statement stringlengths 1 2.88k | proof stringlengths 0 13.9k | type stringclasses 10
values | symbolic_name stringlengths 1 131 | library stringclasses 417
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_root_.homeomorph.to_local_homeomorph (e : α ≃ₜ β) :
local_homeomorph α β | { open_source := is_open_univ,
open_target := is_open_univ,
continuous_to_fun := by { erw ← continuous_iff_continuous_on_univ, exact e.continuous_to_fun },
continuous_inv_fun := by { erw ← continuous_iff_continuous_on_univ, exact e.continuous_inv_fun },
..e.to_equiv.to_local_equiv } | def | homeomorph.to_local_homeomorph | topology | src/topology/local_homeomorph.lean | [
"logic.equiv.local_equiv",
"topology.sets.opens"
] | [
"continuous_iff_continuous_on_univ",
"is_open_univ",
"local_homeomorph"
] | A homeomorphism induces a local homeomorphism on the whole space | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
replace_equiv (e : local_homeomorph α β) (e' : local_equiv α β) (h : e.to_local_equiv = e') :
local_homeomorph α β | { to_local_equiv := e',
open_source := h ▸ e.open_source,
open_target := h ▸ e.open_target,
continuous_to_fun := h ▸ e.continuous_to_fun,
continuous_inv_fun := h ▸ e.continuous_inv_fun } | def | local_homeomorph.replace_equiv | topology | src/topology/local_homeomorph.lean | [
"logic.equiv.local_equiv",
"topology.sets.opens"
] | [
"local_equiv",
"local_homeomorph"
] | Replace `to_local_equiv` field to provide better definitional equalities. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
replace_equiv_eq_self (e : local_homeomorph α β) (e' : local_equiv α β)
(h : e.to_local_equiv = e') :
e.replace_equiv e' h = e | by { cases e, subst e', refl } | lemma | local_homeomorph.replace_equiv_eq_self | topology | src/topology/local_homeomorph.lean | [
"logic.equiv.local_equiv",
"topology.sets.opens"
] | [
"local_equiv",
"local_homeomorph"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
source_preimage_target : e.source ⊆ e ⁻¹' e.target | e.maps_to | lemma | local_homeomorph.source_preimage_target | topology | src/topology/local_homeomorph.lean | [
"logic.equiv.local_equiv",
"topology.sets.opens"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_of_local_equiv_eq {e e' : local_homeomorph α β}
(h : e.to_local_equiv = e'.to_local_equiv) : e = e' | by { cases e, cases e', cases h, refl } | lemma | local_homeomorph.eq_of_local_equiv_eq | topology | src/topology/local_homeomorph.lean | [
"logic.equiv.local_equiv",
"topology.sets.opens"
] | [
"local_homeomorph"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eventually_left_inverse (e : local_homeomorph α β) {x} (hx : x ∈ e.source) :
∀ᶠ y in 𝓝 x, e.symm (e y) = y | (e.open_source.eventually_mem hx).mono e.left_inv' | lemma | local_homeomorph.eventually_left_inverse | topology | src/topology/local_homeomorph.lean | [
"logic.equiv.local_equiv",
"topology.sets.opens"
] | [
"local_homeomorph"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eventually_left_inverse' (e : local_homeomorph α β) {x} (hx : x ∈ e.target) :
∀ᶠ y in 𝓝 (e.symm x), e.symm (e y) = y | e.eventually_left_inverse (e.map_target hx) | lemma | local_homeomorph.eventually_left_inverse' | topology | src/topology/local_homeomorph.lean | [
"logic.equiv.local_equiv",
"topology.sets.opens"
] | [
"local_homeomorph"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eventually_right_inverse (e : local_homeomorph α β) {x} (hx : x ∈ e.target) :
∀ᶠ y in 𝓝 x, e (e.symm y) = y | (e.open_target.eventually_mem hx).mono e.right_inv' | lemma | local_homeomorph.eventually_right_inverse | topology | src/topology/local_homeomorph.lean | [
"logic.equiv.local_equiv",
"topology.sets.opens"
] | [
"local_homeomorph"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eventually_right_inverse' (e : local_homeomorph α β) {x} (hx : x ∈ e.source) :
∀ᶠ y in 𝓝 (e x), e (e.symm y) = y | e.eventually_right_inverse (e.map_source hx) | lemma | local_homeomorph.eventually_right_inverse' | topology | src/topology/local_homeomorph.lean | [
"logic.equiv.local_equiv",
"topology.sets.opens"
] | [
"local_homeomorph"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eventually_ne_nhds_within (e : local_homeomorph α β) {x} (hx : x ∈ e.source) :
∀ᶠ x' in 𝓝[≠] x, e x' ≠ e x | eventually_nhds_within_iff.2 $ (e.eventually_left_inverse hx).mono $
λ x' hx', mt $ λ h, by rw [mem_singleton_iff, ← e.left_inv hx, ← h, hx'] | lemma | local_homeomorph.eventually_ne_nhds_within | topology | src/topology/local_homeomorph.lean | [
"logic.equiv.local_equiv",
"topology.sets.opens"
] | [
"local_homeomorph"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nhds_within_source_inter {x} (hx : x ∈ e.source) (s : set α) :
𝓝[e.source ∩ s] x = 𝓝[s] x | nhds_within_inter_of_mem (mem_nhds_within_of_mem_nhds $ is_open.mem_nhds e.open_source hx) | lemma | local_homeomorph.nhds_within_source_inter | topology | src/topology/local_homeomorph.lean | [
"logic.equiv.local_equiv",
"topology.sets.opens"
] | [
"is_open.mem_nhds",
"mem_nhds_within_of_mem_nhds",
"nhds_within_inter_of_mem"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nhds_within_target_inter {x} (hx : x ∈ e.target) (s : set β) :
𝓝[e.target ∩ s] x = 𝓝[s] x | e.symm.nhds_within_source_inter hx s | lemma | local_homeomorph.nhds_within_target_inter | topology | src/topology/local_homeomorph.lean | [
"logic.equiv.local_equiv",
"topology.sets.opens"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
image_eq_target_inter_inv_preimage {s : set α} (h : s ⊆ e.source) :
e '' s = e.target ∩ e.symm ⁻¹' s | e.to_local_equiv.image_eq_target_inter_inv_preimage h | lemma | local_homeomorph.image_eq_target_inter_inv_preimage | topology | src/topology/local_homeomorph.lean | [
"logic.equiv.local_equiv",
"topology.sets.opens"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
image_source_inter_eq' (s : set α) :
e '' (e.source ∩ s) = e.target ∩ e.symm ⁻¹' s | e.to_local_equiv.image_source_inter_eq' s | lemma | local_homeomorph.image_source_inter_eq' | topology | src/topology/local_homeomorph.lean | [
"logic.equiv.local_equiv",
"topology.sets.opens"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
image_source_inter_eq (s : set α) :
e '' (e.source ∩ s) = e.target ∩ e.symm ⁻¹' (e.source ∩ s) | e.to_local_equiv.image_source_inter_eq s | lemma | local_homeomorph.image_source_inter_eq | topology | src/topology/local_homeomorph.lean | [
"logic.equiv.local_equiv",
"topology.sets.opens"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
source_inter_preimage_inv_preimage (s : set α) :
e.source ∩ e ⁻¹' (e.symm ⁻¹' s) = e.source ∩ s | e.to_local_equiv.source_inter_preimage_inv_preimage s | lemma | local_homeomorph.source_inter_preimage_inv_preimage | topology | src/topology/local_homeomorph.lean | [
"logic.equiv.local_equiv",
"topology.sets.opens"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
source_inter_preimage_target_inter (s : set β) :
e.source ∩ (e ⁻¹' (e.target ∩ s)) = e.source ∩ (e ⁻¹' s) | e.to_local_equiv.source_inter_preimage_target_inter s | lemma | local_homeomorph.source_inter_preimage_target_inter | topology | src/topology/local_homeomorph.lean | [
"logic.equiv.local_equiv",
"topology.sets.opens"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
image_source_eq_target (e : local_homeomorph α β) : e '' e.source = e.target | e.to_local_equiv.image_source_eq_target | lemma | local_homeomorph.image_source_eq_target | topology | src/topology/local_homeomorph.lean | [
"logic.equiv.local_equiv",
"topology.sets.opens"
] | [
"local_homeomorph"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
symm_image_target_eq_source (e : local_homeomorph α β) : e.symm '' e.target = e.source | e.symm.image_source_eq_target | lemma | local_homeomorph.symm_image_target_eq_source | topology | src/topology/local_homeomorph.lean | [
"logic.equiv.local_equiv",
"topology.sets.opens"
] | [
"local_homeomorph"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ext (e' : local_homeomorph α β) (h : ∀x, e x = e' x)
(hinv : ∀x, e.symm x = e'.symm x) (hs : e.source = e'.source) : e = e' | eq_of_local_equiv_eq (local_equiv.ext h hinv hs) | lemma | local_homeomorph.ext | topology | src/topology/local_homeomorph.lean | [
"logic.equiv.local_equiv",
"topology.sets.opens"
] | [
"local_equiv.ext",
"local_homeomorph"
] | Two local homeomorphisms are equal when they have equal `to_fun`, `inv_fun` and `source`.
It is not sufficient to have equal `to_fun` and `source`, as this only determines `inv_fun` on
the target. This would only be true for a weaker notion of equality, arguably the right one,
called `eq_on_source`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ext_iff {e e' : local_homeomorph α β} : e = e' ↔ (∀ x, e x = e' x) ∧
(∀ x, e.symm x = e'.symm x) ∧ e.source = e'.source | ⟨by { rintro rfl, exact ⟨λ x, rfl, λ x, rfl, rfl⟩ }, λ h, e.ext e' h.1 h.2.1 h.2.2⟩ | lemma | local_homeomorph.ext_iff | topology | src/topology/local_homeomorph.lean | [
"logic.equiv.local_equiv",
"topology.sets.opens"
] | [
"local_homeomorph"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
symm_symm : e.symm.symm = e | eq_of_local_equiv_eq $ by simp | lemma | local_homeomorph.symm_symm | topology | src/topology/local_homeomorph.lean | [
"logic.equiv.local_equiv",
"topology.sets.opens"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_at {x : α} (h : x ∈ e.source) : continuous_at e x | (e.continuous_on x h).continuous_at (e.open_source.mem_nhds h) | lemma | local_homeomorph.continuous_at | topology | src/topology/local_homeomorph.lean | [
"logic.equiv.local_equiv",
"topology.sets.opens"
] | [
"continuous_at"
] | A local homeomorphism is continuous at any point of its source | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous_at_symm {x : β} (h : x ∈ e.target) : continuous_at e.symm x | e.symm.continuous_at h | lemma | local_homeomorph.continuous_at_symm | topology | src/topology/local_homeomorph.lean | [
"logic.equiv.local_equiv",
"topology.sets.opens"
] | [
"continuous_at"
] | A local homeomorphism inverse is continuous at any point of its target | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
tendsto_symm {x} (hx : x ∈ e.source) :
tendsto e.symm (𝓝 (e x)) (𝓝 x) | by simpa only [continuous_at, e.left_inv hx] using e.continuous_at_symm (e.map_source hx) | lemma | local_homeomorph.tendsto_symm | topology | src/topology/local_homeomorph.lean | [
"logic.equiv.local_equiv",
"topology.sets.opens"
] | [
"continuous_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_nhds_eq {x} (hx : x ∈ e.source) : map e (𝓝 x) = 𝓝 (e x) | le_antisymm (e.continuous_at hx) $
le_map_of_right_inverse (e.eventually_right_inverse' hx) (e.tendsto_symm hx) | lemma | local_homeomorph.map_nhds_eq | topology | src/topology/local_homeomorph.lean | [
"logic.equiv.local_equiv",
"topology.sets.opens"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
symm_map_nhds_eq {x} (hx : x ∈ e.source) :
map e.symm (𝓝 (e x)) = 𝓝 x | (e.symm.map_nhds_eq $ e.map_source hx).trans $ by rw e.left_inv hx | lemma | local_homeomorph.symm_map_nhds_eq | topology | src/topology/local_homeomorph.lean | [
"logic.equiv.local_equiv",
"topology.sets.opens"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
image_mem_nhds {x} (hx : x ∈ e.source) {s : set α} (hs : s ∈ 𝓝 x) :
e '' s ∈ 𝓝 (e x) | e.map_nhds_eq hx ▸ filter.image_mem_map hs | lemma | local_homeomorph.image_mem_nhds | topology | src/topology/local_homeomorph.lean | [
"logic.equiv.local_equiv",
"topology.sets.opens"
] | [
"filter.image_mem_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_nhds_within_eq (e : local_homeomorph α β) {x} (hx : x ∈ e.source) (s : set α) :
map e (𝓝[s] x) = 𝓝[e '' (e.source ∩ s)] (e x) | calc map e (𝓝[s] x) = map e (𝓝[e.source ∩ s] x) :
congr_arg (map e) (e.nhds_within_source_inter hx _).symm
... = 𝓝[e '' (e.source ∩ s)] (e x) :
(e.left_inv_on.mono $ inter_subset_left _ _).map_nhds_within_eq (e.left_inv hx)
(e.continuous_at_symm (e.map_source hx)).continuous_within_at
(e.continuous_at hx... | lemma | local_homeomorph.map_nhds_within_eq | topology | src/topology/local_homeomorph.lean | [
"logic.equiv.local_equiv",
"topology.sets.opens"
] | [
"continuous_within_at",
"local_homeomorph"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_nhds_within_preimage_eq (e : local_homeomorph α β) {x} (hx : x ∈ e.source) (s : set β) :
map e (𝓝[e ⁻¹' s] x) = 𝓝[s] (e x) | by rw [e.map_nhds_within_eq hx, e.image_source_inter_eq', e.target_inter_inv_preimage_preimage,
e.nhds_within_target_inter (e.map_source hx)] | lemma | local_homeomorph.map_nhds_within_preimage_eq | topology | src/topology/local_homeomorph.lean | [
"logic.equiv.local_equiv",
"topology.sets.opens"
] | [
"local_homeomorph"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eventually_nhds (e : local_homeomorph α β) {x : α} (p : β → Prop)
(hx : x ∈ e.source) : (∀ᶠ y in 𝓝 (e x), p y) ↔ ∀ᶠ x in 𝓝 x, p (e x) | iff.trans (by rw [e.map_nhds_eq hx]) eventually_map | lemma | local_homeomorph.eventually_nhds | topology | src/topology/local_homeomorph.lean | [
"logic.equiv.local_equiv",
"topology.sets.opens"
] | [
"local_homeomorph"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eventually_nhds' (e : local_homeomorph α β) {x : α} (p : α → Prop)
(hx : x ∈ e.source) : (∀ᶠ y in 𝓝 (e x), p (e.symm y)) ↔ ∀ᶠ x in 𝓝 x, p x | begin
rw [e.eventually_nhds _ hx],
refine eventually_congr ((e.eventually_left_inverse hx).mono $ λ y hy, _),
rw [hy]
end | lemma | local_homeomorph.eventually_nhds' | topology | src/topology/local_homeomorph.lean | [
"logic.equiv.local_equiv",
"topology.sets.opens"
] | [
"local_homeomorph"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eventually_nhds_within (e : local_homeomorph α β) {x : α} (p : β → Prop) {s : set α}
(hx : x ∈ e.source) : (∀ᶠ y in 𝓝[e.symm ⁻¹' s] (e x), p y) ↔ ∀ᶠ x in 𝓝[s] x, p (e x) | begin
refine iff.trans _ eventually_map,
rw [e.map_nhds_within_eq hx, e.image_source_inter_eq', e.nhds_within_target_inter (e.maps_to hx)]
end | lemma | local_homeomorph.eventually_nhds_within | topology | src/topology/local_homeomorph.lean | [
"logic.equiv.local_equiv",
"topology.sets.opens"
] | [
"local_homeomorph"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eventually_nhds_within' (e : local_homeomorph α β) {x : α} (p : α → Prop) {s : set α}
(hx : x ∈ e.source) : (∀ᶠ y in 𝓝[e.symm ⁻¹' s] (e x), p (e.symm y)) ↔ ∀ᶠ x in 𝓝[s] x, p x | begin
rw [e.eventually_nhds_within _ hx],
refine eventually_congr ((eventually_nhds_within_of_eventually_nhds $
e.eventually_left_inverse hx).mono $ λ y hy, _),
rw [hy]
end | lemma | local_homeomorph.eventually_nhds_within' | topology | src/topology/local_homeomorph.lean | [
"logic.equiv.local_equiv",
"topology.sets.opens"
] | [
"eventually_nhds_within_of_eventually_nhds",
"local_homeomorph"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
preimage_eventually_eq_target_inter_preimage_inter
{e : local_homeomorph α β} {s : set α} {t : set γ} {x : α}
{f : α → γ} (hf : continuous_within_at f s x) (hxe : x ∈ e.source) (ht : t ∈ 𝓝 (f x)) :
e.symm ⁻¹' s =ᶠ[𝓝 (e x)] (e.target ∩ e.symm ⁻¹' (s ∩ f ⁻¹' t) : set β) | begin
rw [eventually_eq_set, e.eventually_nhds _ hxe],
filter_upwards [(e.open_source.mem_nhds hxe),
mem_nhds_within_iff_eventually.mp (hf.preimage_mem_nhds_within ht)],
intros y hy hyu,
simp_rw [mem_inter_iff, mem_preimage, mem_inter_iff, e.maps_to hy, true_and, iff_self_and,
e.left_inv hy, iff_true_in... | lemma | local_homeomorph.preimage_eventually_eq_target_inter_preimage_inter | topology | src/topology/local_homeomorph.lean | [
"logic.equiv.local_equiv",
"topology.sets.opens"
] | [
"continuous_within_at",
"iff_self_and",
"local_homeomorph"
] | This lemma is useful in the manifold library in the case that `e` is a chart. It states that
locally around `e x` the set `e.symm ⁻¹' s` is the same as the set intersected with the target
of `e` and some other neighborhood of `f x` (which will be the source of a chart on `γ`). | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
preimage_open_of_open {s : set β} (hs : is_open s) : is_open (e.source ∩ e ⁻¹' s) | e.continuous_on.preimage_open_of_open e.open_source hs | lemma | local_homeomorph.preimage_open_of_open | topology | src/topology/local_homeomorph.lean | [
"logic.equiv.local_equiv",
"topology.sets.opens"
] | [
"is_open"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_local_equiv (h : e.is_image s t) : e.to_local_equiv.is_image s t | h | lemma | local_homeomorph.is_image.to_local_equiv | topology | src/topology/local_homeomorph.lean | [
"logic.equiv.local_equiv",
"topology.sets.opens"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
symm (h : e.is_image s t) : e.symm.is_image t s | h.to_local_equiv.symm | lemma | local_homeomorph.is_image.symm | topology | src/topology/local_homeomorph.lean | [
"logic.equiv.local_equiv",
"topology.sets.opens"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
symm_apply_mem_iff (h : e.is_image s t) (hy : y ∈ e.target) : (e.symm y ∈ s ↔ y ∈ t) | h.symm hy | lemma | local_homeomorph.is_image.symm_apply_mem_iff | topology | src/topology/local_homeomorph.lean | [
"logic.equiv.local_equiv",
"topology.sets.opens"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
maps_to (h : e.is_image s t) : maps_to e (e.source ∩ s) (e.target ∩ t) | h.to_local_equiv.maps_to | lemma | local_homeomorph.is_image.maps_to | topology | src/topology/local_homeomorph.lean | [
"logic.equiv.local_equiv",
"topology.sets.opens"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
image_eq (h : e.is_image s t) : e '' (e.source ∩ s) = e.target ∩ t | h.to_local_equiv.image_eq | lemma | local_homeomorph.is_image.image_eq | topology | src/topology/local_homeomorph.lean | [
"logic.equiv.local_equiv",
"topology.sets.opens"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
iff_preimage_eq : e.is_image s t ↔ e.source ∩ e ⁻¹' t = e.source ∩ s | local_equiv.is_image.iff_preimage_eq | lemma | local_homeomorph.is_image.iff_preimage_eq | topology | src/topology/local_homeomorph.lean | [
"logic.equiv.local_equiv",
"topology.sets.opens"
] | [
"local_equiv.is_image.iff_preimage_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
iff_symm_preimage_eq' :
e.is_image s t ↔ e.target ∩ e.symm ⁻¹' (e.source ∩ s) = e.target ∩ t | by rw [iff_symm_preimage_eq, ← image_source_inter_eq, ← image_source_inter_eq'] | lemma | local_homeomorph.is_image.iff_symm_preimage_eq' | topology | src/topology/local_homeomorph.lean | [
"logic.equiv.local_equiv",
"topology.sets.opens"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
iff_preimage_eq' : e.is_image s t ↔ e.source ∩ e ⁻¹' (e.target ∩ t) = e.source ∩ s | symm_iff.symm.trans iff_symm_preimage_eq' | lemma | local_homeomorph.is_image.iff_preimage_eq' | topology | src/topology/local_homeomorph.lean | [
"logic.equiv.local_equiv",
"topology.sets.opens"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_image_eq (h : e '' (e.source ∩ s) = e.target ∩ t) : e.is_image s t | local_equiv.is_image.of_image_eq h | lemma | local_homeomorph.is_image.of_image_eq | topology | src/topology/local_homeomorph.lean | [
"logic.equiv.local_equiv",
"topology.sets.opens"
] | [
"local_equiv.is_image.of_image_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_symm_image_eq (h : e.symm '' (e.target ∩ t) = e.source ∩ s) : e.is_image s t | local_equiv.is_image.of_symm_image_eq h | lemma | local_homeomorph.is_image.of_symm_image_eq | topology | src/topology/local_homeomorph.lean | [
"logic.equiv.local_equiv",
"topology.sets.opens"
] | [
"local_equiv.is_image.of_symm_image_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
left_inv_on_piecewise {e' : local_homeomorph α β} [∀ i, decidable (i ∈ s)]
[∀ i, decidable (i ∈ t)] (h : e.is_image s t) (h' : e'.is_image s t) :
left_inv_on (t.piecewise e.symm e'.symm) (s.piecewise e e') (s.ite e.source e'.source) | h.to_local_equiv.left_inv_on_piecewise h' | lemma | local_homeomorph.is_image.left_inv_on_piecewise | topology | src/topology/local_homeomorph.lean | [
"logic.equiv.local_equiv",
"topology.sets.opens"
] | [
"local_homeomorph"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inter_eq_of_inter_eq_of_eq_on {e' : local_homeomorph α β} (h : e.is_image s t)
(h' : e'.is_image s t) (hs : e.source ∩ s = e'.source ∩ s) (Heq : eq_on e e' (e.source ∩ s)) :
e.target ∩ t = e'.target ∩ t | h.to_local_equiv.inter_eq_of_inter_eq_of_eq_on h' hs Heq | lemma | local_homeomorph.is_image.inter_eq_of_inter_eq_of_eq_on | topology | src/topology/local_homeomorph.lean | [
"logic.equiv.local_equiv",
"topology.sets.opens"
] | [
"local_homeomorph"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
symm_eq_on_of_inter_eq_of_eq_on {e' : local_homeomorph α β} (h : e.is_image s t)
(hs : e.source ∩ s = e'.source ∩ s) (Heq : eq_on e e' (e.source ∩ s)) :
eq_on e.symm e'.symm (e.target ∩ t) | h.to_local_equiv.symm_eq_on_of_inter_eq_of_eq_on hs Heq | lemma | local_homeomorph.is_image.symm_eq_on_of_inter_eq_of_eq_on | topology | src/topology/local_homeomorph.lean | [
"logic.equiv.local_equiv",
"topology.sets.opens"
] | [
"local_homeomorph"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_nhds_within_eq (h : e.is_image s t) (hx : x ∈ e.source) :
map e (𝓝[s] x) = 𝓝[t] (e x) | by rw [e.map_nhds_within_eq hx, h.image_eq, e.nhds_within_target_inter (e.map_source hx)] | lemma | local_homeomorph.is_image.map_nhds_within_eq | topology | src/topology/local_homeomorph.lean | [
"logic.equiv.local_equiv",
"topology.sets.opens"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
closure (h : e.is_image s t) : e.is_image (closure s) (closure t) | λ x hx, by simp only [mem_closure_iff_nhds_within_ne_bot, ← h.map_nhds_within_eq hx, map_ne_bot_iff] | lemma | local_homeomorph.is_image.closure | topology | src/topology/local_homeomorph.lean | [
"logic.equiv.local_equiv",
"topology.sets.opens"
] | [
"closure",
"mem_closure_iff_nhds_within_ne_bot"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
interior (h : e.is_image s t) : e.is_image (interior s) (interior t) | by simpa only [closure_compl, compl_compl] using h.compl.closure.compl | lemma | local_homeomorph.is_image.interior | topology | src/topology/local_homeomorph.lean | [
"logic.equiv.local_equiv",
"topology.sets.opens"
] | [
"closure_compl",
"compl_compl",
"interior"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
frontier (h : e.is_image s t) :
e.is_image (frontier s) (frontier t) | h.closure.diff h.interior | lemma | local_homeomorph.is_image.frontier | topology | src/topology/local_homeomorph.lean | [
"logic.equiv.local_equiv",
"topology.sets.opens"
] | [
"frontier"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_open_iff (h : e.is_image s t) :
is_open (e.source ∩ s) ↔ is_open (e.target ∩ t) | ⟨λ hs, h.symm_preimage_eq' ▸ e.symm.preimage_open_of_open hs,
λ hs, h.preimage_eq' ▸ e.preimage_open_of_open hs⟩ | lemma | local_homeomorph.is_image.is_open_iff | topology | src/topology/local_homeomorph.lean | [
"logic.equiv.local_equiv",
"topology.sets.opens"
] | [
"is_open"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
restr (h : e.is_image s t) (hs : is_open (e.source ∩ s)) :
local_homeomorph α β | { to_local_equiv := h.to_local_equiv.restr,
open_source := hs,
open_target := h.is_open_iff.1 hs,
continuous_to_fun := e.continuous_on.mono (inter_subset_left _ _),
continuous_inv_fun := e.symm.continuous_on.mono (inter_subset_left _ _) } | def | local_homeomorph.is_image.restr | topology | src/topology/local_homeomorph.lean | [
"logic.equiv.local_equiv",
"topology.sets.opens"
] | [
"is_open",
"local_homeomorph"
] | Restrict a `local_homeomorph` to a pair of corresponding open sets. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_image_source_target : e.is_image e.source e.target | e.to_local_equiv.is_image_source_target | lemma | local_homeomorph.is_image_source_target | topology | src/topology/local_homeomorph.lean | [
"logic.equiv.local_equiv",
"topology.sets.opens"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_image_source_target_of_disjoint (e' : local_homeomorph α β)
(hs : disjoint e.source e'.source) (ht : disjoint e.target e'.target) :
e.is_image e'.source e'.target | e.to_local_equiv.is_image_source_target_of_disjoint e'.to_local_equiv hs ht | lemma | local_homeomorph.is_image_source_target_of_disjoint | topology | src/topology/local_homeomorph.lean | [
"logic.equiv.local_equiv",
"topology.sets.opens"
] | [
"disjoint",
"local_homeomorph"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
preimage_interior (s : set β) :
e.source ∩ e ⁻¹' (interior s) = e.source ∩ interior (e ⁻¹' s) | (is_image.of_preimage_eq rfl).interior.preimage_eq | lemma | local_homeomorph.preimage_interior | topology | src/topology/local_homeomorph.lean | [
"logic.equiv.local_equiv",
"topology.sets.opens"
] | [
"interior"
] | Preimage of interior or interior of preimage coincide for local homeomorphisms, when restricted
to the source. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
preimage_closure (s : set β) :
e.source ∩ e ⁻¹' (closure s) = e.source ∩ closure (e ⁻¹' s) | (is_image.of_preimage_eq rfl).closure.preimage_eq | lemma | local_homeomorph.preimage_closure | topology | src/topology/local_homeomorph.lean | [
"logic.equiv.local_equiv",
"topology.sets.opens"
] | [
"closure"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
preimage_frontier (s : set β) :
e.source ∩ e ⁻¹' (frontier s) = e.source ∩ frontier (e ⁻¹' s) | (is_image.of_preimage_eq rfl).frontier.preimage_eq | lemma | local_homeomorph.preimage_frontier | topology | src/topology/local_homeomorph.lean | [
"logic.equiv.local_equiv",
"topology.sets.opens"
] | [
"frontier"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
preimage_open_of_open_symm {s : set α} (hs : is_open s) :
is_open (e.target ∩ e.symm ⁻¹' s) | e.symm.continuous_on.preimage_open_of_open e.open_target hs | lemma | local_homeomorph.preimage_open_of_open_symm | topology | src/topology/local_homeomorph.lean | [
"logic.equiv.local_equiv",
"topology.sets.opens"
] | [
"is_open"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
image_open_of_open {s : set α} (hs : is_open s) (h : s ⊆ e.source) : is_open (e '' s) | begin
have : e '' s = e.target ∩ e.symm ⁻¹' s :=
e.to_local_equiv.image_eq_target_inter_inv_preimage h,
rw this,
exact e.continuous_on_symm.preimage_open_of_open e.open_target hs
end | lemma | local_homeomorph.image_open_of_open | topology | src/topology/local_homeomorph.lean | [
"logic.equiv.local_equiv",
"topology.sets.opens"
] | [
"is_open"
] | The image of an open set in the source is open. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
image_open_of_open' {s : set α} (hs : is_open s) : is_open (e '' (e.source ∩ s)) | image_open_of_open _ (is_open.inter e.open_source hs) (inter_subset_left _ _) | lemma | local_homeomorph.image_open_of_open' | topology | src/topology/local_homeomorph.lean | [
"logic.equiv.local_equiv",
"topology.sets.opens"
] | [
"is_open",
"is_open.inter"
] | The image of the restriction of an open set to the source is open. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_continuous_open_restrict (e : local_equiv α β) (hc : continuous_on e e.source)
(ho : is_open_map (e.source.restrict e)) (hs : is_open e.source) :
local_homeomorph α β | { to_local_equiv := e,
open_source := hs,
open_target := by simpa only [range_restrict, e.image_source_eq_target] using ho.is_open_range,
continuous_to_fun := hc,
continuous_inv_fun := e.image_source_eq_target ▸
ho.continuous_on_image_of_left_inv_on e.left_inv_on } | def | local_homeomorph.of_continuous_open_restrict | topology | src/topology/local_homeomorph.lean | [
"logic.equiv.local_equiv",
"topology.sets.opens"
] | [
"continuous_on",
"is_open",
"is_open_map",
"local_equiv",
"local_homeomorph"
] | A `local_equiv` with continuous open forward map and an open source is a `local_homeomorph`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_continuous_open (e : local_equiv α β) (hc : continuous_on e e.source)
(ho : is_open_map e) (hs : is_open e.source) :
local_homeomorph α β | of_continuous_open_restrict e hc (ho.restrict hs) hs | def | local_homeomorph.of_continuous_open | topology | src/topology/local_homeomorph.lean | [
"logic.equiv.local_equiv",
"topology.sets.opens"
] | [
"continuous_on",
"is_open",
"is_open_map",
"local_equiv",
"local_homeomorph"
] | A `local_equiv` with continuous open forward map and an open source is a `local_homeomorph`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
restr_open (s : set α) (hs : is_open s) :
local_homeomorph α β | (@is_image.of_symm_preimage_eq α β _ _ e s (e.symm ⁻¹' s) rfl).restr
(is_open.inter e.open_source hs) | def | local_homeomorph.restr_open | topology | src/topology/local_homeomorph.lean | [
"logic.equiv.local_equiv",
"topology.sets.opens"
] | [
"is_open",
"is_open.inter",
"local_homeomorph"
] | Restricting a local homeomorphism `e` to `e.source ∩ s` when `s` is open. This is sometimes hard
to use because of the openness assumption, but it has the advantage that when it can
be used then its local_equiv is defeq to local_equiv.restr | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
restr_open_to_local_equiv (s : set α) (hs : is_open s) :
(e.restr_open s hs).to_local_equiv = e.to_local_equiv.restr s | rfl | lemma | local_homeomorph.restr_open_to_local_equiv | topology | src/topology/local_homeomorph.lean | [
"logic.equiv.local_equiv",
"topology.sets.opens"
] | [
"is_open"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
restr_open_source (s : set α) (hs : is_open s) :
(e.restr_open s hs).source = e.source ∩ s | rfl | lemma | local_homeomorph.restr_open_source | topology | src/topology/local_homeomorph.lean | [
"logic.equiv.local_equiv",
"topology.sets.opens"
] | [
"is_open"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
restr (s : set α) : local_homeomorph α β | e.restr_open (interior s) is_open_interior | def | local_homeomorph.restr | topology | src/topology/local_homeomorph.lean | [
"logic.equiv.local_equiv",
"topology.sets.opens"
] | [
"interior",
"is_open_interior",
"local_homeomorph"
] | Restricting a local homeomorphism `e` to `e.source ∩ interior s`. We use the interior to make
sure that the restriction is well defined whatever the set s, since local homeomorphisms are by
definition defined on open sets. In applications where `s` is open, this coincides with the
restriction of local equivalences | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
restr_to_local_equiv (s : set α) :
(e.restr s).to_local_equiv = (e.to_local_equiv).restr (interior s) | rfl | lemma | local_homeomorph.restr_to_local_equiv | topology | src/topology/local_homeomorph.lean | [
"logic.equiv.local_equiv",
"topology.sets.opens"
] | [
"interior"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
restr_source' (s : set α) (hs : is_open s) : (e.restr s).source = e.source ∩ s | by rw [e.restr_source, hs.interior_eq] | lemma | local_homeomorph.restr_source' | topology | src/topology/local_homeomorph.lean | [
"logic.equiv.local_equiv",
"topology.sets.opens"
] | [
"is_open"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
restr_to_local_equiv' (s : set α) (hs : is_open s):
(e.restr s).to_local_equiv = e.to_local_equiv.restr s | by rw [e.restr_to_local_equiv, hs.interior_eq] | lemma | local_homeomorph.restr_to_local_equiv' | topology | src/topology/local_homeomorph.lean | [
"logic.equiv.local_equiv",
"topology.sets.opens"
] | [
"is_open"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
restr_eq_of_source_subset {e : local_homeomorph α β} {s : set α} (h : e.source ⊆ s) :
e.restr s = e | begin
apply eq_of_local_equiv_eq,
rw restr_to_local_equiv,
apply local_equiv.restr_eq_of_source_subset,
exact interior_maximal h e.open_source
end | lemma | local_homeomorph.restr_eq_of_source_subset | topology | src/topology/local_homeomorph.lean | [
"logic.equiv.local_equiv",
"topology.sets.opens"
] | [
"interior_maximal",
"local_equiv.restr_eq_of_source_subset",
"local_homeomorph"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
restr_univ {e : local_homeomorph α β} : e.restr univ = e | restr_eq_of_source_subset (subset_univ _) | lemma | local_homeomorph.restr_univ | topology | src/topology/local_homeomorph.lean | [
"logic.equiv.local_equiv",
"topology.sets.opens"
] | [
"local_homeomorph"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
restr_source_inter (s : set α) : e.restr (e.source ∩ s) = e.restr s | begin
refine local_homeomorph.ext _ _ (λx, rfl) (λx, rfl) _,
simp [e.open_source.interior_eq, ← inter_assoc]
end | lemma | local_homeomorph.restr_source_inter | topology | src/topology/local_homeomorph.lean | [
"logic.equiv.local_equiv",
"topology.sets.opens"
] | [
"local_homeomorph.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
refl (α : Type*) [topological_space α] : local_homeomorph α α | (homeomorph.refl α).to_local_homeomorph | def | local_homeomorph.refl | topology | src/topology/local_homeomorph.lean | [
"logic.equiv.local_equiv",
"topology.sets.opens"
] | [
"homeomorph.refl",
"local_homeomorph",
"topological_space"
] | The identity on the whole space as a local homeomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
refl_local_equiv :
(local_homeomorph.refl α).to_local_equiv = local_equiv.refl α | rfl | lemma | local_homeomorph.refl_local_equiv | topology | src/topology/local_homeomorph.lean | [
"logic.equiv.local_equiv",
"topology.sets.opens"
] | [
"local_equiv.refl",
"local_homeomorph.refl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
refl_symm : (local_homeomorph.refl α).symm = local_homeomorph.refl α | rfl | lemma | local_homeomorph.refl_symm | topology | src/topology/local_homeomorph.lean | [
"logic.equiv.local_equiv",
"topology.sets.opens"
] | [
"local_homeomorph.refl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_set (s : set α) (hs : is_open s) : local_homeomorph α α | { open_source := hs,
open_target := hs,
continuous_to_fun := continuous_id.continuous_on,
continuous_inv_fun := continuous_id.continuous_on,
..local_equiv.of_set s } | def | local_homeomorph.of_set | topology | src/topology/local_homeomorph.lean | [
"logic.equiv.local_equiv",
"topology.sets.opens"
] | [
"is_open",
"local_equiv.of_set",
"local_homeomorph"
] | The identity local equiv on a set `s` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_set_to_local_equiv :
(of_set s hs).to_local_equiv = local_equiv.of_set s | rfl | lemma | local_homeomorph.of_set_to_local_equiv | topology | src/topology/local_homeomorph.lean | [
"logic.equiv.local_equiv",
"topology.sets.opens"
] | [
"local_equiv.of_set"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_set_symm : (of_set s hs).symm = of_set s hs | rfl | lemma | local_homeomorph.of_set_symm | topology | src/topology/local_homeomorph.lean | [
"logic.equiv.local_equiv",
"topology.sets.opens"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_set_univ_eq_refl :
of_set univ is_open_univ = local_homeomorph.refl α | by ext; simp | lemma | local_homeomorph.of_set_univ_eq_refl | topology | src/topology/local_homeomorph.lean | [
"logic.equiv.local_equiv",
"topology.sets.opens"
] | [
"is_open_univ",
"local_homeomorph.refl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
trans' (h : e.target = e'.source) : local_homeomorph α γ | { open_source := e.open_source,
open_target := e'.open_target,
continuous_to_fun := begin
apply e'.continuous_to_fun.comp e.continuous_to_fun,
rw ← h,
exact e.to_local_equiv.source_subset_preimage_target
end,
continuous_inv_fun := begin
apply e.continuous_inv_fun.comp e'.continuous_i... | def | local_homeomorph.trans' | topology | src/topology/local_homeomorph.lean | [
"logic.equiv.local_equiv",
"topology.sets.opens"
] | [
"local_equiv.trans'",
"local_homeomorph"
] | Composition of two local homeomorphisms when the target of the first and the source of
the second coincide. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
trans : local_homeomorph α γ | local_homeomorph.trans' (e.symm.restr_open e'.source e'.open_source).symm
(e'.restr_open e.target e.open_target) (by simp [inter_comm]) | def | local_homeomorph.trans | topology | src/topology/local_homeomorph.lean | [
"logic.equiv.local_equiv",
"topology.sets.opens"
] | [
"local_homeomorph",
"local_homeomorph.trans'"
] | Composing two local homeomorphisms, by restricting to the maximal domain where their
composition is well defined. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
trans_to_local_equiv :
(e.trans e').to_local_equiv = e.to_local_equiv.trans e'.to_local_equiv | rfl | lemma | local_homeomorph.trans_to_local_equiv | topology | src/topology/local_homeomorph.lean | [
"logic.equiv.local_equiv",
"topology.sets.opens"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
trans_source : (e.trans e').source = e.source ∩ e ⁻¹' e'.source | local_equiv.trans_source e.to_local_equiv e'.to_local_equiv | lemma | local_homeomorph.trans_source | topology | src/topology/local_homeomorph.lean | [
"logic.equiv.local_equiv",
"topology.sets.opens"
] | [
"local_equiv.trans_source"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
trans_source' : (e.trans e').source = e.source ∩ e ⁻¹' (e.target ∩ e'.source) | local_equiv.trans_source' e.to_local_equiv e'.to_local_equiv | lemma | local_homeomorph.trans_source' | topology | src/topology/local_homeomorph.lean | [
"logic.equiv.local_equiv",
"topology.sets.opens"
] | [
"local_equiv.trans_source'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
trans_source'' : (e.trans e').source = e.symm '' (e.target ∩ e'.source) | local_equiv.trans_source'' e.to_local_equiv e'.to_local_equiv | lemma | local_homeomorph.trans_source'' | topology | src/topology/local_homeomorph.lean | [
"logic.equiv.local_equiv",
"topology.sets.opens"
] | [
"local_equiv.trans_source''"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
image_trans_source : e '' (e.trans e').source = e.target ∩ e'.source | local_equiv.image_trans_source e.to_local_equiv e'.to_local_equiv | lemma | local_homeomorph.image_trans_source | topology | src/topology/local_homeomorph.lean | [
"logic.equiv.local_equiv",
"topology.sets.opens"
] | [
"local_equiv.image_trans_source"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
trans_target : (e.trans e').target = e'.target ∩ e'.symm ⁻¹' e.target | rfl | lemma | local_homeomorph.trans_target | topology | src/topology/local_homeomorph.lean | [
"logic.equiv.local_equiv",
"topology.sets.opens"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
trans_assoc (e'' : local_homeomorph γ δ) :
(e.trans e').trans e'' = e.trans (e'.trans e'') | eq_of_local_equiv_eq $ local_equiv.trans_assoc e.to_local_equiv e'.to_local_equiv e''.to_local_equiv | lemma | local_homeomorph.trans_assoc | topology | src/topology/local_homeomorph.lean | [
"logic.equiv.local_equiv",
"topology.sets.opens"
] | [
"local_equiv.trans_assoc",
"local_homeomorph"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
trans_refl : e.trans (local_homeomorph.refl β) = e | eq_of_local_equiv_eq $ local_equiv.trans_refl e.to_local_equiv | lemma | local_homeomorph.trans_refl | topology | src/topology/local_homeomorph.lean | [
"logic.equiv.local_equiv",
"topology.sets.opens"
] | [
"local_equiv.trans_refl",
"local_homeomorph.refl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
refl_trans : (local_homeomorph.refl α).trans e = e | eq_of_local_equiv_eq $ local_equiv.refl_trans e.to_local_equiv | lemma | local_homeomorph.refl_trans | topology | src/topology/local_homeomorph.lean | [
"logic.equiv.local_equiv",
"topology.sets.opens"
] | [
"local_equiv.refl_trans",
"local_homeomorph.refl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
trans_of_set {s : set β} (hs : is_open s) :
e.trans (of_set s hs) = e.restr (e ⁻¹' s) | local_homeomorph.ext _ _ (λx, rfl) (λx, rfl) $
by simp [local_equiv.trans_source, (e.preimage_interior _).symm, hs.interior_eq] | lemma | local_homeomorph.trans_of_set | topology | src/topology/local_homeomorph.lean | [
"logic.equiv.local_equiv",
"topology.sets.opens"
] | [
"is_open",
"local_equiv.trans_source",
"local_homeomorph.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
trans_of_set' {s : set β} (hs : is_open s) :
e.trans (of_set s hs) = e.restr (e.source ∩ e ⁻¹' s) | by rw [trans_of_set, restr_source_inter] | lemma | local_homeomorph.trans_of_set' | topology | src/topology/local_homeomorph.lean | [
"logic.equiv.local_equiv",
"topology.sets.opens"
] | [
"is_open"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_set_trans {s : set α} (hs : is_open s) :
(of_set s hs).trans e = e.restr s | local_homeomorph.ext _ _ (λx, rfl) (λx, rfl) $
by simp [local_equiv.trans_source, hs.interior_eq, inter_comm] | lemma | local_homeomorph.of_set_trans | topology | src/topology/local_homeomorph.lean | [
"logic.equiv.local_equiv",
"topology.sets.opens"
] | [
"is_open",
"local_equiv.trans_source",
"local_homeomorph.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_set_trans' {s : set α} (hs : is_open s) :
(of_set s hs).trans e = e.restr (e.source ∩ s) | by rw [of_set_trans, restr_source_inter] | lemma | local_homeomorph.of_set_trans' | topology | src/topology/local_homeomorph.lean | [
"logic.equiv.local_equiv",
"topology.sets.opens"
] | [
"is_open"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_set_trans_of_set
{s : set α} (hs : is_open s) {s' : set α} (hs' : is_open s') :
(of_set s hs).trans (of_set s' hs') = of_set (s ∩ s') (is_open.inter hs hs') | begin
rw (of_set s hs).trans_of_set hs',
ext; simp [hs'.interior_eq]
end | lemma | local_homeomorph.of_set_trans_of_set | topology | src/topology/local_homeomorph.lean | [
"logic.equiv.local_equiv",
"topology.sets.opens"
] | [
"is_open",
"is_open.inter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
restr_trans (s : set α) :
(e.restr s).trans e' = (e.trans e').restr s | eq_of_local_equiv_eq $ local_equiv.restr_trans e.to_local_equiv e'.to_local_equiv (interior s) | lemma | local_homeomorph.restr_trans | topology | src/topology/local_homeomorph.lean | [
"logic.equiv.local_equiv",
"topology.sets.opens"
] | [
"interior",
"local_equiv.restr_trans"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
trans_homeomorph (e' : β ≃ₜ γ) : local_homeomorph α γ | { to_local_equiv := e.to_local_equiv.trans_equiv e'.to_equiv,
open_source := e.open_source,
open_target := e.open_target.preimage e'.symm.continuous,
continuous_to_fun := e'.continuous.comp_continuous_on e.continuous_on,
continuous_inv_fun := e.symm.continuous_on.comp e'.symm.continuous.continuous_on (λ x h, h)... | def | local_homeomorph.trans_homeomorph | topology | src/topology/local_homeomorph.lean | [
"logic.equiv.local_equiv",
"topology.sets.opens"
] | [
"local_homeomorph"
] | Postcompose a local homeomorphism with an homeomorphism.
We modify the source and target to have better definitional behavior. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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