statement stringlengths 1 2.88k | proof stringlengths 0 13.9k | type stringclasses 10
values | symbolic_name stringlengths 1 131 | library stringclasses 417
values | filename stringlengths 17 80 | imports listlengths 0 16 | deps listlengths 0 64 | docstring stringlengths 0 10.2k | source_url stringclasses 1
value | commit stringclasses 1
value |
|---|---|---|---|---|---|---|---|---|---|---|
continuous_on.zpow {f : α → G} {s : set α} (hf : continuous_on f s) (z : ℤ) :
continuous_on (λ x, f x ^ z) s | λ x hx, (hf x hx).zpow z | lemma | continuous_on.zpow | topology.algebra.group | src/topology/algebra/group/basic.lean | [
"group_theory.group_action.conj_act",
"group_theory.group_action.quotient",
"group_theory.quotient_group",
"topology.algebra.monoid",
"topology.algebra.constructions"
] | [
"continuous_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_inv_nhds_within_Ioi {a : H} :
tendsto has_inv.inv (𝓝[>] a) (𝓝[<] (a⁻¹)) | (continuous_inv.tendsto a).inf $ by simp [tendsto_principal_principal] | lemma | tendsto_inv_nhds_within_Ioi | topology.algebra.group | src/topology/algebra/group/basic.lean | [
"group_theory.group_action.conj_act",
"group_theory.group_action.quotient",
"group_theory.quotient_group",
"topology.algebra.monoid",
"topology.algebra.constructions"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_inv_nhds_within_Iio {a : H} :
tendsto has_inv.inv (𝓝[<] a) (𝓝[>] (a⁻¹)) | (continuous_inv.tendsto a).inf $ by simp [tendsto_principal_principal] | lemma | tendsto_inv_nhds_within_Iio | topology.algebra.group | src/topology/algebra/group/basic.lean | [
"group_theory.group_action.conj_act",
"group_theory.group_action.quotient",
"group_theory.quotient_group",
"topology.algebra.monoid",
"topology.algebra.constructions"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_inv_nhds_within_Ioi_inv {a : H} :
tendsto has_inv.inv (𝓝[>] (a⁻¹)) (𝓝[<] a) | by simpa only [inv_inv] using @tendsto_inv_nhds_within_Ioi _ _ _ _ (a⁻¹) | lemma | tendsto_inv_nhds_within_Ioi_inv | topology.algebra.group | src/topology/algebra/group/basic.lean | [
"group_theory.group_action.conj_act",
"group_theory.group_action.quotient",
"group_theory.quotient_group",
"topology.algebra.monoid",
"topology.algebra.constructions"
] | [
"inv_inv",
"tendsto_inv_nhds_within_Ioi"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_inv_nhds_within_Iio_inv {a : H} :
tendsto has_inv.inv (𝓝[<] (a⁻¹)) (𝓝[>] a) | by simpa only [inv_inv] using @tendsto_inv_nhds_within_Iio _ _ _ _ (a⁻¹) | lemma | tendsto_inv_nhds_within_Iio_inv | topology.algebra.group | src/topology/algebra/group/basic.lean | [
"group_theory.group_action.conj_act",
"group_theory.group_action.quotient",
"group_theory.quotient_group",
"topology.algebra.monoid",
"topology.algebra.constructions"
] | [
"inv_inv",
"tendsto_inv_nhds_within_Iio"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_inv_nhds_within_Ici {a : H} :
tendsto has_inv.inv (𝓝[≥] a) (𝓝[≤] (a⁻¹)) | (continuous_inv.tendsto a).inf $ by simp [tendsto_principal_principal] | lemma | tendsto_inv_nhds_within_Ici | topology.algebra.group | src/topology/algebra/group/basic.lean | [
"group_theory.group_action.conj_act",
"group_theory.group_action.quotient",
"group_theory.quotient_group",
"topology.algebra.monoid",
"topology.algebra.constructions"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_inv_nhds_within_Iic {a : H} :
tendsto has_inv.inv (𝓝[≤] a) (𝓝[≥] (a⁻¹)) | (continuous_inv.tendsto a).inf $ by simp [tendsto_principal_principal] | lemma | tendsto_inv_nhds_within_Iic | topology.algebra.group | src/topology/algebra/group/basic.lean | [
"group_theory.group_action.conj_act",
"group_theory.group_action.quotient",
"group_theory.quotient_group",
"topology.algebra.monoid",
"topology.algebra.constructions"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_inv_nhds_within_Ici_inv {a : H} :
tendsto has_inv.inv (𝓝[≥] (a⁻¹)) (𝓝[≤] a) | by simpa only [inv_inv] using @tendsto_inv_nhds_within_Ici _ _ _ _ (a⁻¹) | lemma | tendsto_inv_nhds_within_Ici_inv | topology.algebra.group | src/topology/algebra/group/basic.lean | [
"group_theory.group_action.conj_act",
"group_theory.group_action.quotient",
"group_theory.quotient_group",
"topology.algebra.monoid",
"topology.algebra.constructions"
] | [
"inv_inv",
"tendsto_inv_nhds_within_Ici"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_inv_nhds_within_Iic_inv {a : H} :
tendsto has_inv.inv (𝓝[≤] (a⁻¹)) (𝓝[≥] a) | by simpa only [inv_inv] using @tendsto_inv_nhds_within_Iic _ _ _ _ (a⁻¹) | lemma | tendsto_inv_nhds_within_Iic_inv | topology.algebra.group | src/topology/algebra/group/basic.lean | [
"group_theory.group_action.conj_act",
"group_theory.group_action.quotient",
"group_theory.quotient_group",
"topology.algebra.monoid",
"topology.algebra.constructions"
] | [
"inv_inv",
"tendsto_inv_nhds_within_Iic"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pi.topological_group {C : β → Type*} [∀ b, topological_space (C b)]
[∀ b, group (C b)] [∀ b, topological_group (C b)] : topological_group (Π b, C b) | { continuous_inv := continuous_pi (λ i, (continuous_apply i).inv) } | instance | pi.topological_group | topology.algebra.group | src/topology/algebra/group/basic.lean | [
"group_theory.group_action.conj_act",
"group_theory.group_action.quotient",
"group_theory.quotient_group",
"topology.algebra.monoid",
"topology.algebra.constructions"
] | [
"continuous_apply",
"continuous_pi",
"group",
"topological_group",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nhds_one_symm : comap has_inv.inv (𝓝 (1 : G)) = 𝓝 (1 : G) | ((homeomorph.inv G).comap_nhds_eq _).trans (congr_arg nhds inv_one) | lemma | nhds_one_symm | topology.algebra.group | src/topology/algebra/group/basic.lean | [
"group_theory.group_action.conj_act",
"group_theory.group_action.quotient",
"group_theory.quotient_group",
"topology.algebra.monoid",
"topology.algebra.constructions"
] | [
"homeomorph.inv",
"inv_one",
"nhds"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nhds_one_symm' : map has_inv.inv (𝓝 (1 : G)) = 𝓝 (1 : G) | ((homeomorph.inv G).map_nhds_eq _).trans (congr_arg nhds inv_one) | lemma | nhds_one_symm' | topology.algebra.group | src/topology/algebra/group/basic.lean | [
"group_theory.group_action.conj_act",
"group_theory.group_action.quotient",
"group_theory.quotient_group",
"topology.algebra.monoid",
"topology.algebra.constructions"
] | [
"homeomorph.inv",
"inv_one",
"nhds"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_mem_nhds_one {S : set G} (hS : S ∈ (𝓝 1 : filter G)) : S⁻¹ ∈ (𝓝 (1 : G)) | by rwa [← nhds_one_symm'] at hS | lemma | inv_mem_nhds_one | topology.algebra.group | src/topology/algebra/group/basic.lean | [
"group_theory.group_action.conj_act",
"group_theory.group_action.quotient",
"group_theory.quotient_group",
"topology.algebra.monoid",
"topology.algebra.constructions"
] | [
"filter",
"nhds_one_symm'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
homeomorph.shear_mul_right : G × G ≃ₜ G × G | { continuous_to_fun := continuous_fst.prod_mk continuous_mul,
continuous_inv_fun := continuous_fst.prod_mk $ continuous_fst.inv.mul continuous_snd,
.. equiv.prod_shear (equiv.refl _) equiv.mul_left } | def | homeomorph.shear_mul_right | topology.algebra.group | src/topology/algebra/group/basic.lean | [
"group_theory.group_action.conj_act",
"group_theory.group_action.quotient",
"group_theory.quotient_group",
"topology.algebra.monoid",
"topology.algebra.constructions"
] | [
"continuous_mul",
"continuous_snd",
"equiv.mul_left",
"equiv.prod_shear",
"equiv.refl"
] | The map `(x, y) ↦ (x, xy)` as a homeomorphism. This is a shear mapping. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
homeomorph.shear_mul_right_coe :
⇑(homeomorph.shear_mul_right G) = λ z : G × G, (z.1, z.1 * z.2) | rfl | lemma | homeomorph.shear_mul_right_coe | topology.algebra.group | src/topology/algebra/group/basic.lean | [
"group_theory.group_action.conj_act",
"group_theory.group_action.quotient",
"group_theory.quotient_group",
"topology.algebra.monoid",
"topology.algebra.constructions"
] | [
"homeomorph.shear_mul_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
homeomorph.shear_mul_right_symm_coe :
⇑(homeomorph.shear_mul_right G).symm = λ z : G × G, (z.1, z.1⁻¹ * z.2) | rfl | lemma | homeomorph.shear_mul_right_symm_coe | topology.algebra.group | src/topology/algebra/group/basic.lean | [
"group_theory.group_action.conj_act",
"group_theory.group_action.quotient",
"group_theory.quotient_group",
"topology.algebra.monoid",
"topology.algebra.constructions"
] | [
"homeomorph.shear_mul_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inducing.topological_group {F : Type*} [group H]
[topological_space H] [monoid_hom_class F H G] (f : F) (hf : inducing f) :
topological_group H | { to_has_continuous_mul := hf.has_continuous_mul _,
to_has_continuous_inv := hf.has_continuous_inv (map_inv f) } | lemma | inducing.topological_group | topology.algebra.group | src/topology/algebra/group/basic.lean | [
"group_theory.group_action.conj_act",
"group_theory.group_action.quotient",
"group_theory.quotient_group",
"topology.algebra.monoid",
"topology.algebra.constructions"
] | [
"group",
"inducing",
"map_inv",
"monoid_hom_class",
"topological_group",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
topological_group_induced {F : Type*} [group H]
[monoid_hom_class F H G] (f : F) :
@topological_group H (induced f ‹_›) _ | by { letI := induced f ‹_›, exact inducing.topological_group f ⟨rfl⟩ } | lemma | topological_group_induced | topology.algebra.group | src/topology/algebra/group/basic.lean | [
"group_theory.group_action.conj_act",
"group_theory.group_action.quotient",
"group_theory.quotient_group",
"topology.algebra.monoid",
"topology.algebra.constructions"
] | [
"group",
"inducing.topological_group",
"monoid_hom_class",
"topological_group"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
subgroup.topological_closure (s : subgroup G) : subgroup G | { carrier := closure (s : set G),
inv_mem' := λ g m, by simpa [←set.mem_inv, inv_closure] using m,
..s.to_submonoid.topological_closure } | def | subgroup.topological_closure | topology.algebra.group | src/topology/algebra/group/basic.lean | [
"group_theory.group_action.conj_act",
"group_theory.group_action.quotient",
"group_theory.quotient_group",
"topology.algebra.monoid",
"topology.algebra.constructions"
] | [
"closure",
"inv_closure",
"subgroup"
] | The (topological-space) closure of a subgroup of a space `M` with `has_continuous_mul` is
itself a subgroup. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
subgroup.topological_closure_coe {s : subgroup G} :
(s.topological_closure : set G) = closure s | rfl | lemma | subgroup.topological_closure_coe | topology.algebra.group | src/topology/algebra/group/basic.lean | [
"group_theory.group_action.conj_act",
"group_theory.group_action.quotient",
"group_theory.quotient_group",
"topology.algebra.monoid",
"topology.algebra.constructions"
] | [
"closure",
"subgroup"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
subgroup.le_topological_closure (s : subgroup G) :
s ≤ s.topological_closure | subset_closure | lemma | subgroup.le_topological_closure | topology.algebra.group | src/topology/algebra/group/basic.lean | [
"group_theory.group_action.conj_act",
"group_theory.group_action.quotient",
"group_theory.quotient_group",
"topology.algebra.monoid",
"topology.algebra.constructions"
] | [
"subgroup",
"subset_closure"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
subgroup.is_closed_topological_closure (s : subgroup G) :
is_closed (s.topological_closure : set G) | by convert is_closed_closure | lemma | subgroup.is_closed_topological_closure | topology.algebra.group | src/topology/algebra/group/basic.lean | [
"group_theory.group_action.conj_act",
"group_theory.group_action.quotient",
"group_theory.quotient_group",
"topology.algebra.monoid",
"topology.algebra.constructions"
] | [
"is_closed",
"is_closed_closure",
"subgroup"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
subgroup.topological_closure_minimal
(s : subgroup G) {t : subgroup G} (h : s ≤ t) (ht : is_closed (t : set G)) :
s.topological_closure ≤ t | closure_minimal h ht | lemma | subgroup.topological_closure_minimal | topology.algebra.group | src/topology/algebra/group/basic.lean | [
"group_theory.group_action.conj_act",
"group_theory.group_action.quotient",
"group_theory.quotient_group",
"topology.algebra.monoid",
"topology.algebra.constructions"
] | [
"closure_minimal",
"is_closed",
"subgroup"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dense_range.topological_closure_map_subgroup [group H] [topological_space H]
[topological_group H] {f : G →* H} (hf : continuous f) (hf' : dense_range f) {s : subgroup G}
(hs : s.topological_closure = ⊤) :
(s.map f).topological_closure = ⊤ | begin
rw set_like.ext'_iff at hs ⊢,
simp only [subgroup.topological_closure_coe, subgroup.coe_top, ← dense_iff_closure_eq] at hs ⊢,
exact hf'.dense_image hf hs
end | lemma | dense_range.topological_closure_map_subgroup | topology.algebra.group | src/topology/algebra/group/basic.lean | [
"group_theory.group_action.conj_act",
"group_theory.group_action.quotient",
"group_theory.quotient_group",
"topology.algebra.monoid",
"topology.algebra.constructions"
] | [
"continuous",
"dense_iff_closure_eq",
"dense_range",
"group",
"set_like.ext'_iff",
"subgroup",
"subgroup.coe_top",
"subgroup.topological_closure_coe",
"topological_group",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
subgroup.is_normal_topological_closure {G : Type*} [topological_space G] [group G]
[topological_group G] (N : subgroup G) [N.normal] :
(subgroup.topological_closure N).normal | { conj_mem := λ n hn g,
begin
apply map_mem_closure (topological_group.continuous_conj g) hn,
exact λ m hm, subgroup.normal.conj_mem infer_instance m hm g
end } | lemma | subgroup.is_normal_topological_closure | topology.algebra.group | src/topology/algebra/group/basic.lean | [
"group_theory.group_action.conj_act",
"group_theory.group_action.quotient",
"group_theory.quotient_group",
"topology.algebra.monoid",
"topology.algebra.constructions"
] | [
"group",
"map_mem_closure",
"normal",
"subgroup",
"subgroup.topological_closure",
"topological_group",
"topological_group.continuous_conj",
"topological_space"
] | The topological closure of a normal subgroup is normal. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_mem_connected_component_one {G : Type*} [topological_space G]
[mul_one_class G] [has_continuous_mul G] {g h : G} (hg : g ∈ connected_component (1 : G))
(hh : h ∈ connected_component (1 : G)) : g * h ∈ connected_component (1 : G) | begin
rw connected_component_eq hg,
have hmul: g ∈ connected_component (g*h),
{ apply continuous.image_connected_component_subset (continuous_mul_left g),
rw ← connected_component_eq hh,
exact ⟨(1 : G), mem_connected_component, by simp only [mul_one]⟩ },
simpa [← connected_component_eq hmul] using (mem_... | lemma | mul_mem_connected_component_one | topology.algebra.group | src/topology/algebra/group/basic.lean | [
"group_theory.group_action.conj_act",
"group_theory.group_action.quotient",
"group_theory.quotient_group",
"topology.algebra.monoid",
"topology.algebra.constructions"
] | [
"connected_component",
"connected_component_eq",
"continuous.image_connected_component_subset",
"continuous_mul_left",
"has_continuous_mul",
"mem_connected_component",
"mul_one",
"mul_one_class",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_mem_connected_component_one {G : Type*} [topological_space G] [group G]
[topological_group G] {g : G} (hg : g ∈ connected_component (1 : G)) :
g⁻¹ ∈ connected_component (1 : G) | begin
rw ← inv_one,
exact continuous.image_connected_component_subset continuous_inv _
((set.mem_image _ _ _).mp ⟨g, hg, rfl⟩)
end | lemma | inv_mem_connected_component_one | topology.algebra.group | src/topology/algebra/group/basic.lean | [
"group_theory.group_action.conj_act",
"group_theory.group_action.quotient",
"group_theory.quotient_group",
"topology.algebra.monoid",
"topology.algebra.constructions"
] | [
"connected_component",
"continuous.image_connected_component_subset",
"group",
"inv_one",
"set.mem_image",
"topological_group",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
subgroup.connected_component_of_one (G : Type*) [topological_space G] [group G]
[topological_group G] : subgroup G | { carrier := connected_component (1 : G),
one_mem' := mem_connected_component,
mul_mem' := λ g h hg hh, mul_mem_connected_component_one hg hh,
inv_mem' := λ g hg, inv_mem_connected_component_one hg } | def | subgroup.connected_component_of_one | topology.algebra.group | src/topology/algebra/group/basic.lean | [
"group_theory.group_action.conj_act",
"group_theory.group_action.quotient",
"group_theory.quotient_group",
"topology.algebra.monoid",
"topology.algebra.constructions"
] | [
"connected_component",
"group",
"inv_mem_connected_component_one",
"mem_connected_component",
"mul_mem_connected_component_one",
"subgroup",
"topological_group",
"topological_space"
] | The connected component of 1 is a subgroup of `G`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
subgroup.comm_group_topological_closure [t2_space G] (s : subgroup G)
(hs : ∀ (x y : s), x * y = y * x) : comm_group s.topological_closure | { ..s.topological_closure.to_group,
..s.to_submonoid.comm_monoid_topological_closure hs } | def | subgroup.comm_group_topological_closure | topology.algebra.group | src/topology/algebra/group/basic.lean | [
"group_theory.group_action.conj_act",
"group_theory.group_action.quotient",
"group_theory.quotient_group",
"topology.algebra.monoid",
"topology.algebra.constructions"
] | [
"comm_group",
"subgroup",
"t2_space"
] | If a subgroup of a topological group is commutative, then so is its topological closure. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
exists_nhds_split_inv {s : set G} (hs : s ∈ 𝓝 (1 : G)) :
∃ V ∈ 𝓝 (1 : G), ∀ (v ∈ V) (w ∈ V), v / w ∈ s | have ((λp : G × G, p.1 * p.2⁻¹) ⁻¹' s) ∈ 𝓝 ((1, 1) : G × G),
from continuous_at_fst.mul continuous_at_snd.inv (by simpa),
by simpa only [div_eq_mul_inv, nhds_prod_eq, mem_prod_self_iff, prod_subset_iff, mem_preimage]
using this | lemma | exists_nhds_split_inv | topology.algebra.group | src/topology/algebra/group/basic.lean | [
"group_theory.group_action.conj_act",
"group_theory.group_action.quotient",
"group_theory.quotient_group",
"topology.algebra.monoid",
"topology.algebra.constructions"
] | [
"div_eq_mul_inv",
"nhds_prod_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nhds_translation_mul_inv (x : G) : comap (λ y : G, y * x⁻¹) (𝓝 1) = 𝓝 x | ((homeomorph.mul_right x⁻¹).comap_nhds_eq 1).trans $ show 𝓝 (1 * x⁻¹⁻¹) = 𝓝 x, by simp | lemma | nhds_translation_mul_inv | topology.algebra.group | src/topology/algebra/group/basic.lean | [
"group_theory.group_action.conj_act",
"group_theory.group_action.quotient",
"group_theory.quotient_group",
"topology.algebra.monoid",
"topology.algebra.constructions"
] | [
"homeomorph.mul_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_mul_left_nhds (x y : G) : map ((*) x) (𝓝 y) = 𝓝 (x * y) | (homeomorph.mul_left x).map_nhds_eq y | lemma | map_mul_left_nhds | topology.algebra.group | src/topology/algebra/group/basic.lean | [
"group_theory.group_action.conj_act",
"group_theory.group_action.quotient",
"group_theory.quotient_group",
"topology.algebra.monoid",
"topology.algebra.constructions"
] | [
"homeomorph.mul_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_mul_left_nhds_one (x : G) : map ((*) x) (𝓝 1) = 𝓝 x | by simp | lemma | map_mul_left_nhds_one | topology.algebra.group | src/topology/algebra/group/basic.lean | [
"group_theory.group_action.conj_act",
"group_theory.group_action.quotient",
"group_theory.quotient_group",
"topology.algebra.monoid",
"topology.algebra.constructions"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_mul_right_nhds (x y : G) : map (λ z, z * x) (𝓝 y) = 𝓝 (y * x) | (homeomorph.mul_right x).map_nhds_eq y | lemma | map_mul_right_nhds | topology.algebra.group | src/topology/algebra/group/basic.lean | [
"group_theory.group_action.conj_act",
"group_theory.group_action.quotient",
"group_theory.quotient_group",
"topology.algebra.monoid",
"topology.algebra.constructions"
] | [
"homeomorph.mul_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_mul_right_nhds_one (x : G) : map (λ y, y * x) (𝓝 1) = 𝓝 x | by simp | lemma | map_mul_right_nhds_one | topology.algebra.group | src/topology/algebra/group/basic.lean | [
"group_theory.group_action.conj_act",
"group_theory.group_action.quotient",
"group_theory.quotient_group",
"topology.algebra.monoid",
"topology.algebra.constructions"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
filter.has_basis.nhds_of_one {ι : Sort*} {p : ι → Prop} {s : ι → set G}
(hb : has_basis (𝓝 1 : filter G) p s) (x : G) : has_basis (𝓝 x) p (λ i, {y | y / x ∈ s i}) | begin
rw ← nhds_translation_mul_inv,
simp_rw [div_eq_mul_inv],
exact hb.comap _
end | lemma | filter.has_basis.nhds_of_one | topology.algebra.group | src/topology/algebra/group/basic.lean | [
"group_theory.group_action.conj_act",
"group_theory.group_action.quotient",
"group_theory.quotient_group",
"topology.algebra.monoid",
"topology.algebra.constructions"
] | [
"div_eq_mul_inv",
"filter",
"nhds_translation_mul_inv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_closure_iff_nhds_one {x : G} {s : set G} :
x ∈ closure s ↔ ∀ U ∈ (𝓝 1 : filter G), ∃ y ∈ s, y / x ∈ U | begin
rw mem_closure_iff_nhds_basis ((𝓝 1 : filter G).basis_sets.nhds_of_one x),
refl
end | lemma | mem_closure_iff_nhds_one | topology.algebra.group | src/topology/algebra/group/basic.lean | [
"group_theory.group_action.conj_act",
"group_theory.group_action.quotient",
"group_theory.quotient_group",
"topology.algebra.monoid",
"topology.algebra.constructions"
] | [
"closure",
"filter",
"mem_closure_iff_nhds_basis"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_of_continuous_at_one {M hom : Type*} [mul_one_class M] [topological_space M]
[has_continuous_mul M] [monoid_hom_class hom G M] (f : hom) (hf : continuous_at f 1) :
continuous f | continuous_iff_continuous_at.2 $ λ x,
by simpa only [continuous_at, ← map_mul_left_nhds_one x, tendsto_map'_iff, (∘),
map_mul, map_one, mul_one] using hf.tendsto.const_mul (f x) | lemma | continuous_of_continuous_at_one | topology.algebra.group | src/topology/algebra/group/basic.lean | [
"group_theory.group_action.conj_act",
"group_theory.group_action.quotient",
"group_theory.quotient_group",
"topology.algebra.monoid",
"topology.algebra.constructions"
] | [
"continuous",
"continuous_at",
"has_continuous_mul",
"map_mul",
"map_mul_left_nhds_one",
"map_one",
"monoid_hom_class",
"mul_one",
"mul_one_class",
"topological_space"
] | A monoid homomorphism (a bundled morphism of a type that implements `monoid_hom_class`) from a
topological group to a topological monoid is continuous provided that it is continuous at one. See
also `uniform_continuous_of_continuous_at_one`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
topological_group.ext {G : Type*} [group G] {t t' : topological_space G}
(tg : @topological_group G t _) (tg' : @topological_group G t' _)
(h : @nhds G t 1 = @nhds G t' 1) : t = t' | eq_of_nhds_eq_nhds $ λ x, by
rw [← @nhds_translation_mul_inv G t _ _ x , ← @nhds_translation_mul_inv G t' _ _ x , ← h] | lemma | topological_group.ext | topology.algebra.group | src/topology/algebra/group/basic.lean | [
"group_theory.group_action.conj_act",
"group_theory.group_action.quotient",
"group_theory.quotient_group",
"topology.algebra.monoid",
"topology.algebra.constructions"
] | [
"eq_of_nhds_eq_nhds",
"group",
"nhds",
"nhds_translation_mul_inv",
"topological_group",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
topological_group.ext_iff {G : Type*} [group G] {t t' : topological_space G}
(tg : @topological_group G t _) (tg' : @topological_group G t' _) :
t = t' ↔ @nhds G t 1 = @nhds G t' 1 | ⟨λ h, h ▸ rfl, tg.ext tg'⟩ | lemma | topological_group.ext_iff | topology.algebra.group | src/topology/algebra/group/basic.lean | [
"group_theory.group_action.conj_act",
"group_theory.group_action.quotient",
"group_theory.quotient_group",
"topology.algebra.monoid",
"topology.algebra.constructions"
] | [
"group",
"nhds",
"topological_group",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_continuous_inv.of_nhds_one {G : Type*} [group G] [topological_space G]
(hinv : tendsto (λ (x : G), x⁻¹) (𝓝 1) (𝓝 1))
(hleft : ∀ (x₀ : G), 𝓝 x₀ = map (λ (x : G), x₀ * x) (𝓝 1))
(hconj : ∀ (x₀ : G), tendsto (λ (x : G), x₀ * x * x₀⁻¹) (𝓝 1) (𝓝 1)) :
has_continuous_inv G | begin
refine ⟨continuous_iff_continuous_at.2 $ λ x₀, _⟩,
have : tendsto (λ x, x₀⁻¹ * (x₀ * x⁻¹ * x₀⁻¹)) (𝓝 1) (map ((*) x₀⁻¹) (𝓝 1)),
from (tendsto_map.comp $ hconj x₀).comp hinv,
simpa only [continuous_at, hleft x₀, hleft x₀⁻¹, tendsto_map'_iff, (∘), mul_assoc,
mul_inv_rev, inv_mul_cancel_left] using t... | lemma | has_continuous_inv.of_nhds_one | topology.algebra.group | src/topology/algebra/group/basic.lean | [
"group_theory.group_action.conj_act",
"group_theory.group_action.quotient",
"group_theory.quotient_group",
"topology.algebra.monoid",
"topology.algebra.constructions"
] | [
"continuous_at",
"group",
"has_continuous_inv",
"inv_mul_cancel_left",
"mul_assoc",
"mul_inv_rev",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
topological_group.of_nhds_one' {G : Type u} [group G] [topological_space G]
(hmul : tendsto (uncurry ((*) : G → G → G)) ((𝓝 1) ×ᶠ 𝓝 1) (𝓝 1))
(hinv : tendsto (λ x : G, x⁻¹) (𝓝 1) (𝓝 1))
(hleft : ∀ x₀ : G, 𝓝 x₀ = map (λ x, x₀*x) (𝓝 1))
(hright : ∀ x₀ : G, 𝓝 x₀ = map (λ x, x*x₀) (𝓝 1)) : topological_grou... | { to_has_continuous_mul := has_continuous_mul.of_nhds_one hmul hleft hright,
to_has_continuous_inv := has_continuous_inv.of_nhds_one hinv hleft $ λ x₀, le_of_eq
begin
rw [show (λ x, x₀ * x * x₀⁻¹) = (λ x, x * x₀⁻¹) ∘ (λ x, x₀ * x), from rfl, ← map_map,
← hleft, hright, map_map],
simp [(∘)]
... | lemma | topological_group.of_nhds_one' | topology.algebra.group | src/topology/algebra/group/basic.lean | [
"group_theory.group_action.conj_act",
"group_theory.group_action.quotient",
"group_theory.quotient_group",
"topology.algebra.monoid",
"topology.algebra.constructions"
] | [
"group",
"has_continuous_inv.of_nhds_one",
"has_continuous_mul.of_nhds_one",
"topological_group",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
topological_group.of_nhds_one {G : Type u} [group G] [topological_space G]
(hmul : tendsto (uncurry ((*) : G → G → G)) ((𝓝 1) ×ᶠ 𝓝 1) (𝓝 1))
(hinv : tendsto (λ x : G, x⁻¹) (𝓝 1) (𝓝 1))
(hleft : ∀ x₀ : G, 𝓝 x₀ = map (λ x, x₀*x) (𝓝 1))
(hconj : ∀ x₀ : G, tendsto (λ x, x₀*x*x₀⁻¹) (𝓝 1) (𝓝 1)) : topologica... | begin
refine topological_group.of_nhds_one' hmul hinv hleft (λ x₀, _),
replace hconj : ∀ x₀ : G, map (λ x, x₀ * x * x₀⁻¹) (𝓝 1) = 𝓝 1,
from λ x₀, map_eq_of_inverse (λ x, x₀⁻¹ * x * x₀⁻¹⁻¹) (by { ext, simp [mul_assoc] })
(hconj _) (hconj _),
rw [← hconj x₀],
simpa [(∘)] using hleft _
end | lemma | topological_group.of_nhds_one | topology.algebra.group | src/topology/algebra/group/basic.lean | [
"group_theory.group_action.conj_act",
"group_theory.group_action.quotient",
"group_theory.quotient_group",
"topology.algebra.monoid",
"topology.algebra.constructions"
] | [
"group",
"mul_assoc",
"topological_group",
"topological_group.of_nhds_one'",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
topological_group.of_comm_of_nhds_one {G : Type u} [comm_group G] [topological_space G]
(hmul : tendsto (uncurry ((*) : G → G → G)) ((𝓝 1) ×ᶠ 𝓝 1) (𝓝 1))
(hinv : tendsto (λ x : G, x⁻¹) (𝓝 1) (𝓝 1))
(hleft : ∀ x₀ : G, 𝓝 x₀ = map (λ x, x₀*x) (𝓝 1)) : topological_group G | topological_group.of_nhds_one hmul hinv hleft (by simpa using tendsto_id) | lemma | topological_group.of_comm_of_nhds_one | topology.algebra.group | src/topology/algebra/group/basic.lean | [
"group_theory.group_action.conj_act",
"group_theory.group_action.quotient",
"group_theory.quotient_group",
"topology.algebra.monoid",
"topology.algebra.constructions"
] | [
"comm_group",
"topological_group",
"topological_group.of_nhds_one",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
quotient_group.quotient.topological_space {G : Type*} [group G] [topological_space G]
(N : subgroup G) : topological_space (G ⧸ N) | quotient.topological_space | instance | quotient_group.quotient.topological_space | topology.algebra.group | src/topology/algebra/group/basic.lean | [
"group_theory.group_action.conj_act",
"group_theory.group_action.quotient",
"group_theory.quotient_group",
"topology.algebra.monoid",
"topology.algebra.constructions"
] | [
"group",
"subgroup",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
quotient_group.is_open_map_coe : is_open_map (coe : G → G ⧸ N) | begin
intros s s_op,
change is_open ((coe : G → G ⧸ N) ⁻¹' (coe '' s)),
rw quotient_group.preimage_image_coe N s,
exact is_open_Union (λ n, (continuous_mul_right _).is_open_preimage s s_op)
end | lemma | quotient_group.is_open_map_coe | topology.algebra.group | src/topology/algebra/group/basic.lean | [
"group_theory.group_action.conj_act",
"group_theory.group_action.quotient",
"group_theory.quotient_group",
"topology.algebra.monoid",
"topology.algebra.constructions"
] | [
"continuous_mul_right",
"is_open",
"is_open_Union",
"is_open_map",
"quotient_group.preimage_image_coe"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
topological_group_quotient [N.normal] : topological_group (G ⧸ N) | { continuous_mul := begin
have cont : continuous ((coe : G → G ⧸ N) ∘ (λ (p : G × G), p.fst * p.snd)) :=
continuous_quot_mk.comp continuous_mul,
have quot : quotient_map (λ p : G × G, ((p.1 : G ⧸ N), (p.2 : G ⧸ N))),
{ apply is_open_map.to_quotient_map,
{ exact (quotient_group.is_open_map_coe N)... | instance | topological_group_quotient | topology.algebra.group | src/topology/algebra/group/basic.lean | [
"group_theory.group_action.conj_act",
"group_theory.group_action.quotient",
"group_theory.quotient_group",
"topology.algebra.monoid",
"topology.algebra.constructions"
] | [
"cont",
"continuous",
"continuous_mul",
"continuous_quot_mk",
"is_open_map.to_quotient_map",
"prod_map",
"quotient_group.is_open_map_coe",
"quotient_map",
"quotient_map.continuous_iff",
"surjective_quot_mk",
"topological_group"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
quotient_group.nhds_eq (x : G) : 𝓝 (x : G ⧸ N) = map coe (𝓝 x) | le_antisymm ((quotient_group.is_open_map_coe N).nhds_le x) continuous_quot_mk.continuous_at | lemma | quotient_group.nhds_eq | topology.algebra.group | src/topology/algebra/group/basic.lean | [
"group_theory.group_action.conj_act",
"group_theory.group_action.quotient",
"group_theory.quotient_group",
"topology.algebra.monoid",
"topology.algebra.constructions"
] | [
"quotient_group.is_open_map_coe"
] | Neighborhoods in the quotient are precisely the map of neighborhoods in the prequotient. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
topological_group.exists_antitone_basis_nhds_one :
∃ (u : ℕ → set G), (𝓝 1).has_antitone_basis u ∧ (∀ n, u (n + 1) * u (n + 1) ⊆ u n) | begin
rcases (𝓝 (1 : G)).exists_antitone_basis with ⟨u, hu, u_anti⟩,
have := ((hu.prod_nhds hu).tendsto_iff hu).mp
(by simpa only [mul_one] using continuous_mul.tendsto ((1, 1) : G × G)),
simp only [and_self, mem_prod, and_imp, prod.forall, exists_true_left, prod.exists,
forall_true_left] at this,
have... | lemma | topological_group.exists_antitone_basis_nhds_one | topology.algebra.group | src/topology/algebra/group/basic.lean | [
"group_theory.group_action.conj_act",
"group_theory.group_action.quotient",
"group_theory.quotient_group",
"topology.algebra.monoid",
"topology.algebra.constructions"
] | [
"and_imp",
"exists_true_left",
"forall_true_left",
"mul_one"
] | Any first countable topological group has an antitone neighborhood basis `u : ℕ → set G` for
which `(u (n + 1)) ^ 2 ⊆ u n`. The existence of such a neighborhood basis is a key tool for
`quotient_group.complete_space` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
quotient_group.nhds_one_is_countably_generated : (𝓝 (1 : G ⧸ N)).is_countably_generated | (quotient_group.nhds_eq N 1).symm ▸ map.is_countably_generated _ _ | instance | quotient_group.nhds_one_is_countably_generated | topology.algebra.group | src/topology/algebra/group/basic.lean | [
"group_theory.group_action.conj_act",
"group_theory.group_action.quotient",
"group_theory.quotient_group",
"topology.algebra.monoid",
"topology.algebra.constructions"
] | [
"quotient_group.nhds_eq"
] | In a first countable topological group `G` with normal subgroup `N`, `1 : G ⧸ N` has a
countable neighborhood basis. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_continuous_sub (G : Type*) [topological_space G] [has_sub G] : Prop | (continuous_sub : continuous (λ p : G × G, p.1 - p.2)) | class | has_continuous_sub | topology.algebra.group | src/topology/algebra/group/basic.lean | [
"group_theory.group_action.conj_act",
"group_theory.group_action.quotient",
"group_theory.quotient_group",
"topology.algebra.monoid",
"topology.algebra.constructions"
] | [
"continuous",
"topological_space"
] | A typeclass saying that `λ p : G × G, p.1 - p.2` is a continuous function. This property
automatically holds for topological additive groups but it also holds, e.g., for `ℝ≥0`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_continuous_div (G : Type*) [topological_space G] [has_div G] : Prop | (continuous_div' : continuous (λ p : G × G, p.1 / p.2)) | class | has_continuous_div | topology.algebra.group | src/topology/algebra/group/basic.lean | [
"group_theory.group_action.conj_act",
"group_theory.group_action.quotient",
"group_theory.quotient_group",
"topology.algebra.monoid",
"topology.algebra.constructions"
] | [
"continuous",
"topological_space"
] | A typeclass saying that `λ p : G × G, p.1 / p.2` is a continuous function. This property
automatically holds for topological groups. Lemmas using this class have primes.
The unprimed version is for `group_with_zero`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
topological_group.to_has_continuous_div [topological_space G] [group G]
[topological_group G] : has_continuous_div G | ⟨by { simp only [div_eq_mul_inv], exact continuous_fst.mul continuous_snd.inv }⟩ | instance | topological_group.to_has_continuous_div | topology.algebra.group | src/topology/algebra/group/basic.lean | [
"group_theory.group_action.conj_act",
"group_theory.group_action.quotient",
"group_theory.quotient_group",
"topology.algebra.monoid",
"topology.algebra.constructions"
] | [
"div_eq_mul_inv",
"group",
"has_continuous_div",
"topological_group",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
filter.tendsto.div' {f g : α → G} {l : filter α} {a b : G} (hf : tendsto f l (𝓝 a))
(hg : tendsto g l (𝓝 b)) : tendsto (λ x, f x / g x) l (𝓝 (a / b)) | (continuous_div'.tendsto (a, b)).comp (hf.prod_mk_nhds hg) | lemma | filter.tendsto.div' | topology.algebra.group | src/topology/algebra/group/basic.lean | [
"group_theory.group_action.conj_act",
"group_theory.group_action.quotient",
"group_theory.quotient_group",
"topology.algebra.monoid",
"topology.algebra.constructions"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
filter.tendsto.const_div' (b : G) {c : G} {f : α → G} {l : filter α}
(h : tendsto f l (𝓝 c)) : tendsto (λ k : α, b / f k) l (𝓝 (b / c)) | tendsto_const_nhds.div' h | lemma | filter.tendsto.const_div' | topology.algebra.group | src/topology/algebra/group/basic.lean | [
"group_theory.group_action.conj_act",
"group_theory.group_action.quotient",
"group_theory.quotient_group",
"topology.algebra.monoid",
"topology.algebra.constructions"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
filter.tendsto.div_const' {c : G} {f : α → G} {l : filter α}
(h : tendsto f l (𝓝 c)) (b : G) : tendsto (λ k : α, f k / b) l (𝓝 (c / b)) | h.div' tendsto_const_nhds | lemma | filter.tendsto.div_const' | topology.algebra.group | src/topology/algebra/group/basic.lean | [
"group_theory.group_action.conj_act",
"group_theory.group_action.quotient",
"group_theory.quotient_group",
"topology.algebra.monoid",
"topology.algebra.constructions"
] | [
"filter",
"tendsto_const_nhds"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous.div' (hf : continuous f) (hg : continuous g) :
continuous (λ x, f x / g x) | continuous_div'.comp (hf.prod_mk hg : _) | lemma | continuous.div' | topology.algebra.group | src/topology/algebra/group/basic.lean | [
"group_theory.group_action.conj_act",
"group_theory.group_action.quotient",
"group_theory.quotient_group",
"topology.algebra.monoid",
"topology.algebra.constructions"
] | [
"continuous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_div_left' (a : G) : continuous (λ b : G, a / b) | continuous_const.div' continuous_id | lemma | continuous_div_left' | topology.algebra.group | src/topology/algebra/group/basic.lean | [
"group_theory.group_action.conj_act",
"group_theory.group_action.quotient",
"group_theory.quotient_group",
"topology.algebra.monoid",
"topology.algebra.constructions"
] | [
"continuous",
"continuous_id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_div_right' (a : G) : continuous (λ b : G, b / a) | continuous_id.div' continuous_const | lemma | continuous_div_right' | topology.algebra.group | src/topology/algebra/group/basic.lean | [
"group_theory.group_action.conj_act",
"group_theory.group_action.quotient",
"group_theory.quotient_group",
"topology.algebra.monoid",
"topology.algebra.constructions"
] | [
"continuous",
"continuous_const"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_at.div' {f g : α → G} {x : α} (hf : continuous_at f x) (hg : continuous_at g x) :
continuous_at (λx, f x / g x) x | hf.div' hg | lemma | continuous_at.div' | topology.algebra.group | src/topology/algebra/group/basic.lean | [
"group_theory.group_action.conj_act",
"group_theory.group_action.quotient",
"group_theory.quotient_group",
"topology.algebra.monoid",
"topology.algebra.constructions"
] | [
"continuous_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_within_at.div' (hf : continuous_within_at f s x)
(hg : continuous_within_at g s x) :
continuous_within_at (λ x, f x / g x) s x | hf.div' hg | lemma | continuous_within_at.div' | topology.algebra.group | src/topology/algebra/group/basic.lean | [
"group_theory.group_action.conj_act",
"group_theory.group_action.quotient",
"group_theory.quotient_group",
"topology.algebra.monoid",
"topology.algebra.constructions"
] | [
"continuous_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_on.div' (hf : continuous_on f s) (hg : continuous_on g s) :
continuous_on (λx, f x / g x) s | λ x hx, (hf x hx).div' (hg x hx) | lemma | continuous_on.div' | topology.algebra.group | src/topology/algebra/group/basic.lean | [
"group_theory.group_action.conj_act",
"group_theory.group_action.quotient",
"group_theory.quotient_group",
"topology.algebra.monoid",
"topology.algebra.constructions"
] | [
"continuous_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
homeomorph.div_left (x : G) : G ≃ₜ G | { continuous_to_fun := continuous_const.div' continuous_id,
continuous_inv_fun := continuous_inv.mul continuous_const,
.. equiv.div_left x } | def | homeomorph.div_left | topology.algebra.group | src/topology/algebra/group/basic.lean | [
"group_theory.group_action.conj_act",
"group_theory.group_action.quotient",
"group_theory.quotient_group",
"topology.algebra.monoid",
"topology.algebra.constructions"
] | [
"continuous_const",
"continuous_id",
"equiv.div_left"
] | A version of `homeomorph.mul_left a b⁻¹` that is defeq to `a / b`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_open_map_div_left (a : G) : is_open_map ((/) a) | (homeomorph.div_left _).is_open_map | lemma | is_open_map_div_left | topology.algebra.group | src/topology/algebra/group/basic.lean | [
"group_theory.group_action.conj_act",
"group_theory.group_action.quotient",
"group_theory.quotient_group",
"topology.algebra.monoid",
"topology.algebra.constructions"
] | [
"homeomorph.div_left",
"is_open_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_closed_map_div_left (a : G) : is_closed_map ((/) a) | (homeomorph.div_left _).is_closed_map | lemma | is_closed_map_div_left | topology.algebra.group | src/topology/algebra/group/basic.lean | [
"group_theory.group_action.conj_act",
"group_theory.group_action.quotient",
"group_theory.quotient_group",
"topology.algebra.monoid",
"topology.algebra.constructions"
] | [
"homeomorph.div_left",
"is_closed_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
homeomorph.div_right (x : G) : G ≃ₜ G | { continuous_to_fun := continuous_id.div' continuous_const,
continuous_inv_fun := continuous_id.mul continuous_const,
.. equiv.div_right x } | def | homeomorph.div_right | topology.algebra.group | src/topology/algebra/group/basic.lean | [
"group_theory.group_action.conj_act",
"group_theory.group_action.quotient",
"group_theory.quotient_group",
"topology.algebra.monoid",
"topology.algebra.constructions"
] | [
"continuous_const",
"equiv.div_right"
] | A version of `homeomorph.mul_right a⁻¹ b` that is defeq to `b / a`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_open_map_div_right (a : G) : is_open_map (λ x, x / a) | (homeomorph.div_right a).is_open_map | lemma | is_open_map_div_right | topology.algebra.group | src/topology/algebra/group/basic.lean | [
"group_theory.group_action.conj_act",
"group_theory.group_action.quotient",
"group_theory.quotient_group",
"topology.algebra.monoid",
"topology.algebra.constructions"
] | [
"homeomorph.div_right",
"is_open_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_closed_map_div_right (a : G) : is_closed_map (λ x, x / a) | (homeomorph.div_right a).is_closed_map | lemma | is_closed_map_div_right | topology.algebra.group | src/topology/algebra/group/basic.lean | [
"group_theory.group_action.conj_act",
"group_theory.group_action.quotient",
"group_theory.quotient_group",
"topology.algebra.monoid",
"topology.algebra.constructions"
] | [
"homeomorph.div_right",
"is_closed_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_div_nhds_one_iff
{α : Type*} {l : filter α} {x : G} {u : α → G} :
tendsto (λ n, u n / x) l (𝓝 1) ↔ tendsto u l (𝓝 x) | begin
have A : tendsto (λ (n : α), x) l (𝓝 x) := tendsto_const_nhds,
exact ⟨λ h, by simpa using h.mul A, λ h, by simpa using h.div' A⟩
end | lemma | tendsto_div_nhds_one_iff | topology.algebra.group | src/topology/algebra/group/basic.lean | [
"group_theory.group_action.conj_act",
"group_theory.group_action.quotient",
"group_theory.quotient_group",
"topology.algebra.monoid",
"topology.algebra.constructions"
] | [
"filter",
"tendsto_const_nhds"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nhds_translation_div (x : G) : comap (/ x) (𝓝 1) = 𝓝 x | by simpa only [div_eq_mul_inv] using nhds_translation_mul_inv x | lemma | nhds_translation_div | topology.algebra.group | src/topology/algebra/group/basic.lean | [
"group_theory.group_action.conj_act",
"group_theory.group_action.quotient",
"group_theory.quotient_group",
"topology.algebra.monoid",
"topology.algebra.constructions"
] | [
"div_eq_mul_inv",
"nhds_translation_mul_inv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_open.smul_left (ht : is_open t) : is_open (s • t) | by { rw ←bUnion_smul_set, exact is_open_bUnion (λ a _, ht.smul _) } | lemma | is_open.smul_left | topology.algebra.group | src/topology/algebra/group/basic.lean | [
"group_theory.group_action.conj_act",
"group_theory.group_action.quotient",
"group_theory.quotient_group",
"topology.algebra.monoid",
"topology.algebra.constructions"
] | [
"is_open",
"is_open_bUnion"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
subset_interior_smul_right : s • interior t ⊆ interior (s • t) | interior_maximal (set.smul_subset_smul_left interior_subset) is_open_interior.smul_left | lemma | subset_interior_smul_right | topology.algebra.group | src/topology/algebra/group/basic.lean | [
"group_theory.group_action.conj_act",
"group_theory.group_action.quotient",
"group_theory.quotient_group",
"topology.algebra.monoid",
"topology.algebra.constructions"
] | [
"interior",
"interior_maximal",
"interior_subset",
"set.smul_subset_smul_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_mem_nhds (a : α) {x : β} (ht : t ∈ 𝓝 x) :
a • t ∈ 𝓝 (a • x) | begin
rcases mem_nhds_iff.1 ht with ⟨u, ut, u_open, hu⟩,
exact mem_nhds_iff.2 ⟨a • u, smul_set_mono ut, u_open.smul a, smul_mem_smul_set hu⟩,
end | lemma | smul_mem_nhds | topology.algebra.group | src/topology/algebra/group/basic.lean | [
"group_theory.group_action.conj_act",
"group_theory.group_action.quotient",
"group_theory.quotient_group",
"topology.algebra.monoid",
"topology.algebra.constructions"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
subset_interior_smul : interior s • interior t ⊆ interior (s • t) | (set.smul_subset_smul_right interior_subset).trans subset_interior_smul_right | lemma | subset_interior_smul | topology.algebra.group | src/topology/algebra/group/basic.lean | [
"group_theory.group_action.conj_act",
"group_theory.group_action.quotient",
"group_theory.quotient_group",
"topology.algebra.monoid",
"topology.algebra.constructions"
] | [
"interior",
"interior_subset",
"set.smul_subset_smul_right",
"subset_interior_smul_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_open.mul_left : is_open t → is_open (s * t) | is_open.smul_left | lemma | is_open.mul_left | topology.algebra.group | src/topology/algebra/group/basic.lean | [
"group_theory.group_action.conj_act",
"group_theory.group_action.quotient",
"group_theory.quotient_group",
"topology.algebra.monoid",
"topology.algebra.constructions"
] | [
"is_open",
"is_open.smul_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
subset_interior_mul_right : s * interior t ⊆ interior (s * t) | subset_interior_smul_right | lemma | subset_interior_mul_right | topology.algebra.group | src/topology/algebra/group/basic.lean | [
"group_theory.group_action.conj_act",
"group_theory.group_action.quotient",
"group_theory.quotient_group",
"topology.algebra.monoid",
"topology.algebra.constructions"
] | [
"interior",
"subset_interior_smul_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
subset_interior_mul : interior s * interior t ⊆ interior (s * t) | subset_interior_smul | lemma | subset_interior_mul | topology.algebra.group | src/topology/algebra/group/basic.lean | [
"group_theory.group_action.conj_act",
"group_theory.group_action.quotient",
"group_theory.quotient_group",
"topology.algebra.monoid",
"topology.algebra.constructions"
] | [
"interior",
"subset_interior_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
singleton_mul_mem_nhds (a : α) {b : α} (h : s ∈ 𝓝 b) :
{a} * s ∈ 𝓝 (a * b) | by { have := smul_mem_nhds a h, rwa ← singleton_smul at this } | lemma | singleton_mul_mem_nhds | topology.algebra.group | src/topology/algebra/group/basic.lean | [
"group_theory.group_action.conj_act",
"group_theory.group_action.quotient",
"group_theory.quotient_group",
"topology.algebra.monoid",
"topology.algebra.constructions"
] | [
"smul_mem_nhds"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
singleton_mul_mem_nhds_of_nhds_one (a : α) (h : s ∈ 𝓝 (1 : α)) :
{a} * s ∈ 𝓝 a | by simpa only [mul_one] using singleton_mul_mem_nhds a h | lemma | singleton_mul_mem_nhds_of_nhds_one | topology.algebra.group | src/topology/algebra/group/basic.lean | [
"group_theory.group_action.conj_act",
"group_theory.group_action.quotient",
"group_theory.quotient_group",
"topology.algebra.monoid",
"topology.algebra.constructions"
] | [
"mul_one",
"singleton_mul_mem_nhds"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_open.mul_right (hs : is_open s) : is_open (s * t) | by { rw ←bUnion_op_smul_set, exact is_open_bUnion (λ a _, hs.smul _) } | lemma | is_open.mul_right | topology.algebra.group | src/topology/algebra/group/basic.lean | [
"group_theory.group_action.conj_act",
"group_theory.group_action.quotient",
"group_theory.quotient_group",
"topology.algebra.monoid",
"topology.algebra.constructions"
] | [
"is_open",
"is_open_bUnion"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
subset_interior_mul_left : interior s * t ⊆ interior (s * t) | interior_maximal (set.mul_subset_mul_right interior_subset) is_open_interior.mul_right | lemma | subset_interior_mul_left | topology.algebra.group | src/topology/algebra/group/basic.lean | [
"group_theory.group_action.conj_act",
"group_theory.group_action.quotient",
"group_theory.quotient_group",
"topology.algebra.monoid",
"topology.algebra.constructions"
] | [
"interior",
"interior_maximal",
"interior_subset",
"set.mul_subset_mul_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
subset_interior_mul' : interior s * interior t ⊆ interior (s * t) | (set.mul_subset_mul_left interior_subset).trans subset_interior_mul_left | lemma | subset_interior_mul' | topology.algebra.group | src/topology/algebra/group/basic.lean | [
"group_theory.group_action.conj_act",
"group_theory.group_action.quotient",
"group_theory.quotient_group",
"topology.algebra.monoid",
"topology.algebra.constructions"
] | [
"interior",
"interior_subset",
"set.mul_subset_mul_left",
"subset_interior_mul_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_singleton_mem_nhds (a : α) {b : α} (h : s ∈ 𝓝 b) :
s * {a} ∈ 𝓝 (b * a) | begin
simp only [←bUnion_op_smul_set, mem_singleton_iff, Union_Union_eq_left],
exact smul_mem_nhds _ h,
end | lemma | mul_singleton_mem_nhds | topology.algebra.group | src/topology/algebra/group/basic.lean | [
"group_theory.group_action.conj_act",
"group_theory.group_action.quotient",
"group_theory.quotient_group",
"topology.algebra.monoid",
"topology.algebra.constructions"
] | [
"smul_mem_nhds"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_singleton_mem_nhds_of_nhds_one (a : α) (h : s ∈ 𝓝 (1 : α)) :
s * {a} ∈ 𝓝 a | by simpa only [one_mul] using mul_singleton_mem_nhds a h | lemma | mul_singleton_mem_nhds_of_nhds_one | topology.algebra.group | src/topology/algebra/group/basic.lean | [
"group_theory.group_action.conj_act",
"group_theory.group_action.quotient",
"group_theory.quotient_group",
"topology.algebra.monoid",
"topology.algebra.constructions"
] | [
"mul_singleton_mem_nhds",
"one_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_open.div_left (ht : is_open t) : is_open (s / t) | by { rw ←Union_div_left_image, exact is_open_bUnion (λ a ha, is_open_map_div_left a t ht) } | lemma | is_open.div_left | topology.algebra.group | src/topology/algebra/group/basic.lean | [
"group_theory.group_action.conj_act",
"group_theory.group_action.quotient",
"group_theory.quotient_group",
"topology.algebra.monoid",
"topology.algebra.constructions"
] | [
"is_open",
"is_open_bUnion",
"is_open_map_div_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_open.div_right (hs : is_open s) : is_open (s / t) | by { rw ←Union_div_right_image, exact is_open_bUnion (λ a ha, is_open_map_div_right a s hs) } | lemma | is_open.div_right | topology.algebra.group | src/topology/algebra/group/basic.lean | [
"group_theory.group_action.conj_act",
"group_theory.group_action.quotient",
"group_theory.quotient_group",
"topology.algebra.monoid",
"topology.algebra.constructions"
] | [
"is_open",
"is_open_bUnion",
"is_open_map_div_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
subset_interior_div_left : interior s / t ⊆ interior (s / t) | interior_maximal (div_subset_div_right interior_subset) is_open_interior.div_right | lemma | subset_interior_div_left | topology.algebra.group | src/topology/algebra/group/basic.lean | [
"group_theory.group_action.conj_act",
"group_theory.group_action.quotient",
"group_theory.quotient_group",
"topology.algebra.monoid",
"topology.algebra.constructions"
] | [
"interior",
"interior_maximal",
"interior_subset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
subset_interior_div_right : s / interior t ⊆ interior (s / t) | interior_maximal (div_subset_div_left interior_subset) is_open_interior.div_left | lemma | subset_interior_div_right | topology.algebra.group | src/topology/algebra/group/basic.lean | [
"group_theory.group_action.conj_act",
"group_theory.group_action.quotient",
"group_theory.quotient_group",
"topology.algebra.monoid",
"topology.algebra.constructions"
] | [
"interior",
"interior_maximal",
"interior_subset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
subset_interior_div : interior s / interior t ⊆ interior (s / t) | (div_subset_div_left interior_subset).trans subset_interior_div_left | lemma | subset_interior_div | topology.algebra.group | src/topology/algebra/group/basic.lean | [
"group_theory.group_action.conj_act",
"group_theory.group_action.quotient",
"group_theory.quotient_group",
"topology.algebra.monoid",
"topology.algebra.constructions"
] | [
"interior",
"interior_subset",
"subset_interior_div_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_open.mul_closure (hs : is_open s) (t : set α) : s * closure t = s * t | begin
refine (mul_subset_iff.2 $ λ a ha b hb, _).antisymm (mul_subset_mul_left subset_closure),
rw mem_closure_iff at hb,
have hbU : b ∈ s⁻¹ * {a * b} := ⟨a⁻¹, a * b, set.inv_mem_inv.2 ha, rfl, inv_mul_cancel_left _ _⟩,
obtain ⟨_, ⟨c, d, hc, (rfl : d = _), rfl⟩, hcs⟩ := hb _ hs.inv.mul_right hbU,
exact ⟨c⁻¹, ... | lemma | is_open.mul_closure | topology.algebra.group | src/topology/algebra/group/basic.lean | [
"group_theory.group_action.conj_act",
"group_theory.group_action.quotient",
"group_theory.quotient_group",
"topology.algebra.monoid",
"topology.algebra.constructions"
] | [
"closure",
"inv_mul_cancel_left",
"is_open",
"mem_closure_iff",
"subset_closure"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_open.closure_mul (ht : is_open t) (s : set α) : closure s * t = s * t | by rw [←inv_inv (closure s * t), mul_inv_rev, inv_closure, ht.inv.mul_closure, mul_inv_rev, inv_inv,
inv_inv] | lemma | is_open.closure_mul | topology.algebra.group | src/topology/algebra/group/basic.lean | [
"group_theory.group_action.conj_act",
"group_theory.group_action.quotient",
"group_theory.quotient_group",
"topology.algebra.monoid",
"topology.algebra.constructions"
] | [
"closure",
"inv_closure",
"inv_inv",
"is_open",
"mul_inv_rev"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_open.div_closure (hs : is_open s) (t : set α) : s / closure t = s / t | by simp_rw [div_eq_mul_inv, inv_closure, hs.mul_closure] | lemma | is_open.div_closure | topology.algebra.group | src/topology/algebra/group/basic.lean | [
"group_theory.group_action.conj_act",
"group_theory.group_action.quotient",
"group_theory.quotient_group",
"topology.algebra.monoid",
"topology.algebra.constructions"
] | [
"closure",
"div_eq_mul_inv",
"inv_closure",
"is_open"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_open.closure_div (ht : is_open t) (s : set α) : closure s / t = s / t | by simp_rw [div_eq_mul_inv, ht.inv.closure_mul] | lemma | is_open.closure_div | topology.algebra.group | src/topology/algebra/group/basic.lean | [
"group_theory.group_action.conj_act",
"group_theory.group_action.quotient",
"group_theory.quotient_group",
"topology.algebra.monoid",
"topology.algebra.constructions"
] | [
"closure",
"div_eq_mul_inv",
"is_open"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_group_with_zero_nhd (G : Type u) extends add_comm_group G | (Z [] : filter G)
(zero_Z : pure 0 ≤ Z)
(sub_Z : tendsto (λp:G×G, p.1 - p.2) (Z ×ᶠ Z) Z) | class | add_group_with_zero_nhd | topology.algebra.group | src/topology/algebra/group/basic.lean | [
"group_theory.group_action.conj_act",
"group_theory.group_action.quotient",
"group_theory.quotient_group",
"topology.algebra.monoid",
"topology.algebra.constructions"
] | [
"add_comm_group",
"filter"
] | additive group with a neighbourhood around 0.
Only used to construct a topology and uniform space.
This is currently only available for commutative groups, but it can be extended to
non-commutative groups too. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
topological_group.t1_space (h : @is_closed G _ {1}) : t1_space G | ⟨assume x, by { convert is_closed_map_mul_right x _ h, simp }⟩ | lemma | topological_group.t1_space | topology.algebra.group | src/topology/algebra/group/basic.lean | [
"group_theory.group_action.conj_act",
"group_theory.group_action.quotient",
"group_theory.quotient_group",
"topology.algebra.monoid",
"topology.algebra.constructions"
] | [
"is_closed",
"is_closed_map_mul_right",
"t1_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
topological_group.regular_space : regular_space G | begin
refine regular_space.of_exists_mem_nhds_is_closed_subset (λ a s hs, _),
have : tendsto (λ p : G × G, p.1 * p.2) (𝓝 (a, 1)) (𝓝 a),
from continuous_mul.tendsto' _ _ (mul_one a),
rcases mem_nhds_prod_iff.mp (this hs) with ⟨U, hU, V, hV, hUV⟩,
rw [← image_subset_iff, image_prod] at hUV,
refine ⟨closur... | instance | topological_group.regular_space | topology.algebra.group | src/topology/algebra/group/basic.lean | [
"group_theory.group_action.conj_act",
"group_theory.group_action.quotient",
"group_theory.quotient_group",
"topology.algebra.monoid",
"topology.algebra.constructions"
] | [
"closure",
"interior",
"interior_subset",
"is_closed_closure",
"mul_one",
"regular_space",
"regular_space.of_exists_mem_nhds_is_closed_subset",
"subset_closure"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
topological_group.t3_space [t0_space G] : t3_space G | ⟨⟩ | lemma | topological_group.t3_space | topology.algebra.group | src/topology/algebra/group/basic.lean | [
"group_theory.group_action.conj_act",
"group_theory.group_action.quotient",
"group_theory.quotient_group",
"topology.algebra.monoid",
"topology.algebra.constructions"
] | [
"t0_space",
"t3_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
topological_group.t2_space [t0_space G] : t2_space G | by { haveI := topological_group.t3_space G, apply_instance } | lemma | topological_group.t2_space | topology.algebra.group | src/topology/algebra/group/basic.lean | [
"group_theory.group_action.conj_act",
"group_theory.group_action.quotient",
"group_theory.quotient_group",
"topology.algebra.monoid",
"topology.algebra.constructions"
] | [
"t0_space",
"t2_space",
"topological_group.t3_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
subgroup.t3_quotient_of_is_closed
(S : subgroup G) [subgroup.normal S] [hS : is_closed (S : set G)] : t3_space (G ⧸ S) | begin
rw ← quotient_group.ker_mk S at hS,
haveI := topological_group.t1_space (G ⧸ S) (quotient_map_quotient_mk.is_closed_preimage.mp hS),
exact topological_group.t3_space _,
end | instance | subgroup.t3_quotient_of_is_closed | topology.algebra.group | src/topology/algebra/group/basic.lean | [
"group_theory.group_action.conj_act",
"group_theory.group_action.quotient",
"group_theory.quotient_group",
"topology.algebra.monoid",
"topology.algebra.constructions"
] | [
"is_closed",
"quotient_group.ker_mk",
"subgroup",
"subgroup.normal",
"t3_space",
"topological_group.t1_space",
"topological_group.t3_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
subgroup.properly_discontinuous_smul_of_tendsto_cofinite
(S : subgroup G) (hS : tendsto S.subtype cofinite (cocompact G)) :
properly_discontinuous_smul S G | { finite_disjoint_inter_image := begin
intros K L hK hL,
have H : set.finite _ := hS ((hL.prod hK).image continuous_div').compl_mem_cocompact,
rw [preimage_compl, compl_compl] at H,
convert H,
ext x,
simpa only [image_smul, mem_image, prod.exists] using set.smul_inter_ne_empty_iff',
end } | lemma | subgroup.properly_discontinuous_smul_of_tendsto_cofinite | topology.algebra.group | src/topology/algebra/group/basic.lean | [
"group_theory.group_action.conj_act",
"group_theory.group_action.quotient",
"group_theory.quotient_group",
"topology.algebra.monoid",
"topology.algebra.constructions"
] | [
"compl_compl",
"properly_discontinuous_smul",
"set.finite",
"set.smul_inter_ne_empty_iff'",
"subgroup"
] | A subgroup `S` of a topological group `G` acts on `G` properly discontinuously on the left, if
it is discrete in the sense that `S ∩ K` is finite for all compact `K`. (See also
`discrete_topology`.) | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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