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
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|---|---|---|---|---|---|---|---|---|---|---|
uniform_group.mk' {α} [uniform_space α] [group α]
(h₁ : uniform_continuous (λp:α×α, p.1 * p.2))
(h₂ : uniform_continuous (λp:α, p⁻¹)) : uniform_group α | ⟨by simpa only [div_eq_mul_inv] using
h₁.comp (uniform_continuous_fst.prod_mk (h₂.comp uniform_continuous_snd))⟩ | theorem | uniform_group.mk' | topology.algebra | src/topology/algebra/uniform_group.lean | [
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.uniform_embedding",
"topology.uniform_space.complete_separated",
"topology.uniform_space.compact",
"topology.algebra.group.basic",
"tactic.abel"
] | [
"div_eq_mul_inv",
"group",
"uniform_continuous",
"uniform_continuous_snd",
"uniform_group",
"uniform_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uniform_continuous_div : uniform_continuous (λp:α×α, p.1 / p.2) | uniform_group.uniform_continuous_div | lemma | uniform_continuous_div | topology.algebra | src/topology/algebra/uniform_group.lean | [
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.uniform_embedding",
"topology.uniform_space.complete_separated",
"topology.uniform_space.compact",
"topology.algebra.group.basic",
"tactic.abel"
] | [
"uniform_continuous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uniform_continuous.div [uniform_space β] {f : β → α} {g : β → α}
(hf : uniform_continuous f) (hg : uniform_continuous g) : uniform_continuous (λx, f x / g x) | uniform_continuous_div.comp (hf.prod_mk hg) | lemma | uniform_continuous.div | topology.algebra | src/topology/algebra/uniform_group.lean | [
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.uniform_embedding",
"topology.uniform_space.complete_separated",
"topology.uniform_space.compact",
"topology.algebra.group.basic",
"tactic.abel"
] | [
"uniform_continuous",
"uniform_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uniform_continuous.inv [uniform_space β] {f : β → α}
(hf : uniform_continuous f) : uniform_continuous (λx, (f x)⁻¹) | have uniform_continuous (λx, 1 / f x),
from uniform_continuous_const.div hf,
by simp * at * | lemma | uniform_continuous.inv | topology.algebra | src/topology/algebra/uniform_group.lean | [
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.uniform_embedding",
"topology.uniform_space.complete_separated",
"topology.uniform_space.compact",
"topology.algebra.group.basic",
"tactic.abel"
] | [
"uniform_continuous",
"uniform_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uniform_continuous_inv : uniform_continuous (λx:α, x⁻¹) | uniform_continuous_id.inv | lemma | uniform_continuous_inv | topology.algebra | src/topology/algebra/uniform_group.lean | [
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.uniform_embedding",
"topology.uniform_space.complete_separated",
"topology.uniform_space.compact",
"topology.algebra.group.basic",
"tactic.abel"
] | [
"uniform_continuous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uniform_continuous.mul [uniform_space β] {f : β → α} {g : β → α}
(hf : uniform_continuous f) (hg : uniform_continuous g) : uniform_continuous (λx, f x * g x) | have uniform_continuous (λx, f x / (g x)⁻¹), from hf.div hg.inv,
by simp * at * | lemma | uniform_continuous.mul | topology.algebra | src/topology/algebra/uniform_group.lean | [
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.uniform_embedding",
"topology.uniform_space.complete_separated",
"topology.uniform_space.compact",
"topology.algebra.group.basic",
"tactic.abel"
] | [
"uniform_continuous",
"uniform_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uniform_continuous_mul : uniform_continuous (λp:α×α, p.1 * p.2) | uniform_continuous_fst.mul uniform_continuous_snd | lemma | uniform_continuous_mul | topology.algebra | src/topology/algebra/uniform_group.lean | [
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.uniform_embedding",
"topology.uniform_space.complete_separated",
"topology.uniform_space.compact",
"topology.algebra.group.basic",
"tactic.abel"
] | [
"uniform_continuous",
"uniform_continuous_snd"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uniform_continuous.pow_const [uniform_space β] {f : β → α}
(hf : uniform_continuous f) : ∀ n : ℕ, uniform_continuous (λ x, f x ^ n) | | 0 := by { simp_rw pow_zero, exact uniform_continuous_const }
| (n + 1) := by { simp_rw pow_succ, exact hf.mul (uniform_continuous.pow_const n) } | lemma | uniform_continuous.pow_const | topology.algebra | src/topology/algebra/uniform_group.lean | [
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.uniform_embedding",
"topology.uniform_space.complete_separated",
"topology.uniform_space.compact",
"topology.algebra.group.basic",
"tactic.abel"
] | [
"pow_succ",
"pow_zero",
"uniform_continuous",
"uniform_continuous_const",
"uniform_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uniform_continuous_pow_const (n : ℕ) :
uniform_continuous (λx:α, x ^ n) | uniform_continuous_id.pow_const n | lemma | uniform_continuous_pow_const | topology.algebra | src/topology/algebra/uniform_group.lean | [
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.uniform_embedding",
"topology.uniform_space.complete_separated",
"topology.uniform_space.compact",
"topology.algebra.group.basic",
"tactic.abel"
] | [
"uniform_continuous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uniform_continuous.zpow_const [uniform_space β] {f : β → α}
(hf : uniform_continuous f) : ∀ n : ℤ, uniform_continuous (λ x, f x ^ n) | | (n : ℕ) := by { simp_rw zpow_coe_nat, exact hf.pow_const _, }
| -[1+ n] := by { simp_rw zpow_neg_succ_of_nat, exact (hf.pow_const _).inv } | lemma | uniform_continuous.zpow_const | topology.algebra | src/topology/algebra/uniform_group.lean | [
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.uniform_embedding",
"topology.uniform_space.complete_separated",
"topology.uniform_space.compact",
"topology.algebra.group.basic",
"tactic.abel"
] | [
"uniform_continuous",
"uniform_space",
"zpow_coe_nat",
"zpow_neg_succ_of_nat"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uniform_continuous_zpow_const (n : ℤ) :
uniform_continuous (λx:α, x ^ n) | uniform_continuous_id.zpow_const n | lemma | uniform_continuous_zpow_const | topology.algebra | src/topology/algebra/uniform_group.lean | [
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.uniform_embedding",
"topology.uniform_space.complete_separated",
"topology.uniform_space.compact",
"topology.algebra.group.basic",
"tactic.abel"
] | [
"uniform_continuous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uniform_group.to_topological_group : topological_group α | { continuous_mul := uniform_continuous_mul.continuous,
continuous_inv := uniform_continuous_inv.continuous } | instance | uniform_group.to_topological_group | topology.algebra | src/topology/algebra/uniform_group.lean | [
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.uniform_embedding",
"topology.uniform_space.complete_separated",
"topology.uniform_space.compact",
"topology.algebra.group.basic",
"tactic.abel"
] | [
"continuous_mul",
"topological_group"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uniformity_translate_mul (a : α) :
(𝓤 α).map (λx:α×α, (x.1 * a, x.2 * a)) = 𝓤 α | le_antisymm
(uniform_continuous_id.mul uniform_continuous_const)
(calc 𝓤 α =
((𝓤 α).map (λx:α×α, (x.1 * a⁻¹, x.2 * a⁻¹))).map (λx:α×α, (x.1 * a, x.2 * a)) :
by simp [filter.map_map, (∘)]; exact filter.map_id.symm
... ≤ (𝓤 α).map (λx:α×α, (x.1 * a, x.2 * a)) :
filter.map_mono (uniform_continuo... | lemma | uniformity_translate_mul | topology.algebra | src/topology/algebra/uniform_group.lean | [
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.uniform_embedding",
"topology.uniform_space.complete_separated",
"topology.uniform_space.compact",
"topology.algebra.group.basic",
"tactic.abel"
] | [
"filter.map_map",
"filter.map_mono",
"uniform_continuous_const"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uniform_embedding_translate_mul (a : α) : uniform_embedding (λx:α, x * a) | { comap_uniformity := begin
rw [← uniformity_translate_mul a, comap_map] {occs := occurrences.pos [1]},
rintros ⟨p₁, p₂⟩ ⟨q₁, q₂⟩,
simp [prod.eq_iff_fst_eq_snd_eq] {contextual := tt}
end,
inj := mul_left_injective a } | lemma | uniform_embedding_translate_mul | topology.algebra | src/topology/algebra/uniform_group.lean | [
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.uniform_embedding",
"topology.uniform_space.complete_separated",
"topology.uniform_space.compact",
"topology.algebra.group.basic",
"tactic.abel"
] | [
"mul_left_injective",
"prod.eq_iff_fst_eq_snd_eq",
"uniform_embedding",
"uniformity_translate_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uniform_group_Inf {us : set (uniform_space β)}
(h : ∀ u ∈ us, @uniform_group β u _) :
@uniform_group β (Inf us) _ | { uniform_continuous_div := uniform_continuous_Inf_rng (λ u hu, uniform_continuous_Inf_dom₂ hu hu
(@uniform_group.uniform_continuous_div β u _ (h u hu))) } | lemma | uniform_group_Inf | topology.algebra | src/topology/algebra/uniform_group.lean | [
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.uniform_embedding",
"topology.uniform_space.complete_separated",
"topology.uniform_space.compact",
"topology.algebra.group.basic",
"tactic.abel"
] | [
"uniform_continuous_Inf_dom₂",
"uniform_continuous_Inf_rng",
"uniform_continuous_div",
"uniform_group",
"uniform_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uniform_group_infi {ι : Sort*} {us' : ι → uniform_space β}
(h' : ∀ i, @uniform_group β (us' i) _) :
@uniform_group β (⨅ i, us' i) _ | by {rw ← Inf_range, exact uniform_group_Inf (set.forall_range_iff.mpr h')} | lemma | uniform_group_infi | topology.algebra | src/topology/algebra/uniform_group.lean | [
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.uniform_embedding",
"topology.uniform_space.complete_separated",
"topology.uniform_space.compact",
"topology.algebra.group.basic",
"tactic.abel"
] | [
"Inf_range",
"uniform_group",
"uniform_group_Inf",
"uniform_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uniform_group_inf {u₁ u₂ : uniform_space β}
(h₁ : @uniform_group β u₁ _) (h₂ : @uniform_group β u₂ _) :
@uniform_group β (u₁ ⊓ u₂) _ | by {rw inf_eq_infi, refine uniform_group_infi (λ b, _), cases b; assumption} | lemma | uniform_group_inf | topology.algebra | src/topology/algebra/uniform_group.lean | [
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.uniform_embedding",
"topology.uniform_space.complete_separated",
"topology.uniform_space.compact",
"topology.algebra.group.basic",
"tactic.abel"
] | [
"inf_eq_infi",
"uniform_group",
"uniform_group_infi",
"uniform_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uniform_group_comap {γ : Type*} [group γ] {u : uniform_space γ}
[uniform_group γ] {F : Type*} [monoid_hom_class F β γ] (f : F) :
@uniform_group β (u.comap f) _ | { uniform_continuous_div :=
begin
letI : uniform_space β := u.comap f,
refine uniform_continuous_comap' _,
simp_rw [function.comp, map_div],
change uniform_continuous ((λ p : γ × γ, p.1 / p.2) ∘ (prod.map f f)),
exact uniform_continuous_div.comp
(uniform_continuous_comap.prod_m... | lemma | uniform_group_comap | topology.algebra | src/topology/algebra/uniform_group.lean | [
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.uniform_embedding",
"topology.uniform_space.complete_separated",
"topology.uniform_space.compact",
"topology.algebra.group.basic",
"tactic.abel"
] | [
"group",
"map_div",
"monoid_hom_class",
"uniform_continuous",
"uniform_continuous_comap",
"uniform_continuous_comap'",
"uniform_continuous_div",
"uniform_group",
"uniform_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uniformity_eq_comap_nhds_one : 𝓤 α = comap (λx:α×α, x.2 / x.1) (𝓝 (1:α)) | begin
rw [nhds_eq_comap_uniformity, filter.comap_comap],
refine le_antisymm (filter.map_le_iff_le_comap.1 _) _,
{ assume s hs,
rcases mem_uniformity_of_uniform_continuous_invariant uniform_continuous_div hs
with ⟨t, ht, hts⟩,
refine mem_map.2 (mem_of_superset ht _),
rintros ⟨a, b⟩,
simpa [su... | lemma | uniformity_eq_comap_nhds_one | topology.algebra | src/topology/algebra/uniform_group.lean | [
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.uniform_embedding",
"topology.uniform_space.complete_separated",
"topology.uniform_space.compact",
"topology.algebra.group.basic",
"tactic.abel"
] | [
"filter.comap_comap",
"mem_uniformity_of_uniform_continuous_invariant",
"nhds_eq_comap_uniformity",
"uniform_continuous_div",
"uniform_continuous_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uniformity_eq_comap_nhds_one_swapped :
𝓤 α = comap (λx:α×α, x.1 / x.2) (𝓝 (1:α)) | by { rw [← comap_swap_uniformity, uniformity_eq_comap_nhds_one, comap_comap, (∘)], refl } | lemma | uniformity_eq_comap_nhds_one_swapped | topology.algebra | src/topology/algebra/uniform_group.lean | [
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.uniform_embedding",
"topology.uniform_space.complete_separated",
"topology.uniform_space.compact",
"topology.algebra.group.basic",
"tactic.abel"
] | [
"comap_swap_uniformity",
"uniformity_eq_comap_nhds_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uniform_group.ext {G : Type*} [group G] {u v : uniform_space G}
(hu : @uniform_group G u _) (hv : @uniform_group G v _)
(h : @nhds _ u.to_topological_space 1 = @nhds _ v.to_topological_space 1) :
u = v | uniform_space_eq $
by rw [@uniformity_eq_comap_nhds_one _ u _ hu, @uniformity_eq_comap_nhds_one _ v _ hv, h] | lemma | uniform_group.ext | topology.algebra | src/topology/algebra/uniform_group.lean | [
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.uniform_embedding",
"topology.uniform_space.complete_separated",
"topology.uniform_space.compact",
"topology.algebra.group.basic",
"tactic.abel"
] | [
"group",
"nhds",
"uniform_group",
"uniform_space",
"uniform_space_eq",
"uniformity_eq_comap_nhds_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uniform_group.ext_iff {G : Type*} [group G] {u v : uniform_space G}
(hu : @uniform_group G u _) (hv : @uniform_group G v _) :
u = v ↔ @nhds _ u.to_topological_space 1 = @nhds _ v.to_topological_space 1 | ⟨λ h, h ▸ rfl, hu.ext hv⟩ | lemma | uniform_group.ext_iff | topology.algebra | src/topology/algebra/uniform_group.lean | [
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.uniform_embedding",
"topology.uniform_space.complete_separated",
"topology.uniform_space.compact",
"topology.algebra.group.basic",
"tactic.abel"
] | [
"group",
"nhds",
"uniform_group",
"uniform_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uniform_group.uniformity_countably_generated
[(𝓝 (1 : α)).is_countably_generated] :
(𝓤 α).is_countably_generated | by { rw uniformity_eq_comap_nhds_one, exact filter.comap.is_countably_generated _ _ } | theorem | uniform_group.uniformity_countably_generated | topology.algebra | src/topology/algebra/uniform_group.lean | [
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.uniform_embedding",
"topology.uniform_space.complete_separated",
"topology.uniform_space.compact",
"topology.algebra.group.basic",
"tactic.abel"
] | [
"filter.comap.is_countably_generated",
"uniformity_eq_comap_nhds_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uniformity_eq_comap_inv_mul_nhds_one : 𝓤 α = comap (λx:α×α, x.1⁻¹ * x.2) (𝓝 (1:α)) | begin
rw [← comap_uniformity_mul_opposite, uniformity_eq_comap_nhds_one, ← op_one, ← comap_unop_nhds,
comap_comap, comap_comap],
simp [(∘)]
end | lemma | uniformity_eq_comap_inv_mul_nhds_one | topology.algebra | src/topology/algebra/uniform_group.lean | [
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.uniform_embedding",
"topology.uniform_space.complete_separated",
"topology.uniform_space.compact",
"topology.algebra.group.basic",
"tactic.abel"
] | [
"comap_uniformity_mul_opposite",
"uniformity_eq_comap_nhds_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uniformity_eq_comap_inv_mul_nhds_one_swapped :
𝓤 α = comap (λx:α×α, x.2⁻¹ * x.1) (𝓝 (1:α)) | by { rw [← comap_swap_uniformity, uniformity_eq_comap_inv_mul_nhds_one, comap_comap, (∘)], refl } | lemma | uniformity_eq_comap_inv_mul_nhds_one_swapped | topology.algebra | src/topology/algebra/uniform_group.lean | [
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.uniform_embedding",
"topology.uniform_space.complete_separated",
"topology.uniform_space.compact",
"topology.algebra.group.basic",
"tactic.abel"
] | [
"comap_swap_uniformity",
"uniformity_eq_comap_inv_mul_nhds_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
filter.has_basis.uniformity_of_nhds_one {ι} {p : ι → Prop} {U : ι → set α}
(h : (𝓝 (1 : α)).has_basis p U) :
(𝓤 α).has_basis p (λ i, {x : α × α | x.2 / x.1 ∈ U i}) | by { rw uniformity_eq_comap_nhds_one, exact h.comap _ } | lemma | filter.has_basis.uniformity_of_nhds_one | topology.algebra | src/topology/algebra/uniform_group.lean | [
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.uniform_embedding",
"topology.uniform_space.complete_separated",
"topology.uniform_space.compact",
"topology.algebra.group.basic",
"tactic.abel"
] | [
"uniformity_eq_comap_nhds_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
filter.has_basis.uniformity_of_nhds_one_inv_mul
{ι} {p : ι → Prop} {U : ι → set α} (h : (𝓝 (1 : α)).has_basis p U) :
(𝓤 α).has_basis p (λ i, {x : α × α | x.1⁻¹ * x.2 ∈ U i}) | by { rw uniformity_eq_comap_inv_mul_nhds_one, exact h.comap _ } | lemma | filter.has_basis.uniformity_of_nhds_one_inv_mul | topology.algebra | src/topology/algebra/uniform_group.lean | [
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.uniform_embedding",
"topology.uniform_space.complete_separated",
"topology.uniform_space.compact",
"topology.algebra.group.basic",
"tactic.abel"
] | [
"uniformity_eq_comap_inv_mul_nhds_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
filter.has_basis.uniformity_of_nhds_one_swapped
{ι} {p : ι → Prop} {U : ι → set α} (h : (𝓝 (1 : α)).has_basis p U) :
(𝓤 α).has_basis p (λ i, {x : α × α | x.1 / x.2 ∈ U i}) | by { rw uniformity_eq_comap_nhds_one_swapped, exact h.comap _ } | lemma | filter.has_basis.uniformity_of_nhds_one_swapped | topology.algebra | src/topology/algebra/uniform_group.lean | [
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.uniform_embedding",
"topology.uniform_space.complete_separated",
"topology.uniform_space.compact",
"topology.algebra.group.basic",
"tactic.abel"
] | [
"uniformity_eq_comap_nhds_one_swapped"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
filter.has_basis.uniformity_of_nhds_one_inv_mul_swapped
{ι} {p : ι → Prop} {U : ι → set α} (h : (𝓝 (1 : α)).has_basis p U) :
(𝓤 α).has_basis p (λ i, {x : α × α | x.2⁻¹ * x.1 ∈ U i}) | by { rw uniformity_eq_comap_inv_mul_nhds_one_swapped, exact h.comap _ } | lemma | filter.has_basis.uniformity_of_nhds_one_inv_mul_swapped | topology.algebra | src/topology/algebra/uniform_group.lean | [
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.uniform_embedding",
"topology.uniform_space.complete_separated",
"topology.uniform_space.compact",
"topology.algebra.group.basic",
"tactic.abel"
] | [
"uniformity_eq_comap_inv_mul_nhds_one_swapped"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
group_separation_rel (x y : α) :
(x, y) ∈ separation_rel α ↔ x / y ∈ closure ({1} : set α) | have embedding (λa, a * (y / x)), from (uniform_embedding_translate_mul (y / x)).embedding,
show (x, y) ∈ ⋂₀ (𝓤 α).sets ↔ x / y ∈ closure ({1} : set α),
begin
rw [this.closure_eq_preimage_closure_image, uniformity_eq_comap_nhds_one α, sInter_comap_sets],
simp [mem_closure_iff_nhds, inter_singleton_nonempty, sub_eq... | lemma | group_separation_rel | topology.algebra | src/topology/algebra/uniform_group.lean | [
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.uniform_embedding",
"topology.uniform_space.complete_separated",
"topology.uniform_space.compact",
"topology.algebra.group.basic",
"tactic.abel"
] | [
"closure",
"embedding",
"mem_closure_iff_nhds",
"separation_rel",
"uniform_embedding_translate_mul",
"uniformity_eq_comap_nhds_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uniform_continuous_of_tendsto_one {hom : Type*} [uniform_space β] [group β]
[uniform_group β] [monoid_hom_class hom α β] {f : hom} (h : tendsto f (𝓝 1) (𝓝 1)) :
uniform_continuous f | begin
have : ((λx:β×β, x.2 / x.1) ∘ (λx:α×α, (f x.1, f x.2))) = (λx:α×α, f (x.2 / x.1)),
{ simp only [map_div] },
rw [uniform_continuous, uniformity_eq_comap_nhds_one α, uniformity_eq_comap_nhds_one β,
tendsto_comap_iff, this],
exact tendsto.comp h tendsto_comap
end | lemma | uniform_continuous_of_tendsto_one | topology.algebra | src/topology/algebra/uniform_group.lean | [
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.uniform_embedding",
"topology.uniform_space.complete_separated",
"topology.uniform_space.compact",
"topology.algebra.group.basic",
"tactic.abel"
] | [
"group",
"map_div",
"monoid_hom_class",
"uniform_continuous",
"uniform_group",
"uniform_space",
"uniformity_eq_comap_nhds_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uniform_continuous_of_continuous_at_one {hom : Type*}
[uniform_space β] [group β] [uniform_group β] [monoid_hom_class hom α β]
(f : hom) (hf : continuous_at f 1) :
uniform_continuous f | uniform_continuous_of_tendsto_one (by simpa using hf.tendsto) | lemma | uniform_continuous_of_continuous_at_one | topology.algebra | src/topology/algebra/uniform_group.lean | [
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.uniform_embedding",
"topology.uniform_space.complete_separated",
"topology.uniform_space.compact",
"topology.algebra.group.basic",
"tactic.abel"
] | [
"continuous_at",
"group",
"monoid_hom_class",
"uniform_continuous",
"uniform_continuous_of_tendsto_one",
"uniform_group",
"uniform_space"
] | A group homomorphism (a bundled morphism of a type that implements `monoid_hom_class`) between
two uniform groups is uniformly continuous provided that it is continuous at one. See also
`continuous_of_continuous_at_one`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
monoid_hom.uniform_continuous_of_continuous_at_one
[uniform_space β] [group β] [uniform_group β]
(f : α →* β) (hf : continuous_at f 1) :
uniform_continuous f | uniform_continuous_of_continuous_at_one f hf | lemma | monoid_hom.uniform_continuous_of_continuous_at_one | topology.algebra | src/topology/algebra/uniform_group.lean | [
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.uniform_embedding",
"topology.uniform_space.complete_separated",
"topology.uniform_space.compact",
"topology.algebra.group.basic",
"tactic.abel"
] | [
"continuous_at",
"group",
"uniform_continuous",
"uniform_continuous_of_continuous_at_one",
"uniform_group",
"uniform_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uniform_group.uniform_continuous_iff_open_ker {hom : Type*} [uniform_space β]
[discrete_topology β] [group β] [uniform_group β] [monoid_hom_class hom α β] {f : hom} :
uniform_continuous f ↔ is_open ((f : α →* β).ker : set α) | begin
refine ⟨λ hf, _, λ hf, _⟩,
{ apply (is_open_discrete ({1} : set β)).preimage (uniform_continuous.continuous hf) },
{ apply uniform_continuous_of_continuous_at_one,
rw [continuous_at, nhds_discrete β, map_one, tendsto_pure],
exact hf.mem_nhds (map_one f) }
end | lemma | uniform_group.uniform_continuous_iff_open_ker | topology.algebra | src/topology/algebra/uniform_group.lean | [
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.uniform_embedding",
"topology.uniform_space.complete_separated",
"topology.uniform_space.compact",
"topology.algebra.group.basic",
"tactic.abel"
] | [
"continuous_at",
"discrete_topology",
"group",
"is_open",
"is_open_discrete",
"map_one",
"monoid_hom_class",
"nhds_discrete",
"uniform_continuous",
"uniform_continuous.continuous",
"uniform_continuous_of_continuous_at_one",
"uniform_group",
"uniform_space"
] | A homomorphism from a uniform group to a discrete uniform group is continuous if and only if
its kernel is open. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
uniform_continuous_monoid_hom_of_continuous {hom : Type*} [uniform_space β]
[group β] [uniform_group β] [monoid_hom_class hom α β] {f : hom} (h : continuous f) :
uniform_continuous f | uniform_continuous_of_tendsto_one $
suffices tendsto f (𝓝 1) (𝓝 (f 1)), by rwa map_one at this,
h.tendsto 1 | lemma | uniform_continuous_monoid_hom_of_continuous | topology.algebra | src/topology/algebra/uniform_group.lean | [
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.uniform_embedding",
"topology.uniform_space.complete_separated",
"topology.uniform_space.compact",
"topology.algebra.group.basic",
"tactic.abel"
] | [
"continuous",
"group",
"map_one",
"monoid_hom_class",
"uniform_continuous",
"uniform_continuous_of_tendsto_one",
"uniform_group",
"uniform_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cauchy_seq.mul {ι : Type*} [semilattice_sup ι] {u v : ι → α}
(hu : cauchy_seq u) (hv : cauchy_seq v) : cauchy_seq (u * v) | uniform_continuous_mul.comp_cauchy_seq (hu.prod hv) | lemma | cauchy_seq.mul | topology.algebra | src/topology/algebra/uniform_group.lean | [
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.uniform_embedding",
"topology.uniform_space.complete_separated",
"topology.uniform_space.compact",
"topology.algebra.group.basic",
"tactic.abel"
] | [
"cauchy_seq",
"semilattice_sup"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cauchy_seq.mul_const {ι : Type*} [semilattice_sup ι]
{u : ι → α} {x : α} (hu : cauchy_seq u) : cauchy_seq (λ n, u n * x) | (uniform_continuous_id.mul uniform_continuous_const).comp_cauchy_seq hu | lemma | cauchy_seq.mul_const | topology.algebra | src/topology/algebra/uniform_group.lean | [
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.uniform_embedding",
"topology.uniform_space.complete_separated",
"topology.uniform_space.compact",
"topology.algebra.group.basic",
"tactic.abel"
] | [
"cauchy_seq",
"semilattice_sup",
"uniform_continuous_const"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cauchy_seq.const_mul {ι : Type*} [semilattice_sup ι]
{u : ι → α} {x : α} (hu : cauchy_seq u) : cauchy_seq (λ n, x * u n) | (uniform_continuous_const.mul uniform_continuous_id).comp_cauchy_seq hu | lemma | cauchy_seq.const_mul | topology.algebra | src/topology/algebra/uniform_group.lean | [
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.uniform_embedding",
"topology.uniform_space.complete_separated",
"topology.uniform_space.compact",
"topology.algebra.group.basic",
"tactic.abel"
] | [
"cauchy_seq",
"semilattice_sup",
"uniform_continuous_id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cauchy_seq.inv {ι : Type*} [semilattice_sup ι]
{u : ι → α} (h : cauchy_seq u) : cauchy_seq (u⁻¹) | uniform_continuous_inv.comp_cauchy_seq h | lemma | cauchy_seq.inv | topology.algebra | src/topology/algebra/uniform_group.lean | [
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.uniform_embedding",
"topology.uniform_space.complete_separated",
"topology.uniform_space.compact",
"topology.algebra.group.basic",
"tactic.abel"
] | [
"cauchy_seq",
"semilattice_sup"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
totally_bounded_iff_subset_finite_Union_nhds_one {s : set α} :
totally_bounded s ↔ ∀ U ∈ 𝓝 (1 : α), ∃ (t : set α), t.finite ∧ s ⊆ ⋃ y ∈ t, y • U | (𝓝 (1 : α)).basis_sets.uniformity_of_nhds_one_inv_mul_swapped.totally_bounded_iff.trans $
by simp [← preimage_smul_inv, preimage] | lemma | totally_bounded_iff_subset_finite_Union_nhds_one | topology.algebra | src/topology/algebra/uniform_group.lean | [
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.uniform_embedding",
"topology.uniform_space.complete_separated",
"topology.uniform_space.compact",
"topology.algebra.group.basic",
"tactic.abel"
] | [
"totally_bounded"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_uniformly_on_filter.mul (hf : tendsto_uniformly_on_filter f g l l')
(hf' : tendsto_uniformly_on_filter f' g' l l') :
tendsto_uniformly_on_filter (f * f') (g * g') l l' | λ u hu, ((uniform_continuous_mul.comp_tendsto_uniformly_on_filter
(hf.prod hf')) u hu).diag_of_prod_left | lemma | tendsto_uniformly_on_filter.mul | topology.algebra | src/topology/algebra/uniform_group.lean | [
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.uniform_embedding",
"topology.uniform_space.complete_separated",
"topology.uniform_space.compact",
"topology.algebra.group.basic",
"tactic.abel"
] | [
"tendsto_uniformly_on_filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_uniformly_on_filter.div (hf : tendsto_uniformly_on_filter f g l l')
(hf' : tendsto_uniformly_on_filter f' g' l l') :
tendsto_uniformly_on_filter (f / f') (g / g') l l' | λ u hu, ((uniform_continuous_div.comp_tendsto_uniformly_on_filter
(hf.prod hf')) u hu).diag_of_prod_left | lemma | tendsto_uniformly_on_filter.div | topology.algebra | src/topology/algebra/uniform_group.lean | [
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.uniform_embedding",
"topology.uniform_space.complete_separated",
"topology.uniform_space.compact",
"topology.algebra.group.basic",
"tactic.abel"
] | [
"tendsto_uniformly_on_filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_uniformly_on.mul (hf : tendsto_uniformly_on f g l s)
(hf' : tendsto_uniformly_on f' g' l s) : tendsto_uniformly_on (f * f') (g * g') l s | λ u hu, ((uniform_continuous_mul.comp_tendsto_uniformly_on (hf.prod hf')) u hu).diag_of_prod | lemma | tendsto_uniformly_on.mul | topology.algebra | src/topology/algebra/uniform_group.lean | [
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.uniform_embedding",
"topology.uniform_space.complete_separated",
"topology.uniform_space.compact",
"topology.algebra.group.basic",
"tactic.abel"
] | [
"tendsto_uniformly_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_uniformly_on.div (hf : tendsto_uniformly_on f g l s)
(hf' : tendsto_uniformly_on f' g' l s) : tendsto_uniformly_on (f / f') (g / g') l s | λ u hu, ((uniform_continuous_div.comp_tendsto_uniformly_on (hf.prod hf')) u hu).diag_of_prod | lemma | tendsto_uniformly_on.div | topology.algebra | src/topology/algebra/uniform_group.lean | [
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.uniform_embedding",
"topology.uniform_space.complete_separated",
"topology.uniform_space.compact",
"topology.algebra.group.basic",
"tactic.abel"
] | [
"tendsto_uniformly_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_uniformly.mul (hf : tendsto_uniformly f g l)
(hf' : tendsto_uniformly f' g' l) : tendsto_uniformly (f * f') (g * g') l | λ u hu, ((uniform_continuous_mul.comp_tendsto_uniformly (hf.prod hf')) u hu).diag_of_prod | lemma | tendsto_uniformly.mul | topology.algebra | src/topology/algebra/uniform_group.lean | [
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.uniform_embedding",
"topology.uniform_space.complete_separated",
"topology.uniform_space.compact",
"topology.algebra.group.basic",
"tactic.abel"
] | [
"tendsto_uniformly"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_uniformly.div (hf : tendsto_uniformly f g l)
(hf' : tendsto_uniformly f' g' l) : tendsto_uniformly (f / f') (g / g') l | λ u hu, ((uniform_continuous_div.comp_tendsto_uniformly (hf.prod hf')) u hu).diag_of_prod | lemma | tendsto_uniformly.div | topology.algebra | src/topology/algebra/uniform_group.lean | [
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.uniform_embedding",
"topology.uniform_space.complete_separated",
"topology.uniform_space.compact",
"topology.algebra.group.basic",
"tactic.abel"
] | [
"tendsto_uniformly"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uniform_cauchy_seq_on.mul (hf : uniform_cauchy_seq_on f l s)
(hf' : uniform_cauchy_seq_on f' l s) : uniform_cauchy_seq_on (f * f') l s | λ u hu, by simpa using ((uniform_continuous_mul.comp_uniform_cauchy_seq_on (hf.prod' hf')) u hu) | lemma | uniform_cauchy_seq_on.mul | topology.algebra | src/topology/algebra/uniform_group.lean | [
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.uniform_embedding",
"topology.uniform_space.complete_separated",
"topology.uniform_space.compact",
"topology.algebra.group.basic",
"tactic.abel"
] | [
"uniform_cauchy_seq_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uniform_cauchy_seq_on.div (hf : uniform_cauchy_seq_on f l s)
(hf' : uniform_cauchy_seq_on f' l s) : uniform_cauchy_seq_on (f / f') l s | λ u hu, by simpa using ((uniform_continuous_div.comp_uniform_cauchy_seq_on (hf.prod' hf')) u hu) | lemma | uniform_cauchy_seq_on.div | topology.algebra | src/topology/algebra/uniform_group.lean | [
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.uniform_embedding",
"topology.uniform_space.complete_separated",
"topology.uniform_space.compact",
"topology.algebra.group.basic",
"tactic.abel"
] | [
"uniform_cauchy_seq_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
topological_group.to_uniform_space : uniform_space G | { uniformity := comap (λp:G×G, p.2 / p.1) (𝓝 1),
refl :=
by refine map_le_iff_le_comap.1 (le_trans _ (pure_le_nhds 1));
simp [set.subset_def] {contextual := tt},
symm :=
begin
suffices : tendsto (λp:G×G, (p.2 / p.1)⁻¹) (comap (λp:G×G, p.2 / p.1) (𝓝 1)) (𝓝 1⁻... | def | topological_group.to_uniform_space | topology.algebra | src/topology/algebra/uniform_group.lean | [
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.uniform_embedding",
"topology.uniform_space.complete_separated",
"topology.uniform_space.compact",
"topology.algebra.group.basic",
"tactic.abel"
] | [
"comp_rel",
"exists_nhds_one_split",
"exists₂_congr",
"forall₂_congr",
"is_open",
"is_open_iff_mem_nhds",
"is_open_uniformity",
"monotone_id",
"nhds_translation_div",
"pure_le_nhds",
"set.subset.trans",
"set.subset_def",
"uniform_space",
"uniformity"
] | The right uniformity on a topological group (as opposed to the left uniformity).
Warning: in general the right and left uniformities do not coincide and so one does not obtain a
`uniform_group` structure. Two important special cases where they _do_ coincide are for
commutative groups (see `topological_comm_group_is_un... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
uniformity_eq_comap_nhds_one' :
𝓤 G = comap (λp:G×G, p.2 / p.1) (𝓝 (1 : G)) | rfl | lemma | uniformity_eq_comap_nhds_one' | topology.algebra | src/topology/algebra/uniform_group.lean | [
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.uniform_embedding",
"topology.uniform_space.complete_separated",
"topology.uniform_space.compact",
"topology.algebra.group.basic",
"tactic.abel"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
topological_group_is_uniform_of_compact_space
[compact_space G] : uniform_group G | ⟨begin
apply compact_space.uniform_continuous_of_continuous,
exact continuous_div',
end⟩ | lemma | topological_group_is_uniform_of_compact_space | topology.algebra | src/topology/algebra/uniform_group.lean | [
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.uniform_embedding",
"topology.uniform_space.complete_separated",
"topology.uniform_space.compact",
"topology.algebra.group.basic",
"tactic.abel"
] | [
"compact_space",
"compact_space.uniform_continuous_of_continuous",
"uniform_group"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
subgroup.is_closed_of_discrete [t2_space G]
{H : subgroup G} [discrete_topology H] : is_closed (H : set G) | begin
obtain ⟨V, V_in, VH⟩ : ∃ (V : set G) (hV : V ∈ 𝓝 (1 : G)), V ∩ (H : set G) = {1},
from nhds_inter_eq_singleton_of_mem_discrete H.one_mem,
haveI : separated_space G := separated_iff_t2.mpr ‹_›,
have : (λ p : G × G, p.2 / p.1) ⁻¹' V ∈ 𝓤 G, from preimage_mem_comap V_in,
apply is_closed_of_spaced_out th... | instance | subgroup.is_closed_of_discrete | topology.algebra | src/topology/algebra/uniform_group.lean | [
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.uniform_embedding",
"topology.uniform_space.complete_separated",
"topology.uniform_space.compact",
"topology.algebra.group.basic",
"tactic.abel"
] | [
"discrete_topology",
"eq_of_div_eq_one",
"is_closed",
"is_closed_of_spaced_out",
"nhds_inter_eq_singleton_of_mem_discrete",
"separated_space",
"set.mem_inter",
"subgroup",
"t2_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
topological_group.tendsto_uniformly_iff
{ι α : Type*} (F : ι → α → G) (f : α → G) (p : filter ι) :
@tendsto_uniformly α G ι (topological_group.to_uniform_space G) F f p
↔ ∀ u ∈ 𝓝 (1 : G), ∀ᶠ i in p, ∀ a, F i a / f a ∈ u | ⟨λ h u hu, h _ ⟨u, hu, λ _, id⟩, λ h v ⟨u, hu, hv⟩,
mem_of_superset (h u hu) (λ i hi a, hv (by exact hi a))⟩ | lemma | topological_group.tendsto_uniformly_iff | topology.algebra | src/topology/algebra/uniform_group.lean | [
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.uniform_embedding",
"topology.uniform_space.complete_separated",
"topology.uniform_space.compact",
"topology.algebra.group.basic",
"tactic.abel"
] | [
"filter",
"tendsto_uniformly",
"topological_group.to_uniform_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
topological_group.tendsto_uniformly_on_iff
{ι α : Type*} (F : ι → α → G) (f : α → G) (p : filter ι) (s : set α) :
@tendsto_uniformly_on α G ι (topological_group.to_uniform_space G) F f p s
↔ ∀ u ∈ 𝓝 (1 : G), ∀ᶠ i in p, ∀ a ∈ s, F i a / f a ∈ u | ⟨λ h u hu, h _ ⟨u, hu, λ _, id⟩, λ h v ⟨u, hu, hv⟩,
mem_of_superset (h u hu) (λ i hi a ha, hv (by exact hi a ha))⟩ | lemma | topological_group.tendsto_uniformly_on_iff | topology.algebra | src/topology/algebra/uniform_group.lean | [
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.uniform_embedding",
"topology.uniform_space.complete_separated",
"topology.uniform_space.compact",
"topology.algebra.group.basic",
"tactic.abel"
] | [
"filter",
"tendsto_uniformly_on",
"topological_group.to_uniform_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
topological_group.tendsto_locally_uniformly_iff
{ι α : Type*} [topological_space α] (F : ι → α → G) (f : α → G) (p : filter ι) :
@tendsto_locally_uniformly α G ι (topological_group.to_uniform_space G) _ F f p
↔ ∀ (u ∈ 𝓝 (1 : G)) (x : α), ∃ (t ∈ 𝓝 x), ∀ᶠ i in p, ∀ a ∈ t, F i a / f a ∈ u | ⟨λ h u hu, h _ ⟨u, hu, λ _, id⟩, λ h v ⟨u, hu, hv⟩ x, exists_imp_exists (by exact λ a,
exists_imp_exists (λ ha hp, mem_of_superset hp (λ i hi a ha, hv (by exact hi a ha)))) (h u hu x)⟩ | lemma | topological_group.tendsto_locally_uniformly_iff | topology.algebra | src/topology/algebra/uniform_group.lean | [
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.uniform_embedding",
"topology.uniform_space.complete_separated",
"topology.uniform_space.compact",
"topology.algebra.group.basic",
"tactic.abel"
] | [
"filter",
"tendsto_locally_uniformly",
"topological_group.to_uniform_space",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
topological_group.tendsto_locally_uniformly_on_iff
{ι α : Type*} [topological_space α] (F : ι → α → G) (f : α → G) (p : filter ι) (s : set α) :
@tendsto_locally_uniformly_on α G ι (topological_group.to_uniform_space G) _ F f p s
↔ ∀ (u ∈ 𝓝 (1 : G)) (x ∈ s), ∃ (t ∈ 𝓝[s] x), ∀ᶠ i in p, ∀ a ∈ t, F i a / f a ∈ u | ⟨λ h u hu, h _ ⟨u, hu, λ _, id⟩, λ h v ⟨u, hu, hv⟩ x, exists_imp_exists (by exact λ a,
exists_imp_exists (λ ha hp, mem_of_superset hp (λ i hi a ha, hv (by exact hi a ha)))) ∘ h u hu x⟩ | lemma | topological_group.tendsto_locally_uniformly_on_iff | topology.algebra | src/topology/algebra/uniform_group.lean | [
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.uniform_embedding",
"topology.uniform_space.complete_separated",
"topology.uniform_space.compact",
"topology.algebra.group.basic",
"tactic.abel"
] | [
"filter",
"tendsto_locally_uniformly_on",
"topological_group.to_uniform_space",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
topological_comm_group_is_uniform : uniform_group G | have tendsto
((λp:(G×G), p.1 / p.2) ∘ (λp:(G×G)×(G×G), (p.1.2 / p.1.1, p.2.2 / p.2.1)))
(comap (λp:(G×G)×(G×G), (p.1.2 / p.1.1, p.2.2 / p.2.1)) ((𝓝 1).prod (𝓝 1)))
(𝓝 (1 / 1)) :=
(tendsto_fst.div' tendsto_snd).comp tendsto_comap,
begin
constructor,
rw [uniform_continuous, uniformity_prod_eq_prod, t... | lemma | topological_comm_group_is_uniform | topology.algebra | src/topology/algebra/uniform_group.lean | [
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.uniform_embedding",
"topology.uniform_space.complete_separated",
"topology.uniform_space.compact",
"topology.algebra.group.basic",
"tactic.abel"
] | [
"div_eq_mul_inv",
"mul_comm",
"mul_left_comm",
"uniform_continuous",
"uniform_group",
"uniformity_eq_comap_nhds_one'",
"uniformity_prod_eq_prod"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
topological_group.t2_space_iff_one_closed :
t2_space G ↔ is_closed ({1} : set G) | begin
haveI : uniform_group G := topological_comm_group_is_uniform,
rw [← separated_iff_t2, separated_space_iff, ← closure_eq_iff_is_closed],
split; intro h,
{ apply subset.antisymm,
{ intros x x_in,
have := group_separation_rel x 1,
rw div_one at this,
rw [← this, h] at x_in,
change... | lemma | topological_group.t2_space_iff_one_closed | topology.algebra | src/topology/algebra/uniform_group.lean | [
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.uniform_embedding",
"topology.uniform_space.complete_separated",
"topology.uniform_space.compact",
"topology.algebra.group.basic",
"tactic.abel"
] | [
"closure_eq_iff_is_closed",
"div_eq_one",
"div_one",
"group_separation_rel",
"is_closed",
"separated_iff_t2",
"separated_space_iff",
"subset_closure",
"t2_space",
"topological_comm_group_is_uniform",
"uniform_group"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
topological_group.t2_space_of_one_sep
(H : ∀ x : G, x ≠ 1 → ∃ U ∈ nhds (1 : G), x ∉ U) : t2_space G | begin
rw [topological_group.t2_space_iff_one_closed, ← is_open_compl_iff, is_open_iff_mem_nhds],
intros x x_not,
have : x ≠ 1, from mem_compl_singleton_iff.mp x_not,
rcases H x this with ⟨U, U_in, xU⟩,
rw ← nhds_one_symm G at U_in,
rcases U_in with ⟨W, W_in, UW⟩,
rw ← nhds_translation_mul_inv,
use [W, W... | lemma | topological_group.t2_space_of_one_sep | topology.algebra | src/topology/algebra/uniform_group.lean | [
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.uniform_embedding",
"topology.uniform_space.complete_separated",
"topology.uniform_space.compact",
"topology.algebra.group.basic",
"tactic.abel"
] | [
"is_open_compl_iff",
"is_open_iff_mem_nhds",
"nhds",
"nhds_one_symm",
"nhds_translation_mul_inv",
"t2_space",
"topological_group.t2_space_iff_one_closed"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uniform_group.to_uniform_space_eq {G : Type*} [u : uniform_space G]
[group G] [uniform_group G] : topological_group.to_uniform_space G = u | begin
ext : 1,
rw [uniformity_eq_comap_nhds_one' G, uniformity_eq_comap_nhds_one G]
end | lemma | uniform_group.to_uniform_space_eq | topology.algebra | src/topology/algebra/uniform_group.lean | [
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.uniform_embedding",
"topology.uniform_space.complete_separated",
"topology.uniform_space.compact",
"topology.algebra.group.basic",
"tactic.abel"
] | [
"group",
"topological_group.to_uniform_space",
"uniform_group",
"uniform_space",
"uniformity_eq_comap_nhds_one",
"uniformity_eq_comap_nhds_one'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_div_comap_self (x₀ : α) :
tendsto (λt:β×β, t.2 / t.1) (comap (λp:β×β, (e p.1, e p.2)) $ 𝓝 (x₀, x₀)) (𝓝 1) | begin
have comm : (λx:α×α, x.2/x.1) ∘ (λt:β×β, (e t.1, e t.2)) = e ∘ (λt:β×β, t.2 / t.1),
{ ext t,
change e t.2 / e t.1 = e (t.2 / t.1),
rwa ← map_div e t.2 t.1 },
have lim : tendsto (λ x : α × α, x.2/x.1) (𝓝 (x₀, x₀)) (𝓝 (e 1)),
{ simpa using (continuous_div'.comp (@continuous_swap α α _ _)).tendsto ... | lemma | tendsto_div_comap_self | topology.algebra | src/topology/algebra/uniform_group.lean | [
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.uniform_embedding",
"topology.uniform_space.complete_separated",
"topology.uniform_space.compact",
"topology.algebra.group.basic",
"tactic.abel"
] | [
"comm",
"continuous_swap",
"lim",
"map_div"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
extend_Z_bilin_aux (x₀ : α) (y₁ : δ) :
∃ U₂ ∈ comap e (𝓝 x₀), ∀ x x' ∈ U₂, Φ (x' - x, y₁) ∈ W' | begin
let Nx := 𝓝 x₀,
let ee := λ u : β × β, (e u.1, e u.2),
have lim1 : tendsto (λ a : β × β, (a.2 - a.1, y₁)) (comap e Nx ×ᶠ comap e Nx) (𝓝 (0, y₁)),
{ have := tendsto.prod_mk (tendsto_sub_comap_self de x₀)
(tendsto_const_nhds : tendsto (λ (p : β × β), y₁) (comap ee $ 𝓝 (x₀, x₀)) (𝓝 y₁)),
rw [n... | lemma | dense_inducing.extend_Z_bilin_aux | topology.algebra | src/topology/algebra/uniform_group.lean | [
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.uniform_embedding",
"topology.uniform_space.complete_separated",
"topology.uniform_space.compact",
"topology.algebra.group.basic",
"tactic.abel"
] | [
"ball_mem_comm",
"lim",
"nhds_prod_eq",
"tendsto_const_nhds"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
extend_Z_bilin_key (x₀ : α) (y₀ : γ) :
∃ U ∈ comap e (𝓝 x₀), ∃ V ∈ comap f (𝓝 y₀),
∀ x x' ∈ U, ∀ y y' ∈ V, Φ (x', y') - Φ (x, y) ∈ W' | begin
let Nx := 𝓝 x₀,
let Ny := 𝓝 y₀,
let dp := dense_inducing.prod de df,
let ee := λ u : β × β, (e u.1, e u.2),
let ff := λ u : δ × δ, (f u.1, f u.2),
have lim_φ : filter.tendsto Φ (𝓝 (0, 0)) (𝓝 0),
{ simpa using hφ.tendsto (0, 0) },
have lim_φ_sub_sub : tendsto (λ (p : (β × β) × (δ × δ)), Φ (p.... | lemma | dense_inducing.extend_Z_bilin_key | topology.algebra | src/topology/algebra/uniform_group.lean | [
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.uniform_embedding",
"topology.uniform_space.complete_separated",
"topology.uniform_space.compact",
"topology.algebra.group.basic",
"tactic.abel"
] | [
"continuous",
"continuous_swap",
"dense_inducing.prod",
"filter.prod_mono",
"filter.tendsto",
"nhds_prod_eq",
"prod.swap"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
extend_Z_bilin : continuous (extend (de.prod df) Φ) | begin
refine continuous_extend_of_cauchy _ _,
rintro ⟨x₀, y₀⟩,
split,
{ apply ne_bot.map,
apply comap_ne_bot,
intros U h,
rcases mem_closure_iff_nhds.1 ((de.prod df).dense (x₀, y₀)) U h with ⟨x, x_in, ⟨z, z_x⟩⟩,
existsi z,
cc },
{ suffices : map (λ (p : (β × δ) × (β × δ)), Φ p.2 - Φ p.1)
... | theorem | dense_inducing.extend_Z_bilin | topology.algebra | src/topology/algebra/uniform_group.lean | [
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.uniform_embedding",
"topology.uniform_space.complete_separated",
"topology.uniform_space.compact",
"topology.algebra.group.basic",
"tactic.abel"
] | [
"continuous",
"dense",
"exists_prop",
"extend",
"filter.map_map",
"mem_map",
"nhds_prod_eq",
"set.mem_preimage",
"set.prod_mk_mem_set_prod_eq"
] | Bourbaki GT III.6.5 Theorem I:
ℤ-bilinear continuous maps from dense images into a complete Hausdorff group extend by continuity.
Note: Bourbaki assumes that α and β are also complete Hausdorff, but this is not necessary. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
quotient_group.complete_space' (G : Type u) [group G] [topological_space G]
[topological_group G] [first_countable_topology G] (N : subgroup G) [N.normal]
[@complete_space G (topological_group.to_uniform_space G)] :
@complete_space (G ⧸ N) (topological_group.to_uniform_space (G ⧸ N)) | begin
/- Since `G ⧸ N` is a topological group it is a uniform space, and since `G` is first countable
the uniformities of both `G` and `G ⧸ N` are countably generated. Moreover, we may choose a
sequential antitone neighborhood basis `u` for `𝓝 (1 : G)` so that `(u (n + 1)) ^ 2 ⊆ u n`, and
this descends to an a... | instance | quotient_group.complete_space' | topology.algebra | src/topology/algebra/uniform_group.lean | [
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.uniform_embedding",
"topology.uniform_space.complete_separated",
"topology.uniform_space.compact",
"topology.algebra.group.basic",
"tactic.abel"
] | [
"cauchy_seq",
"cauchy_seq_tendsto_of_complete",
"complete_space",
"continuous",
"continuous_coinduced_rng",
"div_eq_mul_inv",
"div_mul_cancel'",
"div_mul_div_cancel'",
"div_self'",
"forall_true_left",
"ge_iff_le",
"group",
"inv_div",
"inv_inv",
"le_rfl",
"lt_add_one",
"mem_of_mem_nhd... | The quotient `G ⧸ N` of a complete first countable topological group `G` by a normal subgroup
is itself complete. [N. Bourbaki, *General Topology*, IX.3.1 Proposition 4][bourbaki1966b]
Because a topological group is not equipped with a `uniform_space` instance by default, we must
explicitly provide it in order to cons... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
quotient_group.complete_space (G : Type u) [group G] [us : uniform_space G]
[uniform_group G] [first_countable_topology G] (N : subgroup G) [N.normal]
[hG : complete_space G] : @complete_space (G ⧸ N) (topological_group.to_uniform_space (G ⧸ N)) | by { unfreezingI { rw ←@uniform_group.to_uniform_space_eq _ us _ _ at hG }, apply_instance } | instance | quotient_group.complete_space | topology.algebra | src/topology/algebra/uniform_group.lean | [
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.uniform_embedding",
"topology.uniform_space.complete_separated",
"topology.uniform_space.compact",
"topology.algebra.group.basic",
"tactic.abel"
] | [
"complete_space",
"group",
"subgroup",
"topological_group.to_uniform_space",
"uniform_group",
"uniform_group.to_uniform_space_eq",
"uniform_space"
] | The quotient `G ⧸ N` of a complete first countable uniform group `G` by a normal subgroup
is itself complete. In constrast to `quotient_group.complete_space'`, in this version `G` is
already equipped with a uniform structure.
[N. Bourbaki, *General Topology*, IX.3.1 Proposition 4][bourbaki1966b]
Even though `G` is equ... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_uniform_continuous_const_vadd [has_vadd M X] : Prop | (uniform_continuous_const_vadd : ∀ (c : M), uniform_continuous ((+ᵥ) c : X → X)) | class | has_uniform_continuous_const_vadd | topology.algebra | src/topology/algebra/uniform_mul_action.lean | [
"topology.algebra.uniform_group",
"topology.uniform_space.completion"
] | [
"has_vadd",
"uniform_continuous"
] | An additive action such that for all `c`, the map `λ x, c +ᵥ x` is uniformly continuous. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_uniform_continuous_const_smul [has_smul M X] : Prop | (uniform_continuous_const_smul : ∀ (c : M), uniform_continuous ((•) c : X → X)) | class | has_uniform_continuous_const_smul | topology.algebra | src/topology/algebra/uniform_mul_action.lean | [
"topology.algebra.uniform_group",
"topology.uniform_space.completion"
] | [
"has_smul",
"uniform_continuous"
] | A multiplicative action such that for all `c`, the map `λ x, c • x` is uniformly continuous. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
add_monoid.has_uniform_continuous_const_smul_nat [add_group X] [uniform_add_group X] :
has_uniform_continuous_const_smul ℕ X | ⟨uniform_continuous_const_nsmul⟩ | instance | add_monoid.has_uniform_continuous_const_smul_nat | topology.algebra | src/topology/algebra/uniform_mul_action.lean | [
"topology.algebra.uniform_group",
"topology.uniform_space.completion"
] | [
"add_group",
"has_uniform_continuous_const_smul",
"uniform_add_group"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_group.has_uniform_continuous_const_smul_int [add_group X] [uniform_add_group X] :
has_uniform_continuous_const_smul ℤ X | ⟨uniform_continuous_const_zsmul⟩ | instance | add_group.has_uniform_continuous_const_smul_int | topology.algebra | src/topology/algebra/uniform_mul_action.lean | [
"topology.algebra.uniform_group",
"topology.uniform_space.completion"
] | [
"add_group",
"has_uniform_continuous_const_smul",
"uniform_add_group"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_uniform_continuous_const_smul_of_continuous_const_smul [monoid R] [add_comm_group M]
[distrib_mul_action R M] [uniform_space M] [uniform_add_group M] [has_continuous_const_smul R M] :
has_uniform_continuous_const_smul R M | ⟨λ r, uniform_continuous_of_continuous_at_zero (distrib_mul_action.to_add_monoid_hom M r)
(continuous.continuous_at (continuous_const_smul r))⟩ | lemma | has_uniform_continuous_const_smul_of_continuous_const_smul | topology.algebra | src/topology/algebra/uniform_mul_action.lean | [
"topology.algebra.uniform_group",
"topology.uniform_space.completion"
] | [
"add_comm_group",
"continuous.continuous_at",
"distrib_mul_action",
"distrib_mul_action.to_add_monoid_hom",
"has_continuous_const_smul",
"has_uniform_continuous_const_smul",
"monoid",
"uniform_add_group",
"uniform_space"
] | A `distrib_mul_action` that is continuous on a uniform group is uniformly continuous.
This can't be an instance due to it forming a loop with
`has_uniform_continuous_const_smul.to_has_continuous_const_smul` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ring.has_uniform_continuous_const_smul [ring R] [uniform_space R]
[uniform_add_group R] [has_continuous_mul R] : has_uniform_continuous_const_smul R R | has_uniform_continuous_const_smul_of_continuous_const_smul _ _ | instance | ring.has_uniform_continuous_const_smul | topology.algebra | src/topology/algebra/uniform_mul_action.lean | [
"topology.algebra.uniform_group",
"topology.uniform_space.completion"
] | [
"has_continuous_mul",
"has_uniform_continuous_const_smul",
"has_uniform_continuous_const_smul_of_continuous_const_smul",
"ring",
"uniform_add_group",
"uniform_space"
] | The action of `semiring.to_module` is uniformly continuous. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ring.has_uniform_continuous_const_op_smul [ring R] [uniform_space R]
[uniform_add_group R] [has_continuous_mul R] : has_uniform_continuous_const_smul Rᵐᵒᵖ R | has_uniform_continuous_const_smul_of_continuous_const_smul _ _ | instance | ring.has_uniform_continuous_const_op_smul | topology.algebra | src/topology/algebra/uniform_mul_action.lean | [
"topology.algebra.uniform_group",
"topology.uniform_space.completion"
] | [
"has_continuous_mul",
"has_uniform_continuous_const_smul",
"has_uniform_continuous_const_smul_of_continuous_const_smul",
"ring",
"uniform_add_group",
"uniform_space"
] | The action of `semiring.to_opposite_module` is uniformly continuous. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_uniform_continuous_const_smul.to_has_continuous_const_smul
[has_uniform_continuous_const_smul M X] : has_continuous_const_smul M X | ⟨λ c, (uniform_continuous_const_smul c).continuous⟩ | instance | has_uniform_continuous_const_smul.to_has_continuous_const_smul | topology.algebra | src/topology/algebra/uniform_mul_action.lean | [
"topology.algebra.uniform_group",
"topology.uniform_space.completion"
] | [
"has_continuous_const_smul",
"has_uniform_continuous_const_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uniform_continuous.const_smul [has_uniform_continuous_const_smul M X]
{f : Y → X} (hf : uniform_continuous f) (c : M) :
uniform_continuous (c • f) | (uniform_continuous_const_smul c).comp hf | lemma | uniform_continuous.const_smul | topology.algebra | src/topology/algebra/uniform_mul_action.lean | [
"topology.algebra.uniform_group",
"topology.uniform_space.completion"
] | [
"has_uniform_continuous_const_smul",
"uniform_continuous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_uniform_continuous_const_smul.op [has_smul Mᵐᵒᵖ X] [is_central_scalar M X]
[has_uniform_continuous_const_smul M X] : has_uniform_continuous_const_smul Mᵐᵒᵖ X | ⟨mul_opposite.rec $ λ c, begin
change uniform_continuous (λ m, mul_opposite.op c • m),
simp_rw op_smul_eq_smul,
exact uniform_continuous_const_smul c,
end⟩ | instance | has_uniform_continuous_const_smul.op | topology.algebra | src/topology/algebra/uniform_mul_action.lean | [
"topology.algebra.uniform_group",
"topology.uniform_space.completion"
] | [
"has_smul",
"has_uniform_continuous_const_smul",
"is_central_scalar",
"mul_opposite.op",
"uniform_continuous"
] | If a scalar action is central, then its right action is uniform continuous when its left action
is. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_opposite.has_uniform_continuous_const_smul
[has_uniform_continuous_const_smul M X] : has_uniform_continuous_const_smul M Xᵐᵒᵖ | ⟨λ c, mul_opposite.uniform_continuous_op.comp $ mul_opposite.uniform_continuous_unop.const_smul c⟩ | instance | mul_opposite.has_uniform_continuous_const_smul | topology.algebra | src/topology/algebra/uniform_mul_action.lean | [
"topology.algebra.uniform_group",
"topology.uniform_space.completion"
] | [
"has_uniform_continuous_const_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uniform_group.to_has_uniform_continuous_const_smul
{G : Type u} [group G] [uniform_space G] [uniform_group G] :
has_uniform_continuous_const_smul G G | ⟨λ c, uniform_continuous_const.mul uniform_continuous_id⟩ | instance | uniform_group.to_has_uniform_continuous_const_smul | topology.algebra | src/topology/algebra/uniform_mul_action.lean | [
"topology.algebra.uniform_group",
"topology.uniform_space.completion"
] | [
"group",
"has_uniform_continuous_const_smul",
"uniform_group",
"uniform_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_def (c : M) (x : completion X) : c • x = completion.map ((•) c) x | rfl | lemma | uniform_space.completion.smul_def | topology.algebra | src/topology/algebra/uniform_mul_action.lean | [
"topology.algebra.uniform_group",
"topology.uniform_space.completion"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_smul (c : M) (x : X) : ↑(c • x) = (c • x : completion X) | (map_coe (uniform_continuous_const_smul c) x).symm | lemma | uniform_space.completion.coe_smul | topology.algebra | src/topology/algebra/uniform_mul_action.lean | [
"topology.algebra.uniform_group",
"topology.uniform_space.completion"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_one : ((1 : α) : completion α) = 1 | rfl | lemma | uniform_space.completion.coe_one | topology.algebra | src/topology/algebra/uniform_ring.lean | [
"algebra.algebra.basic",
"topology.algebra.group_completion",
"topology.algebra.ring.ideal"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_mul (a b : α) : ((a * b : α) : completion α) = a * b | ((dense_inducing_coe.prod dense_inducing_coe).extend_eq
((continuous_coe α).comp (@continuous_mul α _ _ _)) (a, b)).symm | lemma | uniform_space.completion.coe_mul | topology.algebra | src/topology/algebra/uniform_ring.lean | [
"algebra.algebra.basic",
"topology.algebra.group_completion",
"topology.algebra.ring.ideal"
] | [
"continuous_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_mul : continuous (λ p : completion α × completion α, p.1 * p.2) | begin
let m := (add_monoid_hom.mul : α →+ α →+ α).compr₂ to_compl,
have : continuous (λ p : α × α, m p.1 p.2),
from (continuous_coe α).comp continuous_mul,
have di : dense_inducing (to_compl : α → completion α),
from dense_inducing_coe,
convert di.extend_Z_bilin di this,
ext ⟨x, y⟩,
refl
end | lemma | uniform_space.completion.continuous_mul | topology.algebra | src/topology/algebra/uniform_ring.lean | [
"algebra.algebra.basic",
"topology.algebra.group_completion",
"topology.algebra.ring.ideal"
] | [
"add_monoid_hom.mul",
"continuous",
"continuous_mul",
"dense_inducing"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous.mul {β : Type*} [topological_space β] {f g : β → completion α}
(hf : continuous f) (hg : continuous g) : continuous (λb, f b * g b) | continuous_mul.comp (hf.prod_mk hg : _) | lemma | uniform_space.completion.continuous.mul | topology.algebra | src/topology/algebra/uniform_ring.lean | [
"algebra.algebra.basic",
"topology.algebra.group_completion",
"topology.algebra.ring.ideal"
] | [
"continuous",
"continuous.mul",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_ring_hom : α →+* completion α | ⟨coe, coe_one α, assume a b, coe_mul a b, coe_zero, assume a b, coe_add a b⟩ | def | uniform_space.completion.coe_ring_hom | topology.algebra | src/topology/algebra/uniform_ring.lean | [
"algebra.algebra.basic",
"topology.algebra.group_completion",
"topology.algebra.ring.ideal"
] | [] | The map from a uniform ring to its completion, as a ring homomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous_coe_ring_hom : continuous (coe_ring_hom : α → completion α) | continuous_coe α | lemma | uniform_space.completion.continuous_coe_ring_hom | topology.algebra | src/topology/algebra/uniform_ring.lean | [
"algebra.algebra.basic",
"topology.algebra.group_completion",
"topology.algebra.ring.ideal"
] | [
"continuous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
extension_hom [complete_space β] [separated_space β] :
completion α →+* β | have hf' : continuous (f : α →+ β), from hf, -- helping the elaborator
have hf : uniform_continuous f, from uniform_continuous_add_monoid_hom_of_continuous hf',
{ to_fun := completion.extension f,
map_zero' := by rw [← coe_zero, extension_coe hf, f.map_zero],
map_add' := assume a b, completion.induction_on₂ a b
... | def | uniform_space.completion.extension_hom | topology.algebra | src/topology/algebra/uniform_ring.lean | [
"algebra.algebra.basic",
"topology.algebra.group_completion",
"topology.algebra.ring.ideal"
] | [
"complete_space",
"continuous",
"continuous_fst",
"continuous_mul",
"continuous_snd",
"is_closed_eq",
"separated_space",
"uniform_continuous"
] | The completion extension as a ring morphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
top_ring_compl : topological_ring (completion α) | { continuous_add := continuous_add,
continuous_mul := continuous_mul } | instance | uniform_space.completion.top_ring_compl | topology.algebra | src/topology/algebra/uniform_ring.lean | [
"algebra.algebra.basic",
"topology.algebra.group_completion",
"topology.algebra.ring.ideal"
] | [
"continuous_mul",
"topological_ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_ring_hom (hf : continuous f) : completion α →+* completion β | extension_hom (coe_ring_hom.comp f) (continuous_coe_ring_hom.comp hf) | def | uniform_space.completion.map_ring_hom | topology.algebra | src/topology/algebra/uniform_ring.lean | [
"algebra.algebra.basic",
"topology.algebra.group_completion",
"topology.algebra.ring.ideal"
] | [
"continuous"
] | The completion map as a ring morphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map_smul_eq_mul_coe (r : R) :
completion.map ((•) r) = (*) (algebra_map R A r : completion A) | begin
ext x,
refine completion.induction_on x _ (λ a, _),
{ exact is_closed_eq (completion.continuous_map) (continuous_mul_left _) },
{ rw [map_coe (uniform_continuous_const_smul r) a, algebra.smul_def, coe_mul] },
end | lemma | uniform_space.completion.map_smul_eq_mul_coe | topology.algebra | src/topology/algebra/uniform_ring.lean | [
"algebra.algebra.basic",
"topology.algebra.group_completion",
"topology.algebra.ring.ideal"
] | [
"algebra.smul_def",
"algebra_map",
"continuous_mul_left",
"is_closed_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
algebra_map_def (r : R) :
algebra_map R (completion A) r = (algebra_map R A r : completion A) | rfl | lemma | uniform_space.completion.algebra_map_def | topology.algebra | src/topology/algebra/uniform_ring.lean | [
"algebra.algebra.basic",
"topology.algebra.group_completion",
"topology.algebra.ring.ideal"
] | [
"algebra_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
algebra' : algebra R (completion R) | by apply_instance | instance | uniform_space.completion.algebra' | topology.algebra | src/topology/algebra/uniform_ring.lean | [
"algebra.algebra.basic",
"topology.algebra.group_completion",
"topology.algebra.ring.ideal"
] | [
"algebra"
] | A shortcut instance for the common case | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ring_sep_rel (α) [comm_ring α] [uniform_space α] [uniform_add_group α] [topological_ring α] :
separation_setoid α = submodule.quotient_rel (ideal.closure ⊥) | setoid.ext $ λ x y, (add_group_separation_rel x y).trans $
iff.trans (by refl) (submodule.quotient_rel_r_def _).symm | lemma | uniform_space.ring_sep_rel | topology.algebra | src/topology/algebra/uniform_ring.lean | [
"algebra.algebra.basic",
"topology.algebra.group_completion",
"topology.algebra.ring.ideal"
] | [
"comm_ring",
"ideal.closure",
"setoid.ext",
"submodule.quotient_rel",
"submodule.quotient_rel_r_def",
"topological_ring",
"uniform_add_group",
"uniform_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ring_sep_quot
(α : Type u) [r : comm_ring α] [uniform_space α] [uniform_add_group α] [topological_ring α] :
quotient (separation_setoid α) = (α ⧸ (⊥ : ideal α).closure) | by rw [@ring_sep_rel α r]; refl | lemma | uniform_space.ring_sep_quot | topology.algebra | src/topology/algebra/uniform_ring.lean | [
"algebra.algebra.basic",
"topology.algebra.group_completion",
"topology.algebra.ring.ideal"
] | [
"closure",
"comm_ring",
"ideal",
"topological_ring",
"uniform_add_group",
"uniform_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sep_quot_equiv_ring_quot (α)
[r : comm_ring α] [uniform_space α] [uniform_add_group α] [topological_ring α] :
quotient (separation_setoid α) ≃ (α ⧸ (⊥ : ideal α).closure) | quotient.congr_right $ λ x y, (add_group_separation_rel x y).trans $
iff.trans (by refl) (submodule.quotient_rel_r_def _).symm | def | uniform_space.sep_quot_equiv_ring_quot | topology.algebra | src/topology/algebra/uniform_ring.lean | [
"algebra.algebra.basic",
"topology.algebra.group_completion",
"topology.algebra.ring.ideal"
] | [
"closure",
"comm_ring",
"ideal",
"quotient.congr_right",
"submodule.quotient_rel_r_def",
"topological_ring",
"uniform_add_group",
"uniform_space"
] | Given a topological ring `α` equipped with a uniform structure that makes subtraction uniformly
continuous, get an equivalence between the separated quotient of `α` and the quotient ring
corresponding to the closure of zero. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
comm_ring [comm_ring α] [uniform_space α] [uniform_add_group α] [topological_ring α] :
comm_ring (quotient (separation_setoid α)) | by rw ring_sep_quot α; apply_instance | instance | uniform_space.comm_ring | topology.algebra | src/topology/algebra/uniform_ring.lean | [
"algebra.algebra.basic",
"topology.algebra.group_completion",
"topology.algebra.ring.ideal"
] | [
"comm_ring",
"topological_ring",
"uniform_add_group",
"uniform_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
topological_ring
[comm_ring α] [uniform_space α] [uniform_add_group α] [topological_ring α] :
topological_ring (quotient (separation_setoid α)) | begin
convert topological_ring_quotient (⊥ : ideal α).closure; try {apply ring_sep_rel},
simp [uniform_space.comm_ring]
end | instance | uniform_space.topological_ring | topology.algebra | src/topology/algebra/uniform_ring.lean | [
"algebra.algebra.basic",
"topology.algebra.group_completion",
"topology.algebra.ring.ideal"
] | [
"closure",
"comm_ring",
"ideal",
"topological_ring",
"topological_ring_quotient",
"uniform_add_group",
"uniform_space",
"uniform_space.comm_ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dense_inducing.extend_ring_hom {i : α →+* β} {f : α →+* γ}
(ue : uniform_inducing i) (dr : dense_range i) (hf : uniform_continuous f):
β →+* γ | { to_fun := (ue.dense_inducing dr).extend f,
map_one' := by { convert dense_inducing.extend_eq (ue.dense_inducing dr) hf.continuous 1,
exacts [i.map_one.symm, f.map_one.symm], },
map_zero' := by { convert dense_inducing.extend_eq (ue.dense_inducing dr) hf.continuous 0,
exacts [i.map_zero.symm, f.map... | def | dense_inducing.extend_ring_hom | topology.algebra | src/topology/algebra/uniform_ring.lean | [
"algebra.algebra.basic",
"topology.algebra.group_completion",
"topology.algebra.ring.ideal"
] | [
"continuous",
"continuous.comp",
"continuous_fst",
"continuous_mul",
"continuous_snd",
"dense_inducing.extend_eq",
"dense_range",
"dense_range.induction_on₂",
"extend",
"is_closed_eq",
"uniform_continuous",
"uniform_continuous_uniformly_extend",
"uniform_inducing"
] | The dense inducing extension as a ring homomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
subgroups_basis :
ring_subgroups_basis (λ γ : Γ₀ˣ, (v.lt_add_subgroup γ : add_subgroup R)) | { inter := begin
rintros γ₀ γ₁,
use min γ₀ γ₁,
simp [valuation.lt_add_subgroup] ; tauto
end,
mul := begin
rintros γ,
cases exists_square_le γ with γ₀ h,
use γ₀,
rintro - ⟨r, s, r_in, s_in, rfl⟩,
calc (v (r*s) : Γ₀) = v r * v s : valuation.map_mul _ _ _
... < γ₀*γ₀ : mul_... | lemma | valuation.subgroups_basis | topology.algebra | src/topology/algebra/valuation.lean | [
"topology.algebra.nonarchimedean.bases",
"topology.algebra.uniform_filter_basis",
"ring_theory.valuation.basic"
] | [
"add_subgroup",
"exists_square_le",
"group_with_zero.eq_zero_or_unit",
"left_mul",
"mul_comm",
"mul_inv_lt_of_lt_mul₀",
"mul_lt_mul₀",
"mul_zero",
"right_mul",
"ring_subgroups_basis",
"units.coe_mul",
"units.zero_lt",
"valuation.lt_add_subgroup",
"valuation.map_mul",
"zero_mul"
] | The basis of open subgroups for the topology on a ring determined by a valuation. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
valued (R : Type u) [ring R] (Γ₀ : out_param (Type v))
[linear_ordered_comm_group_with_zero Γ₀] extends uniform_space R, uniform_add_group R | (v : valuation R Γ₀)
(is_topological_valuation : ∀ s, s ∈ 𝓝 (0 : R) ↔ ∃ (γ : Γ₀ˣ), { x : R | v x < γ } ⊆ s) | class | valued | topology.algebra | src/topology/algebra/valuation.lean | [
"topology.algebra.nonarchimedean.bases",
"topology.algebra.uniform_filter_basis",
"ring_theory.valuation.basic"
] | [
"linear_ordered_comm_group_with_zero",
"ring",
"uniform_add_group",
"uniform_space",
"valuation"
] | A valued ring is a ring that comes equipped with a distinguished valuation. The class `valued`
is designed for the situation that there is a canonical valuation on the ring.
TODO: show that there always exists an equivalent valuation taking values in a type belonging to
the same universe as the ring.
See Note [forget... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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