s = e'.source ∩ frontier s)
(Heq : EqOn e e' (e.source ∩ frontier s)) :
(e.piecewise e' s t H H' Hs Heq).symm =
... | rfl | theorem | OpenPartialHomeomorph.symm_piecewise | Topology.OpenPartialHomeomorph | Mathlib/Topology/OpenPartialHomeomorph/Constructions.lean | [] | [
"OpenPartialHomeomorph",
"Set",
"frontier",
"symm"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
disjointUnion (e e' : OpenPartialHomeomorph X Y) [∀ x, Decidable (x ∈ e.source)]
[∀ y, Decidable (y ∈ e.target)] (Hs : Disjoint e.source e'.source)
(Ht : Disjoint e.target e'.target) : OpenPartialHomeomorph X Y | (e.piecewise e' e.source e.target e.isImage_source_target
(e'.isImage_source_target_of_disjoint e Hs.symm Ht.symm)
(by rw [e.open_source.inter_frontier_eq, (Hs.symm.frontier_right e'.open_source).inter_eq])
(by
rw [e.open_source.inter_frontier_eq]
exact eqOn_empty _ _)).repla... | def | OpenPartialHomeomorph.disjointUnion | Topology.OpenPartialHomeomorph | Mathlib/Topology/OpenPartialHomeomorph/Constructions.lean | [] | [
"Disjoint",
"OpenPartialHomeomorph",
"PartialEquiv.disjointUnion_eq_piecewise",
"symm"
] | Combine two `OpenPartialHomeomorph`s with disjoint sources and disjoint targets. We reuse
`OpenPartialHomeomorph.piecewise` then override `toPartialEquiv` to `PartialEquiv.disjointUnion`.
This way we have better definitional equalities for `source` and `target`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
transHomeomorph (e : OpenPartialHomeomorph X Y) (f' : Y ≃ₜ Z) : OpenPartialHomeomorph X Z | where
toPartialEquiv := e.toPartialEquiv.transEquiv f'.toEquiv
open_source := e.open_source
open_target := e.open_target.preimage f'.symm.continuous
continuousOn_toFun := f'.continuous.comp_continuousOn e.continuousOn
continuousOn_invFun := e.symm.continuousOn.comp f'.symm.continuous.continuousOn fun _ => id | def | OpenPartialHomeomorph.transHomeomorph | Topology.OpenPartialHomeomorph | Mathlib/Topology/OpenPartialHomeomorph/Constructions.lean | [] | [
"OpenPartialHomeomorph"
] | Postcompose an open partial homeomorphism with a homeomorphism.
We modify the source and target to have better definitional behavior. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
transHomeomorph_eq_trans (e : OpenPartialHomeomorph X Y) (f' : Y ≃ₜ Z) :
e.transHomeomorph f' = e.trans f'.toOpenPartialHomeomorph | toPartialEquiv_injective <| PartialEquiv.transEquiv_eq_trans _ _ | theorem | OpenPartialHomeomorph.transHomeomorph_eq_trans | Topology.OpenPartialHomeomorph | Mathlib/Topology/OpenPartialHomeomorph/Constructions.lean | [] | [
"OpenPartialHomeomorph",
"PartialEquiv.transEquiv_eq_trans"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
transHomeomorph_transHomeomorph (e : OpenPartialHomeomorph X Y) (f' : Y ≃ₜ Z)
(f'' : Z ≃ₜ Z') :
(e.transHomeomorph f').transHomeomorph f'' = e.transHomeomorph (f'.trans f'') | by
simp only [transHomeomorph_eq_trans, trans_assoc, Homeomorph.trans_toOpenPartialHomeomorph] | theorem | OpenPartialHomeomorph.transHomeomorph_transHomeomorph | Topology.OpenPartialHomeomorph | Mathlib/Topology/OpenPartialHomeomorph/Constructions.lean | [] | [
"Homeomorph.trans_toOpenPartialHomeomorph",
"OpenPartialHomeomorph"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
trans_transHomeomorph (e : OpenPartialHomeomorph X Y) (e' : OpenPartialHomeomorph Y Z)
(f'' : Z ≃ₜ Z') :
(e.trans e').transHomeomorph f'' = e.trans (e'.transHomeomorph f'') | by
simp only [transHomeomorph_eq_trans, trans_assoc] | theorem | OpenPartialHomeomorph.trans_transHomeomorph | Topology.OpenPartialHomeomorph | Mathlib/Topology/OpenPartialHomeomorph/Constructions.lean | [] | [
"OpenPartialHomeomorph"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
subtypeRestr : OpenPartialHomeomorph s Y | (s.openPartialHomeomorphSubtypeCoe hs).trans e | def | OpenPartialHomeomorph.subtypeRestr | Topology.OpenPartialHomeomorph | Mathlib/Topology/OpenPartialHomeomorph/Constructions.lean | [] | [
"OpenPartialHomeomorph",
"trans"
] | The restriction of an open partial homeomorphism `e` to an open subset `s` of the domain type
produces an open partial homeomorphism whose domain is the subtype `s`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
subtypeRestr_def : e.subtypeRestr hs = (s.openPartialHomeomorphSubtypeCoe hs).trans e | rfl | theorem | OpenPartialHomeomorph.subtypeRestr_def | Topology.OpenPartialHomeomorph | Mathlib/Topology/OpenPartialHomeomorph/Constructions.lean | [] | [
"trans"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
subtypeRestr_coe :
((e.subtypeRestr hs : OpenPartialHomeomorph s Y) : s → Y) = Set.restrict ↑s (e : X → Y) | rfl | theorem | OpenPartialHomeomorph.subtypeRestr_coe | Topology.OpenPartialHomeomorph | Mathlib/Topology/OpenPartialHomeomorph/Constructions.lean | [] | [
"OpenPartialHomeomorph",
"Set.restrict"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
subtypeRestr_source : (e.subtypeRestr hs).source = (↑) ⁻¹' e.source | by
simp only [subtypeRestr_def, mfld_simps] | theorem | OpenPartialHomeomorph.subtypeRestr_source | Topology.OpenPartialHomeomorph | Mathlib/Topology/OpenPartialHomeomorph/Constructions.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
map_subtype_source {x : s} (hxe : (x : X) ∈ e.source) :
e x ∈ (e.subtypeRestr hs).target | by
refine ⟨e.map_source hxe, ?_⟩
rw [s.openPartialHomeomorphSubtypeCoe_target, mem_preimage, e.leftInvOn hxe]
exact x.prop | theorem | OpenPartialHomeomorph.map_subtype_source | Topology.OpenPartialHomeomorph | Mathlib/Topology/OpenPartialHomeomorph/Constructions.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
subtypeRestr_target_subset (hs : Nonempty s) : (e.subtypeRestr hs).target ⊆ e.target | by
rw [← e.image_source_eq_target, ← OpenPartialHomeomorph.image_source_eq_target,
e.subtypeRestr_source]
rintro z ⟨z₀, hz₀, rfl⟩
use z₀.val
simpa | lemma | OpenPartialHomeomorph.subtypeRestr_target_subset | Topology.OpenPartialHomeomorph | Mathlib/Topology/OpenPartialHomeomorph/Constructions.lean | [] | [
"OpenPartialHomeomorph.image_source_eq_target"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
subtypeRestr_symm_trans_subtypeRestr (f f' : OpenPartialHomeomorph X Y) :
(f.subtypeRestr hs).symm.trans (f'.subtypeRestr hs) ≈
(f.symm.trans f').restr (f.target ∩ f.symm ⁻¹' s) | by
simp only [subtypeRestr_def, trans_symm_eq_symm_trans_symm]
have openness₁ : IsOpen (f.target ∩ f.symm ⁻¹' s) := f.isOpen_inter_preimage_symm s.2
rw [← ofSet_trans _ openness₁, ← trans_assoc, ← trans_assoc]
refine EqOnSource.trans' ?_ (eqOnSource_refl _)
-- f' has been eliminated !!!
have set_identity : ... | theorem | OpenPartialHomeomorph.subtypeRestr_symm_trans_subtypeRestr | Topology.OpenPartialHomeomorph | Mathlib/Topology/OpenPartialHomeomorph/Constructions.lean | [] | [
"IsOpen",
"OpenPartialHomeomorph",
"Set"
] | This lemma characterizes the transition functions of an open subset in terms of the transition
functions of the original space. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
subtypeRestr_symm_apply {U : Opens X} (hU : Nonempty U)
{y : Y} (hy : y ∈ (e.subtypeRestr hU).target) :
(Subtype.val ∘ (e.subtypeRestr hU).symm) y = e.symm y | by
rw [e.eq_symm_apply _ hy.1]
· change restrict _ e _ = _
rw [← e.subtypeRestr_coe hU, (e.subtypeRestr hU).right_inv hy]
· have := OpenPartialHomeomorph.map_target _ hy
rwa [e.subtypeRestr_source] at this | theorem | OpenPartialHomeomorph.subtypeRestr_symm_apply | Topology.OpenPartialHomeomorph | Mathlib/Topology/OpenPartialHomeomorph/Constructions.lean | [] | [
"OpenPartialHomeomorph.map_target",
"symm"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
subtypeRestr_symm_eqOn {U : Opens X} (hU : Nonempty U) :
EqOn e.symm (Subtype.val ∘ (e.subtypeRestr hU).symm) (e.subtypeRestr hU).target | fun _y hy ↦ (e.subtypeRestr_symm_apply hU hy).symm | theorem | OpenPartialHomeomorph.subtypeRestr_symm_eqOn | Topology.OpenPartialHomeomorph | Mathlib/Topology/OpenPartialHomeomorph/Constructions.lean | [] | [
"symm"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
subtypeRestr_symm_eqOn_of_le {U V : Opens X} (hU : Nonempty U) (hV : Nonempty V)
(hUV : U ≤ V) : EqOn (e.subtypeRestr hV).symm (Set.inclusion hUV ∘ (e.subtypeRestr hU).symm)
(e.subtypeRestr hU).target | by
set i := Set.inclusion hUV
intro y hy
dsimp [OpenPartialHomeomorph.subtypeRestr_def] at hy ⊢
have hyV : e.symm y ∈ (V.openPartialHomeomorphSubtypeCoe hV).target := by
rw [Opens.openPartialHomeomorphSubtypeCoe_target] at hy ⊢
exact hUV hy.2
refine (V.openPartialHomeomorphSubtypeCoe hV).injOn ?_ triv... | theorem | OpenPartialHomeomorph.subtypeRestr_symm_eqOn_of_le | Topology.OpenPartialHomeomorph | Mathlib/Topology/OpenPartialHomeomorph/Constructions.lean | [] | [
"OpenPartialHomeomorph.subtypeRestr_def",
"Set.inclusion",
"symm"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
lift_openEmbedding (e : OpenPartialHomeomorph X Z) (hf : IsOpenEmbedding f) :
OpenPartialHomeomorph X' Z | where
toFun := extend f e (fun _ ↦ (Classical.arbitrary Z))
invFun := f ∘ e.invFun
source := f '' e.source
target := e.target
map_source' := by
rintro x ⟨x₀, hx₀, hxx₀⟩
rw [← hxx₀, hf.injective.extend_apply e]
exact e.map_source' hx₀
map_target' z hz := mem_image_of_mem f (e.map_target' hz)
le... | def | OpenPartialHomeomorph.lift_openEmbedding | Topology.OpenPartialHomeomorph | Mathlib/Topology/OpenPartialHomeomorph/Constructions.lean | [] | [
"Classical.arbitrary",
"ContinuousOn",
"ContinuousOn.congr",
"OpenPartialHomeomorph",
"by_cases",
"symm"
] | Extend an open partial homeomorphism `e : X → Z` to `X' → Z`, using an open embedding
`ι : X → X'`. On `ι(X)`, the extension is specified by `e`; its value elsewhere is arbitrary
(and uninteresting). | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
lift_openEmbedding_toFun (e : OpenPartialHomeomorph X Z) (hf : IsOpenEmbedding f) :
(e.lift_openEmbedding hf) = extend f e (fun _ ↦ (Classical.arbitrary Z)) | rfl | lemma | OpenPartialHomeomorph.lift_openEmbedding_toFun | Topology.OpenPartialHomeomorph | Mathlib/Topology/OpenPartialHomeomorph/Constructions.lean | [] | [
"Classical.arbitrary",
"OpenPartialHomeomorph"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
lift_openEmbedding_apply (e : OpenPartialHomeomorph X Z) (hf : IsOpenEmbedding f) {x : X} :
(lift_openEmbedding e hf) (f x) = e x | by
simp_rw [e.lift_openEmbedding_toFun]
apply hf.injective.extend_apply | lemma | OpenPartialHomeomorph.lift_openEmbedding_apply | Topology.OpenPartialHomeomorph | Mathlib/Topology/OpenPartialHomeomorph/Constructions.lean | [] | [
"OpenPartialHomeomorph"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
lift_openEmbedding_source (e : OpenPartialHomeomorph X Z) (hf : IsOpenEmbedding f) :
(e.lift_openEmbedding hf).source = f '' e.source | rfl | lemma | OpenPartialHomeomorph.lift_openEmbedding_source | Topology.OpenPartialHomeomorph | Mathlib/Topology/OpenPartialHomeomorph/Constructions.lean | [] | [
"OpenPartialHomeomorph"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
lift_openEmbedding_target (e : OpenPartialHomeomorph X Z) (hf : IsOpenEmbedding f) :
(e.lift_openEmbedding hf).target = e.target | rfl | lemma | OpenPartialHomeomorph.lift_openEmbedding_target | Topology.OpenPartialHomeomorph | Mathlib/Topology/OpenPartialHomeomorph/Constructions.lean | [] | [
"OpenPartialHomeomorph"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
lift_openEmbedding_symm (e : OpenPartialHomeomorph X Z) (hf : IsOpenEmbedding f) :
(e.lift_openEmbedding hf).symm = f ∘ e.symm | rfl | lemma | OpenPartialHomeomorph.lift_openEmbedding_symm | Topology.OpenPartialHomeomorph | Mathlib/Topology/OpenPartialHomeomorph/Constructions.lean | [] | [
"OpenPartialHomeomorph",
"symm"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
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